The leverage effect puzzle revisited: Identification in discrete time

The term “leverage effect,” as coined by Black (1976), refers to the tendency of an asset’s volatility to be negatively correlated with the asset’s return. Aït-Sahalia et al. (2013) refer to the “leverage effect puzzle” as the fact that, in spite of a broad agreement that the effect should be present, it is hard to identify empirically. For this purpose, we propose an extension with leverage effect of the discrete time stochastic volatility model of Darolles et al. (2006). This extension is shown to be the natural discrete time analog of the Heston (1993) option pricing model. It shares with Heston (1993) the advantage of structure preserving change of measure: with an exponentially affine stochastic discount factor, the historical and the risk neutral models belong to the same family of joint probability distributions for return and volatility processes. The discrete time approach allows us to make the role of various parameters more transparent: leverage versus volatility feedback effect, connection with daily realized volatility, impact of leverage on the volatility smile, etc. Even more importantly it sheds some new light on the identification of leverage effect and of the various risk premium parameters through link functions in closed form. The price of volatility risk is identified from underlying asset return data, even without option price data, if and only if leverage effect is present. However, the link functions are almost flat if the leverage effect is close to zero, making estimation of the volatility risk price difficult and paving the way for identification robust inference.