(with Torben G. Andersen, Viktor Todorov, and Rasmus Varneskov)
We provide a new estimator for the parameter vector and the latent stat variables of an option pricing models governed by a pure jump process. The estimator takes the form of a penalized least square where the penalization term takes into account for both jump intensity and jump activity index estimated from high-frequency data.
We develop parametric inference procedures for large panels of noisy option data in a setting, where the underlying process is of pure-jump type, i.e., evolves only through a sequence of jumps. The panel consists of options written on the underlying asset with a (different) set of strikes and ma- turities available across the observation times. We consider an asymptotic setting in which the cross-sectional dimension of the panel increases to infinity, while the time span remains fixed. The information set is augmented with high-frequency data on the underlying asset. Given a parametric specification for the risk-neutral asset return dynamics, the option prices are nonlinear functions of a time-invariant parameter vector and a time-varying latent state vector (or factors). Further- more, no-arbitrage restrictions impose a direct link between some of the quantities that may be identified from the return and option data. These include the so-called jump activity index as well as the time-varying jump intensity. We propose penalized least squares estimation in which we minimize the L2 distance between observed and model-implied options. In addition, we penalize for the deviation of the model-implied quantities from their model-free counterparts, obtained from the high-frequency returns. We derive the joint asymptotic distribution of the parameters, factor realizations and high-frequency measures, which is mixed Gaussian. The different components of the parameter and state vector exhibit different rates of convergence, depending on the relative (asymptotic) informativeness of the high-frequency return data and the option panel.
(with Torben G. Andersen and Viktor Todorov)
We show evidence that a tail factor extracted from the option market predicts future market returns in both US and many of the european markets. Most importantly this factor is embedded in the dynamics of option prices but not in the time series of stock prices, showing the importance of the derivatives market in embedding beliefs about possible future market downturns.
The paper explores the global pricing of market tail risk as manifest in equity-index options. We document the presence of a left tail factor that displays large persistent shifts, largely unrelated to the corresponding dynamics of return volatility. This left tail factor is a potent predictor of future excess equity-index returns, while the implied volatility only forecasts future equity return variation, not the expected returns. We conclude that the option surface embeds separate equity risk and risk premium factors which are successfully disentangled by our simple two-factor affine model for all the equity indices explored. The systematic deviations across countries speak to the differential risk and its pricing during the great recession and the European sovereign debt crises. Most strikingly, the relative tail risk pricing displays pronounced heterogeneity for the Southern European countries. During the sovereign debt crisis, their stock markets react almost identically to Euro-wide systematic shock, but these events are priced very differently in the respective option markets, indicating differences in crash beliefs across countries which are hard to detect with stock market data alone.