Delta hedging in discrete time under stochastic interest rate

One of the most discussed assumptions of financial models, especially criticized in periods of financial turmoils, is that of market completeness, that is the perfect replication of any contingent claim by a suitable dynamic trading strategy. Theoretically, this is often achieved by ruling out any market imperfection, like illiquidity, credit risks, transactions costs, taxes, etc and by assuming the possibility of continuous time trading. Of course, real markets usually fail to satisfy most, if not all of such assumptions. One of the main challenges for financial economics is therefore to address such issues, by proposing models with less stringent hypotheses or by studying what happens when they do not hold. In this talk we want to focus on the impossibility of trading continuously in time. Even if all other assumptions of the model are satisfied, the inherent discreteness of trading times is a source of market incompleteness in the real world. The aim of the paper is to efficiently evaluate the impact of trading in discrete time on the final goal of the strategy. The object of our investigation is the ex-ante assessment of the performances of dynamic trading strategies. Probably, the most notable instance of such problem is measuring the hedging error of a strategy, based on a liquid assets, that tries to hedge a future liability. Problems of such kind arise when replicating either a claim using futures contracts, or a payoff of a derivative security with a delta hedging strategy based on the underlying asset, and in any case when a dynamic strategy is adopted. Ex-ante, a possible way to measure the performance of a strategy is by evaluating expected value and variance of its hedging error. This is usually done by approximations or by Monte Carlo simulations. The approach we propose, based on Laplace transforms, allows to efficiently perform such computations for a very general class of models. We consider a market model driven by continuous time affine processes, in which by definition the conditional characteristic function is an exponential of an affine function of the state variables (see Duffie et al. (2000) for a formal definition and properties of affine models). In this framework, Angelini and Herzel (2009,