Delayed Black And Scholes Formulas Driven By Two Stochastic Processes

The Black and Scholes Formula has been one of the most important consequences of the study of continuous time models in fmance([5, 18, 51, 52]) with many developmentunder it. On the other hand, there is a development of dynamic models that take into consideration the influence of past events on the the current and future states of the system([40, 44, 43, 56, 57, 55, 27]).This view is specially appropriate in the study of financial variables, since predictions about their evolution take strongly into account the knowledge of the their past ([39, 72]).Arriojas et al [4] considered the effect of the past in the determination of the fair price of a call option, and assume that the stock price satisfies a stochastic differential delay equation(sdde) driven by a standard Brownian motion. They derive an explicit formula for the valuation of a European call option on a given stock, and prove that this financial market is complete with no arbitrage. In this dissertation, we extend the delayed financial model to the financial models driven by a fractional Brownian motion and a Levy process.This dissertation mainly discuss with these two types of delayed financial models with following conclusions:In the first model, we extend the delayed financial model to the financial model driven by a fractional Brownian motion B_H with Hurst index H∈((?), 1), consider the evolution of the stock price satisfies the following fractional stochastic differential delay equation: where the integral with respect to B_H is of the Wick-It(?) type,μ, a, b > 0,L = max{a, b}, and G,φsatisfy some suitable conditions.We prove the model admits a pathwise unique solution S_H such that S_H(t) > 0 a.s. for all t≥0 wheneverφ(0) > 0. Furthermore, the price process S_H is H(?)lder continuous of order H-ε(0 < e < H). Then, we show that such market model is arbitrage-free by using fractional Girsanov theorem. Finally, we apply these resluts to study the European options and obtain an an explicit formula for a European call option.In the second model, we extend the delayed financial model to the financial model driven by a L(?)vy process Y, assuming that the stock price S satisfies the following stochastic differential delay equation:whereμ,a,b > 0, L = max{a, b}, and Y, g,φsatisfy some suitable conditions. We also prove this model has a pathwise unique solution S such that S(t) > 0 a.s. for all t≥0 when these conditions are satsified. Then, we apply the method in Chan [17], and find out a Follmer-Schweizer minimal measure which could be used to price a European option.