Speeding up Value-At-Risk Calculations for Option Portfolios

(1)

Speeding up Value-At-Risk

Calculations for Option Portfolios

(2)

(3)

(4)

(5)

Abstract

This thesis analyzes the speed versus the accuracy of Monte Carlo Value-at-Risk (VaR) calcu-lations, applied to a portfolio of options on the AEX, DAX and CAC indices. The returns are modelled according to a multivariate student t distribution. It is shown that large improve-ments in speed can be achieved by using the quadratic (‘delta-gamma’) approximating pricing technique to revalue the options. The related VaR proves to be accurate, when compared to the VaR which is calculated by revaluing the options by means of the exact model and under the assumption that the volatilities are modelled according to the Sticky Strike model. On the other hand, when the market is assumed to behave in a Sticky Delta way, the VaR is much more negative. The choice of volatility model is therefore important. Moreover, we consider the widely used normality assumption for the multivariate returns distribution. The data is fat-tailed and the multivariate student t distribution appears to fit the data better. However, the VaR resulting from the multivariate normal and the VaR which is based on the multivariate t distribution are not significantly different. This is caused by the non-linear characteristics of the portfolio not especially having losses in the tails of the return distri-bution. The 1% quantile of the P&L distribution of the non-linear portfolio is not directly linked to the quantiles of the underlyings. Finally, it is shown that neglecting the multivariate dependency of the dataseries gives a too pessimistic VaR estimate.

(6)

(7)

Preface

This thesis is the result of my graduation project in order to obtain the master’s degree in Econometrics, Operations Research and Actuarial Studies at the University of Gronin-gen. The research has been conducted during an internship at the Quant department of All Options.

I would like to thank the following people for their help. First of all, I would like to thank my supervisors at All Options. My main supervisor, Michel Vellekoop, guided me through this project and provided me with valuable insights during the course of this research. His suggestions and comments were very helpful. I would also like to thank my other supervisors at All Options, Julien Gosme and Christoforos Yiannacos. Regardless of there tight schedules, I could always count on their excellent support. “You know where I am” was a sentence I frequently heard. Moreover, I want to thank Francois Myburg for his involvement in the start of my thesis.

Second, I would like to express my thanks to Laura Spierdijk for her help during the process of writing this master’s thesis. After getting to know Laura during the writing of my Bachelor’s thesis, she has again proven herself as a very valuable contact. Again, I have experienced our cooperation as very pleasant. I would also like to mention Ruud Koning here. He was the co-assessor of this thesis and I have learned a lot during his courses. Furthermore, I have really appreciated his involvement and supportive advises during my studies.

In addition, I would like to express my appreciation to All Options for making this inter-esting research project possible. I want to thank all colleagues of All Options for giving me a great time. My thanks also goes to Kasper Duivenvoorden and Bj¨orn Wijbenga for being my ‘latex-helplines’ during my stay in Amsterdam.

Finally, I would like to express my deepest gratitude towards my parents, who made my studies possible. They have always supported me in every way they could.

Margriet van der Wal

(8)

(9)

Introduction and Problem

formulation

“How bad can things get?” is a question all senior managers want answered. A financial insti-tution that is involved in option trading is faced with the problem of managing its market risk. In a typical arrangement at a trading company, the responsibility for a portfolio of derivatives which depends on one particular underlying asset, is assigned to one trader or to a group of traders working together. They typically measure the risk of their portfolio in terms of the Greeks. The Greeks are sensitivities of the option values with respect to important model parameters like the underlying’s price, interest rate or volatility. Each trader and group of traders try to have an acceptable risk level by managing their Greeks, but the Greeks do not provide a complete picture of the total market risk.

Value-at-Risk (VaR) is an attempt to provide an answer to the question: ‘what is the max-imum loss of a portfolio, under a given level of probability over a specified period of time?’. The overnight VaR is a quantile of the profit-and-loss (P&L) distribution of the portfolio, over the specified horizon of one night, i.e. the period in between closure of the market in the afternoon and the opening next morning. In the empirical part of this thesis we focus on measuring the 99% overnight VaR for a portfolio containing European-style options on the AEX, DAX and CaC40 index. We choose to follow a Monte Carlo approach. One difficulty in estimating the Monte Carlo VaR lies in estimating the P&L distribution itself, especially the tail which is associated with large losses. In this estimation process two types of modelling considerations need to be made. First of all, assumptions need to be made regarding the changes in the underlying risk factors to which a portfolio is exposed. Secondly, a mecha-nism needs to be selected for translating these simulated risk factor changes into option prices.

(12)

Chapter 1. Introduction and Problem formulation

calculations are, compared to the Black-Scholes VaR.

In the pricing of options, it is important to have an accurate description of the relationship between different stochastic factors which are used to model the market. The value of an option depends on many different risk factors, such as the underlying stock price and the volatility. In the Black-Scholes model, the volatility is assumed to be constant, but in reality it varies over time and it varies for different options on the same underlying. The models which we will consider to estimate the volatility, are the Sticky Strike and Sticky Delta model (e.g., see Derman (1999)). Both models use an explicit formula for the implied volatilities. Under the Sticky Strike model the implied volatility is fixed per strike, and under the Sticky Delta model the implied volatility is fixed per delta. To model the stock price levels, both the multivariate normal and multivariate student t distribution will be fitted to the return series. The student t distribution could be an appropriate choice, since it happens to be that financial data in practice have fatter tails than is assumed by the normal distribution. We will compare the fit of both distributions and the resulting VaRs and we will investigate how the VaR changes, when the multivariate dependency between the series is not taken into account.

Once the option pricing techniques and the models for the risk factors have been determined, a set of possible option values can be obtained by means of a Monte Carlo simulation. The simulated next morning portfolio values easily follow. The overnight 99% VaR then is the 1%-quantile of this P&L distribution. The overnight VaR will be calculated several times by using the earlier mentioned pricing techniques and risk factor models.

(13)

Chapter 1. Introduction and Problem formulation

1.1 Problem Formulation

The main research questions of this thesis are:

• Is the Black-Scholes-based VaR well approximated when the options in a portfolio are revalued by means of the linear and quadratic approximating pricing techniques? • Is the VaR better approximated when it is assumed that the returns on the underlying

indices are multivariate t distributed instead of multivariate normal?

• How does the VaR change, when the multivariate dependency between the series is not taken into account?

(14)

(15)

Chapter 2

Value-at-Risk and Derivatives

2.1 Value-at-Risk

Value-at-Risk (VaR) has become the standard measure of market risk in financial institutions. Market risk is one type of financial risk and is brought on by changes in the prices of financial assets and liabilities. The main reason for the popularity of VaR, is that it shows the total risk of a portfolio in one single number. Internally, banks and other financial institutions use VaR as a risk management tool by setting VaR limits for individual activities. Externally, regulators impose capital requirements on financial institutions based on their VaR values.1

Therefore, accurately estimating the VaR is of great importance.

The VaR is the maximum loss that might occur from holding a portfolio over a certain period of time, given a specified confidence level. In mathematical terms the α ∗ 100% h-day VaRh,t,α

is implicitly defined by

P (∆Pt+h≥ VaRh,t,α) = α, (2.1)

where ∆Pt+h = Pt+h− Pt, i.e. the profit or loss (P&L) over h days and Pt is the value of

the portfolio at day t. However, in the following chapters we will usually make explicit only the dependence of the VaR on the basic parameter α (the significance level) and we drop the dependence on t and h (the risk horizon).

In a trading environment one-day VaR figures at 99% confidence levels are widely used, hence we focus on estimating the one-day V aR0.99. In this estimation process today’s value of the

portfolio Pt is known. However, tomorrow’s value Pt+1 is a random variable. It depends on

a set of random variables, called risk factors, that influence the price of each instrument in the portfolio. These risk factors can be for example interest rates, equity prices and volatility rates. The main advantage in estimating the VaR lies in the modelling of the relevant risk factors. Only after this, the distribution of the changes in the portfolio value and the quantiles can be calculated. As an illustration we plot a P&L distribution in Figure 2.1. The V aR0.99

and the 1% probability level are marked.

1_{The Basel II Accord for banks (see http://www.bis.org/publ/bcbs128.htm ) and Solvency II for insurance}

(16)

Chapter 2. Value-at-Risk and Derivatives

Figure 2.1: P&L distribution: illustration of Value-at-Risk

In the first section of this chapter we set out three approaches which can be used to model risk factors. Next, we discuss their advantages and disadvantages and explain why we will focus on the Monte Carlo simulation approach. For a more extensive discussion on the approaches, we refer to the literature, e.g. Alexander (2008, Chap. 1) and Crouchy, Galai, and Mark (2001, Chap. 4). Furthermore, we discuss a risk measure which is closely associated with the VaR.

2.1.1 Three approaches to estimate the VaR

The estimation of the VaR can be performed by essentially three methods: the analytic variance-covariance approach, the historical simulation approach and the Monte Carlo ap-proach. These approaches share the preliminary step of selecting the relevant risk factors. When the relevant risk factors are defined, modelling assumptions regarding the risk factors need to be made, such that the P&L distribution at the risk horizon can be estimated.

Analytic Variance-Covariance approach

We start with the analytic variance-covariance approach. The analytic variance-covariance approach assumes that the assets are log-normally distributed and that their joint distribution is multivariate lognormal. This is equivalent to saying that the asset returns are multivariate normally distributed, hence the covariance matrix of the asset returns is all that is required to capture the dependency between the risk factor returns and there exists an explicit formula for the VaR (see Crouchy et al. (2001, Chap. 1)).

Monte Carlo approach

(17)

Say the amount of simulations is N , then each path generates a set of possible values for the J risk factors. Given the pricing function f of the instrument, tomorrow’s value of the instrument in the ith scenario is

ˆ

Pt+1,i= f (v1,i, ..., vJ,i), i = 1, ..., N

where vj,i is the value of the jth risk factor, in the ith scenario at time t + 1. The resulting

P&L from the ith scenario is

∆ ˆPt+1,i= ˆPt+1,i− Pt, i = 1, ..., N

where Pt is the value of the instrument at time t. By sorting the P&L vector ∆ ˆPt+1 in

increasing order, the one-day VaRα is simply the (1 − α)N ’th value. The approach can easily

be extended when a portfolio consists of several instruments.

Historical Simulation approach

The historical simulation approach is conceptually simple. Like the other approaches, first the relevant risk factors need to be selected. Then, the changes that have been seen in the relevant risk factors are analyzed over a specified historical period, say one to four years. The portfolio under examination is then revalued using the changes in the risk factors derived from the historical data, to create the distribution of the portfolio returns. Each daily simulated change in the value of the portfolio is considered as an observation in the h-day P&L distribution. The last step consists of sorting the values to identify the (1 − α)100% quantile.

2.1.2 Advantages and disadvantages of the different approaches

Each of the approaches we have described has advantages and disadvantages. The major attraction of historical simulation is that the method is completely nonparametric and does not depend on any assumption about the distribution of the risk factors. What has to be done is to calculate the synchronous risk factor changes over a given historical period. Compared to the analytic variance-covariance approach it has no problem accommodating fat tails, since the historical returns already reflect actual synchronous moves in the market across all risk factors. The main drawback of the historical simulation is that it assumes that all possible future variations have been experienced in the past and that the historically simulated distribution is identical to the returns distribution over the forward looking risk horizon. Another practical limitation of historical simulation is data availability. The number of data points used to construct the historical distribution is equal to the number of observations on each risk factor return in the simulation. One year of data corresponds to only 250 data points on average, i.e. 250 scenarios. By contrast Monte Carlo simulations usually involve thousands of simulations.

(18)

In its most basic form the Monte Carlo VaR model uses the same assumptions as the normal linear VaR model, i.e. that the risk factor returns are i.i.d. with a multivariate normal distri-bution. Applied to a linear portfolio the analytical derived VaR should be equal to the Monte Carlo VaR which results from simulating the risk factors. However, the Monte Carlo VaR model is extremely flexible and can be based on virtually any multivariate distribution for risk factor returns. Also, it can be applied to non-linear portfolios. It does have a drawback: it can be computationally intensive and therefore it takes a lot of time to calculate the VaRs of very large and complex portfolios.

Summarizing, the analytic variance-covariance will not be our method of choice, since it can not be applied to option portfolios. Both Monte Carlo and historical simulation methods can be applied to portfolios containing options and also allow the calculation of the confidence in-terval of the VaR. As mentioned before, the underlying assumption of the historical simulation approach is that the past, as captured in the historical data set, is a reliable representation of the future. We consider this as a major drawback since the economic environment is quickly changing. Also, the number of data points is usually much smaller than the number of simu-lations in the Monte Carlo approach. As a consequence of this, the number of points in the lower tails of the P&L distribution is much smaller for the historical approach giving a less precise estimate of the VaR.

It is possible to tackle the most important drawback of the Monte Carlo approach, so therefore it is this approach upon which our research will be based. The Monte Carlo VaR can be computationally intensive for large portfolios because of two factors. First, computing the value of an individual instrument may itself be costly. Second, a large number of simulated sets (portfolio evaluations) is required to obtain accurate estimates of the loss distribution. Variance reduction techniques were developed to handle this second issue. They are set out in Glasserman (2004, Chap. 4) and applied in (Glasserman, Heidelberger, and Schahabuddin 2000). In this project we will focus on the first issue: speeding up the revaluation of the instruments. We will employ a linear (‘delta’) and quadratic (‘delta-gamma’) approximation to the change in portfolio value.

2.1.3 VaR for non-linear portfolios and other risk measures

There is a fundamental distinction between linear and non-linear portfolios. The P&L of a linear portfolio may be expressed as a linear function of its risk factors. Portfolios containing only stocks fall into the category of linear portfolios. Option portfolios, on the other hand, are non-linear portfolios since the value of an option is a non-linear function of its input variables. Identifying if either up or down movements in the risk factors cause a reduction in the portfolio value more difficult than if the portfolio is linear. This will become clear in the empirical part of this research.

(19)

how much we could lose if we are ’hit‘ beyond the VaR. Another undesirable property of the VaR is that it is not a coherent risk measure, while the expected shortfall is coherent.2 The VaR is not coherent because quantiles do not obey simple rules such as sub-additivity, unless the returns have an elliptical distribution and the portfolio is linear.

Even though the VaR has some disadvantages, it is applied a lot in practice and research. For reasons of comparrison, we therefore choose to estimate the VaR. A disadvantage of the Expected Shortfall not mentioned yet is that it is difficult to estimate since it completely depends on the events in the tail of the P&L distribution, i.e. it integrates the tail events.

2.2 Derivatives

To be able to calculate the VaR of a certain portfolio of options, some knowledge about op-tions themselves is needed. Opop-tions fall in the category of derivatives and a derivative can be defined as a financial instrument whose value depends on (or derives from) the values of other, more basic underlying variables (Hull 2006). Derivatives have a long history in commerce since they can be used to control risk. A classical example is associated with the Dutch tulip mania in the seventeenth century. The growers bought put options to guarantee a minimum price for their bulbs and on the other hand the dealers bought call options to assure maximum future prices. In the end, the market crashed in 1636 due to the burst of the arised bubble.

Nowadays, there is a wide assortment of derivatives on financial instruments through regulated exchanges. The underlyings can be equity indices, interest rates or exchange rates. How the derivative is related to the underlying asset depends on the type of derivative. Examples are futures, forwards, swaps and options. Since the current financial crisis started, some of these derivatives have been discussed a lot and regulations are adjusted. We will give a short overview on this still ongoing discussion. After that, we focus on stock options; we will explain how this market is structured and what options exactly are. In the end, we will discuss what factors affect the prices of options and handle the well known Black-Scholes formula.

2.2.1 Background on the derivatives market (2008-2009)

As explained, there are many types of derivatives. Some are standard products, such as options, while others are not. These products are standardized contracts that have been defined by the derivatives exchanges and their prices are publicly displayed. Non-standard products, such as for example swaps, are tailored to the specific needs of the user. They offer full flexibility in deciding what for example the end of the contract will be (maturity), while the standard products only offer a limited choice of maturities that can be traded. The non-standardised derivatives are traded off-exchange, as commonly called over-the-counter (OTC), where prices remain private. One other big difference between the OTC and regular derivatives exchanges is that since the derivatives exchanges act as an intermediary to all

2

(20)

related transactions, they take an initial margin from both sides of the trade to guarantee the trades. On the other hand, when a trade on the OTC market is done, both parties are subject to counterparty risk since they depend on the performance of eachother.

It is believed that this counterpary risk on the OTC derivatives markets played a role in the propagation of the current financial crisis. The failure of AIG3 on September 16, 2008 was linked to improper hedging of the credit default swaps they were selling. People argue that due to the central role AIG played in the complex network of OTC derivative markets and the counterparty risks that come with these trades, many banks were exposed to the bankruptcy of AIG.

Many institutions have done research on the role of OTC derivatives in the crisis, including

The European Commission.4 Commission of the European Communities (2009) conclude

that the bilateral nature of the OTC markets creates a complex web of mutual dependence between counterparties. Coupled with the markets’ opaqueness this creates a situation in which it is difficult for market participants and regulators to fully understand the true na-ture and level of risks that market participants are exposed to. This increases uncertainty in times of market stress and may undermine financial stability as has been illustrated by the ongoing financial crisis. Therefore, the European Commission and several other institutions are working on ways to strengthen the derivatives markets.

2.2.2 Market structure of stock option markets

Earlier we explained the existence of OTC markets and stock option markets. We will now focus on the stock option markets and explain how they are structured. Stock options are bought and sold at several options exchanges such as the United States Chicago Board Op-tions Exchange5 _{and the NYSE Euronext}6_{. These option exchanges use market makers to}

facilitate trading. A market maker is an institution that quotes both a bid and an ask price on options. The ask price is the price at which the market maker is prepared to sell and a bid price is the price at which it is prepared to buy. The bid is always lower than the ask, and the difference between the two is called the bid-ask spread. Market makers are called liquidity providers since they provide liquidity to the market. They will often be the counterparty when an individual wants to buy or sell an option.

2.2.3 Options

An option is a contract between a buyer and a seller. There are two basic types of options: call options and put options. A call option gives the buyer the right, but not the obligation,

3_{American International Group (AIG), is an American insurance corporation and is listed on the New York}

Stock Exchange (NYSE). AIG suffered from a liquidity crisis and therefore the United States Federal Reserve Bank created an 85 billion credit facility on September 16, 2008.

4

The European Commission, formally known as the Commission of the European Communities, is the executive branch of the European Union. One of its tasks is the responsibility for proposing legislation.

5_{The Chicago Board Options Exchange (CBOE) is the largest U.S. options exchange (www.cboe.com).} 6

(21)

to buy the underlying asset by a certain date T for an agreed price. A put option gives the buyer the right to sell the underlying asset by a certain date T for an agreed price. This agreed price is known as the strike price K and the expiration date T is also called the ma-turity. When the options can only be used (exercised) at maturity, the options are called European-style options. American options on the other hand, can be exercised at any time before and including the expiration date. If the buyer of an option chooses to exercise his right, the seller is obliged to sell or buy the asset at the strike price. The buyer may also choose not to exercise the right and let it expire. In return for granting the option, the seller receives a payment from the buyer, which is called a premium. In other words, the premium is the price of the option. Note that when an option is bought, the position in the option is said to be long and when it is sold the position is called short.

2.2.4 Pricing options

In this section we will focus on European-style equity options and first look at the value of options at the expiration date. Next we will give the Black-Scholes formula which can be used at any time before and including expiration. We choose to focus on this pricing technique since it is a closed-form formula.

The value of an option at maturity T , can easily be determined. The value of a call option at maturity time T depends on the price of the underlying stock (ST) at this time. Therefore,

the value of the call option at time T can be denoted by C(ST, T ). At maturity there are

two scenarios possible. The underlying’s price at maturity can be higher than the exercise price, i.e. ST > K. The call option will be exercised. The holder buys the underlying at price

K and could sell it immediately, realizing a profit of ST − K. In the second scenario, the

underlying’s price at maturity is equal or lower than the exercise price, i.e. ST ≤ K. It now

can be easily reasoned that the value of the call option at maturity is given by C(ST, T ) = max(0, ST − K).

A call option is said to be in the money, at the money, or out of the money, depending on whether St> K, St= K or St< K, respectively. This terminology applies at any time t, so

also before expiration.

Similar arguments as above show that the payoff for the put option is given by P (ST, T ) =

max(0, K − ST), where P (ST, T ) is the value of the put at maturity.

The preceding analysis focused on the value of an option at expiration. This value is derived from the payoff structure of an option. Even though European options can not be exercised before expiration, they do have a value at earlier times, since they provide the potential for future exercise. In 1973, Fischer Black and Myron Scholes made a major breakthrough in the pricing of stock options. This involved the development of what has become known as the Black-Scholes model. The Black-Scholes model can value European call and put options on a non-dividend paying stock. To be able to apply the Black-Scholes model some assumptions need to be made. The stock price St is assumed to follow a geometric Brownian motion and

(22)

liquid, i.e. it is always possible to buy and/or sell unlimited quantities on the market. If a market is arbitrage free, it is assumed that it is not possible to take advantage of two or more securities being mispriced relative to each other. These are the most important assumptions of the Black-Scholes model and for more information on these assumptions we refer to Bj¨ork (2004).

Black and Scholes assume that several factors affect the value of options. One of these factors is the time to maturity T − t. The current time in years is here denoted by the letter t, and t ∈ [0, T ]. Therefore, the time to maturity is the time measured in years until expiration. Both put and call options become more valuable as the time to maturity increases. This pos-itive effect of time to maturity is called the time value of options. Other factors that affect the value of an option are the current Stock price St, the strike price K, the implied volatility

σ, the risk-free interest rate r and the dividends expected during the life of the option. Now that the factors that influence the value of an option have been discussed shortly, we will show the relationship between these factors and the option value by giving the Black-Scholes formula (Black and Scholes 1973):

Black-Scholes formula for call and put options At time t ≤ T , the price of a European call option (put option) with strike price K, expiration time T , interest rate r and volatility σ, is defined by the formula for C(St, t) (P (St, t)) :

C(St, t) = StN [d1(St, t)] − e−r(T −t)KN [d2(St, t)]

P (St, t) = Ke−r(T −t)N [−d2(St, t)] − StN [−d1(St, t)].

Here N is the cumulative distribution function for the standard normal distribution and

d1(St, t) = ln(St/K) + (r +1₂σ2)(T − t) σ√T − t d2(St, t) = d1(St, t) − σ √ T − t.

The following notation is more complete for the value of a call: C(St, t, T, r, σ, K), but for

convenience we will use the notation C(St, t), or even shorten it to C in this thesis. For the

value of puts, P (St, t, T, r, σ, K), we will mostly use the notation P (St, t) and P . We want to

end this section by noting that when t → T , the solutions of the Black-Scholes formulas are consistent with the payoffs given in the previous subsection.

A portfolio is a particular combination of certain assets. Consider a portfolio of options on one and the same underlying. To determine the value of this portfolio, one needs to know the positions taken in each option under consideration. Assume now that we have a portfolio on n different type of options, where wi represents the number of options of type i held at

time t, where i = 1, ..., n. A positive wi implies a long position in option i, while a negative

wi implies a short position. The value of the ith option will be denoted by Vi(St, t), where

Vi(St, t) is the same as C(St, t) when option i is a call, and P (St, t) when it is a put. The

portfolio value Π now is given by:

Π =

n

X

i=1

(23)

Chapter 3

Approximating pricing techniques

and modelling risk factors

3.1 Approximating pricing techniques with respect to the

Black-Scholes model

In Chapter 2 it has been explained that to calculate the Monte Carlo VaR, each option in the portfolio needs to be revaluated for a large number of simulated values of the underlying fac-tors. In a portfolio with many instruments this procedure can be quite time consuming, even when you use the Black-Scholes method. Therefore we consider two approximating pricing techniques, with respect to the Black-Scholes. An important feature of these approximations is that they require the first and second-order sensitivities of the option price with respect to the underlying value, i.e. the deltas and gammas. These are usually already calculated, hence we could say that they are part of the portfolio characteristics. The delta and gamma are examples of the Greeks and they are called the Greeks since they are named after letters in the Greek alphabet.

3.1.1 The Greeks

The Greeks are the sensitivities of options to a change in the underlying parameters on which the value of the option depends. The delta and gamma are the first and second partial deriva-tives with respect to the underlying price respectively. We will discuss these Greeks one by one, where we will sometimes omit the expression for the put option since it can be derived in the same way as the Greek for the call option.

The delta (∆) of an option is the partial derivative of the option price C with respect to the price of the underlying asset S. In other words, the delta measures the degree to which the value of an option is affected by a small change in the price of the underlying asset:

∆ = ∂C

∂S. (3.1)

(24)

Chapter 3. Approximating pricing techniques and modelling risk factors

derivation, we refer to Hull (2006). We also want to point out that the formal notation for the delta is ∆(St, t, T, r, σ, K), but for convenience we will omit the dependence structure in

the notation for all Greeks.

In case you have a portfolio of options on one single asset, the delta of the portfolio can easily be calculated from the deltas of the individual options (∆i) in the portfolio. If the portfolio

consists out of n different options, where the quantity of option i (1 ≤ i ≤ n) is wi, then the

delta of the portfolio (∆Σ) is given by

∆Σ= n

X

i=1

wi∆i. (3.2)

The gamma (Γ) of an option is the partial derivative of the option’s delta with respect to the price of the underlying asset. It is the second partial derivative with respect to the underlying price:

Γ = ∂

2_C

∂S2. (3.3)

Note that gamma defines the curvature of the option price curve as a function of the un-derlying. In fact, it is the second derivative of the option price curve at the point under consideration. Under the Black-Scholes formula and European options on non-dividend pay-ing stocks, the gamma of both call and put options is equal to:

Γ = N 0_(d 1) S0σ √ T (3.4)

The gamma of a portfolio of options on one underlying, where the gamma of the ith option is Γi, simply is ΓΣ= n X i=1 wiΓi. (3.5)

Another useful number is theta (Θ):

Θ = ∂C

∂t. (3.6)

It measures the time decay of an option, so it is the partial derivative of the option’s value with respect to the passage of time with all else remaining the same. Under the Black-Scholes formula, Equation (3.6) can be written alternatively for European options on non-divided paying stocks as follows:

Θ(call) = −S0N 0_(d 1)σ 2√T − rKe −rT_{N (d} 2), (3.7)

and the theta for a put option is equal to

Θ(put) = −S0N 0_(d 1)σ 2√T + rKe −rT_{N (−d} 2). (3.8)

(25)

that buy and sell orders can always be executed. Market makers therefore provide liquidity to the market and individuals and institutions can buy options from them indirectly. Usually, market makers trade huge amounts of options every day. The Greeks are directly used by trading desks to make sure that the risks that come with these trades are acceptable. In practice, trading desks usually try to have a zero delta and zero gamma position by the end of the day.

3.1.2 Linear approximation model based on the delta of options

The linear approximating technique approximates the value of options as a linear function of the underlying price. Stated in other words, the linear approximating pricing method approximates the value of options as first-order Taylor expansions in the changes in the underlying security prices. Notice that we only take into account the changes in the underlying security prices and time and not the changes in other risk factors such as the volatility. If we again say that V (St, t) is the value of the option at time t on a certain underlying with price

St, then the value of the option at time t + δt can be approximated by

V (St+ δS, t + δt) = V (St, t) + ∂V ∂S St,t ∗δS + ∂V ∂t St,t ∗δt, (3.9)

where δS = St+δt− St. If you want to estimate the next day opening value of an option,

you know exactly how much time will pass so that the last term in Formula 3.9 is known. Therefore we will omit it in the empirical part of this thesis. Note that the formula will be referred to as the linear approximating model, or the Delta-Approximation. This last name is used due to the term ∂V_∂S inside the formula, which is obviously the delta of the option. Once more note that ∆ is the Greek delta, and δ is used to denote a ’difference’.

3.1.3 Quadratic approximation model based on the delta and gamma of options

Formula (3.9) can easily be extended to the quadratic approximating formula by including a second term of the Taylor expansion too. Including a quadratic term in the approximation of V (St+ δS, t + δt), implies using the gamma of an option as well as the delta:

V (St+ δS, t + δt) = V (St, t) + ∂V ∂S St,t ∗δS + 1 2 ∗ ∂2V ∂S2 St,t ∗(δS)2+∂V ∂t St,t ∗δt. (3.10)

In the empirical part in the next section, we will also omit the last term in Formula (3.10) and the formula will be referred to as the quadratic approximation model or the delta-gamma approximation, since the term ∂_∂S2V2 in fact is the gamma (Γ) of the option. It is important to

notice that all the terms in Formula (3.10) are known at time t. The delta and the gamma are calculated as a part of the option pricing model at time t, the old value of the option V (St, t)

is of course known too, and the difference in the underlying δS follows from the simulation of the stock returns.

(26)

Π(St+ δS, t + δt) = Π(St, t) + ∆Σ(St, t) ∗ δS +

1

2 ∗ ΓΣ(St, t) ∗ (δS)

2_. _(3.11)

Since the portfolio value Π, the portfolio’s delta ∆Σ and the portfolio’s gamma ΓΣ are all

known at time t, the approximated value can be calculated fast and the calculation time is independent of the number of options in the portfolio. If you would use the Black-Scholes formula to revalue the portfolio, this would be way more time consuming, since you would have to revalue every option separately. The calculation time using the Black-Scholes formula to revalue your portfolio is therefore proportional to the number of options in the portfolio. So the calculation time using the Black-Scholes formula is of order n, while the calculation time using the quadratic or linear approximating pricing technique is of order 1.

3.2 Modelling risk factors

3.2.1 Modelling the underlying stock prices

Many stocks do not move completely independently from each other. One way to capture the movement and dependence of asset returns is by fitting a multivariate distribution. Quite a while ago, Mandelbrot (1967) and Fama (1965) already presented some evidence that most empirical distributions are more peaked (leptokurtic) and skewed (asymmetric) than the tra-ditionally hypothesized Gaussian distribution. However, classical multivariate analysis has been rigidly tilted toward the multivariate normal distribution, while the multivariate t dis-tribution offers a more viable alternative with respect to real-world data. The multivariate t distribution has fatter tails than the normal one; it assigns more weight to joint extreme outcomes. There are many more heavy tailed distributions, but in this thesis we will investi-gate only the fit of the multivariate student t, since it is the most widely used heavy tailed distribution. The Black-Scholes model on the other hand assumes that the underlying stock prices are lognormally distributed. Therefore, we will also fit the classical multivariate nor-mal distribution, so that in the end we can compare the VaR based on the multivariate t and multivariate normal distribution. The univariate student t and normal distribution will also be fitted to the data in the empirical part of this thesis. They can easily be derived from the multivariate distributions, hence we will only state down the multivariate t and multivariate normal distribution.

Multivariate normal distribution

When a k-dimensional vector X0 = (X1, ..., Xk) with mean vector µ and covariance matrix

C is multivariate normally distributed, this is denoted as Nk(x|µ, C). The density function

is given by: Nk(x|µ, C) = (2π)− k 2|C|− 1 2exp −1 2(x − µ) 0 C−1(x − µ) . (3.12)

(27)

Multivariate t distribution

A k-dimensional random vector X0 is said to have a multivariate t distribution with ν degrees of freedom, mean vector µ and k-by-k matrix C, if its joint probability density function is given by: Tk(x|ν, µ, C) = Γ((ν + k)/2) (πν)k/2_Γ(ν/2)|C|1/2 1 +(x − µ) 0_C−1_{(x − µ)} ν −(ν+k)/2 , (3.13)

where Γ is the gamma function. Unfortunately, in contrast to the normal distribution, no analytical expressions are available for the maximum likelihood estimates of µ, C and ν. Like-lihood optimisation has to be done numerically. Therefore C is not necessarily equal to the covariance matrix. The covariance matrix of X is (ν/(ν −2))C, and it is only defined if ν > 2.

The multivariate t distribution is able to capture dependence between the several compo-nents. This is a major advantage compared to estimating the univariate t distribution for each component separately. Note that if C appears to be diagonal, there is still dependency between the marginal distributions, due to the joint estimation of the degrees of freedom parameter. This is different for the multivariate normal distribution; when C is diagonal, the multivariate normal distribution is similar to the k separate univariate normal distributions.

The degrees of freedom parameter ν is also referred to as the shape parameter; it is a measure for the peakedness of (3.13). By varying ν the peakedness may be diminished, preserved or increased. When ν is increased a lot, the t-distribution approaches the standard normal distribution. The distribution is said to be central if µ = 0. Note that if k = 1, µ = 0 and C = 1, Equation (3.13) reduces to the univariate Student t distribution. Figure 3.1 shows the probability density function for three univariate t distributions all with µ = 0 and σ = 1, but with varying degrees of freedom. The dotted line (ν = 1) has very fat tails and as we move to the dashed line (ν = 4), the tails become less fat. When the degrees of freedom are increased to 100, the solid line seems to get the same shape of a standard normal distribution.

Another way to capture dependency between random variables is for example by using cop-ulas. A copula enables the univariate marginal behavior of random variables to be modeled separately from their dependence. Usually it is easier to estimate multivariate distributions than copulas, hence we choose to estimate the normal and t distribution in the empirical part of this thesis.

3.2.2 Modelling the implied volatilities

People in the market often talk about how volatility changed. There are different kinds of volatility such as the realized volatility and the implied volatility. The realized volatility of the returns on an index Si gives you information about how risky the index was in the past.

In fact, it is the standard deviation of changes in the logreturns over a certain period of time. It can be defined by stating that the realized daily volatility s, of an index Si over a period

(28)

Figure 3.1: Comparison of the univariate student t distributions with ν = 1, ν = 4 and ν = 100, where for all three distributions µ = 0 and σ = 1

s2 = 1 N − 1 X i (ri− µ)2, where µ = 1 N X i ri.

The implied volatility σ of a stock’s option is defined differently. The implied volatility is the value of the volatility parameter in the option pricing model that makes the theoretical value of the option match the market price Cobs of the option. When the Black-Scholes pricing

model is used, the implied volatility is that σ for which the following equation holds:

CBS(St, t, T, r, σ, K) = Cobs. (3.14)

Notice that at time t for each option (with a certain strike K and maturity T ), the underling stock price St, the interest rate r and the market price Cobs can be observed. Therefore, the

volatility parameter can be obtained. In the classical Black-Scholes world, the stock’s return volatility is assumed to be a constant, i.e. it is independent of strike and time to expiration. In other words, all options on the same underlying are assumed to have the same constant volatility. When you now would calculate the theoretical prices of let’s say ten options on the same underlying index, you plug in every option’s own K and T and one constant σ for all of them. Unfortunately, it is highly unlikely that these theoretical Black-Scholes prices would match the prices of these options observed in the market. In reality, options with different strikes or different maturities require different volatilities to make their theoretical Black-Scholes price match with their market price.

Implied volatility patterns: smile and skew

(29)

higher than at-the-money volatilities. In other markets, the volatility pattern looks more like a skewed curve (see right graph in Figure 3.2). This has motivated the name volatility skew.

Figure 3.2: Two examples of volatility curves

There are several explanations for why volatilities exhibit a smile. The one we consider most important relates to the idealized assumptions of the Black-Scholes model. The returns are assumed to be normally distributed. However, often the returns appear to be more leptokurtic than is assumed by the normal distribution. Leptokurtosis would make way out-of-the-money or in-the-money options more expensive than if they were priced with the Black-Scholes for-mula under the assumption of lognormality. By increasing the volatility for such options, the volatility smile could be the markets’ indirect way of achieving such higher prices within the imperfect Balck-Scholes model.

A reason for the volatility skew traders sometimes come up with is one that relates to supply and demand for options. In equity markets, the volatility skew could reflect investors’ fear of market crashes. This would cause them to bid up the options which have a strike below current market level. A more theoretical reason for the skew is the following. The implied distribution of the underlying of many indices and assets often has a heavier left tail than right tail. Very negative returns happen then more than very positive returns. Therefore, deep-out-of-the-money puts have a relatively high price and deep-out-of-the-money calls have a relatively low price. A relatively high price leads to a relatively high implied volatility and a relatively low price leads to relatively low implied volatility. Both explained arguments show that the implied volatility of options can be a decreasing function of the price of the asset and is consistent with the right graph in Figure 3.2.

(30)

As explained above, traders still use the Black-Scholes model, but not in exactly the way Black and Scholes originally intended. The two most important assumptions of the Black-Scholes formula are that the price of the asset follows a geometric Brownian motion and that the volatility of the asset is constant. These two assumptions imply that the price of the asset is lognormally distributed. In reality, the price of the asset is heavier tailed than the lognormal distribution. Nevertheless, the Black-Scholes model is still used, however, in a slightly different way since the implied volatilities can exhibit a smile or a skew.

Implied volatility models

Given the current volatility skew, one wants to know how it will vary with index level in the future. We will discuss three implied volatility models, from which two will be used in the empirical part of this theses. All models can be easiest explained by stating what does not change. Traders refer to what does not change as ‘sticky’. The most simple model is the Sticky Strike model (e.g., see Derman(1999)). It assumes that given the current smile, each option of definite strike will maintain its initial implied volatility. The implied volatility of an option on a certain index with a specific maturity depends only on the strike K at any point in time. Therefore, the dependence structure could be denoted by σ(K).

A second model is the Sticky Moneyness Model. The Sticky Moneyness model is a more subtle view of what quantity remains invariant as the index moves. The Sticky Moneyness model assumes that an option’s volatility depends only on its moneyness K/S. The implied volatility of an option on a certain index can be denoted at time t as σ(K/St). As the price

of the underlying asset changes, moneyness of an option changes and therefore the implied volatility of the option will change. Note that the Sticky Moneyness model quantifies the intuition that the current level of at-the-money volatility should stay constant as the in-dex moves. Similarly, for example, the option that is 5% in-the-money after the inin-dex moved should have the same implied volatility as the 5% in-the-money option before the index move.

The Sticky Moneyness Model can be seen as a simple version of the Sticky Delta model. The Sticky Delta model assumes that an option’s volatility depends only on its delta, so that the dependence structure can be denoted by σ(∆). At a certain point in time, the volatility as a function of delta can be calibrated from the market. It can be reasoned that each delta is then associated with one specific option (which has a specific strike). However, when the underlying changes, this has an effect on the delta of options, since the delta depends on the underlying value St and other variables. Therefore, the resulting implied volatility for a

certain option will be changed according to its change in delta. In Appendix B.1 this process is written down in more detail.

(31)

both models depend on one stochastic source, we will call the implied volatility a risk factor throughout this thesis.

Under the Sticky Delta model, volatilities of options can change over time. Each option has its own volatility and therefore its own sensitivity with respect to the implied volatility. This means that the value of the option is liable to a change because of movements in volatility as well to changes in the underlying and the passage of time. The rate of change of the value of an option with respect to its volatility is named vega:

V = ∂C

∂σ. (3.15)

(32)

(33)

Chapter 4

Empirical results

This chapter addresses the problem of estimating the Monte Carlo VaR for an option portfolio using both exact valuation and the approximating pricing techniques handled in Chapter 3. We will empirically test how close the VaRs resulting from the linear and quadratic approximating pricing technique reach the VaR under which the options are revalued by means of the Black-Scholes formula. Also, we will observe to what extent the computational time is reduced. The portfolio we consider contains various options on the AEX1, DAX2 and CAC3indices. Throughout this entire chapter we will ignore the impact of dividend payments on the option prices.

We start with a description of the three indices in Section 4.1. The returns on the indices are important risk factors for the portfolio of options. Therefore, we will fit multivariate distributions to the return series in Section 4.2. In most research projects, returns are assumed to be multivariate normally distributed (e.g., see El-Jahel, Perraudin, and Sellin (1999)). Besides the normal distribution, we will fit the student t distribution to the data to capture leptokurtosis in the underlying returns. The reader will be introduced to Monte Carlo VaR and the approximating pricing techniques in Section 4.3. There we will perform several VaR calculations for a simple delta-hedged test-portfolio containing options on only the AEX. Note that the implied volatility models are not taken into account here.

In Section 4.4 we will carry out VaR calculations for the real-life portfolio containing options on the AEX, DAX and CAC. The accuracy and speed of the VaR calculation based on the quadratic approximating technique will be investigated by comparing it with the VaR based on the exact formula.4 In calculating the latter, the implied volatilities are considered as risk factors, but they are modelled according to the Sticky Strike or Sticky Delta model so implied volatilities follow directly from stock movements (Sticky Delta) or are constant (Sticky Strike). Also, the consequence of the choice of multivariate distribution on the VaR is analyzed. Furthermore, the effect that multivariate dependency has on the VaR is shown.

1

The AEX index stands for the Amsterdam Exchange Index and it is a capitalization-weighted measure of a maximum of twenty five most traded Dutch companies that trade on the Euronext Amsterdam.

2

The DAX index, i.e. the Deutscher Aktien Index is a stock market index consisting of the thirty major German companies listed on the Frankfurt Stock Exchange.

3_{The CAC40, in France known as the CAC quarante is the most important French stock market index. It}

is a capitalization-weighted measure of the 40 most traded French companies.

(34)

Chapter 4. Empirical results

4.1 Data description

The data on the AEX, DAX and CAC cover the period from January 2, 2004 to July 24, 2009. For all three indices we have their opening and closing values, but some days are missing due to holidays and other market closures. When at least one of the three indices is closed on a certain day, we delete it, so that we end up with all three indices having 1382 observations.5 In Figure 4.1 the indices are plot, where the AEX and CAC index are scaled. It can be observed that the three indices move in the same direction most of the time.

Figure 4.1: Closing values of the DAX, scaled AEX and scaled CAC index: 2004 to 2009

Notes: 1382 observations. The AEX and CAC index are scaled such that they have the same value at the first observation point; the AEX index is multiplied by 11.72 and the CAC index by 1.12.

The movements of all three indices from June 2007 to October 2008, indicate that it was a period of economic uncertainty. In the middle of October 2008 the indices eventually dropped dramatically which can be explained by the financial crisis hitting the stock markets. In the beginning of 2009 the stock markets show some signs of recovery and an up-going trend is observed.

Since all three indices show a certain trend, their price levels are non-stationary and therefore they will not be used in our further research directly. Instead, we will use the overnight logreturns, also referred to as returns, or overnight returns. The overnight returns are the

5_{Consider one market to be closed on day t + 1. We delete day t + 1 for all indices and calculate the}

(35)

returns on the index over the horizon of one night, i.e. the period in between the closure of the market in the afternoon and the opening the next morning. The formula for the overnight return on the t’th day therefore is

rt= Ln

St,open

S_t−1,_close !

, (4.1)

where St,open is the opening value on day t and S_t−1,_{close is the closing value of the index}

on day t − 1. Note now that t is measured in days, but if we talk about maturity T or time to maturity T − t they are in years.

As mentioned before, we choose to ignore the impact of dividend payments on the returns, because we assume the dividends to be small. The graphs of the returns are shown in Figure 4.2 and all three plots show a changing amplitude of the returns. The returns on the DAX index seem to be lower than the AEX and CAC returns, but overall the three graphs show a similar pattern of returns. The peak at January 22, 2008 is striking in all graphs. During this night the financial crisis hit the Asian markets hard and many Asian indices dropped significantly, causing the AEX, DAX and CAC opening almost 11% lower in the morning of January 22. We also note that the return series exhibit volatility clustering. During the peak of the Financial Crisis, i.e. from September 2008 up to March 2009, large returns tend to be followed by large returns in all three series. These observations match the observed correlations between the series; the correlation between the log returns of the AEX and DAX is 0.77, AEX and CAC is 0.96 and the correlation of the DAX and CAC is 0.80. These high correlation figures are obviously caused by the Netherlands, Germany and France being geographically close to each other and the still ongoing globalisation of many companies and markets.

Descriptive statistics are shown in Table 4.1. The table shows that the average overnight returns are about zero percent. All indices exhibit significant excess kurtosis. The kurtosis is a measure for the peakedness of a variable. This means that the univariate distributions are leptokurtic, i.e. they are more peaked than the normal distribution which has a kurtosis of 3. So extreme movements in the indices are more likely than a normal distribution would predict. The skewness is a measure of the asymmetry of the data around the sample mean. It is negative for all the indices, implying that the data are spread out more to the left of the mean than to the right. Especially the DAX has a very negative skewness, i.e. −5.13. Both the measures of the skewness and the kurtosis, indicate that univariate distributions are heavy-tailed.

4.2 Estimating the distribution of the returns

(36)

Figure 4.2: Overnight logreturns (in %) on the AEX, DAX and CAC from 2004 to 2009

4.2.1 Univariate estimations

(37)

Table 4.1: Descriptive statistics for overnight returns on the AEX, DAX and CAC: 2004-2009

Variable AEX DAX CAC Mean1) 0.0337 0.00404 0.0290 Standard deviation1) _0.94 _0.65 _0.95 Minimum1) _-10.8 _-10.4 _-11.4 Maximum1) 6.93 3.75 6.57 Kurtosis 27.9 76.4 30.9 Skewness -1.10 -5.13 -1.28

Notes: 1382 observations. 1)in percentages

since they represent the sample means and the standard deviations. To now evaluate the fit of both distributions, we plot in Figure 4.3, for each index, the estimated distributions on top of the histogram.

As observed, the normal distribution does not assign enough probability mass to the central part and the tails of the datasets. It assigns too much probability mass to the region which is intermediate between the tails and the center. For all three indices, the t distribution seems to fit best. To check this observation, we also plot some QQplots. A QQplot (quantile-quantile plot) is a standard visual tool for showing the relationship between the empirical quantiles of the data and the theoretical quantiles of a reference distribution (McNeil, Frey, and Embrechts 2005). A lack of linearity would be evidence against the hypothesized reference distribution . In Figure 4.4 we show two QQplots; one of the overnight AEX returns with the t distribution as reference, and one with the normal distribution as reference. The QQplots for the DAX and CAC returns are shown in Table A.1.

Table 4.2: Estimated parameters for the univariate normal and student t distributions

AEX DAX CAC

Student t µ 0.0611 0.0375 0.0561 σ 0.412 0.280 0.436 ν 2.104 2.38 2.261 Normal µ 0.0337 0.00404 0.0290 σ 0.945 0.646 0.952

Notes: For the AEX, DAX and CAC index, the univariate student t distribution (upper part table) and the univariate normal distribution (lower part table) are estimated. µ - mean; σ - standard deviation; ν - degrees of freedom. The distributions are estimated using 1382 overnight returns on the three indices, i.e. from January 2, 2004 to July 24, 2009. All estimates, except for ν are in percentages.

(38)

Figure 4.3: Histograms of overnight logreturns, with estimated distributions

Notes: The histograms of the overnight returns on the AEX (left), DAX (middle) and CAC (right) together with the estimated normal distribution (dashed line) and student t distribution (solid line). The data consists of 1382 observations, i.e. from January 2, 2004 to July 24, 2009. The overnight returns are in percentages.

distribution. Looking at more extreme events, we see that the tails are underestimated by the normal distribution. Summarizing, we think that the normal distribution does not approximate the quantiles of the returns series well; neither in the tails nor in the middle.

Figure 4.4: QQplots of the overnight AEX returns against a t reference distribution (left) and against the normal reference distribution (right)

(39)

The student t QQplots all show that the t distribution models the returns well in the middle of the distribution. Take for example the QQplot of the AEX (left graph in Figure 4.4). The t distribution fits the data well for returns in the interval of −3% to 3%, but for more extreme events, it either slightly overestimates or underestimates the quantile. This means that in the tails, the t distribution is a bit ’too fat’ for the data. It is striking that the t dis-tribution fits the AEX and the CAC better than the DAX. The t disdis-tribution underestimates the lower and the upper tail for the DAX. This underestimation of the lower tail means that the t distribution is not fat enough for the DAX returns. Remember that the very negative skewness already pointed in the direction of a very fat lower tail, since a negative skewness means that the data are spread out more to the left of the mean. Nevertheless, we think that the t distribution fits all three indices quite well.

Supplementary to the visual tests just performed, we will perform two goodness-of-fit tests to assess the fit of the estimated normal and t distributions on the three datasets of returns. The Kolmogorov-Smirnoff (KS) test and the Anderson-Darling (AD) test are goodness-of-fit tests to compare a sample with a reference probability distribution. In both tests, the null-hypothesis is that the sample is drawn from the reference distribution. The discrete Kolmogorov distance (e.g., Massey, 1951) is defined by the greatest distance between the empirical distribution and the reference distribution, for all possible values:

DKS = max

1≤i≤n|FR(xi) − FE(xi)|, (4.2)

where x1, ..., xn represent the sample, FR is the cumulative distribution function of the

ref-erence distribution and FE is the empirical distribution function. In our case, the reference

distribution is defined by the estimated parameters and these estimated parameters are based on the dataset of returns. Since the asymptotic distribution of the KS and AD goodness-of-fit tests are only known when based on a reference distribution with fixed parameters, a para-metric bootstrap can now be performed to obtain critical values.

The focus of this project is on Value-at-Risk calculations and therefore the tails of the es-timated distributions should receive special attention. As well as the discrete Kolmogorov-Smirnov test, we perform the discrete Anderson-Darling test (Anderson and Darling, 1952), which attaches more weight to the tails of the distribution, where the extreme values are located. This can be seen by observing that DAD is larger than DKS for values when FR→ 1

or when FR→ 0. DAD= max 1≤i≤n |FR(xi) − FE(xi)| pFR(xi) ∗ (1 − FR(xi)) . (4.3)

(40)

Summarizing, the t distribution seems to fit all three return series well. The normal distribu-tion shows a poor fit for all three distribudistribu-tions, especially in the tails. We base this conclusion on both the visual tests we performed and the outcome of the Kolmogorov Smirnoff and An-derson Darling test.

Table 4.3: p-values of the Kolmogorov Smirnoff and Anderson Darling test

student t distribution normal distribution p-value p-value AEX Kolmogorov-Smirnov test 0.194 0.000 Anderson-Darling test 0.545 0.000 DAX Kolmogorov-Smirnov test 0.065 0.000 Anderson-Darling test 0.243 0.000 CAC Kolmogorov-Smirnov test 0.239 0.000 Anderson-Darling test 0.476 0.000

Notes: The p-values of the KS and AD test are tabulated. For the AEX, DAX and CAC the Kolmogorov Smirnoff and Anderson Darling goodness of fit test are performed for the estimated student t distribution and estimated normal distributions (see Table 4.2). Next, it will be explained how these values are derived by taking as an example the p-value of the KS test, for the AEX return set with the student t distribution as a reference distribution. Since we want to perform the KS test for a reference distribution which is estimated, we first perform a parametric bootstrap. Exactly 1382 returns are generated from the estimated student t distribution. Based on this sample the student t distribution is estimated again and then the KS test can be performed. This procedure is repeated 1000 times so that we end up with 1000 KS values. The KS distance is also calculated for the original dataset of returns and original estimated student t. To find the p-value of the KS test, the number of times the 1000 KS values exceed the ’original KS value’ is divided by 1000. In a similar fashion the p-value of the Anderson Darling test can be calculated.

4.2.2 Multivariate estimations

As in the univariate t distribution, there are no standard estimators for the multivariate t distribution. That is most likely why programs like Matlab have no function available to estimate the parameters of a multivariate t distribution. Therefore we have written a pro-gram ourselves which numerically performs the likelihood optimisation. The code is provided in Appendix A. We experience that the loglikelihood is very sensitive for local minima and therefore we do not stop the estimation process as long as the estimate can be improved. The results of the estimation process are shown in the upper part of Table 4.4.

The estimates of µ and σ of the multivariate t distribution in Table 4.4 are quite close to the univariate estimates (see Table 4.2) and also the estimated correlation parameters ρAEX,DAX,

ρAEX,CAC and ρDAX,CAC are close the correlations of 0.77, 0.96 and 0.80 respectively. The

(41)

Table 4.4: Estimated parameters for the multivariate t and normal distribution

Multivariate t distribution

µAEX 0.0518 σAEX 0.95 ρAEX,DAX 0.8788 ν 2.54

µDAX 0.0314 σDAX 0.63 ρAEX,CAC 0.9359

µCAC 0.0494 σCAC 0.98 ρDAX,CAC 0.9041

Multivariate normal distribution

µAEX 0.0374 σAEX 0.94 ρAEX,DAX 0.777

µDAX 0.00404 σDAX 0.65 ρAEX,CAC 0.967

µCAC 0.0290 σCAC 0.95 ρDAX,CAC 0.802

Notes: The estimated parameters of the multivariate student t and normal distribution are tabulated; µ’s - estimated means, σ’s - estimated standard deviations, ρ’s - estimated correlations, ν - estimated degrees of freedom. Estimates are based on 1382 overnight returns, i.e. from January 2, 2004 to July 24, 2009. The first and second column are displayed in percentages.

As mentioned before, we will also estimate the parameters of the multivariate normal distri-bution to be able to base the Monte Carlo VaR calculations on it later on. The multivariate normal distribution has standard estimators and they are displayed in the lower part of Table 4.4. Since it is shown in Section 4.2.1 that the univariate normal distributions do not fit the data well, it is highly unlikely that the multivariate normal distribution will fit the data well.

We are interested in applying a goodness-of-fit test to the fitted multivariate t distribution, to see if it fits as well as the univariate t distributions. Unfortunately, there are no easily applicable goodness-of-fit tests available for trivariate multivariate t distributions. Justel, Pena, and Zamar (1997) show how to perform a Kolmogorov-Smirnov test in the bivariate case, but argue that there appear considerable computational difficulties when they tried to extend their algorithm to higher dimensions. Kole, Koedijk, and Verbeek (2007) extend stan-dard goodness-of-fit tests to copulas of any dimension, but extending this test to multivariate distributions could be a separate research project in itself. Therefore we choose to rely on the fit of the univariate distributions. In the previous subsection we observed that the univariate student t distributions fit the data the best. We also saw that the estimates of the multi-variate t distribution are of the same order of magnitude as the unimulti-variate estimates and the correlations.

4.2.3 Summary

Speeding up Value-At-Risk Calculations for Option Portfolios