# Option Pricing, Historical Volatility and Tail Risks

**Authors:** Samuel E. Vazquez  
**arXiv:** [1402.1255](https://arxiv.org/abs/1402.1255)  
**Date:** 2014-02-06  
**Summary page:** https://www.summarizepaper.com/en/paper/1402.1255v1/

## Key points

- The paper "Option Pricing, Historical Volatility and Tail Risks" by Samuel E. Vazquez revisits the problem of pricing options using historical volatility estimators in the context of a generalized GARCH model with multiple time scales and asymmetry.
- The observed volatility risk premium is due to tail risk aversion and is parametrized in terms of three coefficients: convexity, skew, and kurtosis risk premium.
- Option prices under the real-world measure are not martingales but their drift is governed by such tail risk premia.
- A fair-pricing equation for options is derived and solutions can be written in terms of a stochastic volatility model in continuous time and under a martingale probability measure.
- The convexity risk premium not only shifts the overall implied volatility level but also changes its term structure.
- The skew risk premium makes the skewness of the volatility smile steeper than a pure historical estimate.
- Analytical formulas for certain implied moments are derived using the Bergomi-Guyon expansion, allowing for very fast calibrations of models.
- Examples are provided of a particular model that can reproduce the observed SPX volatility surface using very few parameters.

## Summary

In the paper "Option Pricing, Historical Volatility and Tail Risks," Samuel E. Vazquez revisits the problem of pricing options using historical volatility estimators in the context of a generalized GARCH model with multiple time scales and asymmetry. The paper argues that the observed volatility risk premium is due to tail risk aversion and parametrizes such risk aversion in terms of three coefficients: convexity, skew, and kurtosis risk premium. The paper proposes that option prices under the real-world measure are not martingales but that their drift is governed by such tail risk premia. The paper derives a fair-pricing equation for options and shows that solutions can be written in terms of a stochastic volatility model in continuous time and under a martingale probability measure. This gives a precise connection between pricing and real-world probability measures, which cannot be obtained using Girsanov Theorem. The paper finds that the convexity risk premium not only shifts the overall implied volatility level but also changes its term structure. Moreover, the skew risk premium makes the skewness of the volatility smile steeper than a pure historical estimate. The paper derives analytical formulas for certain implied moments using the Bergomi-Guyon expansion, allowing for very fast calibrations of models. The paper provides examples of a particular model that can reproduce the observed SPX volatility surface using very few parameters. Overall, this paper provides insights into how tail risks affect option pricing and offers practical implications for modeling financial markets with multiple time scales and asymmetry.