Option Pricing Research Under Riemann-Liouville Fractional Poisson Process

Black-Scholes model is the most commonly used option pricing model in the market,the volatility of the model is the only parameter that we cannot directly obtain in the market.Implicit volatility is the root of the volatility parameter if we use the Black-Scholes formula to solve the market option price.However,for the same time expiry and stock price,if we fit the implicit volatility and the strike price,a smiling or skewing shaped curve will be generate rather than the horizontal line we expected,this phenomena is named implicit volatility smile.Implicit volatility smile shows that Black-Scholes model is flawed,for the most common stochastic volatility models including Heston model and Hull-White model,the implicit volatility smile curves can be generated,which means the assumptions of Heston model and Hull-White model are more correct in the option market comparing to Black-Scholes model.However,the relationships between implicit volatility and the strike price include not only implicit volatility smile,but also inverted smile,increasing skew and decreasing skew,even they are not as common as the implicit volatility smile in the option market,they do exist.Heston model and Hull-White model cannot interpret all of these relationships between implicit volatility and the strike price,which means they still have limitations.This paper proposed a new option pricing model with assumptions that the volatility is under Riemann-Liouville Poisson Process,and derived its European call option pricing formula.The option pricing formula demonstrate that the historical sample path of the volatility plays an important role in option pricing,option price systematically depends on the complex structure of the historical volatility.Comparing to other stochastic volatility models,this model can explain more implicit volatility curve forms,including: smile,inverted smile,convex increasing skew,concave increasing skew,convex decreasing skew and concave decreasing skew.It shows that the assumption of the stochastic volatility model under Riemann-Liouville fractional Poisson process is more reasonable and the model is more flexible.