# Jumps and Diffusions in Volatility: It Takes Two to Tango

1 OMEGA - Probabilistic numerical methods
CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, CNRS - Centre National de la Recherche Scientifique : UMR7502
Abstract : We consider the following stochastic differential equation for describing financial data evolution: $dX_t = b(t, X_t)¸dt + \sigma(t) h(X_t)¸dW_t, \qquad X(0)= x.$ with a stochastic volatility $\sigma(t)$ (e.g. the combination of a diffusion and a jump process). First we prove the existence and positiveness of the solution of a Cox-Ingersoll-Ross type stochastic differential equation with time varying coefficients, which is a special case of our model. Next, from observation on $X_t$ at times $t_i$ (with non regular sampling scheme), we propose a non-parametric estimator for the volatility (optimal in a certain way). We show its point-wise convergence and its asymptotic normality. After, we propose an estimator for the volatility jump times and prove a Central Limit Theorem. Finally the application of these estimators to the BTP futures (Italian ten year bond futures) and Lira 1 month deposit Eurorates seems to confirm the adequacy of the proposed model.
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Submitted on : Wednesday, May 24, 2006 - 1:53:12 PM
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• HAL Id : inria-00073843, version 1

### Citation

Renzo G. Avesani, Pierre Bertrand. Jumps and Diffusions in Volatility: It Takes Two to Tango. [Research Report] RR-2848, INRIA. 1996, pp.34. ⟨inria-00073843⟩

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