We present empirical evidence for considering volatility of Eurodollar futures as a stochastic process, requiring a generalization of the standard Black-Scholes (BS) model which treats volatility as a constant. We use a previous development of a statistical mechanics of financial markets (SMFM) to model these issues. Ke ywords: options; eurodollar; volatility; statistical mechanics

A method for online estimation of the volatility when observing a stock price is proposed. This is based on modeling the volatility dynamics as a stochastic differential equation that is constructed using a technique from the control theory [1]. Identification of the model parameters using the observations is proposed afterwards [2]. It is based on some stochastic calculus. Volatility estimatio...

We present a positivity preserving numerical scheme for the pathwise solution of nonlinear stochastic differential equations driven by a multi-dimensional Wiener process and governed by non-commutative linear and non-Lipschitz vector fields. This strong order one scheme uses: (i) Strang exponential splitting, an approximation that decomposes the stochastic flow separately into the drift flow, a...

Stochastic Volatility models represent a well-known framework for the analysis of nancial time series data, together with the other important class of ARCH-type models. The main diierence between them, at least from a statistical point of view, relies on the possibility of obtaining exact inference, in particular with regard to the estimation issue. While for ARCH-type models the standard resul...

In this paper we propose to use a combination of regular and singular perturbations to analyze parabolic PDEs that arise in the context of pricing options when the volatility is a stochastic process that varies on several characteristic time scales. The classical Black-Scholes formula gives the price of call options when the underlying is a geometric Brownian motion with a constant volatility. ...

In this article, we propose an analytical approximation for the pricing of European options for some lognormal stochastic volatility models. This approximation is a second-order Taylor series expansion of the Fourier transform with respect to the "volatility of volatility". We give, using these formulas, a new method of variance reduction for the Monte-Carlo simulation of the trajectories of th...

For general time-dependent local volatility models, we propose new approximation formulas for the price of call options. This extends previous results of [BGM10b] where stochastic expansions combined with Malliavin calculus were performed to obtain approximation formulas based on the local volatility At The Money. Here, we derive alternative expansions involving the local volatility at strike. ...

We consider th e relationships between ARCH-type and stochast ic volatility models. A new class of volatility models, called generalized bilinear stochastic volatility, is described following an approach that tr ansforms an init ial GARCH-M process. Th e focus here is on th e interpretation of some simulation results, with a special care devoted to model misspecification.