In this talk, I consider the problem of

hedging European and Bermudan option with a given probability. This

question is

more generally linked to portfolio optimisation problems under weak

stochastic target constraints.

I will recall, in a Markovian framework, the characterisation of the

solution by

non-linear PDEs. I will then discuss various numerical algorithms

to compute in practice the quantile hedging price.

This presentation is based on joint works with B. Bouchard (Université

Paris Dauphine), G. Bouveret (University of Oxford) and ongoing work

with C. Benezet (Université Paris Diderot).

# Mathematical and Computational Finance Seminar

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In this talk, I consider the problem of pricing and (statically)

hedging short-term contingent claims written on illiquid or

non-tradable assets.

In a first part, I show how to find the best European payoff written

on a given set of underlying assets for hedging (under several

metrics) a given European payoff written on another set of underlying

assets -- some of them being illiquid or non-tradable. In particular,

I present new results in the case of the Expected Shortfall risk

measure. I also address the associated pricing problem by using

indifference pricing and its link with entropy.

In a second part, I consider the more classic case of hedging with a

finite set of simple payoffs/instruments and I address the associated

pricing problem. In particular, I show how entropic methods (Davis

pricing and indifference pricing à la Rouge-El Karoui) can be used in

conjunction with recent results of extreme value theory (in dimension

higher than 1) for pricing and hedging short-term out-of-the-money

options such as those involved in the definition of Daily Cliquet

Crash Puts.

The solution to the standard cost efficiency problem depends crucially on the fact that a single real-world measure P is available to the investor pursuing a cost-efficient approach. In most applications of interest however, a historical measure is neither given nor can it be estimated with accuracy from available data. To incorporate the uncertainty about the measure P in the cost efficient approach we assume that, instead of a single measure, a class of plausible prior models is available. We define the notion of robust cost-efficiency and highlight its link with the maxmin expected utility setting of Gilboa and Schmeidler (1989) and more generally with robust preferences in a possibly non expected utility setting.

This is joint work with Thibaut Lux and Steven Vanduffel (VUB)