Enhanced Monte Carlo Methods for Pricing and Hedging Exotic Options

Monte Carlo simulation is a widely used tool in finance for computing the prices of options as well as their price sensitivities, which are known as Greeks. The disadvantage of the Monte Carlo simulation, in its standard form, is its slow convergence rate. In the first part of this thesis, we review several methods that they have been proposed, in order to improve the convergence rate of Monte Carlo simulation. These methods find applicability in pricing exotic options such as barrier and lookback options. In the second part of this thesis, we study the applicability of Monte Carlo in estimating price sensitivities. In general, the estimation of Greeks is not as straightforward as that of option prices. Difficulties may arise by discontinuities in the option payoff function, as in the cases of barrier and digital options. The Monte Carlo methods for estimating Greeks can be divided in the following three categories: a) Finite-difference , b) Likelihood Ratio1 and c) Pathwise methods. In this thesis, we focus on the third method, which usually gives better estimates than the other two methods, when it is applicable. A Pathwise estimator is derived by differentiating the payoff function inside the expectation operator. Thus, the interchange between differentiation and expectation is required. However, this interchange is not applicable in several cases such as the computation of delta and gamma of digital and barrier options. To overcome this obstacle we apply a smoothing technique, i.e. we approximate the discontinuous payoff through a smooth function and then we apply the Pathwise method. Although, additional error is introduced from this smoothing approximation, we can show that sufficiently good estimates of the Greeks can be obtained. Numerical results from computation of both prices and Greeks of several exotic options, are given. Thesis Supervisor: Prof. Michael Giles In cases in which the transition density of the underlying price process is not explicitly known, ideas from Malliavin Calculus can be used to extend this method.