pricing of path-dependent basket options using a copula ... · pricing of path-dependent basket...

Pricing of path-dependent basketoptions using a copula approach

Christ Church

University of Oxford

A thesis submitted in partial fulfillment of the requirements for the MSc inMathematical Finance

i

Abstract

The pricing of basket options is usually a difficult task as assets of a basket usually showsignificant dependence structures which have to be incorporated appropriately in mathe-matical models. This becomes especially important if a derivative depends on the wholepath of an option. In general pricing approaches linear correlation between the differentassets are used to describe the dependence structure between them. This does not takeinto account that empirical multivariate distributions tend to show fat tails. One tool toconstruct multivariate distributions to impose a nonlinear dependence structure is the useof copula functions.

In the thesis the general framework of the use of copulas and pricing of basket optionsusing Monte Carlo simulation is presented. On the base of the general framework analgorithm for the pricing of path-dependent basket options with copulas is developed andimplemented. This algorithm conducts the calibration of the model to market data andperforms a simulation and estimates the fair price of a basket option. In order to investigatethe impact of the use of different copulas and marginals the algorithm is applied to aselection of basket options. It is analyzed how the proposed alternative approach affectsthe fair price of the option. In particular, a comparison to standard approaches assumingmultivariate normal distributions is made. The results show that the use and the choiceof copulas and especially the choice of alternative marginals can have a significant impacton the price of the options.

Contents

1 Introduction 1

2 Basket Options 22.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Valuing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3.1 Black-Scholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.2 Finite Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.3 Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Standard Pricing using Monte Carlo 9

4 Copulas 124.1 Definitions and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Sklar’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 Measures of Association . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4 Copula Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.4.1 Elliptical Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4.2 Archimedean Copulas . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.5 Estimation and Calibration from Market Data . . . . . . . . . . . . . . . . 274.6 Simulation Methods for Copulas . . . . . . . . . . . . . . . . . . . . . . . . 29

4.6.1 Elliptical Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.6.2 Archimedean Copulas . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Monte Carlo Simulations with Copulas 33

6 Numerical Experiments 376.1 Examined Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6.2.1 Estimation of Parameters . . . . . . . . . . . . . . . . . . . . . . . 446.2.2 Pricing of the Options . . . . . . . . . . . . . . . . . . . . . . . . . 45

7 Conclusions 53

A Student’s t-distribution 55

ii

CONTENTS iii

B Maximum Likelihood Method 56

List of Figures

4.1 Gaussian copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Random draws from a Gaussian copula . . . . . . . . . . . . . . . . . . . . 194.3 Student’s t-copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.4 Random draws from a Student’s t-copula . . . . . . . . . . . . . . . . . . . 214.5 Gumbel copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.6 Random draws from a Gumbel copula . . . . . . . . . . . . . . . . . . . . . 244.7 Clayton copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.8 Random draws from a Clayton copula . . . . . . . . . . . . . . . . . . . . . 254.9 Frank copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.10 Random draws from a Frank copula . . . . . . . . . . . . . . . . . . . . . . 26

6.1 Scatterplot of the returns in the observed time series . . . . . . . . . . . . 416.2 Histograms of the observed returns for the underlying assets . . . . . . . . 426.3 Q-Q plots of the observed returns against the normal distribution . . . . . 43

iv

List of Tables

4.1 Selected conditional transforms for copula generation . . . . . . . . . . . . 32

6.1 Statistics on historical returns of the Bayer/BASF basket . . . . . . . . . . 386.2 Statistics on historical returns of the BMW/VW basket . . . . . . . . . . . 396.3 Parameters of examined options . . . . . . . . . . . . . . . . . . . . . . . . 406.4 Parameters Φmarginals of the underlyings Bayer(Φmarginal

1 )/BASF (Φmarginal2 )(daily

monitoring) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.5 Parameters Φmarginals of the underlyings Bayer (Φmarginal

1 )/BASF (Φmarginal2 )

(weekly monitoring) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.6 Parameters Φmarginals of the underlyings BMW (Φmarginal

1 )/VW (Φmarginal2 )

(weekly monitoring) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.7 Parameters Φmarginals of the underlyings BMW (Φmarginal

1 ) /VW (Φmarginal2

) (daily monitoring) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.8 Parameters Φcopula of the basket with underlyings BMW/VW (weekly mon-

itoring) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.9 Parameters Φcopula of the basket with underlyings BMW/VW (daily mon-

itoring) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.10 Parameters Φcopula of the basket with underlyings Bayer/BASF (weekly

monitoring) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.11 Parameters Φcopula of the basket with underlyings Bayer/BASF (daily mon-

itoring) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.12 Prices of basket options for underlyings Bayer/BASF in Euro (daily mon-

itoring) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.13 Differences of the prices in % compared to the Gaussian/Gaussian approach

for underlyings Bayer/BASF in Euro (daily monitoring) . . . . . . . . . . . 496.14 Prices of basket options for underlyings BMW/VW in Euro (daily moni-

toring) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.15 Differences of the prices in % compared to the Gaussian/Gaussian approach

for underlyings BMW/VW in Euro (daily monitoring) . . . . . . . . . . . 506.16 Prices of basket options for underlyings BMW/VW in Euro (weekly mon-

itoring) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.17 Differences of the prices in % compared to the Gaussian/Gaussian approach

for underlyings BMW/VW in Euro (weekly monitoring) . . . . . . . . . . 516.18 Prices of basket options for underlyings Bayer/BASF in Euro (daily mon-

itoring) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

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LIST OF TABLES vi

6.19 Differences of the prices in % compared to the Gaussian/Gaussian approachfor underlyings Bayer/BASF in Euro (weekly monitoring) . . . . . . . . . . 52

Chapter 1

Introduction

Many models in finance assume that asset returns are normally distributed. Usually, lin-ear correlation is the chosen measure of dependence between risky assets. The problemwith linear correlation is that equities’ prices exhibit a greater tendency to crash togetherthan to boom simultaneously. A number of empirical papers have shown that Gaussiandistributions do not fit return data well (Mashal and Zeevi [2002], Dobric and Schmid[2005]). This implies that the standard models face some problems when used to calculatefair prices, sensitivities, hedge ratios, etc. for financial derivatives.

Copulas are a proposed framework to model dependence between random variables (RVs),which are able to capture different properties of dependence structures. A copula gener-alizes linear correlation as a measure of dependence. If returns are normally distributed,then variance of the returns is a commonly used measure of risk, and linear correlationdescribes dependence. Copulas allow the construction of joint distributions which specifythe distributions of individual returns separately from each other and separate from thedependence structure. This increases the flexibility in specifying distributions of multiplerandom variables.

The aim of this thesis is to show how copulas can be incorporated in Monte Carlo simula-tions and to study the impact of this amendment in comparison to the standard models.To study the impact path-dependent basket options are used. Therefore, the second Chap-ter is devoted to basket options, as it gives definitions and outlines the standard methodsfor dealing with theses kind of options. The standard pricing approach with Monte Carlosimulations is described in more depth in the third Chapter. The fourth Chapter is ageneral introduction to the theory behind copulas. The term copula is defined and basicproperties of copulas are described. Some examples of common copulas are given andmathematical methods used for copulas are outlined. On the basis of the general frame-work a modification of the standard Monte Carlo simulation which enables to use copulasto model different dependence structures of the returns is proposed in the fifth Chapter.The developed model is applied to different basket options in the sixth Chapter and theimpact on the fair price of the option when pricing path-dependent basket options on theprice of the option is investigated. In the seventh Chapter the results are summarized.

1

Chapter 2

Basket Options

2.1 Definition

Definitions and classifications of basket options overlap one another and in the literaturenumerous definitions and classifications of basket options can be found. They often overlapother options such as Mountain Range options and Rainbow options because of theirmulti-asset characteristic. This thesis builds on the following general definition:

Definition 2.1 (Basket Option) A Basket Option is an option whose payoff dependson the value of a portfolio (or basket) of assets. In general, the corresponding assets arerelated.

The payoff p of a path-dependent basket options depends on the underlying assets atspecified points in time tj. Therefore, the payoff depends on time t and on the values ofthe underlying assets Si, i = 1, . . . , n at the monitored points in time tj, j = 1, . . . ,m.

Let ~Si = (Si(t0), Si(t1) . . . , Si(tm)) denote the set of prices of the ith underlying at the

monitored points in time, then the payoff can be written by p(t, ~S1, ~S2, . . . , ~Sn).The following notation is chosen for the following thesis:

• Number of assets in a basket: n

• Strike of the basket: K

• Number of points in time to monitor a basket: m

• Points in time to monitor a basket: tj, j = 1, . . . ,m

• Time of expiry: T = tm

• Time of valuation: t0

• Weights of the asset i in the basket: αi

• Prices of the ith underlying at time t: Si(t)

• Set of all prices of the ith underlying at the monitored points in time tj: ~Si =Si(t1), Si(t2), . . . , Si(tm).

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CHAPTER 2. BASKET OPTIONS 3

• Undiscounted payoff of an option at time t and realized paths of the underlyings(~S1, ~S2, . . . , ~Sn): p(t, ~S1, ~S2, . . . , ~Sn)

• Volatility of the ith underlying at time t: σi(t)

• Risk-free rate in the market: r

• Dividend yield of the ith underlying: di

• Expected return of the ith asset: µi

• Number of simulations performed in a Monte Carlo simulation: L

• Price of a basket option calculated by a Monte Carlo simulation at time t: V MC(t).

• Fair price of an basket option at time t: V (t).

Basket options are popular because they allow to hedge the risk of a portfolio consistingof several asset. The advantage of buying basket options is that they are usually cheaperthan options on the individual components. Thus, a basket option is considered as acheaper alternative to hedge a risky position consisting of several assets. In addition, abasket option is able to replicate the changes in a portfolio’s value more precisely thanany combination of options on the underlying assets. The typical underlyings of a basketoption are several stocks, indices or currencies. Less frequently, interest rates are alsopossible.

2.2 Examples

There is a huge variety of exotic multi-asset options traded on the markets. Multi-assetoptions features vary from simple basket options whose payoff is linked to the overall per-formance of the basket of stocks to cases where the investor receives a fix coupon providedthat none of a basket’s stocks trespasses a certain barrier. For this thesis’ investigationthe focus is on path-dependent basket options. In the following, several types of liquidlytraded path-dependent basket options are given, and they are used for examination later.1

• Asian Option

Asian Options are commonly traded basket options. Their payoffs depend on theaverage price of the underlying assets 1

T

∫ T

0(∑n

i=1 αiSi(t))dt, where αi is the weightin the basket of the ith asset. In practice the integrals are approximated by sums bysampling at discrete points in time tj ∈ [t0, T = tm]. The payoff at expiry for thediscrete case is therefore:

p(T, ~S1, . . . , ~Sn) =

(

1

m

m∑

j=1

(n∑

i=1

αiSi(tj)) − K

)+

(2.1)

1source: http://www.fincad.com/support/developerFunc/mathref/Basket.htm


where the αi satisfy∑n

i=1 αi = 1. European options with this payoff functions arecalled arithmetic weighted average options or simply arithmetic Asian Options.

• Average Spread Option

The payoff at expiry T of a Basket Average Spread Option is the difference betweenthe average spread 1

m

∑mj=1(S1(tj)− S2(tj)) and the strike K, if the spread is larger

than the difference of average spread and strike, and zero otherwise. The payoff atexpiry T is therefore:

p(T, ~S1, ~S2) =

(

(1

m

m∑

j=1

(S1(tj) − S2(tj))) − K

)+

(2.2)

.

• Average Strike Option

An Average Strike Option depends on the difference of the spread at expiry S1(T )−S2(T ) and the average spread K = 1

m

∑mj=1(S1(tj) − S2(tj)) during the life of the

option. The payoff at expiry is therefore

p(T, ~S1, ~S2) =

(

(S1(T ) − S2(T )) − (1

m

m∑

j=1

S1(tj) − S2(tj))

)+

(2.3)

.

• Lookback Options

A Lookback Option is a derivative product whose payoff depend on the maximumU = max

j=1,...,m(∑n

i=1 αiSi(tj)) or the minimum D = minj=1,...,m

(∑n

i=1 αiSi(tj)) of the real-

ized basket price over the lifetime of the option. For example, a Lookback Put has apayoff at expiry that is the difference between the maximum realized price and thespot price of the basket at expiry T . Therefore, the payoff is:

Lookback Put Option:

p(T, ~S1, . . . , ~Sn) =

(

U −n∑

i=1

αiSi(T )

)+

(2.4)

Lookback Call Option:

p(T, ~S1, . . . , ~Sn) =

(

n∑

i=1

αiSi(T ) − D

)+

(2.5)

• Double Average Rate Option

A Double Average Rate Option’s payoff is the difference between the arithmeticaverage of the underlying spot prices of the sample points in the first sampling


period and the arithmetic average of the underlying in the second sampling period,if it is positive.

Therefore, the payoff can be written as follows:

p(T, ~S1, . . . , ~Sn) =

(

1

j′

j′∑

j=1

(n∑

i=1

αiSi(tj)) −1

m − j′

m∑

j=j′+1

(n∑

i=1

αiSi(tj))

)+

(2.6)

The above described options are only some out of for a high quantity of examples oftraded path-dependent basket options. The above made selection should cover most ofthe liquidly traded basket options on the market. Nevertheless even the liquidly tradedbasket options show rather big spreads which can be partly ascribed to the correlationrisk, which is hard to hedge. That is why the bid-ask spread of the prices of the optionson the market is very high. These prices indicate just a range within which the exact pricecan be found.

2.3 Valuing

One challenge when dealing with exotic basket options is to find the fair price of an option,which is competitive and still generates profit to the trading desk. Under the assumptionof a complete market with no arbitrage opportunities the formula of basket option’s priceis a conditional expected value:

V (t) = er(T−t)E [p(T, s(ω))|Ft] = er(T−t)

∫

ω∈Ω

p(T, s(ω))dp(ω) (2.7)

where p(T, s(ω)) denotes the payoff of an option at expiry T with the realized paths ofthe underlyings s(ω), where p(ω), ω ∈ Ω denotes the risk-neutral realization probabilityand Ω the sample space.

There are different methods to calculate the price of a basket option in Equation (2.7).Three commonly used methods are

• Black-Scholes

• Finite Differences

• Monte Carlo

In the following the three models are described briefly and the advantages and disadvan-tages are sketched.


2.3.1 Black-Scholes

With Ito’s lemma a Black-Scholes formula for a specific class of basket options can bederived. For European basket options whose payoff depends only on the values of theunderlyings at expiry T , the derivation of the Black-Scholes formula is basically analogousto the one dimensional case. The partial differential equation (PDE) of the value of thesetypes of basket option is:

∂V

∂t+

1

2

n∑

i,j=1

ρijσiσjSiSj∂2V

∂Si∂Sj

+n∑

i=1

(r − di) Si∂V

∂Si

− rV = 0 (2.8)

Literature suggests various pricing methods to solve the differential Equation (2.8). Forspecific payout structures, analytical formulas are known (see Margrabe [1987] and John-son [1987]). By using analytical formulas no computational intense numerical methodshave to be applied. The disadvantage is that the Black-Scholes formula has an analyticsolution for only a limited set of basket option and the calculation of the joint cumulativenormal function for more than two variates is numerically very intense and no efficientmethods exist to compute this function.

2.3.2 Finite Differences

Finite difference methods are a means to obtain numerical solutions to partial differentialequations like Equation (2.8). It is a very powerful and flexible technique and is capableof generating accurate numerical solutions to many differential equations used for optionpricing. The finite difference method is based on a convenient and correct discretizationof the partial differential equation associated to the risk neutral pricing formula via theFeynmann-Kac representation. The finite difference method returns the price for all timesand values of the underlying assets of the analyzed basket option. Therefore, this approachis appropriate to price many types of basket options, including options with Americanexercise features.

The disadvantage is that the pricing of basket options via finite difference methods isof practical use for lower dimensional problems only, i.e. the number n of the underly-ing assets is limited. For higher dimensional problems the computational expense growsexponentially with the lattice dimension of the numerical method. This leads to highcomputational effort.

2.3.3 Monte Carlo

An alternative for high dimensional problems are Monte Carlo simulation methods. MonteCarlo methods are easy to use and cope with any type of basket options. The problemabout Monte Carlo simulations is the computational costs they already demand for lowdimensions. Monte Carlo simulations in general provide unbiased estimates with a con-vergence rate not dependent on the dimension of the problem. In contrast to the finite


difference technique, the Monte Carlo method returns the estimate for a single point intime. It is a flexible approach but usually requires refinements, such as variance reductiontechniques, in order to improve its efficiency.

2.4 Hedging

A trader’s typical activity is to price and then sell an exotic option. Then she has to hedgethe position. Therefore, the correct hedging strategy is essential to secure the profit. Todelta hedge a basket option sensitivities have to be calculated to find the hedging coeffi-cients. A poor estimation of correlation may lead to a poor hedge of the derivative, sincethe hedging coefficients would be wrong as well. Therefore, the dependencies between as-sets have an effect on the correct hedging strategy, and models should be able to capturethese dependencies.

Under the assumption that the market is complete all assets including basket optionscan be perfectly hedged by a self-financing portfolio. In practice, hedging basket optionswhen the number of underlying assets is large is a challenge in quantitative finance. Inthe presence of complex dependencies Monte Carlo and PDE methods have difficultiescomputing prices and hedge ratios. A portfolio has to be set up whose value tracks thebasket option. Therefore, sensitivities of the basket option with respect to its risk-factorshave to be found. In this case, hedging with all the underlying assets is not only com-putationally expensive, but also creates high transaction costs which greatly reduce thehedging efficiency. This difficulty on the numerical side is present even in the traditionalBlack-Scholes framework where it is assumed that the logarithms of stocks evolve accord-ing to correlated diffusions with constant correlation matrices. Therefore, one approachto overcome the previously mentioned problems is to use a subset of the basket’s assets tohedge the option. This becomes more practical and essential when some of the underlyingassets are illiquid or not even available for trading. The idea of using only several under-lying assets for basket options hedging was first introduced in Lamberton and Lapeyre[1992].

Apart from the sophisticated payout structures, the inherent challenge of pricing andhedging multi-asset equity options is the illiquidity of implied correlations due to thelack of standardized multi-asset contracts. Correlation risk stems particularly from twosources: First, correlations cannot directly be observed, but must be estimated. Second,even if one is able to estimate correlation exactly it may change over the time.

Equity correlation risk cannot be hedged as precisely as volatility risk. This is unlike for-eign exchange markets (see Kholodnyi and Price [1998]): Here, hedging correlation riskis possible, since volatilities and correlations of currency pairs are linked together via theexchange rate mechanism, as has been shown in a geometric interpretation by Wystup[2002]. Unfortunately, this does not hold for equity markets, as stocks are traded for cashand not in pairs like currencies.


This thesis contains a theoretical study of the impact of the use of copulas on the fairprice of path-dependent basket options. Despite the practical issues which arise whenhedging basket options it is assumed that the observed options can be perfectly hedged.Therefore, a risk-neutral approach is followed when using copulas to perform Monte Carlosimulations.

Chapter 3

Standard Pricing using Monte Carlo

In the following the standard approach of pricing basket options via Monte Carlo simula-tion is sketched. As this approach is the commonly used pricing technique it is a startingpoint comparing the pricing with copulas. Details about Monte Carlo Techniques can befound in Glassermann [2003].

As mentioned in Section 2.4 it is assumed that a perfect hedge can be formed. Therefore,the option value is the discounted risk-neutral expectation of its payoff (see Equation(2.7)). Hence the price can be estimated by Monte Carlo methods, simulating paths of theunderlying assets and taking the discounted mean of the generated payoffs. In principle,this can be done even if complex distributions or payoffs are involved, provided that thepath generating process of the assets is known. As a process for the underlying stocks thegeometric Brownian motion is assumed in standard Monte Carlo simulations:

dSi = µiSidt + σi(t)SidWi(t) (3.1)

ρijdt = dWi(t)dWj(t) (3.2)

where dWi(t) and dWi(t) are correlated increments of a Wiener process. This is the usu-ally used stochastic process and implies that the returns of the assets in the basket arenormally distributed with correlation ρij. For a risk-free pricing approach the drift µi ofthe ith asset is set to µi = r − di, where di denotes the dividend yield of the ith asset andr is the risk-free rate.

Under the assumption of constant volatility σi(t) = σi the solution is still a geometricBrownian motion of the following form:

Si(t) = Si(0) exp

[(

µi −σ2

i

2

)

t + σiWi(t)

]

(3.3)

Pricing path-dependent options requires to monitor the processes (3.3) at a finite set ofpoints in time t1, . . . , tm for each asset Si. This sampling procedure yields the followingexpressions for constant volatility:

Si(tj) = Si(tj−1) exp

[(

µi −σ2

i

2

)

(tj − tj−1) + Wi(tj − tj−1)

]

(3.4)

9

CHAPTER 3. STANDARD PRICING USING MONTE CARLO 10

where the vectors ~W (tj − tj−1) = (W1(tj − tj−1),W2(tj − tj−1), . . . ,Wn(tj − tj−1))T ,

j = 1, . . . ,m, are n-dimensional normal random variables with zero mean vector andcovariance matrix Σn to model the increments.

Sometimes Equation (3.1) is approximated in the discretized case by the Euler forwardmethod for example to solve it numerically

Si(tj) = µiSi(tj−1) · (tj − tj−1) + Si(tj−1)σ√

tj − tj−1Zij (3.5)

where Zij denotes a correlated standard normal distributed random variable. Then theprocess in Equation (3.5) describes the movement of each asset in the model.

The calculation of the price V (t) can be formulated as an integral on the set of all possiblepaths S in the following way (see Dahl and Benth [2001]):

V (t) = exp (−r (T − t))

∫

Sp(T, ~S)fS(~S)dz (3.6)

where ~S = (~S1, . . . , ~Sn) ∈ S denotes a set of n realized paths of the underlying assetsand fS is the probability density on S. This integral can be evaluated approximately byMonte Carlo simulation:

V MC(t) = exp (−r(T − t))1

L

L∑

l=1

p(T, ~Sl1, . . . ,

~Sln) (3.7)

where L is the number of conducted simulations and the vectors ~Sli, l = 1, . . . , L, denote

the realized paths of the ith underlyings in the lth simulation.

The Law of Large Numbers ensures that V MC(t) converges to the expected value in Equa-tion (2.7) in probability a.s. and the Central Limit Theorem 1 states that the differenceV MC(t)−V (t) converges in distribution to a normal with mean 0 and standard deviationσ√L. The convergence rate is O( 1√

L) for all dimensions n.

The expected error of a valuation via Monte Carlo method can be estimated by using thesampled standard deviation or root mean square error (RMSE). For a random variable Xit is definied as follows

RMSE =√

E((E(X) − X)2) = (σ√L

) (3.8)

I our case the RMSE is estimated by:

√

√

√

√

1

L − 1

L∑

k=1

(

pdisc(t, ~Sl1, . . . , ~Sl

n) − V MC(t))2

(3.9)

1under the condition that the random varialbes are identically distributed with finite mean and vari-ance

CHAPTER 3. STANDARD PRICING USING MONTE CARLO 11

where pdisc(t, ~Sl1, . . . , ~Sl

n) = er(T−t)p(T, ~Sl1, . . . , ~Sl

n) denotes the discounted payoff of the lth

simulation with realized path (~Sl1, . . . , ~Sl

n) at time t. Refinements in Monte Carlo methodsconsist in finding techniques whose aim is to reduce the RMSE. From Equation (3.8) itcan be seen that two ways to reduce the RMSE are to reduce σ, known as variance reduc-tion techniques like antithetic variates, control variates, importance sampling or stratifiedsampling, or to increase the number of simulations L.

The main steps when performing a Monte Carlo simulation to valuate a European typeoption are the following:

• Simulate L risk-neutral sample paths ~S1, . . . , ~Sn for the underlying assets.

• Calculate the payoff p(T, ~S1, . . . , ~Sn) at expiry T .

• Approximate the expected value of the option value at the expiry T by calculationthe mean over the simulations.

• Use the risk-free discount rate r and calculate the present value of the option valueV MC .

Chapter 4

Copulas

In this Chapter the basic concepts of copulas are presented. I restrict myself to the twodimensional case. The generalization to n dimensions can be found in Nelson [1999].

The idea behind the copula concept is demonstrated by an example of the Gaussiandistribution. Its bivariate density with correlation coefficient ρ is given by

f(ζ1, ζ2)) =1

2π√

1 − ρ2exp (− 1

2(1 − ρ2)(ζ2

1 + ζ22 − 2ρζ1ζ2)). (4.1)

The cumulative distribution function (CDF) of the marginal distribution FX(x) can bederived from the bivariate CDF by basic calculus:

F (x,∞) =

∫ x

−∞

∫ ∞

−∞f(ζ1, ζ2)dζ1dζ2 (4.2)

=

∫ x

−∞

1√2π

exp (−ζ21

2)dζ1 (4.3)

= FX(x) (4.4)

By deriving Equation 4.3 the marginal densities can be obtained:

F′

X(x) = fX(x) =1√2π

exp (−x2

2) (4.5)

The idea behind copulas is to go the other direction. Here the joint distribution shall bederived from the marginals. If one looks at a bivariate CDF F (x, y) = P (X < x, Y < y) oftwo random variables X and Y with marginal CDFs FX(x) and FY (y), then the randomvariables UX = FX(X) and UY = FY (Y ) are uniformly distributed on [0, 1] and therandom variables X and Y can be expressed by the inverse CDFs:

X = F−1X (UX) (4.6)

Y = F−1Y (UY ) (4.7)

By these equations the bivariate CDF can be expressed as follows:

F (x, y) = P (X < x, Y < y) (4.8)

= P (F−1X (UX) < x, F−1

Y (UY ) < y) (4.9)

= P (UX < FX(x), UY < FY (y)) (4.10)

12

CHAPTER 4. COPULAS 13

Equation (4.10) can be interpreted as a CDF of a bivariate random vector (UX , UY ), whosecomponents are uniformly distributed on [0, 1]. If one defines a function C : [0, 1]2 → [0, 1],with C(FX(x), FY (y)) := P (UX < FX(x), UY < FY (y)), this function still describes thebivariate CDF F (x, y) and contains explicitly the marginal CDFs. If one defines a function

C(u, v) = F (F−1X (u), F−1

Y (v)) (4.11)

then this function can be used to couple the marginal distributions to obtain the jointdistribution:

C(FX(x), FY (y)) =

∫ F−1

X(FX(x))

−∞

∫ F−1

Y(FY (y))

−∞f(ζ1, ζ2)dζ1dζ2 (4.12)

=

∫ x

−∞

∫ y

−∞f(ζ1, ζ2)dζ1dζ2 (4.13)

= F (x, y) (4.14)

This is basically the idea to define copula functions, because by the means of copulas themarginal CDFs can be decoupled from the dependence structure. If one looks at the resultof a bivariate standard normal CDF for two values x = 0.3 and y = 0.6 with ρ = 0.8 onegets

F (0.3, 0.6) = 0.57 (4.15)

The same result can be obtained by performing two steps. First, one evaluates the marginalCDFs

F (0.3) = 0.6179 (4.16)

F (0.6) = 0.7257 (4.17)

Second, one applies the Gaussian copula to the obtained results:

C(0.6179, 0.7257) = 0.57 (4.18)

4.1 Definitions and Basic Properties

A copula function is defined in the bivariate case as follows:

Definition 4.1 (Copula) A two-dimensional copula is a function C : [0, 1] × [0, 1] →[0, 1] with the following properties:

(i) For every u, v ∈ [0, 1]:

C(u, 0) = C(0, v) = 0.

(ii) For every u, v ∈ [0, 1]:

C(u, 1) = u and C(1, v) = v.


(iii) For every u1, u2, v1, v2 ∈ [0, 1] with u1 ≤ u2 and v1 ≤ v2:

C(u2, v2) − C(u2, v1) − C(u1, v2) + C(u1, v1) ≥ 0.

In the following some properties of copulas are presented. The next theorem establishesthe continuity of copulas via a Lipschitz condition on [0, 1] × [0, 1].

Theorem 4.2 (Continuity) Let C be a copula. Then for every u1, u2, v1, v2 ∈ [0, 1]:

| C (u2, v2) − C (u1, v1) | ≤ | u2 − u1 | + | v2 − v1 | .

From Theorem 4.2 it follows that every copula C is uniformly continuous on its domain.

A further important property of copulas concerns the partial derivatives of a copula withrespect to its variables:

Theorem 4.3 Let C be a copula. For almost every 1 u ∈ [0, 1], the partial derivative ∂C∂v

exists for almost all v ∈ [0, 1]. For such u and v one has

∂

∂vC (u, v) ≥ 0 (4.19)

The analogous statement is true for the partial derivative ∂C∂u

.

In addition, the functions u → cv (u) = ∂C (u, v) /∂v and v → cu (v) = ∂C (u, v) /∂u aredefined and non-decreasing almost everywhere on [0,1].

Hence a copula may be considered as a CDF. It is quite typical that their graphs arehard to interpret. Therefore, typically, plots of densities are used to illustrate distribu-tions, rather than plots of CDF. Examples of this are given in Section 4.4. If a copula issufficiently differentiable the copula density can be defined as follows.

Definition 4.4 Copula Density

c (u, v) =∂2C(u, v)

∂u ∂v(4.20)

If F (u, v) is a joint distribution with margins Fu(u), Fv(v) and density f(u, v), then thecopula density is related to the density fi of the margins by the canonical representation(see Cherubini et al. [2004]):

f(u, v) = c(Fu(u), Fv(v)) · fu(u) · fv(v) (4.21)

where fu(u) and fv(v) are the densities of the margins

fu(u) =∂Fu(u)

∂u. (4.22)

1The expression ”‘almost all” is used in the sense of the Lebesgue measure.


and fv(v) defined analogously. The copula density is therefore equal to the ratio of thejoint density f and the product of all marginal densities fj.

c(Fu(u), Fv(v)) =f(u, v)

fu(u) · fv(v)(4.23)

From this expression it is clear that the copula density takes a value equal to 1 everywherewhere the original random variables are independent.The canonical representation is veryuseful in statistical estimation, in order to have a flexible representation for joint densitiesand to determine the copula, if one knows the joint and marginal distribution.

4.2 Sklar’s Theorem

It is well known, that if a random variable U is uniformly distributed on [0, 1], thenfollowing expression holds:

Corollary 4.5 If U ∼ U [0, 1] and F is a CDF, then

P(

F−1 (U) ≤ x)

= F (x) (4.24)

Like wise, if the real-valued random variable Y has a distribution function F and F iscontinuous, then

F (Y ) ∼ U [0, 1] (4.25)

Given this result, it is not surprising that every distribution function on Rn embodies a

copula function. On the other hand, if one chooses a copula and some marginal distri-butions and entangle them in the right way, she will end up with a proper multivariatedistribution function. This result is known as Sklar’s Theorem:

Theorem 4.6 (Sklar’s theorem) Let F be a joint distribution function with margins F1

and F2. Then there exists a copula C with

F (x1, x2) = C (F1(x1), F2(x2)) (4.26)

for every x1, x2 ∈ R. If F1 and F2 are continuous, then C is unique. Otherwise, C isuniquely determined on Ran(F1)×Ran(F2), where Ran(Fi) denotes the range of the CDFFi, i ∈ 1; 2. On the other hand, if C is a copula and F1 and F2 are distribution functions,then the function F defined by (4.26) is a joint distribution function with margins F1 andF2.

The name Copula was chosen by Sklar to describe a function that links a multidimensionaldistribution to its one-dimensional margins and appeared in mathematical literature forthe first time in Sklar [1959]. Usually, the term Copula is used in grammar to describean expression that links a subject and a predicate. The origin of the word is Latin.

If one looks at the expression Fi F−1i (x) = x, the following equation is obtained:

C (u1, u2) = F(

F−11 (u1) , F−1

2 (u2))

(4.27)

Equation (4.27) provides a theoretical tool for the derivation of the copula from a multi-variate, in our case two-dimensional, distribution function. This equation also allows theextraction of a copula directly from a multivariate distribution function.


4.3 Measures of Association

Copulas may be used to couple random variables with distinctive relationships. Tworandom variates X and Y are said to be associated when they are not independent.However, there are many concepts to describe how random variates are associated. Threeof commonly used concepts to measure association are

• Kendall’s tau (Kendall’s τ)

• Spearman’s rho (Spearman’s ρ)

• Pearson’s linear correlation coefficient (Pearson’s ρ)

Kendall’s tauKendall’s tau was first introduced by Fechner in 1897 (see Fechner [1897]) and redis-covered by Kendall [1938]. It is a normalized expected value. The Kendall tau rankcorrelation coefficient is a non-parametric statistic used to measure the degree of cor-respondence between two rankings and to the significance of this correspondence. It isdefined as follows:

Definition 4.7 (Kendall’s τ) Let X and Y be random variables with Copula C andlet F−1

X (u) and F−1X (u) denote the quantile functions and u and v the quantiles. Then

Kendall’s tau (τ) with copula C is defined as

τ = 4

∫ ∫

I2

C(u, v)dC(u, v) − 1 (4.28)

It can be demonstrated that it measures the difference between the probability of con-cordance and the one of discordance for two independent random vectors, (X1, Y1) and(X2, Y2), each with the same joint distribution function F and copula C. The vectors aresaid to be concordant if X1 > X2 whenever Y1 > Y2, and X1 < X2 whenever Y1 < Y2.They are discordant in the opposite case. An unbiased estimator of Kendall’s tau for ann-dimensional sample is:

τ =nc − nd

12n(n − 1)

(4.29)

where nc is the number of concordant pairs and nd is the number of discordant pairs inthe sample.

Spearman’s rhoSpearman’s rho was first proposed in 1904 (seen Spearman [1904]). It is a non-parametricmeasure of correlation - that is, it assesses how well an arbitrary monotonic function coulddescribe the relationship between two variables.

Definition 4.8 (Spearman’s ρS) Let F−1X (u) and F−1

X (u) denote the quantile functionsand u and v the quantiles of two random variables X and Y . Then Spearman’s rho (ρS)with copula C is defined as

ρS = 12

∫ ∫

I2

C(u, v) − uv du dv (4.30)


In practice usually a simpler procedure is used to calculate ρS. According to Myers andWell [2003] for a random sample of n pairs (xi, yi), with i = 1, . . . , n n-dimensional sampleof data ρS is given by :

ρS =n(∑

xiyi) − (∑

xi)(∑

yi))√

n(∑

x2i ) − (

∑

xi)2√

n(∑

y2i ) − (

∑

yi)2(4.31)

In comparison to Kendall’s tau Spearman’s rho exploits probabilities of concordance anddiscordance. Between Kendall’s tau and Spearman’s rho the following relationship holds(see Durbin and Stuart [1951]):

32τ − 1

2≤ ρS ≤ 1

2+ τ − 1

2τ 2 τ ≥ 0

−12

+ τ + 12τ 2 ≤ ρS ≤ 3

2τ + 1

2τ < 0

(4.32)

Pearson’s linear correlation coefficientThe Pearson’s linear correlation coefficient has been introduced for random variablesbelonging to L2. It is probably the best known coefficient to measure association. Pearson’scorrelation coefficient is a parametric statistic and when distributions are not normal itmay be less useful than non-parametric correlation methods.

Definition 4.9 (Pearson’s linear correlation) Let F−1X (u) and F−1

X (u) denote the quan-tile functions and u and v the quantiles of two random variables X and Y . The Pearsonlinear correlation coefficient ρXY th is defined by

ρXY =cov(X,Y )

√

var(X)var(Y )(4.33)

or equivalently in terms of the copula C:

ρXY =12∫

I2(C(F1(u), F2(v)) − F1(u)F2(v))du dv√

var(X)var(Y )(4.34)

The formulation of Pearson’s linear correlation coefficient contains always the variancesof the marginal distributions. Therefore, the measure is not independent of the choice ofthe marginals unlike the previously introduced measures.

Starting from a random sample of n pairs (xi, yi), with i = 1, . . . , n the Pearson correlationcoefficient can be written as:

ρXY =n∑

xiyi −∑

xi

∑

yi√

n∑

x2i − (

∑

xi)2√

n∑

y2i − (

∑

yi)2(4.35)

For elliptical copulas the relationship between the Pearson’s linear correlation coefficientρ and Kendall’s τ is given by:

ρ(u, v) = sin(π

2τ)

(4.36)

and

ρS =6

πarcsin(ρ/2) (4.37)


4.4 Copula Families

Two common families of copulas are the Elliptical and the Archimedean copulas, whichare introduced in the following.

4.4.1 Elliptical Copulas

The Gaussian and Student-t copula are frequently used copulas. They belong to the familyof elliptical copulas. Elliptical copulas are simply the copulas of elliptically contoureddistributions. An advantage of elliptical copula is that one can specify different levels ofcorrelation between the marginals. They are characterized by a range of parameters andcan be fitted flexibly to data. A disadvantage is that elliptical copulas do not have closedform expressions and are restricted to have radial symmetry.

• The bivariate Gaussian copulaOne important elliptical copula is the Gaussian or normal copula which was de-scribed by Embrechts et al. [1999]. In the two-dimensional case it is defined asfollows.

Definition 4.10 (Bivariate Gaussian copula (BGC)) Let ρ ∈ [−1, 1] and Φρ

the standardized bivivariate normal distribution with Pearsons’s linear correlationρ. The BGC is defined as

CGaussρ (u1, u2) = Φρ

(

Φ−1 (u1) , Φ−1 (u2))

(4.38)

where Φ−1 denotes the inverse of the standard univariate normal distribution func-tion Φ. If the correlation ρ = 0, the Gaussian copula becomes the independentcopula,

CGauss (u1, u2) =2∏

i=1

ui = CIndependence (u1, u2) (4.39)

As ui ∈ [0, 1], one can replace ui in 4.38 by Φi (ri). If one considers ri in a probabilisticsense, i.e. ri being values of random variables Ri one obtains from Equation (4.38)

CGauss(Φ1 (r1) , Φ2 (r2)) = P (R1 ≤ r1, R2 ≤ r2)

In other words: CGauss(Φ1 (r1) , Φ2 (r2)) is the two-dimensional CDF.

From the definition of the Gaussian copula one can easily determine the correspond-ing density cGauss

R (ρ2 6= 1):

cGaussρ (u1, u2) =

1

2π√

1 − ρ2exp

(

2ρΦ−1 (u1) Φ−1 (u2) − Φ−1 (u1)2 − Φ−1 (u2)

2

2 (1 − ρ2)

)

(4.40)


0 0.2 0.4 0.6 0.8 1

0

0.5

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u1

u2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(a) Gaussian copula (ρ = 0.7)

0 0.2 0.4 0.6 0.8 1

0

0.5

1

0

0.05

0.1

0.15

0.2

0.25

u1

u2 0

0.05

0.1

0.15

0.2

(b) Density of the Gaussian copula (ρ =0.7)

Figure 4.1: Gaussian copula

Figures 4.1 shows the plot of a Gaussian copula and its density. In Figure 4.2 1000samples of two uniformly distributed random variables coupled with a Gaussian cop-ula are shown. The graph gives an impression on how the choice of the parameter ρinfluences the dependencies between the variables.

One reason for the importance of the Gaussian copula is that it generates the jointnormal standard distributions function, if and only if the margins are standard nor-mal. A proof can be found in Cherubini et al. [2004]. The Gaussian copula areusually used to model linear correlation dependencies.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u1

u2

(a) Gaussian copula (ρ = 0.7)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u1

u2

(b) Gaussian copula (ρ = 0.9)

Figure 4.2: Random draws from a Gaussian copula

• The bivariate Student’s t copula

Definition 4.11 (Bivariate Student’s t copula (BTC)) Let tρ,ν : R2 → R be


the bivariate Student’s distribution function, with ρ ∈ [−1, 1] and ν degrees of free-dom (d.o.f.):

tρ,ν(x, y) =

∫ x

−∞

∫ y

−∞

1

2π√

1 − ρ2

(

1 +s2 + t2 − 2ρst

ν(1 − ρ2)

)− ν+2

2

ds dt (4.41)

then the BTC is defined as follows:

Ctρ,ν(u1, u2) = tρ,ν(t

−1ν (u1), t

−1ν (u2))

=

∫ t−1ν (u1)

−∞

∫ t−1ν (u2)

−∞

1

2π√

1 − ρ2

(

1 +s2 + t2 − 2ρst

ν(1 − ρ2)

)− ν+2

2

ds

where t−1ν is the inverse of the univariate CDF of Student’s t distribution with ν

degrees of freedom.

It turns out hat the copula density for the BTC is (see Cherubini et al. [2004]):

ctρ,ν(u1, u2) = |ρ|−

1

2

Γ(

ν+22

)

Γ(

ν2

)

Γ(

ν+12

)2

(

1 +ζ21+ζ2

2−2ρζ1ζ2

ν(1−ρ2)

)− ν+2

2

∏2j=1

(

1 +ζ2j

ν

)− ν+1

2

(4.42)

0 0.2 0.4 0.6 0.8 1

0

0.5

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u1

u2 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(a) Student’s t-copula (ρ = 0.5, ν = 3)

0 0.2 0.4 0.6 0.8 1

0

0.5

10

1

2

3

4

5

6

7

8

9

10

u1

u2

1

2

3

4

5

6

7

8

9

10

(b) Density of the Student’s t-copula (ρ =0.5, ν = 3)

Figure 4.3: Student’s t-copula

Figure 4.3 shows the plot of a Student’s t-copula and its density. In Figure 4.4 1000samples of two uniformly distributed random variables coupled with a Student’st-copula are shown.

When the number of the degrees of freedom diverges the Student’s t-copula con-verges to the Gaussian copula. This is one reason that the Student’s t-copula is fastgrowing in usage because the weight in the tail dependency can be set by changingthe degrees of freedom parameter ν (see Nystrom and Skoglund [2002]). Large val-ues for ν approximate a Gaussian distribution. Conversely small values for ν increasethe tail mass. For ν = 1 it simulates a bivariate Cauchy distribution.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u1

u2

(a) Student’s t-copula (ρ = 0.5, ν = 3)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u1

u2

(b) Student’s t-copula (ρ = 0.95, ν = 1)

Figure 4.4: Random draws from a Student’s t-copula

4.4.2 Archimedean Copulas

The class of Archimedean copulas has been named by Ling [1965], but it was recognizedby Schweizer and Sklar [1961] in the study of t-norms. From the practical point of viewArchimedean copulas are useful because it is possible to generate a number of copulasfrom interpolating between certain copulas.

Archimedean copulas may be constructed using a continous, decreasing, convex functionφ : I → R

+, such that φ(1) = 0. Such a function φ is called a generator. It is called astrict generator whenever φ(0) = +∞.

The pseudo-inverse of φ is defined as follows:

φ−1(v) =

φ−1(v) 0 ≤ v < φ(0)

0 φ(0) ≤ v ≤ ∞(4.43)

Hence the function φ−1 is continuous and not increasing on [0,∞] and strictly decreasingon [0, φ(0)]. This inverse is such that, by composition with the generator function, it givesthe identity as ordinary inverses do:

φ−1(φ(v)) = v (4.44)

Given a generator and its inverse, an Archimedean copula CArchimedean is generated ac-cording to the Kimberling theorem. A proof can be found in Kimberling [1974]:

Theorem 4.12 (Kimberling theorem) Let φ be a generator. The function C : [0, 1]2 →[0, 1] defined by

C(u1, u2) = φ−1(φ(u1) + φ(u2)) (4.45)

is a copula if φ−1 is strictly monotone on [0,∞].

This theorem allows the definition of many Archimedean copulas. One important sourceof generators for Archimedean copulas is the inverses of the Laplace transforms of CDFs


(see Feller [1971]).

By the generator of an Archimedean copula its density can be derived by the followingequation:

cArchimedean(u1, u2) =−φ

′′

(C(u1, u2))φ′

(u1)φ′

(u2)

(φ′(C(u1, u2)))3(4.46)

For dimensions d > 2 the Archimedean copulas can represent only positive dependencies(see Embrechts et al. [2003]). This constraint is of less concern for practical applicationsin equity markets since negative dependencies are usually rare in that context. A moreimportant property of the most frequently used Archimedean copulas is that they dependon one parameter. Therefore, each two variables have the same degree of dependence,which turns out to be very restrictive for more than two dimensions.

Attention must be paid to the fact that for various modeling purposes Archimedean cop-ulas are flexible enough to capture various dependence structures, e.g. concordance andtail dependence, which makes them suitable for modeling extreme events. Usually theyare used to model a strong dependence in the tail (see Nelson [1999]).

Among Archimedean copulas in particular the one-parameter ones, which are constructedusing a generator φα(t) with one parameter α ∈ R are of importance. In practice, fre-quently used copulas are the Gumbel copula, the Clayton copula and the Frank copula:

• Gumbel CopulaThe Gumbel or Gumbel-Hougaard family of copulas was described by Hutchinsonand Lai [1990]. Their generator is given by φGu(u) = (− ln(u))α, hence (φGu)−1(t) =

exp(−t1

α ). It is completely monotonic if α > 1. The Gumbel copula is therefore:

Definition 4.13 (Gumbel Copula)

CGu (u1, u2) = exp

−[

2∑

i=1

(− ln ui)α

]1/α

with α ∈ [1,∞) . (4.47)

Nelson [1999] showed that the Gumbel-Houghaard Copula CGu can describe mul-tivariate extreme value distributions. For α = 1, Equation (4.47) reduces to

CGu(u1, u2) =2∏

i=1

ui (4.48)

the independent copula in Equation (4.39).

Figures 4.5 show the plot of a Gumbel copula and its density. In Figure 4.6 1000samples of two uniformly distributed random variables coupled with a Gumbel cop-ula are shown. The graph gives an impression how the choice of the parameter α


0 0.2 0.4 0.6 0.8 1

0

0.5

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u1

u2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) Gumbel copula (α = 2)

0 0.2 0.4 0.6 0.8 1

0

0.5

10

2

4

6

8

10

12

14

16

18

20

u1

u2 0

2

4

6

8

10

12

14

16

18

20

(b) Density of the Gumbel copula (α = 2)

Figure 4.5: Gumbel copula

influences the dependencies between the variables.

Gumbel copulas are often used to model extreme distributions. They are asymmet-ric Archimedean copula, exhibiting greater dependence in the positive than in thenegative tail.

The relationship between Kendall’s tau τ and the Gumbel copula parameter α isgiven by:

α =1

1 − τ(4.49)

The density of the bivariate Gumbel copula is given by

cGumbel(u1, u2) = exp

− [(− ln u1)α + (− ln u2)

α]1

α

(− ln u1)α−1 (− ln u2)

α−1

1

u1

1

u2

[(− ln u1)α + (− ln u2)

α]1

α−2(

[(− ln u1)α + (− ln u2)

α]1

α + α − 1)(4.50)

• Clayton CopulaA second example of Archimedean copulas is the Clayton copula. The generator isgiven by φCl(u) = u−α − 1, hence φ−1(t) = (t + 1)−

1

α . It is completely monotonic ifα > 0. The Clayton copula is therefore

Definition 4.14 (Clayton Copula)

CClα (u) =

[

n∑

i=1

u−αi − n + 1

]−1/α

with α ∈ (0,∞) (4.51)

For the limit α → 0 the independent copula is obtained again. In Figure 4.7 thegraph of a Clayton copula CCl

α for α = 5.0 and its corresponding density is shown.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u1

u2

(a) Gumbel copula (α = 2)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u1

u2(b) Gumbel copula (α = 20)

Figure 4.6: Random draws from a Gumbel copula

0 0.2 0.4 0.6 0.8 1

0

0.5

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u1

u2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(a) Clayton copula (α = 6)

0 0.2 0.4 0.6 0.8 1

0

0.5

10

20

40

60

80

100

120

u1

u2 0

10

20

30

40

50

60

70

80

90

100

110

(b) Density of the Clayton copula (α = 6)

Figure 4.7: Clayton copula


Figure 4.8 displays 1000 samples of two uniformly distributed random variables cou-pled with a Clayton copula.

The Clayton copula is an asymmetric Archimedean copula and exhibits greater de-pendence in the negative tail than in the positive, which can be seen from the densityplot in Figure 4.7(b). The Clayton copula is suitable for describing dependencies inthe left tail and that there is empirical evidence of increasing dependence in fallingmarkets (see Longin and Solnik [2001]).

The relationship between Kendall’s tau τ and the Clayton copula parameter α isgiven by:

α =2τ

1 − τ(4.52)

The density of the bivariate Clayton copula is given by:

cClayton = (1 + α)(1 + 2α)(

(u1u2)α−1) (

u−α1 + u−α

2 − 1)− 1

α−2

(4.53)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u1

u2

(a) Clayton copula (α = 2)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u1

u2

(b) Clayton copula (α = 20)

Figure 4.8: Random draws from a Clayton copula

• Frank Copula

The generator of the Frank copula is given by φFrankα (u) = ln

(

exp(−αu)−1exp(−α)−1

)

, hence

φ−1(t) = − 1α

ln (1 + et(e−α − 1)). It is completely monotonic if α > 0. The bivariateFrank copula is therefore given by:

Definition 4.15 (Frank Copula)

C(u1, u2) = − 1

αln

1 +

∏2i=1(e

−αui − 1)

e−α − 1

with α > 0 (4.54)


The relationship between Kendall’s tau τ and the Frank copula parameter α is givenby:

D1(α) − 1

α=

1 − τ

4(4.55)

where

D1(α) =1

α

∫ α

0

t

et − 1dt (4.56)

00.2

0.40.6

0.81

0

0.5

10

0.2

0.4

0.6

0.8

1

1.2

1.4

u1

u2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) Frank copula (α = 14.14)

0 0.2 0.4 0.6 0.8 1

0

0.5

10

2

4

6

8

10

12

14

16

18

20

u1

u2

2

4

6

8

10

12

14

16

18

20

(b) Density of the Frank copula (α = 14.14)

Figure 4.9: Frank copula

The Frank copula is a symmetric Archimedean copula. The density of the bivariateFrank copula is given by:

cFrank(u1, u2) = −α(w1 + 1)(w2 + 1)(e−α − 1)

[(e−α − 1) + w1w2]2 (4.57)

where wi = e−αui − 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u1

u2

(a) Frank copula (α = 2)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u1

u2

(b) Frank copula (α = 20)

Figure 4.10: Random draws from a Frank copula

In the presented copula families, the parameter α respectively ρ of the joint distributionassociated to a random couple (U1, U2) measures the degree of dependence between U1 and


U2. The larger |α| respectively |ρ| the stronger is the dependence. Therefore the param-eters obtained by calibrating to real data represent a measure of dependence. The mostcommonly used copulas are the Gumbel copula for extreme distributions, the Gaussiancopula for linear correlation, and the Archimedean copula (Frank, Gumbel, Clayton), andthe Student’s t-copula for dependence in the tail.

4.5 Estimation and Calibration from Market Data

Copulas represent a powerful tool for tackling the problem of how to describe a joint dis-tribution. From Sklar’s theorem (4.2) follows that the representation in Equation (4.27)can be reconstructed from the marginal and the joint distribution separately. A high flexi-bility in modeling random variables follows out of the separation of marginal distributionsand the dependency. This advantage makes the use of copulas a powerful tool to overcomeproblems which arise when using the standard Gaussian distributions.

The question is of how a copula model can be calibrated to observed data. The calibrationcan be divided into two separate steps. First one has to extract the distribution of themargins. After extracting the distribution of the margins the dependence structure mustbe identified. Since multivariate equity options are usually traded over-the-counter it isnot easily possible to obtain option prices in order to extract the risk-neutral copula frommarket prices. Due to this lack of data it is assumed that the risk-neutral and the realworld copulas are identical. Rosenberg [2003], for instance, argues that under generalconditions this is a reasonable assumption. Relying on historical asset returns differentcopula families can then be fitted to the data. Since returns are strictly monotone trans-formations of asset prices both have identical copulas.

The question is of how one can fit the marginals and the copula to observed data. Sincea copula function is a multivariate model much of the classical statistical theory is notapplicable. Therefore, usually the asymptotic maximum likelihood estimation (MLE) isapplied. In the Appendix B the idea of the method is outlined briefly. (For details seeCherubini et al. [2004]). There are three commonly used methods to fit a copula modelto observed data:

• Exact maximum likelihood method: The exact maximum likelihood method esti-mates all parameters of the marginals and the copula simultaneously. Therefore, it’sa computationally intense method and is of minor use for practical applications.

• IFM method: The inference for the margins (IFM) method estimates the parametersof the copula model in two steps. First, the parameters of the univariate marginaldistributions are estimated by MLE. Second, the copula parameter is estimated viaMLE. This approach is a fully parametric approach.

• CML method: The Canonical Maximum Likelihood (CML) follows the steps of theIFM method. The difference to the IFM method lies in the choice of the marginaldistribution. The CML method uses the empirical distribution and avoids any as-sumptions about the marginal distributions. The advantage of the CML method


is, that the copula can be estimated without specifying the marginals. In this caseone would use the empirical distribution to transform the sample data into uniformvariates.

The IFM method is a commonly used approach and is used for the calculation of thisthesis described in what follows. Given a data set of observed returns at discrete pointsin time µij, i = 1, . . . , n, j = 1, . . . ,m the IFM is performed in the following three steps.

(i) Identification and estimation of the marginal parameters Φmarginalsi

In the first step one has to choose an appropriate marginal distribution. The pa-rameters Φmarginals

i of the chosen marginal distributions Fi with i = 1, . . . , n areestimated via the MLE.

Φmarginalsi = ArgMaxΦmarginals

i

m∑

j=1

ln fi(µij, Φmarginalsi ) (4.58)

where fi denote the density of the ith margin.

As a tool for identifying an appropriate marginal distribtuion Genest and Rivest[1993] propose QQ-plots of the parametric versus the empirical estimations of thecopulas’ distribution functions. The better the fit to the straight line from (0, 0) and(1, 1) the better the fit of the copula to the data. An example for a QQ-plot can befound in Figure 6.3.

(ii) Transformation of the observed data to the unit hypercube

The parameters Φi

marginalsestimated in step (i) are used to transform the data to the

unit hypercube by parametric estimates of their marginal CDF Fi. The sample dataµ1j, µ2j, . . . , µnjm

j=1 are transformed into uniform variates u1j, u2j, . . . , unjmj=1 by

setting uij = Fi(µij; Φimarginals

).

(iii) Definition of the appropriate copula function and estimation of the copula parame-ters Φcopula

By the transformed sample data u1j, u2j, . . . , unjmj=1 the estimation of the copula

parameters with a maximum likelihood algorithm is performed:

Φcopula = ArgMaxΦcopula

m∑

j=1

ln c(

(u1j), (u2j), . . . , unj); Φcopula, Φmarginals

)

(4.59)

where c denotes the copula density of the chosen copula (see Chapter 4.4).

The obtained parameters Φmarginals and Φcopula are the parameters which describe thestatistical distribution of the observed returns best in the sense of a maximum likelihoodapproach.


Joe [1997] proved that, like the MLE, the IFM estimator satisfies the property of asymp-totic normality under regular conditions:

√T (Φ − Φ) → N(0, G−1(Φ)) (4.60)

where G(Φ) denotes the Godambe information matrix. Joe [1997] points out that theIFM method is highly efficient compared with the MLE method.

4.6 Simulation Methods for Copulas

In order to perform a Monte Carlo simulation with the use of copulas one has to generaterandom draws, which are distributed like the chosen copulas. There are several methodsdescribed in literature to generate draws from copulas. The parameters of the randomvariables estimated in Section 4.5 are the basis to generate random scenarios from thecopula set up. The question is how a simulation of paths by the estimated parameters canbe done.

First elliptical copulas are described where the simulations are obtained easily even if theircopula is not in closed form. As for other copulas, like the Archimedean ones, I describea more flexible method, the conditional method. This method may be applied for everychosen copula, but is numerically more intense.

4.6.1 Elliptical Copulas

The random number generator of an elliptical copula is straightforward given a randomnumber generator of the corresponding elliptical distribution. Sklar’s theorem implies thatrandom numbers from a copula can be generated by transforming each margin of ran-dom numbers from a multivariate distribution with its probability integral transformation.

An easy algorithm for random variate generation from the n-dimensional Gaussian copulawith correlation matrix Σn is given by the following algorithm (see Embrechts et al. [2003])

• Find a Cholesky decomposition A of Σn

• Simulate n independent random variates ~z = (z1, z2, . . . , zn)T from N(0, 1)

• Set x = Az

• Set ui = Φ(xi) with i = 1, 2, . . . , n, where Φ denotes the univariate standard normaldistribution function

(u1, . . . , un) are the desired random variates.

An easy algorithm for random variate generation from the n-dimensional Student’s t-copula with correlation matrix Σn and ν degrees of freedom is given by (see Embrechtset al. [2003])


• Find a Cholesky decomposition A of Σn

• Simulate n i.i.d. ~z = (z1, z2, . . . , zn)T from N(0, 1)

• Simulate a random variate s from χ2ν independent of z.

• Set y = Az

• x =√

(νs)y

• Set ui = tν(xi) with i = 1, 2, . . . , n, where tν denotes the univariate Student’s tdistribution function

(u1, . . . , un) are the desired random variates.

Usually statistical programs offer functions to generate independent random variates froma N(0, 1) distribution. For the comparison of the result of this thesis it is helpful to startwith uniformly distributed variates ~w = (w1, w2 . . . , wn). By applying the inverse cumu-lative standard normal distribution function F−1 to the vector ~w a N(0, 1)-distributedvector of random variates can be generated. This vector can be used as the input vector ~zof the two previously described methods. By this method it is possible to generate variatesfrom a chosen copula by the same input uniform variates.

4.6.2 Archimedean Copulas

Melchiori [2006] gives some direct sampling algorithms for some Archimedean copulas, i.e.sampling algorithms based on numerical inversion of Laplace transforms are suggested. Forsome commonly used Archimedean copulas fast algorithm exit when the inverse generatorfunction φ−1 is known to be the Laplace transform of some positive random variable(see Marshall and Olkin [1998]). If the generator φ−1 equals the inverse of the Laplacetransform of a distribution function G on R+ satisfying G(0) = 0, the following algorithmcan be used for simulating random draws from the copula:

• Simulate a variate x with distribution function G such that the Laplace transformof G is the inverse of the generator.

• Simulate n independent variates z1, . . . , zn.

• Return (u1, u2, . . . , un) = (φ1(−log(z1/x), . . . , φ−1(−log(zn/x)).

This procedure can be applied to the Clayton and the Gumbel. However, it is not appli-cable to the Frank copula.

As a result of the flexibility often another method is used to sample from a chosen copula.The most frequently used approach when implementing Archimedean copulas is the con-ditional sampling one. This approach involves differentiation steps for each dimension. Forthis reason, it is not feasible in higher dimensions. Marshall and Olkin [1998] proposed an


alternative method, which is computationally more straightforward than the conditionaldistribution approach.

In order to compare the results of the use of different copulas and to exclude differencescaused by sampling differences it is helpful to use a method, which transforms uniformdistributed random variables to a chosen copula. This allows for the recycling of thesame sample of univariate variates for each simulation, which then are transformed intothe relevant copula distributions. Therefore, I use the conditional approach to simulatedraws from the Archimedean copula. For each simulation I use the same pairs of uniformlydistributed random variables as input to generate the random draws of a chosen copula C.

In the bivariate case the idea of the conditional sampling method is to use the conditionaldistribution. The task is to generate pairs (u1, u2) of observations of [0, 1] uniformly dis-tributed random variables U1 and U2 whose joint distribution function is C. To reachthis goal I use the conditional distribution Cu1

(u2) = F (U2 ≤ u2|U1 = u1). From def-inition copulas are joint distribution functions of standard uniform random variables.C(u1, u2) = F (U1 ≤ u1, U2 ≤ u2). From the following equation

Cu1(u2) = P (U2 ≤ u2|U1 = u1) (4.61)

= limδu→0

C(u1 + δu1, u2) − C(u1, u2)

δu1

(4.62)

=∂C

∂u1

(4.63)

= cu1(u2) (4.64)

where cu1(u2) := ∂C(u1,u2)

∂u1, one knows that cu1

(u2) is a non-decreasing function and existsfor almost all u2 ∈ [0, 1]. The following theorem provides a tool to generate random drawsfrom an Archimedean copula with generator φ. Then

Theorem 4.16 Let C(u1, u2) = φ−1(φ(u1) + φ(u2)) be an Archimedean bivariate copulawith generator φ:

Cu1(u2) =

φ−1(φ(u1) + φ(u2))

u1

(4.65)

This theorem can be formulated and proven for the n-dimensional case as well. The n-dimensional case can be found in Cherubini et al. [2004]. This result can be applied tothe most used Archimedean copulas and can be used to calculate cu1

(u2) without differ-entiating C(u1, u2), if the generator of the copula is known.

The previously derived result can be used to sample random variables from a bivariateArchimedean copula by a conditionals approach. It can be formulated in the followingway:

• Generate two independent uniform random variables (u1, v) ∈ [0, 1]; u1 is the firstdraw of the desired random vector.

• Compute the inverse function of cu1(v). Set u2 = c−1

u1(v) to obtain the second desired

draw.


Table 4.1 shows the inverses for the three in Chapter 4.4.2 introduced Archimedean copu-las. To make draws from the Gumbel copula using conditional sampling, the calculation of

Copula Conditional Copula

Clayton u2 =(

u−α1

(

v−α/(α+1) − 1)

+ 1)−1/α

Frank u2 = − 1α

ln

(

1 +v(1−e−α)

v(e−αu1−1)−e−αu1

)

Gumbel no closed solution available:

v = φ−1((− ln(u1))α+(− ln(u2))α)φ−1((− ln(u1))α)

with φ−1(w) = −e−w 1αw1−α

Table 4.1: Selected conditional transforms for copula generation

cu1(v) requires an iterative approach. This is computationally expensive for applications

with many simulated draws. Here numerical methods like Newton or the secant methodcan be applied. Marshall and Olkin [1998] suggest an alternative algorithm based onmixtures of powers.As it can be seen in case of the Gumbel copula the drawback of the conditional approachis that it might be not possible to calculate an inverse function analytically. To be con-sistent in generating the random samples and obtain results which are comparable thisdisadvantage is accepted in the following simulations.The by the previously methods obtained random variates u1, . . . , un can then be used togenerate the random retures by applying the inverse CDF F−1

i of the chosen marginaldistribution. This calculation can be done in different ways. There are different methodsdescribed in the literature to do this calculation neatly. For the Student’s t distributionfor example Shaw [2006] proposes iterative techniques to approximate the inverse CDFfor different degrees of freedom.

Chapter 5

Monte Carlo Simulations withCopulas

Performing a Monte Carlo simulation by using copulas is basically the same procedure asthe standard Monte Carlo procedure described in Chapter 3. The process described byEquation (3.1) is the usually used price process for assets, when pricing equity derivatives.The Brownian motion Wi implies that returns are normally distributed in the model. Toamend the standard Monte Carlo method to take copulas into account one has to changeassumed normal distributions and the dependence structure formulated in Equation (3.2).

As described in the previous chapters copulas allow a flexibility to model the marginals aswell as the dependence structures. To take this advantage into account I (beside the useof different copula) assume different marginal distributions than in the standard approachin Equation (3.1) as well. The Student’s t-distribution is often used to capture observedfat-tail features. To study the impact of the alternative choice of the marginal distributionon the simulations, the Student’s t-distribution is used as one alternative to the standardGaussian approach. To follow the risk-free pricing approach I use the t-distribution withscale parameters. This is a family of univariate probability distributions parameterizedby a location parameter µ and a scale parameter σ ≥ 0. If X is a Student’s t-distributedrandom variable with ν degrees of freedom then Y = µ + σX is the t location-scaledistribution. It has the following density

f(x; µ, σ, ν) =Γ(ν+1

2)

σ√

νπΓ(ν2)

ν +(

x−µσ

2)

ν

(

ν + 1

2

)

(5.1)

SinceE(Y ) = E(µ + σX) = E(µ) + σE(X) = µ (5.2)

the use of the t location-scale distribution enables to easily adjust the expected value tothe risk-free rate by setting the parameter µ appropriately and captures fat-tail featuresof a sample of data.In order to generalize Equation (3.5) in the discrete case the returns of the ith marginal

33

CHAPTER 5. MONTE CARLO SIMULATIONS WITH COPULAS 34

is modeled via a random variable Xdti :

Xdti =

Si(tj) − Si(tj−1)

Si(tj−1)(5.3)

where Xdti denotes a random variable with the chosen distribution of the returns. This

formulation allows for the flexible choice of the distribution of the returns. By generatingthe vector

(

Xdt1 , . . . , Xdt

n

)

by the chosen copula and marginal distributions the desiredmarginal distributions and dependencies between the returns can be modeled. In case ofGaussian marginals and Gaussian copula, Xdt

i is normally distributed with mean E(Xdti ) =

µdti , standard deviation

√

V ar(Xi)dt = σdti and the random vector (Xdt

1 , . . . , Xdtn ) has a

correlation matrix Σn. This is equivalent to the distribution of the returns in Equations(3.1) and (3.2). The Markoff property of the paths is furthermore maintained and Xdt

t (tj),j = 1, . . . ,m − 1, are independent. Therefore, the expectation of a path simulated withm steps of the length dt is

E(m−1∏

j=0

(1 + Xdti (tj))) = E((1 + Xdt

i ))m) (5.4)

= (E(1 + Xdti ))m (5.5)

= (1 + µdti )m (5.6)

To model the price paths of assets one needs to discretize the paths at certain pointsin time. The chosen modeled points in time t0, t1, . . . , tn have to be the points in timewhen the underlyings of the option are monitored. A weekly monitoring of a basket forexample requires a weekly simulation of the asset prices Si. There are several ways tosimulate the risk-neutral random walk discretely. For the simulations in this thesis Isimulate discrete returns for the given time steps dt by a random variable ~Xdt, where~Xdt = (Xdt

1 , Xdt2 , . . . , Xdt

n ) denotes an n-dimensional random vector, which is distributedlike the returns for the time period dt of the underlyings of the basket. Further let

~µl(tj) = (µl1(tj), . . . , µ

ln(tj)) (5.7)

denote the realization of the random variable ~Xdt at time tj for the lth simulation. To findthe distribution of Xdt

i the according observed returns are used to calibrate the distribu-tion, i.e. for a daily monitoring observed daily returns for a weekly monitoring observedweekly returns are used.

In order to be consistent with the pricing approach of the standard Monte Carlo approach,one has to ensure by setting E(Xdt

i ), i = 1, . . . , n appropriately that the expected valueequals the adjusted risk-free drift (see Section 2.4). In other words one has to amend toreal word measure, to a risk-free measure. This is done by the amendment of the expectedvalue of the simulated discretized returns (see Equation (5.6)) for the time period dt.

E(Xdti ) = (1 + r − di)

dt − 1 (5.8)

As for the specification of the distributions distributions, one could use implied informa-tion or a time series analysis based on historical data. Due to the usually big bid-ask-spread


for most of the traded basket options (see Section 2.4) the simultaneous calculation of theimplied volatilities and correlations is not possible. Therefore, historical returns are usedto calibrate the copula parameters.

Let µdti (t1), µ

dti (t2), . . . , µ

dti (tm) denote the observed returns and Sobs

i (tj) denote the ob-served prices of the financial underlying assets i of our basket at the times t1, . . . , tm:

µdti (tj) :=

Sobsi (tj) − Sobs

i (tj−1)

Sobsi (tj−1)

(5.9)

where (tj − tj−1) = dt. For the estimation of the parameters Φcopula and Φmarginals theIFM method (see Section 4.5) is used. The chosen observed market data are observedclosing quotes over the same period as the lifetime of the chosen option, i.e. for an optionwith a lifetime of one year (T=1) a history of one year of returns is used to estimatethe parameters of the distribution. Therefore, the same number of market observationsto calibrate the model are used as to be simulated.

The estimated parameters Φcopula and Φmarginals are used to generate m−1 returns for eachof the n assets at the discrete points in time tj, j = 1, . . . ,m−1 for each of the simulationl, l = 1, . . . , L. These returns are used to generate paths of assets Sl

i(tj), i = 1, . . . , n.Therefore, one obtains a three dimensional cube with the dimension m × n × L, whichstores all samples paths.

As described in Section 4.5 copulas allow to deal separately with marginal and jointdistribution modeling. Thus, one can choose for each data series the marginal distributionthat fits best the sample, and afterwards put everything together using a copula functionwith some desirable properties. Therefore, the number of combinations is large and one caneasily get lost in looking for the best combination of marginal and joint distributions. Inthis thesis I concentrate on the frequently used marginal distributions. Gaussian and theStudent’s t-distribution. The Student’s t- distribution is able to capture a high kurtosis.I use a non-central Student’s t- distribution (see Equation 5.1), so that one can allowa negative skewness 1. To have a greater flexibility the degrees of freedom are allowedto have non-integer values ν ∈ R

+. To model the dependencies I use elliptical copulas(Gaussian and Student’s t-copula) and Archimedean copulas (Gumbel, Frank, Clayton).

For each simulation L paths ~Sl1, . . . , ~Sl

n, l = 1, . . . , L are simulated. The calculation of theprice of the observed option is done by:

V MC(t) = er(T−t) 1

L

L∑

l=1

p(T, ~Sl1, . . . ,

~SLn ) (5.10)

For each simulation the following steps are performed:

(i) Estimation of the marginal distribution parameters Φmarginalsi by IFM method with

the observed returns µdti (tj) via MLE for each asset i.

1Details on the Student’s t-distribution are shown in the Appendix A


(ii) Transformation of the observed returns µdti (tj) with the estimated marginals param-

eters Φmarginalsi into uniformly distributed random variables.

(iii) Estimation of the copula parameters Φcopula via MLE.

(iv) Changing the real world measure to a risk-neutral measure by amending the ex-pected value of the marginals E(Xdt

i ), i = 1, . . . , n of the random variables Xdti .

(v) Draw of (m − 1) × L random vectors ~µl(tj) defined according to Equation (5.7) of

the random vector ~Xdt by transformation of uniformly distributed random variables(identical for each simulation).

(vi) Generate paths ~Sl1, ~Sl

2, . . . , Sln from the simulated returns ~µl(tj).

(vii) Calculation of payoff p(T, ~Sl1, . . . , ~Sl

n) at expiry T for each simulation.

(viii) Discounting the payoff p(T, ~Sl1, . . . , ~Sl

n) by the risk-free rate r.

(ix) Calculation of the mean of the payoffs of all simulations.

For this thesis the MATLAB optimizaton function fminsearch is used.

Chapter 6

Numerical Experiments

6.1 Examined Options

I calculated fair values of the options described in Chapter 2.2 for two different basketsin order to test the influence of the use of copulas on the prices. The two baskets containGerman DAX-companies from two different industries. For the chemical industry BayerAG (Bayer) and the BASF AG (BASF) were chosen, from the automotive industry theshares of Volkswagen AG (VW) and Bayerische Motorenwerke AG (BMW) were chosen.Of course, the underlyings of a basket don’t necessarily have to belong to the same in-dustry, but their dependence structure is more likely to show special features such as fattails for example.

To calibrate the model to observed market data the parameters of the marginals Φmarginals

and the chosen copula Φcopulas were estimated. The calibration of the model was done onthe basis of returns of an interval which corresponds to the monitoring interval of the op-tion: If the basket is monitored daily, historical daily returns are the basis for the fittingalgorithm, if the baskets are monitored weekly, historical weekly return are used. There-fore, the historical daily (weekly) returns µdaily

i (tj) respectively µweeklyi (tj), j = 1, . . . ,m

of each asset i were calculated as percentage differences per time period dt according toEquation 5.9 from the historical price data. The correlations from the historic returnswere calculated for the time window of past returns corresponding to the life time of theoption, i.e. 250 return data for the daily monitoring of the option and 52 return data forthe weekly monitoring. This is in line with market conventions. Statistically speaking,this convention implies that the market assumes stationarity of the model over the lifetime of the option.

The risk-free drift for the ith asset per time period dt was deduced from Equation (5.8).For the weekly returns one obtains the following equation with the assumption of 52 weeksper year:

rweeklyi = (1 + r − di)

1

52 − 1 (6.1)

where r denotes the risk-free rate per year. With the assumption of 250 business days one

37

CHAPTER 6. NUMERICAL EXPERIMENTS 38

Bayer (daily) BASF (daily) Bayer(weekly) BASF (weekly)

spot price Si(t0) 54.40 EUR 87.00 EUR 54.40 EUR 87.00 EURmean of µi 0.1057 % 0.0591 % 0.5130 % 0.2127 %standard deviation 1.823 1.549 3.732 3.4149skew -0.1374 -0.61231 -0.8052 -0.5552dividend yield p.a. 2.16 % 3.85 % 2.16 % 3.85 %Pearson linear correlation 0.7002 0.7374Kendall’s τ 0.5012 0.4449Spearmann’s ρ 0.6806 0.6038

Table 6.1: Statistics on historical returns of the Bayer/BASF basket

gets for the daily returns a risk-free drift for the ith asset of

rdailyi = (1 + r − di)

1

250 − 1 (6.2)

In order to examine the impact of the use of copulas when pricing a basket option I usethe basket options described in Chapter 2.2. In each case basket options with a maturityof one year (T=1) are chosen. Options with weekly and daily monitoring interval of theunderlying processes are studied. Table 6.3 shows the parameters of the chosen options.All options were priced as at-the-money options. Therefore, the strike K of the Aver-age Spread Option was set to the initial spread of the spot prices, K = |S2(t0) − S1(t0)|,the strike K of the Asian Option was set to the mean of the spot prices, K = |S2(t0)−S1(t0)|

2.

To capture different market movements two different pricing dates were considered. Thefirst date was the 19th February 2008 as pricing date for the Bayer/BASF-basket. Onthe 19th February 2008 the 1-year-EURIBOR-rate was r = 4, 39%. This rate is regardedas the risk-free rate. The studied options have a lifetime of one year (T=1), thereforetheir expiry date is on 18th February 2009. They can be exercised only at the expiry(European feature). The closing spot prices on the 18th February 2008 of the stocks were54.40 Euro (Bayer) and 87.00 Euro (BASF). Table 6.1 shows the statistical data of theobserved returns.

Table 6.1 shows some statistical data of the observed historical returns of the underlyingsin the Bayer/BASF basket. Both stocks showed a positive drift. Apparently the mean andthe standard deviation of the weekly returns have to be larger than corresponding data ofthe daily returns. All returns show a negative skew, i.e. the left tail is longer. The Gaussiandistribution is symmetric, therefore a Gaussian distribution does not capture this featureof the historical distribution. All dependence measures show a significant association ofthe two assets.

For the BMW/VW-basket the 11th October 2008 was chosen as pricing date. The 1-year-EURIBOR-rate was r = 5, 489% at the 11th October 2008. The studied options have alifetime of one year (T=1). Therefore, the options expire on 10th October 2009. They canbe exercised only at the expiry (European feature). Equal weights for the stocks in the


basket were chosen. The closing prices on the 10th October 2008 of the stocks were 19.63Euro (BMW) and 342.00 Euro (VW). Table 6.2 shows statistical data of the observedreturns.

Table 6.2 shows some statistical data of the observed historical returns of the underlyingsin the BMW/VW basket. The stock of BMW had a negative performance in the observedtime. Especially VW shows a significant positive skew. All dependence measures showa weaker dependence structure than the chemical basket. Interestingly the linear corre-lation coefficient shows different signs of linear correlation between the observed dailyreturns and the observed weekly returns. However, the two additional association mea-sures, Kendall’s τ and Spearman’s ρ show positive association between the returns.

BMW (daily) VW (daily) BMW (weekly) VW (weekly)

spot price Si(t0) 19.63 EUR 342.00 EUR 19.63 EUR 342.00 EURmean of µi -0.2757 % 0.2409 % -1.3844 % 1.3354 %standard deviation 2.378 2.8011 4.8307 8.036skew 0.5803 3.1597 -0.3885 2.7903dividend yield p.a. 2.16 % 3.85 % 5.32 % 0.86 %Pearson linear correlation 0.1162 -0.1286Kendall’s τ 0.2354 0.0298Spearmann’s ρ 0.3327 0.0508

Table 6.2: Statistics on historical returns of the BMW/VW basket

The histograms of the observed returns are illustrated in Figure 6.2. From the graphicsthe histograms resemble a Gaussian distribution, but especially the returns of VW showstrong fat-tail features, which is not captured by a standard approach by modeling Gaus-sian marginals.

In order to determine whether a data set comes from a certain distribution Q-Q plotscan be used. Figure 6.3 provides the Q-Q plots of the observed returns versus the normaldistribution. If the points of the plot, which are formed from the quantiles of the data, areroughly on a line with a slope of 1, then the data set comes from the distributions. TheGaussian distribution shows quite good results for the daily returns. However, especiallyat the tails of the distribution the fit is less accurate. The Q-Q-plot shows the fat tails atthe positive end of the distribution of the VW returs which could already be derived inFigure 6.2. The feature of tail dependence should for example be captures better by theStudent’s t-distribution with low degrees of freedom.

Graphik 6.1 shows the scatter plot of the daily/weekly returns of the chosen stocks ofthe Bayer/BASF and BMW/VW shares. A significant dependence structure between theassets can be observed from the concentration of the points in certain areas/shapes. Allplots show graphically the existence of a dependence structure between the returns of thestocks.


Option Type Maturity Strike in EUR Weight of assets MonitoringAverage Spread 1 year 32.60 (Bayer/BASF) equally weighted weekly/daily

322.37 (BMW/VW)Average Strike 1 year n/a equally weighted weekly/dailyLookback Put 1 year n/a equally weighted weekly/dailyLookback Call 1 year n/a equally weighted weekly/dailyDouble Average rate 1 year n/a 1 equally weighted weekly/dailyAsian 1 year 70.70 (Bayer/BASF) equally weighted weekly/daily

180.82 (BMW/VW)

Table 6.3: Parameters of examined options


−8 −6 −4 −2 0 2 4 6−8

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VW

(d) weekly returns (BMW/VW)

Figure 6.1: Scatterplot of the returns in the observed time series


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Figure 6.2: Histograms of the observed returns for the underlying assets


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Figure 6.3: Q-Q plots of the observed returns against the normal distribution


6.2 Results

6.2.1 Estimation of Parameters

Tables 6.4 - 6.7 show the results of the estimated parameters Φmarginal of the margins.For the Gaussian distribution the parameter ν is not relevant.

The parameter σi of the Gaussian distribution equals the standard deviation of the ob-served returns in Tables 6.1 and 6.2. This is not surprising because the estimated standarddeviation should equal the standard deviation of the observed data sample. Apparentlythis observation allows us to avoid a maximum likelihood estimation and could acceleratethe calibration procedure. However, due to comparability reasons the same methods ofcalibration was used for each margin.

The parameter µ equals the risk-free rate calculated by Equation 6.1 respectively 6.2. Bythe amendment of the drift the real-word measure is transformed to the risk-free measure,in accordance with the standard pricing procedure.

As can be seen in the table, most of the degrees of freedom are quite low. Keeping inmind that the Student’s t-distribution converges against the Gaussian distribution, onesees that the marginal behaviour is not close to normal. Only the weekly returns of theBASF and BMW displayed in Tables 6.5 and 6.7 show a rather high degree of freedom.

Φmarginal1 Φmarginal

2

Margin µ1 σ1 ν1 µ2 σ2 ν2

Gaussian 0,00882239 1,82365848 n/a 0,00215421 1,54939361 n/at 0,00882239 1,47403185 5,6586597 0,00215421 1,11025711 3,89036227

Table 6.4: Parameters Φmarginals of the underlyings Bayer(Φmarginal1 )/BASF

(Φmarginal2 )(daily monitoring)


2


Gaussian 0,04242244 3,73272305 n/a 0,01035721 3,41494546 n/at 0,04242244 3,04964362 5,90035118 0,01035721 3,19472055 18,1581225

Table 6.5: Parameters Φmarginals of the underlyings Bayer (Φmarginal1 )/BASF (Φmarginal

2 )(weekly monitoring)

Tables 6.8 - 6.11 show the results of the copula parameters Φcopula via IFM method. De-pending on the choice of the copula not all parameters are relevant. The parameter ρof the Gaussian/Gaussian combination equals the Pearson linear correlation coefficient ofthe baskets as reported in Table 6.2 and 6.1. It can be seen that the choice of the marginal



2


Gaussian 0,00324731 4,83071986 n/a 0,08705821 8,03600256 n/at 0,00324731 4,33917837 11,0284845 0,08705821 3,55800396 2,15705559

Table 6.6: Parameters Φmarginals of the underlyings BMW (Φmarginal1 )/VW (Φmarginal

2 )(weekly monitoring)


2


Gaussian 0,00067543 2,37877625 n/a 0,01810187 2,80111153 n/at 0,00067543 1,9068844 5,67968544 0,01810187 1,26899819 2,3207351

Table 6.7: Parameters Φmarginals of the underlyings BMW (Φmarginal1 ) /VW (Φmarginal

2 )(daily monitoring)

distribution has an impact on the estimated parameters of the copula. Especially the de-grees of freedom ν of Student’s t-copula show a large difference when using t-distributedmargins. This effect can be observed for all baskets and monitoring frequencies. An expla-nation is that the t-marginal is able to capture tail dependence better than the Gaussianmarginal. The fact that the t-distribution keeps the fat tail feature when transformingthe sampled data into the uniform random variables strengthens the impact of fat tailsin the dependence structure as well.

As the dependence measures in Table 6.3 already suggested, the parameters of the BMW/VWbasket (see Table 6.8) are close to the independent copula. The stronger dependence ofthe Bayer/BASF basket are reflected by the larger values of the the parameters of allstudied copula.

6.2.2 Pricing of the Options

In order to obtain the fair price of the option 25 000 Monte Carlo runs were performed.Tables 6.16 and 6.18 report the fair prices with the different simulation methods for thereturns. The figures show that the choice of the copula has an not neglectalbe effect onthe estimated prices of the studied options. In order to compare the results the Tables6.13, 6.19, 6.15 and 6.17 show the differences in percentage terms to the multivariatenormal assumption (Gaussian/Gaussian). It can be seen that using copulas to modela dependence structure can have a remarkable effect on the estimated fair prices (upto almost 18 % compared to the standard Gaussian/Gaussian approach). In particularit can be seen that the choice of the marginal distribution changes the results significantly.

The results for the weekly monitored BMW/VW basket differ for Gaussian marginals onlyby a maximum of 1.43 %. This should not surprise keeping in mind that the dependencemeasures showed almost independence of the returns and so the copulas do not capturefat tails. However, the use of Student’s t-marginals has a significant impact on the prices.


Margin Copula α ρ ν

gaussian gaussian n/a -0,13116115 n/agaussian t n/a -0,12698872 197,151426gaussian clayton -0,2593965 n/a n/agaussian frank 0,0001049 n/a n/agaussian gumbel 1,00000623 n/a n/a

t gaussian n/a -0,05676578 n/at t n/a -0,04584793 66,7800311t clayton -0,34314095 n/a n/at frank 0,30886649 n/a n/at gumbel 1,0000314 n/a n/a

Table 6.8: Parameters Φcopula of the basket with underlyings BMW/VW (weekly moni-toring)


Gaussian Gaussian n/a 0,12635575 n/aGaussian t n/a 0,41826006 5,99717817Gaussian Clayton 0,39660205 n/a n/aGaussian Frank 3,26702699 n/a n/aGaussian Gumbel 1,09313132 n/a n/a

t Gaussian n/a 0,29301827 n/at t n/a 0,36095964 3,14937499t Clayton 0,54538044 n/a n/at Frank 2,32916236 n/a n/at Gumbel 1,24193053 n/a n/a

Table 6.9: Parameters Φcopula of the basket with underlyings BMW/VW (daily monitor-ing)




Table 6.10: Parameters Φcopula of the basket with underlyings Bayer/BASF (weekly mon-itoring)





Table 6.11: Parameters Φcopula of the basket with underlyings Bayer/BASF (daily moni-toring)

Especially the prices for Averaging Options (Average Spread / Average Strike /DoubleAverage Rate / Asian), which do have an averaging feature in their payoff differ up to8.54 % compared to the Gaussian/Gaussian approach.

The dependence measures showed a significant dependence within the Bayer/BASF-basket. This is also reflected by the price differences when using copulas. The prices differsignificantly when using Archimedean or the Student’s t-copula instead of the Gaussian.

The obtained results suggest two major implications of the use of copulas when pricingbasket options. Firstly the model choice is relevant (copulas or multivariate normalityassumptions) as input choice in a multivariate framework. As a consequence, input choiceissues should be considered as crucial as model issues by traders or by risk managers.

Secondly, the assumption of marginal distributions plays a relevant role. The copula ap-proach with normal marginals produced fair values which were quite distant from usingcopulas with Student’s t-marginals, since the marginal return distribution is allowed tobe fat-tailed. Therefore, it should be questioned whether it is reasonable to use copulas inorder to model the dependence structure among assets while maintaining the much sim-pler assumptions on marginal return distributions. The simulation with normal marginalreturns seems from a theoretical point of view to miss some of the opportunities forgreater flexibility that the use of copulas gives. Using copulas is clearly different from asimple change in correlation parameters, since it implies a completely different mechanicsof price movements. Due to the flexibility when choosing the marginal distribution it en-ables to more properly model the marginal distributions. Even the use of non-parametricmarginals and of empirical distribution function can be considered. This avoids any as-sumptions about the distribution of the margins, but requires sufficient data since everynon-parametric method performs much better when data are not scarce.

For the calculation of the results no performance enhancing techniques like variance reduc-


tion were used. Those techniques are well studied for the standard Monte Carlo approachassuming normal distributed returns. Those techniques cannot be applied without amend-ment to a Monte Carlo simulation with copulas. One reason is the asymmetric distributionwhich copulas produce. There are a few papers which developed similar techniques forcommonly used copulas (see Kang and Shahabuddin [2005]). Methods for importancesampling for example in connection with normally distributed random variables do notcarry over to the case of other copula. Successful application of importance sampling inheavy-tailed settings is notoriously delicate. References for this are Asmussen and Bin-swanger [1997] or Binswanger [1997].

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Gaussian Gaussian 3,43 3,75 10,36 11,94 3,18 3,54Gaussian t 3,32 3,66 10,34 11,85 3,16 3,54Gaussian Clayton 3,63 3,96 11,32 12,94 3,48 3,83Gaussian Frank 3,13 3,48 11,64 13,40 3,59 3,94Gaussian Gumbel 3,30 3,65 11,48 13,19 3,54 3,89

t Gaussian 3,84 4,14 10,34 12,14 3,22 3,60t t 3,39 3,71 10,24 11,77 3,15 3,54t Clayton 3,68 4,02 11,43 13,08 3,55 3,90t Frank 3,78 4,14 11,36 13,02 3,52 3,87t Gumbel 3,48 3,83 11,49 13,18 3,57 3,93

Table 6.12: Prices of basket options for underlyings Bayer/BASF in Euro (daily monitor-ing)


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Gaussian Gaussian 0,00 % 0,00 % 0,00 % 0,00 % 0,00 % 0,00 %Gaussian t -3,24 % -2,45 % -0,23 % -0,74 % -0,46 % -0,18 %Gaussian Clayton 5,78 % 5,67 % 9,22 % 8,36 % 9,56 % 8,22 %Gaussian Frank -8,85 % -7,09 % 12,33 % 12,18 % 13,18 % 11,32 %Gaussian Gumbel -3,81 % -2,64 % 10,84 % 10,50 % 11,55 % 9,91 %

t Gaussian 11,94 % 10,30 % -0,15 % 1,63 % 1,39 % 1,49 %t t -1,19 % -0,95 % -1,13 % -1,46 % -0,67 % 0,01 %t Clayton 7,22 % 7,09 % 10,31 % 9,57 % 11,86 % 10,13 %t Frank 10,21 % 10,49 % 9,65 % 9,01 % 10,75 % 9,23 %t Gumbel 1,35 % 2,05 % 10,86 % 10,41 % 12,44 % 10,92 %

Table 6.13: Differences of the prices in % compared to the Gaussian/Gaussian approachfor underlyings Bayer/BASF in Euro (daily monitoring)

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Table 6.14: Prices of basket options for underlyings BMW/VW in Euro (daily monitoring)


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Gaussian Gaussian 0,00 % 0,00 % 0,00 % 0,00 % 0,00 % 0,00 %Gaussian t -6,81 % -9,49 % -4,30 % -6,33 % -6,22 % -4,34 %Gaussian Clayton 0,79 % -0,05 % 1,55 % 1,68 % 2,03 % 1,92 %Gaussian Frank 0,22 % -1,23 % 2,66 % 2,71 % 2,98 % 3,01 %Gaussian Gumbel 0,99 % 0,72 % 0,85 % 1,01 % 1,17 % 1,02 %

t Gaussian -10,18 % -12,42 % -11,79 % -15,40 % -10,04 % -9,01 %t t -2,62 % -4,39 % -6,09 % -10,16 % -2,07 % -1,54 %t Clayton 3,40 % 3,65 % 0,26 % -1,42 % 7,95 % 4,36 %t Frank 3,78 % 3,44 % -0,16 % -1,52 % 6,43 % 4,24 %t Gumbel 5,22 % 3,49 % 0,40 % -0,93 % 6,87 % 5,91 %

Table 6.15: Differences of the prices in % compared to the Gaussian/Gaussian approachfor underlyings BMW/VW in Euro (daily monitoring)

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Gaussian Gaussian 47,42 39,45 57,57 77,63 19,77 23,49Gaussian t 47,46 39,47 57,62 77,69 19,79 23,52Gaussian clayton 47,60 39,63 58,29 78,70 20,05 23,78Gaussian frank 47,60 39,63 58,29 78,70 20,05 23,78Gaussian gumbel 47,60 39,63 58,29 78,70 20,05 23,78

t Gaussian 51,25 42,34 58,04 74,11 21,30 24,74t t 51,01 42,11 57,88 74,38 21,00 24,77t clayton 51,48 42,54 58,45 74,65 21,47 24,90t frank 51,37 42,49 58,51 74,72 21,51 24,90t gumbel 51,48 42,54 58,45 74,65 21,47 24,90

Table 6.16: Prices of basket options for underlyings BMW/VW in Euro (weekly monitor-ing)


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Gaussian Gaussian 0,00 % 0,00 % 0,00 % 0,00 % 0,00 % 0,00 %Gaussian t 0,09 % 0,05 % 0,09 % 0,07 % 0,09 % 0,12 %Gaussian clayton 0,38 % 0,47 % 1,25 % 1,38 % 1,43 % 1,20 %Gaussian frank 0,38 % 0,47 % 1,25 % 1,38 % 1,43 % 1,20 %Gaussian gumbel 0,38 % 0,47 % 1,25 % 1,38 % 1,43 % 1,20 %

t Gaussian 8,07 % 7,34 % 0,82 % -4,54 % 7,76 % 5,29 %t t 7,56 % 6,76 % 0,54 % -4,19 % 6,25 % 5,44 %t clayton 8,56 % 7,84 % 1,53 % -3,85 % 8,63 % 6,00 %t frank 8,33 % 7,72 % 1,64 % -3,75 % 8,79 % 5,99 %t gumbel 8,56 % 7,84 % 1,53 % -3,85 % 8,63 % 6,01 %

Table 6.17: Differences of the prices in % compared to the Gaussian/Gaussian approachfor underlyings BMW/VW in Euro (weekly monitoring)

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Table 6.18: Prices of basket options for underlyings Bayer/BASF in Euro (daily monitor-ing)


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Gaussian Gaussian 0,00 % 0,00 % 0,00 % 0,00 % 0,00 % 0,00 %Gaussian t -0,08 % 0,00 % 0,02 % -0,03 % 0,00 % 0,06 %Gaussian Clayton 17,89 % 16,93 % 10,51 % 9,77 % 11,01 % 9,76 %Gaussian Frank 7,28 % 7,38 % 13,20 % 12,81 % 13,36 % 12,06 %Gaussian Gumbel 1,39 % 2,15 % 14,01 % 13,63 % 14,58 % 13,11 %

t Gaussian 12,83 % 11,31 % 4,10 % 4,82 % 6,10 % 4,86 %t t 6,55 % 6,13 % 2,54 % 2,48 % 3,63 % 3,16 %t Clayton 16,09 % 15,03 % 9,83 % 9,01 % 10,85 % 9,40 %t Frank 15,29 % 14,43 % 10,35 % 9,62 % 10,68 % 9,39 %t Gumbel 4,01 % 4,30 % 12,29 % 11,69 % 13,31 % 11,70 %

Table 6.19: Differences of the prices in % compared to the Gaussian/Gaussian approachfor underlyings Bayer/BASF in Euro (weekly monitoring)

Chapter 7

Conclusions

This thesis analyzed the impact of the use of copula functions when valuating basketoptions. The multivariate normality assumption for the underlyings’ returns was droppedand copula functions were applied to generate the Monte Carlo paths. The proposed ap-proach enables the separate modeling of margins and dependence structure.

The experiments suggests that the use of copulas can have a substantial effect on theprices Average Spread, Average Strike, Lookback Put, Lookback Call and Asian basketoptions. The flexibility which comes with the copula approach enables us to use anymarginal distribution for the underlying assets. Especially the use of alternative marginaldistributions (in this case Student’s t-distributed) shows a rather strong impact on theprices.

The advantage of the copula approach is the flexibility of the choice of the marginal distri-butions which helps to overcome the usual problems when assuming normally distributedreturns. By means of copula function one is able to capture fat-tail features of the under-lying and dependence structures. A poor estimate of the dependence structure may leadthe trader to misprice the option and to hedge it poorly. The standard Gaussian approachdoes not display fat-tails. Therefore, the use of copulas can greatly improve the modelingof dependencies in practice and therefore leads to better hedging strategies and leads toa consistent way of modeling dependence.

Furthermore the copula approach is very flexible and can be extended in numerous waysbecause it is possible to construct the right amount of tail dependence by using linearcombination of copulas (see Wei and Hu [2002]). Hence the question arises which copulais the ’right’ one to choose. The property of the Archimedean copulas is that each twovariables have the same degree of dependence. This leads to less flexibility when modelingmore than two assets. Therefore, Archimedean copulas are not well suited to model thedependence in more than two dimensions due to their restrictive characteristics, whereaselliptical t-copulas should provide a better fit. Other copula families could be consid-ered such as hierarchical Archimedean copulas, which are less restrictive than the simpleArchimedean copulas and might achieve a better fit which takes into account asymmetri-cal dependencies and lead to a more accurate valuation.

53

CHAPTER 7. CONCLUSIONS 54

Another promising direction is the consideration of dynamic dependence. By applyingmeasures of goodness of fit a measure for the choice of a copula itself and its parametersmay be introduced. This implies that depending on the input data a copula might changeover time. Allowing the copula and the copula parameters to be time-varying (see Goor-bergh et al. [2005] for such an approach in two dimensions) should further improve theperformance of the model.

Despite the mentioned advantages one should not ignore the disadvantages of a copulaapproach. The great flexibility of a copula-based model can also be seen as a critical point,because the flexibility requires some arbitrary choices which have to be made and whichhave a significant impact on the results. This makes the model itself to a factor which hasto be agreed upon between the involved groups (traders, risk managers). Another problemabout the concept is the practical implementation. The problem is that the estimation ofcopulas and their marginals requires the application of a maximum likelihood algorithm.Such a procedure is computationally intense. In addition to the effort to fit the distribu-tions via MLE an issue arises when doing Monte Carlo simulation without the assumptionof normally distributed returns. Without this assumption many of the standard variancereduction techniques cannot be used to enhance the performance of the Monte Carlo sim-ulation. This makes the already computationally intense simulation method even moredemanding, which makes the method not applicable for practical purposes.

In order to overcome the mentioned disadvantages more future research is necessary. Espe-cially more efficient techniques to do the calibration and the generation of random drawsof copulas have to be developed. This would certainly help to spread the use of copulamethod in practice.

Clearly, the observed results have to be taken as an academic example, and direct applica-tion of the model would call for specification of the risk-premia for example. However, thecomparison highlighted the power of copula functions to effectively separate informationabout the marginal distributions and the dependence structure among the assets and canbe a helpful tool to enhance the pricing and the hedging of basket options.

Appendix A

Student’s t-distribution

The density of the Student’s t-distribution with ν- degrees of freedom is given by

gν(x) =Γ(ν+1

2)

Γ(ν2)Γ(1

2)√

ν

(

1 + x2

ν

)− ν+1

2

(A.1)

The generalized Student’s t-distribution is given by

fν(x; µ, σ) =1

σgν(

x − µ

σ) =

Γ(ν+12

)

Γ(ν2)Γ(1

2)√

νσ

(

1 +(x − µ)2

νσ2

)− ν+1

2

(A.2)

By construction µ is the mean of the distribution. However, σ2 does not represent thevariance of fν . The link to the variance v of the distribution is given by the followingequation:

v =

√

ν

ν − 2σ (A.3)

The CDF Fν(x) of the generalized Student’s t-distribution with ν degrees of freedom isgiven by

Fν(x) =Γ(ν+1

2)

Γ(ν2)Γ(1

2)√

νσ(νσ2)

ν+1

2 I(x;−2µ, νσ2 + µ2, ν) (A.4)

where I is defined recursively

I(x; a, b, ν) =

∫ x

−∞((x

′

)2 + ax′

+ b)dx′

(A.5)

55

Appendix B

Maximum Likelihood Method

The Maximum likelihood estimation (MLE) is a statistical method used for fitting amathematical model to some data. The modeling of real world data using estimationby maximum likelihood enables to tune free parameters of a model to provide a goodfit. For a fixed set of data r1, . . . , rn and underlying probability density function f withparameters p1, . . . , pk, maximum likelihood sets the values parameters that make the data”more likely” than any other parameter values would make them. Therefore, one can writef in the form f(p1, . . . , pn; r1, . . . , rn). Let rj = (r1,j, . . . , rn,j), j = 1, . . . , N be a set ofn data points. The task is now to estimate the parameters p1, . . . , pk given the observeddata points. Therefore, the probability described by f is maximized. This is done via themaximum likelihood function L:

L(r1, . . . , rn; p1, . . . , pk) =∏

f(rj, p1, . . . , pk) (B.1)

The maximum likelihood methods assumes that the k-tupel (p1, . . . , pk) at which L has amaximum is the best estimator for the parameters (p1, . . . , pk).To ease the calculation of the function L usually the logarithm of L is calculated.

l(r1, . . . , rn; p1, . . . , pk) = ln(L(r1, . . . , rn; p1, . . . , pk)) (B.2)

=N∑

j=1

ln f(rj, p1, . . . , pk)) (B.3)

Since the logarithm is a strictly monotone increasing function the maximum doesn’tchange by this transformation. To find the maximum of l analytical methods can beapplied. Therefore, one has to find a vector p1, . . . , pk such that

∂l

∂pi

|(p1,...,pn) = 0 for i = 1, . . . , k (B.4)

In general the analytical form of f is too complicated to find a solution for Equation B.4.In this case numerical methods have to be applied to find the maximum.

56

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