A multiscale correction to the Black-Scholes formula
Based on a new multiscale hybrid structure of the volatility of the underlying asset price, we study the pricing of a European option in such a way that the resultant option price has a desirable correction to the Black–Scholes formula. The correction effects are obtained by asymptotic analysis based upon the Ornstein–Uhlenbeck diffusion that decorrelates rapidly while fluctuating on a fast time-scale. The subsequent implied volatilities demonstrate a smile effect (right geometry), which overcomes the major drawback of the Black–Scholes model as well as local volatility models, and move to a right direction as the underlying asset price increases (right dynamics), which fits the observed market behavior and removes the possible instability of hedging that the local volatility models may cope with. Further, we demonstrate that our correction brings significant improvement in terms of fitting to the implied volatility surface through a calibration exercise.