Model Calibration, Risk Measurement, and the Hedging of Derivatives

Abstract

A derivatives contract is a redundant security when there exists a self-financing strategy involving other traded securities that can replicate the payoff of this contract. The initial const of the replicating strategy equals the price of the contract if no arbitrage is allowed in the market. Due to the complexity of derivatives contracts or portfolios, mathematical pricing models are often used to help forming hedging strategies. Such models usually involves three stops: (1) calibrating model parameters to a set of calibrating instruments that are liquid in the market and contain the necessary market information to make the derivatives redundant securities; (2) calculating sensitivities of the derivatives portfolio with respect to model parameters; and (3) translating the parameter sensitivities into hedge ratios with respect to a given set of hedging instruments. This paper presents a methodology for analyzing these steps. Our analysis applies to models in general where exact calibration does not exist and the objective is to minimize calibration errors using numerical optimization techniques. We derive analytic solutions for model parameter sensitivities with respect to prices of calibrating instruments. This eliminates the need for time-consuming model re-calibrations required to calculate these sensitivities numerically. Since hedges are not unique we present two optimal hedges. One of them is additive in the sense that the sum of two optimal hedges of two portfolios is also an optimal hedge for the sum of the two portfolios. The other is self-recoverable which means the optimal hedge of a hedging instrument is the instrument itself. We also study the use of calibrating instruments as hedges by utilizing information such as model parameter sensitivities with respect to calibrating instruments obtained in the model calibration process. The methodology we developed for mapping the parameter sensitivities to their delta or vega equivalents helps not only the hedging but also the risk management of derivatives portfolios on a consistent basis.

Keywords: Derivatives, Model Calibration, Replication, Hedging, Risk Management

JEL Classification: G13