C. Satke:

"A Numerical Solver for the Multivariate Black-Scholes Problem Using the Multigrid Method";

Supervisor: D. Praetorius; Institute for Analysis and Scientific Computing, 2009; final examination: 06-16-2009.

Options are derivative financial products that involve the right but not the obligation to trade in one or several risky assets, i.e. assets whose future price is not deterministic, like for instance, shares, commodities, or currencies, at a fixed predefined price. We will only consider so-called European-style options where the option can only be exercised at one specified date in future, the so-called maturity or expiry date. On the financial markets, increasingly sophisticated and complex financial instruments are being developed and need to be correctly priced. A considerable variety of derivative products can be interpreted as involving an option on several underlying assets, hence the pricing of multi-asset options is of both theoretical and practical interest in mathematical finance. Let us present some examples. [Margrabe, 1978] shows that performance incentive fees, margin accounts, exchange offers, and standby commitments constitute options to exchange one asset for another, i.e. options contingent on two underlying assets. [Stulz, 1982] discusses applications of options on the maximum or the minimum of two assets, including the valuation of foreign currency bonds, option-bonds,risk-sharing and incentive contracts, secured debt, and certain types of investment opportunities. Furthermore, [Boyle et al., 1989] and [Boyle and Tse, 1990] consider the so-called quality option or "cheapest to deliver" option which occurs in future contracts when the short position can deliver any one of a set of deliverable assets, and which can be expressed as an option on the minimum of this (arbitrarily large) set. [Barraquand, 1995] and [Barraquand and Martineau, 1995] list applications of multidimensional option pricing in assets and liability management, corporate capital budgeting, risk management, property/liability insurance, and the pricing of Over The Counter warrants, multidimensional interest rate term structure contingent claims, mortgage-backed securities, and life insurance policies.

In 1973, Fisher Black, Myron Scholes, and Robert Merton published a pioneering model allowing to uniquely determine the price of derivative products, which has since been known as the Black-Scholes model. Since then, a vast literature on option pricing theory has emerged. There are two equivalent ways to derive the option price according to the Black-Scholes model (see, e.g., [Wilmott et al., 1995]): one approach derives a partial differential equation (PDE) that governs the option price depending on the price(s) of the underlying asset(s) and time, and the other represents the option price as the discounted expected value of the option's pay-off under an equivalent martingale measure, the so-called risk neutral measure. There are only few cases where the resulting PDE or multidimensional integral, respectively, can be solved analytically, hence one must resort to numerical valuation methods.

In this thesis, we will concentrate on the PDE approach and design a numerical algorithm based on finite differences to solve the PDE for multi-asset options. In particular, our objective is that the algorithm should be applicable to options on an arbitrary number of underlying assets, or in mathematical terms, to PDEs in arbitrary space dimensions.

In Chapter 1, we present the Black-Scholes PDE and its multi-dimensional generalisation, transform the problem to a more convenient form, and prove the existence and uniqueness of a solution based on theorems by [Friedman, 1964]. However, this problem is set on an unbounded domain, which is not suitable for a finite-difference method. Thus, for the numerical solution, we have to truncate the problem to a bounded computational domain. In Section 1.4, we analyse the truncated problem, its solvability, and the localisation error caused by the truncation. Based on results of [Kangro and Nicolaides, 2000] and [Hilber et al., 2004], we derive an estimate of this error in Section 1.4.3 that proves convergence of the solution of the truncated problem to the original solution and hence justifies the truncation. In Chapter 2, we present the finite-difference discretisation that we use, and give a brief discussion of the convergence of such numerical schemes. The finite-difference method results in a set of linear systems of equations which are in general very large and sparse. In order to solve them, we use the highly efficient iterative multigrid method whose basic principles we explain in Chapter 3. After that, we turn to the practical implementation of our numerical solver in Chapter 4. In particular, we show how to implement all required algorithms in MATLAB in a vectorised way and for an a-priori arbitrary number of spatial dimensions. Subsequently, we test the performance of our solver in a number of numerical experiments presented in Chapter 5. Our findings are finally summarised in Chapter 6, and the pros and cons of our numerical method with respect to other numerical valuation strategies found in literature are briefly discussed.

http://www.asc.tuwien.ac.at/~dirk/download/thesis/msc/satke2009.pdf

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