Martingale Theory And Its Applications In Some Financial Models

Martingale theory is one of the most important branches of probability theory, andit is also the basis of probability theory and random process. It has a lot of practicalsignificance, for example, in the process of making pricing decisions or controllingmodels. We can simplify some of the complicate problems just by constructing theframework of martingale theory. That’s the reason why martingale theory is widely used.In the meantime, martingale theory also has connections with statistical analysis,harmonic analysis, geometry of Banach space and random analysis, and has getvarious of achievements.This paper has two parts. The first part is a sketch of martingale theory which isdivided into two parts of classical martingale theory and modern martingale theory. Thecontent of classical martingale theory includes the definitions and properties of theparametric martingale of discrete time and the parametric martingale of continuous time,sets of inequalities, convergence theorem, stopping theorem and so on. The content ofmodern martingale theory includes the decomposition of the martingale, integrablequadric variation martingale, square integrable martingale, partial martingale, halfmartingale, theory of limits of martingale and stochastic integral problem under theviewpoint of martingale theory. The method of martingale is form from the combinationof martingale theory and stochastic integral knowledge which has valuable significancein the financial field.The primary coverage of the second part is the application of martingale theory. Inthis part, we have draw single session model, multi-session model and classical Black—Scholes option pricing model into the theory. The paper has analyzed these modelsthrough the knowledge of martingale theory and get corresponding results. From then,we create the option model with change exercise price on the basis of the exponentialOrnstein-Uhlenbeck process with underlying stock paying dividends and calculatedifferent prices under European style option and American style option. It is said thatthe content of martingale theory has strong practical values in the aspect of optionvaluation in the financial field. The martingale theory provides fundamental basis forpricing interrelated spin-offs in the financial field with its continuous improvements.