Nikolay Nikolaev, Goldsmiths College, University of London

Volatility Arbitrage and Option Pricing

SV

Stochastic Volatility Inference

Accurate pricing of financial instruments and sucessfull trading can be achieved using estimates of the volatility, that is the deviation of particular stock returns in the market. Among the existing approaches to describing the volatility, the Stochastic Volatility (SV) models are considered more successfull than standard GARCH models as the SV capture better second-order properties of the returns as well as leverage effects. The main difference is that SV models allow random perturbations, that do not depend on past information, to influence the variance process. Finding the stochastic volatility from series of stock prices involves learning of nonlinear models. Current research actively investigates Particle Filters (PF) for nonlinear volatility inference. The particle filter is a Bayesian mechanism that uses a population of Monte Carlo samples, called particles, whose averaging allows to compute the characteristics (mean and variance) of the posterior distribution of the unknown parameters. A distinguishing advantage of PF is that they can work reliably in relaxed circumstances, like non-Gaussianity and non-stationarity.

PF are typically applied to find the volatility considered as an unobsrved component in the SV model. Our current research investigates Gaussian particle filters (GPF) which use mixtures of Gaussians to represent the distributions of interest. Experiments with GPF are conducted on:

- stochastic volatility inference using the Hull-White model;

- stochastic volatility inference using the Heston model.

Modelling the volatilities in business cycles is a problem whose solutions have important implications for various financial activities, like asset pricing, risk management, porfolio selection, etc..

OP

Option Pricing

The option price theory assumes that stock prices follow a geometric Brownian motion. The available mathematical models treat the stock price as depending on the current price, the risk-free interest rate, the volatility, the strike price and the time to maturity. Among these factors only the volatility (variance of the stock return) can not be observed directly. One popular strategy is to estimate the volatility implied by the prices using some available model. The format of the model determines the corresponding learning algorithm. Adopting the Black-Scholes model for pricing of European put and call options, for example, one may attempt to compute the volatility using Kalman filters through a state-space model formulation. The problem is that the returns and the volatility become uncorrelated when the linear Kalman equations are considered. This requires the elaboration of derivative-free filters that can estimate the implied volatility with the this model.

Our research developed fully derivative-free Unscented Grid Filters (UGF) that sample the volatility deterministically to achieve more efficient approximation to its posterior probability distribution. Currently applications to several financial engineering tasks are investigated:

- implied volatility modelling according to the Black-Scholes formula;

- modelling the dynamics of statistical arbitrage for risk control.

The UGF filter is also extended to estimate stochastic volatility models with leverage effects and jumps. It has been found empirically that such filters, considered within the maximum likelihood framework, have the potential for accurate prediction of various price movements.