Abstract
We focus on two particular aspects of model risk: the inability of a chosen model to fit observed market prices at a given point in time (calibration error) and the model risk due to the recalibration of model parameters (in contradiction to the model assumptions). In this context, we use relative entropy as a pre-metric in order to quantify these two sources of model risk in a common framework, and consider the trade-offs between them when choosing a model and the frequency with which to recalibrate to the market. We illustrate this approach by applying it to the seminal Black/Scholes model and its extension to stochastic volatility, while using option data for Apple (AAPL) and Google (GOOG). We find that recalibrating a model more frequently simply shifts model risk from one type to another, without any substantial reduction of aggregate model risk. Furthermore, moving to a more complicated stochastic model is seen to be counterproductive if one requires a high degree of robustness, for example, as quantified by a 99% quantile of aggregate model risk.
Highlights
IntroductionEach type of model risk manifests itself as some form of ambiguity about the “true” probability measures that should be used for these purposes, and being able to quantify different types of model risk in a unified setting while using a pre-metric for the divergence between distributions (like relative entropy) allows for one to make an informed choice about the trade-offs between different sources of model risk
We focus on the model risk that is inherent in the calibration and recalibration of option pricing models, and to illustrate our approach we consider the models of Black and Scholes (1973) and Heston (1993), comparing the most classical option pricing model with its popular extension incorporating stochastic volatility
In (4) and (18), we are deliberately prioritising the minimisation of calibration error, as this is congruent to the focus of practitioners on calibration error
Summary
Each type of model risk manifests itself as some form of ambiguity about the “true” probability measures that should be used for these purposes, and being able to quantify different types of model risk in a unified setting while using a pre-metric for the divergence between distributions (like relative entropy) allows for one to make an informed choice about the trade-offs between different sources of model risk. We invert this problem by noting that higher relative entropy between model distributions indicates higher model risk, and propose a method to jointly evaluate model risk of two types, based on how this model risk manifests itself when option pricing models are calibrated and recalibrated to liquid market instruments.
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