Applications of hidden Markov models in financial modelling

This thesis was submitted for the degree of Doctor of Philosophy and was awarded by Brunel University.Various models driven by a hidden Markov chain in discrete or continuous time are developed to capture the stylised features of market variables whose levels or values constitute as the underliers of financial derivative contracts or investment portfolios. Since the parameters are switching regimes, the changes and developments in the economy as soon as they arise are readily reflected in these models. The change of probability measure technique and the EM algorithm are fundamental techniques utilised in the optimal parameter estimation. Recursive adaptive filters for the state of the Markov chain and other auxiliary processes related to the Markov chain are derived which in turn yield self-tuning dynamic financial models. A hidden Markov model (HMM)-based modelling set-up for commodity prices is developed and the predictability of the gold market under this setting is examined. An Ornstein-Uhlenbeck (OU) model with HMM parameters is proposed and under this set-up, we address two statistical inference issues: the sensitivity of the model to small changes in parameter estimates and the selection of the optimal number of states. The extended OU model is implemented on a data set of 30-day Canadian T-bill yields. An exponential of a Markov-switching OU process plus a compound Poisson process is put forward as a model for the evolution of electricity spot prices. Using a data set compiled by Nord Pool, we illustrate the vast improvements gained in incorporating regimes in the model. A multivariate HMM is employed as a framework in providing the solutions of two asset allocation problems; one involves the mean-variance utility function and the other entails the CVaR constraint. Finally, the valuation of credit default swaps highlights the important considerations necessitated by pricing in a regime-switching environment. Certain numerical schemes are applied to obtain approximations for the default probabilities and swap rates.Brunel Research Initiative and Enterprise Fund (BRIEF) and European Union (Marie Curie Fellowship

HMM filtering and parameter estimation of an electricity spot price model

In this paper we develop a model for electricity spot price dynamics. The spot price is assumed to follow an exponential Ornstein-Uhlenbeck (OU) process with an added compound Poisson process, therefore the model allows for mean-reversion and possible jumps. A sinusoidal factor is also introduced to capture the seasonality component of prices. The mean-reverting level, speed of adjustment and volatility of the OU process as well as the mean and variance of the normally distributed jump sizes of the compound Poisson process are all modulated by a hidden Markov chain in discrete time. The parameters are able to switch between different economic regimes representing various levels of supply and demand. Through the application of reference probability technique, adaptive filters are derived, which in turn, provide optimal estimates for the state of the Markov chain and related quantities of the observation process. The EM algorithm is applied to find optimal estimates of the model parameters in terms of the recursive filters. Since the parameters are updated everytime a new information is available, the model is self-calibrating. We implement the model on a deseasonalized series of daily spot electricity prices from the Nordic exchange Nord Pool. On the basis of one-step ahead forecasts, we found that the model is able to capture the stylised features of Nord Pool spot prices

Applications of hidden Markov models in financial modelling

Various models driven by a hidden Markov chain in discrete or continuous time are developed to capture the stylised features of market variables whose levels or values constitute as the underliers of financial derivative contracts or investment portfolios. Since the parameters are switching regimes, the changes and developments in the economy as soon as they arise are readily reflected in these models. The change of probability measure technique and the EM algorithm are fundamental techniques utilised in the optimal parameter estimation. Recursive adaptive filters for the state of the Markov chain and other auxiliary processes related to the Markov chain are derived which in turn yield self-tuning dynamic financial models. A hidden Markov model (HMM)-based modelling set-up for commodity prices is developed and the predictability of the gold market under this setting is examined. An Ornstein-Uhlenbeck (OU) model with HMM parameters is proposed and under this set-up, we address two statistical inference issues: the sensitivity of the model to small changes in parameter estimates and the selection of the optimal number of states. The extended OU model is implemented on a data set of 30-day Canadian T-bill yields. An exponential of a Markov-switching OU process plus a compound Poisson process is put forward as a model for the evolution of electricity spot prices. Using a data set compiled by Nord Pool, we illustrate the vast improvements gained in incorporating regimes in the model. A multivariate HMM is employed as a framework in providing the solutions of two asset allocation problems; one involves the mean-variance utility function and the other entails the CVaR constraint. Finally, the valuation of credit default swaps highlights the important considerations necessitated by pricing in a regime-switching environment. Certain numerical schemes are applied to obtain approximations for the default probabilities and swap rates.EThOS - Electronic Theses Online ServiceBrunel Research Initiative and Enterprise Fund (BRIEF) : European Union (Marie Curie Fellowship)GBUnited Kingdo

HMM based scenario generation for an investment optimisation problem

This is the post-print version of the article. The official published version can be accessed from the link below - Copyright @ 2012 Springer-Verlag.The Geometric Brownian motion (GBM) is a standard method for modelling financial time series. An important criticism of this method is that the parameters of the GBM are assumed to be constants; due to this fact, important features of the time series, like extreme behaviour or volatility clustering cannot be captured. We propose an approach by which the parameters of the GBM are able to switch between regimes, more precisely they are governed by a hidden Markov chain. Thus, we model the financial time series via a hidden Markov model (HMM) with a GBM in each state. Using this approach, we generate scenarios for a financial portfolio optimisation problem in which the portfolio CVaR is minimised. Numerical results are presented.This study was funded by NET ACE at OptiRisk Systems

Absence of XMRV and Closely Related Viruses in Primary Prostate Cancer Tissues Used to Derive the XMRV-Infected Cell Line 22Rv1

The 22Rv1 cell line is widely used for prostate cancer research and other studies throughout the world. These cells were established from a human prostate tumor, CWR22, that was serially passaged in nude mice and selected for androgen independence. The 22Rv1 cells are known to produce high titers of xenotropic murine leukemia virus-related virus (XMRV). Recent studies suggested that XMRV was inadvertently created in the 1990's when two murine leukemia virus (MLV) genomes (pre-XMRV1 and pre-XMRV-2) recombined during passaging of the CWR22 tumor in mice. The conclusion that XMRV originated from mice and not the patient was based partly on the failure to detect XMRV in early CWR22 xenografts. While that deduction is certainly justified, we examined the possibility that a closely related virus could have been present in primary tumor tissue. Here we report that we have located the original prostate tumor tissue excised from patient CWR22 and have assayed the corresponding DNA by PCR and the tissue sections by fluorescence in situ hybridization for the presence of XMRV or a similar virus. The primary tumor tissues lacked mouse DNA as determined by PCR for intracisternal A type particle DNA, thus avoiding one of the limitations of studying xenografts. We show that neither XMRV nor a closely related virus was present in primary prostate tissue of patient CWR22. Our findings confirm and reinforce the conclusion that XMRV is a recombinant laboratory-generated mouse virus that is highly adapted for human prostate cancer cells

The Pricing of credit default swaps under a Markov-modulated Merton's structural model

We consider the valuation of credit default swaps (CDSs) under an extended version of Merton’s structural model for a firm’s corporate liabilities. In particular, the interest rate process of a money market account, the appreciation rate, and the volatility of the firm’s value have switching dynamics governed by a finite-state Markov chain in continuous time. The states of the Markov chain are deemed to represent the states of an economy. The shift from one economic state to another may be attributed to certain factors that affect the profits or earnings of a firm; examples of such factors include changes in business conditions, corporate decisions, company operations, management strategies, macroeconomic conditions, and business cycles. In this article, the Esscher transform, which is a well-known tool in actuarial science, is employed to determine an equivalent martingale measure for the valuation problem in the incomplete market setting. Systems of coupled partial differential equations (PDEs) satisfied by the real-world and risk-neutral default probabilities are derived. The consequences for the swap rate of a CDS brought about by the regimeswitching effect of the firm’s value are investigated via a numerical example for the case of a two-state Markov chain. We perform sensitivity analyses for the real-world default probability and the swap rate when different model parameters vary. We also investigate the accuracy and efficiency of the PDE approach by comparing the numerical results from the PDE approach to those from the Monte Carlo simulation.28 page(s