Localizing Volatilities

Marc N. Atlan (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We propose two main applications of Gy\"{o}ngy (1986)'s construction of inhomogeneous Markovian stochastic differential equations that mimick the one-dimensional marginals of continuous It\^{o} processes. Firstly, we prove Dupire (1994) and Derman and Kani (1994)'s result. We then present Bessel-based stochastic volatility models in which this relation is used to compute analytical formulas for the local volatility. Secondly, we use these mimicking techniques to extend the well-known local volatility results to a stochastic interest rates framework.

Mots clés: Local volatility; Stochastic Volatility; Stochastic Interest Rates; Bessel Processes; Local Martingales

Domaines: Probabilités [math.PR]

Marc N. Atlan. Localizing Volatilities. 2006. ⟨hal-00022702⟩