Maurizio Pratelli

Mean-Variance Hedging for Stochastic Volatility Models

In this paper we discuss the tractability of stochastic volatility models for pricing and hedging options with the mean-variance hedging approach. We characterize the variance-optimal measure as the solution of an equation between Doleans exponentials; explicit examples include both models where volatility solves a diffusion equation and models where it follows a jump process. We further discuss the closedness of the space of strategies.

Functional convergence of Snell envelopes: Applications to American options approximations

Abstract. The main result of the paper is a stability theorem for the Snell envelope under convergence in distribution of the underlying processes: more precisely, we prove that if a sequence

On the use of measure-valued strategies in bond markets

(X^n)

A theory of stochastic integration for bond markets

of stochastic processes converges in distribution for the Skorokhod topology to a process

On a Lemma by Ansel and Stricker

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Functional Cramér-Rao bounds and Stein estimators in Sobolev spaces, for Brownian motion and Cox processes

and satisfies some additional hypotheses, the sequence of Snell envelopes converges in distribution for the Meyer–Zheng topology to the Snell envelope of

Presentation Unifiee de Certaines Inegalites de la Theorie des Martingales

X

Super-replication and utility maximization in large financial markets

(a brief account of this rather neglected topology is given in the appendix). When the Snell envelope of the limit process is continuous, the convergence is in fact in the Skorokhod sense. This result is illustrated by several examples of approximations of the American options prices; we give moreover a kind of robustness of the optimal hedging portfolio for the American put in the Black and Scholes model.

Stochastic Integration with Respect to a Sequence of Semimartingales

Abstract.We propose here a theory of cylindrical stochastic integration, recently developed by Mikulevicius and Rozovskii, as mathematical background to the theory of bond markets. In this theory, since there is a continuum of securities, it seems natural to define a portfolio as a measure on maturities. However, it turns out that this set of strategies is not complete, and the theory of cylindrical integration allows one to overcome this difficulty. Our approach generalizes the measure-valued strategies: this explains some known results, such as approximate completeness, but at the same time it also shows that either the optimal strategy is based on a finite number of bonds or it is not necessarily a measure-valued process.

Sur certains espaces de martingales localement de carre integrable

We introduce a theory of stochastic integration with respect to a family of semimartingales depending on a continuous parameter, as a mathematical background to the theory of bond markets. We apply our results to the problem of super-replication and utility maximization from terminal wealth in a bond market. Finally, we compare our approach to those already existing in literature.