Quantifying the Model Risk Inherent in the Calibration and Recalibration of Option Pricing Models

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.

Non-asymptotic convergence rates for the plug-in estimation of risk measures

“…Along the same lines some authors have investigated risk measures (and other stochastic maximization problems) under model uncertainty to account for the effect of possible misspecification of the estimated model, see e.g. [4,19,17,20,26] where it is often assumed that the true model belongs to a Wasserstein ball. At this point, we should also mention Pichler [39] who studies the continuity of risk measures (in Wasserstein distance).…”

Limits of random walks with distributionally robust transition probabilities

“…Finally, let us highlight the connection to distributionally robust optimization (DRO) using the Wasserstein distance. In DRO, the basic task consists of computing inf λ S(s)f λ , where (f λ ) λ is a parametrized family of function; we refer to [3,6,11,12] for recent results and applications. Here duality arguments often help to compute the (infinite dimensional) optimization problem appearing in the definition of S(s).…”