The realized local volatility surface

Yuming Ma, Shintaro Sengoku and Kazuhide Nakata

Abstract

Estimating the distribution of realized volatility is vital for formulating options trading strategies as well as for other portfolio risk management purposes. The main contribution of this paper is the proposition of a Bayesian nonparametric estimation method to reconstruct a counterfactual generalized Wiener measure from historical price data. To do this, a stick-breaking Gaussian mixture model is applied to compensate for the return distribution over both time and price dimensions. For the fitting of model parameters, the Hamiltonian Monte Carlo method is applied to obtain the stick-breaking Gaussian mixture model parameters that maximize a posteriori probability. Once the posterior distribution is obtained, we draw samples from it to compute the standard deviation and construct the realized local volatility surface to within a 95% credible interval. Then, a numerical experiment result using tickerlevel high-frequency data for Tesla, Inc. is used to construct a hypothetical realized local volatility surface. Finally, we address the possible use of the realized volatility surface to compensates for implied volatility in hedging by managing risks caused by abrupt movements in the underlying price.