A study of the Hobson and Rogers volatility model

We firstly examine the model of Hobson and Rogers for the volatility of a financial asset such as a stock or share. The main feature of this model is the specification of volatility in terms of past price returns. The volatility process and the underlying price process share the same source of randomness and so the model is said to be complete. Complete models are advantageous as they allow a unique, preference independent price for options on the underlying price process. One of the main objectives of the model is to reproduce the `smiles' and `skews' seen in the market implied volatilities and this model produces the desired effect. In the first main piece of work we numerically calibrate the model of Hobson and Rogers for comparison with existing literature. We also develop parameter estimation methods based on the calibration of a GARCH model. We examine alternative specifications of the volatility and show an improvement of model fit to market data based on these specifications. We also show how to process market data in order to take account of inter-day movements in the volatility surface. In the second piece of work, we extend the Hobson and Rogers model in a way that better reflects market structure. We extend the model to take into account both first and second order effects. We derive and numerically solve the pde which describes the price of options under this extended model. We show that this extension allows for a better fit to the market data. Finally, we analyse the parameters of this extended model in order to understand intuitively the role of these parameters in the volatility surface.

Correcting the errors: Volatility forecast evaluation using high-frequency data and realized volatilities

We develop general model-free adjustment procedures for the calculation of unbiased volatility loss functions based on practically feasible realized volatility benchmarks. The procedures, which exploit recent nonparametric asymptotic distributional results, are both easy-to-implement and highly accurate in empirically realistic situations. We also illustrate that properly accounting for the measurement errors in the volatility forecast evaluations reported in the existing literature can result in markedly higher estimates for the true degree of return volatility predictability.

Equity trading volume and volatility: Latent information arrivals and common long-run dependencies

This article examines the behavior of equity trading volume and volatility for the individual firms composing the Standard & Poor's 100 composite index. Using multivariate spectral methods, we find that fractionally integrated processes best describe the long-run temporal dependencies in both series. Consistent with a stylized mixture-of-distributions hypothesis model in which the aggregate "news"-arrival process possesses long-memory characteristics, the long-run hyperbolic decay rates appear to be common across each volume-volatility pair.

Estimating stochastic volatility diffusion using conditional moments of integrated volatility

We exploit the distributional information contained in high-frequency intraday data in constructing a simple conditional moment estimator for stochastic volatility diffusions. The estimator is based on the analytical solutions of the first two conditional moments for the latent integrated volatility, the realization of which is effectively approximated by the sum of the squared high-frequency increments of the process. Our simulation evidence indicates that the resulting GMM estimator is highly reliable and accurate. Our empirical implementation based on high-frequency five-minute foreign exchange returns suggests the presence of multiple latent stochastic volatility factors and possible jumps. © 2002 Elsevier Science B.V. All rights reserved.

Long-term equity anticipation securities and stock market volatility dynamics

Recent empirical findings suggest that the long-run dependence in U.S. stock market volatility is best described by a slowly mean-reverting fractionally integrated process. The present study complements this existing time-series-based evidence by comparing the risk-neutralized option pricing distributions from various ARCH-type formulations. Utilizing a panel data set consisting of newly created exchange traded long-term equity anticipation securities, or leaps, on the Standard and Poor's 500 stock market index with maturity times ranging up to three years, we find that the degree of mean reversion in the volatility process implicit in these prices is best described by a Fractionally Integrated EGARCH (FIEGARCH) model. © 1999 Elsevier Science S.A. All rights reserved.

Volume, volatility, and leverage: A dynamic analysis

This paper uses dynamic impulse response analysis to investigate the interrelationships among stock price volatility, trading volume, and the leverage effect. Dynamic impulse response analysis is a technique for analyzing the multi-step-ahead characteristics of a nonparametric estimate of the one-step conditional density of a strictly stationary process. The technique is the generalization to a nonlinear process of Sims-style impulse response analysis for linear models. In this paper, we refine the technique and apply it to a long panel of daily observations on the price and trading volume of four stocks actively traded on the NYSE: Boeing, Coca-Cola, IBM, and MMM.

Bid-ask spreads and volatility in the foreign exchange market. An empirical analysis

Consistent with the implications from a simple asymmetric information model for the bid-ask spread, we present empirical evidence that the size of the bid-ask spread in the foreign exchange market is positively related to the underlying exchange rate uncertainty. The estimation results are based on an ordered probit analysis that captures the discreteness in the spread distribution, with the uncertainty of the spot exchange rate being quantified through a GARCH type model. The data sets consists of more than 300,000 continuously recorded Deutschemark/dollar quotes over the period from April 1989 to June 1989. © 1994.

Multigrid for American option pricing with stochastic volatility

The paper describes an implicit finite difference approach to the pricing of American options on assets with a stochastic volatility. A multigrid procedure is described for the fast iterative solution of the discrete linear complementarity problems that result. The accuracy and performance of this approach is improved considerably by a strike-price related analytic transformation of asset prices and adaptive time-stepping.

A distributed algorithm for European options with nonlinear volatility

A distributed algorithm is developed to solve nonlinear Black-Scholes equations in the hedging of portfolios. The algorithm is based on an approximate inverse Laplace transform and is particularly suitable for problems that do not require detailed knowledge of each intermediate time steps.