Section: New Results

Financial Mathematics

Participants : Madalina Deaconu, Antoine Lejay, Paolo Pigato, Khaled Salhi, Etienne Tanré.

Published works and preprints

• When the underlying asset price is given by a exponential Lévy model, the market is almost incomplete. Under this hypothesis, M. Deaconu, A. Lejay and K. Salhi worked on derivatives hedging under a budget constraint on the initial capital. He considers, as criterion of optimization, the CVaR of the terminal hedging risk. First, he rewrites the problem an optimisation problem on the random fraction of the payoff that permits to respect the budget constraint. Then, he approximates the problem by relaxing the constraint and considering only a specific equivalent martingale measure. This approximate problem is solved using Neyman-Pearson's Lemma and, in the case of European options, a numerical valuation of the approximated minimal CVaR based on fast Fourier transform [28].

• A. Lejay and P. Pigato studied the estimation of the coefficients of the Geometric Oscillating Brownian motion on financial data. This stochastic process is a modification of the Black & Scholes model that takes into account leverage effect and other sudden changes in the volatility [58], [41].

• V. Reutenauer and E. Tanré have worked on extensions of the exact simulation algorithm introduced by Beskos et al.  [63]. They propose an unbiased algorithm to approximate the two first derivatives with respect to the initial condition $x$ of quantities with the form $𝔼\Psi \left(X_T^x\right)$, where $X$ is a one-dimensional diffusion process and $\Psi$ any test-function. They also propose an efficient modification of Beskos et al. algorithm ([59], paper in revision).