Inversion Of Implied Volatility Of Black-Seholes Mode

Black-Scholes option pricing model describes the relationship between option price and time,underlying asset price,expiration time,strike price and other variables.Implied volatility in Black-Scholes option pricing model is an important parameter to determine option prices,but volatility cannot be measured directly from market data,so it is necessary to use market data to identify implied volatility.The identification of implied volatility is a nonlinear ill-posed problem and regularization method should be introduced to overcome the instability of the problem itself.In the market,due to the impact of emergencies,volatility has a large jump,so it is more reasonable to assume the discontinuity of implied volatility in mathematics,and the regularization method with Total Variation penalty(TV penalty for short)is an effective method to identify the piecewise constant.Therefore,based on the measured data,this paper uses the regularization method with Total Variation penalty to identify the discontinuous implied volatility,and then calculates the option price.The main content of the paper is divided into the following parts:The first chapter introduces the background and significance of Black-Scholes option pricing model,SSE 50 ETF options and the identification of implied volatility.At the same time,this paper introduces some knowledge needed to identify implied volatility.In chapter 2,based on the Black-Scholes option pricing model and the mathematical model of identifying implied volatility,the nonlinear inverse problem is approximately transformed into a linear operator in the local area under the discrete scheme.Then,based on the regularization method with Total Variation penalty,the identification of implied volatility is transformed into the problem of finding the minimum of the cost functional.Then the existence of minimum is proved and the convergence of approximate solution is analyzed under Bregman distance.Finally,based on the fixed point principle,a lag fixed point algorithm is designed to identify implied volatility.In chapter 3,based on simulated data,the lag fixed point algorithm is used to identify implied volatility,and the effectiveness of the algorithm is verified numerically.Then,based on the SSE 50 ETF option market data,the lag fixed point algorithm is used to identify the implied volatility,and then the option price is calculated based on the identified implied volatility and compared with the option market price to verify the effectiveness of the algorithm.The fourth chapter summarizes the innovations and shortcomings of this paper,and prospects the pricing research of other financial derivatives.