Hidden Markov Models in Finance

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1 Hidden Markov Models in Finance Edited by, Rogemar S. Mamon Robert J. Elliott B fya Springer

2 Contents 1 An Exact Solution of the Term Structure of Interest Rate under Regime-Switching Risk Shu Wu, Yong Zeng Introduction A new representation for modeling regime shift The model Two state variables Pricing kernel The risk-neutral probability measure 5 1.3:4 The term structure of interest rates A tractable specification with exact solution Affine regime-switching models Conclusions 13 References 13 2 The Term Structure of Interest Rates in a Hidden Markov Setting Robert J. Elliott, Craig A. Wilson Introduction ; The Model The Markov chain The short-term interest rate The zero-coupon bond value Implementation Results Conclusion 30 References 30

3 VI Contents 3 On Fair Valuation of Participating Life Insurance Policies With Regime Switching Tak Kuen Siu ' Introduction The model dynamics _ Dimension reduction to regime-switching PDE Further investigation 42 References \ 42 4 Pricing Options and Variance Swaps iri Markov-Modulated Brownian Markets Robert J. Elliott, Anatoliy V. Swishchuk Introduction Literature review Martingale characterization of Markov processes Pricing options for Markov-modulated security markets Incompleteness of Markov-modulated Brownian security markets The Black-Scholes formula for pricing options in a Markov-modulated Brownian market Pricing options for Markov-modulated Brownian markets with jumps 58' Incompleteness of Markov-modulated Brownian (B, S')-security markets with jumps Black-Scholes formula for pricing options in Markovmodulated Brownian (B, S)-security market with jumps Pricing of Variancev swaps for stochastic volatility driven by Markov process Stochastic volatility driven by Markov process Pricing of variance swaps for stochastic volatility driven by Markov process Example of variance swap for stochastic volatility driven by two-state continuous Markov chain 64 A Some auxiliary results 64 A.I A Feynmann-Kac formula for the Markov-modulated process (y s (t),x s (t)) t > s 64 A.2 Formula for the option price /T(ST) for the market combined Markov-modulated (B, 5)-security market and compound geometric Poisson process (see Section 4.4.2) References 67

4 Contents 5 Smoothed Parameter Estimation for a Hidden Markov Model of Credit Quality Malgorzata W. Korolkiewicz, Robert J. Elliott Introduction Dynamics of the Markov chain and observations Reference probability Recursive filter Parameter estimates Smoothed estimates 75 A Appendix 80 References 90 6 Expected Shortfall Under a Model With Market and Credit Risks Kin Bong Siu, Hailiang Yang Introduction Markov regime-switching model Weak Markov-regime switching model Concluding remarks 99 References, 99 7 Filtering of Hidden Weak Markov Chain -Discrete Range Observations Shangzhen Luo, Allanus H. Tsoi : Introduction Basic Settings Change of Measure A general unnormalized recursive filter Estimation of states, transitions and occupation times State estimation Estimators for the number of jumps Estimators for 1-state occupation times Estimators for 2-state occupation times Ill Estimators for state to observation transitions Ill 7.6 Parameter re-estimations Error analysis Conclusion 117 References Filtering of a Partially Observed Inventory System Lakhdar Aggoun Introduction Model description Reference probability Filtering Filters for G H, and Si* 128 VII

5 VIII Contents 8.6 Parameter re-estimation 131 References An empirical investigation of the unbiased forward exchange rate hypothesis in a regime switching market Emilio Russo, Fabio Spagnolo and Rogemar Mamon ^-Introduction Stylised features and statistical properties of foreign exchange rates Stationary and nonstationary time series Cointegration and the unbiased forward exchange rate (UFER) hypothesis Evidence from exchange rate market via a Markov regime-switching model Concluding remarks 151 References 151 s 10 Early Warning Systems for Currency Crises: A Regime-Switching Approach Abdul Abiad Introduction A Markov-switching approach to early warning systems Data description and transformation Estimation results Indonesia Korea Malaysia The Philippines Thailand Forecast assessment Conclusions 180 References 182

Contents 1 An Exact Solution of the Term Structure of Interest Rate under Regime-Switching Risk

Contents 1 An Exact Solution of the Term Structure of Interest Rate under Regime-Switching Risk Shu Wu, Yong Zeng... 1 1.1 Introduction... 1 1.2 Anewrepresentationformodelingregimeshift... 3 1.3 Themodel...