We study a dynamic multi-asset economy with private information, a stock and a derivative. There are informed and uninformed investors as well as bounded rational investors trading on noise. The noisy rational expectations equilibrium is obtained in closed form. The equilibrium stock price follows a non-Markovian process, is positive and has stochastic volatility. The derivative cannot be replicated, except at rare endogenous times. At any point in time, the derivative price adds information relative to the stock price, but the pair of prices is less informative than volatility, the residual demand or the history of prices. The rank of the asset span drops at endogenous times causing turbulent trading activity. The effects of financial innovation are discussed. The equilibrium is fully revealing if the derivative is not traded: financial innovation destroys information.

# Past Mathematical and Computational Finance Seminar

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.

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).

We formulate and solve a class of Backward Stochastic Differential Equations (BSDEs) driven by the compensated random measure associated to a given marked point process on a general state space. We present basic well-posedness results in L 2 and in L 1 . We show that in the setting of point processes it is possible to solve the equation recursively, by replacing the BSDE by an ordinary differential equation in between jumps. Finally we address applications to optimal control of marked point processes, where the solution of a suitable BSDE allows to identify the value function and the optimal control. The talk is based on joint works with Marco Fuhrman and Jean Jacod.

This paper formulates an utility indifference pricing model for investors trading in a discrete time financial market under non-dominated model uncertainty.

The investors preferences are described by strictly increasing concave random functions defined on the positive axis. We prove that under suitable

conditions the multiple-priors utility indifference prices of a contingent claim converge to its multiple-priors superreplication price. We also

revisit the notion of certainty equivalent for random utility functions and establish its relation with the absolute risk aversion.

In the first part of the talk, we present some recent and new developments in the theory of control and optimal stopping with nonlinear expectations. We first introduce an optimal stopping game with nonlinear expectations (Generalized Dynkin Game) in a non-Markovian framework and study its links with nonlinear doubly reflected BSDEs. We then present some new results (which are part of an ongoing work) on mixed stochastic stochastic control/optimal stopping problems (as well as stochastic control/optimal stopping game problems) in a non-Markovian framework and their relation with constrained reflected BSDEs with lower obstacle (resp. upper obstacle). These results are obtained using some technical tools of stochastic analysis. In the second part of the talk, we discuss applications to the $\cal{E}^g$ pricing of American options and Game options in complete and incomplete markets (based on joint works with M.C.Quenez and Agnès Sulem).

We derive sharp bounds for the prices of VIX futures using the full information of S&P 500 smiles. To that end, we formulate the model-free sub/superreplication of the VIX by trading in the S&P 500 and its vanilla options as well as the forward-starting log-contracts. A dual problem of minimizing/maximizing certain risk-neutral expectations is introduced and shown to yield the same value. The classical bounds for VIX futures given the smiles only use a calendar spread of log-contracts on the S&P 500. We analyze for which smiles the classical bounds are sharp and how they can be improved when they are not. In particular, we introduce a tractable family of functionally generated portfolios which often improves the classical spread while still being tractable, more precisely, determined by a single concave/convex function on the line. Numerical experiments on market data and SABR smiles show that the classical lower bound can be improved dramatically, whereas the upper bound is often close to optimal.

This talk explains how to formulate the now classical problem of optimal liquidation (or optimal trading) inside a Mean Field Game (MFG). This is a noticeable change since usually mathematical frameworks focus on one large trader in front of a " background noise " (or " mean field "). In standard frameworks, the interactions between the large trader and the price are a temporary and a permanent market impact terms, the latter influencing the public price. Here the trader faces the uncertainty of fair price changes too but not only. He has to deal with price changes generated by other similar market participants, impacting the prices permanently too, and acting strategically. Our MFG formulation of this problem belongs to the class of " extended MFG ", we hence provide generic results to address these " MFG of controls ", before solving the one generated by the cost function of optimal trading. We provide a closed form formula of its solution, and address the case of " heterogenous preferences " (when each participant has a different risk aversion). Last but not least we give conditions under which participants do not need to instantaneously know the state of the whole system, but can " learn " it day after day, observing others' behaviors.

We consider the problem of modelling the term structure of defaultable bonds, under minimal assumptions on the default time. In particular, we do not assume the existence of a default intensity and we therefore allow for the possibility of default at predictable times. It turns out that this requires the introduction of an additional term in the forward rate approach by Heath, Jarrow and Morton (1992). This term is driven by a random measure encoding information about those times where default can happen with positive probability. In this framework, we derive necessary and sufficient conditions for a reference probability measure to be a local martingale measure for the large financial market of credit risky bonds, also considering general recovery schemes. This is based on joint work with Thorsten Schmidt.