Partial Uncertainty and Applications to Risk-Averse Valuation

TL;DR

This paper introduces R-conditioning, an intermediary between expectation types, to evaluate risk-averse values of derivatives, providing a new numerical algorithm and applications in finance and high-dimensional estimation.

Contribution

It defines R-conditioning as a novel concept bridging expectation types, proves its well-posedness, and develops a convergent numerical algorithm for practical computation.

Findings

R-conditioning approximates sublinear expectations arbitrarily closely

The algorithm for R-conditioning is strongly convergent

Application to Black-Scholes-Merton shows differences in risk-averse vs. risk-neutral valuation

Abstract

This paper introduces an intermediary between conditional expectation and conditional sublinear expectation, called R-conditioning. The R-conditioning of a random-vector in $L^2$ is defined as the best $L^2$-estimate, given a $\sigma$-subalgebra and a degree of model uncertainty. When the random vector represents the payoff of derivative security in a complete financial market, its R-conditioning with respect to the risk-neutral measure is interpreted as its risk-averse value. The optimization problem defining the optimization R-conditioning is shown to be well-posed. We show that the R-conditioning operators can be used to approximate a large class of sublinear expectations to arbitrary precision. We then introduce a novel numerical algorithm for computing the R-conditioning. This algorithm is shown to be strongly convergent. Implementations are used to compare the risk-averse value of a Vanilla option to its traditional risk-neutral value, within the Black-Scholes-Merton framework. Concrete connections to robust finance, sensitivity analysis, and high-dimensional estimation are all treated in this paper.