The Importance of the Loss Function in Option Valuation

Abstract

Which loss function should be used when estimating and evaluating option valuation models? Many different functions have been suggested, but no standard has emerged. We do not promote a particular function, but instead emphasize that consistency in the choice of loss functions is crucial. First, for any given model, the loss function used in parameter estimation and model evaluation should be identical, otherwise suboptimal parameter estimates may be obtained. Second, when comparing models, the estimation loss function should be identical across models, otherwise unfair comparisons will be made. We illustrate the importance of these issues in an application of the so-called Practitioner Black-Scholes (PBS) model to S&P500 index options. We find reductions of over 50 percent in the root mean squared error of the PBS model outperforms a benchmark structural model when the estimation and evaluation loss functions are aligned. We also find that the PBS model outperforms a benchmark structural model when the estimation loss functions are identical across models, but otherwise not. The new PBS model with aligned loss functions thus represents a much tougher benchmark against which future structural models can be compared.