master thesis excl. appendix

8/11/2019 Master Thesis Excl. Appendix

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Pricing and hedging of FX plain vanilla options

An empirical study on the hedging performance

of a dynamic Black-Scholes delta hedge with updating implied volatility

under the assumption of Heston and Black-Scholes underlying dynamics, respectively,

in the interpolation/extrapolation of option prices.

Jannik Nrgaard

MSc Finance Thesis

Supervisor:

Elisa Nicolato

Department of Business Studies

Aarhus School of Business,

University of Aarhus

August 2011


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c Jannik Nrgaard 2011

The thesis has been typed with Computer Modern 12ptLayout and typography is made by the author using L ATEX

The author wish to thank the following: My supervisor Elisa Nicolato, Researcher atAarhus School of Business in the Finance Research Group, Aarhus, Denmark for advice.A thanks to Matthias Thul, PhD Candidate in Finance at Australian School of Business,New South Wales, Sydney, Australia for answering questions. I thank the people whovehelped me gain access to the Bloomberg terminals at the University of Aarhus as well asthe employees at the Bloomberg service desk for answering my questions. Lastly, thanks

to Nordea for providing me the access to the Nordea Markets platform, Nordea Analytics,from where Ive gathered supplementary data.


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I want to take the opportunity to thank my parents

for their unconditional support during my years of study.


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Abstract

The thesis shows evidence against the Black-Scholes assumption of a diffu-sion process for the log asset price that has stationary and independent normal

increments resulting in a log-normal distribution of asset returns by consider-ing a time-series of spot rates on the EURUSD and the USDJPY covering aperiod of recent years. Observations of distributions exhibiting high peaknessand "fat tails" as well as observations of volatility clustering are supported byempirical evidence of heteroscedasticity, implying that the volatility of returnsis not constant over time, and evidence of autocorrelation.

In order to calibrate The Heston model and the Black-Scholes model tomarket prices on plain vanilla call options the thesis deals with the foreign ex-change specic quoting conventions and considers the difference here between

the EURUSD and the USDJPY. A data set of 371 recent trading days arecollected from published quotes on Bloomberg where each model is calibratedto a set of option prices on each day to obtain an overall goodness of t mea-sure that shows the superior performance of the Heston model. In the caseof both underlying FX pairs the volatility surface is negatively skew shapedthroughout the period considered.

Based on the calibrations a large scale hedging experiment is set up wherea number of plain vanilla call options with different maturities and strikes issold on each day. A dynamic BS Delta hedge with updating implied volatility

simulated in each of the models results in a better hedging performance whenthe underlying dynamics follows the Heston model. Furthermore we observethat the hedging error is correlated with the underlying returns.


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Contents

Contents i

List of Figures iii

List of Tables v

1 Introduction 1

2 Problem Statement 4

2.1 Research Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Delimitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 The FX Market 8

3.1 FX rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 FX forward contract . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 FX options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 The Black-Scholes model 12

4.1 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . 124.2 The Black-Scholes equation . . . . . . . . . . . . . . . . . . . . . . . 134.3 The Garman-Kohlhagen formula . . . . . . . . . . . . . . . . . . . . . 154.4 Simulation of the Black-Scholes model . . . . . . . . . . . . . . . . . 18

5 Empirical facts 19

5.1 The distribution of FX returns . . . . . . . . . . . . . . . . . . . . . . 19

6 The Heston model 24

6.1 The process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.2 The solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.3 Simulation of the Heston model . . . . . . . . . . . . . . . . . . . . . 27

7 Market data 29

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7.1 Quoting conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.2 Retrieving the implied volatility . . . . . . . . . . . . . . . . . . . . . 31

8 Data description 35

9 Calibration of the models 36

9.1 Building the market implied volatility surface . . . . . . . . . . . . . 369.2 Calibration of the Heston model . . . . . . . . . . . . . . . . . . . . . 379.3 Calibration of the Black-Scholes Model . . . . . . . . . . . . . . . . . 389.4 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.5 Calibration results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

10 Empirical study on the hedging performances 49

10.1 Size of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.2 Strike levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010.3 The hedging portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

11 Conclusion 55

Bibliography 57

A Retrieving the strike price corresponding to a premium included

Delta 60

B Building the market implied volatility surface 63

C Calibration of the Heston model 76

D Calibration of the Black-Scholes model 82

E Simulation of the Heston model 85

F Simulation of the Black-Scholes model 89

G No hedge 92

H Dynamic BS Delta Hedge with updating imp. vol. from the

Heston model 97

I Dynamic BS Delta Hedge with updating imp. vol. from the

Black-Scholes model 109

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List of Figures

5.1 Empirical sample frequency for EURUSD . . . . . . . . . . . . . . . . . . 205.2 Empirical sample frequency for USDJPY . . . . . . . . . . . . . . . . . . 205.3 Q-Q plot for EURUSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.4 Q-Q plot for USDJPY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.5 Daily log returns for EURUSD . . . . . . . . . . . . . . . . . . . . . . . 205.6 Daily log returns for USDJPY . . . . . . . . . . . . . . . . . . . . . . . . 205.7 Autocorrelation for EURUSD . . . . . . . . . . . . . . . . . . . . . . . . 205.8 Autocorrelation for USDJPY . . . . . . . . . . . . . . . . . . . . . . . . 205.9 Rolling historic volatility for EURUSD . . . . . . . . . . . . . . . . . . . 215.10 Rolling historic volatility for USDJPY . . . . . . . . . . . . . . . . . . . 21

9.1 One week moving average of . . . . . . . . . . . . . . . . . . . . . . . . 439.2 One week moving average of . . . . . . . . . . . . . . . . . . . . . . . . 439.3 One week moving average of . . . . . . . . . . . . . . . . . . . . . . . . 439.4 One week moving average of . . . . . . . . . . . . . . . . . . . . . . . . 439.5 One week moving average of vt . . . . . . . . . . . . . . . . . . . . . . . 439.6 Call prices 1M on EURUSD 1/4/2010 . . . . . . . . . . . . . . . . . . . 459.7 Call prices 1Y on EURUSD 1/4/2010 . . . . . . . . . . . . . . . . . . . . 459.8 Imp. vol. 1M on EURUSD 1/4/2010 . . . . . . . . . . . . . . . . . . . . 459.9 Imp. vol. 1Y on EURUSD 1/4/2010 . . . . . . . . . . . . . . . . . . . . 45

9.10 Call prices 1M on EURUSD 6/1/2010 . . . . . . . . . . . . . . . . . . . 459.11 Call prices 1Y on EURUSD 6/1/2010 . . . . . . . . . . . . . . . . . . . . 459.12 Imp. vol. 1M on EURUSD 6/1/2010 . . . . . . . . . . . . . . . . . . . . 459.13 Imp. vol. 1Y on EURUSD 6/1/2010 . . . . . . . . . . . . . . . . . . . . 459.14 Call prices 1M on USDJPY 1/4/2010 . . . . . . . . . . . . . . . . . . . . 469.15 Call prices 1Y on USDJPY 1/4/2010 . . . . . . . . . . . . . . . . . . . . 469.16 Imp. vol. 1M on USDJPY 1/4/2010 . . . . . . . . . . . . . . . . . . . . 469.17 Imp. vol. 1Y on USDJPY 1/4/2010 . . . . . . . . . . . . . . . . . . . . . 46

9.18 Call prices 1M on USDJPY 6/1/2010 . . . . . . . . . . . . . . . . . . . . 469.19 Call prices 1Y on USDJPY 6/1/2010 . . . . . . . . . . . . . . . . . . . . 46

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9.20 Imp. vol. 1M on USDJPY 6/1/2010 . . . . . . . . . . . . . . . . . . . . 469.21 Imp. vol. 1Y on USDJPY 6/1/2010 . . . . . . . . . . . . . . . . . . . . . 46

10.1 Development in EURUSD spot rate . . . . . . . . . . . . . . . . . . . . . 53

10.2 Development in USDJPY spot rate . . . . . . . . . . . . . . . . . . . . . 53

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List of Tables

5.1 Jarque-Bera test on normality . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Levenes test on equality of variances . . . . . . . . . . . . . . . . . . . . 23

7.1 Premium included Delta . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.2 Conversion of a Premium Included Delta to Strike . . . . . . . . . . . . . 34

9.1 Quarterly mean and standard deviation of the goodness of t of Hestonparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

9.2 Quarterly mean and standard deviation of the goodness of t of theBlack-Scholes parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

9.3 Heston parameter values on 1/4/2010 and 6/1/2010 on EURUSD . . . . 449.4 Heston parameter values on 1/4/2010 and 6/1/2010 on USDJPY . . . . 44

9.5 Black-Scholes parameter values on 1/4/2010 and 6/1/2010 on EURUSDand USDJPY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

10.1 Number of options under investigation . . . . . . . . . . . . . . . . . . . 5010.2 Number of option expirations in quarterly periods . . . . . . . . . . . . . 5010.3 Delta level on average of shorted EURUSD call options at initiation . . . 5110.4 Delta level on average of shorted USDJPY call options at initiation . . . 5110.5 Number of EURUSD call options expiring in-the-money . . . . . . . . . . 5310.6 Number of USDJPY call options expiring in-the-money . . . . . . . . . . 53

10.7 The mean prot and loss and standard deviation on the hedging errorwith Black-Scholes and Heston pricing . . . . . . . . . . . . . . . . . . . 54

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1

Introduction

In a nancial world which have experienced market crashes starting with Black Mon-day in 1987 the introduction of extreme market movements have given rise to thereconsideration of the assumptions behind the pricing of nancial instruments suchas options on stocks as well as foreign exchange. In the past market participantsand practitioners have relied more on the Black- Scholes model and its assump-tion about asset returns whereas today the market prices of options do not reectthose predicted by the Black-Scholes model. Instead a family of stochastic volatility

models has emerged, with the Heston model being the most well known, with morerealistic assumptions about the probability distribution of asset returns today. Stillthough, the Black-Scholes model are applied by market participants and practition-ers in circumvention that avoid its aws. This thesis incorporates the applicationof both types of models and tries to uncover pricing misspecications and, in anempirical study, investigates if one is preferable to the other given a specic pricingand hedging setting.

In chapter 3, we start by given an introduction to the FX market and FX plainvanilla options, which are traded over-the-counter (OTC). This fact inuences thedata collected to represent the market prices, which in this case is retrieved fromBloomberg where an arbitrage free volatility surface is reported from a collectionof option quotes from several contributors representing the worlds largest nancialinstitutions. As opposite to exchange traded options that are quoted with a xedmaturity date and with the initiation of new options only on xed dates, fromBloomberg we are provided with a full set of new options everyday covering thesame range of maturities just with the expiration one day later than the previous

days quoted options.

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Chapter 4 covers the Black-Scholes (BS) model and its assumptions about log-normally distributed asset returns. With specic interest in the pricing of FX op-tions we present the Garman-Kohlhagen formula, which is a simple extension tothe BS model. In this chapter we furthermore introduce the concept of the im-

plied probability density function and risk neutral valuation. Finally we present thesimulation of the BS model.

In Chapter 5 we analyse the distribution of FX log returns considering a sampleof recent years spot FX rates and compare this with the assumption of log-normaldistributed returns in the BS model. The ndings here inspire to consider differ-ent assumptions on the distribution of log returns, which leads us to introduce astochastic volatility model in the next chapter.

Chapter 6 then introduces the process and the closed form solution to the Heston

model. In the calibration of the Heston model we calibrate to this closed formsolution by numerical integration. Furthermore we present the simulation of theHeston model that is carried out in a mixing solution framework simulated in aMilstein scheme.

Before the empirical study we present chapter 7, which explains the very FXspecic quoting conventions. More comprehensive than other option markets theFX option market has a wide range of possible conventions which need to be properlyhandled in order to be able to build a volatility surface based on the quotes in themarket. More specically the volatilities are quoted in trading structures that needsto be converted. Moreover the options are quoted in terms of Delta in the moneynessdimension. Depending on the Delta convention of the specic FX pair, we need touse a numerical estimation technique to retrieve the strike level.

Chapter 8 consists of an overview of the data used in the empirical study.

In chapter 9 we then calibrate the BS model and the Heston model to eachday of 371 trading days in the period from 1/4/2010 - 06/22/2011. We presentthe objective function and its inherited weighting scheme that is common for bothmodels. We furthermore analyse the sensitivity of the volatility surface to the changein Heston parameters by looking at two different days. Also a comparison betweenthe ability of the two models to t the observed market prices is done by calculatingthe goodness of t for each model.

In chapter 10 we lay out the hedging strategy consisting of a dynamic BS Deltahedge with updating implied volatility simulated in the BS model and simulated inthe Heston model. More specically we hedge a number of shorted call options withdifferent maturities and strike levels. We then identify which elements that change

the value of the hedging portfolio. Finally we present the ndings of the studycomparing the BS model as a tool in the interpolation/extrapolation of the updating

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implied volatility to the Heston model by comparing the hedging performance of thesame BS Delta hedge.

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2

Problem Statement

In this study we consider the two FX pairs EURUSD and USDJPY. We start bythe following introductory research questions:

I. How are FX returns distributed considering a period of recent years?

II. How does the distribution of FX returns compare to the assumptions aboutlog-normal distributed asset returns in the Black-Scholes model?

As pointed out by (Reiswich and Wystrup, 2010), the smile construction proce-dure and the volatility quoting mechanisms are FX specic and differ signicantly from other markets...Market participants entering the FX OTC derivative market are confronted with the fact that the volatility smile is usually not directly observable in the market...Unlike in other markets, the FX smile is given implicitly as a set of restrictions implied by market instruments. This lead us to the question:

III. How do we handle the FX specic quoting conventions in order to end up withmarket prices on plain vanilla option.

In a very recent paper "Applying hedging strategies to estimate model risk andprovision calculation" (Elices, 2011) the authors study the hedging performance of the BS model and the Vanna-Volga method by assuming that the market volatilitysurface is driven by Hestons dynamics calibrated to market for a given time horizon.The hedging strategy is then built in order to neutralize the uncertain factors in theHeston model which consist of the spot and the volatility.

In the same way, we rely on a model dependant building of the volatility surface

by calibration of the BS model and the Heston model, respectively, to the observedmarket prices.

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IV. How well does the Black-Scholes and Heston model, respectively, reect a setof market prices on plain vanilla options over a recent period?

Then we use these calibrations in order to investigate how well a pure Delta

hedging strategy, with the Delta calculated as a BS Delta, is able to replicate thepayoff of a plain vanilla FX call option contract. We create a setting where aset of European plain vanilla FX options with different maturities and strikes aresold every day during a period of 371 trading days. By delta hedging each optioncontract individually until its expiry, we obtain the hedging error that we express asthe difference between the payoff of the option contract and the hedging portfolio.Two experiments are set up where we calculate the BS Delta dynamically with anupdating volatility from the Black-Scholes model and an updating implied volatilityfrom the Heston model. This leads to the nal research questions:

V. Applying a dynamic BS Delta hedge with updating implied volatility underthe assumption of Black-Scholes underlying dynamics, what is the standarddeviation of the hedging error for each option contract?

VI. Applying a dynamic BS Delta hedge with updating implied volatility underthe assumption of Heston underlying dynamics, what is the standard deviationof the hedging error for each option contract?

VII. Are the outcome of the hedging correlated with the market return?

2.1 Research Approach

We point out and argue for our choice of research approach in three areas of thethesis: The inclusion of two different FX pairs, the building of the implied volatilitysurface and the range of option prices used to build the implied volatility surface.

We choose to include both the EURUSD and the USDJPY in the study be-cause of mainly one reason. The quoting conventions for the two pairs aredifferent and by including both we show how to handle these different quot-ing conventions. In addition to this reason, the volatility surface of thesetwo pairs has historically had different shapes with the EURUSD exhibit-ing more of a symmetrical smile and the USDJPY exhibiting a step skew(Bossens, Rayee, Skantzos, and Deelstra, 2010), (Beneder and Elkenbracht-Huizing, 2003), (Chalamandaris and Tsekrekos, 2008). Like other studies thisis an attempt to cover a different set of market conditions (Bossens, Rayee,

Skantzos, and Deelstra, 2010).

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We calibrate to raw data where no interpolation or extrapolation has takenplace beforehand. Alternatively we could have used a SVI parametrisation(Gatheral, 2006) or some other functional form to rst build the surface andthen calibrate to a set of interpolated/extrapolated prices.

We calibrate to only a few number of options counting 5 different maturi-ties and 5 different strike levels. This is done because of two reasons. First,we want to calibrate only to raw data that has not yet been interpolatedin Bloombergs own interpolation scheme, which can be seen in (Bloomberg,2011). Bloombergs interpolation is based on ATM, 25 Delta and 10 Deltaquotes and if available also 35 25 and 5 Delta (Bloomberg, 2009). This factensures us that we only calibrate to raw data. Second, a lot of effort has goneinto the development of methods that are able to build the full implied volatil-ity surface with only a few set of option prices (Malz, 1997), (Castagna andMercurio, 2006), (Reiswich and Wystrup, 2010). On an OTC option market,often only a few prices is available and we want to restrict this study to includeonly the prices that are most often available.

This thesis uses the same range of option prices from the same source asin U. Wystrup and D. Reiswichs article "FX Volatility Smile Construction" (Reiswich and Wystrup, 2010) by using the ATM, 10D RR, 25D RR, 10DVWB and 25D VWB quotes published on Bloomberg.

2.2 Delimitation

The thesis is limited in areas where additions would bring more accuracy and detailinto the study.

In order to test a pricing model for its misspecications a classical hedgingexperiment like the one carried out in (Bakshi, Cao, and Chen, 1997) and(Elices, 2011) could be done. Here, they test a models ability to replicatean option payoff by taking positions in all assets necessary to neutralize riskwith that number depending on the assumption of the given pricing model.For the Heston model this implies taking a position in both the underlyingand another option in order to obtain a delta-neutral hedge. In this thesis werestrict ourselves to only take a position in one asset, the underlying. So thisstudy cannot be classied under this type of conventional approach.

The interest rate setting in this study is simplied. There has been no buildingof an interest rate term structure to use in the simulation of the option pricing6


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3

The FX Market

3.1 FX rate

A foreign exchange rate (FX rate) is the price of one currency in terms of anothercurrency. The two currencies make a currency pair. As an example this could bethe currency pair labelled EURUSD. This is the euro/US dollars exchange rate andby the end of the trading day on the 1th of May 2011 this was quoted at 1.3196.This is the convention on how to quote this particular currency cross, but it is

equivalent to USDEUR 0.7578, which is just the reciprocal value of the rst FXrate. The exchange rate EURUSD denotes how many US dollars are worth 1 euro.The domestic (numeraire) currency is the US dollar and the foreign (base) currencyis the euro. So generally speaking, the exchange rate is the price of the base currencyin terms of the numeraire currency. The last time a US dollar was worth more thana euro was on the 4th of December 2002 on which day the exchange rate was quotedat 0.9997. Since after the introduction of euro coins and banknotes on the 1th of January 2002 this has been the only year that the US dollar has been worth morethan the euro, reected in an exchange rate less than 1.0000.

3.2 FX forward contract

The forward contract provides a hedge for someone who wants to lock in the ex-change rate for a future transaction. The buyer of a forward contract is then guar-anteed a future exchange rate. The forward price is decided as

F 0 = S 0e(rdr

f )T (3.1)

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The underlying asset in such contracts is a certain number of units of the foreigncurrency. The variable S 0 is dened as the spot price in domestic currency of oneunit of the foreign currency and equivalently F 0 is the forward price in domestic

currency of one unit of the foreign currency. Both domestic and foreign interestrates are the continuously compounded risk-free interest rates per annum.

3.2.1 Interest rate parity

Equation 3.1 is exactly the interest rate parity, which in its continuous compoundingform is often equated as

F (t, T ) = S ter

f (T t )

er d (T t ) (3.2)

or by its money market conventions for capitalization and discounting, i.e simplecompounding (Castagna, 2010, p. 7)

F (t, T ) = S t(1 + r f )(T t)(1 + rd)(T t)

(3.3)

where r f and r d are the risk-free interest rates per annum and (T-t) follow the time

convention of 360 trading days in a year.According to the interest rate parity, the forward exchange rate of a given cur-

rency pair is determined by the respective risk-free interest rates. As an example,we consider a holder of one unit of foreign currency. There are two ways that thiscan be converted into domestic currency at time T . One is by investing it for (T t)years at r f and at the same time selling a forward contract. Then at time T youwould be obligated to sell the proceeds from the investment to collect domestic cur-rency. The other possibility is to exchange the foreign currency to domestic in the

spot market and then invest these at rd

for (T-t) years. In the absence of arbitrageopportunities equation 3.4 should then hold (Hull, 2008, p. 113), which is exactlyequation 3.2 rewritten.

erf (T t )F 0 = S 0er

d (T t ) (3.4)

The interest rate parity presented here is also called the covered interest rate parityas opposite to the uncovered interest rate parity (Oldeld and Messina, 1977). The

former comes from the fact that the trading strategy is risk-free. This is opposite tothe latter where you as a holder of the foreign currency still invest in r f , but instead

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of simultaneously entering into a forward contract, you instead keep your positionin foreign currency uncovered and exposed to the movement in the exchange ratefrom t to (T t).

Empirical research shows that for developed countries, the covered interest rate

parity holds fairly well. Prior to the dismantling of capital controls, and in manyemerging markets today (interpreted as political risk associated with the possibil-ity of governmental authorities placing restrictions on deposits located in different jurisdictions), the covered interest rate parity is unlikely to hold (Chinn, 2007).

From an option pricing point of view the covered interest parity is an underlyingassumption in one of the option pricing models introduced later on here.

3.3 FX options

FX options are traded Over-The-Counter (OTC) as opposite to exchange tradedoptions. As a trading platform an exchange serves as a link between a buyer anda seller. The exchange will be providing bid and ask quotes and will be on eitherone or the other end of the transaction. The market making is in this case carriedout by the exchange. In the case of FX options there is no exchange involved inthe transaction. A trade will be processed directly between buyer and seller. Inone setting, one might think of a buyer being a corporation that is trading from

a hedging or speculative point of view and the seller being a bank. On the FXoptions market one might think of the banks as market makers providing the priceson options and other FX derivatives.

In order to hedge a foreign exchange exposure FX options are an alternative to FXforward contracts.The payoff from a long position in a European call option is

max( S T

K, 0) (3.5)

and the payoff from a long position in a European put option is

max( K S T , 0) (3.6)

with S T being the spot exchange rate at maturity T of the option and K the agreedupon strike price.

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Assuming we have the pair EURUSD, two counterparties entering into a plainvanilla FX option contract can agree on the following, according to the type of option traded:

Type EUR call USD put: The buyer has the right to enter at expiry into aspot contract to buy (sell) the notional amount of EUR (USD), at the strikeFX rate level K .

Type EUR put USD call: The buyer has the right to enter at expiry into aspot contract to sell (buy) the notional amount of EUR (USD), at the strikeFX rate level K .

Considering, as an example, the last type listed above, an American company

due to receive euro at a known time in the future can hedge its risk by buying putoptions on euro that mature at that time. This strategy guarantees that the valueof the euros will not be less than the strike price while still allowing the company tobenet from any favorable upward movements in the exchange rate. Similarly, if thecompany where to pay euros in the future they could hedge their expose to upwardmovements in the exchange rate by buying calls on euros, the rst type listed above.

whereas forward contracts locks in the exchange rate for a future transaction andguarantees the parties an exchange rate, as described above, an option provides atype of insurance. It costs nothing to enter into a forward contract, whereas optionsrequire a premium to be paid paid up front in order to be insured.

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4

The Black-Scholes model

This chapter reviews the most well-known option pricing model, The Black-Scholesmodel (Black and Scholes, 1973), because of its inclusion in the empirical study.Also it remains the building block of present option pricing models, including theHeston model and the Bates model.

4.1 Geometric Brownian Motion

Black-Scholes assumes the underlying spot price to follow a geometric Brownianmotion generating log-normally distributed returns, the spot price in this case beingthe exchange rate on any given FX pair. The process is stochastic by including aWiener process that introduces the randomness to the spot price.

dS t = S t dt + S t dW = S t (dt + dW ) (4.1)

The spot price S t depends on S t itself, a constant drift, , a constant volatility term,

, and a standard Wiener process, W t , where dt is denoting a time differential. Inorder to obtain the explicit solution to this stochastic differential equation (SDE) weconsider equation 4.2 the process of logS, i.e. the process describing the log-returns.

dlogS t = ( 12

2)dt + dZ (4.2)

i.elogS T = logS 0 + (

12

2)T + dZ (4.3)

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and the explicit solution is then obtained by taking the exponential of logS

S T = S 0e(12

2 )T + Z T (4.4)

4.2 The Black-Scholes equation

With the empirical study of this thesis in mind we have a look at the derivation of the Black-Scholes (BS) equation which is governing the BS option pricing formula.This will tell us the principle of delta hedging. Furthermore we take a look at thenecessary adjustments to the Black Scholes equation in order to be able to price FXoptions in particular. As a note it is not in the interest of this thesis to go through

the derivation of the solution to the BS equation that will lead to the BS formula.The Black-Scholes equation can be derived in many alternative ways i.e. using

empirically established nancial theories such as the CAPM and Arbitrage Pricingtheory. The most general derivation assumes an economy with only the underlyingasset and a risk-free money market deposit/risk-free bond which together makesup the replicating portfolio of the value of the derivative. Meanwhile, the originalderivation uses what is known as the hedging argument, and that is the derivationthat we will outline here (Rouah, 2011).

The derivation follows from imposing the condition that a risk-free portfoliomade up of a position in the underlying asset and the option on that asset mustreturn the same interest rate as other risk-free assets. As a result of this Blackand Scholes propose that if it is possible to hedge an option position by dynamicallyrebalancing a stock position, then the price of a European call option should dependon the underlying spot price, S t (i.e. the FX rate), and the time to maturity on theoption, T .

In order to perform such a hedge Black and Scholes assumes a set of conditionsto hold that they call the ideal market condition:

The FX rate, S t , follows the geometric Brownian motion with known constantdrift, , and volatility, .

The option can be exercised only at maturity.

Trading takes place continuously in time.

Money can be borrowed and lend at the same risk-free interest rate.

Short selling is allowed.

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Short-term risk-free interest rates (r d and r f ) are known and constant.

The underlying asset pays no dividends. (This assumption is relaxed in thecase of FX options.)

We consider a portfolio made up of a quantity of the risky asset (i.e. the FXpair) and short one option on the FX pair (a put or a call, not yet specied). Letf (S, t ) denote the value of the option and (t) the value of the portfolio.

(t) = S f (S, t ) (4.5) is chosen at every time t so as to make the portfolio riskless. The self-nancingassumption implies that

d(t) = dS df (S, t ) (4.6)In order to decide the quantity to meet this condition we want to know the dynam-ics of f (S, t ). Here we use Itos Lemma, which is a rule for calculating differentialsof quantities dependent on stochastic processes.

df (S, t ) = f t

dt + f S

dS + 12

2S 2 2f S 2

dt (4.7)

and by plugging in 4.7 into 4.6 we get

d = dS (f t

dt + f S

dS + 12

2S 2 2f S 2

)dt

( f S

)dS (f t

+ 12

2S 2 2f S 2

)dt (4.8)

observing that the term dS is the only risky element to the portfolio value, we caneliminate this by setting

( f S ) = 0which is satised if

= f S

(4.9)

Then we have constructed a risk-free portfolio with the dynamics given in the lastpart of 4.8 and by a no arbitrage argument the portfolio must yield the risk-free

interest rate, i.e.

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d = rdt (4.10)

Plugging the risk-free dynamics of the option value in 4.8 and the rst equation 4.5into 4.10 and rewrittin, we get the BS equation in 4.11.

(f t

+ 12

2S 2 2f S 2

)dt = r (f S

S f (S, t ))dt

f t

12

2S 2 2f S 2

= r (f S

S f (S, t ))f t

+ 12

2S 2 2f S 2

+ rf S

S rf = 0 (4.11)

The derivation stipulates that in order to hedge the single option, we need to holda quantity of the FX pair, which turns out to be the quantity f S . This is theprinciple behind delta hedging. Any price of a derivative with the same assumedprocess for the underlying as in equation 4.1 has to follow the BS equation.Theequation has many solutions for the derivative price, f , where the particular pricethat is obtained depends on the payoff function of the given derivative. In thecase of a European call/put the solution is obtained in the BS formula, but formore complex payoff functions accompanied by more exotic options the analyticalsolution may be hard to obtain.

4.3 The Garman-Kohlhagen formula

In the same year 1973 as the Black and Scholes paper was published the pricingmodel was quickly adjusted to include dividend paying stocks by Merton (1973).Robert C. Merton further concludes in this paper that the assumption of log-normally distributed returns and continuous trading is critical to the model. With-out these, the delta hedge would not give a perfect hedge, thus making the arbi-

trage argument invalid. Many years later after the FX options was rst listed onthe Philadelphia Stock Exchange in 1982 (Exchange, 2004), the pricing model wasadjusted to also be able to price FX plain vanilla options (Garman and Kohlhagen,1983). Under similar assumptions as in Black-Scholes, that it is possible to operatea perfect local hedge between a FX option and underlying foreign exchange, Gar-man and Kohlhagen derive a PDE. One of the insights is that the risk-free interestrate of foreign currency r f has the same impact on the FX option price as the con-tinuous dividend yield on the stock option. The main contribution is to combinethe Black-Scholes model with the interest rate parity theory, as presented in the

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beginning of this thesis. More precisely, by assuming the covered interest rate par-ity to hold and the underlying FX rate to follow a geometric brownian motion, thelogarithmic difference between the forward, F (t, T ), and the spot, S (t), FX ratescan be explained by the spread between the domestic risk-free interest rate, r d, and

the foreign risk-free interest rate, r f .The resulting pricing formula for a call option in equation 4.12 is presented in its

forward rate form, where the forward rate is explicitly present in the formula. Thisis a Black model (Black, 1976) (adjusted to price FX options), which is a variationof the original BS model and can be generalized into a class of models known aslog-normal forward models. The adaption of the covered interest rate parity intothe option pricing formula becomes apparent when we compare the calculation of the forward rate in Equation 4.12 to Equation 3.2.

c = erd (t,T ) )[F (t, T )(d1) K(d2)] (4.12)

d1 = ln( F (t,T )K ) +

12

2

d2 = d1 F (t, T ) = S t

erf (t,T )

er d (t,T )

with the the equivalent spot rate form of the Garman-Kohlhagen formula

c = S 0erf (t,T ) (d1) Ker

d (t,T ) (d2) (4.13)

d1 = ln( S 0K ) + ( r

d(t, T ) r f (t, T ) + 12 2)

d2 = d1

The foreign and domestic interest rates are risk-free and constant over the term of the options life. All interest rates are expressed as continuously compounded rates.

4.3.1 Implied Probability Density Functions

In order to establish a link between the observed option prices in the market and thecharacteristic shapes of the volatility surface we mention the implied risk-neutral

density function (RND).

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The RND in the Black-Scholes model is assumed to be lognormal with mean(r d r f v2/ 2)(T t) and variance v2(T t).

The price of an undiscounted call option is given by

C (S 0,K ,T ) = E [max{S T K, 0}] (4.14)= K (s K ) (s; T, S 0)ds (4.15)

where (s; T, S 0) in (4.15) is the probability density function of S T . This is a generalpricing formula independent of the choice of pricing model. Pricing an option inthis framework requires the knowledge of the probability density function, which isthe distribution of the future spot prices.

(Breeden and Litzenberger, 1978) found that provided a continuum of Europeancall options with same maturity and a strike range going from zero to innity writ-ten on a single underlying FX pair, we can recover the RND in a unique way bydifferentiating (4.15) with respect to K twice

C K

= K (s; T, S 0)ds (4.16) 2C K 2

= (s; T, S 0)ds (4.17)

4.3.2 Risk-neutral valuation

Another approach to nd the price of a derivative is by risk neutral valuation orequivalently by the Martingale approach. The equivalence between the PDE ap-proach and the risk neutral valuation is guaranteed by Feynman-Kac by establishinga link between PDEs and stochastic processes.

The solution to the Garman-Kohlhagen equation can also be expressed in termsof an expectation. By the Feynman-Kac theorem we have

V (S t , t ) = E Q e T t r

ds ds V (S T , T ) (4.18)

where S t is the solution to the SDE (4.1) with = rd r f . The drift is risk neutral and consists of the continuously compounded domestic interest rate net of the foreign interest rate. What (4.18) says is that the value of a contingent claim(a claim that is dependant on the underlying value) like a European option, can becalculated by nding the risk neutral expectation of the discounted terminal payoff.The terminal payoff is discounted by the domestic interest rate and the risk neutral

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expectation and the Q measure involves the process of S T to evolve not as originalbut risk neutrally.

To recapitulate the general pricing framework above, there is a connection be-tween the existence of a replication portfolio replicating the nal value of the op-

tion, and the existence of a equivalent martingale measure. They both guarantee anarbitrage-free price. This can be calculated as the current value of the replicationportfolio, or as the expected value of the discounted terminal payoff of the optioncalculated under the risk-neutral probability measure.

4.4 Simulation of the Black-Scholes model

We consider the risk neutral process in Equation (4.19) and compute the risk neutral

expectation of the terminal payoff as suggested by the Feyman-Kac theorem.

dS t = ( r dt rf t )S t dt + t S t dW (4.19)

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29/118

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Figure 5.2: Empirical sample frequencyfor USDJPY

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0

0.01

0.02

0.03

0.04

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Q u a n t i

l e s o

f I n p u t

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l e

QQ Plot of Sample Data versus St andard Normal

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l e s o

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QQ Plot of Sample Data versus St andard Normal

Figure 5.3: Q-Q plot for EURUSD Figure 5.4: Q-Q plot for USDJPY

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Figure 5.6: Daily log returns for USD-JPY

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0.6

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S a m p

l e A u t o c o r r e

l a t i o n

Sample Autocorrelation Function

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l e A u t o c o r r e

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Sample Autocorrelation Function

Figure 5.7: Autocorrelation for EU-RUSD

Figure 5.8: Autocorrelation for USD-JPY

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Figure 5.10: Rolling historic volatilityfor USDJPY

5.1.1 Jarque-Bera

To conrm our results and to nd further evidence against the normality assumptionunderlying the Black-Scholes model we make use of the Jarque-Bera test (Jarque andBera, 1987). Based on the sample kurtosis and skewness we test the null hypothesisthat the data is drawn from a normal distribution. The null hypothesis is a jointhypothesis of the skewness being 0 and the excess kurtosis being 0, which in thelatter case is the same as a kurtosis of 3.

The overall conclusion by looking into tabel 5.1, when considering the full sam-ple of log returns, is that we clearly reject the null hypothesis, that the sample datais from a normal distribution, in both the EURUSD and USDJPY case. This con-

clusion comes with a high degree of certainty with a signicance level below 0.1%.When we then have a look at the separate years considering rst the EURUSD,we are able to reject in 3 out of 6 years at a signicance level of 5.0%, whereasfor the USDJPY case this is 4 out of 6 years. When looking into the estimates of the overall skewness and kurtosis and comparing the two pairs, one observes thatin terms of skewness the EURUSD deviates the most from a normal, whereas interms of kurtosis it is the USDJPY that deviates the most from the normal. Thesedifferences in skewness and kurtosis between the two pairs is somewhat visual in

gures 5.1 and 5.2 from before. Comparing the tails of the frequency distributionsone might see that the EURUSD log returns has a longer right tail exhibiting morepositive skewness whereas the USDJPY log returns has a longer left tail exhibitingmore negative skewness (Even though apparently not enough for the full sample tobe negatively skewed). Both distributions though are on an overall scale slightlypositively distributed meaning that most values are concentrated on the left of themean, with extreme values to the right (as opposite to negatively skewed distribu-tions, where most values are concentrated on the right of the mean, with extremevalues to the left). The difference in the kurtosis of the two pairs of log returns isalso somewhat visual from the gures 5.3 and 5.4 from before, where the USDJPY

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Table 5.1: Jarque-Bera test on normalityEURUSD USDJPY

period skewness excess kurtosis JB sign. level skewness excess JB sign. level

2006 0.145 0 .001 > 50 .000% 0 .242 0 .164 20 .473%

2007 0 .331 3 .573 2 .186% 0 .712 4 .675 < 0 .100%

2008 0.330 1 .500 < 0 .100% 0 .269 3 .807 < 0 .100%2009 0.193 1 .091 0 .540% 0 .154 0 .392 21 .547%2010 0 .014 0 .237 > 50 .000% 0 .073 5 .270 < 0 .100%2011 0 .120 0 .007 > 50 .000% 0 .410 0 .941 4 .660%

2006-2011

0 .124 1 .937 < 0 .100% 0 .044 4 .378 < 0 .100%

log returns seems to exhibit the most kurtosis.The test statistic JB is dened as

JB = n6 (S 2 + 14K

2) (5.1)

where n is the number of observations, S is the sample skewness in Equation 5.2and K is the sample excess kurtosis in Equation 5.3.

S = 33

=1n

ni=1 (xi x)3

( 1nni=1 (x i x)2)

32

(5.2)

K = 44 3 =

1n

ni=1 (xi

x)4

( 1n ni=1 (xi x)2)2 3 (5.3)where 3 and 4 are the estimates of the third and fourth central moments, respec-tively, x is the sample mean and is the estimate of the second central moment,the variance.

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5.1.2 Levene

Excess kurtosis might indicate heteroscedastic returns, where homoscedastic returnsis the assumption underlying the Black & Scholes model. We therefore perform theLevenes test of homoscedatic returns, where the null hypothesis is that the varianceof two successive subsamples are equal as well as the variances of all subsamples.Considering the latter we strongly reject the hypothesis that the variance in thesubsamples are constant thus violating the assumption in the Black Scholes model.Comparing the individual successive yearly subsamples, in the case of the EURUSDwe are able to reject in 2 out of 5 cases at a signicance level of 5%. In the caseof the USDJPY this is 4 out of 5 cases in correspondence with the superior excesskurtosis compared to the EURUSD case.

Table 5.2: Levenes test on equality of variancesEURUSD USDJPY

period 1 period 2 volatility 1 volatility 2 Levene sig. level volatility 1 volatility 2 Levene sig. level

2006 2007 7.35% 6 .16% 0 .859% 7 .83% 9 .62% 1 .244%2007 2008 6.16% 13 .78% 0 .000% 9 .62% 16 .18% 0 .000%2008 2009 13 .78% 12 .03% 9 .691% 16 .18% 12 .68% 1 .659%2009 2010 12 .03% 11 .76% 75 .421% 12 .68% 10 .36% 2 .458%2010 2011 11 .76% 9 .85% 7 .890% 10 .36% 9 .87% 74 .257%

2006 2011 0.000% 0 .000%

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6

The Heston model

The most well-known and popular of all stochastic volatility models is the Hestonmodel (Gatheral, 2006) and was presented in (Heston, 1993).

6.1 The process

The process followed by the underlying asset in the Heston model is

dS t = S t dt + vt S t dW (1)t (6.1)

dvt = (vt )dt + vt dW (2)t (6.2)

withdW (1)t dW

(2)t = dt

where is the rate of reversion of vt to the long run variance, , is the volatilityof volatility and is the correlation between the two stochastic increments of theprocesses dW (1)t and dW (2)t . The process of the underlying in (6.1) is the same

process assumed in the Black Scholes model presented in (4.1) only now the volatilityis stochastic. That is, another random factor is introduced by dW (2)t . What denesthe specic process of the underlying in the Heston model compared to the generalcase of stochastic volatility models is

dvt = (S t , vt , t )dt + (S t , vt , t ) vt dW (2)t (6.3)

(S t , vt , t ) = (vt ) (S t , vt , t ) = 1

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where the process followed by the instantaneous variance, vt , can be categorized asa version of the square root process (CIR) in (Cox, Ingersoll Jr, and Ross, 1985).Given that the Feller condition in equation (6.4) is satised the variance process isalways strictly positive. (Anderson, 2005) shows that this condition is often violated

when calibrating the Heston model to market data.

2 2 (6.4)What makes the Heston stochastic volatility model stand out from other stochasticvolatility models can be adressed to two reasons. First, the volatility process isnon-negative and mean reverting which is what we observe in the market. Secondly,The Heston model has a semi-analytical closed form solution for European option,which is fast and relatively easy to implement. The closed form solution is especiallyuseful when calibrating the parameters in the model to the observed vanilla optionmarket. This efficient computational ability of the model is characterised as thegreatest advantage of the model over other potentially more realistic SV models(Janek, Kluge, Weron, and Wystup, 2010).

Furthermore, after adapting the model to a FX setting, the model is describedas being particular useful in explaining the volatility smile found in FX marketsoften characterised by a more symmetrical smile when comparing to equity marketswhere the structure is a strongly asymmetric skew as a consequence of the leverage

effect on these markets(Janek, Kluge, Weron, and Wystup, 2010).

6.2 The solution

The PDE of the Heston model can be derived using the same approach as when wederive the PDE for the BS model where standard arbitrage arguments is used. Inaddition to the replication portfolio used to derive the BS model another asset inthe form of an option is added in order to hedge the randomness introduced by the

stochastic volatility. The following PDE can then be derived

V t

+ 12

vS 2 2V S 2

+ vS 2V vS

+ 12

2v 2V v2

+ rS V S rV

+ {( V ) (S,v,t ) v}V v

= 0 (6.5)

where (S,v,t ) is the market price of volatility risk. The closed-form solution of aEuropean call option on an FX pair for the Heston model is

S t P 1 Ke (r d r f )( T t )P 2 (6.6)

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where

P j = 12

+ 1 0 Re e

i ln( K )f j (x, vt , ,)i

d (6.7)

f j = eC (, )+ D (, )vt + ix

C (, ) = ( r d r f )i + a2

(b j i + d) 2ln1 ged

1 g

D(, ) = b j i + d

2 1ed 1

ged

g = b j i + db j i d

d = (i b j )2 2(2u j i 2)u1 =

12

, u2 = 12

, a = , b1 = , b2 = +

for j = {1, 2} and x = lnS 0 and where put options can be solved by the put-callparity in equation (6.8).

C (K, T ) = P (K, T ) + S t e(rf (T t ) e(r

d (T t )K (6.8)

The integral part of (6.7) with the integration of f j is the reason why it is only a semianalytical closed-form solution, because the integral only can be evaluated approx-imately. This can be done with reasonable accuracy using a numerical integrationtechnique. This includes the calculation of the complex logarithm in C (, ) inequation (6.7), which can cause numerical instabilities. The problem can be solvedalmost entirely if d is replaced by d = d (Albrecher, Mayer, Schoutens, and Tis-taert, 2005). Meanwhile this correction of the original pricing formula is likely tohave the most impact of long maturities above 3-5 years, where the problem forthe short and middle term options might not even be detectable (Albrecher, Mayer,Schoutens, and Tistaert, 2005), (Janek, Kluge, Weron, and Wystup, 2010). We later

calibrate the Heston model to the closed form solution presented in Equation (6.6).The code of the pricing formula is an exact replicate from (Moodley, 2005).

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6.3 Simulation of the Heston model

The simulation of the Heston model can be done with the mixing solution approachto stochastic volatility models by (). The mixing theorem is a way of expressing

the option price in a more complicated model like a stochastic volatility model asa weighted sum of the option prices in a simpler base model, in this case the BSmodel. This can also be expressed as in

C (S t , vt , ) = E [C BS (S eff , veff , ] (6.9)where the option price in the Heston model is expressed as an expectation of the BSvalue with a so-called effective spot, S eff and effective volatility

veff . In order

to get to these effective values we start by writing the explicit solution to equation(6.1) at maturity, T, in a risk neutral setting and with an FX pair as the underlyingasset:

S T = S 0e(rdr

f )T 12

T 0 vs ds +

T 0 vs dB s (6.10)

where

B t = W t +

1

2Z t , dW t dZ t = 0 (6.11)

and plugging (6.11) into (6.10) and re-arranging

S T = S ef f T exp (rd r f )T

12

T

0(1 2)vs ds +

T

0 (1 2)vs dZ s (6.12)where the effective spot is

S ef f T = S 0 exp 12

T

02vs ds +

T

0 vs dW s (6.13)

and we express the effective variance as

veff T = 1T

T

0(1 2)vs ds (6.14)

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with V t evolving as the stochastic volatility process specic for the Heston model asdescribed in equation (6.2).

The benet of this approach is to improve Monte Carlo techniques, where the

mixing theorems increase the efficiency of Monte Carlo evaluation of option pricesin stochastic volatility models.

6.3.1 The Milstein scheme

In equations (6.13) - (6.14) two integrals over the process vt have to be estimated.In this relation we use the Milstein scheme in the discretization of the process of vt ,which in general terms (Glasserman, 2003) can be written as

X (i +1) X (i)+ a( X (i))h + b( X (i)) hZ i+1 + 12

b( X (i))b ( X (i))h(Z 2i+1 1) (6.15)

which in the specic case of The Heston model translates to

vt+1 = vi (vi ) t + vt tZ t+1 + 2

4 t(Z 2t+1 1) (6.16)

where if vt = 0 and 4/2

> 1 then vt+1 > 0, which means that the occurrence of a negative variance should be substantially reduced compared to the Euler scheme(Gatheral, 2006, p. 22).

In addition to this we introduce antithetic variables that runs the same calcula-tions but with an opposite sign of the random value to speed up the converge of thesimulated price to the true model price. Furthermore we introduce the absorbingassumption that if v < 0 then v = 0 (Gatheral, 2006, p. 22).

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7

Market data

As noted in the chapter 2, the volatility quoting mechanisms are FX specic andthe FX smile is given implicitly as a set of restrictions implied by market structures.

7.1 Quoting conventions

It is possible to identify some structures that are very popular amongst professionalmarket participants:

7.1.1 ATM straddle

The quotes for this structure on standard maturities are the most liquid ones(Castagna, 2010, p. 16).

I. ATM straddle (ATM STDL): The sum of a (base currency) call and a (basecurrency) put struck at the at-the-money level.

This structure here is traded in different ways depending on which ATM conventionis used. The rst kind is

ATM spot: The strike is set equal to the FX spot rate. (The strike is inde-pendent of the maturity on the options.)

ATM forward : The strike is set equal to the FX forward rate. (In this casethe strike is dependent on the maturity of the options, c.f. 3.1.)

ATM Delta neutral STDL : The strike is chosen so that, given the expiry, aput and a call have the same Delta but with different signs. (In this case the

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strike is also dependent on the maturity of the options because of its inuenceon Delta on the two options.)

The ATM STDL implied volatility quoted in the FX option market uses the last

listed ATM convention; the ATM Delta neutral STDL (ATM DNS). So the impliedvol. in this ATM structure is dependent on the strike level implied by the Deltason the two options and should not be confused by the simpler implied vol. basedon just setting the strike equal to the spot rate i.e. the ATM spot convention.

AT M STDL = AT M DNS (7.1)

7.1.2 Risk Reversal

Besides the ATM STDL, there are at least two other structures frequently traded.One of them is the risk reversal.

2. Risk reversal (RR): One buys a (base currency) call and sells a (base currency)put both featured with the same absolute Delta (symmetric).

Delta can be chosen equal to any level, but the 25% is the most liquid one (Castagna,2010, p. 17). So the call and the put entering into the RR will have a strike levelyielding a 25% Delta without considering the sign, which for puts will be negative.The RR is quoted as the difference between the implied volatilities of the two op-tion prices and we indicate this price in volatility as rr. Equation 7.2 denotes thisrelationship.

RR = Call P utrr (t, T ; 25) = 25C (t.T ) 25P (t, T ) (7.2)

where (t, T ) is the implied vol. at t for an option expiring in T and with a strikecorresponding to the Delta level indicated in the subscript.

7.1.3 Vega-weighted buttery

The other structure that is traded frequently is the Vega-weighted buttery.

3. Vega weighted buttery (VWB): One sells an ATM STDL and buys a strangle.The strangle is the sum of a (base currency) call and put both featured withthe same absolute Delta (symmetric).

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The 25% Delta VWB is the most traded VWB (Castagna, 2010, p. 18). The VWBis quoted as the implied volatilities of the two options in the strangle, which areevenly split, and the implied vol. of an ATM DNS.

V W B = 0.5(Call + P ut ) AT M DNS vwb(t, T ;25) = 0.5(25C (t, T ) + 25P (t, T )) ATM DNS (t, T ) (7.3)

In the interbank market the quotations of the VWB appear as in 7.3 (Bloomberg).

7.2 Retrieving the implied volatility

From the three main structures dealt on the FX options market, ATM STDL, RR

and the VWB it is possible, from the relationships in equations 7.1, 7.2 and 7.3to immediately retrieve the implied volatilities. As for the at-the-money impliedvol. it is synonymous with the quoted ATM STDL (from now on instead of writingATM DNS we write simply AT M ), whereas the implied vol. on a 25 Delta call andput (actually -25 Delta for the put) can be calculated as in equations 7.5 and 7.6,respectively.

AT M = AT M ST DL (7.4)

25C (t, T ) = AT M (t, T ) + vwb(t, T ; 25) + 0.5rr (t, T ;25) (7.5)

25P (t, T ) = AT M (t, T ) + vwb(t, T ;25) 0.5rr (t, T ;25) (7.6)

These examples show the calculations of the implied volatility on a 25D call andput, but applies to any Delta level assuming the corresponding VWB and RR areavailable.

7.2.1 The Delta-sticky convention

FX derivatives markets quote the strike prices in terms of Delta of the option. Thisis referred to as the so-called Delta-sticky convention. Its implications concerns thefollowing:

The sticky Delta: Once the deal is closed, given the level of the FX spot rateand the implied vol. agreed upon (where the interest rate levels will be taken

from the money market), the strike will be set at a level yielding the BS Deltathat the two counterparties were dealing.

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Practically this means that, if the FX spot rate moves, all other things being equal,the curve of implied vol. vs. Delta will remain unchanged, while the curve of theimplied vol. vs. strike will shift.

When the option is quoted with reference to a strike level expressed in terms of

Delta, once the option is traded and the FX spot reference rate and the implied vol.are xed, then the absolute level of the strike can be retrieved by setting

wP f (t, T )(wd1) = (7.7)

which can be expressed as

K = F (t, T ) exp w

(T t)1(| |/P f (t, T )) + 0 .52(T t) (7.8)

where the at-the-money strike reduces to

K AT M (t, T ) = F (t, T ) exp 0.52AT M (T t) (7.9)

where w = 1 (respectively, w = -1) if a call (put) option. 1 is the inverse of thecumulative normal distribution function, and the values and are the requiredinputs. So if one wants to nd the corresponding strike level to a 25 Delta call, the

inputs into equation 7.8 must be the at the 25 Delta call level and would be0.25. The latter enters into the formula as its absolute value, which is relevant when

considering put options.

7.2.1.1 Premium included Delta

The Delta convention used in the market for EURUSD and USDJPY has impli-cations on which method to use when converting Deltas into strikes. In the caseof the EURUSD a regular Delta is quoted whereas in the case of the USDJPY a

premium included/adjusted Delta is the market convention on how to quote Delta(Bloomberg), (Reiswich and Wystrup, 2010).In order to explain the difference between a regular premium excluded Delta

and a premium included Delta we use numbers from the example in (Reiswich andWystrup, 2010, p. 4) to create Tabel 7.1.The relationship between the premium included Delta and Delta is

P I = V S 0

(7.10)

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Table 7.1: Premium included DeltaEURUSD Call

Notional 1,000,000 EURS 0 1.3900K 1.3500

-T -

Premium 102,400 USD 73,669 EUR

60% 600,000 EUR PI 52.63% 526,331 EUR and T is not provided, but is also irrelevant in the example.

where the amount of foreign currency units to buy in a hedge of a short position is

F OR = N ( V S 0 ) (7.11)

If we go short in a EURUSD call option with notional EUR 1 mio. we receive apremium of USD 102,400, which corresponds to EUR 73,669. Lets say the Deltaon the option is 60%. Then we have to buy EUR 0.6 mio. in order to keep alocal Delta hedge. But considering that we receive something from the trade of theoption, the EUR amount to hedge is only EUR 526,331, which corresponds to apremium included Delta of 52.63%.

As just mentioned in the EURUSD case, the market convention is actually toquote the regular premium excluded Delta. So in this case we can rely on equations7.8 - 7.9 and directly retrieve the strike. This is not the case for the USDJPY wherewe have to resort to a numerical procedure (based on the Newton-Raphson scheme),when we want to retrieve the strike, since the option premium entering into Deltais a function of the strike itself (Castagna, 2010, p. 35-36). The procedure used inthis study follows this scheme and is outlined in the Appendix.

An example of the conversion of the premium included delta to strike is presentedin Tabel 9.1 where we calculate both the strike retrieved directly from equations7.8 - 7.9 and the strike adjusted for the premium included (PI) Delta, calculatednumerically, which in this case for the USDJPY is the correct way to retrieve thestrike. This is done both for a call and a put.

The premium included Delta will always be less than the regular premium ex-cluded Delta. This is true both when we consider calls and puts (where the Delta ona call is measured along the scale from 0% - 100% and the Delta on a put is measuredalong the scale from 0% - (-100%) going from OTM - ITM on both scales). This fact

would imply that, in the case of a call, the strike retrieved from a PI Delta, withoutmaking any adjustments (calculated from 7.8 - 7.9), will always be higher than the

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Table 7.2: Conversion of a Premium Included Delta to StrikeUSDJPY Call USDJPY Put

S 0 81.54 S 0 81.54 25 c 11.97% 25 p 13.13%T-t 0.5 T-t 0.5

P d

0.99828 P d

0.99828P f 0.99786 P f 0.99786

K 86.596 K 76.905K adjusted for PI D 86.302 K adjusted for PI D 76.5974/28/2011

strike adjusted for the PI Delta, calculated numerically. This implies a strike thatis more ITM. In case of a put, the strike retrieved from a PI Delta, without makingany adjustments, will always be higher than the strike adjusted for the PI Delta,which implies a strike that is less ITM as opposite to the case of the call. Thisrelationship is apparent in Tabel 9.1 where K > K adjusted for PI D in both thecall and put cases.

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8

Data description

I. In Chapter 5, Empirical facts:

Daily spot FX rates from 1/6/2006 - 5/3/2011 on 1.388 weekdays on theEURUSD and the USDJPY.

2. In Chapter 9, Calibration of the models and Chapter 10, Empirical study onthe hedging performance:

Bid and Ask prices on D10 RR, D10 VWB, D25 RR, D25 VWB and the ATMDNS with maturites 1M, 2M, 3M, 6M and 1Y from 1/4/2010 - 06/22/2011consisting of 18.550 quotes on 371 trading days for both the EURUSD and theUSDJPY. The conventions used in the quoting is for the ATM setting: the ATMDNS, for the Delta Premium: Excluded in the case of the EURUSD and Included inthe case of the USDJPY, for the Delta style: Spot Delta (up to < 1Y then forwardDelta), for the RR: Call P ut and for the VWB: (Call + P ut )/ 2 ATM DNS .Preference: Bloomberg BGN and Cutoff: New York 10:00.

Daily spot FX rates from 1/4/2010 - 6/22/2011 on 371 trading days.Source : Bloomberg

Domestic and Foreign interest rates on 1M, 2M, 3M, 6M and 1Y from 1/4/2010- 6/22/2011 on 371 trading days. More specically, we use the deposit interestrates Euribor for the EUR, Libor USD for USD and Libor JPY for JPY.

Source : Nordea Analytics

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9

Calibration of the models

9.1 Building the market implied volatility surface

We want to go from the data as described in Chapter 8, consisting of 5 structureson each of 5 maturities, to a set of market call option prices with correspondingstrike prices. We calibrate to prices and not imp. vols. Furthermore we recognisethat we have only the OTM part of the IVS for call options after retrieving the imp.vol. on a 25D call and 10D call by Equation (7.5). In order to get the ITM part of

the IVS for call options we have to make an assumption. The following steps arecarried out in the procedure to convert our market data into market prices on calloptions covering a wide range of the IVS:

1. We apply Equations 7.5 - 7.6 in order to get from D10 RR, D10 VWB, D25RR, D25 VWB and the ATM DNS to 10C , 10P , 25C , 25P and AT M losingthe time subscript.

2. From here we retrieve the strikes from Equations (7.8) - (7.9) in case of the

EURUSD and use the numerical method applied in Table 9.1 in case of theUSDJPY in order to get from a Delta moneyness to strike prices. We useEquation (7.9) in both the EURUSD and USDJPY case to retrieve AT M asprescribed in ?? .

3. We then assume the put-call parity in 6.8 to hold all though research showsthat this is rarely the case (Chalamandaris and Tsekrekos, 2008). Under thisassumption we calculate the BS call prices on the 5 pairs of 10P / K 10 P , 25P / K 25 P , AT M / K AT M , 25C / K 25 C , 10C / K 10 C . It is the convention on

every option market that the imp. vol. quoted is the BS imp. vol. whichallows us to use the BS model to calculate the prices.

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We then end up with market call option prices with the Delta moneyness 90D,75D, (50)D, 25D and 10D with corresponding strike prices.

9.2 Calibration of the Heston model"The price to pay for more realistic models is the increased complexity of model calibration: as noted by [Jacquier & Jarrow (2000)], in presence of complex models the estimation method becomes as crucial as the model itself" (Hamida and Cont,2005, p. 3) .

The calibration of the parameters in a SV model can be done in two conceptuallydifferent ways.

First, one can choose to look at the historical time series of asset returns recordingonly the historical spot prices and on this basis try to estimate the model parametersthat would yield the current observed option prices. Applied estimation methodshere have been Generalized, Simulated and Efficient Methods of Moments (Janek,Kluge, Weron, and Wystup, 2010). In this case 6 parameters incl. has to beestimated. It has been showed that those parameters that produce the currentobserved prices and their time-series estimate counterparties are in fact different(Bakshi, Cao, and Chen, 1997), which makes this way of calibrating a less attractive

choice.The second way is to calibrate the model parameters to current observed vanillaoption prices. In this case we only have to estimate 5 parameters with the ex-clusion of . This way of calibrating is also categorized as an indirect method of approximating the RND by tting the parameters, driving the stochastic process of the underlying, to observed option prices (Jackwerth, 1999), (Brunner and Hafner,2003).

We calibrate to the semi analytical closed-form solution in equation (6.7) wherethe price of a call option is calculated by numerical integration by approximation of the integral with the MATLAB function quad, that uses a low order method usingan adaptive recursive Simpsons rule. Alternatively the pricing formula could besolved by the Fast Fourier Transform (Carr and Madan, 1999).

9.2.1 The calibration problem

The minimization of the sum of squared errors, S () , is not a straight forward prob-

lem. Minimizing the objective function in (9.3) is a nonlinear problem subject to thenonlinear constraint in (6.4) and is far from being convex. It turns out that usually

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there exist many local minima (Mikhailov and Nogel, 2003, p. 3). The optimizationoptimizers in MATLAB are all local minimum optimizers as fminsearch and lsqnon-lin . So when applying these functions one is not sure that the solution obtained isa global minimum. Meanwhile we use the function lsqnonlin to minimize the objec-

tive function, while still trying to get around the potential problem of the existenceof many local minima. This function uses the Trust-Region-Reective Algorithm.Using the function acquires an initial guess on the parameters at which values theoptimizer will start its search for a local minimum. This has the implication thatthe solution is dependant on the initial guess.

In order to try to get around the problem we divide each of the 5 parameters ina number of values which will create a 5-dimensional space of initial guesses. As anexample is divided into the 3 values = [0.9, 0.45, 0] (which is based on theknowledge that the spot price and the volatility are in fact negatively correlated). Inthis way, by given each parameter 3 values that is tested up against each combinationof the 3 possible values of the 4 other parameters, we will be equipped with 35 = 243initial guesses. This procedure is only carried out for the 1. sample day. Theparameters in the solution with the smallest S () is then chosen as the solution onthe 1. day and carried over as the initial guess at the next day. The solution on the2. day is then carried over as the initial guess at the 3. day in continuation untilthe last day. The function lsqnonlin also allows for an upper and lower bound onparameters in the solution.

9.2.2 Implementing the Feller condition

The Feller condition in (6.4) is reformulated into

2 2 > 0. (9.1)The LHS of equation (9.1) is then the rst parameter that goes into the functionthat need to be optimized by the 5 parameters. Then, from the solution, we canretrieve the parameter by setting

= (2 2) + 2

2 (9.2)

9.3 Calibration of the Black-Scholes Model

In the BS model there is only one parameter to calibrate, the volatility . Thisparameter is by the model specication a xed parameter across the moneyness-

maturity level suggesting a at and constant imp. vol. surface. We here use the

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exact same objective function in Equation (9.3) as when calibrating the Hestonmodel, including the weighting scheme of bid and ask prices. That is, we minimizethe objective function in order to nd the best t to the 25 option prices observedon each day to get one volatility parameter per day.

9.4 Objective Function

The optimization algorithm applied in this study is shown in (9.3).

= arg min 125

5

i=1

5

j =1

[wi,j (C market (K i,j , T i ) C model (K i,j , T i |))]2 (9.3)

We minimize the sum of squared errors between observed option prices covering asurface in the strike and maturity dimension. More specically we choose to calibrateto call (mid) prices with strikes corresponding to 90D, 75D, ATM, 25D, 10D withina given maturity, where the specic maturities are 1 month, 2 months, 3 months,6 months and 1 year. This gives us 25 (mid) prices on each day governed by theparameter set , ,v0, and in the Heston model and in the BS model, which weestimate by the calibration. Furthermore we implement a weighting scheme, which

we will introduce in the next section.

9.4.1 Weighting scheme

We implicitly introduce weights when choosing which strikes and maturities to cali-brate the model to. In the case of the strike points things to consider in this respectare the coverage of the strike dimension. By choosing strike points with a range inthe moneyness dimension from D90% to D10% we cover 90 % of the smile, whichwould imply to us the curvature and skewness of the smile. Also the strikes D75,

ATM and D25 are the most traded as mentioned earlier and including the D90%and D10% these are the once quoted by Nordea and most easily retrievable onBloomberg. Considering the maturity points, a prior analysis of the liquidity onthese could be done, if this information was retrievable. Meanwhile we stick to thevery commonly quoted maturities that we have chosen.

The weighting scheme implemented in (9.3) considers the bid-ask spread on theprices observed in the market like in (Moodley, 2005). What we want to do isto incorporate the extra information that we get from a bid-ask spread into our

estimation. So our objective is to t better the price with a smaller bid-ask spread

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compared to a price with a bigger bid-ask spread. The way this is implemented isapparent in (9.4).

wi,j = 1

|C bid

market(K i,j , T i )

C ask

market(K i,j , T i )

| (9.4)

So a smaller bid-ask spread will introduce a bigger weight making the error cal-culated bigger and vica versa, a higher bid-ask spread will give a smaller weightresulting in a smaller error. The implications of this scheme is that we are ttingbetter the prices with smaller bid-ask spreads by not allowing the t to get too farfrom the mid price, while at the same time allowing the t to move further awayfrom the mid price with a larger bid-ask spread.

9.5 Calibration results

9.5.1 Goodness of t

In order to get an idea of just how well our calibrations of the models are, wecalculate a goodness of t measure according to equation (9.3). The mean andstandard deviation of the sum of squared errors, S () is calculated on a quarterlybasis as well as overall in Tables 9.1 - 9.2. The numbers are not directly comparable

between the two FX pair calibrations because of the difference in the level of op-tion prices as a consequence of very different levels of underlying FX exchange rates.

Considering rst the overall Heston calibration to the EURUSD from the Figures9.6 - 9.7 and Figures 9.10 - 9.11 the calibration performance looks reasonable, allthough not great, when comparing the market prices with the resulting closed formmodel prices. Equivalently the Figures from 9.14 - 9.15 and 9.18 - 9.19 provides uswith an idea about the Heston calibration to the USDJPY, which also in this caseseem reasonable but not great. This problem is partly caused by the likeliness thatsome or all calibrations have not been able to nd the true global minimum of theoptimization. This problem could be overcome by more advanced methods such asintroducing a heuristic optimization optimizer such as Differential Evolution as usedin (Gilli, Groe, and Schumann, 2010), also used here in a minimization problem,allthough a different one.

When comparing the goodness of t of the Heston model to that of the BS modelfrom Tables 9.1 - 9.2, we observe a 10 times better t of the Heston model in theEURUSD case with almost similar standard deviation around this measure. In the

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case of the USDJPY the difference between the t of the Heston model and the BSmodel is much more. Here we observe a 50 times better t of the Heston modelwith a lot of deviation around the poor t of the BS model suggesting days of evenworse t.

From looking at the volatility smiles through Figures 9.6 - 9.21, we would expectthe BS model to have a better t to the EURUSD than the USDJPY. The formerhas a curve closer to a constant at curve which is the only shape that the BS modelcan generate.

Table 9.1: Quarterly mean and standard deviation of the goodness of t of Hestonparameters

EURUSD USDJPY

mean std. dev. mean std. dev.2010 Q1 0.0046 0 .00080 0 .2577 0 .084182010 Q2 0.0055 0 .00148 0 .2278 0 .044382010 Q3 0.0037 0 .00010 0 .3524 0 .398852010 Q4 0.0059 0 .00400 0 .2094 0 .079752011 Q1 0.0039 0 .00130 0 .1293 0 .080292011 Q2 0.0842 0 .07644 0 .0963 0 .01747

0 .0169 0 .05188 0 .2144 0 .19531

2011 Q2 contains the trading days up to 06/22/2011

Table 9.2: Quarterly mean and standard deviation of the goodness of t of theBlack-Scholes parameter

EURUSD USDJPYmean std. dev. mean std. dev.

2010 Q1 0.1163 0 .0264 8 .334 2 .67982010 Q2 0.0996 0 .0269 8 .984 3 .39212010 Q3 0.0922 0 .0282 9 .333 2 .56912010 Q4 0.0738 0 .0404 12 .530 3 .97712011 Q1 0.1428 0 .0669 15 .206 4 .71382011 Q2 0.1960 0 .0760 15 .331 2 .8921

0 .1186 0 .0626 11 .564 4 .5013

2011 Q2 contains the trading days up to 06/22/2011

Supported by the results in (Bakshi, Cao, and Chen, 1997) we also nd that theimplied parameters in the stochastic volatility model vary through time. This ndingwould lead to the conclusion that the model is misspecied. In the case of the BSmodel we nd that this model is misspecied because of the presence of a volatilitysurface as we see it if we as an example compare Figures 9.8 and Figures 9.9. Inboth gures we observe a strike dependence of the imp. vol. and by comparingthe two different maturities we further observe a maturity dependence with higherimp. vol. in the long maturity of 1Y. Furthermore this surface changes dynamically

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through time, which can be observed by comparing the set of Figures 9.8 and 9.9to the set of Figures 9.12 and 9.13, where the surface is at a much higher level.The stochastic volatility model provides us with a more realistic representation of the underlying dynamics by introducing the possibility of a return distribution that

provides different levels of skewness and kurtosis. But the fact that the impliedparameters change through time and therefore has to be re-calibrated is evidence of its misspecication. These changes in the calibrated parameter values can be seenin Figures 9.1 - 9.5.

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The Figures from 9.6 - 9.21 represents two different days of option prices

master thesis excl. appendix

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