Pricing of a “worst of” option using a

Copula method

M A X I M E M A L G R A T

Master of Science Thesis Stockholm, Sweden 2013

Pricing of a “worst of” option using a Copula method

M A X I M E M A L G R A T

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-100 44 Stockholm, Sweden URL: www.kth.se/sci

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.

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.

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

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

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.

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:

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:

(

)

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.

(1.1)

(1.2)

Consequently, we can rewrite the equations (1.1) and (1.2) under the risk neutral world as follow:

and

(

)

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.

(1.3)

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:

( ) {

}

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:

(

)

The return of such a portfolio should be equal to the risk free interest rate. Then, we can write:

From equations (1.5) and (1.6), we have:

(

) (

)

Hence,

The equation (1.7) is the Black-Scholes differential equation!

(1.5)

(1.6)

(1.7)

(1.4)

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:

( ) ( ) ( )

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:

( ) ( ) ( )

( ) ( ) ( )

(1.8)

After deriving the expectation, we get the famous Black-Scholes formula for a call/put option:

( ) ( ) ( ) ( )

( ) ( )

where and are defined by :

( ⁄ ) ( ⁄ )

√

√

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:

( )

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.

(1.9)

(1.10)

If one supposes independent of time, the solution of equation (1.10) is still a Geometric Brownian

motion given by:

{(

) ( ) (

)}

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:

∑ ( )

∑ ( )

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 :

( ) ( ) ∫ ( ) ( )

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.

(1.11)

(1.13)

(1.12)

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:

{(

) √ )}

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.

(1.14)

Then, the price of the basket option given by equation (1.13) can be estimated by a Monte Carlo

simulation. Hence,

( ) ( )

∑ ( )

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:

( ) { ( ( )

( ) ( )

( )) }

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

(1.15)

(1.16)

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 0,88938484

SocGen 0,88938484 1

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 0,562108

LVMH 0,562107833 1

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.

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

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

distributed on [0,1]. In the same way, if is uniformly distributed, then ( ) has the distribution

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 10 000 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.

Figure 2.1: Samples of size 10 000 from two bivariate distributions

-4 -2 0 2 4

-4-2

02

4

Bivariate Gaussian with correl 0.5

-2 0 2 4

-4-2

02

4

Bivariate Gaussian with correl 0.9

-4 -2 0 2 4

-20

24

Bivariate Student's t dist with df 1

-4 -2 0 2 4

-3-1

13

Bivariate Student's t dist with df 3

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:

( ) ( )

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:

( ) ( )

To obtain the density of the n-dimensional distribution F, we use the following relationship:

( ) ( )∏ ( )

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:

( ) ( )

∏ ( )

(2.1)

(2.2)

(2.3)

(2.4)

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

( ) ( ( ) ( ))

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 , ( ) ( (

)

( )

( ))

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.

(2.5)

(2.6)

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:

( ) (

( ) ( ))

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:

( )

| |

⁄(

( ) )

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 0.9.

Figure 2.2: Random draws from a bivariate Gaussian Copula

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Bivariate Gaussian Copula with correlation 0.6

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Bivariate Gaussian Copula with correlation 0.9

(2.7)

(2.8)

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:

( ) ( ( ))

( ))

∫ ∫ ∫ (

)

(

)( )

(

)

( )

( )

( )

where is the inverse of the univariate distribution function of the Student’s t distribution with degrees

of freedom.

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

3 dim Gaussian Copula with correlation 0.6

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

3 dim Gaussian Copula with correlation 0.9

(2.9)

Definition 2.6: Multivariate Student’s t copula density

The density of a multivariate Student’s Copula is given as follows:

( ) | |

(

)

(

)

( (

)

(

))

(

)

∏ (

)

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 5.

Figure 2.4 Random draws from a bivariate Student’s t Copula

Figure 2.5 Random draws from a 3 dimensional Student’s t Copula

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Bivariate Student's t Copula with 1 degree of freedom

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Bivariate Student's t Copula with 3 degree of freedom

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

3 dim Student s t Copula with 1 degree of freedom

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

3 dim Student s t Copula with 3 degree of freedom

(2.10)

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

( ( ))

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.

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 ( ( ) ) ( ( ))

(2.11)

(2.12)

Definition 2.10: Archimedean Copula density

Let C be a Copula defined by Kimberling theorem. Then, by for all u and v in , the density of C is

given:

( )

( ( )) ( ) ( )

[ ( ( ))]

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:

( ) ( ( ) ( ))

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

( ) ( )

Definition 2.11: Clayton Copula

Let ( ) be a vector in Then, the multivariate Clayton Copula is defined by:

( ) (∑

)

⁄

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:

( ) ( )( )( )

(

)

(2.13)

(2.14)

(2.15)

(2.16)

(2.17)

(2.18)

Figure 2.6 shows a sample of 3 000 random draws from a bivariate Clayton Copula with and

Figure 2.7 shows a sample of 3 000 random draws from a 3 dimensional Clayton Copula with

and

Figure 2.6 Random draws from a bivariate Clayton Copula

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

( )

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Bivariate Clayton Copula with alpha = 2

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Bivariate Clayton Copula with alpha = 10

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

3 dim Clayton Copula with alpha = 2

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

3 dim Clayton Copula with alpha = 10

(2.19)

(2.20)

Definition 2.12: Gumbel Copula

Let ( ) be a vector in and . Then, the multivariate Gumbel Copula is defined

by:

( ) {( ∑( )

)

⁄

}

If , then the previous equation can be written as follows:

( ) ∏

Figure 2.8 shows a sample of 3 000 random draws from a bivariate Gumbel Copula with and

Figure 2.9 shows a sample of 3 000 random draws from a 3 dimensional Gumbel Copula with

and

Figure 2.8 Random draws from a bivariate Gumbel Copula

Figure 2.9 Random draws from a 3 dimensional Gumbel Copula

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Bivariate Gumbel Copula with alpha = 2

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Bivariate Gumbel Copula with alpha = 10

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

3 dim Gumbel Copula with alpha = 2

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

3 dim Gumbel Copula with alpha = 10

(2.21)

(2.22)

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:

( ) (

)

with and also

( )

( ( ))

Definition 2.12: Frank Copula

Let ( ) be a vector in and . Then, the multivariate Frank Copula is defined by:

( )

{

∏ ( )

}

From this definition, we can give the bivariate Frank Copula density as follows:

( ) ( )

( )

( ) ( )( )

Figure 2.10 shows a sample of 3 000 random draws from a bivariate Frank Copula with and

Figure 2.11 shows a sample of 3 000 random draws from a 3 dimensional Frank Copula with

and

Figure 2.10 Random draws from a bivariate Frank Copula

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Bivariate Frank Copula with alpha = 2

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Bivariate Frank Copula with alpha = 10

(2.23)

(2.24)

(2.25)

(2.26)

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:

( )

√ ( ) ( )

With these definitions, we can now define the concepts of Kendall’s Tau and

Spearman’s Rho

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

3 dim Frank Copula with alpha = 2

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

3 dim Frank Copula with alpha = 10

(2.27)

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

∬ ( ) ( )

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

( )

( )

It is also possible to define an unbiased estimator of the Kendall’s Tau for an n-dimensional

sample (Cherubini [2004]):

( )∑∑

with {( )( )}

From formula (2.29), we can obtain an estimator of the correlation ρ given as follow:

(

)

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:

∫ ( )

( )

(2.28)

(2.29)

(2.30)

(2.31)

(2.32)

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:

∬ ( )

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

( ) ( )

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 1 + 4 ( ( ) )

( ) ( )

Where,

( )

∫

(2.33)

(2.34)

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.

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 ith asset and the parameters of the copula density. Let ( )be the vector to be

estimated. Then the likelihood function is given by:

( ) ∑ { ( ) ( ) } ∑∑ ( )

Hence, the maximum likelihood estimator is given by:

( )

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:

∑ (

)

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

(3.1)

(3.2)

(3.3)

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. Estimation of copula parameters

:

The parameters defined in equation (3.4) are now used to estimate the parameters copula with another

Maximum Likelihood estimation:

∑ (( )

)

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.

(3.4)

(3.5)

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 ith 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.

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 ith 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:

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:

Theorem 3.1

Let ( ) ( ( ) ( )) be a n-dimensional Archimedean function Copula

with generator function . Then for k=2,…, n, we have:

( | ) ( )( ( ) ( ))

( )( ( ) ( ))

where ( ) denotes the ( ) derivatives of the inverse of the generator function

For the bivariate case, we have:

( | ) ( ( ) ( ))

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 also

( ) ( )

and

( )

( )

( )

The bivariate Clayton Copula is defined by:

( ) (∑

)

⁄

(3.6)

(3.7)

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.

1. Simulation of risk free returns

In the discrete case, we define the return of an asset by the following formula:

( ) ( )

( )

where .

Our simulated returns will be modeled by random variables issued from the ith 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 ith marginal.

Then, from the last formula, one can easily get the simulated returns :

(

)

(

( )

( )

( ))

If one wants daily returns over 1 year, one has to simulate 252 vectors, (

)

If one

wants weekly returns over 1 year, one has to simulate 52 vectors (

)

We will denote the

time points where T is the maturity of the option.

Hence, from all the simulated returns and using formula (3.8), it is easy to compute the asset price by:

∏( ( ))

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.

(3.8)

(3.9)

(3.10)

According to the pricing theory, we know that under the risk neutral probability the expected

return of a risky portfolio (here an asset) is equal to the annual risk free rate (or in case of

dividend). Then, one can write,

[

]

From equation (3.10), we can rewrite this equation as follows with m the number of times steps:

[

] [∏(

( ))

]

But,

[∏( ( ))

] [( ( ))

] [ [

( )]]

( ( ))

Hence, from equation (3.11), we can write:

[

] [∏(

( ))

] ( )

and,

[

] [

]

under the risk neutral probability.

Finally, inverting equation (3.12), we get:

( )

where ⁄

The equation (3.13) represents the expected risk free return of an asset on the period dt under the

risk neutral probability.

Concretely, will be used in order to generate random draws with expected return

from the random

variates issued from the fitted copula. This means that the historical returns are only used to estimate the

copula parameters.

(3.11)

(3.12)

(3.13)

2. Choice of the marginals

As we mentioned before, copulas offer a large choice in order to model the dependence structure

of the asset. One can choose the marginals and the family of the copula. In the Black Scholes model, the

normal distribution is used in order to model distribution of the asset.

Often, if one examines financial data, one can see that the returns show fatter tails than the normal

distribution. The Student’s t distribution is often used to model fat tail distribution because this

distribution has more probability mass in the tail than the standard normal distribution. The standard

Student’s t distribution has mean 0 and a variance determined by the degree of freedom. When the degree

of freedom is large, the Student’s t looks like the standard normal distribution. However, we know that the

financial data depend essentially on the volatility (a scale parameter σ) and a location parameter .

Then, it comes interesting to use location scale Student’s t distribution.

Let X be a Student’s t distribution with degrees of freedom. Then, is the location

scale Student’s t distribution with density:

( ) (

)

√ ( ){ (

)

}(

)

The location scale Student’s t distribution enables to capture fat tail distribution and easily model

the distribution under the risk neutral probability by choosing the location parameter µ like in equation

(3.13).

Consequently, in this thesis, we will choose two families of marginal distribution: the standard

normal distribution which is used in the classical Black Scholes approach and the location scale Student’s

t distribution as defined before in order to capture fat tail features. The price obtained by the two methods

will be compared.

3. Monte Carlo Method with Copulas

This part will sum up all the methods presented in the different part of this thesis and present how

to use the Monte Carlo method in order to compute the price of an option using copulas.

From observed daily returns over three year (769 observations, m = 769), the parameters of the

marginals and the copulas will be estimated via the IFM method (see chapter 3 part A). Standard normal

distributions and location scale Student’s t distributions will be used for the marginal distributions.

Different families of copulas will be used: Elliptical Copulas (Gaussian and Student’s t copula) and

Archimedean Copula (Clayton, Gumbel and Frank Copula). These are the main copulas used in the

financial world and that we have presented in chapter 2.

The estimated parameters

and

are used to generate the random variates from the

fitted copula (see Chapter 3 part B) and then generate the m returns and calculate the paths of the asset

(equation 3.10).

The Monte Carlo method will be done with a number of simulations P (P =10 000 for example).

Then, we will generate P times m returns for each asset. We will obtain a tri-dimensional matrix with

dimension

(3.14)

We will define by

the pth Monte Carlo simulation (p = 1,2, …, P) for the i

th asset. Then, the

price of an option with maturity T and strike K with payoff ( ) is given, according to Chapter

1 by:

( ) ( )

∑ (

( ) ( )

( ))

In order to recap the Monte Carlo method with copula, one can find below the steps of the method:

a. Estimate the marginal distribution parameters

and the Copula parameter

via the

IFM method from observed daily returns (Chapter 3 part A),

b. Compute the expected return of the assets under the risk neutral probability (equation (3.10)),

c. For each Monte Carlo simulation p, generate m random variates for each asset from the fitted

Copula (Chapter 3 Part B),

d. For each Monte Carlo simulation, transform the random variates into returns via equation (3.9),

e. Generate paths for each asset

for each Monte Carlo simulation p from the estimated

returns,

f. Compute the payoff ( ( )

( ) ( )),

g. Compute the price of the option by discounting the mean of the P simulated payoffs with the risk

free rate r (equation 3.15)

In this thesis, all the calculus are done under the software R. We used the packages Copula, sn and

mvtnorm.

(3.15)

Chapter 4 Numerical Results

In this part, we will apply the Monte Carlo method with copula in order to price a worst of basket.

The option will be composed of two assets. We will consider two worst of options: one with two

correlated assets and another one with two uncorrelated assets.

We will first present the properties of the market data used in order to estimate the parameters.

Then, we will show the results of my estimations and analyze them. Finally, we will give the prices of the

options according to the chosen marginal distributions and the chosen copulas.

A. Market data

The underlying assets of the two options will be BNP Paribas, Société Générale and LVMH

(Louis Vuitton). The first option will be composed of BNP Paribas and Société Générale and the second

one will be composed of BNP Paribas and LVMH.

In order to price an option with maturity 3 years, we will use 769 daily returns from the 3 assets

from the 31st May 2010 to the 31

st May 2013. We will price the option At The Money (ATM) on the 31

st

May 2013, so the spot price at the 31st May 2013 will be used to generate the path of the assets. One can

find in the table below, all the properties of these daily returns.

BNP Paribas Société Générale LVMH

Spot as of 31/05/2013 45,325 30,86 136,65

mean 0,0003738 0,000376789 0,000734312

Standard Deviation 0,02950114 0,034984177 0,017759952

Yearly Dividend Yield 2.043% 1.043% 4.02%

On Figure 4.1, one can see the scatterplots of the daily returns of the stocks Société Générale/BNP

Paribas and LVMH/BNP Paribas. These plots show a dependence structure between the assets and one

can see that Société Générale and BNP Paribas are highly correlated instead of BNP Paribas and LVMH.

This is in accordance with the correlation presented in chapter 1 which shows that BNP Paribas is more

correlated to Société Générale than to LVMH.

Figure 4.1: Scatterplot of the daily returns

-0.10 -0.05 0.00 0.05 0.10 0.15

-0.1

0.0

0.1

0.2

Daily returns Société Générale/BNP Paribas

BNP

SG

-0.10 -0.05 0.00 0.05 0.10 0.15

-0.0

6-0

.04

-0.0

20

.00

0.0

20

.04

Daily returns LVMH/BNP Paribas

BNP

LV

MH

On the histograms on Figure 4.2, one can see that Société Générale and BNP Paribas have fatter

tails than LVMH. These histograms, and especially the one of LVMH daily returns, are distributed like a

Gaussian distribution but the fat tails of the Société Générale and BNP Paribas histograms are not

modeling by the Gaussian distribution. Then, the Student’s t distribution can be used. On Figure 4.3, one

can see the histograms of a 1 000 sample of a random variable from a normal distribution with standard

deviation 0.02 and a location scale Student’s t distribution with 3 degrees of freedom and scale parameter

0.02. One can compare the histograms obtained with the histograms in Figure 4.2: histograms of Société

Générale and BNP Paribas are closer to the Student’s t distribution than to the normal distribution whereas

the LVMH’s histogram stays close the normal distribution.

Figure 4.2: Histograms of daily returns

BNP Paribas Daily returns over 3 years

value

Fre

qu

en

cy

-0.10 -0.05 0.00 0.05 0.10 0.15

02

04

06

08

0

LVMH Daily returns over 3 years

value

Fre

qu

en

cy

-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06

05

10

15

20

25

Société Générale Daily returns over 3 years

value

Fre

qu

en

cy

-0.1 0.0 0.1 0.2

01

02

03

04

05

06

0

Figure 4.3: Histograms of samples from normal and Student’s t distribution

One can use Quantile-Quantile Plots (QQ-plots) in order to guess the distribution verified by the

returns. QQ-plot is a plot of empirical quantiles against quantile from a known distribution function. If the

data are generated by a probability distribution similar to the known distribution, then QQ-plot is almost

linear. QQ-plots are useful to study the tails of the distribution. In Figure 4.4, the QQ-plots of the daily

returns against the standard normal distribution have been drawn. In Figure 4.5, the QQ- plots of the daily

returns against the standard Student’s t distribution have been drawn.

When the quantiles of daily returns from Société Générale and BNP Paribas are facing the

quantiles of the normal distribution, the QQ-plot is not linear because of the heavy weight of the right and

left tail of the distribution of Société Générale and BNP Paribas. However, the QQ-plot with LVMH

quantiles is linear when facing quantiles of the standard normal distribution. We can say that the observed

returns from LVMH are normally distributed.

QQ-plots become more linear when it comes to face quantile of daily returns from Société

Générale and BNP Paribas to the quantile of a standard Student’s t distribution with three degrees of

freedom. This fact confirms that the distributions of the observed returns from Société Générale and BNP

Paribas are closer to a Student’s t distribution than a normal distribution.

Now, one can understand the choice of Gaussian and location scale Student’s t marginal

distributions.

Sample from location scale Student's t distribution

Observations

Fre

qu

en

cy

-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15

02

04

06

08

01

00

Sample from a normal distribution

Observations

Fre

qu

en

cy

-0.05 0.00 0.05

01

02

03

04

05

0

Figure 4.4: QQ-plots of quantiles returns against standard normal quantiles

Figure 4.5: QQ-plots of quantiles returns against Student’s t quantiles

-3 -2 -1 0 1 2

-0.1

0-0

.05

0.0

00

.05

0.1

00

.15

Quantiles of BNP Paribas daily returns

Standard Normal Quantiles

BN

P

-3 -2 -1 0 1 2 3

-0.1

0.0

0.1

0.2

Quantiles of Société Générale daily returns

Standard Normal Quantiles

SG

-3 -2 -1 0 1 2 3

-0.0

6-0

.04

-0.0

20

.00

0.0

20

.04

Quantiles of LVMH daily returns

Standard Normal Quantiles

LV

MH

-10 -5 0 5 10

-0.1

0-0

.05

0.0

00

.05

0.1

00

.15

Quantiles of BNP Paribas daily returns

Standard Student's t Quantiles

BN

P

-10 -5 0 5 10

-0.1

0.0

0.1

0.2

Quantiles of Société Générale daily returns

Standard Student's t Quantiles

SG

B. Numerical Results

1. Estimated marginal distribution parameters

We have chosen two families of marginal distributions: Gaussian and location scale Student’s t.

Then, we will estimate for each of the three assets the parameters of their marginal distributions using the

observed daily returns. The estimated parameters for the marginal distributions are shown in the following

table for each asset:

BNP Paribas Société Générale

Marginals

Gaussian 0.0003778449 0.0294827365 / 0.000381584 0.034962349 /

Student's t -0.000183703 0.0211336847 3.9051767572 -0.000238248 0.0260879230 4.4824510339

LVMH

Marginals

Gaussian 0.000736746 0.017748871 /

Student's t

0.0009507444

0.0154059161 7.9450875276

One can see that with the estimated parameters for a Gaussian margin distributions are quite

closed to the observed parameters. However, these parameters are not so closed when one uses Student’s t

distribution. Nevertheless, the estimated degrees of freedom for the LVMH distribution are higher than the

others. This shows that LVMH distribution is closed to a normal distribution because when the degrees of

freedom of a Student’s t distribution goes up, the Student’s t is distributed like a normal distribution.

2. Estimated copulas parameters

According to the IFM method, the copulas parameters have to be estimated by using the estimated

parameters of the marginal distributions. Consequently, we will use the previous results, to estimate

copulas parameters.

The results are presented in the following page. The results of the two first tables represent the

estimated parameters for the elliptical copulas whereas the two last are the results for Archimedean

Copulas.

The results show that the estimated parameters are quite closed to the observed parameters. The

use of different marginal distributions (Normal or Student’s t) do not impact a lot the copula estimated

parameters.

The estimated degrees of freedom are larger when the basket option is composed of LVMH asset.

This was an expected result according to what we said in Part A.

Estimated Copula Parameters for the basket composed of BNP Paribas and Société Générale

Copula Marginal Gaussian Normal 0.834031 /

Gaussian Student’s t 0.820243 /

Student’s t Normal 0.866485 2.942990

Student’s t Student’s t 0.83987 1.99180

Estimated Copula Parameters for the basket composed of BNP Paribas and LVMH

Copula Marginal

Gaussian Normal 0.55680 /

Gaussian Student’s t 0.55293 /

Student’s t Normal 0.57513 9.05778

Student’s t Student’s t 0.5535 3.6957

Estimated Copula Parameters for the basket composed of BNP Paribas and Société Générale

Copula Marginal Copula Marginal

Clayton Gaussian 2.7186 Calyton Student’s t 2.6915

Gumbel Gaussian 2.65129 Gumbel Student’s t 2.61672

Frank Gaussian 10.6979 Frank Student’s t 9.1047

Estimated Copula Parameters for the basket composed of BNP Paribas and LVMH

Copula Marginal Copula Marginal

Clayton Gaussian 0.73245 Clayton Student’s t 0.87594

Gumbel Gaussian 1.54444 Gumbel Student’s t 1.57949

Frank Gaussian 4.5452 Frank Student’s t 3.9158

According to Part C in chapter 3, Monte Carlo method has to be done under the risk free

probability. Then, we have now to compute the expected risk free return of an asset with formula (3.13).

The risk free rate (Libor rate) over 3 years on the 31st May 2013 is 0.923%. We price an option over three

years, which corresponds to 769 simulations for the daily returns ( ⁄ ). From formula (3.13), we

derived the following expected risk free rate for each asset:

Underlying ( )

⁄

BNP Paribas -0.000025617

Société générale -0.00001242

LVMH -0.00005210

Due to the low risk free rate, the expected risk free returns are under 0. This is quite surprising but if one

looks at the expected parameters of the marginals, one can see in case of Student’s t marginal distribution

that expected returns are lower than 0.

Now, we have all the inputs needed to generate daily returns from the fitted copula. One can see

below an example of 769 simulated daily returns from the fitted Gaussian copula with normal and

Student’s t marginal distributions.

Figure 4.6: Simulated daily returns from a fitted Gaussian Copula with Normal marginals

-0.05 0.00 0.05

-0.1

0-0

.05

0.0

00

.05

0.1

0

Simulated Daily returns Société Générale/BNP Paribas

BNP

SG

-0.05 0.00 0.05

-0.0

4-0

.02

0.0

00

.02

0.0

40

.06

Simulated Daily returns LVMH/BNP Paribas

BNP

LV

MH

Simulated BNP daily returns over 3 years

value

Fre

qu

en

cy

-0.05 0.00 0.05

05

10

15

20

25

30

Simulated LVMH Daily returns over 3 years

value

Fre

qu

en

cy

-0.04 -0.02 0.00 0.02 0.04 0.06

05

10

15

20

Figure 4.7: Histograms of Simulated daily returns from a fitted Gaussian Copula with Normal marginals

Figure 4.8: Simulated daily returns from a fitted Gaussian Copula with Student’s t marginals

Figure 4.9: Simulated daily returns from a fitted Gaussian Copula with Student’s t marginals

One can see that the dependence structure has been respected if one compares the scatter plot of

the observed returns and the scatter plot of the simulated returns. Société Générale and BNP Paribas

conserve their fat tails.

In the Normal marginal distribution case, the histograms show that both BNP Paribas and LVMH

are distributed like normal distributions. This does not respect exactly the observed distribution of BNP

Paribas which is more Student’s t distributed. However, when we use Student’s t marginal distributions,

the scatter plots show fatter tails for Société Générale and BNP Paribas which is in accordance with the

observed returns. Moreover, if one looks at the simulated histograms, one can see that the BNP Paribas

distribution is closed to a Student’t distribution with fat tails and that the LVMH distribution is closed to a

Gaussian distribution with no fat tails. This corresponds exactly to the description we made from the

observed daily returns in Part A. Consequently, we expect that the price of the option with Student’s t

marginal distributions will be better than the case with Gaussian marginal distributions.

Others observations can be done for each case (Gaussian, Student’s t, Gumbel, Clayton or Frank

Copula and Gaussian or Student’s t marginal distribution), but it will be very long and boring for the

lector. Now, it is time to price our options with Monte Carlo Simulations.

-0.2 -0.1 0.0 0.1 0.2

-0.2

-0.1

0.0

0.1

0.2

Simulated Daily returns Société Générale/BNP Paribas

BNP

SG

-0.2 -0.1 0.0 0.1

-0.0

6-0

.04

-0.0

20

.00

0.0

20

.04

0.0

6

Simulated Daily returns LVMH/BNP Paribas

BNP

LV

MH

Simulated BNP daily returns over 3 years

value

Fre

qu

en

cy

-0.2 -0.1 0.0 0.1

01

02

03

04

05

06

07

0

Simulated LVMH Daily returns over 3 years

value

Fre

qu

en

cy

-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06

05

10

15

20

25

3. Pricing Results

The Monte Carlo method is running with 50 000 simulations for the Gaussian marginal

distribution case and 2 000 simulations for the Student’s t distribution case. Indeed, with software R, the

computation time for the simulations of Copulas with Student’s t marginal distribution is very long. The R

codes used to price the options can be found in Appendix A.

The Basket 1 is composed of BNP Paribas and Société Générale and Basket 2 is composed of

BNP Paribas and LVMH. The strike of the option is 0.5 and the maturity three years. The worst of payoff

is given by:

( ) { ( ( )

( ) ( )

( )) }

The Pricing Partners price is obtained by using the company software. We used a Black Scholes

model and a Dupire model for the asset and a Monte Carlo method with 10 000 simulations in order to get

the price.

The following table gives the price of the option in all cases:

Copulas Marginals Price Basket 1 (in %) Price Basket 2 (in %)

Gaussian Gaussian 37.61% 27.78%

Gaussian Student’s t 36.78% 27.03%

Student’s t Gaussian 38.39% 28.15%

Student’s t Student’s t 36.97% 27.46%

Clayton Gaussian 34.18% 24.90%

Clayton Student’s t 34.12% 24.77%

Gumbel Gaussian 37.39% 27.29%

Gumbel Student’s t 37.77% 27.48%

Frank Gaussian 37.19% 28.49%

Frank Student’s t 36.06% 26.61%

Pricing Partners (Black Scholes) 33.01% 25.26%

Pricing Partners (Dupire) 33.43% 27.02%

For the basket 1, the obtained prices are higher than the price obtained with Pricing Partners

software. We have a difference of 4% except for the case with the Clayton Copula where the price is

around 34%. This can be explained by the fact that Clayton copula is modeling better dependence in the

negative tail than in the positive tail of distribution function. That means that there will be more events in

the right tail and consequently the price of the option decreases.

However, for basket 2, our prices are closer to the price of Pricing Partners. The prices stay almost

all in the same percentage except for the case of the Clayton Copula. That means that the Copula method

is better when the underlying assets are not correlated than when they are correlated.

The prices obtained with Student’s t distribution are generally lower than the prices obtained with

Gaussian marginal distributions. Indeed, we know that Student’s t distributions have fatter tails and

consequently they generate more extreme events. Then, with more extreme events, the price of an option

is decreasing.

The cases with a Student’s t Copula are not really precise because of the low numbers of Monte

Carlo simulations. One has to know that in this case it takes almost 1 hour to compute the price with only

2 000 simulations whereas, with Gaussian marginal distributions, we simulated the 50 000 Monte Carlo

simulations in only 10 minutes. Consequently, the Student’s t case is very computationally intense under

software R. These low numbers of simulations explain the fact that the results with Student’s t marginal

distributions cannot be analyzed with precision. We only can say that it is seems logical to obtain lower

prices because the Student’s t distribution has fat tails.

As we said before (Part A), the daily returns of BNP Paribas are distributed like a Student’s t

distribution whereas the daily returns of LVMH are distributed like a normal distribution. When we

presented the Copulas in chapter 2, we said that one of the advantages of Copulas is that one can easily

choose the dependence structure. The idea now is to use a Gaussian or Student’s t Copula with the

specified marginals for each asset. In this case, we have to estimate the new Copula parameters and then

simulate random variates from the fitted copula with a Student’s t marginal distribution for BNP Paribas

and a Gaussian marginal distribution for LVMH.

The estimated parameters for the marginals stay the same. The estimated parameters for the

Copulas and the prices of the basket 2 are presented in the table below. The prices stay acceptable and

lightly better (closer to the Pricing Partners prices).

Estimated Copula parameters

Copula Price Basket 2 (in %)

Gaussian 0.5538 / 26.76%

Student’s t 0.56426 4.52672 27.27%

The differences between our prices and the market price can be explained by the assumptions

made in this thesis. In fact, when we price with the Pricing Partners software, we use different methods

and data to price the option. We use a volatility surface (implied volatility) whereas in the copula method

we use an historical volatility. The same thing appears with the correlation because in Pricing Partners we

are using an implied correlation. The risk free is supposed to be not constant under Pricing Partners

software.

The use of Gaussian marginals seems to miss some of the extreme events that the

Student’s t distribution allows. In the major cases, we overestimate the market price but we do not have

extreme prices. When we mix the Gaussian and the Student’s t distributions the prices are closer to the

price because in this case we take into account the fact that the Student’s t distribution has fat tails and is

closer to the distribution of BNP Paribas.

To conclude this chapter, the Copula method seems to be an acceptable method to approximate

the price of an option like a “worst of” option. But this method seems computationally intense especially

in the case of Student’s t marginal distributions. The choice of Copulas has not a very important impact on

the price of this kind of option, except in the case of the Clayton Copula (see above). One can see that the

prices obtained are quite stable. So, the Copula method can be seen as an alternative method to the

classical method with Black Scholes method and Monte Carlo simulations. We can say that the Copula

Method is a statistical method based only on past observations.

Conclusion

This thesis aimed at using copulas methods to price basket options. We reviewed the concept of

Copulas to model the dependence structure between the assets of a basket options. We considered two

types of Copulas (Elliptical and Archimedean) and two types of marginal distributions (Gaussian and

Student’s t).

From the observed daily returns, we estimated the copulas parameters in order to generate the

simulated paths of the assets. Then, with a Monte Carlo method, we priced the option.

The Monte Carlo simulations showed that the Copula Method is an intense computational method

regarding the time to compute an option. This time is acceptable when it comes to price the option using a

Copula with Gaussian marginal distributions but it comes very bad with Student’s t marginal distributions.

In fact, this comes from the fact that we are using location scale Student’s t distributions which are, under

the software R, quite long to simulate compared to the standard Student’s t distribution. Consequently, the

results obtained using Student’s t marginal distributions are difficult to analyze because of the law number

of Monte Carlo simulations.

According to our prices displayed in chapter 4, the influence of the Copula family has not an

important impact on the price of the option except for the case of the Clayton Copula where the price is

much lower than the others. This is explained by the fact that the Clayton Copula is modeling better

dependence in the negative tail than in the positive tail of distribution function. Nevertheless, if one

compares our price with the PricingPartners prices, one can see that our price is overestimated when the

two underlying assets are strongly correlated. This means that, when the assets are correlated, the Copulas

see the assets more correlated that they are in reality. However, when the assets are not correlated, our

prices are matching the PricingPartners prices except for the Clayton Copula. Then, the idea was to use a

Student’s t marginal distribution for the asset for which the observed distribution is closed to a Student’s t

distribution and a Gaussian marginal distribution for the asset for which the observed distributions is

closed to a Gaussian distribution. It appears that the prices are lightly better in this case. This fact is in

accordance with the properties of the tail of Student’s t distribution which corresponds to the distribution

of the observed returns (from BNP Paribas).

Consequently, one of the advantages of the Copula Method is that one can easily choose the

dependence structure by choosing which marginal distribution corresponds the best to the observed

distribution.

Nevertheless, there exist disadvantages with the Copula Method. Indeed, this method is

computationally intense because of the Maximum Likelihood estimation. Moreover, the Copula Method is

only based on historical data and consequently, we only use an historical volatility and an historical

correlation (which are estimated) at the contrary that in reality we use an implied volatility and an implied

correlation which are directly computed from options already traded on the market. For example, we do

not take into account the volatility smile and we do not use a volatility surface which is commonly used on

the market.

Consequently, Copula method should be used with a lot of observed historical returns, and

therefore this method could be more useful and accurate to price basket options with long maturity.

This method is not used for the moment by practitioners because of the drawbacks presented

previously. The future researches on this subject should focus on more efficient calibration methods and

how to use implied volatility and implied correlation. This pricing method can be seen as an alternative

method to the classical pricing method due to the low difference between the computed prices.

Apendix A : R Codes ##Market Data## data <- read.table("data.txt",header=TRUE) BNP <- data[,1] LVMH <- data[,3] SG <- data[,2]

mean1 <- mean(BNP) stdev1<- sd(BNP) b1 <- c(mean1,stdev1) mean2 <- mean(SG) stdev2 <- sd(SG) b2 <- c(mean2,stdev2) tical Copula mean3 <- mean(LVMH) stdev3 <- sd(LVMH) b3 <- c(mean3,stdev3) a1 <- cor(SG,BNP) a2 <- cor(BNP,LVMH)

###### IFM Method ###### ## First MLE with Gaussian Marginals## loglik.marg <- function(b,x) {sum(dnorm(x, mean = b[1], sd = b[2], log = TRUE))} ctrl <- list(fnscale =-100) b1hat <- optim(b1, fn = loglik.marg, x= BNP, control = ctrl)$par b2hat <- optim(b2, fn = loglik.marg, x= SG,control = ctrl)$par b3hat <- optim(b3, fn = loglik.marg, x= LVMH,control = ctrl)$par ## First MLE with Student’s t Marginals## loglik.margt <- function(b,x) {sum(dst(x, location = b[1], scale = b[2], df = b[3],log = TRUE))} ctrl <- list(fnscale =-1) b1hatt <- optim(b1t, fn = loglik.margt, x = BNP, control = ctrl)$par b2hatt <- optim(b2t, fn = loglik.margt, x= data[,2],control = ctrl)$par b3hatt <- optim(b3t, fn = loglik.margt, x= data[,3],control = ctrl)$par

## Transformation in the Hypercube## #For Gaussian Marginals: udat1 <- pnorm(BNP, mean = b1hat[1], sd = b1hat[2]) udat2 <- pnorm(SG, mean = b2hat[1], sd = b2hat[2]) udat3 <- pnorm(LVMH, mean = b3hat[1], sd = b3hat[2]) udat <- cbind(udat1,udat2) #For Student’st Marginals: udat1 <- pst(data[,1],location = b1hatt[1], scale = b1hatt[2], df = b1hatt[3]) udat2 <- pst(data[,2],location = b2hatt[1], scale = b2hatt[2], df = b2hatt[3]) udat3 <- pst(data[,3],location = b3hatt[1], scale = b3hatt[2], df = b3hatt[3]) udatt<- cbind(udat1,udat2) ## Second MLE for Gaussian Marginals## mycopulamvd <- mvdc(copula = ellipCopula(family = "normal", dim = 2, param = a1, dispstr = "un"), marginals = +c("norm","norm"), paramMarginals = list(list(mean = b1hat[1], sd = b1hat[2]), list(mean = b3hat[1], sd = +b3hat[2]))) mycopula <- mycopulamvd@copula fit.ml <- fitCopula(mycopula,udat,method = "ml") fit.ml ## Second MLE for Student’s t Marginals## mycopulamvdt <- mvdc(copula = ellipCopula(family = "normal", dim = 2, param = a2, dispstr = "un"), marginals = +c("st","st"), paramMarginals = list(list(location = b1hatt[1], scale = b1hatt[2], df = b1hatt[3]), list(location = +b2hatt[1], scale = b2hatt[2], df = b2hatt[3]))) mycopulat <- mycopulamvdt@copula fit.ml <- fitCopula(mycopulat,udatt,method = "ml") fit.ml ## Second MLE for Archimedean Copulas ## mycopulamvdcl <- mvdc(copula = archmCopula(family = "Frank", dim = 2, param = 9, dispstr = "ex"), marginals = +c("norm","norm"), paramMarginals = list(list(mean = b1hat[1], sd = b1hat[2]), list(mean = b3hat[1], sd = +b3hat[2]))) mycopulacl <- mycopulamvdcl@copula fit.ml <- fitCopula(mycopulacl,udat,method = "ml") fit.ml ## Creation of the fitted Copula (one case only)## finalcopula<- mvdc(copula = ellipCopula(family = "t", dim = 2, param = ahatt, dispstr = "ex", df = dfhat), marginals = c("norm","norm"), paramMarginals = list(list(mean = -0.000025617 , sd = b1hat[2]), list(mean = -0.00001242, sd = b2hat[2])))

####Monte Carlo Simulation#### p <- 50000 BNPspot <- 45.325 SGspot <- 30.86 LVMHspot <- 136.65 BNPsim <- (1:p) LVMHsim <- (1:p) SGsim <- (1:p) Payoff <- (1:p) strike <- 0.5 for ( m in (1:p)) { sim <- rMvdc(769, finalcopula) u <- 1 v <- 1 for (i in (1:769)) { u <- (sim[i,1]+1)*u v <- (sim[i,2]+1)*v } BNPsim[m] <- u*BNPspot LVMHsim[m] <- v*LVMHspot Payoff[m] <-max(min(u,v)-strike,0) } price <-(exp(-0.000923*3))*mean(Payoff) price

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