Section: New Results

Financial Mathematics

Participants : Maxime Bonelli, Mireille Bossy, Nicolas Champagnat, Madalina Deaconu, Antoine Lejay, Sylvain Maire, Khaled Salhi, Denis Talay, Etienne Tanré.

Published works and preprints

• K. Salhi, M. Deaconu, A. Lejay and N. Champagnat worked with N. Navet (University of Luxembourg) [28]. They construct a regime switching model for the univariate Value-at-Risk estimation. Extreme value theory (EVT) and hidden Markov models (HMM) are combined to estimate a hybrid model that takes volatility clustering into account. In the first stage, HMM is used to classify data in crisis and steady periods, while in the second stage, EVT is applied to the previously classified data to rub out the delay between regime switching and their detection. This new model is applied to prices of numerous stocks exchanged on NYSE Euronext Paris over the period 2001-2011. The relative performance of the regime switching model is benchmarked against other well-known modeling techniques, such as stable, power laws and GARCH models.

• K. Salhi wrote a survey paper about option pricing and risk management under exponential Lévy models [55]. He detailed some notions that are not well explained in the literature and he proposed new trends in the risk management of derivatives.

• In [26], D. Talay, E. Tanré, Christophe Michel (CA-CIB) and Victor Reutenauer (fotonower) have studied a model in financial mathematics including bid-ask spread cost. They study the optimal strategy to hedge an interest rate swap that pays a fixed rate against a floating rate. They present a methodology using a stochastic gradient algorithm to optimize strategies.

Other works in progress

• M. Bossy and M. Bonelli (Koris International) are working on the optimal portfolio investment problem under the drawdown constraint that the wealth process never falls below a fixed fraction of its running maximum. They derive optimal allocation programs by solving numerically the Hamilton-Jacobi-Bellman equation that characterizes the finite horizon expected utility maximization problem, for investors with power utility as well as S-shape utility. Using numerical experiments they show that implementing the drawdown constraint can be gainful in optimal portfolios for the power utility, for some market configurations and investment horizons. However, their study reveals different results in a prospect theory context.

• When the underlying asset price is given by a exponential Lévy model, the market is almost incomplete. Under this hypothesis, K. Salhi works 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. The article will be submitted soon.