A Strike-related Implied Volatility Model Based On The Inverse Problem

In recent years,financial derivatives have developed rapidly,especially in the size and the variety of the transaction.The option is one of the most important financial derivatives and the option pricing has became a hot issue among the field of financial engineering.For option pricing,Black-Scholes model(B-S model)is the common used model.In this model,implied volatility was assumed as a constant.But the empirical analysis show that the implied volatility is not a constant.The implied volatility values by inverting option prices in the B-S model by using the market data vary with the underlying asset price,maturity date,time to maturity,and the strike price,that is,the implied volatility is a function of these variables instead of a constant.Therefore,it is important to model implied volatility function for accurate option pricing.At present,the modeling of implied volatility can be classified into two types:one is using statistical regression method,such as the parameter model,non-parametric model and semi-parametric model;the other is using the methods of the inverse problem to model implied volatility.Using the statistical regression method,modeling and solving is simple.But it is assumed that the form of implied volatility function or the distribution of the weights of model is given which make the model lack feasibility.The model based on the inverse problem avoids to assume the form of implied volatility function,instead,it fits the function by using lots of market data which makes it more feasible and better forecast performance.In this paper,based on the Osher model and the Chiarella model,a new model based on inverse problem is proposed.Different implied volatility models are used for different strike prices,which may show the volatility smile.The weights based on the distance to the given strike price is used to decide the importance of the option price deviations which make the model a good fitting effect.The model is transferred to a Poisson equation by using the Euler-Lagrange method which is solved by using the boundary conditions fitted by the real data.The experimental results show that,compared with the traditional models,the improved model shows better feasibility and accuracy in the simulation test data and the market data.The simulation data is similar to the experimental data in Chiarella's paper.The market data is downloaded from the APPL option data.