Risk about future payoffs is at the center of asset pricing, asset allocation, and option pricing. Prices at the stock and bond market, and in particular option prices, can react significantly to a change in uncertainty. Over the last years, the VIX volatility index, which is the square-root of a model-free variance calculated from the cross-section of SPX option prices, became a well-known measure for uncertainty. In the financial crises, the VIX, which has usually been around 20%, has exceeded the level of 80% several times. During this time, but also during former periods of market stress like the stock market crash of 1987, or the collapse of LTCM, volatility derivatives, which allow to trade variance risk, thus naturally attracted attention.
The realized or expected variance of stock and index prices constitutes a further asset class investors can trade in. There is ample empirical evidence that the premium paid on a long position in variance, which can be implemented by buying a variance swap, is negative. The economic question is why there is a negative variance risk premium, and which of the risk factors that drive variance risk the premium is paid on. The answer to this question is of major importance when it comes to the pricing of volatility sensitive derivative contracts. It is also important for portfolio planning, where investors want to know which risk factors they are exposed to, and which risk premia they earn (or have to pay) on these risk factors.
In this paper, we derive the dynamics of the stock price, the dynamics of its variance, and the risk premia in an equilibrium Lucas-tree economy with recursive preferences. This equilibrium setup allows for the consistent pricing of derivatives on the stock and on its variance. Most importantly, we do not have to make any assumptions on the risk premia in particular for variance risk, but derive them endogenously. We can thus identify the main risk factors variance is exposed to, and the main drivers of the significantly negative variance risk premium. Furthermore, we can determine the equilibrium prices of derivatives on variance. We show that our model is indeed able to explain the major stylized facts observed at the market.
We rely on a long-run risk model with a long-run growth factor and stochastic volatility. Since our main focus is on variance risk, we pay special attention to modeling uncertainty in the market. We assume that there are two volatility factors. The first factor describes the long-run level of variance. It has a low volatility, a low mean-reversion speed, and is highly persistent. Jumps in this factor are rare events, but influence the market for several years. The second factor captures the current volatility in the market, which moves around the persistent long-run level. This process is much more volatile and has a higher mean-reversion speed than the long-run level. Jumps in this process happen rather often, while their impact on the future dynamics vanishes quickly over time. This setup allows us to distinguish both long-run and short-run components of volatility. Furthermore, we assume that the uncertainty factors do not only govern the volatility of consumption, but also the volatilities and jump intensities of the factors themselves and thus the level of uncertainty about overall consumption risk.
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What is the Equilibrium Price of Variance Risk? A Long-Run Risk Model with Two Volatility Factors
