Calibration Risk for Exotic Options

Abstract

For academic modelers, the major objective is typically to derive a valuation equation for a derivative instrument, while computing Greek letter risks is only a secondary consideration. For users of models in the real world, however, the market provides the correct value and the parameters of the theoretical pricing model are calibrated to match the market, in order to generate risk assessment and risk management strategies consistent with the way the market is currently pricing the security. But calibration can be done in a number of ways. Do we want parameters that minimize the overall discrepancy between model values and market prices in dollars, or in percent? Alternatively, should we focus on the crucial but unobservable volatility parameter and minimize aggregate differences between the implied volatility from the model and Black-Scholes implied volatility, either in volatility points or in percentages? This might make sense because the latter are often used in setting option price quotes in the market. In this paper, the authors examine what impact this choice has on the results of model calibration, taking as examples the Heston stochastic volatility model and the Bates model, which allows price jumps as well. They find considerable difference among parameters estimated by using absolute price differences versus one of the other techniques. The paper demonstrates that the choice of calibration method can have a substantial impact on pricing of exotic option contracts and on model risk. One result is that calibration based on absolute price discrepancies produces quite different parameter estimates than the other three methods.