Dark Matter in (Volatility and) Equity Option Risk Premiums

Abstract

In their paper, “Dark Matter in (Volatility and) Equity Option Risk Premiums,” Bakshi, Crosby, and Gao ask a provocative question: is there dark matter embedded in volatility and equity options? They consider a theoretical approach that allows them to introduce the constructs of risk premiums on jumps crossing the strike and on local time. The treatment of jumps crossing the strike and local time is integral to their theory because their absence would be counterfactual from an empirical standpoint. They label such abstract uncertainties—driven by unspanned risk components—“dark matter” as these uncertainties can be hard to identify, but their presence is implied in options data, and the workings of dark matter can be economically influential. Their empirical exercises are based on weekly equity index options (the “weeklys”) in addition to the farther-dated (index and futures) options up to 88 days maturity.

Modeling Conditional Factor Risk Premia Implied by Index Option Returns

“…S&P 500 index (SPX) puts are much more sensitive to market variance risk than calls, and calls are more sensitive to market return risk. These results suggest that the index option market seems to be somewhat segmented, consistent with Constantinides, Jackwerth, and Perrakis (2009) and Bakshi, Crosby, and Gao (2022), who show that index call options are priced differently from index put options. To illustrate the value of the estimated conditional model exposures, we compare the hedging performance of our model with standard benchmarks.…”

Pricing levered warrants under the CEV diffusion model

“…We also follow Schroder (1989) and Heston et al (2007) to compute the prices under the CEV model. Although not trivial, our framework is flexible enough and can be extended or modified in order to analyze the existence of dark matter in corporate warrants-see, for instance, Bakshi et al (2022). Second, we propose numerical solutions for pricing warrants with longer or shorter maturity than debt under the CEV diffusion.…”