Research On Options Pricing Problems Of Stock Price Model Driven By Skew Brownian Motion

With the development of the economy,financial derivatives have emerged in the financial market,making the pricing of financial derivatives a crucial issue in financial mathematics research in recent years.In 1973,Black and Scholes proposed the milestone Black-Scholes model,combining mathematical models with option pricing theory,which laid a solid foundation for pricing financial derivatives.For a long time,this model played an important role in the financial market.One of the critical assumptions for the validity of this formula is that the underlying asset follows a geometric Brownian motion.However,extensive empirical evidence shows that stock returns exhibit characteristics of fat tails and negative skewness,which do not conform to the characteristics of geometric Brownian motion in actual markets.Skew Brownian motion is a type of stochastic process that has richer statistical properties than standard Brownian motion.When it has not reached the skew point,skew Brownian motion is the same as standard Brownian motion.However,when it reaches the slant point,it moves upwards with a probability ofand downwards with a probability of 1-,which can describe certain phenomena well.This study investigates the framework of research based on the Black-Scholes model and modifies the underlying assumptions by replacing the standard Brownian motion-driven stock price model with a skew Brownian motion-driven model.The pricing problems are examined under this new model.Firstly,the existence and uniqueness of solutions to stochastic differential equations driven by sticky and skew OU processes are studied,and a new pricing formula is obtained by applying it to the Vasicek interest rate model.Next,a double-skewed Brownian motion is constructed,which has two skew points.Under the stock price model driven by this double-skewed Brownian motion,an equivalent martingale measure is sought,and a new pricing formula for European call options is derived using martingale methods and mathematical tools in stochastic processes.Finally,the double skew Brownian motion is combined with reset options,and a new formula for pricing reset put options is obtained using martingale methods under the risk-neutral measure.