Pricing of Path-Dependent Basket Options Using a Copula Approach

Pricing of path-dependent basket options using a copula approach

Christ Church

University of Oxford

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

Abstract

The pricing of basket options is usually a difficult task as assets of a basket usually show significant dependence structures which have to be incorporated appropriately in mathematical models. This becomes especially important if a derivative depends on the whole path of an option. In general pricing approaches linear correlation between the different assets are used to describe the dependence structure between them. This does not take into account that empirical multivariate distributions tend to show fat tails. One tool to construct multivariate distributions to impose a nonlinear dependence structure is the use of copula functions.

In the thesis the general framework of the use of copulas and pricing of basket options using Monte Carlo simulation is presented. On the base of the general framework an algorithm for the pricing of path-dependent basket options with copulas is developed and implemented. This algorithm conducts the calibration of the model to market data and performs a simulation and estimates the fair price of a basket option. In order to investigate the impact of the use of different copulas and marginals the algorithm is applied to a selection of basket options. It is analyzed how the proposed alternative approach affects the fair price of the option. In particular, a comparison to standard approaches assuming multivariate normal distributions is made. The results show that the use and the choice of copulas and especially the choice of alternative marginals can have a significant impact on the price of the options. Contents

1 Introduction 1

2 Basket Options 2 2.1 Deﬁnition ...... 2 2.2 Examples ...... 3 2.3 Valuing ...... 5 2.3.1 Black-Scholes ...... 6 2.3.2 FiniteDiﬀerences ...... 6 2.3.3 MonteCarlo...... 6 2.4 Hedging ...... 7

3 Standard Pricing using Monte Carlo 9

4 Copulas 12 4.1 DeﬁnitionsandBasicProperties ...... 13 4.2 Sklar’sTheorem...... 15 4.3 MeasuresofAssociation ...... 16 4.4 CopulaFamilies...... 18 4.4.1 EllipticalCopulas...... 18 4.4.2 ArchimedeanCopulas ...... 21 4.5 Estimation and Calibration from Market Data ...... 27 4.6 SimulationMethodsforCopulas...... 29 4.6.1 EllipticalCopulas...... 29 4.6.2 ArchimedeanCopulas ...... 30

5 Monte Carlo Simulations with Copulas 33

6 Numerical Experiments 37 6.1 ExaminedOptions ...... 37 6.2 Results...... 44 6.2.1 EstimationofParameters ...... 44 6.2.2 PricingoftheOptions ...... 45

7 Conclusions 53

A Student’s t-distribution 55

ii CONTENTS iii

B Maximum Likelihood Method 56 List of Figures

4.1 Gaussiancopula...... 19 4.2 RandomdrawsfromaGaussiancopula ...... 19 4.3 Student’st-copula...... 20 4.4 RandomdrawsfromaStudent’st-copula ...... 21 4.5 Gumbelcopula ...... 23 4.6 RandomdrawsfromaGumbelcopula...... 24 4.7 Claytoncopula ...... 24 4.8 Random draws from a Clayton copula ...... 25 4.9 Frankcopula ...... 26 4.10 RandomdrawsfromaFrankcopula...... 26

6.1 Scatterplot of the returns in the observed time series ...... 41 6.2 Histograms of the observed returns for the underlying assets ...... 42 6.3 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 ...... 38 6.2 Statistics on historical returns of the BMW/VW basket ...... 39 6.3 Parametersofexaminedoptions ...... 40 ˆ marginals ˆ marginal ˆ marginal 6.4 Parameters Φ of the underlyings Bayer(Φ1 )/BASF (Φ2 )(daily monitoring) ...... 44 ˆ marginals ˆ marginal ˆ marginal 6.5 Parameters Φ of the underlyings Bayer (Φ1 )/BASF (Φ2 ) (weeklymonitoring)...... 44 ˆ marginals ˆ marginal ˆ marginal 6.6 Parameters Φ of the underlyings BMW (Φ1 )/VW (Φ2 ) (weeklymonitoring)...... 45 ˆ marginals ˆ marginal ˆ marginal 6.7 Parameters Φ of the underlyings BMW (Φ1 ) /VW (Φ2 )(dailymonitoring)...... 45 6.8 Parameters Φcopula of the basket with underlyings BMW/VW (weekly monitoring)...... 46 6.9 Parameters Φcopula of the basket with underlyings BMW/VW (daily monitoring)...... 46 6.10 Parameters Φcopula of the basket with underlyings Bayer/BASF (weekly monitoring) ...... 46 6.11 Parameters Φcopula of the basket with underlyings Bayer/BASF (daily monitoring)...... 47 6.12 Prices of basket options for underlyings Bayer/BASF in Euro (daily monitoring)...... 48 6.13 Differences of the prices in % compared to the Gaussian/Gaussian approach for underlyings Bayer/BASF in Euro (daily monitoring) ...... 49 6.14 Prices of basket options for underlyings BMW/VW in Euro (daily monitoring)...... 49 6.15 Differences of the prices in % compared to the Gaussian/Gaussian approach for underlyings BMW/VW in Euro (daily monitoring) ...... 50 6.16 Prices of basket options for underlyings BMW/VW in Euro (weekly monitoring)...... 50 6.17 Differences of the prices in % compared to the Gaussian/Gaussian approach for underlyings BMW/VW in Euro (weekly monitoring) ...... 51 6.18 Prices of basket options for underlyings Bayer/BASF in Euro (daily monitoring)...... 51

v LIST OF TABLES vi

6.19 Diﬀerences of the prices in % compared to the Gaussian/Gaussian approach for underlyings Bayer/BASF in Euro (weekly monitoring) ...... 52 Chapter 1

Introduction

Many models in finance assume that asset returns are normally distributed. Usually, linear correlation is the chosen measure of dependence between risky assets. The problem with linear correlation is that equities’ prices exhibit a greater tendency to crash together than to boom simultaneously. A number of empirical papers have shown that Gaussian distributions 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 calculate fair 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 correlation describes dependence. Copulas allow the construction of joint distributions which specify the distributions of individual returns separately from each other and separate from the dependence structure. This increases the ﬂexibility in specifying distributions of multiple random variables.

The aim of this thesis is to show how copulas can be incorporated in Monte Carlo simulations 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 methods for dealing with theses kind of options. The standard pricing approach with Monte Carlo simulations is described in more depth in the third Chapter. The fourth Chapter is a general introduction to the theory behind copulas. The term copula is defined and basic properties of copulas are described. Some examples of common copulas are given and mathematical methods used for copulas are outlined. On the basis of the general framework a modification of the standard Monte Carlo simulation which enables to use copulas to 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 the impact on the fair price of the option when pricing path-dependent basket options on the price 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 literature numerous definitions and classifications of basket options can be found. They often overlap other options such as Mountain Range options and Rainbow options because of their multi-asset characteristic. This thesis builds on the following general definition: Definition 2.1 (Basket Option) A Basket Option is an option whose payoff depends on the value of a portfolio (or basket) of assets. In general, the corresponding assets are related. The payoff p of a path-dependent basket options depends on the underlying assets at specified points in time tj. Therefore, the payoff depends on time t and on the values of the underlying assets Si, i = 1,...,n at the monitored points in time tj, j = 1,...,m. th Let S~i = (Si(t0),Si(t1) ...,Si(tm)) denote the set of prices of the i underlying at the monitored points in time, then the payoff can be written by p(t, S~1, S~2,..., S~n). 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: t , j = 1,...,m • j Time of expiry: T = t • m Time of valuation: t • 0 Weights of the asset i in the basket: α • i Prices of the ith underlying at time t: S (t) • i th Set of all prices of the i underlying at the monitored points in time tj: S~i = • S (t ),S (t ),...,S (t ) . { i 1 i 2 i m } 2 CHAPTER 2. BASKET OPTIONS 3

Undiscounted payoﬀ of an option at time t and realized paths of the underlyings • (S~1, S~2,..., S~n): p(t, S~1, S~2,..., S~n) Volatility of the ith underlying at time t: σ (t) • i Risk-free rate in the market: r • Dividend yield of the ith underlying: d • i Expected return of the ith asset: µ • i 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 consisting of several asset. The advantage of buying basket options is that they are usually cheaper than options on the individual components. Thus, a basket option is considered as a cheaper alternative to hedge a risky position consisting of several assets. In addition, a basket option is able to replicate the changes in a portfolio’s value more precisely than any combination of options on the underlying assets. The typical underlyings of a basket option are several stocks, indices or currencies. Less frequently, interest rates are also possible.

2.2 Examples

There is a huge variety of exotic multi-asset options traded on the markets. Multi-asset options features vary from simple basket options whose payoﬀ is linked to the overall performance of the basket of stocks to cases where the investor receives a ﬁx coupon provided that none of a basket’s stocks trespasses a certain barrier. For this thesis’ investigation the focus is on path-dependent basket options. In the following, several types of liquidly traded 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 the 1 T n average price of the underlying assets T 0 ( i=1 αiSi(t))dt, where αi is the weight in the basket of the ith asset. In practice the integrals are approximated by sums by R P sampling at discrete points in time tj [t0, T = tm]. The payoff at expiry for the discrete case is therefore: ∈ + 1 m n p(T, S~ ,..., S~ )= ( α S (t )) K (2.1) 1 n m i i j − j=1 i=1 ! X X 1source: http://www.fincad.com/support/developerFunc/mathref/Basket.htm CHAPTER 2. BASKET OPTIONS 4

n where the αi satisfy i=1 αi = 1. European options with this payoﬀ functions are called arithmetic weighted average options or simply arithmetic Asian Options. P

Average Spread Option • The payoff at expiry T of a Basket Average Spread Option is the difference between 1 m the average spread m j=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 at expiry T is therefore: P

+ 1 m p(T, S~ , S~ )= ( (S (t ) S (t ))) K (2.2) 1 2 m 1 j − 2 j − j=1 ! X . Average Strike Option • An Average Strike Option depends on the difference of the spread at expiry S1(T ) 1 m − S2(T ) and the average spread K = m j=1(S1(tj) S2(tj)) during the life of the option. The payoff at expiry is therefore − P + 1 m p(T, S~ , S~ )= (S (T ) S (T )) ( S (t ) S (t )) (2.3) 1 2 1 − 2 − m 1 j − 2 j j=1 ! X . Lookback Options • A Lookback Option is a derivative product whose payoff depend on the maximum n n U = max ( αiSi(tj)) or the minimum D = min ( αiSi(tj)) of the real- j=1,...,m i=1 j=1,...,m i=1 ized basket priceP over the lifetime of the option. For exampleP , a Lookback Put has a payoff at expiry that is the difference between the maximum realized price and the spot price of the basket at expiry T . Therefore, the payoff is:

Lookback Put Option:

n + p(T, S~ ,..., S~ )= U α S (T ) (2.4) 1 n − i i i=1 ! X Lookback Call Option:

n + p(T, S~ ,..., S~ )= α S (T ) D (2.5) 1 n i i − i=1 ! X Double Average Rate Option • A Double Average Rate Option’s payoff is the difference between the arithmetic average of the underlying spot prices of the sample points in the first sampling CHAPTER 2. BASKET OPTIONS 5

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:

′ + j 1 n 1 m n p(T, S~ ,..., S~ )= ( α S (t )) ( α S (t )) (2.6) 1 n j i i j − m j i i j ′ j=1 i=1 ′ ′ i=1 ! X X − j=Xj +1 X The above described options are only some out of for a high quantity of examples of traded path-dependent basket options. The above made selection should cover most of the liquidly traded basket options on the market. Nevertheless even the liquidly traded basket options show rather big spreads which can be partly ascribed to the correlation risk, which is hard to hedge. That is why the bid-ask spread of the prices of the options on the market is very high. These prices indicate just a range within which the exact price can 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 assumption of a complete market with no arbitrage opportunities the formula of basket option’s price is a conditional expected value:

r(T t) r(T t) V (t)= e − E [p(T,s(ω)) F ]= e − p(T,s(ω))dp(ω) (2.7) | t ωZ Ω ∈ where p(T,s(ω)) denotes the payoﬀ of an option at expiry T with the realized paths of the underlyings s(ω), where p(ω), ω Ω denotes the risk-neutral realization probability and 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 disadvantages are sketched. CHAPTER 2. BASKET OPTIONS 6

2.3.1 Black-Scholes With Itô’s lemma a Black-Scholes formula for a specific class of basket options can be derived. For European basket options whose payoff depends only on the values of the underlyings at expiry T , the derivation of the Black-Scholes formula is basically analogous to the one dimensional case. The partial differential equation (PDE) of the value of these types of basket option is:

∂V 1 n ∂2V n ∂V + ρ σ σ S S + (r d ) S rV = 0 (2.8) ∂t 2 ij i j i j ∂S ∂S − i i ∂S − i,j=1 i j i=1 i X X Literature suggests various pricing methods to solve the differential Equation (2.8). For specific payout structures, analytical formulas are known (see Margrabe [1987] and John- son [1987]). By using analytical formulas no computational intense numerical methods have to be applied. The disadvantage is that the Black-Scholes formula has an analytic solution for only a limited set of basket option and the calculation of the joint cumulative normal function for more than two variates is numerically very intense and no efficient methods exist to compute this function.

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

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

2.3.3 Monte Carlo An alternative for high dimensional problems are Monte Carlo simulation methods. Monte Carlo methods are easy to use and cope with any type of basket options. The problem about Monte Carlo simulations is the computational costs they already demand for low dimensions. Monte Carlo simulations in general provide unbiased estimates with a convergence rate not dependent on the dimension of the problem. In contrast to the finite CHAPTER 2. BASKET OPTIONS 7 difference technique, the Monte Carlo method returns the estimate for a single point in time. It is a flexible approach but usually requires refinements, such as variance reduction techniques, 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 hedge the position. Therefore, the correct hedging strategy is essential to secure the profit. To delta hedge a basket option sensitivities have to be calculated to find the hedging coefficients. A poor estimation of correlation may lead to a poor hedge of the derivative, since the hedging coefficients would be wrong as well. Therefore, the dependencies between assets have an effect on the correct hedging strategy, and models should be able to capture these dependencies.

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

Apart from the sophisticated payout structures, the inherent challenge of pricing and hedging multi-asset equity options is the illiquidity of implied correlations due to the lack of standardized multi-asset contracts. Correlation risk stems particularly from two sources: 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 foreign exchange markets (see Kholodnyi and Price [1998]): Here, hedging correlation risk is possible, since volatilities and correlations of currency pairs are linked together via the exchange 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 cash and not in pairs like currencies. CHAPTER 2. BASKET OPTIONS 8

This thesis contains a theoretical study of the impact of the use of copulas on the fair price of path-dependent basket options. Despite the practical issues which arise when hedging 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 Carlo simulations. Chapter 3

Standard Pricing using Monte Carlo

In the following the standard approach of pricing basket options via Monte Carlo simulation is sketched. As this approach is the commonly used pricing technique it is a starting point comparing the pricing with copulas. Details about Monte Carlo Techniques can be found 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 the underlying 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 the path generating process of the assets is known. As a process for the underlying stocks the geometric 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 usually used stochastic process and implies that the returns of the assets in the basket are normally distributed with correlation ρij. For a risk-free pricing approach the drift µi of th th the i asset is set to µi = r di, where di denotes the dividend yield of the i asset and r is the risk-free rate. −

Under the assumption of constant volatility σi(t) = σi the solution is still a geometric Brownian motion of the following form: σ2 S (t)= S (0) exp µ i t + σ W (t) (3.3) i i i − 2 i i Pricing path-dependent options requires to monitor the processes (3.3) at a ﬁnite set of points in time t1,...,tm for each asset Si. This sampling procedure yields the following expressions for{ constant volatility:}

2 σi Si(tj)= Si(tj 1) exp µi (tj tj 1)+ Wi(tj tj 1) (3.4) − − 2 − − − − 9 CHAPTER 3. STANDARD PRICING USING MONTE CARLO 10

T where the vectors W~ (tj tj 1) = (W1(tj tj 1),W2(tj tj 1),...,Wn(tj tj 1)) , − − − − − − − − j = 1,...,m, are n-dimensional normal random variables with zero mean vector and covariance matrix Σn to model the increments.

Sometimes Equation (3.1) is approximated in the discretized case by the Euler forward method 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 distributedp random variable. Then the process 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 possible paths in the following way (see Dahl and Benth [2001]): S V (t) = exp ( r (T t)) p(T, S~)f (S~)dz (3.6) − − S ZS

where S~ = (S~1,..., S~n) denotes a set of n realized paths of the underlying assets and f is the probability∈ density S on . This integral can be evaluated approximately by S Monte Carlo simulation: S

1 L V MC (t) = exp ( r(T t)) p(T, S~l ,..., S~l ) (3.7) − − L 1 n Xl=1 ~l where L is the number of conducted simulations and the vectors Si, 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 diﬀerence V MC (t) V (t) converges in distribution to a normal with mean 0 and standard deviation σ . The− convergence rate is O( 1 ) for all dimensions n. √L √L

The expected error of a valuation via Monte Carlo method can be estimated by using the sampled standard deviation or root mean square error (RMSE). For a random variable X it is deﬁnied as follows σ RMSE = E((E(X) X)2)=( ) (3.8) − √L p I our case the RMSE is estimated by:

L 1 2 pdisc(t, S~l ,..., S~l ) V MC (t) (3.9) vL 1 1 n − u k=1 u − X t 1under the condition that the random varialbes are identically distributed with ﬁnite mean and variance CHAPTER 3. STANDARD PRICING USING MONTE CARLO 11

disc ~l ~l r(T t) ~l ~l th where p (t, S1,..., Sn)= e − p(T, S1,..., Sn) denotes the discounted payoff of the l ~l ~l simulation with realized path (S1,..., Sn) at time t. Refinements in Monte Carlo methods consist in finding techniques whose aim is to reduce the RMSE. From Equation (3.8) it can be seen that two ways to reduce the RMSE are to reduce σ, known as variance reduction techniques like antithetic variates, control variates, importance sampling or stratified sampling, or to increase the number of simulations L.

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

Simulate L risk-neutral sample paths S~ ,..., S~ for the underlying assets. • 1 n Calculate the payoﬀ p(T, S~ ,..., S~ ) at expiry T . • 1 n Approximate the expected value of the option value at the expiry T by calculation • the mean over the simulations.

Use the risk-free discount rate r and calculate the present value of the option value • V MC . Chapter 4

Copulas

In this Chapter the basic concepts of copulas are presented. I restrict myself to the two dimensional 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 Gaussian distribution. Its bivariate density with correlation coeﬃcient ρ is given by

1 1 2 2 f(ζ1, ζ2)) = exp ( (ζ1 + ζ2 2ρζ1ζ2)). (4.1) 2π 1 ρ2 −2(1 ρ2) − − − The cumulative distribution functionp (CDF) of the marginal distribution FX (x) can be derived from the bivariate CDF by basic calculus: x ∞ F (x, ) = f(ζ , ζ )dζ dζ (4.2) ∞ 1 2 1 2 −∞ −∞ Z x Z 2 1 ζ1 = exp ( )dζ1 (4.3) √2π − 2 Z−∞ = FX (x) (4.4) By deriving Equation 4.3 the marginal densities can be obtained: 2 ′ 1 x F (x)= fX (x)= exp ( ) (4.5) X √2π − 2 The idea behind copulas is to go the other direction. Here the joint distribution shall be derived from the marginals. If one looks at a bivariate CDF F (x, y)= P (X

= 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 ), whose components 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 the bivariate CDF F (x, y) and contains explicitly the marginal CDFs. If one deﬁnes a function

1 1 C(u, v)= F (FX− (u),FY− (v)) (4.11) then this function can be used to couple the marginal distributions to obtain the joint distribution:

−1 −1 FX (FX (x)) FY (FY (y)) C(FX (x),FY (y)) = f(ζ1, ζ2)dζ1dζ2 (4.12)

Z−∞x y Z−∞ = f(ζ1, ζ2)dζ1dζ2 (4.13) Z−∞ Z−∞ = F (x, y) (4.14)

This is basically the idea to deﬁne copula functions, because by the means of copulas the marginal CDFs can be decoupled from the dependence structure. If one looks at the result of a bivariate standard normal CDF for two values x = 0.3 and y = 0.6 with ρ = 0.8 one gets F (0.3, 0.6) = 0.57 (4.15) The same result can be obtained by performing two steps. First, one evaluates the marginal CDFs

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. CHAPTER 4. COPULAS 14

(iii) For every u , u ,v ,v [0, 1] with u u and v v : 1 2 1 2 ∈ 1 ≤ 2 1 ≤ 2 C(u ,v ) C(u ,v ) C(u ,v )+ C(u ,v ) 0. 2 2 − 2 1 − 1 2 1 1 ≥

In the following some properties of copulas are presented. The next theorem establishes the continuity of copulas via a Lipschitz condition on [0, 1] [0, 1]. × Theorem 4.2 (Continuity) Let C be a copula. Then for every u , u ,v ,v [0, 1]: 1 2 1 2 ∈ C (u ,v ) C (u ,v ) u u + v v . | 2 2 − 1 1 | ≤ | 2 − 1 | | 2 − 1 |

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 with respect to its variables:

1 ∂C Theorem 4.3 Let C be a copula. For almost every u [0, 1], the partial derivative ∂v exists for almost all v [0, 1]. For such u and v one has∈ ∈ ∂ C (u, v) 0 (4.19) ∂v ≥ ∂C The analogous statement is true for the partial derivative ∂u .

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

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

Deﬁnition 4.4 Copula Density ∂2C(u, v) c (u, v)= (4.20) ∂u ∂v

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

f(u, v)= c(F (u),F (v)) f (u) f (v) (4.21) u v · u · v where fu(u) and fv(v) are the densities of the margins ∂F (u) f (u)= u . (4.22) u ∂u 1The expression ”‘almost all” is used in the sense of the Lebesgue measure. CHAPTER 4. COPULAS 15

and fv(v) deﬁned analogously. The copula density is therefore equal to the ratio of the joint density f and the product of all marginal densities fj. f(u, v) c(F (u),F (v)) = (4.23) u v f (u) f (v) u · v From this expression it is clear that the copula density takes a value equal to 1 everywhere where the original random variables are independent.The canonical representation is very useful in statistical estimation, in order to have a ﬂexible representation for joint densities and 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], then following expression holds: Corollary 4.5 If U U [0, 1] and F is a CDF, then ∼ 1 P F − (U) x = F (x) (4.24) ≤ Like wise, if the real-valued random variable Y has a distribution function F and F is continuous, 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 distributions and entangle them in the right way, she will end up with a proper multivariate distribution 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 x , x R. If F and F are continuous, then C is unique. Otherwise, C is 1 2 ∈ 1 2 uniquely determined on Ran(F1) Ran(F2), where Ran(Fi) denotes the range of the CDF F , i 1; 2 . On the other hand,× if C is a copula and F and F are distribution functions, i ∈ { } 1 2 then the function F deﬁned by (4.26) is a joint distribution function with margins F1 and F2. The name Copula was chosen by Sklar to describe a function that links a multidimensional distribution to its one-dimensional margins and appeared in mathematical literature for the ﬁrst time in Sklar [1959]. Usually, the term Copula is used in grammar to describe an expression that links a subject and a predicate. The origin of the word is Latin.

1 If one looks at the expression F F − (x)= x, the following equation is obtained: i ◦ i 1 1 C (u1, u2)= F F1− (u1) ,F2− (u2) (4.27) Equation (4.27) provides a theoretical tool for the derivation of the copula from a multivariate, in our case two-dimensional, distribution function. This equation also allows the extraction of a copula directly from a multivariate distribution function. CHAPTER 4. COPULAS 16

4.3 Measures of Association

Copulas may be used to couple random variables with distinctive relationships. Two random 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. Three of 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 tau Kendall’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 rank correlation coefficient is a non-parametric statistic used to measure the degree of correspondence between two rankings and to the significance of this correspondence. It is defined as follows: Definition 4.7 (Kendall’s τ) Let X and Y be random variables with Copula C and 1 1 let FX− (u) and FX− (u) denote the quantile functions and u and v the quantiles. Then Kendall’s tau (τ) with copula C is defined as

τ = 4 C(u, v)dC(u, v) 1 (4.28) 2 − ZZI It can be demonstrated that it measures the diﬀerence between the probability of concordance 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 are said 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 an n-dimensional sample is: n n τ = c − d (4.29) 1 n(n 1) 2 − where nc is the number of concordant pairs and nd is the number of discordant pairs in the sample.

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

1 1 Deﬁnition 4.8 (Spearman’s ρS) Let FX− (u) and FX− (u) denote the quantile functions and 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 C(u, v) uv du dv (4.30) 2 − ZZI CHAPTER 4. COPULAS 17

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

n( xiyi) ( xi)( yi)) ρS = − (4.31) n( x2) ( x )2 n( y2) ( y )2 i P− i P Pi − i In comparison to Kendall’sp tauP Spearman’sP rhop exploitsP probaPbilities of concordance and discordance. Between Kendall’s tau and Spearman’s rho the following relationship holds (see Durbin and Stuart [1951]):

3 τ 1 ρ 1 + τ 1 τ 2 τ 0 2 − 2 ≤ S ≤ 2 − 2 ≥ (4.32) 1 + τ + 1 τ 2 ρ 3 τ + 1 τ < 0 (− 2 2 ≤ S ≤ 2 2 Pearson’s linear correlation coefficient The Pearson’s linear correlation coefficient has been introduced for random variables belonging to L2. It is probably the best known coefficient to measure association. Pearson’s correlation coefficient is a parametric statistic and when distributions are not normal it may be less useful than non-parametric correlation methods.

1 1 Definition 4.9 (Pearson’s linear correlation) Let FX− (u) and FX− (u) denote the quantile functions and u and v the quantiles of two random variables X and Y . The Pearson linear correlation coefficient ρXY th is defined by cov(X,Y ) ρXY = (4.33) var(X)var(Y ) or equivalently in terms of the copula Cp:

12 I2 (C(F1(u),F2(v)) F1(u)F2(v))du dv ρXY = − (4.34) R var(X)var(Y ) The formulation of Pearson’s linear correlationp coeﬃcient contains always the variances of the marginal distributions. Therefore, the measure is not independent of the choice of the marginals unlike the previously introduced measures.

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

n xiyi xi yi ρXY = − (4.35) n x2 ( x )2 n y2 ( y )2 i −P i P Pi − i For elliptical copulas the relationshipp P betweenP thep Pearson’P sP linear correlation coeﬃcient ρ and Kendall’s τ is given by: π ρ(u, v) = sin τ (4.36) 2 and 6 ρ = arcsin(ρ/2) (4.37) S π CHAPTER 4. COPULAS 18

4.4 Copula Families

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

4.4.1 Elliptical Copulas The Gaussian and Student-t copula are frequently used copulas. They belong to the family of elliptical copulas. Elliptical copulas are simply the copulas of elliptically contoured distributions. An advantage of elliptical copula is that one can specify different levels of correlation between the marginals. They are characterized by a range of parameters and can be fitted flexibly to data. A disadvantage is that elliptical copulas do not have closed form expressions and are restricted to have radial symmetry.

The bivariate Gaussian copula • One important elliptical copula is the Gaussian or normal copula which was described by Embrechts et al. [1999]. In the two-dimensional case it is deﬁned as follows.

Deﬁnition 4.10 (Bivariate Gaussian copula (BGC)) Let ρ [ 1, 1] and Φρ the standardized bivivariate normal distribution with Pearsons’s linear∈ − correlation ρ. The BGC is deﬁned as

Gauss 1 1 Cρ (u1, u2) = Φρ Φ− (u1) , Φ− (u2) (4.38) 1 where Φ− denotes the inverse of the standard univariate normal distribution function Φ. If the correlation ρ = 0, the Gaussian copula becomes the independent copula, 2 Gauss Independence C (u1, u2)= ui = C (u1, u2) (4.39) i=1 Y As u [0, 1], one can replace u in 4.38 by Φ (r ). If one considers r in a probabilistic i ∈ i i i i sense, i.e. ri being values of random variables Ri one obtains from Equation (4.38)

CGauss(Φ (r ) , Φ (r )) = P (R r ,R r ) 1 1 2 2 1 ≤ 1 2 ≤ 2 Gauss In other words: C (Φ1 (r1) , Φ2 (r2)) is the two-dimensional CDF.

From the deﬁnition of the Gaussian copula one can easily determine the corresponding density cGauss (ρ2 = 1): R 6 1 1 1 2 1 2 Gauss 1 2ρΦ− (u1) Φ− (u2) Φ− (u1) Φ− (u2) cρ (u1, u2)= exp − 2 − 2π 1 ρ2 2 (1 ρ ) ! − − (4.40) p CHAPTER 4. COPULAS 19

1 0.25 0.9 0.9 0.2

0.8 0.8 0.2 0.7 0.7 0.6 0.15 0.15

0.5 0.6

0.4 0.1 0.5 0.3 0.1 0.2 0.4 0.05

0.1 0.3 0 0 1 0.05 0.2 1

0.5 0.1 0.5 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0 u2 0 0.2 0.2 0 u2 0 u1 u1 (a) Gaussian copula (ρ = 0.7) (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 1000 samples of two uniformly distributed random variables coupled with a Gaussian copula are shown. The graph gives an impression on how the choice of the parameter ρ inﬂuences the dependencies between the variables.

One reason for the importance of the Gaussian copula is that it generates the joint normal standard distributions function, if and only if the margins are standard normal. A proof can be found in Cherubini et al. [2004]. The Gaussian copula are usually used to model linear correlation dependencies.

1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

u2 0.5 u2 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 u1 u1 (a) Gaussian copula (ρ = 0.7) (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 t : R2 R be ρ,ν → CHAPTER 4. COPULAS 20

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

ν+2 x y 1 s2 + t2 2ρst − 2 tρ,ν(x, y)= 1+ − ds dt (4.41) 2π 1 ρ2 ν(1 ρ2) Z−∞ Z−∞ − − then the BTC is deﬁned as follows:p t 1 1 Cρ,ν(u1, u2) = tρ,ν(tν− (u1),tν− (u2)) −1 −1 ν+2 t (u1) t (u2) ν ν 1 s2 + t2 2ρst − 2 = 1+ − ds 2π 1 ρ2 ν(1 ρ2) Z−∞ Z−∞ − − 1 where tν− is the inverse of the univariatep 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]):

ν+2 2 2 ζ +ζ 2ρζ1ζ2 2 ν+2 ν 1 2 − 1 1+ − 2 t Γ 2 Γ 2 ν(1 ρ ) − 2 − cρ,ν(u1, u2)= ρ 2 ν+1 (4.42) | | ν+1 ζ2 2 Γ 2 2 j − j=1 1+ ν Q

1 10 0.9 9 0.9 9

0.8 0.8 8 8

0.7 7 0.7 7 0.6 6

0.5 0.6 5 6

0.4 4 0.5 5 0.3 3

0.2 0.4 2 4

0.1 1 0.3 3 0 0 1 1 0.2 2

0.5 0.1 0.5 1 1 1 0.8 0.8 0.6 0.6 0.4 0 0.4 0 0.2 0 0.2 u2 0 u2 0 u1 u1 (a) Student’s t-copula (ρ = 0.5, ν = 3) (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 1000 samples of two uniformly distributed random variables coupled with a Student’s t-copula are shown.

When the number of the degrees of freedom diverges the Student’s t-copula converges to the Gaussian copula. This is one reason that the Student’s t-copula is fast growing in usage because the weight in the tail dependency can be set by changing the degrees of freedom parameter ν (see Nystrom and Skoglund [2002]). Large values for ν approximate a Gaussian distribution. Conversely small values for ν increase the tail mass. For ν = 1 it simulates a bivariate Cauchy distribution. CHAPTER 4. COPULAS 21

1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

u2 0.5 u2 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 u1 u1 (a) Student’s t-copula (ρ = 0.5, ν = 3) (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 recognized by Schweizer and Sklar [1961] in the study of t-norms. From the practical point of view Archimedean copulas are useful because it is possible to generate a number of copulas from interpolating between certain copulas.

Archimedean copulas may be constructed using a continous, decreasing, convex function φ : I R+, such that φ(1) = 0. Such a function φ is called a generator. It is called a strict→ generator whenever φ(0) = + . ∞ The pseudo-inverse of φ is deﬁned as follows:

1 1 φ− (v) 0 v<φ(0) φ− (v)= ≤ (4.43) 0 φ(0) v ( ≤ ≤ ∞ 1 Hence the function φ− is continuous and not increasing on [0, ] and strictly decreasing on [0,φ(0)]. This inverse is such that, by composition with the generat∞ or function, it gives the identity as ordinary inverses do:

1 φ− (φ(v)) = v (4.44)

Given a generator and its inverse, an Archimedean copula CArchimedean is generated according 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 → 1 C(u1, u2)= φ− (φ(u1)+ φ(u2)) (4.45) 1 is a copula if φ− is strictly monotone on [0, ]. ∞ This theorem allows the deﬁnition of many Archimedean copulas. One important source of generators for Archimedean copulas is the inverses of the Laplace transforms of CDFs CHAPTER 4. COPULAS 22

(see Feller [1971]).

By the generator of an Archimedean copula its density can be derived by the following equation: ′′ ′ ′ Archimedean φ (C(u1, u2))φ (u1)φ (u2) c (u1, u2)= − ′ 3 (4.46) (φ (C(u1, u2))) 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 applications in equity markets since negative dependencies are usually rare in that context. A more important property of the most frequently used Archimedean copulas is that they depend on 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 copulas are ﬂexible enough to capture various dependence structures, e.g. concordance and tail dependence, which makes them suitable for modeling extreme events. Usually they are used to model a strong dependence in the tail (see Nelson [1999]).

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

Gumbel Copula • The Gumbel or Gumbel-Hougaard family of copulas was described by Hutchinson Gu α Gu 1 and Lai [1990]. Their generator is given by φ (u)=( ln(u)) , hence (φ )− (t)= 1 − exp( t α ). It is completely monotonic if α > 1. The Gumbel copula is therefore: − Deﬁnition 4.13 (Gumbel Copula)

2 1/α CGu (u , u ) = exp ( ln u )α with α [1, ) . (4.47) 1 2 − − i  ∈ ∞ " i=1 #  X  Nelson [1999] showed that the Gumbel-Houghaard Copula CGu can describe multivariate extreme value distributions. For α = 1, Equation (4.47) reduces to

2 Gu C (u1, u2)= ui (4.48) i=1 Y the independent copula in Equation (4.39).

Figures 4.5 show the plot of a Gumbel copula and its density. In Figure 4.6 1000 samples of two uniformly distributed random variables coupled with a Gumbel copula are shown. The graph gives an impression how the choice of the parameter α CHAPTER 4. COPULAS 23

1 20

1 20 0.9 18 0.9 18

0.8 0.8 16 16

0.7 14 0.7 14 0.6 12

0.5 0.6 10 12

0.4 8 0.5 10 0.3 6

0.2 0.4 4 8

0.1 2 0.3 6 0 0 1 1 0.2 4

0.5 0.1 0.5 2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0 0 0.2 0 0.2 u2 0 u2 0 u1 u1 (a) Gumbel copula (α = 2) (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 asymmetric Archimedean copula, exhibiting greater dependence in the positive than in the negative tail.

The relationship between Kendall’s tau τ and the Gumbel copula parameter α is given by: 1 α = (4.49) 1 τ − The density of the bivariate Gumbel copula is given by

1 Gumbel α α α 1 α 1 c (u , u )= exp [( ln u ) +( ln u ) ] α ( ln u ) − ( ln u ) − 1 2 − − 1 − 2 − 1 − 2 (4.50) 1 1 n 1 o 1 α α α 2 α α α [( ln u1) +( ln u2) ] − [( ln u1) +( ln u2) ] + α 1 u1 u2 − − − − − Clayton Copula • A second example of Archimedean copulas is the Clayton copula. The generator is Cl α 1 1 given by φ (u)= u− 1, hence φ− (t)=(t + 1)− α . It is completely monotonic if α> 0. The Clayton copula− is therefore

Deﬁnition 4.14 (Clayton Copula)

1/α n − Cl α C (u)= u− n + 1 with α (0, ) (4.51) α i − ∈ ∞ " i=1 # X For the limit α 0 the independent copula is obtained again. In Figure 4.7 the → Cl graph of a Clayton copula Cα for α = 5.0 and its corresponding density is shown. CHAPTER 4. COPULAS 24

1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

u2 0.5 u2 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 u1 u1 (a) Gumbel copula (α = 2) (b) Gumbel copula (α = 20)

Figure 4.6: Random draws from a Gumbel copula

110

1 120 0.9 100 0.9 100 0.8 0.8 90

0.7 80 0.7 80 0.6 70 0.5 0.6 60

0.4 60 0.5 40 0.3 50 0.2 0.4 20 40 0.1 0.3 0 0 30 1 1 0.2 20

0.5 0.5 0.1 10 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0 0 0.2 0 0.2 u2 0 u2 0 u1 u1 (a) Clayton copula (α = 6) (b) Density of the Clayton copula (α = 6)

Figure 4.7: Clayton copula CHAPTER 4. COPULAS 25

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

The Clayton copula is an asymmetric Archimedean copula and exhibits greater dependence in the negative tail than in the positive, which can be seen from the density plot in Figure 4.7(b). The Clayton copula is suitable for describing dependencies in the left tail and that there is empirical evidence of increasing dependence in falling markets (see Longin and Solnik [2001]). The relationship between Kendall’s tau τ and the Clayton copula parameter α is given by: 2τ α = (4.52) 1 τ − The density of the bivariate Clayton copula is given by:

1 Clayton α 1 α α α 2 c = (1+ α)(1+2α) (u u ) − u− + u− 1 − − (4.53) 1 2 1 2 −

1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

u2 0.5 u2 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 u1 u1 (a) Clayton copula (α = 2) (b) Clayton copula (α = 20)

Figure 4.8: Random draws from a Clayton copula

Frank Copula • Frank exp( αu) 1 − − The generator of the Frank copula is given by φα (u) = ln exp( α) 1 , hence − − 1 1 t α φ− (t)= α ln(1 + e (e− 1)). It is completely monotonic if α> 0. The bivariate Frank copula− is therefore given− by:

Deﬁnition 4.15 (Frank Copula)

2 αui 1 (e− 1) C(u , u )= ln 1+ i=1 − with α> 0 (4.54) 1 2 −α e α 1 ( Q − − ) CHAPTER 4. COPULAS 26

The relationship between Kendall’s tau τ and the Frank copula parameter α is given by: D (α) 1 1 τ 1 − = − (4.55) α 4 where 1 α t D (α)= dt (4.56) 1 α et 1 Z0 −

1 20

1.4 20 0.9 18 18 1.2 16 0.8 16 1 14 0.7 14 12 0.8 10 0.6 12 0.6 8 0.5 10 0.4 6

4 0.4 8 0.2 2 0.3 6 0 0 1 1 0.2 4

0.5 0.1 0.5 2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0 0.2 0 0.2 u2 0 u2 0 u1 u1 (a) Frank copula (α = 14.14) (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 bivariate Frank copula is given by:

α Frank (e− 1) c (u1, u2)= α(w1 + 1)(w2 + 1) − (4.57) − [(e α 1) + w w ]2 − − 1 2 αui where w = e− 1 i −

1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

u2 0.5 u2 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 u1 u1 (a) Frank copula (α = 2) (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 distribution associated to a random couple (U1, U2) measures the degree of dependence between U1 and CHAPTER 4. COPULAS 27

U . The larger α respectively ρ the stronger is the dependence. Therefore the param- 2 | | | | eters obtained by calibrating to real data represent a measure of dependence. The most commonly used copulas are the Gumbel copula for extreme distributions, the Gaussian copula for linear correlation, and the Archimedean copula (Frank, Gumbel, Clayton), and the 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 distribution. 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 distributions and the dependency. This advantage makes the use of copulas a powerful tool to overcome problems which arise when using the standard Gaussian distributions.

The question is of how a copula model can be calibrated to observed data. The calibration can be divided into two separate steps. First one has to extract the distribution of the margins. After extracting the distribution of the margins the dependence structure must be identified. Since multivariate equity options are usually traded over-the-counter it is not easily possible to obtain option prices in order to extract the risk-neutral copula from market prices. Due to this lack of data it is assumed that the risk-neutral and the real world copulas are identical. Rosenberg [2003], for instance, argues that under general conditions this is a reasonable assumption. Relying on historical asset returns different copula 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. Since a copula function is a multivariate model much of the classical statistical theory is not applicable. Therefore, usually the asymptotic maximum likelihood estimation (MLE) is applied. In the Appendix B the idea of the method is outlined briefly. (For details see Cherubini et al. [2004]). There are three commonly used methods to fit a copula model to observed data: Exact maximum likelihood method: The exact maximum likelihood method esti- • mates all parameters of the marginals and the copula simultaneously. Therefore, it’s a computationally intense method and is of minor use for practical applications. IFM method: The inference for the margins (IFM) method estimates the parameters • of the copula model in two steps. First, the parameters of the univariate marginal distributions are estimated by MLE. Second, the copula parameter is estimated via MLE. This approach is a fully parametric approach. CML method: The Canonical Maximum Likelihood (CML) follows the steps of the • IFM method. The difference to the IFM method lies in the choice of the marginal distribution. The CML method uses the empirical distribution and avoids any assumptions about the marginal distributions. The advantage of the CML method CHAPTER 4. COPULAS 28

is, that the copula can be estimated without specifying the marginals. In this case one would use the empirical distribution to transform the sample data into uniform variates.

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

marginals (i) Identiﬁcation and estimation of the marginal parameters Φi In the ﬁrst step one has to choose an appropriate marginal distribution. The pa- marginals rameters Φi of the chosen marginal distributions Fi with i = 1,...,n are estimated via the MLE. m marginals marginals Φˆ = ArgMax marginals ln fi(µij, Φ ) (4.58) i Φi i j=1 X th where fi denote the density of the i 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 the copulas’ 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 be found in Figure 6.3.

(ii) Transformation of the observed data to the unit hypercube marginals The parameters Φˆi estimated in step (i) are used to transform the data to the unit hypercube by parametric estimates of their marginal CDF Fi. The sample data m m µ1j,µ2j,...,µnj j=1 are transformed into uniform variates u1j, u2j,...,unj j=1 by { } marginals { } setting uij = Fi(µij; Φˆi ).

(iii) Deﬁnition of the appropriate copula function and estimation of the copula parameters Φcopula m By the transformed sample data u1j, u2j,...,unj j=1 the estimation of the copula parameters with a maximum likelihood{ algorithm} is performed:

m ˆ copula copula ˆ marginals Φ = ArgMaxΦcopula ln c (u1j), (u2j),...,unj); Φ , Φ (4.59) j=1 X where c denotes the copula density of the chosen copula (see Chapter 4.4). The obtained parameters Φˆ marginals and Φˆ copula are the parameters which describe the statistical distribution of the observed returns best in the sense of a maximum likelihood approach. CHAPTER 4. COPULAS 29

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

1 √T (Φˆ Φ) N(0,G− (Φ)) (4.60) − → where G(Φ) denotes the Godambe information matrix. Joe [1997] points out that the IFM 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 generate random draws, which are distributed like the chosen copulas. There are several methods described in literature to generate draws from copulas. The parameters of the random variables estimated in Section 4.5 are the basis to generate random scenarios from the copula set up. The question is how a simulation of paths by the estimated parameters can be done.

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

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

An easy algorithm for random variate generation from the n-dimensional Gaussian copula with correlation matrix Σn is given by the following algorithm (see Embrechts et al. [2003]) Find a Cholesky decomposition A of Σ • n Simulate n independent random variates ~z =(z , z ,...,z )T from N(0, 1) • 1 2 n Set x = Az •

Set ui = Φ(xi) with i = 1, 2,...,n, where Φ denotes the univariate standard normal • distribution 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 Embrechts et al. [2003]) CHAPTER 4. COPULAS 30

Find a Cholesky decomposition A of Σ • n Simulate n i.i.d. ~z =(z , z ,...,z )T from N(0, 1) • 1 2 n Simulate a random variate s from χ2 independent of z. • ν Set y = Az • x = ( ν )y • s

Set upi = tν(xi) with i = 1, 2,...,n, where tν denotes the univariate Student’s t • distribution function

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

Usually statistical programs oﬀer functions to generate independent random variates from a N(0, 1) distribution. For the comparison of the result of this thesis it is helpful to start with uniformly distributed variates ~w = (w1,w2 ...,wn). By applying the inverse cumu- 1 lative standard normal distribution function F − to the vector ~w a N(0, 1)-distributed vector of random variates can be generated. This vector can be used as the input vector ~z of the two previously described methods. By this method it is possible to generate variates from 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. For some commonly used Archimedean copulas fast algorithm exit when the inverse generator 1 function φ− is known to be the Laplace transform of some positive random variable 1 (see Marshall and Olkin [1998]). If the generator φ− equals the inverse of the Laplace transform of a distribution function G on R+ satisfying G(0) = 0, the following algorithm can be used for simulating random draws from the copula:

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

Simulate n independent variates z ,...,z . • 1 n 1 1 Return (u , u ,...,u )=(φ ( log(z /x),...,φ− ( log(z /x)). • 1 2 n − 1 − n This procedure can be applied to the Clayton and the Gumbel. However, it is not applicable 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 conditional sampling one. This approach involves diﬀerentiation steps for each dimension. For this reason, it is not feasible in higher dimensions. Marshall and Olkin [1998] proposed an CHAPTER 4. COPULAS 31

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

In order to compare the results of the use of different copulas and to exclude differences caused by sampling differences it is helpful to use a method, which transforms uniform distributed random variables to a chosen copula. This allows for the recycling of the same sample of univariate variates for each simulation, which then are transformed into the relevant copula distributions. Therefore, I use the conditional approach to simulate draws from the Archimedean copula. For each simulation I use the same pairs of uniformly distributed 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 conditional distribution. The task is to generate pairs (u1, u2) of observations of [0, 1] uniformly distributed random variables U1 and U2 whose joint distribution function is C. To reach this goal I use the conditional distribution Cu1 (u2) = F (U2 u2 U1 = u1). From definition copulas are joint distribution functions of standard uniform≤ | random variables. C(u , u )= F (U u , U u ). From the following equation 1 2 1 ≤ 1 2 ≤ 2 C (u ) = P (U u U = u ) (4.61) u1 2 2 ≤ 2| 1 1 C(u + δu , u ) C(u , u ) = lim 1 1 2 − 1 2 (4.62) δu 0 → δu1 ∂C = (4.63) ∂u1

= cu1 (u2) (4.64) where c (u ) := ∂C(u1,u2) , one knows that c (u ) is a non-decreasing function and exists u1 2 ∂u1 u1 2 for almost all u [0, 1]. The following theorem provides a tool to generate random draws 2 ∈ from an Archimedean copula with generator φ. Then 1 Theorem 4.16 Let C(u1, u2) = φ− (φ(u1)+ φ(u2)) be an Archimedean bivariate copula with generator φ: 1 φ− (φ(u1)+ φ(u2)) Cu1 (u2)= (4.65) u1 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 to

the 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 bivariate Archimedean copula by a conditionals approach. It can be formulated in the following way:

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

1 Compute the inverse function of cu1 (v). Set u2 = cu−1 (v) to obtain the second desired • draw. CHAPTER 4. COPULAS 32

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

Copula Conditional Copula α α/(α+1) 1/α Clayton u = u− v− 1 + 1 − 2 1 − v 1 e−α 1 ( − ) Frank u = ln 1+ − − 2 − α v(e αu1 1) e αu1 − − Gumbel no closed solution available: −1 α α φ (( ln(u1)) +( ln(u2)) ) v = − −1 − α φ (( ln(u1)) ) 1 − w 1 1 α with φ− (w)= e− w − − 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 method can be applied. Marshall and Olkin [1998] suggest an alternative algorithm based on mixtures of powers. As it can be seen in case of the Gumbel copula the drawback of the conditional approach is that it might be not possible to calculate an inverse function analytically. To be consistent in generating the random samples and obtain results which are comparable this disadvantage is accepted in the following simulations. The by the previously methods obtained random variates u1,...,un can then be used to 1 generate the random retures by applying the inverse CDF Fi− of the chosen marginal distribution. This calculation can be done in different ways. There are different methods described in the literature to do this calculation neatly. For the Student’s t distribution for example Shaw [2006] proposes iterative techniques to approximate the inverse CDF for different degrees of freedom. Chapter 5

Monte Carlo Simulations with Copulas

Performing a Monte Carlo simulation by using copulas is basically the same procedure as the standard Monte Carlo procedure described in Chapter 3. The process described by Equation (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. To amend the standard Monte Carlo method to take copulas into account one has to change assumed 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 as well as the dependence structures. To take this advantage into account I (beside the use of different copula) assume different marginal distributions than in the standard approach in Equation (3.1) as well. The Student’s t-distribution is often used to capture observed fat-tail features. To study the impact of the alternative choice of the marginal distribution on the simulations, the Student’s t-distribution is used as one alternative to the standard Gaussian approach. To follow the risk-free pricing approach I use the t-distribution with scale parameters. This is a family of univariate probability distributions parameterized by a location parameter µ and a scale parameter σ 0. If X is a Student’s t-distributed random variable with ν degrees of freedom then Y≥ = µ + σX is the t location-scale distribution. It has the following density

x µ 2 ν+1 Γ( ) ν + −σ ν + 1 f(x; µ,σ,ν)= 2 (5.1) σ√νπΓ( ν )  ν  2 2   Since E(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 to the risk-free rate by setting the parameter µ appropriately and captures fat-tail features of 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

dt is modeled via a random variable Xi :

dt Si(tj) Si(tj 1) Xi = − − (5.3) Si(tj 1) − dt where Xi 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 generating dt dt the vector X1 ,...,Xn by the chosen copula and marginal distributions the desired marginal distributions and dependencies between the returns can be modeled. In case of dt dt Gaussian marginals and Gaussian copula, Xi is normally distributed with mean E(Xi )= dt dt dt dt dt µi , standard deviation V ar(Xi) = σi and the random vector (X1 ,...,Xn ) has a correlation matrix Σn. This is equivalent to the distribution of the returns in Equations p dt (3.1) and (3.2). The Markoﬀ property of the paths is furthermore maintained and Xt (tj), j = 1,...,m 1, are independent. Therefore, the expectation of a path simulated with m steps of the− length dt is

m 1 − dt dt m E( (1 + Xi (tj))) = E((1 + Xi )) ) (5.4) j=0 Y dt m = (E(1 + Xi )) (5.5) dt m = (1+ µi ) (5.6) To model the price paths of assets one needs to discretize the paths at certain points in time. The chosen modeled points in time t0,t1,...,tn have to be the points in time when the underlyings of the option are monitored. A weekly monitoring of a basket for example requires a weekly simulation of the asset prices Si. There are several ways to simulate the risk-neutral random walk discretely. For the simulations in this thesis I simulate discrete returns for the given time steps dt by a random variable X~ dt, where ~ dt dt dt dt X = (X1 , X2 ,...,Xn ) denotes an n-dimensional random vector, which is distributed like the returns for the time period dt of the underlyings of the basket. Further let

l l l ~µ (tj)=(µ1(tj),...,µn(tj)) (5.7)

dt th denote the realization of the random variable X~ at time tj for the l simulation. To ﬁnd dt the distribution of Xi the according observed returns are used to calibrate the distribution, i.e. for a daily monitoring observed daily returns for a weekly monitoring observed weekly returns are used.

In order to be consistent with the pricing approach of the standard Monte Carlo approach, dt one has to ensure by setting E(Xi ), i = 1,...,n appropriately that the expected value equals the adjusted risk-free drift (see Section 2.4). In other words one has to amend to real word measure, to a risk-free measure. This is done by the amendment of the expected value of the simulated discretized returns (see Equation (5.6)) for the time period dt. E(Xdt)=(1+ r d )dt 1 (5.8) i − i − As for the speciﬁcation of the distributions distributions, one could use implied information or a time series analysis based on historical data. Due to the usually big bid-ask-spread CHAPTER 5. MONTE CARLO SIMULATIONS WITH COPULAS 35

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

dt dt dt obs Let µi (t1),µi (t2),...,µi (tm) denote the observed returns and Si (tj) denote the observed prices of the ﬁnancial underlying assets i of our basket at the times t1,...,tm:

obs obs dt Si (tj) Si (tj 1) − µi (tj) := obs− (5.9) Si (tj 1) − copula marginals where (tj tj 1) = dt. For the estimation of the parameters Φ and Φ the − IFM method− (see Section 4.5) is used. The chosen observed market data are observed closing quotes over the same period as the lifetime of the chosen option, i.e. for an option with a lifetime of one year (T=1) a history of one year of returns is used to estimate the parameters of the distribution. Therefore, the same number of market observations to calibrate the model are used as to be simulated.

The estimated parameters Φˆ copula and Φˆ marginals are used to generate m 1 returns for each − of the n assets at the discrete points in time tj, j = 1,...,m 1 for each of the simulation − l l, l = 1,...,L. These returns are used to generate paths of assets Si(tj), i = 1,...,n. Therefore, one obtains a three dimensional cube with the dimension m n L, which stores all samples paths. × ×

As described in Section 4.5 copulas allow to deal separately with marginal and joint distribution modeling. Thus, one can choose for each data series the marginal distribution that ﬁts best the sample, and afterwards put everything together using a copula function with some desirable properties. Therefore, the number of combinations is large and one can easily get lost in looking for the best combination of marginal and joint distributions. In this thesis I concentrate on the frequently used marginal distributions. Gaussian and the Student’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 allow a negative skewness 1. To have a greater ﬂexibility the degrees of freedom are allowed to 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). ~l ~l For each simulation L paths S1,..., Sn, l = 1,...,L are simulated. The calculation of the price of the observed option is done by:

L MC r(T t) 1 l L V (t)= e − p(T, S~ ,..., S~ ) (5.10) L 1 n Xl=1 For each simulation the following steps are performed:

ˆ marginals (i) Estimation of the marginal distribution parameters Φi by IFM method with dt the observed returns µi (tj) via MLE for each asset i.

1Details on the Student’s t-distribution are shown in the Appendix A CHAPTER 5. MONTE CARLO SIMULATIONS WITH COPULAS 36

dt (ii) Transformation of the observed returns µi (tj) with the estimated marginals param- ˆ marginals eters Φi into uniformly distributed random variables. (iii) Estimation of the copula parameters Φˆ copula via MLE.

(iv) Changing the real world measure to a risk-neutral measure by amending the ex- dt dt pected value of the marginals E(Xi ), i = 1,...,n of the random variables Xi . (v) Draw of (m 1) L random vectors ~µl(t ) deﬁned according to Equation (5.7) of − × j the random vector X~ dt by transformation of uniformly distributed random variables (identical for each simulation).

~l ~l l l (vi) Generate paths S1, S2,...,Sn from the simulated returns ~µ (tj).

~l ~l (vii) Calculation of payoff p(T, S1,..., Sn) at expiry T for each simulation. ~l ~l (viii) Discounting the payoff p(T, S1,..., Sn) 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 baskets in order to test the influence of the use of copulas on the prices. The two baskets contain German DAX-companies from two different industries. For the chemical industry Bayer AG (Bayer) and the BASF AG (BASF) were chosen, from the automotive industry the shares 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 industry, but their dependence structure is more likely to show special features such as fat tails 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 on the basis of returns of an interval which corresponds to the monitoring interval of the option: If the basket is monitored daily, historical daily returns are the basis for the ﬁtting algorithm, if the baskets are monitored weekly, historical weekly return are used. There- daily weekly fore, the historical daily (weekly) returns µi (tj) respectively µi (tj), j = 1,...,m of each asset i were calculated as percentage diﬀerences per time period dt according to Equation 5.9 from the historical price data. The correlations from the historic returns were calculated for the time window of past returns corresponding to the life time of the option, i.e. 250 return data for the daily monitoring of the option and 52 return data for the weekly monitoring. This is in line with market conventions. Statistically speaking, this convention implies that the market assumes stationarity of the model over the life time 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 weeks per year: 1 rweekly = (1+ r d ) 52 1 (6.1) i − i − 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 EUR mean of µi 0.1057 % 0.0591 % 0.5130 % 0.2127 % standard deviation 1.823 1.549 3.732 3.4149 skew -0.1374 -0.61231 -0.8052 -0.5552 dividend yield p.a. 2.16 % 3.85 % 2.16 % 3.85 % Pearson linear correlation 0.7002 0.7374 Kendall’s τ 0.5012 0.4449 Spearmann’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

1 rdaily = (1+ r d ) 250 1 (6.2) i − i − In order to examine the impact of the use of copulas when pricing a basket option I use the basket options described in Chapter 2.2. In each case basket options with a maturity of one year (T=1) are chosen. Options with weekly and daily monitoring interval of the underlying 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) , | S2(−t0) S1(t0)| | − | the strike K of the Asian Option was set to the mean of the spot prices, K = 2 .

To capture different market movements two different pricing dates were considered. The first date was the 19th February 2008 as pricing date for the Bayer/BASF-basket. On the 19th February 2008 the 1-year-EURIBOR-rate was r = 4, 39%. This rate is regarded as the risk-free rate. The studied options have a lifetime of one year (T=1), therefore their 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 were 54.40 Euro (Bayer) and 87.00 Euro (BASF). Table 6.1 shows the statistical data of the observed returns.

Table 6.1 shows some statistical data of the observed historical returns of the underlyings in the Bayer/BASF basket. Both stocks showed a positive drift. Apparently the mean and the standard deviation of the weekly returns have to be larger than corresponding data of the daily returns. All returns show a negative skew, i.e. the left tail is longer. The Gaussian distribution is symmetric, therefore a Gaussian distribution does not capture this feature of the historical distribution. All dependence measures show a signiﬁcant association of the 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 a lifetime of one year (T=1). Therefore, the options expire on 10th October 2009. They can be exercised only at the expiry (European feature). Equal weights for the stocks in the CHAPTER 6. NUMERICAL EXPERIMENTS 39

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

Table 6.2 shows some statistical data of the observed historical returns of the underlyings in the BMW/VW basket. The stock of BMW had a negative performance in the observed time. Especially VW shows a significant positive skew. All dependence measures show a weaker dependence structure than the chemical basket. Interestingly the linear correlation coefficient shows different signs of linear correlation between the observed daily returns and the observed weekly returns. However, the two additional association measures, 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 EUR mean of µi -0.2757 % 0.2409 % -1.3844 % 1.3354 % standard deviation 2.378 2.8011 4.8307 8.036 skew 0.5803 3.1597 -0.3885 2.7903 dividend yield p.a. 2.16 % 3.85 % 5.32 % 0.86 % Pearson linear correlation 0.1162 -0.1286 Kendall’s τ 0.2354 0.0298 Spearmann’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 graphics the histograms resemble a Gaussian distribution, but especially the returns of VW show strong 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 plots can be used. Figure 6.3 provides the Q-Q plots of the observed returns versus the normal distribution. If the points of the plot, which are formed from the quantiles of the data, are roughly on a line with a slope of 1, then the data set comes from the distributions. The Gaussian distribution shows quite good results for the daily returns. However, especially at the tails of the distribution the ﬁt is less accurate. The Q-Q-plot shows the fat tails at the positive end of the distribution of the VW returs which could already be derived in Figure 6.2. The feature of tail dependence should for example be captures better by the Student’s t-distribution with low degrees of freedom.

Graphik 6.1 shows the scatter plot of the daily/weekly returns of the chosen stocks of the Bayer/BASF and BMW/VW shares. A signiﬁcant dependence structure between the assets can be observed from the concentration of the points in certain areas/shapes. All plots show graphically the existence of a dependence structure between the returns of the stocks. CHAPTER 6. NUMERICAL EXPERIMENTS 40

Option Type Maturity Strike in EUR Weight of assets Monitoring Average Spread 1 year 32.60 (Bayer/BASF) equally weighted weekly/daily 322.37 (BMW/VW) Average Strike 1 year n/a equally weighted weekly/daily Lookback Put 1 year n/a equally weighted weekly/daily Lookback Call 1 year n/a equally weighted weekly/daily Double Average rate 1 year n/a 1 equally weighted weekly/daily Asian 1 year 70.70 (Bayer/BASF) equally weighted weekly/daily 180.82 (BMW/VW)

Table 6.3: Parameters of examined options CHAPTER 6. NUMERICAL EXPERIMENTS 41

6 8

4 6

4 2

0 0 BASF BASF −2 −2

−4 −4

−6

−6 −8

−8 −10 −8 −6 −4 −2 0 2 4 6 −15 −10 −5 0 5 10 Bayer Bayer (a) daily returns (BASF/Bayer) (b) weekly returns (BASF/Bayer)

30 50

25 40

30 15

20 10 VW VW 5 10

0 0

−5

−10 −10

−15 −20 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 BMW BMW (c) daily returns (BMW/VW) (d) weekly returns (BMW/VW)

Figure 6.1: Scatterplot of the returns in the observed time series CHAPTER 6. NUMERICAL EXPERIMENTS 42

35 8

7 30

6 25

15 3

10 2

5 1

0 0 −25 −20 −15 −10 −5 0 5 10 15 20 25 −50 −40 −30 −20 −10 0 10 20 30 40 50 (a) daily returns (Bayer) (b) weekly returns (Bayer)

45 8

40 7

35 6

30 5

25 4

3 15

2 10

1 5

0 0 −25 −20 −15 −10 −5 0 5 10 15 20 25 −50 −40 −30 −20 −10 0 10 20 30 40 50 (c) daily returns (BASF) (d) weekly returns (BASF)

45 7

40 6

5 30

4 25

20 3

15 2

1 5

0 0 −30 −20 −10 0 10 20 30 −50 −40 −30 −20 −10 0 10 20 30 40 50 (e) daily returns (VW) (f) weekly returns (VW)

35 5

4.5 30

25 3.5

3 20

2.5

15 2

10 1.5

1 5 0.5

0 0 −25 −20 −15 −10 −5 0 5 10 15 20 25 −50 −40 −30 −20 −10 0 10 20 30 40 50 (g) daily returns (BMW) (h) weekly returns (BMW)

Figure 6.2: Histograms of the observed returns for the underlying assets CHAPTER 6. NUMERICAL EXPERIMENTS 43

QQ Plot of Sample Data versus Standard Normal 6 QQ Plot of Sample Data versus Standard Normal 10

0 0

−2 −5 Quantiles of Input Sample Quantiles of Input Sample

−4

−10 −6

−8 −15 −3 −2 −1 0 1 2 3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 Standard Normal Quantiles Standard Normal Quantiles (a) daily returns (Bayer) (b) weekly returns (Bayer)

QQ Plot of Sample Data versus Standard Normal 6 QQ Plot of Sample Data versus Standard Normal 8

4 6

4 2

0 0

−2 −2 Quantiles of Input Sample Quantiles of Input Sample −4 −4

−6

−6 −8

−8 −10 −3 −2 −1 0 1 2 3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 Standard Normal Quantiles Standard Normal Quantiles (c) daily returns (BASF) (d) weekly returns (BASF)

QQ Plot of Sample Data versus Standard Normal QQ Plot of Sample Data versus Standard Normal 30 50

25 40

30 15

10 20

5 10 Quantiles of Input Sample 0 Quantiles of Input Sample 0

−5

−10 −10

−15 −20 −3 −2 −1 0 1 2 3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 Standard Normal Quantiles Standard Normal Quantiles (e) daily returns (VW) (f) weekly returns (VW)

QQ Plot of Sample Data versus Standard Normal QQ Plot of Sample Data versus Standard Normal 15 10

10 5

5 0

0 −5 Quantiles of Input Sample Quantiles of Input Sample

−5 −10

−10 −15 −3 −2 −1 0 1 2 3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 Standard Normal Quantiles Standard Normal Quantiles (g) daily returns (BMW) (h) weekly returns (BMW)

Figure 6.3: Q-Q plots of the observed returns against the normal distribution CHAPTER 6. NUMERICAL EXPERIMENTS 44

6.2 Results

6.2.1 Estimation of Parameters Tables 6.4 - 6.7 show the results of the estimated parameters Φˆ marginal 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 observed returns in Tables 6.1 and 6.2. This is not surprising because the estimated standard deviation should equal the standard deviation of the observed data sample. Apparently this observation allows us to avoid a maximum likelihood estimation and could accelerate the calibration procedure. However, due to comparability reasons the same methods of calibration was used for each margin.

The parameter µ equals the risk-free rate calculated by Equation 6.1 respectively 6.2. By the 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 in mind that the Student’s t-distribution converges against the Gaussian distribution, one sees that the marginal behaviour is not close to normal. Only the weekly returns of the BASF and BMW displayed in Tables 6.5 and 6.7 show a rather high degree of freedom.

ˆ marginal ˆ marginal Φ1 Φ2 Margin µ1 σ1 ν1 µ2 σ2 ν2 Gaussian 0,00882239 1,82365848 n/a 0,00215421 1,54939361 n/a t 0,00882239 1,47403185 5,6586597 0,00215421 1,11025711 3,89036227

ˆ marginals ˆ marginal Table 6.4: Parameters Φ of the underlyings Bayer(Φ1 )/BASF ˆ marginal (Φ2 )(daily monitoring)

ˆ marginal ˆ marginal Φ1 Φ2 Margin µ1 σ1 ν1 µ2 σ2 ν2 Gaussian 0,04242244 3,73272305 n/a 0,01035721 3,41494546 n/a t 0,04242244 3,04964362 5,90035118 0,01035721 3,19472055 18,1581225

ˆ marginals ˆ marginal ˆ marginal Table 6.5: Parameters Φ of the underlyings Bayer (Φ1 )/BASF (Φ2 ) (weekly monitoring)

Tables 6.8 - 6.11 show the results of the copula parameters Φˆ copula via IFM method. De- pending on the choice of the copula not all parameters are relevant. The parameter ρ of the Gaussian/Gaussian combination equals the Pearson linear correlation coeﬃcient of the baskets as reported in Table 6.2 and 6.1. It can be seen that the choice of the marginal CHAPTER 6. NUMERICAL EXPERIMENTS 45

ˆ marginal ˆ marginal Φ1 Φ2 Margin µ1 σ1 ν1 µ2 σ2 ν2 Gaussian 0,00324731 4,83071986 n/a 0,08705821 8,03600256 n/a t 0,00324731 4,33917837 11,0284845 0,08705821 3,55800396 2,15705559

ˆ marginals ˆ marginal ˆ marginal Table 6.6: Parameters Φ of the underlyings BMW (Φ1 )/VW (Φ2 ) (weekly monitoring)

ˆ marginal ˆ marginal Φ1 Φ2 Margin µ1 σ1 ν1 µ2 σ2 ν2 Gaussian 0,00067543 2,37877625 n/a 0,01810187 2,80111153 n/a t 0,00067543 1,9068844 5,67968544 0,01810187 1,26899819 2,3207351

ˆ marginals ˆ marginal ˆ marginal Table 6.7: Parameters Φ of the underlyings BMW (Φ1 ) /VW (Φ2 ) (daily monitoring) distribution has an impact on the estimated parameters of the copula. Especially the degrees of freedom ν of Student’s t-copula show a large diﬀerence when using t-distributed margins. 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 Gaussian marginal. The fact that the t-distribution keeps the fat tail feature when transforming the sampled data into the uniform random variables strengthens the impact of fat tails in the dependence structure as well.

As the dependence measures in Table 6.3 already suggested, the parameters of the BMW/VW basket (see Table 6.8) are close to the independent copula. The stronger dependence of the Bayer/BASF basket are reﬂected by the larger values of the the parameters of all studied 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 the returns. The figures show that the choice of the copula has an not neglectalbe effect on the estimated prices of the studied options. In order to compare the results the Tables 6.13, 6.19, 6.15 and 6.17 show the differences in percentage terms to the multivariate normal assumption (Gaussian/Gaussian). It can be seen that using copulas to model a dependence structure can have a remarkable effect on the estimated fair prices (up to almost 18 % compared to the standard Gaussian/Gaussian approach). In particular it 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 only by a maximum of 1.43 %. This should not surprise keeping in mind that the dependence measures showed almost independence of the returns and so the copulas do not capture fat tails. However, the use of Student’s t-marginals has a signiﬁcant impact on the prices. CHAPTER 6. NUMERICAL EXPERIMENTS 46

Margin Copula α ρ ν gaussian gaussian n/a -0,13116115 n/a gaussian t n/a -0,12698872 197,151426 gaussian clayton -0,2593965 n/a n/a gaussian frank 0,0001049 n/a n/a gaussian gumbel 1,00000623 n/a n/a t gaussian n/a -0,05676578 n/a t t n/a -0,04584793 66,7800311 t clayton -0,34314095 n/a n/a t frank 0,30886649 n/a n/a t gumbel 1,0000314 n/a n/a

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

Margin Copula α ρ ν Gaussian Gaussian n/a 0,12635575 n/a Gaussian t n/a 0,41826006 5,99717817 Gaussian Clayton 0,39660205 n/a n/a Gaussian Frank 3,26702699 n/a n/a Gaussian Gumbel 1,09313132 n/a n/a t Gaussian n/a 0,29301827 n/a t t n/a 0,36095964 3,14937499 t Clayton 0,54538044 n/a n/a t Frank 2,32916236 n/a n/a t Gumbel 1,24193053 n/a n/a

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

Margin Copula α ρ ν Gaussian Gaussian n/a 0,7417179 n/a Gaussian t n/a 0,74092307 197,102384 Gaussian Clayton 1,28258775 n/a n/a Gaussian Frank 5,64416896 n/a n/a Gaussian Gumbel 1,98689539 n/a n/a t Gaussian n/a 0,71507989 n/a t t n/a 0,7020926 11,2631201 t Clayton 1,33267966 n/a n/a t Frank 5,0423586 n/a n/a t Gumbel 1,91872451 n/a n/a

Table 6.10: Parameters Φcopula of the basket with underlyings Bayer/BASF (weekly monitoring) CHAPTER 6. NUMERICAL EXPERIMENTS 47

Margin Copula α ρ ν Gaussian Gaussian n/a 0,70121631 n/a Gaussian t n/a 0,726116 6,3437519 Gaussian Clayton 1,46517326 n/a n/a Gaussian Frank 6,73442538 n/a n/a Gaussian Gumbel 1,90956712 n/a n/a t Gaussian n/a 0,69782645 n/a t t n/a 0,7073016 4,15061453 t Clayton 1,60322313 n/a n/a t Frank 5,81384235 n/a n/a t Gumbel 1,87853714 n/a n/a

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

Especially the prices for Averaging Options (Average Spread / Average Strike /Double Average Rate / Asian), which do have an averaging feature in their payoﬀ diﬀer up to 8.54 % 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 differ significantly 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 pricing basket options. Firstly the model choice is relevant (copulas or multivariate normality assumptions) as input choice in a multivariate framework. As a consequence, input choice issues 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 approach with normal marginals produced fair values which were quite distant from using copulas with Student’s t-marginals, since the marginal return distribution is allowed to be fat-tailed. Therefore, it should be questioned whether it is reasonable to use copulas in order to model the dependence structure among assets while maintaining the much simpler assumptions on marginal return distributions. The simulation with normal marginal returns seems from a theoretical point of view to miss some of the opportunities for greater flexibility that the use of copulas gives. Using copulas is clearly different from a simple change in correlation parameters, since it implies a completely different mechanics of price movements. Due to the flexibility when choosing the marginal distribution it enables to more properly model the marginal distributions. Even the use of non-parametric marginals and of empirical distribution function can be considered. This avoids any assumptions about the distribution of the margins, but requires sufficient data since every non-parametric method performs much better when data are not scarce.

For the calculation of the results no performance enhancing techniques like variance reduc- CHAPTER 6. NUMERICAL EXPERIMENTS 48 tion were used. Those techniques are well studied for the standard Monte Carlo approach assuming normal distributed returns. Those techniques cannot be applied without amendment to a Monte Carlo simulation with copulas. One reason is the asymmetric distribution which copulas produce. There are a few papers which developed similar techniques for commonly used copulas (see Kang and Shahabuddin [2005]). Methods for importance sampling for example in connection with normally distributed random variables do not carry over to the case of other copula. Successful application of importance sampling in heavy-tailed settings is notoriously delicate. References for this are Asmussen and Bin- swanger [1997] or Binswanger [1997].

Margin Copula Average Spread Average Strike Lookback Put Lookback Call Double Average Rate Asian Gaussian Gaussian 3,43 3,75 10,36 11,94 3,18 3,54 Gaussian t 3,32 3,66 10,34 11,85 3,16 3,54 Gaussian Clayton 3,63 3,96 11,32 12,94 3,48 3,83 Gaussian Frank 3,13 3,48 11,64 13,40 3,59 3,94 Gaussian 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 3,60 t t 3,39 3,71 10,24 11,77 3,15 3,54 t Clayton 3,68 4,02 11,43 13,08 3,55 3,90 t Frank 3,78 4,14 11,36 13,02 3,52 3,87 t 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 monitoring) CHAPTER 6. NUMERICAL EXPERIMENTS 49

Margin Copula Average Spread Average Strike Lookback Put Lookback Call Double Average Rate Asian 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 approach for underlyings Bayer/BASF in Euro (daily monitoring)

Margin Copula Average Spread Average Strike Lookback Put Lookback Call Double Average Rate Asian Gaussian Gaussian 37,11 29,49 49,02 63,25 14,77 18,81 Gaussian t 34,58 26,69 46,91 59,24 13,85 17,99 Gaussian Clayton 37,41 29,47 49,78 64,31 15,07 19,17 Gaussian Frank 37,19 29,13 50,33 64,96 15,21 19,38 Gaussian Gumbel 37,48 29,70 49,44 63,88 14,94 19,00 t Gaussian 33,33 25,83 43,24 53,51 13,29 17,12 t t 36,14 28,20 46,04 56,82 14,47 18,52 t Clayton 38,37 30,57 49,15 62,35 15,95 19,63 t Frank 38,51 30,50 48,94 62,29 15,72 19,61 t Gumbel 39,05 30,52 49,22 62,66 15,79 19,92

Table 6.14: Prices of basket options for underlyings BMW/VW in Euro (daily monitoring) CHAPTER 6. NUMERICAL EXPERIMENTS 50

Margin Copula Average Spread Average Strike Lookback Put Lookback Call Double Average Rate Asian 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 approach for underlyings BMW/VW in Euro (daily monitoring)

Margin Copula Average Spread Average Strike Lookback Put Lookback Call Double Average Rate Asian Gaussian Gaussian 47,42 39,45 57,57 77,63 19,77 23,49 Gaussian t 47,46 39,47 57,62 77,69 19,79 23,52 Gaussian clayton 47,60 39,63 58,29 78,70 20,05 23,78 Gaussian frank 47,60 39,63 58,29 78,70 20,05 23,78 Gaussian 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 24,74 t t 51,01 42,11 57,88 74,38 21,00 24,77 t clayton 51,48 42,54 58,45 74,65 21,47 24,90 t frank 51,37 42,49 58,51 74,72 21,51 24,90 t 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 monitoring) CHAPTER 6. NUMERICAL EXPERIMENTS 51

Margin Copula Average Spread Average Strike Lookback Put Lookback Call Double Average Rate Asian 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 approach for underlyings BMW/VW in Euro (weekly monitoring)

Margin Copula Average Spread Average Strike Lookback Put Lookback Call Double Average Rate Asian Gaussian Gaussian 3,19 3,48 9,31 10,52 3,04 3,39 Gaussian t 3,18 3,48 9,31 10,52 3,04 3,40 Gaussian Clayton 3,76 4,07 10,29 11,55 3,37 3,73 Gaussian Frank 3,42 3,74 10,54 11,87 3,44 3,80 Gaussian Gumbel 3,23 3,56 10,61 11,96 3,48 3,84 t Gaussian 3,59 3,88 9,69 11,03 3,22 3,56 t t 3,39 3,70 9,54 10,79 3,15 3,50 t Clayton 3,70 4,01 10,22 11,47 3,37 3,71 t Frank 3,67 3,99 10,27 11,54 3,36 3,71 t Gumbel 3,31 3,63 10,45 11,75 3,44 3,79

Table 6.18: Prices of basket options for underlyings Bayer/BASF in Euro (daily monitoring) CHAPTER 6. NUMERICAL EXPERIMENTS 52

Margin Copula Average Spread Average Strike Lookback Put Lookback Call Double Average Rate Asian 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 approach for underlyings Bayer/BASF in Euro (weekly monitoring) Chapter 7

Conclusions

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

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

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

Furthermore the copula approach is very flexible and can be extended in numerous ways because it is possible to construct the right amount of tail dependence by using linear combination of copulas (see Wei and Hu [2002]). Hence the question arises which copula is the ’right’ one to choose. The property of the Archimedean copulas is that each two variables have the same degree of dependence. This leads to less flexibility when modeling more than two assets. Therefore, Archimedean copulas are not well suited to model the dependence in more than two dimensions due to their restrictive characteristics, whereas elliptical t-copulas should provide a better fit. Other copula families could be considered such as hierarchical Archimedean copulas, which are less restrictive than the simple Archimedean 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 applying measures of goodness of ﬁt a measure for the choice of a copula itself and its parameters may be introduced. This implies that depending on the input data a copula might change over 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 the performance of the model.

Despite the mentioned advantages one should not ignore the disadvantages of a copula approach. 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 which have a significant impact on the results. This makes the model itself to a factor which has to be agreed upon between the involved groups (traders, risk managers). Another problem about the concept is the practical implementation. The problem is that the estimation of copulas 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 distributions via MLE an issue arises when doing Monte Carlo simulation without the assumption of normally distributed returns. Without this assumption many of the standard variance reduction techniques cannot be used to enhance the performance of the Monte Carlo simulation. This makes the already computationally intense simulation method even more demanding, 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 draws of copulas have to be developed. This would certainly help to spread the use of copula method in practice.

Clearly, the observed results have to be taken as an academic example, and direct application of the model would call for speciﬁcation of the risk-premia for example. However, the comparison highlighted the power of copula functions to eﬀectively separate information about the marginal distributions and the dependence structure among the assets and can be 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

ν+1 ν+1 Γ( ) 1+ x2 − 2 g (x)= 2 (A.1) ν Γ( ν )Γ( 1 )√ν ν 2 2 The generalized Student’s t-distribution is given by

ν+1 ν+1 1 x µ Γ( ) (x µ)2 − 2 f (x; µ, σ)= g ( − )= 2 1+ − (A.2) ν σ ν σ Γ( ν )Γ( 1 )√νσ νσ2 2 2 By construction µ is the mean of the distribution. However, σ2 does not represent the variance of fν. The link to the variance v of the distribution is given by the following equation: ν v = σ (A.3) ν 2 r − The CDF Fν(x) of the generalized Student’s t-distribution with ν degrees of freedom is given by ν+1 Γ( ) ν+1 2 2 2 2 2 Fν(x)= ν 1 (νσ ) I(x; 2µ,νσ + µ ,ν) (A.4) Γ( 2 )Γ( 2 )√νσ − where I is deﬁned recursively

x ′ ′ ′ I(x; a, b, ν)= ((x )2 + ax + b)dx (A.5) Z−∞

55 Appendix B

Maximum Likelihood Method

The Maximum likelihood estimation (MLE) is a statistical method used for fitting a mathematical model to some data. The modeling of real world data using estimation by maximum likelihood enables to tune free parameters of a model to provide a good fit. For a fixed set of data r1,...,rn and underlying probability density function f with parameters 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 write f in the form f(p1,...,pn; r1,...,rn). Let rj = (r1,j,...,rn,j), j = 1,...,N be a set of n data points. The task is now to estimate the parameters p1,...,pk given the observed data points. Therefore, the probability described by f is maximized. This is done via the maximum likelihood function L:

L(r1,...,rn; p1,...,pk)= f(rj,p1,...,pk) (B.1)

The maximum likelihood methods assumes thatY the k-tupel (ˆp1,..., pˆk) at which L has a maximum 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

= ln f(rj,p1,...,pk)) (B.3) j=1 X Since the logarithm is a strictly monotone increasing function the maximum doesn’t change by this transformation. To find the maximum of l analytical methods can be applied. Therefore, one has to find a vectorp ˆ1,..., pˆk such that ∂l (ˆp1,...,pˆn) = 0 for i = 1,...,k (B.4) ∂pi | 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 Bibliography

S. Asmussen and K. Binswanger, 1997, Simulation of ruin probabilities for subexponential claims, ASTIN Bulletin 27, no. 2, pages 297-318.

P. Acworth, M. Broadie, and P. Glasserman, 1998, A comparison of some Monte Carlo and quasi-Monte Carlo methods for option pricing, in Monte Carlo and Quasi-Monte Carlo Methods 1996: Proceedings of a conference at the University of Salzburg, Austria, July 9-12, 1996, ed. H. Niederreiter, P. Hellekalek, G. Larcher and P. Zinterhof, Lecture Notes in Statistics 127, Springer-Verlag, New York.

K. Binswanger, 1997, Rare events and insurance, Ph.D. thesis, Swiss Federal Institute of Technology, Zurich.

U. Cherubini, E. Luciano, W. Vecchiato, 2004, Copula Methods in Finance, John Wiley & Sons, Ltd.

N. Cufaro-Petroni, 1996, Lezioni di Calcolo delle Probabilitua Edizioni dal Sud, Modugno, 1996.

L.O. Dahl, F.E. Benth, 2001. Valuation of the Asian Basket Option with Quasi- Monte Carlo Techniques and Singular Value Decomposition, Pure Mathematics, 5.

J. Dewynne, 2007, American Options, Lecture Notes, Department for Continuing Educa- tion, University of Oxford.

J. Dobri´c,F. Schmid, 2005, Testing Goodness of Fit for Parametric Families of copulas - Application to Financial Data, Communications in Statistics: Simulation and Compu- tation, Vol. 34, pages 1053-1068.

J. Durban, A. Stuart, 1951, Inversions and rank correlations, J. Roy. Statist. Soc., Series B., 2, pages 303-309.

P. Embrechts, F. Lindskog, A. McNeil, 2003, Modelling Dependence with copulas and Applications to Risk Management, in: Rachev, S.T. (ed.), Handbook of Heavy Tailed Distributions in Finance, Amsterdam.

P. Embrechts, A. McNeil, D. Straumann, July 1999, Correlation and Dependence in Risk Management: Properties and Pitfalls, ETH Z¨urich.

G.T. Fechner, (1897), Kollektivmasslehre (G.F. Lipps, Ed.), Leipzig.

57 BIBLIOGRAPHY 58

W. Feller, 1971, An Introduction to Probability Theory and Its Applications, Vol.II John Wiley & Sons, New York.

R. Frey and A. McNeil, 2003, Dependent Defaults in Models of Portfolio Credit Risk, Journal of Risk, 6 (1), pages 59-92.

H. Gennheimer, 2002, Model Risk in Copula Based Default Pricing Models

C. Genest, L.-P. Rivest, 1993, Statistical Inference Procedures for Bivariate Archimedean copulas, Journal of the American Statistical Association, Vol. 88, pages 1034-1043.

P. Glassermann, 2003, Monte Carlo Methods in Financial Engineering: v. 53 (Applications of Mathematics), Springer, Berlin.

T. P. Hutchinson, C. D. Lai, 1990, Continuous Bivariate Distributions, Emphasising Ap- plications. Rumsby Scientiﬁc Publishing, Adelaide.

H. Joe, Multivariate Models and Dependence Concepts,1997, Chapman & Hall, London.

H. Johnson, 1987, Options on the maximum or the minimum of several assets. Journal of Financial and Quantitative Analysis, 22, pages 277-283.

M.G. Kendall,1938, A new measure of rank correlation, Biometrika, 30, pages 81-93.

W. Kang, P. Shahabudding, 2005, Fast simulation for multifactor portfolio credit risk in the t-copula model, Winter Simulation Conference.

C.H. Kimberling, 1994, A probablistic interpretation of complete monotonicity, Aequa- tiones Statist., 4, pages 152-164.

KMV Coproration, 2001, Portfolio Management of Default Risk, available from http://www.moodyskmv.com/research/portfolioCreditRisk_wp.html.

V. A. Kholodnyi, J. F. Price, 1998, Foreign Exchange Option Symmetry, AuthorWorld Scientiﬁc Publishing Company.

E. Lamberton, B. Lapeyre (1992) Hedging Index Options with Few Assets, Mathematical Finance, Vol. 3, No. 1, pages 25-41.

F. Longin, B. Solnik, 2001, Extreme Correlations of International Equity Markets, The Journal of Finance, Vol. 56, pages 649-676.

D. Li, 2001, On default Correlation: A Copula Function Approach, Journal of Fixed in- come, 9l, pages 43-54.

C.H. Ling, 1965, Representation of associative functions, Publ. Math. Debrecen, 12, 189- 212

W. Margrabe, 1987, The value of an option to exchange one asset for another. Journal of Finance, 33, pages 177-186. BIBLIOGRAPHY 59

A. W. Marshall, I. Olkin, 1988, Families of multivariate distributions, Journal of the American Statistical Association, 83, pages 834-841.

, R. Mashal, A. Zeevi, 2002, Beyond Correlation: Extreme Co-movements Between Finan- cial Assets, Columbia University, New York.

, M.R. Melchiori Mario, Tools for Sampling Multivariate Archimedean copulas, Yield- Curve, 2006.

R. Merton, On the pricing of Corporate Debt: The Risk structure of Interest Rates, Journal of Finance,29, pages 449-470.

J.L. Myers, A. D. Well, 2003, Research Design and Statistical Analysis (second edition ed.), Lawrence Erlbaum, page 508.

R. B. Nelsen, 1999 An Introduction to copulas. Springer, New York.

K. Nystrom, J. Skoglund, 2002, Univariate Extreme Value Theory, GARCH and Measures of Risk, Swedbank.

RiskMetrics Group, 1997, CreditMetrics - Technical Document, available from http://www.riskmetrics.com/cmtdovv.html.

J.V. Rosenberg, (2003), Nonparametric Pricing of Multivariate Contingent Claims, Jour- nal of Derivatives, Vol. 10, pages 9-26.

B. Schweizer, A. Sklar, 1961, Associative functions and statistical triangle inequalities, Publ. Math. Debrecen, 8, 168-186.

J. Shao, 1999 Mathematical Statistics, Springer, New York.

W.T. Shaw, 2006, Sampling Student’s T distribution - use of the inverse cumulative distribution function., Journal of Computational Finance, Vol 9 Issue 4, pages 37-73.

A. Sklar, 1959 Fonctions de r´epartition `an dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, pages 229-231.

I. Slavchev, S. Wilkens, 2008 The Valuation of Multivariate Equity Options by Means of copulas: Theory and Application to the European Derivatives Market.

C. Spearman, 1904, The proof and measurement of association between two things, Amer- ican Journal of Psychology, 15, pages 334-342.

X. Sue, 2007, Hedging Basket Options by Using a Subset of Underlying Assets, Publ. Bonn Graduate School of Economics, Bonn.

R.W.J. Van den Goorbergh, C. Genest, B.J.M. Werker, 2005, Bivariate Option Pricing Using Dynamic Copula Models, Insurance: Mathematics and Economics, Vol. 37, pages 101-114. BIBLIOGRAPHY 60

G. Wei, T. Hu, 2002, Supermodular dependence ordering on a class of multivariate copulas, Statistics and Probability Letters 57(4), pages 375-385.

P. Wilmott, 1998 Derivatives, John Wiley & Sons.

U. Wystup, 2002, Making the most out of Multiple Currency Exposure: Protection with Basket Options, The Euromoney Foreign Exchange and Treasury Management Hand- book 2002.