nicolas privault notes on stochastic finance · stochastic calculus, optimization, partial...

Nicolas Privault

Notes onStochastic Finance

This version September 17 2020httpswwwntuedusghomenprivaultindexthtml

Notes on Stochastic Finance

Preface

This text is an introduction to pricing and hedging in discrete andcontinuous-time financial models without friction (ie without transactioncosts) with an emphasis on the complementarity between analytical andprobabilistic methods Its contents are mostly mathematical and also aim atmaking the reader aware of both the power and limitations of mathematicalmodels in finance by taking into account their conditions of applicabilityThe book covers a wide range of classical topics including Black-Scholes pric-ing exotic and american options term structure modeling and change ofnumeacuteraire as well as models with jumps It is targeted at the advanced un-dergraduate and graduate level in applied mathematics financial engineeringand economics The point of view adopted is that of mainstream mathemat-ical finance in which the computation of fair prices is based on the absenceof arbitrage hypothesis therefore excluding riskless profit based on arbitrageopportunities and basic (buying lowselling high) trading Similarly this doc-ument is not concerned with any ldquopredictionrdquo of stock price behaviors thatbelong other domains such as technical analysis which should not be con-fused with the statistical modeling of asset prices The text also about 20examples based on actual market data

The descriptions of the asset model self-financing portfolios arbitrage andmarket completeness are first given in Chapter 1 in a simple two time-stepsetting These notions are then reformulated in discrete time in Chapter 2Here the impossibility to access future information is formulated using thenotion of adapted processes which will play a central role in the constructionof stochastic calculus in continuous time

In order to trade efficiently it would be useful to have a formula to esti-mate the ldquofair pricerdquo of a given risky asset helping for example to determinewhether the asset is undervalued or overvalued at a given time Althoughsuch a formula is not available we can instead derive formulas for the pric-ing of options that can act as insurance contracts to protect their holdersagainst adverse changes in the prices of risky assets The pricing and hedgingof options in discrete time particularly in the fundamental example of the

v


N Privault

Cox-Ross-Rubinstein model are considered in Chapter 3 with a descriptionof the passage from discrete to continuous time that prepares the transitionto the subsequent chapters

A simplified presentation of Brownian motion stochastic integrals and theassociated Itocirc formula is given in Chapter 4 with application to stochasticasset price modeling in Chapter 5 The Black-Scholes model is presented fromthe angle of partial differential equation (PDE) methods in Chapter 6 withthe derivation of the Black-Scholes formula by transforming the Black-ScholesPDE into the standard heat equation wich is then solved by a heat kernelargument The martingale approach to pricing and hedging is then presentedin Chapter 7 and complements the PDE approach of Chapter 6 by recover-ing the Black-Scholes formula via a probabilistic argument An introductionto stochastic volatility is given in Chapter 8 followed by a presentation ofvolatility estimation tools including historical local and implied volatilitiesin Chapter 9 This chapter also contains a comparison of the prices obtainedby the Black-Scholes formula with actual option price market data

Exotic options such as barrier lookback and Asian options are treated inChapters 11 12 and 13 respectively following an introduction to the prop-erties of the maximum of Brownian motion given in Chapter 10 Optimalstopping and exercise with application to the pricing of American optionsare considered in Chapter 15 following the presentation of background ma-terial on filtrations and stopping times in Chapter 14 The construction offorward measures by change of numeacuteraire is given in Chapter 16 and is appliedto the pricing of interest rate derivatives in Chapter 19 after an introductionto bond pricing and to the modeling of forward rates in Chapters 17 and 18based on material from Privault (2012)

Stochastic calculus with jumps is dealt with in Chapter 20 and is restrictedto compound Poisson processes which only have a finite number of jumps onany bounded interval Those processes are used for option pricing and hedgingin jump models in Chapter 21 in which we mostly focus on risk minimiz-ing strategies as markets with jumps are generally incomplete Chapter 22contains an elementary introduction to finite difference methods for the nu-merical solution of PDEs and stochastic differential equations dealing withthe explicit and implicit finite difference schemes for the heat equations andthe Black-Scholes PDE as well as the Euler and Milshtein schemes for SDEsThe text is completed with an appendix containing the needed probabilisticbackground

The material in this book has been used for teaching in the Masters ofScience in Financial Engineering at City University of Hong Kong and at theNanyang Technological University in Singapore The author thanks Nickyvan Foreest Kazuhiro Kojima Sijian Lin Sandu Ursu and Ju-Yi Yen for

vi



corrections and improvements

This text contains 224 exercises and 13 problems with solutions Clickingon an exercise number inside the solution section will send to the originalproblem text inside the file Conversely clicking on a problem number sendsthe reader to the corresponding solution however this feature should not bemisused The cover graph represents the time evolution of the HSBC stockprice from January to September 2009 plotted on the price surface of a Eu-ropean put option on that asset expiring on October 05 2009 cf sect 61

The pdf file contains internal and external links and 323 figures includ-ing 47 animated Figures 37 39 46 47 49 410 414 56 65 101 102103 106 1116 131 121 127 1216 152 1713 186 189 1810 18172011 2013 2014 and S14 2 embedded videos in Figures 01 and 93 and3 interacting 3D graphs in Figures 64 611 and 112 that may require us-ing Acrobat Reader for viewing on the complete pdf file It also includes14 Python codes on pages 69 84 87 90 128 138 207 233 319 500 and810 and 54 R codes on pages 137 138 140 143 193 189 207 210 221 215231 233 247 319 320 334 300 370 379 590 637 659 662 675 677 745and 748

Nicolas Privault2020

vii


N Privault

viii


Contents

Introduction 1

1 Assets Portfolios and Arbitrage 1911 Definitions and Notation 1912 Portfolio Allocation and Short Selling 2013 Arbitrage 2114 Risk-Neutral Probability Measures 2715 Hedging Contingent Claims 3016 Market Completeness 3317 Example Binary Market 33Exercises 41

2 Discrete-Time Market Model 4721 Discrete-Time Compounding 4722 Arbitrage and Self-Financing Portfolios 5023 Contingent Claims 5624 Martingales and Conditional Expectation 6025 Market Completeness and Risk-Neutral Measures 6626 The Cox-Ross-Rubinstein (CRR) Market Model 69Exercises 73

3 Pricing and Hedging in Discrete Time 7731 Pricing Contingent Claims 7732 Pricing Vanilla Options in the CRR Model 8333 Hedging Contingent Claims 8834 Hedging Vanilla Options in the CRR model 9035 Hedging Exotic Options in the CRR Model 9836 Convergence of the CRR Model 106Exercises 112

ix

N Privault

4 Brownian Motion and Stochastic Calculus 13141 Brownian Motion 13142 Three Constructions of Brownian Motion 13543 Wiener Stochastic Integral 13944 Itocirc Stochastic Integral 14845 Stochastic Calculus 156Exercises 167

5 Continuous-Time Market Model 17951 Asset price modeling 17952 Arbitrage and Risk-Neutral Measures 18153 Self-Financing Portfolio Strategies 18454 Geometric Brownian Motion 192Exercises 196

6 Black-Scholes Pricing and Hedging 20161 The Black-Scholes PDE 20162 European Call Options 20663 European Put Options 21364 Market Terms and Data 21865 The Heat Equation 22266 Solution of the Black-Scholes PDE 226Exercises 229

7 Martingale Approach to Pricing and Hedging 23971 Martingale Property of the Itocirc Integral 23972 Risk-neutral Probability Measures 24473 Change of Measure and the Girsanov Theorem 24874 Pricing by the Martingale Method 25075 Hedging by the Martingale Method 258Exercises 263

8 Stochastic Volatility 28581 Stochastic Volatility Models 28582 Realized Variance Swaps 28983 Realized Variance Options 29384 European Options - PDE Method 30185 Perturbation Analysis 307Exercises 311

9 Volatility Estimation 31591 Historical Volatility 31592 Implied Volatility 31893 Local Volatility 32694 The VIXreg Index 332Exercises 336

x



10 Maximum of Brownian motion 339101 Running Maximum of Standard Brownian Motion 339102 The Reflection Principle 342103 Density of the Maximum of Brownian Motion 346104 Average of Geometric Brownian Extrema 356Exercises 361

11 Barrier Options 365111 Options on Extrema 365112 Knock-Out Barrier Options 371113 Knock-In Barrier Options 383114 PDE Method 388Exercises 393

12 Lookback Options 397121 The Lookback Put Option 397122 PDE Method 400123 The Lookback Call Option 406124 Delta Hedging for Lookback Options 417Exercises 422

13 Asian Options 425131 Bounds on Asian Option Prices 425132 Pricing by the Hartman-Watson distribution 432133 Laplace Transform Method 435134 Moment Matching Approximations 436135 PDE Method 442Exercises 453

14 Stopping Times and Martingales 457141 Filtrations and Information Flow 457142 Submartingales and Supermartingales 458143 Stopping Times 460144 Application to drifted Brownian motion 467Exercises 473

15 American Options 477151 Perpetual American Options 477152 PDE approach 483153 Finite Expiration American Options 491154 PDE approach 495Exercises 499

xi


N Privault

16 Change of Numeacuteraire and Forward Measures 511161 Notion of Numeacuteraire 511162 Change of Numeacuteraire 514163 Foreign Exchange 523164 Pricing Exchange Options 531165 Hedging by Change of Numeacuteraire 534Exercises 538

17 Short Rates and Bond Pricing 545171 Short-Term Mean-Reverting Models 545172 Calibration of the Vasicek Model 551173 Zero-Coupon and Coupon Bonds 556174 Bond Pricing PDE 559Exercises 574

18 Forward Rates 583181 Construction of Forward Rates 583182 Interest Rate Swaps 593183 The HJM Model 597184 Yield Curve Modeling 603185 Two-Factor Model 607186 The BGM Model 611Exercises 614

19 Pricing of Interest Rate Derivatives 619191 Forward Measures and Tenor Structure 619192 Bond Options 623193 Caplet Pricing 625194 Forward Swap Measures 629195 Swaption Pricing 631Exercises 638

20 Stochastic Calculus for Jump Processes 653201 The Poisson Process 653202 Compound Poisson Process 660203 Stochastic Integrals and Itocirc Formula with Jumps 666204 Stochastic Differential Equations with Jumps 678205 Girsanov Theorem for Jump Processes 683Exercises 691

21 Pricing and Hedging in Jump Models 697211 Market Returns vs Gaussian and Power Tails 697212 Risk-Neutral Probability Measures 701213 Pricing in Jump Models 703214 Black-Scholes PDE with Jumps 704215 Exponential Leacutevy Models 707

xii



216 Mean-Variance Hedging with Jumps 710Exercises 714

22 Basic Numerical Methods 717221 Discretized Heat Equation 717222 Discretized Black-Scholes PDE 720223 Euler Discretization 724224 Milshtein Discretization 725

Appendix Background on Probability Theory 727231 Probability Sample Space and Events 727232 Probability Measures 731233 Conditional Probabilities and Independence 733234 Random Variables 735235 Probability Distributions 737236 Expectation of Random Variables 744237 Conditional Expectation 756Exercises 761

Exercise Solutions 765Chapter 1 765Chapter 2 772Chapter 3 778Chapter 4 811Chapter 5 840Chapter 6 853Chapter 7 869Chapter 8 906Chapter 9 911Chapter 10 916Chapter 11 931Chapter 12 951Chapter 13 961Chapter 14 970Chapter 15 977Chapter 16 1005Chapter 17 1016Chapter 18 1032Chapter 19 1043Chapter 20 1065Chapter 21 1077Background on Probability Theory 1081

Index 1087

Author index 1097 xiii


N Privault

References 1101

xiv


List of Figures

01 ldquoAs if a whole new world was laid out before merdquolowast 302 Comparison of WTI vs Keppel price graphs 503 Hang Seng index 604 Payoff function of a put option 705 Sample price processes simulated by a geometric Brownian motion 706 Payoff function of a call option 807 ldquoInfogramesrdquo stock price curve 1008 Brent and WTI price graphs 1009 Price graph for a four-way collar option 11010 Payoff function of a four-way collar option 11011 Four-way collar option as a combination of call and put optionslowast 12012 Implied probabilities 16013 Implied probabilities according to bookmakers 16014 Implied probabilities according to polling 17

11 Triangular arbitrage 2212 Arbitrage Retail prices around the world 2413 Separation of convex sets 30

21 Illustration of the self-financing condition (27) 5322 Why apply discounting 5523 Oil price graph 5524 Take the quiz 6025 Discrete-time asset price tree in the CRR model 7026 Discrete-time asset price graphs in the CRR model 7127 Function x 7rarr ((1 + x)21 minus (1 + x)10)x 74

31 Discrete-time call option pricing tree 8832 Discrete-time call option hedging strategy (risky component) 9233 Discrete-time call option hedging strategy (riskless component) 9234 Tree of asset prices in the CRR model 97

xv

N Privault

35 Tree of option prices in the CRR model 9836 Tree of hedging portfolio allocations in the CRR model 9837 Galton board simulationlowast 10738 A real-life Galton board 10839 Multiplicative Galton board simulationlowast 109310 Dividend detachment graph on Z74SI 116311 Put spread collar price map 118312 Call spread collar price map 119

41 Sample paths of a one-dimensional Brownian motion 13242 Evolution of the fortune of a poker player vs number of games played 13343 Web traffic ranking 13344 Two sample paths of a two-dimensional Brownian motion 13445 Sample path of a three-dimensional Brownian motion 13446 Scaling property of Brownian motionlowast 13547 Brownian motion as a random walklowast 13648 Statistics of one-dimensional Brownian paths vs Gaussian distribution 13749 Leacutevyrsquos construction of Brownian motionlowast 138410 Construction of Brownian motion by series expansionslowast 139411 Step function 140412 Area under the step function 141413 Squared step function 143414 Step function approximationlowast 144415 Adapted pair trading portfolio strategy 149416 Squared simple predictable process 153417 NGram Viewer output for the term stochastic calculus 156418 Simulated path of (430) with α = 10 and σ = 02 166419 Simulated path of (434) 166420 Simulated path of (435) with micro = 5 and σ = 1 167

52 Why apply discounting 18053 Illustration of the self-financing condition (54) 18454 Illustration of the self-financing condition (510) 18755 Sample paths of geometric Brownian motion 18956 Geometric Brownian motion started at S0 = 1lowast 19357 Statistics of geometric Brownian paths vs lognormal distribution 196

61 Underlying market prices 20262 Simulated geometric Brownian motion 20263 Graph of the Gaussian Cumulative Distribution Function (CDF) 20764 Black-Scholes call price maplowast 20865 Time-dependent solution of the Black-Scholes PDE (call option)lowast 20866 Delta of a European call option 21067 Gamma of a European call option 21168 HSBC Holdings stock price 212

xvi



69 Path of the Black-Scholes price for a call option on HSBC 212610 Time evolution of a hedging portfolio for a call option on HSBC 213611 Black-Scholes put price functionlowast 214612 Time-dependent solution of the Black-Scholes PDE (put option)lowast 215613 Delta of a European put option 216614 Path of the Black-Scholes price for a put option on HSBC 217615 Time evolution of the hedging portfolio for a put option on HSBC 218616 Time-dependent solutions of the Black-Scholes PDElowast 219617 Warrant terms and data 221618 Time-dependent solution of the heat equationlowast 223619 Time-dependent solution of the heat equationlowast 225620 Short rate t 7rarr rt in the CIR model 230621 Option price as a function of the volatility σ 233

71 Drifted Brownian path 24572 Drifted Brownian paths under a shifted Girsanov measure 24773 Payoff functions of bull spread and bear spread options 26574 Butterfly payoff function 26575 Option price as a function of underlying asset price and time to maturity 28076 Delta as a function of underlying asset price and time to maturity 28177 Gamma as a function of underlying asset price and time to maturity 28178 Option price as a function of underlying asset price and time to maturity28279 Delta as a function of underlying asset price and time to maturity 283710 Gamma as a function of underlying asset price and time to maturity 284

81 Euro SGD exchange rate 28682 Variance call option prices with b = 015 30083 Variance call option prices with b = minus005 30084 Option price approximations plotted against v with ρ = minus05 311

91 Underlying asset price vs log returns 31792 Historical volatility graph 31793 The fugazi itrsquos a wazy itrsquos a woozie Itrsquos fairy dustlowast 31894 Option price as a function of the volatility σ 31995 SampP500 option prices plotted against strike prices 32196 Implied volatility of Asian options on light sweet crude oil futures 32297 Market stock price of Cheung Kong Holdings 32398 Market call option price on Cheung Kong Holdings 32399 Black-Scholes call option price on Cheung Kong Holdings 324910 Market stock price of HSBC Holdings 324911 Market call option price on HSBC Holdings 324912 Black-Scholes call option price on HSBC Holdings 325913 Market put option price on HSBC Holdings 325914 Black-Scholes put option price on HSBC Holdings 326915 Call option price vs underlying asset price 326

xvii


N Privault

916 Local volatility estimated from Boeing Co option price data 331917 VIXreg Index vs the SampP 500 335918 VIXreg Index vs historical volatility for the year 2011 335919 Correlation estimates between GSPC and the VIXreg 336920 VIXreg Index vs 30 day historical volatility for the SampP 500 336

101 Brownian motion (Wt)tisinR+ and its running maximum (Xt0)tisinR+lowast 340

102 Running maximum of Brownian motionlowast 340103 Zeroes of Brownian motionlowast 341104 Graph of the Cantor functionlowast 341105 A function with no last point of increase before t = 1 342106 Reflected Brownian motion with a = 1lowast 343107 Probability density of the maximum of Brownian motion 344108 Probability density of the maximum Maxtisin[0T ] St of geometric

Brownian motion 346109 Joint probability density of Brownian motion and its maximum 3481010 Heat map of the joint density of W1 and its maximum 3481011 Probability density of the maximum of drifted Brownian motion 351

111 Probability computed as a volume integral 367112 Up-and-out barrier call option price with B gt Klowast 373113 Up-and-out barrier put option price with K gt B 378114 Up-and-out barrier put option price with B gt K 378115 Pricing data for an up-and-out barrier put option with K = B = $28 379116 Down-and-out barrier call option price with B lt K 380117 Down-and-out barrier call option price with K lt B 381118 Down-and-out barrier call option price as a function of volatility 381119 Down-and-out barrier put option price with K gt B 3831110 Down-and-in barrier call option price with K gt B 3841111 Down-and-in barrier call option price with K lt B 3841112 Up-and-in barrier call option price with K gt B 3851113 Down-and-in barrier put option price with K gt B 3861114 Up-and-in barrier put option price with K gt B 3871115 Up-and-in barrier put option price with K lt B 3871116 Delta of the up-and-out barrier call optionlowast 392

121 Lookback put option price (3D)lowast 398122 Graph of lookback put option prices 399123 Graph of lookback put option prices (2D) 399124 Normalized lookback put option price 404125 Black-Scholes put price in the decomposition (1212) 406126 Correction term hp(τ z) in the decomposition (1212) 406127 Lookback call option pricelowast 410128 Graph of lookback call option prices 410129 Graphs of lookback call option prices (2D) 411

xviii



1210 Normalized lookback call option price 4141211 Underlying asset prices 4141212 Running minimum of the underlying asset price 4151213 Lookback call option price 4151214 Black-Scholes call price in the normalized lookback call price 4161215 Function hc(τ z) in the normalized lookback call option price 4161216 Delta of the lookback call optionlowast 4191217 Rescaled portfolio strategy for the lookback call option 4191218 Delta of the lookback put optionlowast 422

131 Brownian motion and its moving averagelowast 426132 Asian option price vs European option pricelowast 430133 Asian call option prices 435134 Lognormal approximation of probability density 438135 Lognormal approximation to the Asian call option price 439136 Dividend detachment graph on Z74SI 456

141 Drifted Brownian path 459142 Evolution of the fortune of a poker player vs number of games played 459143 Stopped process 462144 Sample paths of a gambling process (Mn)nisinN 466145 Brownian motion hitting a barrier 467146 Drifted Brownian motion hitting a barrier 469147 Hitting probabilities of drifted Brownian motionlowast 470

151 American put prices by exercising at τL for different values of L 481152 Animated graph of American put prices x 7rarr fL(x)

lowast 482153 Option price as a function of L and of the underlying asset price 482154 Path of the American put option price on the HSBC stock 483155 American call prices by exercising at τL for different values of L 489156 Animated graph of American call prices x 7rarr fL(x)

lowast 490157 American call prices for different values of L 490158 Expected Black-Scholes European call option price vs (x t) 7rarr (xminusK)+493159 Black-Scholes put option price map vs (x t) 7rarr (K minus x)+ 4931510 Optimal frontier for the exercise of a put option 4941511 PDE estimates of finite expiration American put option prices 4961512 Longstaff-Schwartz estimates of finite expiration American put prices 4971513 Comparison between Longstaff-Schwartz and finite differences 497

161 Why change of numeacuteraire 513162 Overseas investment opportunity 524

171 Short rate t 7rarr rt in the Vasicek model 547172 CBOE 10 Year Treasury Note (TNX) yield 548173 Short rate t 7rarr rt in the CIR model 549174 Calibrated Vasicek simulation vs market data 555

xix


N Privault

175 Five-dollar 1875 Louisiana bond with 75 biannual coupons 556176 Comparison of Monte Carlo and PDE solutions 565177 Bond price t 7rarr P (tT ) vs t 7rarr eminusr0(Tminust) 566178 Bond price t 7rarr Pc(tT ) with a 5 coupon rate 566179 Bond price with coupon rate 625 5671710 Orange Cnty Calif bond prices 5681711 Orange Cnty Calif bond yields 5681712 Approximation of Dothan bond prices 5721713 Brownian bridge 573

181 Forward rate S 7rarr f(t tS) 584182 Indonesian government securities yield curve 584183 Forward rate process t 7rarr f(tT S) 589184 Instantaneous forward rate process t 7rarr f(tT ) 590185 Federal Reserve yield curves from 1982 to 2012 590186 European Central Bank yield curveslowast 591187 August 2019 Federal Reserve yield curve inversionlowast 592188 Stochastic process of forward curves 598189 Forward instantaneous curve in the Vasicek modellowast 6021810 Forward instantaneous curve x 7rarr f(0x) in the Vasicek modellowast 6021811 Short-term interest rate curve t 7rarr rt in the Vasicek model 6031812 Nelson-Siegel graph 6031813 Svensson model graph 6041814 Fitting of a Svensson curve to market data 6041815 Graphs of forward rates 6051816 Forward instantaneous curve in the Vasicek model 6051817 ECB data vs fitted yield curvelowast 6071818 Bond prices t 7rarr P (tT2)P (tT2)P (tT3) 6081819 Forward rates in a two-factor model 6101820 Random evolution of instantaneous forward rates in a two-factor model 6111821 Roadmap of stochastic interest rate modeling 613

191 Implied swaption volatilities 637

201 Sample path of a Poisson process (Nt)tisinR+ 654202 Sample path of the Poisson process (Nt)tisinR+ 659203 Sample path of the compensated Poisson process (Nt minus λt)tisinR+ 660204 Sample path of a compound Poisson process (Yt)tisinR+ 662205 Sample trajectories of a gamma process 675206 Sample trajectories of a variance gamma process 676207 Sample trajectories of an inverse Gaussian process 676208 Sample trajectories of a negative inverse Gaussian process 677209 Sample trajectories of a stable process 6772010 USDCNY Exchange rate data 6782011 Geometric Poisson processlowast 680

xx



2012 Ranking data 6802013 Geometric compound Poisson processlowast 6812014 Geometric Brownian motion with compound Poisson jumpslowast 6822015 Share price with jumps 683

211 Market returns vs normalized Gaussian returns 698212 Empirical vs Gaussian CDF 699213 Quantile-Quantile plot 699214 Empirical density vs normalized Gaussian density 700215 Empirical density vs power density 700

221 Divergence of the explicit finite difference method 722222 Stability of the implicit finite difference method 724233 Probability computed as a volume integral 740

S1 Strike price as a function of risk-free rate 771S2 Investment graph 772S3 Investment graph 773S4 Put spread collar price map 791S5 Put spread collar payoff function 792S6 Put spread collar option as a combination of call and put optionslowast 792S7 Call spread collar price map 793S8 Call spread collar payoff function 793S9 Call spread collar option as a combination of call and put optionslowast 794S10 Put option pricing 810S11 Function x 7rarr fε(x) 827S12 Derivative x 7rarr f primeε(x) 828S13 Samples of linear interpolations 831S14 Brownian crossings of level 1lowast 849S15 Brownian path 851S16 Risk-neutral pricing of a foreign exchange option 851S17 Delta hedging of a foreign exchange option 851S18 Bitcoin XBTUSD order book 852S19 Time spent by Brownian motion within a given range 853S20 Market data for the warrant 01897 on the MTR Corporation 859S21 Lower bound vs Black-Scholes call price 871S22 Lower bound vs Black-Scholes put option price 872S23 Bull spread option as a combination of call and put optionslowast 872S24 Bear spread option as a combination of call and put optionslowast 873S25 Butterfly option as a combination of call optionslowast 874S26 Delta of a butterfly option 875S27 Price of a binary call option 886S28 Risky hedging portfolio value for a binary call option 887S29 Risk-free hedging portfolio value for a binary call option 887S30 Black-Scholes price of the maximum chooser option 890

xxi


N Privault

S31 Delta of the maximum chooser option 890S32 Black-Scholes price of the minimum chooser option 891S33 Delta of the minimum chooser option 892S34 Implied vs local volatility 914S35 Average return by selling at the maximum vs selling at maturity 922S36 Ratios of average returns 923S37 Black-Scholes call price upper bound 926S38 Black-Scholes put price upper bound 927S39 ldquoOptimal exerciserdquo put price upper bound 929S40 Price of the up-and-in long forward contract 936S41 Delta of the up-and-in long forward contract 937S42 Price of the up-and-out long forward contract 938S43 Delta of up-and-out long forward contract price 939S44 Price of the down-and-in long forward contract 940S45 Delta of down-and-in long forward contract 940S46 Price of the down-and-out long forward contract 941S47 Delta of down-and-out long forward contract 942S48 Payoff function of the European knock-out call option 946S49 Price map of the European knock-out call option 948S50 Payoff function of the European knock-in put option 948S51 Price map of the European knock-in put option 949S52 Payoff function of the European knock-in call option 949S53 Price map of the European knock-in call option 950S54 Payoff function of the European knock-out put option 951S55 Price map of the European knock-out put option 951S56 Expected minimum of geometric Brownian motion 952S57 Black-Scholes put price upper bound 953S58 Time derivative of the expected minimum 953S59 Expected maximum of geometric Brownian motion 955S60 Black-Scholes call price upper bound 956S61 Time derivative of the expected maximum 956S62 Lookback call option price as a function of maturity time T 959S63 Lookback put option price (2D) as a function of Mt

0 960S64 Hitting times of a straight line started at α lt 0 975S65 Hitting times of a straight line started at α lt 0 976S66 Perpetual vs finite expiration American put option price 980S67 American put price approximation 981S68 Perpetual American binary put price map 993S69 Perpetual American binary call price map 994S70 Finite expiration American binary call price map 996S71 Finite expiration American binary put price map 998S72 Log bond prices correlation graph in the two-factor model 1041

lowast Animated figures (work with Acrobat Reader)

xxii


List of Tables

11 Mark Six ldquoInvestment Tablerdquo 24

21 Self-financing portfolio value process 5422 NTRC Input investment plan 7323 Avenda Insurance investment plan 74

41 Itocirc multiplication table 161

61 Black-Scholes Greeks 21862 Variations of Black-Scholes prices 219

111 Barrier option types 370112 Boundary conditions for barrier option prices 392

121 Extended Itocirc multiplication table 401

141 Martingales and stopping times 467142 List of martingales 473

151 Optimal exercise strategies 499

161 Local vs foreign exchange options 530

181 Stochastic interest rate models 612

191 Forward rates arranged according to a tenor structure 619

201 Itocirc multiplication table with jumps 673

211 Market models and their properties 714

241 CRR pricing and hedging table 780242 CRR pricing tree 782

xxiii

N Privault

243 CRR pricing and hedging tree 783244 CRR pricing tree 786245 CRR pricing and hedging tree 787

xxiv


Introduction

Modern quantitative finance requires a strong background in fields such asstochastic calculus optimization partial differential equations (PDEs) andnumerical methods or even infinite dimensional analysis In addition theemergence of new complex financial instruments on the markets makes itnecessary to rely on increasingly sophisticated mathematical tools Not allreaders of this book will eventually work in quantitative financial analysisnevertheless they may have to interact with quantitative analysts and becom-ing familiar with the tools they employ be an advantage In addition despitethe availability of ready made financial calculators it still makes sense to beable oneself to understand design and implement such financial algorithmsThis can be particularly useful under different types of conditions includ-ing an eventual lack of trust in financial indicators possible unreliability ofexpert advice such as buysell recommendations or other factors such asmarket manipulation Instead of relying on predictions of stock price move-ments based on various tools (technical analysis charting ldquo rdquofigures) we acknowledge that predicting the future is a difficult task and werely on the Efficient Market Hypothesis In this framework the time evolutionof the prices of risky assets will be modeled by random walks and stochasticprocesses

Historical sketch

We start with a description of some of the main steps ideas and individualsthat played an important role in the development of the field over the lastcentury

Robert Brown botanist 1827

Brown observed the movement of pollen particles as described in his paperldquoA brief account of microscopical observations made in the months of June 1


N Privault

July and August 1827 on the particles contained in the pollen of plants andon the general existence of active molecules in organic and inorganic bodiesrdquoPhil Mag 4 161-173 1828

Philosophical Magazine first published in 1798 is a journal that ldquopublishesarticles in the field of condensed matter describing original results theoriesand concepts relating to the structure and properties of crystalline materialsceramics polymers glasses amorphous films composites and soft matterrdquo

Louis Bachelier Mathematician PhD 1900

Bachelier (1900) used Brownian motion for the modeling of stock prices inhis PhD thesis ldquoTheacuteorie de la speacuteculationrdquo Annales Scientifiques de lrsquoEcoleNormale Supeacuterieure 3 (17) 21-86 1900

Albert Einstein physicist

Einstein received his 1921 Nobel Prize in part for investigations on the theoryof Brownian motion ldquo in 1905 Einstein founded a kinetic theory to accountfor this movementrdquo presentation speech by S Arrhenius Chairman of theNobel Committee Dec 10 1922

Einstein (1905) ldquoUumlber die von der molekularkinetischen Theorie der Waumlrmegeforderte Bewegung von in ruhenden Fluumlssigkeiten suspendierten TeilchenrdquoAnnalen der Physik 17

Norbert Wiener Mathematician founder of cybernetics

Wiener is credited among other fundamental contributions for the mathe-matical foundation of Brownian motion published in 1923 In particular heconstructed the Wiener space and Wiener measure on C0([0 1]) (the spaceof continuous functions from [0 1] to R vanishing at 0)

Wiener (1923) ldquoDifferential spacerdquo Journal of Mathematics and Physics ofthe Massachusetts Institute of Technology 2 131-174 1923

Kiyoshi Itocirc (伊伊伊藤藤藤清清清) Mathematician CF Gauss Prize 2006

Itocirc constructed the Itocirc integral with respect to Brownian motion cf ItocircKiyoshi Stochastic integral Proc Imp Acad Tokyo 20 (1944) 519-524 Healso constructed the stochastic calculus with respect to Brownian motionwhich laid the foundation for the development of calculus for random pro-cesses see Itocirc (1951) ldquoOn stochastic differential equationsrdquo in Memoirs ofthe American Mathematical Society

2



ldquoRenowned math wiz Itocirc 93 diesrdquo (The Japan Times Saturday Nov 152008)

Kiyoshi Itocirc an internationally renowned mathematician and professoremeritus at Kyoto University died Monday of respiratory failure at a Ky-oto hospital the university said Friday He was 93 Itocirc was once dubbedldquothe most famous Japanese in Wall Streetrdquo thanks to his contributionto the founding of financial derivatives theory He is known for his workon stochastic differential equations and the ldquoItocirc Formulardquo which laid thefoundation for the Black and Scholes (1973) model a key tool for financialengineering His theory is also widely used in fields like physics and biology

Paul Samuelson economist Nobel Prize 1970

Samuelson (1965) rediscovered Bachelierrsquos ideas and proposed geometricBrownian motion as a model for stock prices In an interview he stated ldquoInthe early 1950s I was able to locate by chance this unknown Bachelier (1900)book rotting in the library of the University of Paris and when I opened itup it was as if a whole new world was laid out before merdquo We refer to ldquoRa-tional theory of warrant pricingrdquo by Paul Samuelson Industrial ManagementReview p 13-32 1965

Fig 01 Clark (2000) ldquoAs if a whole new world was laid out before merdquolowast

In recognition of Bachelierrsquos contribution the Bachelier Finance Society wasstarted in 1996 and now holds the World Bachelier Finance Congress everylowast Click on the figure to play the video (works in Acrobat Reader on the entire pdf file)

3


var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton0)ocgs[i]state=false

N Privault

two years

Robert Merton Myron Scholes economists

Robert Merton and Myron Scholes shared the 1997 Nobel Prize in eco-nomics ldquoIn collaboration with Fisher Black developed a pioneering formulafor the valuation of stock options paved the way for economic valuationsin many areas generated new types of financial instruments and facilitatedmore efficient risk management in societyrdquolowast

Black and Scholes (1973) ldquoThe Pricing of Options and Corporate LiabilitiesrdquoJournal of Political Economy 81 (3) 637-654

The development of options pricing tools contributed greatly to the expansionof option markets and led to development several ventures such as the ldquoLongTerm Capital Managementrdquo (LTCM) founded in 1994 The fund yielded an-nualized returns of over 40 in its first years but registered lost US$ 46billion in less than four months in 1998 which resulted into its closure inearly 2000

Oldřich Vašiacuteček economist 1977

Interest rates behave differently from stock prices notably due to the phe-nomenon of mean reversion and for this reason they are difficult to modelusing geometric Brownian motion Vašiacuteček (1977) was the first to suggest amean-reverting model for stochastic interest rates based on the Ornstein-Uhlenbeck process in ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics 5 177-188

David Heath Robert Jarrow Andrew Morton

These authors proposed in 1987 a general framework to model the evolu-tion of (forward) interest rates known as the Heath-Jarrow-Morton (HJM)model see Heath et al (1992) ldquoBond pricing and the term structure of inter-est rates a new methodology for contingent claims valuationrdquo Econometrica(January 1992) Vol 60 No 1 pp 77-105

Alan Brace Dariusz Gatarek Marek Musiela (BGM)

The Brace et al (1997) model is actually based on geometric Brownianmotion and it is specially useful for the pricing of interest rate derivativessuch as interest rate caps and swaptions on the LIBOR market see ldquoThelowast This has to be put in relation with the modern development of risk societies ldquosocietiesincreasingly preoccupied with the future (and also with safety) which generates thenotion of riskrdquo

4



Market Model of Interest Rate Dynamicsrdquo Mathematical Finance Vol 7page 127 Blackwell 1997 by Alan Brace Dariusz Gatarek Marek Musiela

Financial derivatives

The following graphs exhibit a correlation between commodity (oil) pricesand an oil-related asset price

2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910

(b) Graph of Keppel Corp stock price

Fig 02 Comparison of WTI vs Keppel price graphs

The study of financial derivatives aims at finding functional relationshipsbetween the price of an underlying asset (a company stock price a commodityprice etc) and the price of a related financial contract (an option a financialderivative etc)

Option contracts

Option credit contracts appear to have been used as early as the 10th centuryby traders in the Mediterranean Early accounts of option trades can also befound in The Politics Aristotle (0 BC) by Aristotle (384-322 BC) Referringto the philosopher Thales of Miletus (c 624 - c 546 BC) Aristotle writes

ldquoHe (Thales) knew by his skill in the stars while it was yet winter thatthere would be a great harvest of olives in the coming year so having alittle money he gave deposits for the use of all the olive-presses in Chiosand Miletus which he hired at a low price because no one bid against himWhen the harvest-time came and many were wanted all at once and of asudden he let them out at any rate which he pleased and made a quantityof moneyrdquo

As of year 2015 the size of the financial derivatives market is estimatedat over one quadrillion (or one million billions or 1015) USD which is morethan 10 times the size of the total Gross World Product (GWP)

We close this introduction with a description of (European) call and putoptions which are at the basis of risk management

5


N Privault

European put option contracts

As previously mentioned an important concern for the buyer of a stock attime t is whether its price ST can decline at some future date T The buyer ofthe stock may seek protection from a market crash by purchasing a contractthat allows him to sell his asset at time T at a guaranteed price K fixed attime t This contract is called a put option with strike price K and exercisedate T

Fig 03 Graph of the Hang Seng index - holding a put option might be useful here

Definition 01 A (European) put option is a contract that gives its holderthe right (but not the obligation) to sell a quantity of assets at a predefinedprice K called the strike price (or exercise price) and at a predefined date Tcalled the maturity

In case the price ST falls down below the level K exercising the contract willgive the holder of the option a gain equal to K minus ST in comparison to thosewho did not subscribe the option contract and have to sell the asset at themarket price ST In turn the issuer of the option contract will register a lossalso equal to K minus ST (in the absence of transaction costs and other fees)

If ST is above K then the holder of the option contract will not exercisethe option as he may choose to sell at the price ST In this case the profitderived from the option contract is 0

6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x

Put option payoff (K-x)+

Fig 04 Payoff function of a put option with strike price K = 100

See eg httpoptioncreatorcomstwwxvz

In general the payoff of a (so called European) put option contract can bewritten as

φ(ST ) = (K minus ST )+ =

K minus ST ST 6 K

0 ST gt K

Two possible scenarios (ST finishing above K or below K) are illustrated inFigure 05

0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St

Fig 05 Sample price processes simulated by a geometric Brownian motion

Example of put option the lowast in currency exchange isa common example of simple European put optionlowast Right-click to open or save the attachment

7


N Privault

European call option contracts

On the other hand if the trader aims at buying some stock or commodityhis interest will be in prices not going up and he might want to purchase acall option which is a contract allowing him to buy the considered asset attime T at a price not higher than a level K fixed at time t

Here in the event that ST goes above K the buyer of the option contractwill register a potential gain equal to ST minusK in comparison to an agent whodid not subscribe to the call option

Definition 02 A (European) call option is a contract that gives its holderthe right (but not the obligation) to purchase a quantity of assets at a pre-defined price K called the strike price and at a predefined date T called thematurity

Cash settlement vs physical delivery

Cash settlement In the case of a cash settlement the option contract is-suer will satisfy the option contract by selling α = 1 stock at the priceS1 isin $2 $5 refund the initial $2 loan and hand in the remaining amountC = (S1 minusK)+ to the option contract holder

Physical delivery In the case of physical delivery of the underlying asset theoption contract issuer will deliver α = 1 stock to the option contract holderin exchange for K = $2 which will be used to refund the initial $2 loansubscribed by the option contract issuer

0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x

Call option payoff (x-K)+

Fig 06 Payoff function of a call option with strike price K = 100

See eg httpoptioncreatorcomstqhbgn

8



In general the payoff of a (so called European) call option contract can bewritten as

φ(ST ) = (ST minusK)+ =

ST minusK ST gt K

0 ST 6 KIn market practice options are often divided into a certain number n of war-rants the (possibly fractional) quantity n being called the entitlement ratio

Example of call option the lowast in online booking sys-tems are common examples of simple European call options

The derivatives market

As of year 2015 the size of the derivatives market was estimated at more that$12 quadrilliondagger or more than 10 times the Gross World Product (GWP)See here or here for up-to-date data on notional amounts outstanding andgross market value from the Bank for International Settlements

Option pricing

In order for an option contract to be fair the buyer of the option contractshould pay a fee (similar to an insurance fee) at the signature of the contractThe computation of this fee is an important issue which is known as optionpricing

Option hedging

The second important issue is that of hedging ie how to manage a givenportfolio in such a way that it contains the required random payoff (KminusST )+(for a put option) or (ST minusK)+ (for a call option) at the maturity date T

The next Figure 07 illustrates a sharp increase and sharp drop in assetprice making it valuable to hold a call option contract during the first halfof the graph whereas holding a put option contract would be recommendedduring the second halflowast Right-click to open or save the attachmentdagger One thousand trillion or one million billion or 1015

9


N Privault

Fig 07 ldquoInfogramesrdquo stock price curve

Example Fuel hedging and the four-way zero-collar option

5

10

15

20

25WTI Oil Prices [2010minus01minus042015minus11minus27]

Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120

BRENT Oil Prices [2010minus01minus042015minus11minus30]

Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(b) Brent price graph

Fig 08 Brent and WTI price graphs

1 installpackages(Quandl)library(Quandl)library(quantmod)

3 BRENT_Oil_Prices = Quandl(FREDDCOILBRENTEUstart_date=2010-01-01end_date=2015-11-30type=xts)

chartSeries(BRENT_Oil_Pricesupcol=bluetheme=whitename = BRENT OilPriceslw=5)

5 getSymbols(WTI from=2010-01-01 to=2015-11-30)WTI_Oil_Prices lt- Ad(`WTI`)

7 chartSeries(WTI_Oil_Pricesupcol=bluetheme=whitename = WTI Oil Priceslw=5)getSymbols(DCOILBRENTEU src=FRED)

9 chartSeries(DCOILBRENTEUupcol=bluetheme=whitename = BRENT OilPriceslw=5)

(April 2011) lowast

(November 2015) lowast

lowast Right-click to open or save the attachment

10


ALSO READ

MARKETS (HTTPSWWWBUSINESSDAILYAFRICACOMMARKETS539552-539552-UJWMR5INDEXHTML)

Fuel hedge promises Kenya Airways smooth ride in volatileoil marketTUESDAY APRIL 12 2011 000

Kenya Airwaysrsquo fuel hedge contract is expected to shield the airline from soaring oil prices owing to a lower lock-in rate of $90 a barrel against about

$110 today in international markets

The national carrier is expected to make signicant savings on fuel costs mdash its biggest expenditure

The current hedging contract signed in 2010 will save the company from the current oil price rally triggered by the unrest in fuel-producing countries

of North Africa and the Middle East

ldquoThe high fuel costs may not have a major impact on the airlinersquos earnings given its 50-50 hedging structurerdquo said Sterling Investment Bank in a

research note

Fuel costs constituted more than 40 per cent of Kenya Airways (KQrsquos) direct costs last year an indication of how big its exposure is to volatility in

petroleum prices

(httpswwwbusinessdailyafricacom)

aSearch b

ADVERTISEMENT

A research note by Sterling Investment Bank indicates the hedge would give the airline room to contain costs PhotoFILE

Contain costs

Sterling said the contract would help the airline to contain its costs and boost protability two years after declining oil prices cost the carrier some

Sh89 billion as its forward contract tied it to more expensive fuel

The hedging loss wiped off its operating prots for the year pushing overall performance to an after-tax loss of Sh56 billion as KQ like other airlines

had signed forward contracts when oil prices had peaked at $147 a barrel

Owing to the volatility of oil prices airlines generally enter into forward agreements with nanciers to lock in the prices for about half of their oil

requirements in what is meant to limit exposure to higher unexpected prices

As fuel prices rose gradually in the 200910 year Kenya Airways was able to gain over Sh61 billion from the fuel contracts allowing the rm to post

an overall prot of Sh2 billion

Mr Gregory Waweru a research analyst at Kestrel Capital said the forward hedge reduces exposure to rising prices leaving it better-off as it is

cushioned

ldquoThe forward hedge is set to reduce shocks associated with the increasing oil prices in the international markets which would otherwise inate its

expense billrdquo said Mr Waweru

He added that the recent price gains that the counter has recorded arose from increased demand as investors started trooping back to the market after

reduced activity in the rst quarter of the year with most counters heading South over the period

ldquoWe have seen increased investor demand in the market which is expected to lift the airlinersquos stock price at the market as there has not been any new

information specic to KQrdquo he added

The stock held at Sh3275 on Monday after recording a strong rally during most of last week days after the companyrsquos books were closed for the

nancial year ending March 31

KQ has announced plans to acquire a cargo plane by September this year as it seeks to tap into the growing trade volumes in Africa and Asia and

reverse its falling revenues from the segment

The 737-300 aircraft with a capacity of 19 tonnes per trip is to be acquired on a lease agreement to give the rm a share of trade volumes especially

from China which have been growing in double digits over the past ve years

mmichirakenationmediacom

Nyandarua barred from alcohol taxes October bonds trade falls to Sh35bn

Treasury issues 10-year note seeking to raiseSh50bn

Kenya deals decline 25pc in ten months

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A close look at the role of fuel hedging in Kenya AirwaysrsquoSh26bn lossTHURSDAY NOVEMBER 5 2015 1844

A new era of alternative investments is dawning in Kenya as the capital market prepares for the exciting world of derivatives

Derivatives are nancial instruments which solely derive their value from an underlying interest

They are used to manage risks enhance returns and sometimes facilitate market entry and exit The most popular derivatives are in commodities

currencies stocks bonds and interest rates

As our capital market develops and more Kenyan companies become exposed to global nancial market risks derivatives will increasingly be used to

manage risks

The terms futures forwards options and swaps will become household names in our capital market boardrooms and nancial reporting


aSearch b

ADVERTISEMENT

Kanu Bestlady Sh100m land case delayed LeapFrog injects Sh309m into GoodlifePharmacy expansion

A Kenya Airways plane at Jomo Kenyatta International Airport in Nairobi PHOTO | FILE

The rise in jet fuel cost over several years has been putting pressure on Kenya Airways

(httpswwwbusinessdailyafricacomstocks-13224401394190-shkrn9-indexhtml ) to maintain protability and positive cash ows

Most airlines have traditionally embraced hedging as the ultimate protection against fuel price volatility

After all fuel price is the highest cost of operating an airline Hedging against rising fuel prices seemed smart until the trend reversed mid last year

Hedges turned sour when prices started going south in mid 2014 as a result of increased oil supply from the US and decreased demand as the Chinese

economy contracted No one foresaw the rapid fall in oil prices

Even hedge fund managers who are very good in predicting probable price movements using sophisticated quantitative models missed it

As such we should not solely blame KQ management and the board for bets that turned sour

Their predictions were as good as those made by seasoned derivatives experts After all Kenyans are used to perpetually rising pump prices even

when global oil prices are falling

We have to appreciate that KQ is one of a few Kenyan companies which are signicantly exposed to global nancial risks which are beyond their

control

The airline compared to any other local company has the biggest global footprint As such the company is exposed to not only fuel price

uctuations but also foreign currency and interest rate risks

No wonder Safaricom (httpswwwbusinessdailyafricacomstocks-13224401394278-shkse6-indexhtml ) another Kenyan corporate giant does

not have any derivatives designed as hedging instruments When Kenya Airways released its 2014 results the huge loss of Sh26 billion shook the

entire East African nancial markets to the core

READ Kenya Airways reports record Sh26bn net loss (httpswwwbusinessdailyafricacomCorporate-NewsKenya-Airways-reports-record-Sh25-

7bn-loss-5395502813746-d5g10fz-indexhtml)

The loss was partly due to hedge derivatives which accounted Sh75 billion How did this happen and how could it have been avoided

Kenya Airwaysrsquo use of oil hedge derivatives to protect it against sudden increases in prices is not new The only probable change seems to have been a

switch from oil future hedges to other hedge options

Scangroup CEO gains Sh237m in four-day rallyof shares

KCAA issues tough warning on drones use

ADVERTISEMENT

Itrsquos worth noting that the airline has in the past made massive prots from hedges It did not make any losses related to oil hedges between 2012 and

2014

The management got it right their hedges worked effectively The hedges returned a cumulative gain on fuel derivatives of Sh407 billion with Sh25

billion realised in 2012 alone

Prior to 2013 KQ seemed to have used hedging oil prices future contracts a type of forward where two parties commit to do a transaction at a xed

price at a future date

As in all markets where commodity derivatives are available there is a choice of either using exchange traded derivatives which are contract specic

but very liquid or over the counter (OTC) derivatives which can be tailored to the clientrsquos requirements

Leading rms which trade in energy derivatives include Chicago Mercantile Exchange (CME) Intercontinental Exchange (ICE) and Dubai Mercantile

Exchange (DBE)

Soon Nairobi Securities Exchange will join this league once the Single Stocks Futures starts trading It would be fair to assume that KQ had entered

into energy future contracts instead of forwards since the majority of commodity trading has developed on exchange rather than OTC

If future hedges go against the bet a companyrsquos liquidity is affected negatively due to intra-day prot or losses crystallising on a daily basis

Future prices uctuate from day to day and the contract buyers and sellers attempt to prot from these price changes

This marking to market process means that if the long position loses money against daily settlement prices losses must be paid following the close of

the trade and vice versa

These daily payments of variations margins can partly lead to liquidity issues if the bet goes against the buyer In the case of KQ the hedges worked in

its favour in 2012 resulting in gains of Sh25 billion

In 2014 KQ seems to have changed its tactics and started using options instead of oil futures

It employed fuel options to supposedly hedge rising jet oil risk effectively Unlike a futures contract an option is the only derivative instrument which

allows the buyer to walk away from their obligations

In an option contract the buyer is given a right but not an obligation to purchase or sell something at a later date at a price agreed upon today This is

how the option may have resulted in the loss

In an option if an airlinersquos management forecast rising fuel prices they will buy a call option They will pay an upfront premium

If their prediction comes true and the price increases the airline exercises the option and buys the jet fuel at the lower price which was agreed upon

at the inception of the contract

This call option is said to be in the money since the airline will be buying fuel below the current market price

If it were a vanilla call option and it turned out that the strike price is higher than the current fuel price the airline lets the option expire and buys

fuel at the prevailing market price

This call will be said to be out of the money The only cost that the airline will incur is what it paid at the inception of the call option

Vanilla is the simplest form of derivative Itrsquos derived from ice cream the vanilla type being the simplest and easiest to obtain In some cases jet fuel

future contracts are not available to hedge against a rise in price

Other underlying commodities like crude oil can be used to hedge jet fuel in such situations That is why KQ has been using the Brent Crude Oil call

and put options KQ has scored big time for choosing the four ways zero collar options

Unlike the vanilla call or put option where the buyer pays premium upfront the four ways zero collar is an exotic option requiring minimal or

sometime no upfront premiums cost

This strategy is normally used in bullish conditions KQ must have owned the underlying security which in this case is jet fuel or crude oil Exotic is

the opposite of vanilla

Buying jet fuel derivatives was not the only choice the airline had to hedge rising oil prices

In most cases there can be no assurance that derivatives will provide adequate protection against unpredictable changes in jet fuel cost If hedges are

not working the management can opt to do nothing and keep buying jet fuel based on the current market price

This would have been the case with KQ had the risk management committee predicted that oil prices would fall

KQ management didnrsquot want to take that route since going by history their options were in the money in previous years

As such doing nothing to manage risks was out of question After all they had made money from hedging the previous year This time round the

management lost and bets went the wrong way

As a former British Airways CEO said you can run from high fuel prices briey through hedging but you canrsquot run very long

In this era of falling oil prices doing away with hedges may seem like the best route to take In the US American Airlines did away with oil hedges

The no hedge policy has enabled the company to report decent prots while competitors like South Western and Delta are feeling the heat of oil

derivatives gone sour

Hedging is like insurance if the cost of having a policy outweighs benets the prudent thing to do is not to take one

Now that low oil prices will persist for an unforeseeable future breaking the hedge contracts may be cheaper for KQ in the long run instead of piling

up losses

One of the ways of getting out of the contracts is through novation In novation the airline transfers all its obligations and rights under either the call

or put option to a third party

Derivatives markets are active and there is always a party willing to buy out contracts and bet in the opposite direction

As a result of the hedges the airline is currently buying fuel at $80 per barrel This is $35 more than the prevailing market price

The management should look at the cost of innovating from these hedges The airline should discontinue the current hedges if the cost involved in

revoking them is lower than the cost of buying jet fuel at the current inated hedge prices

The savings could be huge given that the airline has fuel hedge contracts for 40 per cent of the anticipated usage

Mr Kiragu is an accounting and nance professor at Mount Saint Vincent University Canada

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70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x

Fig 09 Price graph for a four-way collar option

The four-way collar option contract will result into a positive or negativepayoff depending on current fuel prices as illustrated in Figure 010

-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST

four-way collar payoff

Fig 010 Payoff function of a four-way collar option

The four-way collar payoff can be written as a linear combination

φ(ST ) = (K1 minus ST )+ minus (K2 minus ST )+ + (ST minusK3)+ minus (ST minusK4)

+

of call and put option payoffs with respective strike prices

K1 = 90 K2 = 100 K3 = 120 K4 = 130

see eg httpoptioncreatorcomst5rf51

11


N Privault

Fig 011 Four-way collar option as a combination of call and put optionslowast

Therefore the four-way collar option contract can be synthesized by

1 purchasing a put option with strike price K1 = $90 and2 selling (or issuing) a put option with strike price K2 = $100 and3 purchasing a call option with strike price K3 = $120 and4 selling (or issuing) a call option with strike price K4 = $130

Moreover the collar option contract can be made costless by adjusting theboundaries K1 K2 K3 K4

Example - the 4-5-2 model

We close this introduction with a simplified example of the pricing and hedg-ing technique in a binary model Consider a risky stock valued S0 = $4 attime t = 0 and taking only two possible values

S1 =

$5

$2

at time t = 1 In addition consider an option contract that promises a claimpayoff C whose values are contingent to the market data of S1

C =

$3 if S1 = $5

$0 if S1 = $2

lowast The animation works in Acrobat Reader on the entire pdf file

12



Question Does C represent the payof of a put option contract Of a calloption contract If yes with which strike price K

At time t = 0 the option contract issuer (or writer) chooses to invest α unitsin the risky asset S while keeping $β on our bank account meaning that weinvest a total amount

αS0 + $β at t = 0

Here the amount $β may be positive or negative depending on whether itis corresponds to savings or to debt and is interpreted as a liability

The following issues can be addresseda) Hedging how to choose the portfolio allocation (α $β) so that the value

αS1 + $β

of the portfolio matches the future payoff C at time t = 1

b) Pricing how to determine the amount αS0 + $β to be invested by theoption contract issuer in such a portfolio at time t = 0

S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0

Hedging means that at time t = 1 the portfolio value matches the futurepayoff C ie

αS1 + $β = C

This condition can be rewritten as

C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3

2α+ β = 0which yields

α = 1

$β = minus$2

In other words the option contract issuer purchases 1 (one) unit of the stockS at the price S0 = $4 and borrows $2 from the bank The price of the 13


N Privault

option contract is then given by the portfolio value

αS0 + $β = 1times $4minus $2 = $2

at time t = 0

The above computation is implemented in the attachedthat can be run here This algorithm is scalable and can be extended torecombining binary trees over multiple time steps

Definition 03 The arbitrage price of the option contract is interpreted asthe initial cost αS0 + $β of the portfolio hedging the claim payoff C

Conclusion in order to deliver the random payoff C =

$3 if S1 = $5

$0 if S1 = $2to the option contract holder at time t = 1 the option contract issuer (orwriter) has to

1 charge αS0 + $β = $2 (the option contract price) at time t = 0

2 borrow minus$β = $2 from the bank

3 invest those $2+ $2 = $4 into the purchase of α = 1 unit of stock valuedat S0 = $4 at time t = 0

4 wait until time t = 1 to sell the stock at the price S1 = $5 or S1 = $2and refund the $2 loan

We find that the portfolio value evolved into

C =

αtimes $5 + $β = 1times $5minus $2 = $3 if S1 = $5

αtimes $2 + $β = 1times $2minus $2 = 0 if S1 = $2

so that the option contract and the equality C = αS1 + $β can be fulfilledwhatever the evolution of S and allows us to break even

Here the option contract price αS0 + $β = $2 is interpreted as the cost ofhedging the option In Chapters 2 and 3 we will see that this model is scalableand extends to discrete time

We note that the initial option contract price of $2 can be turned to C = $3(50 profit) or into C = $0 (total ruin)

Thinking further

1) The expected claim payoff at time t = 1 is14


cells [ cell_type code execution_count null metadata outputs [] source [ from IPythoncoredisplay import display HTMLn display(HTML( )) ] cell_type code execution_count null metadata outputs [] source [ pip install networkx==23 ] cell_type code execution_count null metadata outputs [] source [ This code implements hedging and pricing using backward recursionn n matplotlib inlinen import networkx as nx n import numpy as npn import matplotlib n import matplotlibpyplot as plt n n N=1S0=4n n r=00a=-05b=025n n p = (r-a)(b-a)n q = (b-r)(b-a)n n def plot_tree(g)n pltfigure(figsize=(2010))n pos=n lab=n n for n in gnodes()n pos[n]=(n[0]n[1])n if gnode[n][value] is not None lab[n]=float(02fformat(gnode[n][value]))n n elarge=gedges(data=True)n nxdraw_networkx_labels(gposlabfont_size=15)n nxdraw_networkx_nodes(gposnode_color=redalpha=04node_size=1000)n nxdraw_networkx_edges(gposedge_color=bluealpha=07width=3edgelist=elarge)n pltylim(-N+05N+15) n pltxlim(-05N+05)n pltshow()n n def graph_stock()n S=nxGraph()n for k in range(0N)n for l in range(-k+1k+32)n Sadd_edge((kl)(k+1l+1))n Sadd_edge((kl)(k+1l-1))n n for n in Snodes() n k=n[0]n l=n[1]-1n Snode[n][value]=S0((10+b)((k+l)2))((10+a)((k-l)2))n return Sn n plot_tree(graph_stock()) ] cell_type code execution_count null metadata outputs [] source [ def European_call_hedge_risky(K)n n price = nxGraph()n hedge_risky = nxGraph()n S = graph_stock()n n for k in range(0N)n for l in range(-k+1k+32)n priceadd_edge((kl)(k+1l+1))n priceadd_edge((kl)(k+1l-1))n hedge_riskyadd_edge((kl)(k+1l+1))n hedge_riskyadd_edge((kl)(k+1l-1))n n for l in range(-N+1N+32)n pricenode[(Nl)][value] = npmaximum(Snode[(Nl)][value]-K0)n hedge_riskynode[(Nl)][value] = Nonen n for k in reversed(range(0N))n for l in range(-k+1k+32)n pricenode[(kl)][value] = (pricenode[(k+1l+1)][value]p+pricenode[(k+1l-1)][value]q)(1+r)n hedge_riskynode[(kl)][value] = (pricenode[(k+1l+1)][value]-pricenode[(k+1l-1)][value])(b-a)(Snode[(kl)][value])n return hedge_risky ] cell_type code execution_count null metadata outputs [] source [ def European_call_hedge_riskless(K)n n price = nxGraph()n hedge_riskless = nxGraph()n S = graph_stock()n n for k in range(0N)n for l in range(-k+1k+32)n priceadd_edge((kl)(k+1l+1))n priceadd_edge((kl)(k+1l-1))n hedge_risklessadd_edge((kl)(k+1l+1))n hedge_risklessadd_edge((kl)(k+1l-1))n n for l in range(-N+1N+32)n pricenode[(Nl)][value] = npmaximum(Snode[(Nl)][value]-K0)n hedge_risklessnode[(Nl)][value] = Nonen n for k in reversed(range(0N))n for l in range(-k+1k+32)n pricenode[(kl)][value] = (pricenode[(k+1l+1)][value]p+pricenode[(k+1l-1)][value]q)(1+r)n hedge_risklessnode[(kl)][value] = ((1+b)pricenode[(k+1l-1)][value]-(1+a)pricenode[(k+1l+1)][value])(b-a)pow(1+rk+1)n return hedge_riskless ] cell_type code execution_count null metadata outputs [] source [ K = input(Strike Price K=)n n print(Underlying asset prices)n plot_tree(graph_stock())n print(Risky hedging strategy)n plot_tree(European_call_hedge_risky(float(K)))n print(Riskless hedging strategy)n plot_tree(European_call_hedge_riskless(float(K))) ] cell_type code execution_count null metadata outputs [] source [ def Hedge_then_price(K)n n hedge_riskless = European_call_hedge_riskless(K)n hedge_risky = European_call_hedge_risky(K)n S = graph_stock()n hedge_then_price = nxGraph()n n for k in range(0N)n for l in range(-k+1k+32)n hedge_then_priceadd_edge((kl)(k+1l+1))n hedge_then_priceadd_edge((kl)(k+1l-1))n n for l in range(-N+1N+32)n hedge_riskynode[(Nl)][value] = 0n hedge_then_pricenode[(Nl)][value] = npmaximum(Snode[(Nl)][value]-K0)n n for k in reversed(range(0N))n for l in range(-k+1k+32)n hedge_then_pricenode[(kl)][value] = hedge_riskynode[(kl)][value]Snode[(kl)][value]+hedge_risklessnode[(kl)][value](1+r)kn return hedge_then_price ] cell_type code execution_count null metadata outputs [] source [ print(Option prices)n plot_tree(Hedge_then_price(float(K))) ] ] metadata anaconda-cloud kernelspec display_name Python 3 language python name python3 language_info codemirror_mode name ipython version 3 file_extension py mimetype textx-python name python nbconvert_exporter python pygments_lexer ipython3 version 373 widgets state version 112 nbformat 4 nbformat_minor 1


IE[C] = $3timesP(C = $3) + $0timesP(C = $0)= $3timesP(S1 = $5)

In absence of arbitrage opportunities (ldquofair marketrdquo) this expected payoffIE[C] should equal the initial amount $2 invested in the option In that casewe should have IE[C] = $3timesP(S1 = $5) = $2

P(S1 = $5) + P(S1 = $2) = 1from which we can infer the probabilities

P(S1 = $5) = 23

P(S1 = $2) = 13

(01)

which are called risk-neutral probabilities We see that under the risk-neutralprobabilities the stock S has twice more chances to go up than to go downin a ldquofairrdquo market

2) Based on the probabilities (01) we can also compute the expected valueIE[S1] of the stock at time t = 1 We find

IE[S1] = $5timesP(S1 = $5) + $2timesP(S1 = $2)

= $5times 23 + $2times 1

3= $4= S0

Here this means that on average no additional profit can be made from aninvestment on the risky stock In a more realistic model we can assume thatthe riskles bank account yields an interest rate equal to r in which case theabove analysis is modified by letting $β become $(1 + r)β at time t = 1nevertheless the main conclusions remain unchanged

Implied probabilities

By matching the theoretical price IE[C] to an actual market price data $Pas

$P = IE[C] = $3timesP(C = $3) + $0timesP(C = $0) = $3timesP(S1 = $5)

we can infer the probabilities

15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)

which are implied probabilities estimated from market data as illustrated inFigure 012 We note that the conditions 0 lt P(S1 = $5) P(S1 = $2) lt 1are equivalent to 0 lt $P lt 3 which is consistent with financial intuition

Fig 012 Implied probabilities

Note that implied probabilities should also be used with caution as shownin Figures 013-014

Fig 013 Implied probabilities according to bookmakers

16



Fig 014 Implied probabilities according to polling

Implied probabilities can be estimated using eg binary options see for ex-ample Exercise 39The Practitioner expects a good model to be

bull Robust with respect to missing spurious or noisy databull Fast - prices have to be delivered daily in the morningbull Easy to calibrate - parameter estimationbull Stable with respect to re-calibration and the use of new data sets

Typically a medium size bank manages 5000 options and 10000 deals dailyover 1000 possible scenarios and dozens of time steps This can mean ahundred million computations of IE[C] daily or close to a billion such com-putations for a large bank

The Mathematician tends to focus on more theoretical features such as

bull Elegancebull Sophisticationbull Existence of analytical (closed-form) solutions error boundsbull Significance to mathematical finance

This includes

bull Creating new payoff functions and structured productsbull Defining new models for underlying asset pricesbull Finding new ways to compute expectations IE[C] and hedging strategies

The methods involved include

bull Monte Carlo (60)bull PDEs and finite differences (30)bull Other analytic methods and approximations (10)

+ AI and Machine Learning techniques

17


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Preface

This text is an introduction to pricing and hedging in discrete andcontinuous-time financial models without friction (ie without transactioncosts) with an emphasis on the complementarity between analytical andprobabilistic methods Its contents are mostly mathematical and also aim atmaking the reader aware of both the power and limitations of mathematicalmodels in finance by taking into account their conditions of applicabilityThe book covers a wide range of classical topics including Black-Scholes pric-ing exotic and american options term structure modeling and change ofnumeacuteraire as well as models with jumps It is targeted at the advanced un-dergraduate and graduate level in applied mathematics financial engineeringand economics The point of view adopted is that of mainstream mathemat-ical finance in which the computation of fair prices is based on the absenceof arbitrage hypothesis therefore excluding riskless profit based on arbitrageopportunities and basic (buying lowselling high) trading Similarly this doc-ument is not concerned with any ldquopredictionrdquo of stock price behaviors thatbelong other domains such as technical analysis which should not be con-fused with the statistical modeling of asset prices The text also about 20examples based on actual market data

The descriptions of the asset model self-financing portfolios arbitrage andmarket completeness are first given in Chapter 1 in a simple two time-stepsetting These notions are then reformulated in discrete time in Chapter 2Here the impossibility to access future information is formulated using thenotion of adapted processes which will play a central role in the constructionof stochastic calculus in continuous time

In order to trade efficiently it would be useful to have a formula to esti-mate the ldquofair pricerdquo of a given risky asset helping for example to determinewhether the asset is undervalued or overvalued at a given time Althoughsuch a formula is not available we can instead derive formulas for the pric-ing of options that can act as insurance contracts to protect their holdersagainst adverse changes in the prices of risky assets The pricing and hedgingof options in discrete time particularly in the fundamental example of the

v


N Privault






vi







vii


N Privault

viii


Contents

Introduction 1




ix

N Privault







x









xi


N Privault







xii







Index 1087



N Privault

References 1101

xiv


List of Figures





xv

N Privault





xvi







xvii


N Privault







xviii











xix


N Privault





xx







xxi


N Privault




xxii


List of Tables















xxiii

N Privault


xxiv


Introduction


Historical sketch





N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















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70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

N Privault






vi







vii


N Privault

viii


Contents

Introduction 1




ix

N Privault







x









xi


N Privault







xii







Index 1087



N Privault

References 1101

xiv


List of Figures





xv

N Privault





xvi







xvii


N Privault







xviii











xix


N Privault





xx







xxi


N Privault




xxii


List of Tables















xxiii

N Privault


xxiv


Introduction


Historical sketch





N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

ADVERTISEMENT

You May Like



Franklin Templeton



HealthHub




WealthMasteryasia






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Womens Method







AspireAbovecom














11 HOURS AGO



18 HOURS AGO

IN THE HEADLINES








Kingdom of Men





17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






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Food amp Drinks



Health amp Fitness



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Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



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Letters





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Market News








Apps



Mar2515-1)












Follow us


ALSO READ








manage risks



aSearch b

ADVERTISEMENT














control













ADVERTISEMENT


2014








Exchange (DBE)










































up losses









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You May Like




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Kingdom of Men











11 HOURS AGO



17 HOURS AGO

IN THE HEADLINES


fannntastic-woman











Gentsprime






17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



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Letters





Capital Markets



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Apps



Mar2515-1)












Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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01

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anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39






vii


N Privault

viii


Contents

Introduction 1




ix

N Privault







x









xi


N Privault







xii







Index 1087



N Privault

References 1101

xiv


List of Figures





xv

N Privault





xvi







xvii


N Privault







xviii











xix


N Privault





xx







xxi


N Privault




xxii


List of Tables















xxiii

N Privault


xxiv


Introduction


Historical sketch





N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

ADVERTISEMENT

You May Like



Franklin Templeton



HealthHub




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom














11 HOURS AGO



18 HOURS AGO

IN THE HEADLINES








Kingdom of Men





17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us


ALSO READ








manage risks



aSearch b

ADVERTISEMENT














control













ADVERTISEMENT


2014








Exchange (DBE)










































up losses









ADVERTISEMENT

You May Like




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom







Kingdom of Men











11 HOURS AGO



17 HOURS AGO

IN THE HEADLINES


fannntastic-woman











Gentsprime






17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

00

01

02

03

04

05

06

07

08

09

010

011

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0221

0222

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0226

0227

0228

0229

0230

anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

N Privault

viii


Contents

Introduction 1




ix

N Privault







x









xi


N Privault







xii







Index 1087



N Privault

References 1101

xiv


List of Figures





xv

N Privault





xvi







xvii


N Privault







xviii











xix


N Privault





xx







xxi


N Privault




xxii


List of Tables















xxiii

N Privault


xxiv


Introduction


Historical sketch





N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

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70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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0228

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0230

anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

Contents

Introduction 1




ix

N Privault







x









xi


N Privault







xii







Index 1087



N Privault

References 1101

xiv


List of Figures





xv

N Privault





xvi







xvii


N Privault







xviii











xix


N Privault





xx







xxi


N Privault




xxii


List of Tables















xxiii

N Privault


xxiv


Introduction


Historical sketch





N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

ADVERTISEMENT

You May Like



Franklin Templeton



HealthHub




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom














11 HOURS AGO



18 HOURS AGO

IN THE HEADLINES








Kingdom of Men





17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us


ALSO READ








manage risks



aSearch b

ADVERTISEMENT














control













ADVERTISEMENT


2014








Exchange (DBE)










































up losses









ADVERTISEMENT

You May Like




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TopCars






Womens Method







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Kingdom of Men











11 HOURS AGO



17 HOURS AGO

IN THE HEADLINES


fannntastic-woman











Gentsprime






17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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0201

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0217

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0220

0221

0222

0223

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0226

0227

0228

0229

0230

anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

N Privault







x









xi


N Privault







xii







Index 1087



N Privault

References 1101

xiv


List of Figures





xv

N Privault





xvi







xvii


N Privault







xviii











xix


N Privault





xx







xxi


N Privault




xxii


List of Tables















xxiii

N Privault


xxiv


Introduction


Historical sketch





N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

ADVERTISEMENT

You May Like



Franklin Templeton



HealthHub




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom














11 HOURS AGO



18 HOURS AGO

IN THE HEADLINES








Kingdom of Men





17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us


ALSO READ








manage risks



aSearch b

ADVERTISEMENT














control













ADVERTISEMENT


2014








Exchange (DBE)










































up losses









ADVERTISEMENT

You May Like




WealthMasteryasia






TopCars






Womens Method







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Kingdom of Men











11 HOURS AGO



17 HOURS AGO

IN THE HEADLINES


fannntastic-woman











Gentsprime






17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

00

01

02

03

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anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39








xi


N Privault







xii







Index 1087



N Privault

References 1101

xiv


List of Figures





xv

N Privault





xvi







xvii


N Privault







xviii











xix


N Privault





xx







xxi


N Privault




xxii


List of Tables















xxiii

N Privault


xxiv


Introduction


Historical sketch





N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

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Contain costs










cushioned

















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70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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0201

0202

0203

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0205

0206

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0208

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0210

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0213

0214

0215

0216

0217

0218

0219

0220

0221

0222

0223

0224

0225

0226

0227

0228

0229

0230

anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

N Privault







xii







Index 1087



N Privault

References 1101

xiv


List of Figures





xv

N Privault





xvi







xvii


N Privault







xviii











xix


N Privault





xx







xxi


N Privault




xxii


List of Tables















xxiii

N Privault


xxiv


Introduction


Historical sketch





N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

ADVERTISEMENT

You May Like



Franklin Templeton



HealthHub




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom














11 HOURS AGO



18 HOURS AGO

IN THE HEADLINES








Kingdom of Men





17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



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Travel



Book Review





Columnists



Editorials



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Letters





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Apps



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Follow us


ALSO READ








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ADVERTISEMENT














control













ADVERTISEMENT


2014








Exchange (DBE)










































up losses









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You May Like




WealthMasteryasia






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Kingdom of Men











11 HOURS AGO



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IN THE HEADLINES


fannntastic-woman











Gentsprime






17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



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Travel



Book Review





Columnists



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Capital Markets



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Apps



Mar2515-1)












Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

00

01

02

03

04

05

06

07

08

09

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0100

0101

0102

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0111

0112

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0114

0115

0116

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0120

0121

0122

0123

0124

0125

0126

0127

0128

0129

0130

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0135

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0138

0139

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0164

0165

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0168

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0170

0171

0172

0173

0174

0175

0176

0177

0178

0179

0180

0181

0182

0183

0184

0185

0186

0187

0188

0189

0190

0191

0192

0193

0194

0195

0196

0197

0198

0199

0200

0201

0202

0203

0204

0205

0206

0207

0208

0209

0210

0211

0212

0213

0214

0215

0216

0217

0218

0219

0220

0221

0222

0223

0224

0225

0226

0227

0228

0229

0230

anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39






Index 1087



N Privault

References 1101

xiv


List of Figures





xv

N Privault





xvi







xvii


N Privault







xviii











xix


N Privault





xx







xxi


N Privault




xxii


List of Tables















xxiii

N Privault


xxiv


Introduction


Historical sketch





N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

ADVERTISEMENT

You May Like



Franklin Templeton



HealthHub




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom














11 HOURS AGO



18 HOURS AGO

IN THE HEADLINES








Kingdom of Men





17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



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Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



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Global Markets



Market News








Apps



Mar2515-1)












Follow us


ALSO READ








manage risks



aSearch b

ADVERTISEMENT














control













ADVERTISEMENT


2014








Exchange (DBE)










































up losses









ADVERTISEMENT

You May Like




WealthMasteryasia






TopCars






Womens Method







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Kingdom of Men











11 HOURS AGO



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IN THE HEADLINES


fannntastic-woman











Gentsprime






17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

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17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

N Privault

References 1101

xiv


List of Figures





xv

N Privault





xvi







xvii


N Privault







xviii











xix


N Privault





xx







xxi


N Privault




xxii


List of Tables















xxiii

N Privault


xxiv


Introduction


Historical sketch





N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















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70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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anm0

animtiph1

animtiph2

pbsARFix2

ikona2

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pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

List of Figures





xv

N Privault





xvi







xvii


N Privault







xviii











xix


N Privault





xx







xxi


N Privault




xxii


List of Tables















xxiii

N Privault


xxiv


Introduction


Historical sketch





N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

ADVERTISEMENT

You May Like



Franklin Templeton



HealthHub




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom














11 HOURS AGO



18 HOURS AGO

IN THE HEADLINES








Kingdom of Men





17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



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Travel



Book Review





Columnists



Editorials



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Letters





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Apps



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Follow us


ALSO READ








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ADVERTISEMENT














control













ADVERTISEMENT


2014








Exchange (DBE)










































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You May Like




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Kingdom of Men











11 HOURS AGO



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17 HOURS AGO



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Counties



East Africa



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Companies



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Industry



MarketPlace






Technology





BD Life



Art



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Food amp Drinks



Health amp Fitness



Man About Town



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Capital Markets



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Apps



Mar2515-1)












Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

N Privault





xvi







xvii


N Privault







xviii











xix


N Privault





xx







xxi


N Privault




xxii


List of Tables















xxiii

N Privault


xxiv


Introduction


Historical sketch





N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

ADVERTISEMENT

You May Like



Franklin Templeton



HealthHub




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom














11 HOURS AGO



18 HOURS AGO

IN THE HEADLINES








Kingdom of Men





17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us


ALSO READ








manage risks



aSearch b

ADVERTISEMENT














control













ADVERTISEMENT


2014








Exchange (DBE)










































up losses









ADVERTISEMENT

You May Like




WealthMasteryasia






TopCars






Womens Method







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Kingdom of Men











11 HOURS AGO



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IN THE HEADLINES


fannntastic-woman











Gentsprime






17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39






xvii


N Privault







xviii











xix


N Privault





xx







xxi


N Privault




xxii


List of Tables















xxiii

N Privault


xxiv


Introduction


Historical sketch





N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















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Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

N Privault







xviii











xix


N Privault





xx







xxi


N Privault




xxii


List of Tables















xxiii

N Privault


xxiv


Introduction


Historical sketch





N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

ADVERTISEMENT

You May Like



Franklin Templeton



HealthHub




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom














11 HOURS AGO



18 HOURS AGO

IN THE HEADLINES








Kingdom of Men





17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



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Market News








Apps



Mar2515-1)












Follow us


ALSO READ








manage risks



aSearch b

ADVERTISEMENT














control













ADVERTISEMENT


2014








Exchange (DBE)










































up losses









ADVERTISEMENT

You May Like




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Kingdom of Men











11 HOURS AGO



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IN THE HEADLINES


fannntastic-woman











Gentsprime






17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



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Book Review





Columnists



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Capital Markets



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Apps



Mar2515-1)












Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

00

01

02

03

04

05

06

07

08

09

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0221

0222

0223

0224

0225

0226

0227

0228

0229

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anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39










xix


N Privault





xx







xxi


N Privault




xxii


List of Tables















xxiii

N Privault


xxiv


Introduction


Historical sketch





N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

ADVERTISEMENT

You May Like



Franklin Templeton



HealthHub




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom














11 HOURS AGO



18 HOURS AGO

IN THE HEADLINES








Kingdom of Men





17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us


ALSO READ








manage risks



aSearch b

ADVERTISEMENT














control













ADVERTISEMENT


2014








Exchange (DBE)










































up losses









ADVERTISEMENT

You May Like




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom







Kingdom of Men











11 HOURS AGO



17 HOURS AGO

IN THE HEADLINES


fannntastic-woman











Gentsprime






17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

00

01

02

03

04

05

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08

09

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anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

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pbsARFix6

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pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

N Privault





xx







xxi


N Privault




xxii


List of Tables















xxiii

N Privault


xxiv


Introduction


Historical sketch





N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

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Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

00

01

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0221

0222

0223

0224

0225

0226

0227

0228

0229

0230

anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39






xxi


N Privault




xxii


List of Tables















xxiii

N Privault


xxiv


Introduction


Historical sketch





N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

ADVERTISEMENT

You May Like



Franklin Templeton



HealthHub




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom














11 HOURS AGO



18 HOURS AGO

IN THE HEADLINES








Kingdom of Men





17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us


ALSO READ








manage risks



aSearch b

ADVERTISEMENT














control













ADVERTISEMENT


2014








Exchange (DBE)










































up losses









ADVERTISEMENT

You May Like




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Womens Method







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Kingdom of Men











11 HOURS AGO



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IN THE HEADLINES


fannntastic-woman











Gentsprime






17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



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Proles



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Book Review





Columnists



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Capital Markets



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Apps



Mar2515-1)












Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

00

01

02

03

04

05

06

07

08

09

010

011

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0201

0202

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0204

0205

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0207

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0210

0211

0212

0213

0214

0215

0216

0217

0218

0219

0220

0221

0222

0223

0224

0225

0226

0227

0228

0229

0230

anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

N Privault




xxii


List of Tables















xxiii

N Privault


xxiv


Introduction


Historical sketch





N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

ADVERTISEMENT

You May Like



Franklin Templeton



HealthHub




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom














11 HOURS AGO



18 HOURS AGO

IN THE HEADLINES








Kingdom of Men





17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us


ALSO READ








manage risks



aSearch b

ADVERTISEMENT














control













ADVERTISEMENT


2014








Exchange (DBE)










































up losses









ADVERTISEMENT

You May Like




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom







Kingdom of Men











11 HOURS AGO



17 HOURS AGO

IN THE HEADLINES


fannntastic-woman











Gentsprime






17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

00

01

02

03

04

05

06

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08

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anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

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pbsARFix10

pbsARFix11

pbsARFix12

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pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

List of Tables















xxiii

N Privault


xxiv


Introduction


Historical sketch





N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

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Mar2515-1)












Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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01

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0221

0222

0223

0224

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0226

0227

0228

0229

0230

anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

N Privault


xxiv


Introduction


Historical sketch





N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

ADVERTISEMENT

You May Like



Franklin Templeton



HealthHub




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom














11 HOURS AGO



18 HOURS AGO

IN THE HEADLINES








Kingdom of Men





17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us


ALSO READ








manage risks



aSearch b

ADVERTISEMENT














control













ADVERTISEMENT


2014








Exchange (DBE)










































up losses









ADVERTISEMENT

You May Like




WealthMasteryasia






TopCars






Womens Method







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Kingdom of Men











11 HOURS AGO



17 HOURS AGO

IN THE HEADLINES


fannntastic-woman











Gentsprime






17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

00

01

02

03

04

05

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09

010

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0210

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0213

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0216

0217

0218

0219

0220

0221

0222

0223

0224

0225

0226

0227

0228

0229

0230

anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

Introduction


Historical sketch





N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

ADVERTISEMENT

You May Like



Franklin Templeton



HealthHub




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom














11 HOURS AGO



18 HOURS AGO

IN THE HEADLINES








Kingdom of Men





17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us


ALSO READ








manage risks



aSearch b

ADVERTISEMENT














control













ADVERTISEMENT


2014








Exchange (DBE)










































up losses









ADVERTISEMENT

You May Like




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom







Kingdom of Men











11 HOURS AGO



17 HOURS AGO

IN THE HEADLINES


fannntastic-woman











Gentsprime






17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

00

01

02

03

04

05

06

07

08

09

010

011

012

013

014

015

016

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0226

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0230

anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

N Privault













2









3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

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Mar2515-1)












Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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01

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anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39








3



N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

ADVERTISEMENT

You May Like



Franklin Templeton



HealthHub




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom














11 HOURS AGO



18 HOURS AGO

IN THE HEADLINES








Kingdom of Men





17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us


ALSO READ








manage risks



aSearch b

ADVERTISEMENT














control













ADVERTISEMENT


2014








Exchange (DBE)










































up losses









ADVERTISEMENT

You May Like




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom







Kingdom of Men











11 HOURS AGO



17 HOURS AGO

IN THE HEADLINES


fannntastic-woman











Gentsprime






17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

00

01

02

03

04

05

06

07

08

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011

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0226

0227

0228

0229

0230

anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

N Privault

two years











4






2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

ADVERTISEMENT

You May Like



Franklin Templeton



HealthHub




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom














11 HOURS AGO



18 HOURS AGO

IN THE HEADLINES








Kingdom of Men





17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us


ALSO READ








manage risks



aSearch b

ADVERTISEMENT














control













ADVERTISEMENT


2014








Exchange (DBE)










































up losses









ADVERTISEMENT

You May Like




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom







Kingdom of Men











11 HOURS AGO



17 HOURS AGO

IN THE HEADLINES


fannntastic-woman











Gentsprime






17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

00

01

02

03

04

05

06

07

08

09

010

011

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0230

anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39





2012 2013 2014 2015 2016

4060

8010

0

(a) WTI price graph

2012 2013 2014 2015 2016

78

910




Option contracts





5


N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


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research note


petroleum prices


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70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

N Privault







6



0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

ADVERTISEMENT

You May Like



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HealthHub




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Counties



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Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

00

01

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anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39


0

5

10

15

20

80 85 90 95 100 105 110 115 120

(K-x)+

x






K minus ST ST 6 K

0 ST gt K


0

1

2

3

4

5

6

7

8

9

10

01 02 03 04 05 06 07 08 09 1

S01

T=

Strike price

ST-Kgt0

ST-Klt0

St



7


N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

ADVERTISEMENT

You May Like



Franklin Templeton



HealthHub




WealthMasteryasia






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17 HOURS AGO



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Counties



East Africa



World





Companies



Enterprise



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Industry



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Columnists



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Apps



Mar2515-1)












Follow us


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manage risks



aSearch b

ADVERTISEMENT














control













ADVERTISEMENT


2014








Exchange (DBE)










































up losses









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You May Like




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11 HOURS AGO



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fannntastic-woman











Gentsprime






17 HOURS AGO



1 DAY AGO






Counties



East Africa



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Companies



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Apps



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Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

00

01

02

03

04

05

06

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08

09

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anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

N Privault








0

5

10

15

20

80 85 90 95 100 105 110 115 120

(x-K)+

x




8





ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

ADVERTISEMENT

You May Like



Franklin Templeton



HealthHub




WealthMasteryasia






TopCars






Womens Method







AspireAbovecom














11 HOURS AGO



18 HOURS AGO

IN THE HEADLINES








Kingdom of Men





17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



Society



Travel



Book Review





Columnists



Editorials



Ideas amp Debate



Letters





Capital Markets



Commodities



Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us


ALSO READ








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70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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pbsARFix21

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pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

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pbsARFix34

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anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

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ST minusK ST gt K





Option pricing


Option hedging



9


N Privault



5

10

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20


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70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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animtiph1

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pbsARFix2

ikona2

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pbsARFix11

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pbsARFix14

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pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

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N Privault



5

10

15

20


Last 352

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015

(a) WTI price graph

40

60

80

100

120


Last 4373

Jan 042010

Jan 032011

Jan 032012

Jan 022013

Jan 022014

Jan 022015









(April 2011) lowast



10


ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

ADVERTISEMENT

You May Like



Franklin Templeton



HealthHub




WealthMasteryasia






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control













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2014








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Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

ALSO READ









research note


petroleum prices


aSearch b

ADVERTISEMENT


Contain costs










cushioned

















ADVERTISEMENT

ADVERTISEMENT

You May Like



Franklin Templeton



HealthHub




WealthMasteryasia






TopCars






Womens Method







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IN THE HEADLINES








Kingdom of Men





17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



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Fashion






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Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



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Travel



Book Review





Columnists



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Letters





Capital Markets



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Market News








Apps



Mar2515-1)












Follow us


ALSO READ








manage risks



aSearch b

ADVERTISEMENT














control













ADVERTISEMENT


2014








Exchange (DBE)










































up losses









ADVERTISEMENT

You May Like




WealthMasteryasia






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Womens Method







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11 HOURS AGO



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fannntastic-woman











Gentsprime






17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






Gardening



Food amp Drinks



Health amp Fitness



Man About Town



Music



Personal Finance



Proles



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Travel



Book Review





Columnists



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Capital Markets



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Currencies



Global Markets



Market News








Apps



Mar2515-1)












Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

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1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

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1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

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90

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70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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10

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anm1

1EndLeft

1StepLeft

1PauseLeft

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1PlayPauseLeft

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1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

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70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

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pbsARFix24

ikona2

TooltipField

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pbsARFix26

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pbsARFix30

pbsARFix31

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pbsARFix33

pbsARFix34

10

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17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

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pbsARFix39


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Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

00

01

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anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

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pbsARFix5

pbsARFix6

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pbsARFix18

pbsARFix19

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pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

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1PlayRight

1PlayPauseRight

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1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

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pbsARFix38

pbsARFix39



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Apps



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2014








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IN THE HEADLINES


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Gentsprime






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1 DAY AGO






Counties



East Africa



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Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






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Food amp Drinks



Health amp Fitness



Man About Town



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Apps



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Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

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pbsARFix34

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1EndLeft

1StepLeft

1PauseLeft

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1PlayPauseLeft

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1Minus

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70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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pbsARFix17

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pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

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1EndLeft

1StepLeft

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1PlayLeft

1PlayPauseLeft

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70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

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pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

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pbsARFix31

pbsARFix32

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pbsARFix34

10

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anm1

1EndLeft

1StepLeft

1PauseLeft

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1PlayPauseLeft

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1PlayPauseRight

1StepRight

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70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

00

01

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anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

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pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

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1PlayPauseLeft

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pbsARFix36

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2014








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You May Like




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70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

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17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39












control













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70

80

90

100

110

120

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140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

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17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39


2014








Exchange (DBE)










































up losses









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70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39




























up losses









ADVERTISEMENT

You May Like




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IN THE HEADLINES


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70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

00

01

02

03

04

05

06

07

08

09

010

011

012

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anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39



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You May Like




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TopCars






Womens Method







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Kingdom of Men











11 HOURS AGO



17 HOURS AGO

IN THE HEADLINES


fannntastic-woman











Gentsprime






17 HOURS AGO



1 DAY AGO






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






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Health amp Fitness



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Capital Markets



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Mar2515-1)












Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

00

01

02

03

04

05

06

07

08

09

010

011

012

013

014

015

016

017

018

019

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animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

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pbsARFix35

pbsARFix36

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11 HOURS AGO



17 HOURS AGO

IN THE HEADLINES


fannntastic-woman











Gentsprime






17 HOURS AGO



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140

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70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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animtiph1

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pbsARFix2

ikona2

pbsARFix3

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pbsARFix6

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pbsARFix8

pbsARFix9

pbsARFix10

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pbsARFix13

pbsARFix14

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pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

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pbsARFix34

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1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39



17 HOURS AGO



1 DAY AGO






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Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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anm0

animtiph1

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pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

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pbsARFix14

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pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

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pbsARFix34

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anm1

1EndLeft

1StepLeft

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1PlayPauseLeft

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1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39






Counties



East Africa



World





Companies



Enterprise



Health



Industry



MarketPlace






Technology





BD Life



Art



Fashion






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Food amp Drinks



Health amp Fitness



Man About Town



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Personal Finance



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Travel



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Apps



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Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

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pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

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pbsARFix34

10

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17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

Man About Town



Music



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Apps



Mar2515-1)












Follow us



70

80

90

100

110

120

130

140

150

160

70 80 90 100 110 120 130 140 150

x

four-way collary=x



-20

-15

-10

-5

0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

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1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39



Mar2515-1)












Follow us



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x

four-way collary=x



-20

-15

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0

5

10

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20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

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pbsARFix38

pbsARFix39


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x

four-way collary=x



-20

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0

5

10

15

20

70 80 90 100 110 120 130 140 150

K1 K2 K3 K4ST





+


K1 = 90 K2 = 100 K3 = 120 K4 = 130


11


N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

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1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

N Privault







S1 =

$5

$2


C =

$3 if S1 = $5

$0 if S1 = $2


12





αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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animtiph1

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pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

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pbsARFix35

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αS0 + $β at t = 0



αS1 + $β



S1 = 5 and C = 3

S0 = 4

S1 = 2 and C = 0

S0 = 4

S1 = 5 and C = 3

S1 = 2 and C = 0


αS1 + $β = C


C =

$3 = αtimes $5 + $β if S1 = $5

$0 = αtimes $2 + $β if S1 = $2

ie 5α+ β = 3


α = 1

$β = minus$2



N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

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pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

N Privault


αS0 + $β = 1times $4minus $2 = $2

at time t = 0




$3 if S1 = $5







C =






Thinking further








P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

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1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39





P(S1 = $5) = 23

P(S1 = $2) = 13

(01)





3= $4= S0






15


N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

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0220

0221

0222

0223

0224

0225

0226

0227

0228

0229

0230

anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

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17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

N Privault

P(S1 = $5) = $P

3

P(S1 = $2) = 3minus $P3

(02)





16









This includes





17


pbsARFix1

00

01

02

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09

010

011

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0216

0217

0218

0219

0220

0221

0222

0223

0224

0225

0226

0227

0228

0229

0230

anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

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17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39








This includes





17


pbsARFix1

00

01

02

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08

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0201

0202

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0215

0216

0217

0218

0219

0220

0221

0222

0223

0224

0225

0226

0227

0228

0229

0230

anm0

animtiph1

animtiph2

pbsARFix2

ikona2

pbsARFix3

pbsARFix4

pbsARFix5

pbsARFix6

pbsARFix7

pbsARFix8

pbsARFix9

pbsARFix10

pbsARFix11

pbsARFix12

pbsARFix13

pbsARFix14

pbsARFix15

pbsARFix16

pbsARFix17

pbsARFix18

pbsARFix19

pbsARFix20

pbsARFix21

pbsARFix22

pbsARFix23

pbsARFix24

ikona2

TooltipField

TooltipField

pbsARFix25

fdrm0

pbsARFix26

pbsARFix27

pbsARFix28

pbsARFix29

pbsARFix30

pbsARFix31

pbsARFix32

pbsARFix33

pbsARFix34

10

11

12

13

14

15

16

17

anm1

1EndLeft

1StepLeft

1PauseLeft

1PlayLeft

1PlayPauseLeft

1PauseRight

1PlayRight

1PlayPauseRight

1StepRight

1EndRight

1Minus

1Reset

1Plus

pbsARFix35

pbsARFix36

pbsARFix37

pbsARFix38

pbsARFix39

nicolas privault notes on stochastic finance · stochastic calculus, optimization, partial...

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