Option Pricing Under A Class Of Local-stochastic Volatility Models

Based on no-arbitrage principle, Black and Scholes derived the famous Black-Scholes option pricing formula. Indeed, the advent of this formula caused a revolution in financial economics and financial industries. However, under the assumption of constant volatility, the model implied volatility in Black-Scholes’framework is a constant, which is indepen-dent of the strike price K and expiration time T, and does not change over time.In fact, the market implied volatility surface not only depends on the strike price K and expiration time T, but also changes with current time t:(1) For a fixed maturity time T, the market implied volatility curve reflects the volatil-ity smile feature.(2) The implied volatility is a monotonically increasing function of the underlying price.Since these market characteristics have a close link with the option hedge, they are very important. Therefore, the defect of Black-Scholes has motivated many improvements from the industry.At present there are mainly two types of models to reflect the characteristics of volatil-ity:one is the stochastic volatility model, which directly models the instantaneous volatility with a new stochastic process; the other one is the local volatility model, whose instanta-neous volatility is built as a deterministic function of the underlying asset and time t.In my opinion, local and stochastic volatility depict the internal and external factors, respectively, and they both depict the instantaneous volatility partially. It will be more com-prehensive to model the instantaneous volatility as the weighted sum of local and stochastic volatility. Therefore we propose Additive local-stochastic volatility model and specify it to SSCEV. We make use of J-P Fouque’s singular perturbation technique to derive the first-order approximation formula of the European option pricing formula, where the first-order and zero-order terms are approximated by asymptotic expansion about the exponential pa-rameter. Furthermore, we give error estimation of the approximation formula.The empirical results show that SSCEV model can fit quite well the market data of European option whose expirations are greater than90days. Also the dynamics of the market smile predicted by SSCEV is consistent with the observed market behavior. So hedging based on this model will perform more stably.In addition, based on Jim Gatheral’s work, we improve Pierre Henry-Labordere’s ap-proximation result and give a new approximation formula of implied volatility for general local volatility model. Based on the Taylor series expansion, the implied volatility can be approximated by a simple polynomial series. The numerical results show that, our formu-la has high approximation accuracy under the homogeneous and non-homogeneous CEV model, and also has high computational efficiency. Therefore, it could be a good alternative for CEV model’s exact solution. We also compare our approximation formula with Pierre Henry-Labordere’s in non-homogeneous CEV model. Our formula shows not only higher accuracy but also better stability, which illustrates that our improvements are very good and necessary.