Multi-Dimensional Black-Scholes Model Of Option Pricing

Option contracts may be stopped before expire date by many reasons. Forexample, when a company bankrupts or is annexed, its unexpire option should beexercised immediately. In reality, many random factors can affect interest rates, andthe price of underlying asset which may jump due to some important information.In this paper, we take three factors above into account, with stochatic analysis, (1)a multi-dimensional continuous Black-Scholes option pricing model with a stochas-tic life in a complete market of random interest rates is studied; (2) in a marketof random interest rates, an option pricing model on a single stock described by anIto stochastic differential equation (SDE) which is driven by a geometric Brownianmotion (BM) and multi-dimensional Poisson processes is investigated; (3) a multi-dimensional option pricing model with jumps on several stocks in a complete marketof random interest rates is studied and extended to cases of stochastic lives.The main results of this paper are as follows:(1) For a multi-dimensional continuous Black-Scholes option pricing model witha stochastic life in a complete market of random interests, its option pricing formulais obtained by the martingale method and stopping time theory.(2) In a market of random interests, for an option pricing model on a singlestock described by an Ito (SDE) which is driven by a geometric BM and multi-dimensional Poisson processes, European option pricing formula is gotten by thecapital asset pricing theory.(3) For a multi-dimensional option pricing model with jumps on several stocksin a complete market of random interests described by a system of Ito SDEs whichare driven by geometric BMs and multi-dimensional Poisson processes, a multi- dimensional European option pricing formula is obtained by the martingale methodand extended to cases of stochastic lives.