Application Domains
New Software and Platforms
Bilateral Contracts and Grants with Industry
Partnerships and Cooperations
Bibliography
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## Section: New Results

### Risk management in finance and insurance

#### Option pricing in a non-linear incomplete market model with default

Agnès Sulem has studied with Miryana Grigorova (University of Leeds) and Marie-Claire Quenez (Université Paris Denis Diderot) superhedging prices and the associated superhedging strategies for both European and American options (see [33] and [32] in a non-linear incomplete market model with default. The underlying market model consists of a risk-free asset and a risky asset driven by a Brownian motion and a compensated default martingale. The portfolio processes follow non-linear dynamics with a non-linear driver $f$.

#### Neural network regression for Bermudan option pricing

The pricing of Bermudan options amounts to solving a dynamic programming principle, in which the main difficulty, especially in high dimension, comes from the conditional expectation involved in the computation of the continuation value. These conditional expectations are classically computed by regression techniques on a finite dimensional vector space. In [36], Bernard Lapeyre and Jérôme Lelong study neural networks approximations of conditional expectations. They prove the convergence of the well-known Longstaff and Schwartz algorithm when the standard least-square regression is replaced by a neural network approximation. They illustrate the numerical efficiency of neural networks as an alternative to standard regression methods for approximating conditional expectations on several numerical examples.

#### American options

With his PhD student Giulia Terenzi, Damien Lamberton has been working on American options in Heston's model [22]. He is currently preparing his contribution to a winter school on "Theory and practice of optimal stopping and free boundary problems" (cf. https://conferences.leeds.ac.uk/osfbp/).

#### Solvency Capital Requirement 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.

A. Alfonsi and his PhD student A. Cherchali have developed a model of the ALM management of insurance companies that takes into account the regulatory constraints on life-insurance [25]. We now focus on developing Multilevel Monte-Carlo methods to approximate the SCR (Solvency Capital Requirement).