Quasi-Monte Carlo Methods Applied In Sensitivities Of Asian Options

As a kind of financial instrument, the payof of an option relies on its underlyingasset. Asian options, especially weighted arithmetic Asian options, where the payoffunctions are based on the average of the prices of underlying asset over some periodprior to maturity, has become a hot issue in computational finance. Sensitivities areimportant in both practical and theoretical way in financial research. They are impor-tant for hedging, that is, for reducing the risk of a security. Diferent from the prices ofoptions, options’ sensitivities cannot be observed in the market, so accurate estimationof sensitivities is arguably even more than that of prices.There are several papers research using MC methods in sensitivities estimation,but few are about QMC methods. As the development of MC methods, QMC methodsuse sequences which are deterministic instead of random. These sequences improveconvergence rate and give deterministic error bounds. In the present thesis, we useQMC methods and several path generation methods, such as STD, BB and PCA to es-timate Delta and Vega of geometric Asian call options and weighted arithmetic Asiancall options and compare our results with classical MC methods. Moreover, we useorthogonal transformation methods(OT) in this paper: first we obtain a orthogonal ma-trix which can turn a high dimension problem of geometric Asian options into an onedimension one; because the payof function of arithmetic Asian options has a similardimension structure of geometric Asian options, we then use this matrix into the similarproblem of arithmetic Asian options.We take d equals16and64, using QMC and MC methods to estimate Delta andVega of geometric Asian options and seven weighted arithmetic Asian options, whereweights control the dependence of the payof functions on the underlying asset prices atdiferent time steps. We ofer good cases and bad cases of each path generation methodsand compare all the results. We find that in Asian options’ sensitivities calculation,QMC methods are more efcient that MC methods; and in most cases, OT is better than other path generation methods.