On accurate and efficient valuation of financial contracts under models with jumps

The aim of this thesis is to develop efficient valuation methods for nancial contracts under models with jumps and stochastic volatility, and to present their rigorous mathematical underpinning. For efficient risk management, large books of exotic options need to be priced and hedged under models that are exible enough to describe the observed option prices at speeds close to real time. To do so, hundreds of vanilla options, which are quoted in terms of implied volatility, need to be calibrated to market prices quickly and accurately on a regular basis. With this in mind we develop efficient methods for the evaluation of (i) vanilla options, (ii) implied volatility and (iii) common path-dependent options. Firstly, we derive a new numerical method for the classical problem of pricing vanilla options quickly in time-changed Brownian motion models. The method is based on ra- tional function approximations of the Black-Scholes formula. Detailed numerical results are given for a number of widely used models. In particular, we use the variance-gamma model, the CGMY model and the Heston model without correlation to illustrate our results. Comparison to the standard fast Fourier option pricing method with respect to speed appears to favour our newly developed method in the cases considered. Secondly, we use this method to derive a procedure to compute, for a given set of arbitrage-free European call option prices, the corresponding Black-Scholes implied volatility surface. In order to achieve this, rational function approximations of the inverse of the Black-Scholes formula are used. We are thus able to work out implied volatilities more efficiently than is possible using other common methods. Error estimates are presented for a wide range of parameters. Thirdly, we develop a new Monte Carlo variance reduction method to estimate the expectations of path-dependent functionals, such as first-passage times and occupation times, under a class of stochastic volatility models with jumps. The method is based on a recursive approximation of the rst-passage time probabilities and expected oc- cupation times of Levy bridge processes that relies in part on a randomisation of the time- parameter. We derive the explicit form of the recursive approximation in the case of bridge processes corresponding to the class of Levy processes with mixed-exponential jumps, and present a highly accurate numerical realisation. This class includes the linear Brownian motion, Kou's double-exponential jump-di usion model and the hyper-exponential jump- difusion model, and it is dense in the class of all Levy processes. We determine the rate of convergence of the randomisation method and con rm it numerically. Subsequently, we combine the randomisation method with a continuous Euler-Maruyama scheme to es- timate path-functionals under stochastic volatility models with jumps. Compared with standard Monte Carlo methods, we nd that the method is signi cantly more efficient. To illustrate the efficiency of the method, it is applied to the valuation of range accruals and barrier options.Open Acces

The wiener-hopf technique and discretely monitored path-dependent option pricing

Fusai, Abrahams, and Sgarra (2006) employed the Wiener-Hopf technique to obtain an exact analytic expression for discretely monitored barrier option prices as the solution to the Black-Scholes partial differential equation. The present work reformulates this in the language of random walks and extends it to price a variety of other discretely monitored path-dependent options. Analytic arguments familiar in the applied mathematics literature are used to obtain fluctuation identities. This includes casting the famous identities of Baxter and Spitzer in a form convenient to price barrier, first-touch, and hindsight options. Analyzing random walks killed by two absorbing barriers with a modified Wiener-Hopf technique yields a novel formula for double-barrier option prices. Continuum limits and continuity correction approximations are considered. Numerically, efficient results are obtained by implementing Padé approximation. A Gaussian Black-Scholes framework is used as a simple model to exemplify the techniques, but the analysis applies to Lévy processes generally

Efficient Pricing of European-Style Options Under Heston's Stochastic Volatility Model

Heston's stochastic volatility model is frequently employed by finance researchers and practitioners. Fast pricing of European-style options in this setting has considerable practical significance. This paper derives a computationally efficient formula for the value of a European-style put under Heston's dynamics, by utilizing a transform approach based on inverting the characteristic function of the underlying stock's log-price and by exploiting the characteristic function's symmetry. The value of a European-style call is computed using a parity relationship. The required characteristic function is obtained as a special case of a more general solution derived in prior research. Computational advantage of the option value formula is illustrated numerically. The method can help to mitigate the time cost of algorithms that require repeated evaluation of European-style options under Heston's dynamics.characteristic function inversion; Heston's model; European-style option

Australian Asian options

We study European options on the ratio of the stock price to its average and viceversa. Some of these options are traded in the Australian Stock Exchange since 1992, thus we call them Australian Asian options. For geometric averages, we obtain closed-form expressions for option prices. For arithmetic means, we use dierent approximations that produce very similar results.Asian options, arithmetic average, geometric average, edgeworth expansion, lognormal distribution, gamma distribution