We focus our work here on some very recent results obtained by Cherny and Madan on risk measures. They developed a rigorous mathematical framework for the study of coherent risk measures. The first sections mainly review the existing literature. We present it here for sake of completeness as...

The Heston model is one of the most popular stochastic volatility models for Equity and FX modelling. Although it was developed more than fifteen years ago, its understanding is still not complete and many recent publications have addressed deep theoretical and implementation issues. We review...

We rigorize the work of Lewis (2007) and Durrleman (2005) on the small-time asymptotic behavior of the implied volatility under the Heston stochastic volatility model (Theorem 2.1). We apply the Gärtner-Ellis theorem from large deviations theory to the exponential affine closed-form expression...

We build on of the work of Henry-Labordµere and Lewis on the small-time behaviour of the return distribution under a general local-stochastic volatility model with zero correlation. We do this using the Freidlin-Wentzell theory of large deviations for stochastic differential equations, and then...

Using the Gartner-Ellis theorem from large deviation theory, we characterize the leading-order behaviour of call option prices under the Heston model, in a new regime where the maturity is large and the log-moneyness is also proportional to the maturity. Using this result, we then derive the...

We show that if the discounted Stock price process is a continuous martingale, then there is a simple relationship linking the variance of the terminal Stock price and the variance of its arithmetic average. We use this to establish a model-independent upper bound for the price of a continuously...

In this paper we prove an approximate formula expressed in terms of elementary functions for the implied volatility in the Heston model. The formula consists of the constant and first order terms in the large maturity expansion of the implied volatility function. The proof is based on...

We show that the implied volatility has a uniform (in log moneyness x) limit as the maturity tends to infinity, given by an explicit closed-form formula, for x in some compact neighborhood of zero in the class of affine stochastic volatility models. This expression is function of the convex dual...

We study here the large-time behavior of all continuous affine stochastic volatility models (in the sense of Keller-Ressel) and deduce a closed-form formula for the large-maturity implied volatility smile. Based on refinements of the Gartner-Ellis theorem on the real line, our proof reveals...

For any strictly positive martingale S=e^X for which X has an analytically tractable 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 log(K/S 0 ). We illustrate the...