Hedging and Evaluating Tail Risks via Two Novel Options Based on Type II Extreme Value Distribution

Abstract

Tail risk is an important financial issue today, but directly hedging tail risks with an ad hoc option is still an unresolved problem since it is not easy to specify a suitable and asymmetric pricing kernel. By defining two ad hoc underlying “assets”, this paper designs two novel tail risk options (TROs) for hedging and evaluating short-term tail risks. Under the Fréchet distribution assumption for maximum losses, the closed-form TRO pricing formulas are obtained. Simulation examples demonstrate the accuracy of the pricing formulas. Furthermore, they show that, no matter whether at scale level (symmetric “normal” risk, with greater volatility) or shape level (asymmetric tail risk, with a smaller value in tail index), the greater the risk, the more expensive the TRO calls, and the cheaper the TRO puts. Using calibration, one can obtain the TRO-implied volatility and the TRO-implied tail index. The former is analogous to the Black-Scholes implied volatility, which can measure the overall symmetric market volatility. The latter measures the asymmetry in underlying losses, mirrors market sentiment, and provides financial crisis warnings. Regarding the newly proposed TRO and its implied tail index, economic implications can be offered to investors, portfolio managers, and policy-makers.