European Option Pricing Model With Default Risk

Option pricing theory is the core one of financial engineering. In1973, Black andScholes put forward the famous Black-Scholes option pricing model and obtained its pricingformula based on the full capital market assumptions. However, many scholars agree that thefractional Brownian motion is applied better to the mathematical models of financial market,and the default risk is one important factor in the financial market.In this dissertation, the author argues that the stock price obeys the stochasticdifferential equation driven by fractional Brownian motion. The financial market model isbuilt in fractional Brownian motion environment, using stochastic analysis theory forfractional Brownian motion and insurance actuary method. Meanwhile, the discussionfocuses on the pricing problem of vulnerable option.The thesis includes five chapters.Chapter one is a quick review of the history and current researches of option pricing, aswell as an introduction to the background of the topic selection and the coverage of thisdissertation.Chapter two gives a definition to fractional Brownian motion and position process andmakes an analysis of its characteristics. Also what is introduced in this chapter is theinsurance actuary method for European option pricing.Chapter three is a discussion of the pricing problem for the vulnerable option, throughwhich the author puts forward the argument that the stock price obeys the stochasticdifferential equation driven by fractional Brownian motion and obtains the pricing formulafor the vulnerable option. The scholastic analysis theory of fractional Brownian motion andthe insurance actuarial method are adopted in the argumentation.In chapter four, we formulate the fractional jump-diffusion model for mathematicalmodels of financial market, using fractional jump-diffusion process theory and the insuranceactuarial method. Finally, we discusse the pricing problems of the vulnerable option and obtains an explicit pricing formula for vulnerable option.Chapter five is a summarization of the argumentation of this dissertation and points outsome issues which need further investigation.