Quantitative finance and applied probability research

Book

Financial modelling with jump processes (with Rama Cont), CRC Press (2003)

Recent papers

Control and optimal stopping Mean Field Games: a linear programming approach (with Roxana Dumitrescu and Marcos Leutscher), Electronic Journal of Probability 26, 1-49, (2021)

We develop the linear programming approach to mean-field games in a general setting. This relaxed control approach allows to prove existence results under weak assumptions, and lends itself well to numerical implementation. We consider mean-field game problems where the representative agent chooses both the optimal control and the optimal time to exit the game, where the instantaneous reward function and the coefficients of the state process may depend on the distribution of the other agents. Furthermore, we establish the equivalence between mean-field games equilibria obtained by the linear programming approach and the ones obtained via the controlled/stopped martingale approach, another relaxation method used in a few previous papers in the case when there is only control.

Mean-field games of optimal stopping: a relaxed solution approach (with Géraldine Bouveret and Roxana Dumitrescu), SIAM Journal on Control and Optiimization 58.4 (2020): 1795-1821. (preprint on Arxiv)

We consider the mean-field game where each agent determines the optimal time to exit the game by solving an optimal stopping problem with reward function depending on the density of the state processes of agents still present in the game. We place ourselves in the framework of relaxed optimal stopping, which amounts to looking for the optimal occupation measure of the stopper rather than the optimal stopping time. This framework allows us to prove the existence of a relaxed Nash equilibrium and the uniqueness of the associated value of the representative agent under mild assumptions. Further, we prove a rigorous relation between relaxed Nash equilibria and the notion of mixed solutions introduced in earlier works on the subject and provide a criterion under which the optimal strategies are pure strategies, that is, behave in a similar way to stopping times. Finally, we present a numerical method for computing the equilibrium in the case of potential games and show its convergence.

Optimal importance sampling for Lévy processes (with Adrien Genin), Stochastic Processes and their Applications 130.1 (2020): 20-46. (preprint on Arxiv)

We develop generic and efficient importance sampling estimators for Monte Carlo evaluation of prices of single- and multi-asset European and path-dependent options in asset price models driven by Lévy processes, extending earlier works which focused on the Black-Scholes and continuous stochastic volatility models. Using recent results from the theory of large deviations on the path space for processes with independent increments, we compute an explicit asymptotic approximation for the variance of the pay-off under an Esscher-style change of measure. Minimizing this asymptotic variance using convex duality, we then obtain an easy to compite asymptotically efficient importance sampling estimator of the option price. Numerical tests for European baskets and for Asian options in the variance gamma model show consistent variance reduction with a very small computational overhead.

Volatility options in rough volatility models (with Blanka Horvath and Antoine Jacquier), SIAM Journal on Financial Mathematics 11.2 (2020): 437-469. (preprint on Arxiv)

We discuss the pricing and hedging of volatility options in some rough volatility models. First, we develop efficient Monte Carlo methods and asymptotic approximations for computing option prices and hedge ratios in models where log volatility follows a Gaussian Volterra process. While providing a good fit for European options, these models are unable to reproduce the VIX option smile observed in the market and are thus not suitable for VIX products. To accommodate these, we introduce the class of modulated Volterra processes and show that they successfully capture the VIX smile.

Asymptotic lower bounds for optimal tracking: a linear programming approach (with Jiatu Cai and Mathieu Rosenbaum), The Annals of Applied Probability 27.4 (2017): 2455-2514. (open access pdf)

We consider the problem of tracking a target whose dynamics is modeled by a continuous Itô semi-martingale. The aim is to minimize both deviation from the target and tracking efforts. We establish the existence of asymptotic lower bounds for this problem, depending on the cost structure. These lower bounds can be related to the time-average control of Brownian motion, which is characterized as a deterministic linear programming problem. A comprehensive list of examples with explicit expressions for the lower bounds is provided.