Pricing of a worst of option using a Copula method M AXIME MALGRAT

Transcription

1 Pricing of a worst of option using a Copula method M AXIME MALGRAT Master of Science Thesis Stockholm, Sweden 2013

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3 Pricing of a worst of option using a Copula method MAXIME MALGRAT Degree Project in Mathematical Statistics (30 ECTS credits) Degree Programme in Engineering Physics (300 credits) Royal Institute of Technology year 2013 Supervisor at KTH was Boualem Djehiche Examiner was Boualem Djehiche TRITA-MAT-E 2013:52 ISRN-KTH/MAT/E--13/52--SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE Stockholm, Sweden URL:

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5 Abstract In this thesis, we use a Copula Method in order to price basket options and especially worst of options. The dependence structure of the underlying assets will be modeled using different families of copulas. The copulas parameters are estimated via the Maximum Likelihood Method from a sample of observed daily returns. The Monte Carlo method will be revisited when it comes to generate underlying assets daily returns from the fitted copula. Two baskets are priced: one composed of two correlated assets and one composed of two uncorrelated assets. The obtained prices are then compared with the price obtained using the Pricing Partners software.

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7 Acknowledgments First, I would like to thank all my colleagues at Pricing Partners who accepted to take me as an intern in their valuation team during 6 months in Paris. They were always available to answer my questions and I would like to thank them all for the instructive conversations we had. In particular, I would like to thank Benedetta Bartoli who helps me and guides me in this thesis. I also would like to thank KTH and all teachers with whom I have been in contact and taught me a lot in Financial Mathematics and statistics. I am also grateful to my supervisor and professor Boualem Djehiche for his advices and feedbacks on this thesis. Finally, I would like to thank the Ecole Centrale Marseille without which it would not have been possible to do my internship and write my thesis in France.

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9 Contents Introduction I. The Multidimensional Black Scholes model A. The Black Scholes Theory 1. The basic concepts of the Black Scholes model 2. The Black-Scholes PDE and the Black Scholes formula B. The multidimensional case C. The Monte Carlo Method 1. The theoretical principles 2. Applications to pricing of options D. Worst of options II. Bivariate and Multivariate Copulas A. Definitions and Properties B. Sklar s Theorem C. Copulas Families 1. Elliptical Copulas 2. Archimedean Copulas D. Kendall s Tau and Spearman s Rho 1. Kendall s Tau 2. Spearman s Rho

10 III. Modeling dependence with Copulas A. Estimation and Calibration from Market Data 1. Exact Maximum Likelihood method 2. IFM Method 3. The CML Method B. Simulation Methods for Copulas 1. Simulation methods for Elliptical Copulas 2. Simulation Methods for Archimedean Copulas C. Monte Carlo Simulations with Copulas 1. Simulation risk free returns 2. Choice of the marginals 3. Monte Carlo Method with Copulas IV. Numerical Results A. Market data B. Numerical Results 1. Estimated marginal distribution parameters 2. Estimated copulas parameters 3. Pricing Results Conclusion Appendix A: R Codes

11 Introduction Since the last decade, basket options have been often used in the banking industry and are viewed as excellent hedging multi asset contingent claims. The principal reason for using basket options is that they are cheaper than the corresponding portfolio composed of vanilla options on the individual assets. The underlying assets of basket options can be multiple: equities, indexes, currencies, commodities, credit spread, etc Generally, the basket option depends on the performance of its underlying assets. A large variety of basket options can be found on the market like Asian Basket options, worst of / best of options, Credit spread basket (CDO, CDS) and others multi assets options with complicated payoff. The valuation of basket options is generally a difficult task because the dependence structure of the underlying assets can in certain cases be very complex because of the large choice of underlying assets that one can find. In the pricing of the option under the Black Scholes model, we use a multivariate Geometric Brownian motion with a linear correlation. This last parameter represents the dependence structure between the underlying assets. Another way to model dependence structure is the use of Copulas. Indeed, Copulas are often used to model dependence structure between random variables. Copulas can be seen as a multivariate probability distribution with marginal distribution uniformly distributed. In finance, Copulas are used in order to price Credit Spread baskets because using Copulas is an acceptable method for the modeling of the joint distribution between default times. Therefore, Copulas are often used to model the dependence structure of the future default events. This method is used in Pricing Partners software in order to price CDO (Credit Debt Obligation), FTD (First-To-Default) and NTD (N-To-Default) options which are credit derivatives baskets. In fact, using Copulas is a way to correlate the systematic risk to the idiosyncratic risk (risk that is specific to an asset or a small group of assets). Then, it is easy to simulate the times of default and price the credit derivatives via a Monte Carlo method. The framework of this thesis is to expose an alternative method to the classical Black Scholes method using a Monte Carlo simulation. The task here is to adapt the Copula concept to the pricing of a worst of option via a revisited Monte Carlo method. In the first chapter, we will recall the basics of the Black Scholes theory and the pricing of a multi asset product. Especially, we will deal with the multidimensional Black Scholes model and the Monte Carlo method. Then, in the second part, we will present the definitions and the properties of Copulas. We will see that there exist several families of Copulas and that they have different properties and dependence structures. In the third chapter, we will show how to fit a copula. The objective will be to present different methods to show how to estimate Copulas parameters and how to simulate the random returns from the fitted Copula by using the Monte Carlo Method. The last chapter will be dedicated to the numerical results. We will compare the prices obtained by using the Copula method with the prices obtained by using PricingPartners Software. We will analyze these results and present the advantages and drawbacks of the Copula Method.

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13 Chapter 1 The Multidimensional Black Scholes Model In 1997, Robert Merton and Myron Scholes received the Nobel Price for the quality and the importance of their research related to the valuation of European Options. In fact, during the 70 s, they developed the so called Black Scholes model which has been seen as a revolution in the Banking Industry in terms of valuation and hedging of derivatives. In this chapter, we will first present the concept of the Merton Black and Scholes formula, then we will give the Black-Scholes formula and finally we will introduce the Monte Carlo Method and present a Worst of option. A. The Black Scholes Theory 1. The basic concepts of the Black Scholes Model The model proposed by Black and Scholes in order to describe the dynamics of the stocks is a continuous model with a risky asset (stocks for example) and a risk free asset (a bond for example). We suppose the dynamic of the risk free asset by the following differential equation: where r is the constant interest rate. The solution of this equation is given by. We suppose that the dynamics of the risky asset follows the differential equation: (1.1) where µ and σ are constants and is a standard Brownian Motion under the probability P. The dynamic developed by in equation (1.1) is called a Geometric Brownian Motion. If we say that is the stock price at, the solution of equation (1.1) at time t is given by: (1.2) According to the Girsanov Theorem (see Björk[2009]), there exists a probability Q, equivalent to P, under which the actualized price is a martingale. Under Q, is a Brownian Motion. Q is called the risk-neutral probability.

14 Consequently, we can rewrite the equations (1.1) and (1.2) under the risk neutral world as follow: and (1.3) For the rest of this thesis, we suppose that we are under the risk neutral probability and we will note the Brownian motion under this probability as Merton, Black and Scholes developed this model in order to price financial derivatives and especially European options. For example, a European Call option gives the right, at the time of maturity T, to its holder to buy one share of the underlying stock at the strike price K from the issuer of the option. Mathematically, the contract function or payoff of such an option can be written as Merton, Black and Scholes said that the arbitrage free price of any contingent claim is given at time t by: where Q is the risk neutral probability and the follows the dynamic derived in equation (1.3). We will show in the next part where this result comes from. 2. The Black-Scholes PDE and the Black Scholes formula The Black Scholes differential equation In order to obtain the Black Scholes PDE, we need to precise the hypotheses that are made (Hull [2012]): The stock price follows the stochastic process shown in part 1 (equation 1.3), µ and σ are constant, The short selling of securities with full use of proceeds is authorized, There are no transactions costs or taxes, There are no riskless arbitrage opportunities, Security trading is continuous, The risk-free rate of interest r is constant for all the life of the option. The following approach considered by Merton, Black and Scholes leads to the Black Scholes differential equation. The no arbitrage theory is saying that a riskless portfolio is consisting of a position in the derivatives and a position in the stock. Moreover the return from such a portfolio must be the risk free interest rate r otherwise the portfolio s investors will make an arbitrage. The reason that it is possible to build such a riskless portfolio comes from the fact the underlying stock and the derivatives are both exposed to the same random source: the price of the underlying stock. Consequently, on a short period, the derivative is perfectly correlated with the price of the underlying stock which means that, on a short period, the portfolio s value is always known.

15 We recall that the dynamic of the underlying stocks is the dynamic in equation (1.1) Set f the price of a derivative, for example a call option (with underlying S). We will apply the Itô s formula to f. I will first recall the Itô Lemma (Björk, [2009]). Definition 1.1: Itô s Formula: Assume that the process S has the stochastic differential given by Let define the process Z by with f a function. Then Z has a stochastic differential given by: { } (1.4) As we have seen before, we have to define a risk free portfolio composed of one stock and one of its derivatives in order to eliminate the random component. This portfolio can be defined by: Short in one unit of the derivative, Long in stocks. The value of this portfolio is given by: Then, the variation of our portfolio is given by: According to the dynamic of S (see equation (1.1)) and the Itô s formula applied to f (see equation (1.4), we have: (1.5) The return of such a portfolio should be equal to the risk free interest rate. Then, we can write: (1.6) From equations (1.5) and (1.6), we have: Hence, (1.7) The equation (1.7) is the Black-Scholes differential equation!

16 As an example, we have for an European Call option f = max(s K ; 0) and for an European Put option f = max(k S ; 0). To solve the Black - Scholes PDE, we are using the Feymman Kac formula (Björk, [2009]) Feymman-Kac Formula: Assume that F is a solution to the following problem F(T,x) = f(x) If X follows the dynamic with Then F admits the representation In the case of the Black-Scholes PDE, we have and Under the risk neutral probability, we can solve the Black-Scholes differential equation by using the Feymman-Kac formula for the claim in order to get the price of the derivative. The price at time t of a derivative, which is the solution of the Black-Scholes PDE, with payoff where T is the maturity of the derivative and S the underlying stock with dynamic described in equation (1.3) is given by: (1.8) The Black Scholes formula From equation (1.8), we can price any derivatives with payoff and, in particular, the famous Black-Scholes formula which gives the price of any European call/put options. We recall that for a call/put option with strike K and maturity T we have: where the dynamic of is given by equation (1.3): Then the price of a call/put option at time t is given by:

17 After deriving the expectation, we get the famous Black-Scholes formula for a call/put option: where and are defined by : ( ) ( ) (1.9) The function N(x) is the standard normal cumulative distribution function. Figure 1.1: The grey zone represents N(x) To finish this part, we will just mention a useful formula: the put-call parity. This formula is given by: B. The multidimensional case In order to price a basket option, we need to define a multidimensional Black Scholes model. As we have seen in the previous part the option price is the discounted risk neutral expectation of its payoff. We define n risky assets (stocks for example) and one risk free asset defined like in part A.1. The dynamics of the risky assets with t in [0,T] satisfies the stochastic differential equations: (1.10) where the are constant volatilities and the are the drifts which are equals to under the risk neutral probability (where is the yearly dividend yield of asset ). Moreover, and are correlated increments of a Wiener process and denotes the correlation of the normally distributed assets of the basket.

18 If one supposes motion given by: independent of time, the solution of equation (1.10) is still a Geometric Brownian { } (1.11) As in the one dimentional case (see equation (1.7)), there exists also a Black Scholes multiditional differential equation verified by the price of a derrivative. We will not go again throught all the calculus and we will just give the formula. Therefore, for n correlated assets, this equation is given by: (1.12) We call S the vector of the n underlying assets and we write The claim of a basket depends on S and on the maturity of the derivative and it could be writen as. Then, according to Dahl and Benth [2001] and without going through all details, the calculation of the price of a basket can be formulated as an integral on all the possible parths Ω : (1.13) where is the probability density of. In order to compute equation (1.13), a Monte Carlo method will be used. In the next part, the principles and applications to basket option of this method are mentioned. C. The Monte Carlo Method As we have seen in the previous part, the price of a basket is given by the discounted risk-neutral expectation of its payoff. Hence, the price can be estimated by a Monte Carlo method which consists of simulating the paths of the underlying assets and taking the discounted mean of the simulated payoffs. In this part, I will briefly recall the principles of the method and then show the application to the pricing of a basket option. 1. The theoretical principles The Monte Carlo method comes from the fundamental central limit theorem. Roughly, the central limit theorem states that the distribution of the sum of a large number of independent and identically distributed variables will be approximately normal, regardless of the underlying distribution.

19 Namely, suppose that are n independent identically distributed random variables with mean µ and standard deviation σ. Then, the empirical average has the following property (Law Of large Number): ( ) This leads us to the following property: The Monte Carlo method is based on this property and we will see in the next part how to use it. 2. Applications to pricing of options As mentioned before, the valuation of an option is based on the risk neutral expectation of its payoff (see equation (1.13)). The Monte Carlo method is used to estimate this expectation. Suppose that the risk free interest rate and the volatility are constant and that we have a payoff which depends on. The steps followed by a Monte Carlo method are presented below: Simulate L trajectories for under the risk neutral world, Compute the payoff of the option for each simulation, Compute the mean of simulated payoffs in order to obtain an estimate of the risk neutral expectation, Discount the estimated expectation with the risk free interest rate r. Recall from equation (1.10) that for an asset we have: { } with { }. We call the time step We know that. Then we can rewrite equation (1.10) in discrete time as follow: { )} (1.14) where is a correlated standard normal distributed random variable which takes into account the correlation between the underlying assets. This equation describes the dynamic of each asset in the model.

20 Then, the price of the basket option given by equation (1.13) can be estimated by a Monte Carlo simulation. Hence, (1.15) where is the payoff computed with the simulation and L the number of Monte Carlo simulations. From the Law of Large Numbers seen in the previous part, estimates the option price when L becomes large. The advantage of a Monte Carlo method is that it can be used as well with path dependent options as when the payoff depends only on the value of the underlying assets at the maturity of the option. In the banking world, Monte Carlo method is manly used in order to price complex derivatives. American Monte Carlo method is also often used. This method is a sort of backward Monte Carlo method. Also, the price of a derivative can also be calculated directly by solving the Black-Scholes differential equation (equation (1.12) by finite differences method. The chosen method depends principally on the structure of the product. In this thesis, we will use the classical Monte Carlo method in order to price a Worst Of option. D. Worst Of options There are a lot of multivariate options available on the market like Asian Options, Average spread Options, LoockBack Options, Rainbow options and Best/Worst of Options. Basket options are useful when it comes to hedge a portfolio consisting of several assets. One important advantage of using basket options is that the price of a basket is cheaper than the sum of the prices of the options on only one of the underlying assets. The underlying assets of a basket option are multiple: stocks, commodities, credit spread, indices, currencies and sometimes interest rates. In this thesis, we concentrate only on Worst of Options on stocks. As these options are manly traded on the OTC (Over The Counter) Market, prices are not directly available. A worst of option is also called a Call option on the worse performer (Call-on-min). If we consider 2 stocks traded on the market. Then, the payoff of a Worst of option on these 2 stocks, with maturity T and exercise price K (Strike price) is given by: { ( ) } (1.16) where is the price of the asset at the start date (or strike date) of the option. Hence, the strike K is generally around 1. In this thesis, we will focus on two worst of options with a maturity of 3 year. The main difference between the two options will be the correlation between the underlying assets. The first option will be composed of two correlated assets from two French Banks (Société Générale, BNP Paribas). The assumption that these two stocks are correlated is not so bad because they belong to the same business

21 area (bank industry). If we compute the correlation of the daily price of these two stocks over the last year we get the following matrix: BNP SocGen BNP 1 0, SocGen 0, Then, the second option will be composed of two uncorrelated stocks. The two stocks belong to two different business areas: BNP Paribas (Bank industry) and LVMH (Lux industry). The correlation over the last year is given by: BNP LVMH BNP 1 0, LVMH 0, The price of these options will be estimated by using the Monte Carlo method presented before. We will simulate the daily returns over three years using Copulas to model dependence of the two assets and use formulas (1.14) and (1.15) to estimate the option price.

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23 Chapter 2 Bivariate and Multivariate Copulas In this chapter, we will deal with the concept of Copula. We will mention the main properties and interpretations of Copulas. We will voluntary omit the proof of the theorems and properties. More details can be found in Cherubini s book, Copula Methods for Finance [2004]. An easy way to understand Copulas was given by Embrechts and Lindskog [2001]: A copula is a multivariate distribution function defined on the unit cube with uniformly distributed marginal. In this chapter we will first define a copula and then state the Sklar s Theorem. Then we will deal with the two different categories of Copulas: elliptical Copulas and Archimedean Copulas. Finally, we will present the different measures of dependence with copulas such that Kendall s and Spearman s. A. Definitions and Properties Bivariate Gaussian with correl Bivariate Gaussian with correl Before starting with Copulas and in order to better understand the concepts, I would like first to introduce some notions on multivariate distributions. We recall that if is a random variable with distribution function, then is uniformly has the distribution distributed on [0,1]. In the same way, if is uniformly distributed, then function. Then, if ( ) is a multivariate random variable with distribution function, the random vector ( ( ) is a multivariate ( ) ( ) model with known univariate distributions. One can see below the draws of a samples from a bivariate standard normal distribution with linear correlation 0.5 and 0.9 and from a bivariate standard Student s t distribution with 1 and 3 degrees of freedom with standard normal margin distributions Bivariate Student's t dist with df Bivariate Student's t dist with df Figure 2.1: Samples of size from two bivariate distributions 4

24 Now we have that in mind, we can have a first definition of a copula. If we set a multivariate random variable uniformly distributed on [0,1], we can write like: ( ) The vector receives the dependence of the vector. Then, the distribution function of is called a copula and we have: (2.1) Definition 2.1: Copula A n-dimensional Copula is a function C: with the following properties: 1) For every in : 2) For every in Property 2.2 Let C be a multivariate Copula. For every in and for every i in { }, the partial derivative exists for all and we have: Definition 2.3: Density Copula For every in, we define the density copula of the copula by: (2.2) To obtain the density of the n-dimensional distribution F, we use the following relationship: (2.3) Where is the density of the marginal distribution. Therefore, the copula density is equal to the ratio of the joint density f and the product of all marginal densities: (2.4)

25 B. Sklar s Theorem Sklar s theorem is an important theorem in the Copula theory because it provides a way to analyze the dependence structure of multivariate distributions without studying marginal distributions. The Sklar s theorem for multivariate Copulas defined as below (Cherubini, [2004]). Let be given marginal distribution functions from a joint distribution function Then there exists, for every a Copula with ( ) (2.5) If are continuous then C is unique. On the other hand, if C is a Copula and are distribution functions, then the distribution function defined above is a joint distribution function with marginals In practice, using the fact that where is uniformly distributed, we write a Copula as defined below. For every in, ( ) (2.6) Sklar s theorem guarantees that the cumulative joint probability can be written as a function of the cumulative marginal ones and vice versa. We can say that multidimensional Copulas are dependence functions. C. Copulas Families There exist two families of Copulas: the elliptical Copulas and the Archimedean Copula. The Elliptical Copulas is the family the more used in Finance because there are simple to calibrate. 1. Elliptical Copulas Gaussian Copula and Student s t Copula come both from the elliptical Copula family. Elliptical Copulas are simply the Copulas from elliptical distributions. The advantage of elliptical Copulas is that the parameters such as correlation can be easily fitted from market data. a) The Gaussian Copula The multivariate Gaussian Copula was described in Cherubini [2004]. It is defined as follows.

26 Definition 2.4: Multivariate Gaussian Copula Let be the standardized multivariate normal distribution with correlation matrix such as is a n-dimensional, symmetric, positive definite matrix with. Then, the Multivariate Gaussian Copula is defined as follows: ( ) (2.7) where is the inverse of the standard normal distribution. From Sklar s theorem, we can note that the Gaussian Copula generates the standard joint normal distribution function iff the marginals are standard normal. In fact, one can write, ( ) From the definition of the Gaussian Copula, we can easily determine (see Cherubini [2004]), the density of the corresponding copula. Definition 2.5: Density Copula Let define ( ). Then the density of a multivariate Gaussian Copula is given by: ( ) (2.8) where is the unite matrix composed only with 1 s in the diagonal. The Gaussian Copulas are usually used to model linear correlation dependencies. In the figures below, one can find random draws from a bivariate Copula and a 3 dimensional Copula with correlation parameter 0.6 and Bivariate Gaussian Copula with correlation 0.6 Figure 2.2: Random draws from a bivariate Gaussian Copula Bivariate Gaussian Copula with correlation 0.9

27 dim Gaussian Copula with correlation dim Gaussian Copula with correlation 0.9 Figure 2.3: Random draws from a 3 dimensional Gaussian Copula b) The Student t Copula Definition 2.5: Multivariate Student s t Copula Let be the standardized multivariate Student s t distribution with correlation matrix and degrees of freedom, ie Then, the multivariate Student s t Copula is defined as follows: ( ) ) (2.9) where is the inverse of the univariate distribution function of the Student s t distribution with degrees of freedom.

28 Definition 2.6: Multivariate Student s t copula density The density of a multivariate Student s Copula is given as follows: ( ) ( ) ( ( ) ) ( (2.10) ) ( ) where When the degrees of freedom of the Student s t Copula are going to infinity, the Student s t copula converges to the Gaussian Copula. We can say that for a large value the Student s t Copula approximates the Gaussian Copula. For small values of, the tail mass increases. The following draws represent random draws from a bivariate and a 3 dimensional Student s t Copula with 1 and 3 degrees of freedom with correlation Bivariate Student's t Copula with 1 degree of freedom Bivariate Student's t Copula with 3 degree of freedom Figure 2.4 Random draws from a bivariate Student s t Copula dim Student s t Copula with 1 degree of freedom dim Student s t Copula with 3 degree of freedom Figure 2.5 Random draws from a 3 dimensional Student s t Copula 1.0

29 2. Archimedean Copulas The difference between Elliptical and Archimedean copula is in the fact that Archimedean Copulas are not derived from multivariate distribution functions using Sklar s theorem. From a practical point of view, Archimedean Copulas are useful because it is possible to generate a number of copulas from interpolating between certain copulas. There exit a large variety of Archimedean Copulas like Clayton Copula, Gumble Copula and Frank Copula. In this part, we will first give a general definition of an Archimedean Copula by using a function called a generator. Then, we chose to present the Clayton, Gumbel and Frank Copula. If one wants to get more information about this family of copula, one can have a look to Genest and MacKay [1986], Nelsen[1999] and Joe [1997]. This last reference is also a reputed paper for multivariate Archimedean Copulas. a) General Definitions In order to construct Archimedean Copulas, we first have to define what is a generator. Definition 2.7: Generator Let : I=[0,1] be a continuous, decreasing, and convex function such that (1) =0. Such a function is called a generator. If (0) = +, then is called a strict generator. For every u in, we defined the pseudo inverse of by: { Then for every u in I, we can write ( ) (2.11) One generator often used to construct Archimedean Copulas is the inverses of the Laplace transforms. From the definition of a generator, we can now construct an Archimedean Copula from Kimberling Theorem. Theorem 2.8: Kimberling Theorem Let be a generator (see definition 2.7). Let C be the function defined from by: ( ) Then, C is a bivariate Copula if and only if is convex. (2.12) The following properties will allow us to define multivariate Archimedian Copula. Property 2.9 (Embrechts and Lindskog [2001]): Symmetry and association Let C be an Archimedean Copula with generator. Then for all u,v and w in : 1. C is symmetric, ie 2. C is associative, ie ( ) ( )

30 Definition 2.10: Archimedean Copula density Let C be a Copula defined by Kimberling theorem. Then, by for all u and v in given:, the density of C is ( ) [ ( )] (2.13) Remark: We can extend Kimberking Theorem to the n-dimensional case (see Embrechts and Lindskog [2001]). If we take a generator as defined in definition 2.7, then there exists an n dimensional Copula if and only if is convex such that: ( ) (2.14) The Archimedean Copulas have an important disadvantage compared to Gaussian Copula. In fact they have very limited dependence structure because all the marginals are identical in the view of Kimberling theorem. The marginals are random variables. Nevertheless they are flexible enough to capture various dependence structures which makes them suitable for modeling extreme events (see Nelsen [1999]).They are used to model a strong dependence in the tail. b) Clayton Copula The generator of the Clayton Copula is given by: with and also (2.15) (2.16) Definition 2.11: Clayton Copula Let be a vector in Then, the multivariate Clayton Copula is defined by: ( ) (2.17) In the following figures, one can see that Clayton Copula is an asymmetric Archimedean copula modeling better dependence in the negative tail than in the positive tail of distribution function. The density of a bivariate Clayton Copula is given by: (2.18)

31 Figure 2.6 shows a sample of random draws from a bivariate Clayton Copula with and Figure 2.7 shows a sample of random draws from a 3 dimensional Clayton Copula with and Bivariate Clayton Copula with alpha = Bivariate Clayton Copula with alpha = Figure 2.6 Random draws from a bivariate Clayton Copula dim Clayton Copula with alpha = dim Clayton Copula with alpha = 10 Figure 2.6 Random draws from a 3 dimensional Clayton Copula c) Gumbel Copula The generator of the Gumbel Copula is given by: with ( and also ) (2.19) (2.20)

32 Definition 2.12: Gumbel Copula Let be a vector in and. Then, the multivariate Gumbel Copula is defined by: {( ) } (2.21) If, then the previous equation can be written as follows: (2.22) Figure 2.8 shows a sample of random draws from a bivariate Gumbel Copula with and Figure 2.9 shows a sample of random draws from a 3 dimensional Gumbel Copula with and Bivariate Gumbel Copula with alpha = 2 Bivariate Gumbel Copula with alpha = 10 Figure 2.8 Random draws from a bivariate Gumbel Copula dim Gumbel Copula with alpha = 2 3 dim Gumbel Copula with alpha = 10 Figure 2.9 Random draws from a 3 dimensional Gumbel Copula

33 As we can see on the graphs, Gumbel Copulas is an asymmetric Archimedean Copula and it is modeling better dependence in the positive tail. Then, these kinds of Copulas can be used to model extreme scenarios. d) Frank Copula The generator of the Gumbel Copula is given by: (2.23) with and also ( ) (2.24) Definition 2.12: Frank Copula Let be a vector in and. Then, the multivariate Frank Copula is defined by: { } (2.25) From this definition, we can give the bivariate Frank Copula density as follows: (2.26) Figure 2.10 shows a sample of random draws from a bivariate Frank Copula with and Figure 2.11 shows a sample of random draws from a 3 dimensional Frank Copula with and Bivariate Frank Copula with alpha = 2 Bivariate Frank Copula with alpha = 10 Figure 2.10 Random draws from a bivariate Frank Copula

34 dim Frank Copula with alpha = 2 3 dim Frank Copula with alpha = 10 Figure 2.11 Random draws from a 3 dimensional Frank Copula D. Kendall s Tau and Spearman s Rho Kendall s Tau and Spearman s Rho are two parameters that measure dependence. They can be seen as an alternative to the linear correlation coefficient. Before dealing with Kendall s Tau and Spearman s Rho, we would like to recall the definition of concordance and linear correlation. The concept of concordance is defined in Cherubini [2004] and it is said that concordance occurs when probability of having large or small values of two random variables X and Y is high, while the probability of having large (or small ) X together with small (or large ) Y is low. For two random variables X and Y, the linear correlation is defined as follows: (2.27) With these definitions, we can now define the concepts of Kendall s Tau and Spearman s Rho

35 1. Kendall s Tau Definition 2.13: Kendall s Tau Let X and Y be random variables with Copula C and let and denote the quantile functions and u and v the quantiles defined on. Then Kendall s Tau with Copula C is defined as (2.28) One can show that measures the difference between the probability of concordance and the one of discordance for two independent random variables and. Then one can write: ( ) ( ) Lindskog [2012] gives a practical definition of Kendall s Tau for elliptical distribution. Proposition 2.14: Kendall s Tau Let have an elliptical distribution with location parameter and linear correlation ρ. If, then (2.29) It is also possible to define an unbiased estimator of the Kendall s Tau for an n-dimensional sample (Cherubini [2004]): with {} (2.30) From formula (2.29), we can obtain an estimator of the correlation ρ given as follow: ( ) (2.31) For Archimedean Copula, Embrechts and Lindskog [2001] define another way to compute the Kendall s Tau with the generator function. Let X and Y be random variables with an Archimedean copula C generated by. Then, Kendall s Tau of X and Y is given by: (2.32)

36 2. Spearman s Rho Definition 2.15: Spearman s Rho Let X and Y be random variables with Copula C defined on and let and denote the quantile functions and u and v the quantiles. The Spearman s Rho with copula C is defined as: (2.33) One can see that Spearman s Rho is a multiple of the difference between probability of concordance and the probability of discordance for the vectors. Then, if are iid with copula C: ( ) ( ) It is also possible to define an unbiased estimator of the of Spearman s Rho for an n dimensional sample (Cherubini [2004]): where ( )( ) ( ) ( ) (2.34) To conclude this part, we collect in the following table the Kendall s Tau and the Spearman s Rho of the three Archimedean Copulas seen before. Copula Family Kendall s Tau Spearman s Rho Gumbel 1 - Clayton Complicated Frank ( ) Where,

37 Chapter 3 Modeling dependence with Copulas We know now that Copulas are useful to describe the dependence of the marginals of a joint distribution. From chapter 2, we know how to define a Copula and give a definition of the distribution function and the density of a Copula. As we have seen before, Copulas are defined by parameters and parameters of its margin distributions. In this chapter, we will first present methods to estimate and calibrate the Copulas parameters from the market data. Then, we will explain how we can generate random variates from the desired copula with the estimated parameters. Finally, we will explain how to use Monte Carlo simulation with Copulas in order to price an option. A. Estimation and Calibration from Market Data Copulas allow a high flexibility in modeling random variables because one can chose separately the parameters of the marginals and the parameters of the joint distribution. To estimate and calibrate these parameters, one has first to extract the distribution of the marginals and then extract the dependence structure of the joint distribution. These estimations are based on historical returns (say daily returns from a stock for example). But, multi-asset options are generally traded on Over-The-Counter (OTC) markets and the prices of such options are not directly available. Consequently, it is difficult to extract the risk neutral parameters for the copula. Then, one can make the assumption that the risk neutral and real world copula are the same. Estimations and calibration of the parameters are based on Maximum Likelihood Estimation (MLE). Three methods are generally used and are presented in details in Cherubini s book: Exact Maximum Likelihood method: This method estimates the parameters of the marginals and the parameters (dependence structure) of the joint distribution simultaneously. Inference For the Marginals (IFM) method: With this method, one estimates first the parameters of the marginals and then the parameter of the joint distribution using the estimated margin parameters. Canonical Maximum Likelihood (CML) method: This method is similar to the IFM method except the fact that with CML method, one does not have to specify the marginals. It uses the empirical distribution and any assumptions on the marginal distribution.

38 1. Exact Maximum Likelihood method This method is not often used in practice because it is a computationally intense method. In fact, the parameters of the marginal distributions and the parameters of the dependence structure from the copula are estimate together which could be very computationally intense in the case of high dimension. Suppose that we have extracted daily returns from n assets represented in the matrix with i in { }. We denote by the marginal distribution function and by the marginal density function of asset i and by c the density of the copula as defined in definition 2.3. We recall that the density of the copula expressed in terms of the density of the marginal distributions and the density of the associated multivariate distribution is given in equation (2.3). Let be the parameters of the margin distribution of the i th asset and the parameters of the copula density. Let be the vector to be estimated. Then the likelihood function is given by: (3.1) { } Hence, the maximum likelihood estimator is given by: (3.2) This is the Exact Maximum likelihood method for copulas. 2. IFM Method If one looks at equation (3.1) more closely, one can see that the likelihood function is composed of two terms: one term composed of the density function of the Copula and another one composed of the density functions of the marginals. Then, the Maximum likelihood estimation can be divided in two Maximum Likelihood estimations (one for the copula term and another for the marginals distribution term). Suppose we observe daily returns from n assets over a period of p time steps. From these market data, the IFM method can be divided in three steps: 1. Estimation of the marginal parameters The parameters of the marginal distributions are estimated via a MLE: (3.3) In order to choose the appropriate marginal distribution function, one can use QQ-plots of the parametric quantiles versus empirical estimations. In general, the empirical estimations are faced to data simulated from a normal distribution

39 2. Transformation of the market data with the estimated distribution functions: From the estimated parameters in equation (3.3), one can now transform the market in order to use them to estimate the copula s parameters. Then, observed market data are transformed into uniform variates ( ) (3.4) 3. Estimation of copula parameters : The parameters defined in equation (3.4) are now used to estimate the parameters copula with another Maximum Likelihood estimation: ( ) (3.5) where c is the copula density as defined in definition 2.3. The estimated parameters and are the parameters that one have to use in order to create the fitted copula which describes the best the distribution of the observed market data. This estimation method is more efficient that exact MLE method because it is less computationally intense. In this thesis, we will use the IFM method in my experiments. 3. The CML Method This method consists in transforming the sample data into uniform variates and then estimating the copula parameters. That means that the copula parameters can be estimated without choosing specified marginals. This method can be divided into two steps: 1. Estimate the marginal using empirical distribution without specifying the form of each marginals. Then, we have: 2. Estimate the copula parameters via MLE: (( ) ) We will not enter into details for this method but if one needs more details, one can see Cherubini [2004] to see applications. We know now how to estimate and fit a Copula from market data. In the next part, we will show how to simulate random variates from a Copula.

40 B. Simulation Methods for Copulas In order to perform Monte Carlo simulation with the use of Copulas, one has to generate random scenarios which are distributed like the fitted Copula. In this part, we will describe methods in order to generate random variates from the fitted copula. We will first deal with simulation methods from elliptical copulas (Gaussian and Student s t) and then we will see how to generate random variates from Archimedean Copulas. 1. Simulation methods for Elliptical Copulas a) Gaussian Copula The n-dimensional Gaussian copula with linear correlation R is given by: ( ) From Embrechts and Lindskog [2001], we can say that if R is a strictly positive definite matrix and R can be written as where A is a matrix, then where with ) One possible choice of A is the Cholesky decomposition of R. The Cholesky decomposition of R is the unique triangular matrix L with R =L. Then, with the following algorithm, it is possible to generate random variates from n-dimensional Gaussian Copula with linear correlation R: Find the Cholesky decomposition A of R, Simulate n independent random variates from, Set, Set with where is the univariate standard normal distribution function are the desired random variates with: where denotes the i th margin. ( ) b) Student s t Copula The n-dimensional Student s t Copula with correlation matrix R and degrees of freedom is given by, ( ) ) where denotes the distribution function of where and are independent.

41 With the following algorithm, it is possible to generate random variates from n-dimensional Student s t Copula with linear correlation R and degrees of freedom: Find the Cholesky decomposition A of R, Simulate n independent random variates from, Simulate a random variates S from independent of z, Set, Set, Set with where th v t St d t t distribution function, are the desired random variates with ( ) where denotes the i th margin. 2. Simulation Methods for Archimedean Copulas After estimating the parameters of the generator function from market data with IFM method, we can now show how to generate random variates from an Archimedean Copula. There exist two methods which are manly used to simulate random variates from Archimedean Copulas. One can use the Marshall and Olkin method which used Laplace transforms or one can use conditional sampling approach. a) Marshall and Olkin Method This method involves the use of Laplace transform and its inverse function. We recall that the Laplace transform of a random variable X is defined by: Let be a generator function from an Archimedean Copula. For some Archimedean copulas, can be the Laplace transform of a positive random variable. If equals the inverse of the Laplace transform of a distribution function G satisfying then the following algorithm can be used to simulate random variates from Archimedean Copulas:

42 Simulate a random variate x from distribution G such that the Laplace transform of G is the inverse of the generator function, Simulate n independent uniform variates, ( ( ) ( )) are the desired random variates. This algorithm can be applied to the Clayton and Gumbel Copula but it is not applicable to the Frank Copula (see Cherubini [2004]). b) Conditional Sampling Approach This method is the most frequently used approach to generate random variates from an Archimedean Copula. This approach involves the concept of conditional distribution for copula. By definition for a bivariate copula, the conditional distribution is defined by: ( ) where is the partial derivatives of the copula. For an n-dimensional Copula, let be the k- dimensional marginal of C since have the joint distribution function C. Then, the conditional distribution of given the values is given by: ( ) ( ) ( ) ( ) The following theorem gives an easier formula than the previous one:

43 Theorem 3.1 Let ( ) be a n-dimensional Archimedean function Copula with generator function. Then for k=2,, n, we have: ( ) ( ) ( ) (3.6) where denotes the derivatives of the inverse of the generator function For the bivariate case, we have: ( ) ( ) (3.7) With this last theorem and the previous definition we can now write an algorithm to generate random variates from an Archimedean Copula: Generate uniform random variables, Set, Set ( ), Invert the expression in order to find using theorem 3.1, Set ( ), Invert the previous expression in order to find, Simulate a random variate from ( ), are the desired random variates. Let us consider an example in order to better understand this method. We consider a bivariate Clayton Copula with parameter. Then, the generator function can be written as follows: with and and also The bivariate Clayton Copula is defined by: ( )

44 Now, we can apply the previous algorithm: Simulate two independent random variables from Set, Set ( ), Then ( ) ( ) Hence, ( ) Our desired random variates are given by the vector. For more details concerning a multidimensional Clayton Copula, one can look at Cherubini [2004]. Cherubini developed also examples for the Gumbel Copula and the Frank Copula. It is easy to obtain the conditional copula for the Frank Copula but the case of the Gumbel Copula is extremely computational because the computation of the conditional copula in this case requires an iterative method. The difficulty comes from fact that it is extremely difficult to compute analytically the inverse of the generator function. Now, we know how to generate random variates from a fitted copula. The next step in the valuation of an option is to simulate the returns from the random variates. Monte Carlo simulations, as presented in chapter 1, are classically used when the random variates are modeled thanks to joint normal distributions. But when one uses random variates from copula, the Monte Carlo method is slightly different. C. Monte Carlo Simulations with Copula The Monte Carlo method displayed in chapter 1 is no longer correct when one assumed to model returns with Copulas. Especially, equation (1.14) is valid in the case where the returns of the assets are normally distributed. But the Monte Carlo method with copula stays closed to the Monte Carlo method exposed in chapter 1in the sense that it is still based on the Law of Large number and the Central Limit Theorem. The purpose here is to simulate the returns directly from the random variates generated from the fitted copulas (see chapter 2) without using equation (1.14) or any others formulas. In this chapter, we will first show how to extract the desired returns from the random variates computed in Part C of chapter 2 and how to compute them in a risk free world. In a second time, we will present the two marginals used in this thesis. Finally, we will present the all Monte Carlo Method in order to get the price of an option by using Copulas.

45 1. Simulation of risk free returns In the discrete case, we define the return of an asset by the following formula: where. (3.8) Our simulated returns will be modeled by random variables issued from the i th margin of a fitted Copula. The vector ( ) will be generated from a chosen copula and chosen marginal distributions. Then, it is easy to simulate the returns with the desired dependencies. For example, to obtain normal distributed returns like in chapter 1, one has to choose a Gaussian Copula and Gaussian marginal distributions. According to the definition of the Gaussian Copula, the s will be normally distributed with mean standard deviation and linear correlation. This is equivalent to the distribution of the returns in equation (1.10). Concretely, after choosing a Copula and the marginals, estimating the parameters and generating random variates (see previous part), we obtain a vector of random variates as follows (see part B): ( ) where denotes the i th marginal. Then, from the last formula, one can easily get the simulated returns : ( ) (3.9) If one wants daily returns over 1 year, one has to simulate 252 vectors, ( wants weekly returns over 1 year, one has to simulate 52 vectors ( time points where T is the maturity of the option. ) If one ) We will denote the Hence, from all the simulated returns and using formula (3.8), it is easy to compute the asset price by: ( ) (3.10) where is the spot price and In order to be in accordance with the pricing option theory, one has to ensure that we are pricing in a risk free world. In other words, this means that we have to find [ ] in order to price under the risk neutral probability.