The Implied Volatility Model With The Exponential Parameter

Option pricing is one of the hot issues in the field of financial engineering in which the most famous and common used model is the Black-Scholes(or Black-Scholes-Merton)model.The model assumes that the implied volatility is a constant across all strikes and maturities.But,in practice,the values of implied volatilities by inverting option prices with different strikes and maturities are varied.That is,in a three-dimensional coordinate system,the implied volatility is plotted as a function of the strike price and the time-to-maturity which forms a non-flat surface called the implied volatility surface.This surface contains a lot of information of the option market,is an important tool for option pricing and financial risk management.Thus,modeling the implied volatility becomes very important in the field of financial engineering.In recent years,with the option market data becoming abundant,researchers focus on the deterministic modeling methodologies.The deterministic implied volatility models include the parametric model,non-parametric and semi-parametric model,which consider that there is a deterministic functional relationship among the implied volatility,the time-to-maturity and the strike price.In this paper,we mainly discuss the fitting and forecasting performance of the parametric models and semi-parametric models,and propose new implied volatility parametric model with the exponential parameter and Gaussian semi-parametric model with the exponential parameter.In the new models,the Gaussian function is used to construct a smooth function substituting the quadratic term in the polynomial model.The improved models are more flexible to the changeable market data,thus are much closer to the actual application.Secondly,in order to get no-arbitrage implied volatility and option price,the arbitrage-free conditions of the implied volatility surface are used to calibrate the Gaussian semi-parametric model.The calibrated no-arbitrage Gaussian semi-parametric model can forecast the implied volatility and option price without arbitrage which meet the assumption of B-S option pricing model that there is no arbitrage opportunities in financial market.In empirical analysis,the market dataset of AAPL stock options is used to verify the effectiveness of the new models.For Chinese financial market,the only option—50ETF fund option is also used for empirical analysis.The tests show that the new models based on deterministic implied volatility models is more flexible and has a better fitting and forecasting performance,at the same time the new models also have a good applicability for 50ETF option.