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Institutionen for informationsteknologi

Dimension Reduction for the Black-Scholes Equation Alleviating the Curse of Dimensionality Erik Ekedahl, Eric Hansander and Erik Lehto Report in Scienti�c Computing, Advanced Course

June 2007

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Abstract

An important area in computational finance concerns the pricing of options. When options depend on several underlying stocks, the complexity of the problem makes it difficult to solve using conventional finite difference methods. Instead, stochastic approaches are employed despite the extremely slow convergence of these methods.

The objective of this report is to determine if a dimension reduction technique, namely principal component analysis, could be utilized in combination with a finite difference method for these problems. A number of numerical experiments were designed to examine the efficiency under different conditions. In each case the result was compared to a reference solution from a Monte Carlo method.

The results show that the proposed approach performs very well when the cor- relation between the underlying stocks is sufficiently high. An example with an option on the OMXS30-index indicates that this criterium is most likely fulfilled in real-world cases.

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Contents 1 Introduction 3

1.1 Option pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 The curse of dimensionality . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 A different approach – principal component analysis . . . . . . . . . 4

2 Theory 4 2.1 Options and basket options . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The Black-Scholes equation . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Principal component analysis . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 Mathematical derivation . . . . . . . . . . . . . . . . . . . . 7 2.4 Asymptotic expansion . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Monte Carlo methods . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Implementation 9 3.1 Solution algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Finite difference method . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Results 11 4.1 2-dimensional option with highly correlated assets . . . . . . . . . . . 11 4.2 5-dimensional option with highly correlated assets . . . . . . . . . . . 13 4.3 5-dimensional option with weakly correlated assets . . . . . . . . . . 14 4.4 30-dimensional stock option (OMXS30) . . . . . . . . . . . . . . . . 14

5 Discussion 15 5.1 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.2 Possible improvements . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.3 Other features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6 Acknowledgements 17

A Initial data for the numerical experiments 19 A.1 30-dimensional stock option (OMXS30) . . . . . . . . . . . . . . . . 19

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1 Introduction

In financial markets of today, where prices fluctuate by the minute, an increasingly common strategy to keep up with the competition is to use computational models to gain some insight into what the future holds. As the technology advances, these meth- ods are expected to perform even better and are increasingly well trusted by market analysts, further increasing the demands on accuracy and performance.

The focus of this report is to investigate a numerical method for estimating the proper price of basket options using a technique which is quite different from the stan- dard methods.

1.1 Option pricing

We begin with an overview of the terminology. The holder of an option has the right, but not the obligation, to perform a certain act. In finance, a European style option is a contract which grants the holder the right to buy or sell an asset at a given price (the strike price) at a given date in the future (the exercise time). An option to buy an asset is commonly known as a call option and an option to sell is called a put option.

The main reason for trading options is to manage risks in a portfolio (a collection of investments). The value of an option depends on at least five factors. These are the expiry date, the strike price, the interest rate, the price of the underlying asset and the volatility of this asset. The volatility is the standard deviation of the change in value of an asset within a given time period. This is in some sense a measure of the risk associated with the option, since a high volatility indicates a higher uncertainty of the value of the asset at the expiry date.

A model for the pricing of European style options was introduced in 1973 by Fischer Black and Myron Scholes, known as the Black-Scholes equation for option pricing [2]. The Black-Scholes equation is a parabolic partial differential equation, which in combination with the known option value at the expiry date gives rise to a final value problem. This problem will be discussed in detail later on.

1.2 The curse of dimensionality

In this paper a certain kind of European style option called a basket option will be stud- ied. A basket option has two or more underlying assets and the holder of the option has the right to purchase a specified amount of these at the expiry date. When determining the value of such an option using the Black-Scholes model, the dimensionality of the problem grows linearly with the number of underlying assets.

The conventional method for solving partial differential equations numerically is to discretize the continuous variables in space and time and solve the equation in dis- crete form, using e.g. finite difference methods [4]. Since every dimension must be discretized, the number of discrete points where a solution has to be calculated will increase exponentially with the number of dimensions. This is known as “the curse of dimensionality”.

A common approach to deal with this problem is to use a Monte Carlo method1. Monte Carlo methods have the advantage of handling a growing number of dimensions very well – the amount of computational work grows only linearly with the number of

1Monte Carlo methods are described further in section 2.2.

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dimensions. The problem is that these methods generally offer extremely slow conver- gence, requiring huge numbers of simulations to yield a sufficiently accurate result.

1.3 A different approach – principal component analysis The approach that will be tried and evaluated in this project is called principal com- ponent analysis (PCA) in combination with a finite difference method, which is an entirely different strategy compared to the Monte Carlo methods. Since the assets are correlated, the idea is to reduce the number of dimensions by finding a low-dimensional structure that captures the overall behavior of the high-dimensional problem, disregard- ing smaller variations. If the contribution to the solution from these variations is neg- ligible, “the curse of dimensionality” can be alleviated and a finite difference method may be used.

2 Theory First, we will take a quick look at the economic theory concerning options in sec- tion 2.1. In dealing with the pricing of basket options, it is essential to understand the Black-Scholes equation and the final value problem it yields. This equation will be briefly discussed in section 2.2.

The main theoretical concern in this report is principal component analysis (PCA), which is described in section 2.3. This is the method we will implement for reducing the dimension of a partial differential equation in order to make it possible to solve with a finite difference method.

As a reference solver of the same equations, a Monte Carlo method was used, and therefore a very brief overview of the theory behind Monte Carlo simulations is provided in section 2.5.

2.1 Options and basket options Since the theory for the pricing of put options is identical to that of call options, apart from the final value, only the latter will be addressed in the following sections.

The European style call option is a contract which gives the buyer the right to buy a specified number of assets at a specific date. At the expiry time T the value u(S, t) of this option is given by

u(S, T ) = max(S −K, 0), (1) where K is the strike price and S denotes the price of the underlying asset. Since the holder of the option is not obliged to exercise it, the value is never less than zero. The corresponding equation for a call option on a d-dimensional basket (of d assets) is

u(S1, S2, . . . , Sd, T ) = max

( d∑

i=1

µiSi −K, 0 )

, (2)

where Si is the price of underlying asset i and µi is the weight factor of asset i. A thorough introduction to options can be found in [10].

Since the price of the underlying asset at expiry is unknown today (i.e. at time t = 0), a method to estimate the value of the option at this time is required. A famous model for calculating the value of European style options is the Black-Scholes equation [10].

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2.2 The Black-Scholes equation The Black-Scholes partial differential equation describes the evolution of the price of an option and is given by

∂u

∂t +

1 2 σ2S2

∂2u

∂S2 + rS

∂u

∂S − ru = 0, (3)

where r is the interest rate and σ denotes the volatility of the underlying asset. This equation is easy to solve analytically, when r and σ are constant. In Figure 1, u is plotted as a function of S, here with the strike price K = 40. The dashed line is the analytical solution for the price

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