Tiziano Vargiolu

Robustness of the Black-Scholes Approach in the Case of Options on Several Assets

Abstract. In this paper we analyse a stochastic volatility model that is an extension of the traditional Black-Scholes one. We price European options on several assets by using a superstrategy approach. We characterize the Markov superstrategies, and show that they are linked to a nonlinear PDE, called the Black-Scholes-Barenblatt (BSB) equation. This equation is the Hamilton-Jacobi-Bellman equation of an optimal control problem, which has a nice financial interpretation. Then we analyse the optimization problem included in the BSB equation and give some sufficient conditions for reduction of the BSB equation to a linear Black-Scholes equation. Some examples are given.

Invariant Measures for the Musiela Equation with Deterministic Diffusion Term

Abstract. In this article the forward rates equation of the Musiela model is analysed. The equation is studied in the Sobolev spaces

A Bayesian adaptive control approach to risk management in a binomial model

H^1_\gamma({\Bbb R}^+)

Optimal portfolio for HARA utility functions in a pure jump multidimensional incomplete market

and

Optimal Portfolio for CRRA Utility Functions when Risky Assets are Exponential Additive Processes

H^1({\Bbb R}^+)

FINANCIAL MODELS WITH DEPENDENCE ON THE PAST: A SURVEY

. Explicit mild solutions and equivalent conditions for the existence and uniqueness of invariant measures are presented.

Superreplication of European multiasset derivatives with bounded stochastic volatility

We consider the problem of shortfall risk minimization when there is uncertainty about the exact stochastic dynamics of the underlying. Starting from the general discrete time model and the approach described in Runggaldier and Zaccaria (1999), we derive explicit analytic solutions for the particular case of a binomial model when there is uncertainty about the probability of an “up-movement”. The solution turns out to be a rather intuitive extension of that for the classical Cox-Ross-Rubinstein model.

Optimal Exercise of Swing Contracts in Energy Markets: An Integral Constrained Stochastic Optimal Control Problem ∗

In this paper, we analyse a pure jump incomplete market where the risky assets can jump upwards or downwards. In this market we show that, when an investor wants to maximise a HARA utility function of his/her terminal wealth, his/her optimal strategy consists of keeping constant proportions of wealth in the risky assets, thus extending the classical Merton result to this market. Finally, we compare our results with the classical ones in the diffusion case in terms of scalar dependence of portfolio proportions on the risk-aversion coefficient.

Optimal Portfolio in a Regime-switching Model

In this paper, we analyse a market where the risky assets follow exponential additive processes, which can be viewed as time-inhomogeneous generalizations of geometric Levy processes. In this market we show that, when an investor wants to maximize a CRRA utility function of his/her terminal wealth, his/her optimal strategy consists in keeping proportions of wealth in the risky assets which depend only on time but not on the current wealth level or on the prices of the risky assets. In the time-homogeneous case, the optimal strategy is to keep constant proportions of wealth, a result already found by Kallsen which extends the classical Merton’s result to this market. While the one-dimensional case has been extensively treated and the multidimensional case has been treated only in the time-homogeneous case Callegaro and Vargiolu (2009), Kallsen (2000), and Korn et al. (2003) to the authors’ knowledge this is the first time that such results are obtained for exponential additive processes in the multidimensional case. We use these results to show that the optimal solution in the presence of jumps has the form of the analogous one without jumps but with the asset yields vector reduced by suitable quantities: in the one-dimensional case, we extend a result by Benth et al. (2001). We conclude with four examples.

Shortfall risk minimising strategies in the binomial model: characterisation and convergence

In this paper we will examine some models for the financial markets where the evolution of the prices of the assets depends not only on the current value but also from the values assumed in the past. First we characterise the shape of volatility that a “good” financial model, also depending on past values, must have, and prove the completeness of the market in this general framework. Then, we analyse two models taken from the literature, one with “finite memory” which in general does not give nice numerical results, and one with “infinite memory”, where the model reduces to a Markov system with more state variables than the number of risky assets. Finally, we prove that a generalisation of the results of this last model with finite delay horison is not feasible without using anticipative stochastic calculus.