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Since the groundbreaking research of Harry Markowitz into the application of operations research to the optimization of investment portfolios, finance has been one of the most important areas of application of operations research. The use of hidden Markov models (HMMs) has become one of the hottest areas of research for such applications to finance. This handbook offers systemic applications of different methodologies that have been used for decision making solutions to the financial problems of global markets. As the follow-up to the authors’ Hidden Markov Models in Finance (2007), this offers the latest research developments and applications of HMMs to finance and other related fields. Amongst the fields of quantitative finance and actuarial science that will be covered are: interest rate theory, fixed-income instruments, currency market, annuity and insurance policies with option-embedded features, investment strategies, commodity markets, energy, high-frequency trading, credit risk, numerical algorithms, financial econometrics and operational risk.

Hidden Markov Models in Finance: Further Developments and Applications, Volume II presents recent applications and case studies in finance and showcases the formulation of emerging potential applications of new research over the book’s 11 chapters. This will benefit not only researchers in financial modeling, but also others in fields such as engineering, the physical sciences and social sciences. Ultimately the handbook should prove to be a valuable resource to dynamic researchers interested in taking full advantage of the power and versatility of HMMs in accurately and efficiently capturing many of the processes in the financial market.



Chapter 1. Robustification of an On-line EM Algorithm for Modelling Asset Prices Within an HMM

In this paper, we establish a robustification of Elliott’s on-line EM algorithm for modelling asset prices within a hidden Markov model (HMM). In this HMM framework, parameters of the model are guided by a Markov chain in discrete time, parameters of the asset returns are therefore able to switch between different regimes. The parameters are estimated through an on-line algorithm, which utilizes incoming information from the market and leads to adaptive optimal estimates. We robustify this algorithm step by step against additive outliers appearing in the observed asset prices with the rationale to better handle possible peaks or missings in asset returns.
Christina Erlwein-Sayer, Peter Ruckdeschel

Chapter 2. Stochastic Volatility or Stochastic Central Tendency: Evidence from a Hidden Markov Model of the Short-Term Interest Rate

We develop a two-factor model for the short-term interest rate that incorporates additional randomness in both the drift and diffusion components. In particular, the model nests stochastic volatility and stochastic central tendency, and therefore provides a medium for testing the overall importance of both factors. The randomness in the drift and diffusion terms is governed by a hidden Markov chain. The likelihood function is determined through an iterative procedure and maximum likelihood estimates are obtained via numerical maximization. This process allows likelihood ratio testing of nested restrictions. These tests show that stochastic volatility is more important than stochastic central tendency for describing the short rate dynamics.
Craig A. Wilson, Robert J. Elliott

Chapter 3. An Econometric Model of the Term Structure of Interest Rates Under Regime-Switching Risk

This paper develops and estimates a continuous-time model of the term structure of interests under regime shifts. The model uses an analytically simple representation of Markov regime shifts that elucidates the effects of regime shifts on the yield curve and gives a clear interpretation of regime-switching risk premiums. The model falls within the broad class of essentially affine models with a closed form solution of the yield curve, yet it is flexible enough to accommodate priced regime-switching risk, time-varying transition probabilities, regime-dependent mean reversion coefficients as well as stochastic volatilities within each regime. A two-factor version of the model is implemented using Efficient Method of Moments. Empirical results show that the model can account for many salient features of the yield curve in the U.S.
Shu Wu, Yong Zeng

Chapter 4. The LIBOR Market Model: A Markov-Switching Jump Diffusion Extension

This paper demonstrates how the LIBOR Market Model of Brace et al. (Math Financ 7(2):127–147, 1997) and Miltersen et al. (J Financ 52(1):409–430, 1997) may be extended in a way that not only takes into account sudden market shocks without long-term effects, but also allows for structural breaks and changes in the overall economic climate. This is achieved by substituting the simple diffusion process of the original LIBOR Market model by a Markov-switching jump diffusion. Since interest rates of different maturities are modeled under different (forward) measures, we investigate the effects of changes between measures on all relevant quantities. Using the Fourier pricing technique, we derive pricing formula for the most important interest rate derivatives, caps/caplets, and calibrate the model to real data.
Lea Steinrücke, Rudi Zagst, Anatoliy Swishchuk

Chapter 5. Exchange Rates and Net Portfolio Flows: A Markov-Switching Approach

In this paper we investigate the impact of net bond and equity portfolio flows on exchange rate changes. Two-state Markov-switching models are estimated for Canada, the euro area, Japan and the UK exchange rates vis-à-vis the US dollar. Our results suggest that the relationship between net portfolio flows and exchange rate changes is nonlinear for all currencies considered but Canada.
Faek Menla Ali, Fabio Spagnolo, Nicola Spagnolo

Chapter 6. Hedging Costs for Variable Annuities Under Regime-Switching

A general methodology is described in which policyholder behaviour is decoupled from the pricing of a variable annuity based on the cost of hedging it, yielding two weakly coupled systems of partial differential equations (PDEs): the pricing and utility systems. The utility system is used to generate policyholder withdrawal behaviour, which is in turn fed into the pricing system as a means to determine the cost of hedging the contract. This approach allows us to incorporate the effects of utility-based pricing and factors such as taxation. As a case study, we consider the Guaranteed Lifelong Withdrawal and Death Benefits (GLWDB) contract. The pricing and utility systems for the GLWDB are derived under the assumption that the underlying asset follows a Markov regime-switching process. An implicit PDE method is used to solve both systems in tandem. We show that for a large class of utility functions, the pricing and utility systems preserve homogeneity, allowing us to decrease the dimensionality of the PDEs and thus to rapidly generate numerical solutions. It is shown that for a typical contract, the fee required to fund the cost of hedging calculated under the assumption that the policyholder withdraws at the contract rate is an appropriate approximation to the fee calculated assuming optimal consumption. The costly nature of the death benefit is documented. Results are presented which demonstrate the sensitivity of the hedging expense to various parameters.
Parsiad Azimzadeh, Peter A. Forsyth, Kenneth R. Vetzal

Chapter 7. A Stochastic Approximation Approach for Trend-Following Trading

This work develops a feasible computation procedure for trend-following trading under a bull-bear switching market model. In the asset model, the drift of the stock price switches between two parameters corresponding to an uptrend (bull market) and a downtrend (bear market) according to a partially observable Markov chain. The objective is to buy and sell the underlying stock to maximize an expected return. It is shown in Dai et al. (SIAM J Financ Math 1:780–810, 2010; Optimal trend following trading rules. Working paper) that an optimal trading strategy can be obtained in terms of two threshold levels. Finding the threshold levels turns out to be a difficult task. In this paper, we develop a stochastic approximation algorithm to approximate the threshold levels. One of the main advantages of this approach is that one need not solve the associated HJB equations. We also establish the convergence of the algorithm and provide numerical examples to illustrate the results.
Duy Nguyen, George Yin, Qing Zhang

Chapter 8. A Hidden Markov-Modulated Jump Diffusion Model for European Option Pricing

The valuation of a European-style contingent claim is discussed in a hidden Markov regime-switching jump-diffusion market, where the evolution of a hidden economic state process over time is described by a continuous-time, finite-state, hidden Markov chain. A two-stage procedure is used to discuss the option valuation problem. Firstly filtering theory is employed to transform the original market with hidden quantities into a filtered market with complete observations. Then a generalized version of the Esscher transform based on a Doléan-Dade stochastic exponential is employed to select a pricing kernel in the filtered market. A partial-differential-integral equation for the price of a European-style option is presented.
Tak Kuen Siu

Chapter 9. An Exact Formula for Pricing American Exchange Options with Regime Switching

This paper investigates the pricing of American exchange options when the price dynamics of each underlying risky asset are assumed to follow a Markov-modulated Geometric Brownian motion; that is, the appreciation rate and the volatility of each underlying risky asset depend on unobservable states of the economy described by a continuous-time hidden Markov process. We show that the price of an American exchange option can be reduced to the price of an American option. Then, we modify the result of Zhu and Chan (An analytic formula for pricing American options with regime switching. Submitted for publication, 2012), a closed-form analytical pricing formula for the American exchange option is given.
Leunglung Chan

Chapter 10. Parameter Estimation in a Weak Hidden Markov Model with Independent Drift and Volatility

We develop a multivariate higher-order Markov model, also known as weak hidden Markov model (WHMM), for the evolution of asset prices. The means and volatilities of asset’s log-returns are governed by a second-order Markov chain in discrete time. WHMM enriches the usual HMM by incorporating more information from the past thereby capturing presence of memory in the underlying market state. A filtering technique in conjunction with the Expectation-Maximisation algorithm is adopted to develop the optimal estimates of model parameters. To ensure that the errors between the “true” parameters and estimated parameters are due only to the estimation method and not from model uncertainty, recursive filtering algorithms are implemented to a simulated dataset. Using goodness-of-fit metrics, we show that the WHMM-based filtering techniques are able to recover the “true” underlying parameters.
Xiaojing Xi, Rogemar S. Mamon

Chapter 11. Parameter Estimation in a Regime-Switching Model with Non-normal Noise

This paper deals with the estimation of a Markov-modulated regime-switching model for asset prices, where the noise term is assumed non-normal consistent with the well-known observed market phenomena that log-return distributions exhibit heavy tails. Hence, the proposed model augments the flexibility of the current Markov-switching models with normal perturbation whilst still achieving dynamic calibration of parameters. In particular, under the setting where the model’s noise term follows a t-distribution, we employ the method of change of reference probability measure to provide recursive filters for the estimate of the state and transition probabilities of the Markov chain. Although recursive filters are no longer available for the maximum likelihood estimation of the model’s drift and volatility components under the current extension, we show that such estimation is tantamount to solving numerically a manageable system of nonlinear equations. Practical applications with the use of simulated and real-market data are included to demonstrate the implementation of our proposed algorithms.
Luka Jalen, Rogemar S. Mamon
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