Pricing Of Two Exotic Options Based On Subfractional Brownian Motion With Jump Diffusion Process

The option pricing problem is a core component of quantitative finance.In order to re-spond to the various needs and needs of the market,many non-standard options are derived from the market,which are well-known singular options.Asian options and exchange option-s are popular unconventional options.The sub-fractional Brownian motion is non-random when the Hurst exponent>0.5,and has the characteristics of long-term correlation,thick-tailedness,and rapid degradation.Therefore,it is more suitable for describing changes in the financial market than the standard Brownian motion.Considering the volatility caused by uncertain risks in the market,this paper studies two singular option pricing problems based on the combination of sub-fractional Brownian motion and jump-diffusion process.The main work of the thesis is divided into three parts.The first part briefly introduces Asian options.Under the condition that the underlying asset price satisfies the sub-fractional jump-diffusion process,the partial differential equation satisfying the value of Asian options is derived through a self-financing trading strategy,and the pricing model is solved by sec-ondary variable substitution.An analytical solution to the value of Asian call options.The pricing formula for put options and the parity formula for the value of call-put options are also given.Through numerical simulation,the influence of parameters such as Hurst index and jump strength on the value of options is studied.The second part assumes that the interest rate is characterized by a sub-score Vasicek stochastic model,the underlying asset price meets the sub-score jump-diffusion process,and adds transaction fees and dividend payment conditions to study Asian option pricing.By constructing an appropriate investment portfolio and applying the principle of no arbitrage,a partial differential equation for Asian option pricing is obtained.Through three variable substitutions,the partial differential equation is converted into a heat conduction equation,and then the analytical solutions of the value of Asian call options and put options are cal-culated,and the derivation process of the parity formulas of call options and put options is also given.Discuss the relationship between Hurst index,transaction rate,jump intensity and option value through numerical calculation analysis.The third part studies the problem of exchange option pricing.From subfractional Brownian motion to mixed Gaussian model,the mixed Gaussian model has the excellen-t properties of subfractional Brownian motion,and when(?)is a semimartingale.Assuming that the change in the price of the option target is characterized by a mixed Gaus-sian model,the partial differential equation satisfying the value of the exchange option is obtained by the no-arbitrage principle,and the analytical solution of the pricing formula of the exchange option is obtained by solving the Mellin transform.After the jump is added,the Mellin transform method is still applicable,and then the exchange option pricing for-mula for the jump-diffusion process under the hybrid Gaussian model is derived.Through numerical simulation,the relationship between Hurst index,expiration time,jump strength and exchange option value is explored.