Section: New Results
Risk management in finance and insurance
Control of systemic risk in a dynamic framework
Interconnected systems are subject to contagion in time of distress. Recent effort has been dedicated to understanding the relation between network topology and the scope of distress propagation. It is critical to recognize that connectivity is a result of an optimization problem of agents, who derive benefits from connections and view the associated contagion risk as a cost. In our previous works on the control of contagion in financial systems (see e.g. , , ), a central party, for example a regulator or government, seeks to minimize contagion. In , in contrast, the financial institutions themselves are the decision makers, and their decision is made before the shock, with a rational expectation on the way the cascade will evolve following the shock. We are extending these studies in a dynamic framework by allowing a recovery feature in the financial system during the cascade process, captured by introducing certain extent of growth of the banks' assets between each round of contagion.
Option pricing in financial markets with imperfections and default
A. Sulem, M.C. Quenez and R. Dumitrescu have studied robust pricing in an imperfect financial market with default. In this setting, the pricing system is expressed as a nonlinear g-expectation induced by a nonlinear BSDE with nonlinear driver and default jump (see ). The case of American options in this market model is treated in . The incomplete market case is under study.
With Giulia Terenzi, D. Lamberton has been been working on American options in Heston's model. They have some results about existence and uniqueness for the associated variational inequality, in suitable weighted Sobolev spaces (see Feehan and co-authors for recent results on elliptic problems). Their paper "Variational formulation of American option prices in the Heston model"  is now in minor revision for SIAM Journal on Financial Mathematics.
They also have some results on monotonicity and regularity properties of the price function.
D. Lamberton has also a paper on the binomial approximation of the American put, in which a new bound for the rate of convergence of the binomial approximation of the Black-Scholes American put price is derived .
Optimal stopping problems involving the maximum of a diffusion is currently under investigation. Partial results obtained by D. Lamberton and M. Zervos) enable them to treat reward functions with little regularity.
Monte-Carlo methods for the computation of the Solvency Capital Requirement (SCR) in Insurance
A. Alfonsi has obtained a grant from AXA Foundation on a Joint Research Initiative with a team of AXA France working on the strategic asset allocation. This team has to make recommendations on the investment over some assets classes as, for example, equity, real estate or bonds. In order to do that, each side of the balance sheet (assets and liabilities) is modeled in order to take into account their own dynamics but also their interactions. Given that the insurance products are long time contracts, the projections of the company's margins have to be done considering long maturities. When doing simulations to assess investment policies, it is necessary to take into account the SCR which is the amount of cash that has to be settled to manage the portfolio. Typically, the computation of the future values of the SCR involve expectations under conditional laws, which is greedy in computation time. The goal of this project is to develop efficient Monte-Carlo methods to compute the SCR for long investment strategies. A. Cherchali has started his PhD thesis in September 2017 on this topic.
A. Alfonsi and A. Cherchali are developing a model of the ALM management of insurance companies that takes into account the regulatory constraints on life-insurance. We are testing this model. The purpose is then to use this model to develop Monte-Carlo methods to approximate the SCR (Solvency Capital Requirement).