Volatility is a multifaced key concept of modern finance, its importance being acknowledged from both academics and industry practioners; the recent activity in the field exhibits remarkable mathematical depth and makes uses of various aspects of asymptotic (stochastic) analysis as well as (linear and non-linear viscosity) PDE methods.
Surprisingly perhaps, there are essentially model-free results concerning rational shapes of the implied volatility surface (e.g. [BF09]). In the context of parametric stochastic and/or local volatility models, these results can be used upon knowlegde of asymptotic (density / call price) expansions; examples include the widely used SABR formula. In [KR11] Martin Keller-Ressel quantified the (volatility) smile behaviour at extreme strikes in generic affine stochastic volatility models. Precise expansions of implied volatility in the Heston model were recently obtained by Friz et al., [FGGS11].
- [BFS11] M. Beiglboeck, P. Friz, S. Sturm: Is the minimum value of an option on variance generated by local volatility? SIAM J. Finan. Math. 2, pp. 213-220, 2011.
- [BF09] S. Benaim, P. Friz: Regular Variation and Smile Asymptotics, Math. Finance Vol. 19 no 1. pp. 1-12, 2009.
- [KR11] M. Keller-Ressel: Moment Explosions and Long-Term Behavior of Affine Stochastic Volatility Models. Mathematical Finance 21/1, pp. 73-98, 2011.
- [FGGS11] P. Friz, S. Gerhold, A. Gulisashvili, S. Sturm On refined volatility smile expansion in the Heston model, Quantitative Finance, Volume 11, Issue 8, pp. 1151-1164, 2011.