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.
Highlights
In the past two decades, many extreme events, such as the subprime mortgage crisis of the United States in 2008, the European sovereign debt crisis in 2013, the “stock disaster”of China’s stock market in 2015, and, more recently, the crash of the Brent and WTI crude oil futures markets because of COVID-2019 in early 2020, have happened in global financial markets
If Q0,T is defined by Equation (1), i.e., the tail risk options (TROs) is written on Type I ad hoc “asset“, thereby, the TRO can be refined as a high-frequency index option with payoff functions: max{max( X1,0,T, X2,0,T, · · ·, X M−1,0,T, X M,0,T ) − K, 0}
Tail risk hedging is an important issue for both investors and policy-makers
Summary
In the past two decades, many extreme events, such as the subprime mortgage crisis of the United States in 2008, the European sovereign debt crisis in 2013, the “stock disaster”. This paper proposes two novel types of tail risk options (TRO) for tail risk hedging and evaluation to fill the above gaps. This paper introduces two ad hoc non-trading underlying “assets” (the time series and cross-sectional daily maximum losses), corresponding to two types of tail risks. To capture the asymmetric connotations of tail risks and extreme losses, Fréchet distribution is used to model the ad hoc underlying “asset”. This treatment eventually yields a closed-form solution for both TRO European calls and puts. This paper provides a framework for crisis early warning from a risk management perspective using an option-implied tail index.
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