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.