A Class of Nonlinear Stochastic Volatility Models and Its Implications on Pricing Currency Options

This paper proposes a class of stochastic volatility (SV) models which offers an alternative to the one introduced in Andersen (1994). The class encompasses all standard SV models that have appeared in the literature, including the well known lognormal model, and allows us to empirically test all standard specifications in a convenient way. We develop a likelihood-based technique for analyzing the class. Daily dollar/pound exchange rate data reject all the standard models and suggest evidence of nonlinear SV. An efficient algorithm is proposed to study the implications of this nonlinear SV on pricing currency options and it is found that the lognormal model overprices options.Box-Cox transformations, Stochastic volatility, MCMC, Exchange rate volatility, Option pricing.

Estimation of Hidden Markov Models and Their Applications in Finance

Movements of financial variables exhibit extreme fluctuations during periods of economic crisis and times of market uncertainty. They are also affected by institutional policies and intervention of regulatory authorities. These structural changes driving prices and other economic indicators can be captured reasonably by models featuring regime-switching capabilities. Hidden Markov models (HMM) modulating the model parameters to incorporate such regime-switching dynamics have been put forward in recent years, but many of them could still be further improved. In this research, we aim to address some of the inadequacies of previous regime-switching models in terms of their capacity to provide better forecasts and efficiency in estimating parameters. New models are developed, and their corresponding filtering results are obtained and tested on financial data sets. The contributions of this research work include the following: (i) Recursive filtering algorithms are constructed for a regime-switching financial model consistent with no-arbitrage pricing. An application to the filtering and forecasting of futures prices under a multivariate set-up is presented. (ii) The modelling of risk due to market and funding liquidity is considered by capturing the joint dynamics of three time series (Treasury-Eurodollar spread, VIX and S\&P 500 spread-derived metric), which mirror liquidity levels in the financial markets. HMM filters under a multi-regime mean- reverting model are established. (iii) Kalman filtering techniques and the change of reference probability-based filtering methods are integrated to obtain hybrid algorithms. A pairs trading investment strategy is supported by the combined power of both HMM and Kalman filters. It is shown that an investor is able to benefit from the proposed interplay of the two filtering methods. (iv) A zero-delay HMM is devised for the evolution of multivariate foreign exchange rate data under a high-frequency trading environment. Recursive filters for quantities that are functions of a Markov chain are derived, which in turn provide optimal parameter estimates. (v) An algorithm is designed for the efficient calculation of the joint probability function for the occupation time in a Markov-modulated model for asset returns under a general number of economic regimes. The algorithm is constructed with accessible implementation and practical considerations in mind