Extrapolation of implicit numerical methods for stochastic differential equations and stochastic models for multiple assets with application to options
Koskodan, Rachel C.
This dissertation involves two different projects. The first project involves improving the accuracy of approximate solutions to stochastic differential equations. The stochastic theta numerical method forms a family of implicit Euler methods for approximating solutions to Ito stochastic differential equations. It is proved that the weak error for the stochastic theta numerical method is of the correct form for applying Richardson extrapolation to increase the order of accuracy of the approximations. Several computational examples illustrate the improvement in accuracy of the approximations when applying extrapolation. In the second project, consistent Monte Carlo, discrete stochastic, and stochastic differential equation models are constructed from first principles for a mutual fund with multiple assets. The different stochastic models are shown to be consistent in the estimation of mutual fund values. The models are applied to the calculation of European call option prices. It is shown that option prices are insensitive to the form of the stochastic model's diffusion term. In addition, a general n-dimensional Black-Scholes partial differential equation is derived for option prices. Computational examples illustrate that the Black-Scholes partial differential equation and the stochastic differential equation models are consistent in estimating option prices.