Skew and implied leverage effect: smile dynamics revisited

Variance Dynamics - An empirical journey

“…. This martingale property is the basis of numerous stochastic volatility models [3,8,21] ; We also follow that choice, because of the generality and flexibility of the resulting volatility models.…”

“…The weighting functions ω α might depend on the curve ξ t and time, but not on the spot S t -this is the choice followed in [4,8,21]. Quite often, they are chosen as time-invariant decreasing functions, i.e.…”

“…Straightforward computation by iterative conditioning on the filtrations at times t + ∆ and t + δt, followed by the use of Eq. 3 (see [21]) leads to:…”

“…We pursue our exploration of the spot/vol properties by delving into the theoritical implications of the non-linearities. We investigate the impact on the pricing and hedging of derivatives, an area of active research [2,4,10,5,21]. We first focus on standard vanilla options and review the link between skewness of an underlying and the implied skew (section 4.1).…”

“…GARCH models [6,14]) are probably the best known and most used. Recent advances have shown that the dynamics they generate are intrinsically constrained by the class they belong to [2,4,10,5,21]. For instance, Bergomi demonstrates in [4] that the dynamics of ATM implied volatilities generated by a stochastic volatility model are inherently linked to the smile produced by the model.…”