Several Options Pricing Conditions Fractional Brownian Motion Environment

Option pricing plays an important role in financial mathematics research. Assumes the transactions are not continuous, based on historical information and risk-neutral preference, Rostek S. and Schoebel R. have given the analytical expressions of European option prices driven by conditional fractional Brownian motion. The thesis is a continuation of their study,its main works incuding:Firstly,fractional Brownian motion was introduced, the derivation of conditional fractional Ito theorem given by Rostek S. and Schoebel R. was revised,and the new concept of geometric conditional fractional Brownian motion was proposed.Secondly,the analytical expressions of several European type options'prices driven by conditional fractional Brownian motion were given. Because the transactions were not continuous, so the no arbitrage pricing approach based on dynamic hedging method was no longer available, introducing risk preferences becomed a natural need. When the market was in equilibrium, the stock price process was a geometric conditional fractional Brownian motion. Under the risk-neutral preference condition,the analytical expressions of digital options'prices and asset-or-nothing options'prices were given, then by using the replication method with linear combination,the analytical expressions of other related options'prices were given, and the rationality of using fractional Brownian motion in the option pricing model was also discussed, European power option prices were recalculated with the method different from Xiao Yanqing,et al. did.Finally, a conditional fractional-jump-diffusion model was new proposed, and the analytical expressions of several European type options'prices under the model were given. Under the new model, the jump component was added into the stock price process. Still assumed the transactions were not continuous, the risk preference was neutral, at this point the calculations of option prices were similar to the cases of no-jump.The theory system of option pricing have been enriched by the works of the thesis, thereinto the actual financial market can be better described and the scope of the study have been widened by the new proposed conditional fractional-jump-diffusion model.