The relationship between conditional value at risk and option prices with a closed-form solution

Abstract

Options and CVaR (conditional value at risk) are significant areas of research in their own right; moreover, both are important to risk management and understanding of risk. Despite the importance and the overlap of interests in CVaR and options, the literature relating the two is virtually non-existent. In this paper we derive a model-free, simple and closed-form analytic equation that determines the CVaR associated with a put option. This relation is model free and is applicable in complete and incomplete markets. We show that we can account for implied volatility effects using the CVaR risk of options. We show how the relation between options and CVaR has important risk management implications, particularly in terms of integrated risk management and preventing arbitrage opportunities. We conduct numerical experiments to demonstrate obtaining CVaR from empirical options data.