Martingale Analysis On Option Pricing Affected By Random Factors

Option is an important financial derivative in financial markets. Under promotion of the famous Black-Scholes model, option trading soon becomes a main content of global financial markets, and the theory of option pricing occupies an important place in the modern financial statistics. Martingale is a frontier theory of the stochastic process, which has many advantages like profound and concision in research of option pricing. Studying option pricing with the theory of martingale has great practical significance on improving the financial market and promoting the option trading. The paper introduces random factors into the option pricing model, and studies the problems of option pricing affected by random factors.Firstly, the paper reviews the research status of option pricing and martingale analysis. Relative concepts and theories are clarified. Classic Black-Scholes European option pricing model is discussed, and several methods of option pricing and their disadvantages are compared.Secondly, martingale analysis is introduced into the study of the random factor dependent option pricing. The paper investigates the random factors which have influence on option pricing, e. g. the stock price volatility according to random environments, expected rate of return, the stochastic interest rate, and the payment of dividends, etc. Ito? fractional Black-Scholes market model is established under the actual market situation, by assuming equity dividends paid, the expected rate of return and interest rate of stock and dividend yield are all the random functions of time, and assuming the underlying asset price follows fractional Brownian motion. Using equivalent martingale measure, the paper derives the pricing formulas of European option with dividend payments, call put parity and European two-way option pricing formula, which are all affected by the random factors.Finally, the more common power of n-th power European option with payoff function of form V_t =|K-S_T~n|~+ is investigated. Extending the above model to the pricing of n-th power European option obtains the n-th power European call and put option pricing formulas and n-th power European two-way option pricing formula, which are both affected by the above random factors.These results promote the classic Black-Scholes model, and they have more practical significance on studying finance and economy, because they are closer to the actual financial markets.