Monique Pontier

Optimal portfolio for a small investor in a market model with discontinuous prices

A consumption-investment problem is considered for a small investor in the case of a market model in which prices evolve according to a stochastic equation with a jump-process component. The techniques we use include the martingale representation theorem, Lagrange multiplier methods, and Markovian methods for the resolution of stochastic differential equations. We establish a Black-Scholes formula.

Insider Trading in a Continuous Time Market Model

This paper uses the enlargement of Brownian filtrations and a probability change for modelling the observation of a financial market by an insider trader. A characterization of admissible strategies and a criterion for optimization are given. Then a statistical test is proposed to test whether or not the trader is an insider.

Free lunch and arbitrage possibilities in a financial market model with an insider

We consider financial market models based on Wiener space with two agents on different information levels: a regular agent whose information is contained in the natural filtration of the Wiener process W, and an insider who possesses some extra information from the beginning of the trading interval, given by a random variable L which contains information from the whole time interval. Our main concern are variables L describing the maximum of a pricing rule. Since for such L the conditional laws given by the smaller knowledge of the regular trader up to fixed times are not absolutely continuous with respect to the law of L, this class of examples cannot be treated by means of the enlargement of filtration techniques as applied so far. We therefore use elements of a Malliavin and Ito calculus for measure-valued random variables to give criteria for the preservation of the semimartingale property, the absolute continuity of the conditional laws of L with respect to its law, and the absence of arbitrage. The master example, given by supt[set membership, variant][0,1] Wt, preserves the semimartingale property, but allows for free lunch with vanishing risk quite generally. We deduce conditions on drift and volatility of price processes, under which we can construct explicit arbitrage strategies.

ASYMMETRICAL INFORMATION AND INCOMPLETE MARKETS

This paper is an extension of [13] where we have studied a financial market with informed and non-informed agents. The informed agent has an initial insider information. We analyze the completeness of the market and the family of risk-neutral probabilities. Some cases of arbitrage opportunities are done.

A Hull and White formula for a general stochastic volatility jump-diffusion model with applications to the study of the short-time behavior of the implied volatility

In this paper, generalizing results in Alòs, León and Vives (2007b), we see that the dependence of jumps in the volatility under a jump-diffusion stochastic volatility model, has no effect on the short-time behaviour of the at-the-money implied volatility skew, although the corresponding Hull and White formula depends on the jumps. Towards this end, we use Malliavin calculus techniques for Lévy processes based on Løkka (2004), Petrou (2006), and Solé, Utzet and Vives (2007).

Optimal strategies in a risky debt context

This paper analyses structural models for the evaluation of risky debt following Leland (J. Finance 49 (1994), pp. 1213–1252) with an approach of optimal stopping problem. Moreover, we introduce an investment control parameter and we optimize with respect to the failure threshold and coupon rate. We show that the value of the optimal coupon policy decreases if the strict priority rule is removed.

Calcul anticipatif d'ordre deux

An anticipative stochastic calculus for manifold valued processes is defined; if uses both, second-order geometry defined by P.-A. Meyer [8] and Nualart-Pardouxs duality definition of Skorohod Integral [11] This stochastic calculus allows us to integrate non-adapted processes taking their values in the space of second-order I-forms that are above a manifold valued semi-martingale

Backward Stochastic Differential Equations in a Lie Group

We study backward stochastic differential equations where the solution process lives in a finite dimensional Lie group. The group stucture makes this problem easier to deal with than in a general manifold, but the geometry still imposes interesting conditions. The main tools are the stochastic exponential and logarithm of Lie groups, used to change group-valued martingales into \(\mathbb{R}^d\)-valued martingales. We are first interested in getting a group-valued martingale with prescribed terminal value: existence and uniqueness are proved for nilpotent Lie groups by a constructive method; also a recursive construction of the solution is given and uniqueness is obtained for groups where a convex barycenter can be defined. We then study more general backward stochastic equations with a drift term.

Filtering with observations on a Riemannian symmetric space

The point under discussion in this paper refers to a problem of filtering in which observation Y of a system state X is a process with values in a symmetric space M. The observation is a Brownian motion transformed by an isometry of M depending on the state. It takes its values in manifold M and its multiplicative formulation is nonstandard. In many physical situations, e.g., mechanics, robotics, spatial fields, the filtering problems are naturally set up in manifolds as well for the signal and the observation. The reference probability method is used to construct the model. Then filtering equations are deduced; these comply with the conditional law according to its observation. Unique characterization of this conditional law is given. Last, two examples are investigated. First the multivariate case: the observation Y is in

Capital structure with firm’s net cash payouts

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