From characteristic functions to implied volatility expansions

For any strictly positive martingale $S = \exp(X)$ for which $X$ has a characteristic function, we provide an expansion for the implied volatility. This expansion is explicit in the sense that it involves no integrals, but only polynomials in the log strike. We illustrate the versatility of our expansion by computing the approximate implied volatility smile in three well-known martingale models: one finite activity exponential L\'evy model (Merton), one infinite activity exponential L\'evy model (Variance Gamma), and one stochastic volatility model (Heston). Finally, we illustrate how our expansion can be used to perform a model-free calibration of the empirically observed implied volatility surface.Comment: 21 pages, 4 figure

A Fast Mean-Reverting Correction to Heston's Stochastic Volatility Model

We propose a multi-scale stochastic volatility model in which a fast mean-reverting factor of volatility is built on top of the Heston stochastic volatility model. A singular pertubative expansion is then used to obtain an approximation for European option prices. The resulting pricing formulas are semi-analytic, in the sense that they can be expressed as integrals. Difficulties associated with the numerical evaluation of these integrals are discussed, and techniques for avoiding these difficulties are provided. Overall, it is shown that computational complexity for our model is comparable to the case of a pure Heston model, but our correction brings significant flexibility in terms of fitting to the implied volatility surface. This is illustrated numerically and with option data

Asymptotics for dd-dimensional L\'evy-type processes

We consider a general d-dimensional Levy-type process with killing. Combining the classical Dyson series approach with a novel polynomial expansion of the generator A(t) of the Levy-type process, we derive a family of asymptotic approximations for transition densities and European-style options prices. Examples of stochastic volatility models with jumps are provided in order to illustrate the numerical accuracy of our approach. The methods described in this paper extend the results from Corielli et al. (2010), Pagliarani and Pascucci (2013) and Lorig et al. (2013a) for Markov diffusions to Markov processes with jumps.Comment: 20 Pages, 3 figures, 3 table