Mathematical Finance Seminar

March 21, 2002 5:30 PM to 7:00 PM

Paul Malliavin, Universite de Paris VI

Stochastic Calculus of Variations in Mathematical Finance

In a market with high frequency quotations, we implement a new algorithm, based on Fourier analysis, giving an accurate measurement of the volatilities. A hedged Stochastic calculus of variations is introduced where the reduced length of the variation propagate according a linear ODE, defining the instantaneous rate of liquidity. When this rate is negative the market enjoys some internal stability. This rate can be mathematically computed for the risk free process associated to any univariate model where volatility is function of the price. The length of the variation for the historical price process, satisfies a linear SDE which induces another stability indicator. In a model free situation, an approach to get the instantaneous rate of liquidity is proposed through measurements of iterated volatilities. In the multivariate case the instantaneous rate has to be generalized in an elasticity matrix describing the direction of internal stability of the market for the risk free process. Derivation of Monte-Carlo simulations relatively to parameters can be realized by suitable stochastic calculus of variations.