Quadratic Hedging of Basis Risk

Hardy Hulley,Thomas Mcwalter

doi:10.3390/jrfm8010083

Abstract

This paper examines a simple basis risk model based on correlated geometric Brownian motions. We apply quadratic criteria to minimize basis risk and hedge in an optimal manner. Initially, we derive the Föllmer–Schweizer decomposition for a European claim. This allows pricing and hedging under the minimal martingale measure, corresponding to the local risk-minimizing strategy. Furthermore, since the mean-variance tradeoff process is deterministic in our setup, the minimal martingale- and variance-optimal martingale measures coincide. Consequently, the mean-variance optimal strategy is easily constructed. Simple pricing and hedging formulae for put and call options are derived in terms of the Black–Scholes formula. Due to market incompleteness, these formulae depend on the drift parameters of the processes. By making a further equilibrium assumption, we derive an approximate hedging formula, which does not require knowledge of these parameters. The hedging strategies are tested using Monte Carlo experiments, and are compared with results achieved using a utility maximization approach.

Highlights

When a contingent claim is written on an asset or process that is not traded, it is natural to enquire about the effectiveness of hedging with a correlated security
In this paper we used quadratic criteria to derive simple hedging rules for minimizing basis risk. These rules are considerably simpler than the hedging rules based on utility maximization and perform slightly better in terms of minimizing risk. Their simplicity is a consequence of the fact that they are based on the Black–Scholes formula
The utility maximization approach requires the use of series expansions in order to solve the relevant partial differential equation (PDE), which were originally derived using a “distortion”

Summary

Introduction

When a contingent claim is written on an asset or process that is not traded, it is natural to enquire about the effectiveness of hedging with a correlated security. In this situation the market is incomplete, and the risk that arises as a result of imperfect hedging is known as basis risk. Since not all the risk can be hedged, we are dealing with a typical incomplete market situation, in which the appetite for risk must be specified (usually in terms of a utility function). Our approach is similar to that of Schweizer [8] (see Duffie and Richardson [9]), where the application was hedging futures with a correlated asset. For comprehensive reviews on the theory of quadratic hedging, the reader is directed to the works of Pham [10], Schweizer [11] and McWalter [12]

Objectives

Results

Conclusion