Hedging of options under discrete observation on assets with stochastic volatility.

The paper considers the hedging of contingent claims on assets with stoachstic volatilities when the asset price is only observable at discrete time instants. Explicit fomulae are given for risk-minimizing hedging strategies

Interest rates

In this chapter we consider the term structure of interest rates and the interest rate derivatives. The interest rates are closely related to the bond market; we therefore introduce the interest rates in connection with the simplest assets on the bond market, namely the so-called T-bonds which are contracts that guarantee a unitary amount at a given maturity T and their prices express the expectations of the market on the future value of money

A Robustness Result for Stochastic Control

The solution of a stochastic control problem depends on the underlying model, i.e., on the probability measure induced by the model. The real world model may not be known precisely, and so one solves the problem for a hypothetical model that induces a measure generally different but close to the real one. We investigate two ways to derive a bound on the suboptimality of the hypothetical optimal control when it is used in the real problem. Both bounds are in terms of the Radon-Nikodym derivative of the real world measure with respect to the hypothetical one

American options

The American options generalize the European options in the sense that they can be exercised at any moment prior to maturity. They are part of the more general category of American-type derivatives that we shall define in Subsection 3.1 as a sequence X = (Xn) of random variables that are adapted to a given filtration (Fn), typically generated by the prices of the underlyings. The value of Xn is the premium/payoff paid to the holder of the derivative if he/she exercises the option at time tn

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

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

Pathwise optimality for benchmark tracking

We consider the problem of investing in a portfolio in order to track or "beat" a given benchmark. We study this problem from the point of view of almost sure/pathwise optimality. We first obtain a control that is optimal in the mean and this control is then shown to be also pathwise optimal. The standard Merton model leads to lognormality of the value process so that it does not possess the required ergodic properties. We obtain ergodicity by transforming the process so that it remains bounded thereby using a method that can be related to a random time change. We furthermore describe a general approach to solve the Hamilton-Jacobi-Bellman equation corresponding to the given problem setup

Portfolio Optimization in Discontinuous Markets under Incomplete Information

We consider the problem of maximization of expected utility from terminal wealth for log and power utility functions in a market model that leads to purely discontinuous processes. We study this problem as a stochastic control problem both under complete as well as incomplete information. Our contribution consists in showing that the optimal strategy can be obtained by solving a system of equations that in some cases is linear and that a certainty equivalence property holds not only for log-utility but also for a power utility function. For the case of a power utility under incomplete information we also present an independent direct approach based on a Zakai-type equation