Year

2011

Degree Name

Doctor of Philosophy

Department

University of Wollongong. School of Mathematics and Applied Statistics

Abstract

Options and other ¯nancial derivatives have become increasingly important in financial markets ever since Black and Scholes (1973) proposed an analytical and quantitative formula for valuing European options or other similar ¯nancial derivatives of a ¯xed lifetime. However, how to rationally price option derivatives e±ciently and accurately is still one of the major challenges in today's ¯nance industry. This thesis contributes to the literature signi¯cantly by further exploring some quantitative approaches for pricing various derivatives. Classi¯ed by the quantitative methods adopted to price option derivatives, this thesis consists three parts, with each part addressing one of the key quantitative approach. Moreover, these parts are based on ten papers published in or submitted to various top-class international journals. The issue regarding numerically pricing option derivations, particularly, the American puts, is discussed in Part 1, which contains Chapter 2, Chapter 3 and Chapter 4. In this part, we ¯rst introduce a new numerical scheme, based on the ADI (alternating direction implicit) method, to price American put options under a stochastic volatility model. Realizing the fact that the numerical approaches designed for American puts with ¯nite maturities are usually with low accuracy and computational ineciency when applied to deal with the perpetual case, a new numerical scheme, based on the Legendre pseudospectral method, is then introduced to solve for the price of perpetual American puts with stochastic volatility e±ciently and accurately. On the other hand, upon considering the fact that a \convergency-proved" numerical approach has never been proposed for the valuation of American options, we also introduce, in this part, an IFE (inverse ¯nite element) approach to price American puts under the Black-Scholes model. Numerical results show that the IFE approach is quite accurate and e±cient, and can be easily extended to multi-asset or stochastic ii volatility pricing problems. Most remarkably, we have managed to provide a convergence analysis for the IFE approach, which ensures that our numerical solution does indeed converge to the exact one of the original nonlinear system. In Part 2, we concentrate on deriving analytic approximations for option derivatives. Two sub-issues, regarding the asymptotic behaviour of the optimal exercise price near expiry and pricing approximation formulae for vanilla options, are discussed in details in this part. On one hand, we derive two explicit analytical expressions for the optimal exercise price near expiry under the local volatility and the stochastic volatility models, respectively, by using the method of matched asymptotic expansions. The results show that under the local volatility model, if the underlying dividend is greater than the risk-free interest rate, the behavior of the optimal exercise price is parabolic, otherwise, an extra logarithmic factor appears, which agrees with the constant volatility case. We also ¯nd that under the stochastic volatility model, the option prices are quite di®erent from the corresponding Black-Scholes' case, but the leading order term of the optimal exercise price remains almost the same as the constant volatility case if the spot volatility is given the same value as the constant volatility appearing in the Black-Scholes model. On the other hand, a series of approximations for the price of vanilla options are also provided in this part. In particular, by realizing that Heston's formula is problematic, we derive a new approximation for European puts under the Heston model by using singular perturbation method. The newly-obtained formula only involves the standard normal distribution function, and is thus as fast and easy to implement as the Black-Scholes formula. Moreover, we derive three formulae for pricing perpetual American puts under the slow-mean reverting, fast-mean reverting and multi-scale stochastic volatility models, respectively. Based on the formulae, the quantitative e®ects of di®erent stochastic volatility dynamics on the optimal exercise strategies of perpetual American puts are also discussed and compared. It is found that the e®ect of a slowly-varying volatility factor varies with respect to the spot volatility. That is, for certain values of the spot volatility, the stochastic volatility tends to add the value of the contract, but for iii others, it makes the contract less valuable, whereas the fast mean-reversion factor always tends to add the price of a put option contract, had the underlying been assumed to be falling. The last part of this thesis deals with the exact solution approach, which is extremely important in both theoretic and practical sides of option pricing. In particular, we consider the analytical pricing of Parisian-type options, i.e., Parisian and Parasian options, which are barrier options with the knock-in or knock-out feature only activated after the underlying has spent a certain prescribed time beyond or below the barrier. By the reduction of a three-dimensional problem to a two-dimensional problem through a coordinate transform that has elegantly \absorbed" the directional derivative associated with the \barrier time" into the time derivative, we have been able to obtain two closed-form analytic formulae for prices of Parisian and Parasian options, respectively, which can both be easily and e±ciently evaluated numerically.

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