Flavio Angelini

Consistent Initial Curves for Interest Rate Models

It was a significant step forward in modeling the term structure of interest rates to require the model to be arbitrage-free, in the sense that applying the model’s dynamics to the current market yield curve did not produce any arbitrage opportunities, and therefore, internal inconsistency. The Heath-Jarrow-Morton (HJM) class of interest rate models leads to a large family of arbitrage-free term structure processes. Fitting HJM models empirically requires estimating a continuous forward rate curve from the discrete set of bond prices observed in the market. But, as the authors point out, unless this is done with care, the function that is fitted to market rates may not be consistent with the HJM dynamics for the forward curve. That is, current forward rates would be fitted to a functional form that cannot evolve to another forward curve of the same type in the next period under the assumed HJM model specification. This problem is generally overlooked in practice, since the forward rate curve is refitted every period anyway, but it may well show up in the form of parameter instability in the model. In this article, Angelini and Herzel examine this issue both theoretically and empirically, and show that requiring the forward curve to be fitted to a functional form that is consistent with the interest rate process can make a substantial difference in model performance and in parameter stability.

Evaluating discrete dynamic strategies in affine models

We consider the problem of measuring the performance of a dynamic strategy, re-balanced at a discrete set of dates, with the objective of hedging a claim in an incomplete market driven by a general multi-dimensional affine process. The main purpose of the paper is to propose a method to efficiently compute the expected value and variance of the hedging error of the strategy. Representing the payoff of the claim as an inverse Laplace transform, we are able to obtain semi-explicit formulas for strategies satisfying a certain property. The result is quite general and can be applied to a very rich class of models and strategies, including Delta hedging. We provide illustrations for the case of the Heston stochastic volatility model.

Portfolio management with benchmark related incentives under mean reverting processes

We study the problem of a fund manager whose compensation depends on the relative performance with respect to a benchmark index. In particular, the fund manager’s risk-taking incentives are induced by an increasing and convex relationship of fund flows to relative performance. We consider a dynamically complete market with N risky assets and the money market account, where the dynamics of the risky assets exhibit mean reversions, either in the drift or in the volatility. The manager optimizes the expected utility of the final wealth, with an objective function that is non-concave. The optimal solution is found by using the martingale approach and a concavification method. The optimal wealth and the optimal strategy are determined by solving a system of Riccati equations. We provide a semi-closed solution based on the Fourier transform.

On the Effect of Skewness and Kurtosis Misspecification on the Hedging Error

Using a result in Angelini and Herzel (2009a), we measure, in terms of variance, the cost of hedging a contingent claim when the hedging portfolio is re-balanced at a discrete set of dates. We analyse the dependence of the variance of the hedging error on the skewness and kurtosis as modeled by a Normal Inverse Gaussian model. We consider two types of strategies, the standard Black-Scholes Delta strategy and the locally variance-optimal strategy, and we perform some robustness tests. In particular, we investigate the effect of different types of model misspecification on the performance of the hedging, like that of hedging without taking skewness into account. Computations are performed using a Fast Fourier Transform approach.

Delta hedging in discrete time under stochastic interest rate

We propose a methodology based on the Laplace transform to compute the variance of the hedging error due to time discretization for financial derivatives when the interest rate is stochastic. Our approach can be applied to any affine model for asset prices and to a very general class of hedging strategies, including Delta hedging. We apply it in a two-dimensional market model, obtained by combining the models of Black-Scholes and Vasicek, where we compare a strategy that correctly takes into account the variability of interest rates to one that erroneously assumes that they are deterministic. We show that the differences between the two strategies can be very significant. The factors with stronger influence are the ratio between the standard deviations of the equity and that of the interest rate, and their correlation. The methodology is also applied to study the Delta hedging strategy for an interest rate option in the Cox-Ingersoll and Ross model, measuring the variance of the hedging error as a function of the frequency of the rebalancing dates. We compare the results obtained to those coming from a classical Monte Carlo simulation.