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Grégoire Loeper interviendra sur Black and Scholes, Legendre and Sinkhorn
Abstract :
This talk will be a unified overview of some recent and less recent contributions in derivatives pricing.
The financial topics are option pricing with market impact and model calibration.
The mathematical tools are fully non-linear partial differential equations and martingale optimal transport.
Some new and fun (?) results will be a Black-Scholes-Legendre formula for option pricing with market impact, a Measure Preserving Martingale Sinkhorn algorithm for martingale optimal transport, a new point of view on the Bass Martingale, and some algorithms for exact local volatility calibration.
Aymeric Baradat interviendra sur Entropic JKO scheme for the Muskat problem
Abstract :
The Muskat problem is a system of PDEs describing the evolution of two incompressible and immiscible fluids of different densities under the action of gravitation. When the heavier fluid stands above the lighter fluid, the so-called Saffman-Taylor instability makes the system ill-posed in Sobolev spaces. Yet, Otto proposed in the late 90s to study a relaxation of this problem by interpreting it as a Wasserstein gradient flow and then by considering the limit of the corresponding JKO scheme as the time-step goes to zero. This procedure allowed him to draw a link between the Muskat problem and the entropic solutions of one-dimensional conservation laws. In this talk, I will show that an entropic version of the JKO scheme is particularly well adapted to this problem. On the practical side, it can be efficiently computed using a very simple Sinkhorn algorithm. On the theoretical side, it converges towards the solution of a well-posed system, shedding light on Otto’s connection with conservation laws. This is a work in progress with Sofiane Cher.
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