The Term Structure of Implied Volatility in Symmetric Models with Applications to Heston

We study the term structure of the implied volatility in a situation where the smile is symmetric. Starting from the result by Tehranchi that a symmetric smile generated by a continuous martingale necessarily comes from a mixture of normal distributions, we derive representation formulae for the at-the-money (ATM) implied volatility level and curvature in a general symmetric model. As a result, the ATM curve is directly related to the Laplace transform of the quadratic variation of the log price. To deal with the remaining part of the volatility surface, we build a time dependent SVI-type approximation which matches the ATM and extreme moneyness structure. As an instance of a symmetric model, we consider uncorrelated Heston: in this framework, our representation of the ATM volatility takes semiclosed (and easy to implement) form and the time-dependent SVI approximation displays considerable performances in a wide range of maturities and strikes. In addition, we show how to apply our results to a skewed smile by considering a displaced model. Finally, a noteworthy fact is that all along the paper we will deal only with Laplace transforms and not with Fourier transforms, thus avoiding any complex-valued function