A multiscale correction to the Black–Scholes formula
Article first published online: 19 JAN 2014
Copyright © 2014 John Wiley & Sons, Ltd.
Applied Stochastic Models in Business and Industry
How to Cite
Kim, J.-H., Lee, J., Zhu, S.-P. and Yu, S.-H. (2014), A multiscale correction to the Black–Scholes formula. Appl. Stochastic Models Bus. Ind.. doi: 10.1002/asmb.2006
- Article first published online: 19 JAN 2014
- Manuscript Accepted: 22 OCT 2013
- Manuscript Revised: 17 OCT 2013
- Manuscript Received: 30 MAR 2013
- Black–Scholes formula;
- stochastic elasticity of variance;
- implied volatility
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. Copyright © 2014 John Wiley & Sons, Ltd.