Tails, Fears, and Risk Premia

Abstract

We show that the compensation for rare events accounts for a large fraction of the average equity and variance risk premia. Exploiting the special structure of the jump tails and the pricing thereof, we identify and estimate a new Investor Fears index. The index reveals large time-varying compensation for fears of disasters. Our empirical investigations involve new extreme value theory approximations and high-frequency intraday data for estimating the expected jump tails under the statistical probability measure, and short maturity out-of-the-money options and new model-free implied variation measures for estimating the corresponding risk-neutral expectations.So what are policymakers to do? First and foremost, reduce uncertainty. Do so by removing tail risks, and the perception of tail risks.-Olivier Blanchard, chief economist, IMF, The Economist, January 31, 2009.LACK OF INVESTOR CONFIDENCE is frequently singled out as one of the main culprits behind the massive losses in market values in the advent of the Fall 2008 financial crises. At the time, the portrayal in the financial news media of different doomsday scenarios in which the stock market would have declined even further were also quite commonplace. Motivated by these observations, we provide a new theoretical framework for better understanding the way in * Bollerslev is with Duke University and Todorov is with Kellogg School of Management, Northwestern University. We thank Yacine Aït-Sahalia, Torben

General equilibrium pricing of currency and currency options

“…The above mechanism advocates that it is the concern about the (unshared) disaster risks, or "crash-o-phobia", that matters the most to the observed pricing. The same economics also explain the regularities of carry trade returns (Section 5.2.1), the UIP anomaly (Section 5.2.2), the time variations of volatility smirk in equity index option pricing (Appendix B.2), and is consistent with the findings by Bollerslev and Todorov (2011) that investors' fear of jump tail risks play a crucial role for explaining risk premia embedded in aggregate equity.…”

Risk, Uncertainty, and Expected Returns

“…The central empirical variable of this paper, as a proxy for economic uncertainty, is the market variance risk premium (VRP)-which is not directly observable but can be esti-4 Alternatively, our empirical result on VRP may be interpreted as compensating for the rare disaster risk (Gabaix, 2011), jump risk (Todorov, 2010;Drechsler and Yaron, 2011), or tail risk (Bollerslev and Todorov, 2011;Kelly, 2011). Alternatively, VRP can be generated from a habit-formation model with sophisticated consumption dynamics (Bekaert and Engstrom, 2010).…”

Inferring Volatility Dynamics and Risk Premia from the S&P 500 and VIX Markets

“…As a consequence, for all datasets containing options, the estimated volatility-of-volatility parameter σv is considerably smaller for the SVJ than for the SV2 and SVJ2. and sizes were estimated following the method described in Bollerslev and Todorov (2011) Jump size estimates under Q are statistically significant and strongly negative (around -10%), which is due to investors' risk aversion to jumps, and yields a non-zero jump risk premium. Similarly, the volatility of the returns' jump sizes is larger under Q than under P. This finding for jumps in returns seems to contrast with the result for the jumps in volatility.…”