Approximating functions of integrals of log-Gaussian processes : applications in finance
This dissertation looks at various specific applications of stochastic processes in finance. The motivation for this work has been the work on the valuation of the price of an Asian option by Rogers and Shi (1995). Here, we look at functions of integrals of log - Gaussian processes to obtain approximations to the prices of various financial instruments. We look at pricing of bonds and payments contingent on the interest rate. The interest rate is assumed to be log - Gaussian, thus ensuring that it does not go negative. Obtaining the exact price might not be easy in all cases - hence we use of a combination of a conditioning argument and Jensen's inequality to obtain the lower bound to the prices of the bond as well as payments contingent on interest rates. We look at single driver models as well as multi-driver models. We also look at bonds where default is possible. We try to provide a mathematical justification for the choice of the conditioning factor used throughout the thesis to approximate the price of bonds and options. This is similar to the approach used by Rogers and Shi (1995) to valuing an Asian option; but they had provided no mathematical justification. Another part of this dissertation deals with the problem of pricing European call options on stochastically volatile assets. Further, the price and the volatility processes are in general correlated amongst themselves. Obtaining an exact price is quite involved and computation intensive. Most of the previous work in this field has been based on the solution to a system of partial differential equations. As in the case of pricing bonds, here too, we use a conditioning argument to obtain an approximation to the prices. This method is much faster and less computation intensive. We look at the situations of fixed and stochastic interest rates separately and in each case, we look at the volatility process following a simple Brownian motion and an Ornstein Uhlenbeck process. We also look at the value of stop - loss reinsurance contract for the case of a doubly stochastic Poisson process. Finally, we look at an alternative method of pricing bonds and Asian options. This is done by using a direct expansion and thus avoids the numerical integration that is used in the earlier chapters.