3. Scientiﬁc Foundations 3.1. Numerical methods for option pricing and hedging and model calibration

The development of increasingly complex ﬁnancial products requires the use of advanced stochastic and numerical analysis techniques that pose challenging problems to mathematicians. In this ﬁeld, the scientiﬁc skills of the MATHFI research team are focused on • the modeling of the price of assets by stochastic processes (diffusions with jumps, stable processes, fractional Brownian processes, stochastic volatility models), • probabilistic and deterministic numerical methods and their implementation (Monte-Carlo methods, Malliavin calculus, numerical analysis of linear and nonlinear parabolic partial differential equations), • Stochastic control, with applications to calibration of ﬁnancial assets, pricing and hedging of derivative products, dynamic portfolio optimization in incomplete markets (transaction costs, discrete time hedging, portfolio constraints, partial observation). with jumps, local volatility and stochastic volatility. Some calibration algorithms have been implemented. reward function is the maximum of two functions which are also the value functions of two optimal stopping time problems. We give a characterization of the optimal stopping times and prove their existence under some regularity assumptions. In the Markovian case, the value function is a solution of a PDE system. We study two examples: First, we consider the case of an investor who can invest in two risky assets. At time 0, he holds one asset S 1 and one asset S 2 . Then, at time θ 1 ( ∈ T 0 ), he can sell the ﬁrst asset and invest the remain amount in the second asset, or either sell the second asset and invest the remain amount in the ﬁrst one. At time time θ 2 , he transfers his wealth into a bond until time T . The problem is to determine the optimal strategy which maximizes the expected utility of terminal wealth. Second, we study the problem of pricing a call option on this ﬁnancial instrument and show that in the Markovian case, the price is associated to a system of PDEs. We also consider some options like digital options on two or more assets. In this case, the reward function is not regular and we can not apply the previous results. This problem is under study. We study the regularity of the viscosity solution of a quasilinear parabolic partial differential equation with Lipschitz coefﬁcients by using its connection with a forward backward stochastic differential equation (in short FBSDE) and we give a probabilistic representation of the generalized gradient (derivative in the distribution sense) of the viscosity solution. This representation is a kind of nonlinear Feynman-Kac formula. The main idea is to show that the FBSDE admits a unique linearized version interpreted as its distributional derivative with respect to the initial condition. If the diffusion coefﬁcient of the forward equation is uniformly elliptic, we approximate the FBSDE by smooth ones and use Krylov’s estimate to prove the convergenceof the derivatives. In the degenerate case, we use techniques of Bouleau-Hirsch on absolute continuity of probability measures. that E t ⊆ F t for all t ∈ [0 , T ] . The general problem is to maximize a given performance functional of this system over all admissible controls. This is a partial observation stochastic control problem in an anticipating environment . Examples of applications include stochastic volatity models in ﬁnance, insider inﬂuenced ﬁnancial markets and stochastic control of systems with delayed noise effects. Some speciﬁc cases from ﬁnance, involving optimal portfolio with logarithmic utility, are solved explicitly.