From implied to spot volatilities

Abstract

This paper is concerned with the link between spot and implied volatilities. The main result is the derivation of the stochastic differential equation driving the spot volatility based on the shape of the implied volatility surface. This equation is a consequence of no-arbitrage constraints on the implied volatility surface right before expiry. We investigate the regularity of this surface at maturity in the case of the Constant Elasticity of Variance and Heston models. We also show that a simple way to link spot and implied volatilities is to relate the coefficients of the implied volatility surface Taylor expansion to the coefficients of a certain chaos expansion of the spot volatility process. As a byproduct, we give expansions for the implied volatility surface for a general stochastic volatility model.

Option pricing in the moderate deviations regime

“…□ Proposition 2.4 combined with Theorem 2.6 allows to compute skew and curvature (and higher derivatives of the implied volatility smile, if desired) directly from the coefficients of a general stochastic volatility model. Related formulae for "general" (even non-Markovian) models also appear in the work of Durrleman (theorem 3.1.1. in Durrleman, 2004; see also Durrleman, 2010). While not written in the setting of general Markovian diffusion models, and hence not in terms of the energy function Λ, they inevitably give the same results if applied to given parametric stochastic volatility models (see section 3.1 in Durrleman, 2004).…”

Short Maturity Asian Options in Local Volatility Models

“…We present the proof on the case K ≥ S 0 , the case K < S 0 is treated in a similar way, and leads to the same final result (37). For given log-strike x = log(K/S 0 ) ≥ 0, one has to find f 1 by solving the equation (23), and the rate function is obtained by substituting this value for f 1 into (19). We note that as x ↓ 0, we have f 1 ↓ 0.…”

“…Step 3. Express the result (19) for the rate function I(K, S 0 ) for K ≥ S 0 as a series in z 1 . Further, use here the expansion for z 1 in powers of x obtained in Step 2.…”

“…Asymptotics of the option prices, and implied volatility for short maturity European options have been extensively studied in the literature, see e.g. [4,29,30,19,8] for the local volatility models, [22,46,3,18,43] for the exponential Lévy models and [5,36,25,20,21,23,1] for the stochastic volatility models and [28,40] for model-free approaches. To the best of our knowledge, the short maturity Asian options have been much less studied.…”

Forward Variance Dynamics: Bergomi’s Model Revisited

“…So the model will generate the positive skew observed in the VIX options (See for example Durrleman (2005) …”