Incorporating Views on Marginal Distributions in the Calibration of Risk Models

TL;DR

This paper develops a unified entropy-based approach for calibrating risk models that incorporates expert views on tail risks and market data, improving portfolio risk assessment and option pricing accuracy.

Contribution

It introduces a novel methodology that integrates constraints on moments and marginal distributions into risk model calibration, extending traditional entropy methods.

Findings

Incorporating tail risk views results in fatter, more realistic loss tails.

The methodology improves non-traded option pricing using market data.

Enhanced risk models better reflect expert opinions and market information.

Abstract

Entropy based ideas find wide-ranging applications in finance for calibrating models of portfolio risk as well as options pricing. The abstracted problem, extensively studied in the literature, corresponds to finding a probability measure that minimizes relative entropy with respect to a specified measure while satisfying constraints on moments of associated random variables. These moments may correspond to views held by experts in the portfolio risk setting and to market prices of liquid options for options pricing models. However, it is reasonable that in the former settings, the experts may have views on tails of risks of some securities. Similarly, in options pricing, significant literature focuses on arriving at the implied risk neutral density of benchmark instruments through observed market prices. With the intent of calibrating models to these more general stipulations, we develop a unified entropy based methodology to allow constraints on both moments as well as marginal distributions of functions of underlying securities. This is applied to Markowitz portfolio framework, where a view that a particular portfolio incurs heavy tailed losses is shown to lead to fatter and more reasonable tails for losses of component securities. We also use this methodology to price non-traded options using market information such as observed option prices and implied risk neutral densities of benchmark instruments.