The Market Price of Volatility Risk and the Dynamics of Market and Actuarial Implied Volatilities

Abstract

Option prices in the market embed a probability distribution over the option’s payoff, which can be fairly easily extracted. But in this risk-neutral distribution, the market’s true expectations about future returns are distorted by risk preferences, in particular, aversion to volatility risk. To separate the returns forecasts from risk aversion, one needs another way to estimate the market’s probability estimates, which the author calls “actuarial probabilities.” This difficult problem becomes even harder in the swaptions market, because the risk exposure for a swaption depends on two different time horizons. For example, a 2 × 5 swaption has two years (“expiry”) of optionality culminating with the option to enter into a swap with a maturity (“tail”) of five years. The volatility surface is three-dimensional, with two time variables and the strike interest rate (the “volatility cube”), and it is difficult to estimate statistically using available data on long-lived contracts. The author develops a procedure based on extracting a small number of common factors of the volatility process using principal components. Since all swaptions are exposed to these factors, with different loadings, modeling the evolution of the principal components, rather than the specific swaption volatilities, allows a more efficient use of the data. Applying this approach to about 10 years of swaption data, the article reports several interesting results, including the fact that three principal components capture a large proportion of the variation in actuarial volatilities across the volatility cube, while a couple extra PCs are needed to achieve the same level of explanatory power for market implied volatilities. One clear result is that the market price of volatility risk is time-varying, and it moves rather quickly, with rates of mean-reversion that seem much faster than for actuarial volatility, particularly for short expiry–long tail swaptions. Another interesting result is that the volatility risk premium reverses sign, being largely positive (market IV > actuarial IV) for contracts with short expiry, but becoming negative for long expiry contracts.