Pricing Financial Derivatives Based On Fbm Model

Financial asset pricing is the core issue of classical finance. The pricing theory of financial derivatives is the main content of financial asset pricing, also it is one of the most fundamental and substantial areas in mathematical finance. Option pricing theory has a long and illustrious history, but it also underwent a revolutionary change in 1973. The break-through in option valuation theory started with the publication of two seminal papers by Black&Scholes and Merton.The Black-Scholes model has become the most popular method for option pricing and its generalized version has provided mathematically beautiful and powerful results on option pricing. Nevertheless, classical mathematical models of financial assets are far from perfect. One apparent problem exists in the Black-Scholes formulation, namely that financial processes are not Markovian in distribution. In fact, behavioral finance as well as empirical studies shows that there exists long-range dependence in stock returns and verifies that long-range dependence is one of the genuine features of financial markets. Behavioral finance also suggests the return distributions of stocks are leptokurtic and have longer and fatter tails than normal distribution and there exists long-range dependence in stock returns. These features have some differences with the standard brown motion, while are in accordance with the fractional brown motion. The fractional Black-Scholes model is a generalization of the Black-Scholes model, which is based on replacing the standard Brownian motion by a fractional Brownian motion in the Black-Scholes model.This thesis is devoted to the financial derivatives pricing problem in a fractional ltd type financial market, and we want to establish the mathematical model for the financial market in fractional Brownian motion setting, by assuming the underlying asset price obey-ing the stochastic differential equation driven by fractional Brownian motion. Three topics are studied in this thesis. ·The first topic is the option pricing problem in incomplete markets. We focused on option pricing with proportional transaction costs. The problem is completely solved using the fractional Brownian motion theory and PDE approach, and general pricing formula for the European option with transaction costs is derived. Meanwhile, we get the explicit expression for the European option price with transaction costs and the call-put parity. The perpetual American put option pricing problem is also considered.·The second topic is the option pricing problem when the risk-free interest rate is stochastic. In this part, we take fractional Vasicek model as an example of stochastic inter-est rate. We establish the mathematical model for the financial market in fractional Brown-ian motion setting. Using the risk hedge technique, fractional stochastic analysis and PDE method, we obtain the general pricing formula for the European option with stochastic in-terest rate. At the same time, we get a explicit expression for European option price with stochastic interest rate and the call-put parity. As we will show, the results in this part extend as well as improve previously known results.·The third topic is the option pricing problem when underlying asset returns are dis-continuous. In this problem. We use compound Poisson process to characterize the jump, and we assume the underlying asset is driven by a mixture of both continues and jump pro-cesses, where we characterize the continues part by a fractional Browian motion. We call the process as the fractional jump-diffusion model. Using measure transformation technol-ogy and quasi-martingale approach, we derive an option pricing formula under fractional jump-diffusion model.Finally, we apply these results to actual financial markets, including segregated funds pricing problem, convertible bonds pricing problem and credit risk modeling problem.