Pricing European Options Under Multi-time Scale CEV Model

One of the key issues in the field of financial engineering is the classical Black-Scholes option pricing model, which has been attention by the financial and academia circle, but the Black-Scholes option pricing model has many disadvantages, in particular, the instantaneous volatility of the underlying asset returns is assumed to be constant, a large number of empirical studies show that this hypothesis does not accord with the characteristics of the actual market movement, therefore, it has brought great limitations to the application of the model. Then improving Black-Scholes option pricing model has very important theoretical and practical significance.There are two ways to improve Black-Scholes option pricing model, the one way is to suppose that the instantaneous volatility of the underlying asset returns depend on the underlying asset it-self, that is CEV model (model of constant elasticity of variance); the other way is to randomize the instantaneous volatility of the underlying asset returns, and bringing in stochastic volatility model, such as Heston stochastic volatility model. Due to the fast and slow characteristics of fi-nancial market and economic development, some scholars have introduced the multi-time scale stochastic volatility model based on Black-Scholes option pricing model. The multi-time scale stochastic volatility model can describes the external factors of the instantaneous volatility, the CEV model describes the internal factors of the instantaneous volatility, both of them partially de-scribe the instantaneous volatility of the underlying asset returns, if the multi-time scale stochastic volatility model and the CEV model are combined together, it will more fully characterize the instantaneous volatility of the underlying asset returns. Therefore this paper constructs multi-time scale CEV model.This paper studies the pricing of standard European option under multi-time scale CEV model. the partial differential equation of standard European option is derived under multi-time scale CEV model with the use of Ito formula and martingale approach, standard European option is expanded by using asymptotic expansion method, and formula solutions of the leading order term and the first order correction term price of standard European option are obtained with the use of singu-lar perturbation analysis method and multi-scale technique, then the errors between the real price of the option and the price of its first order approximation are analyzed, the aspects of implied volatility and calibration are also analyzed, implied volatility and the correction formula of stan-dard European call option are obtained, confirming that implied volatility move in accordance with market data by numerical analysis, multi-time scale CEV model is more tally with the changing market than multi-time scale stochastic volatility model.Studying the pricing of Asian option based on the study of standard European option, the partial differential equation of Asian option is obtained by using the Feynman-Kac theorem. The pricing formula of Asian option is obtained by using the dimension reduction technique, the sin-gular perturbation analysis method and the multi-scale technique, and the model is corrected and analyzed, the parameter relationship formulas between Asian option and standard European option are obtained, and Asian call-put parity is obtained, using the numerical analysis to verify that the multi-time scale CEV model is better than multi-time scale stochastic volatility model.Studying the option pricing under multi-time scale CEV model has greater practicality, more in line with market rules, more effective to provide a theoretical basis for investment and risk management.