Estimating Volatility from ATM Options with Lognormal Stochastic Variance and Long Memory

In this article we propose a nonlinear state space representation to model At-The-Money (ATM) implied volatilities and to estimate the unobserved Stochastic Volatility (SVOL) for the underlying asset. We derive a polynomial measurement model relating fractionally cointegrated implied and spot volatilities. We then use our state space representation to obtain Maximum Likelihood (ML) estimates of the short-memory model parameters, and for filtering the fractional spot volatility. We are also able to estimate the average volatility risk premia. We applied our methodology to implied volatilities on eurodollar options, from which we filter the unobserved spot local variance. These data arise from Over The Counter (OTC) transactions that account for high liquidity. For these data, we estimated a positive average volatility risk premia, which is consistent with the Intertemporal Capital Asset Pricing Model (ICAPM) setup of Merton (1973). We also had evidence of highly nonlinear relation between eurodollar spot and implied volatilities. From a methodological and computational point of view, the likelihood function, and all the iterative procedures associated with it, converged uniformly in the parameter space at very little computational expense. We illustrated the effectiveness of our approach by evaluating the approximated Information matrix, the Hotelling’s T2 test along with other diagnostic procedures.