Paper review: regime- switching models with applications in finance Matthew Couch Matthew Couch March 5, 2009

Paper review: regime-switching models with applications in finance

Matthew CouchMarch 5, 2009

*Paper review: regime-switching models with applications in finance

Outline:

A Markov Model For Switching Regressions, Stephen M. Goldfeld and Richard E. Quandt (1973)A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle, James D. Hamilton (1989)Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching, Robert J. Elliott, Tak Kuen Siu, Leunglung Chan (2005)

A Markov Model For Switching RegressionsStephen M. Goldfeld and Richard E. Quandt (1973)

*A Markov Model For Switching RegressionsOne of the earliest papers with regime switches based on a Markov ChainApplied to a regression model for housing markets

*A Markov Model For Switching Regressions

*A Markov Model For Switching RegressionsWe consider possible structures for switching regressions:The simplest type of structure consists of the assumption that there is at most one switch in the data series; i.e., that the first m (unknown) observations in a time series are generated by regime 1 and the remaining n-m observations by regime 2. Problems of this type have been analyzed in various ways by Brown and Durbin (1968), Farley and Hinich (1970) and Quandt (1958, 1960). This simple model, permitting only one switch, is clearly unrealistic in some economic contexts. A more complex situation arises if it is assumed that the system may switch back and forth between the two regimes. Accordingly the first m(1) observations may come from regime 1, the next m(2) from regime 2, the next m(3) from regime 1 again, etc., with {m(j)} being unknown. Under this assumption it is theoretically possible for the system to switch between regimes every time that a new observation is generated.

*A Markov Model For Switching RegressionsThe Method


The essence of the -method as stated in the previous section is that the probability that nature selects regime 1 or 2 at the ith trial is independent of what state the system was in on the previous trial. We shall explicitly relax this assumption and introduce the matrix T of transition probabilities, where (r,s) being the probability that the system will make a transition from state r to state s. This interpretation makes the regime switching process a Markov chain.

A New Approach to the Economic Analysis of Nonstationary Time Series and the Business CycleJames D. Hamilton (1989)

*A New Approach to the Economic Analysis of Nonstationary Time Series and the Business CycleEarly financial/economic application of Regime switching (modeling GNP)Helped popularize the Regime Switching Models, (often cited in current papers)

*A New Approach to the Economic Analysis of Nonstationary Time Series and the Business CycleThis paper proposes a very tractable approach to modeling changes in regime. The parameters of an autoregression are viewed as the outcome of a discrete-state Markov process. For example, the mean growth rate of a nonstationary series may be subject to occasional, discrete shifts. The econometrician is presumed not to observe these shifts directly, but instead must draw probabilistic inference about whether and when they may have occurred based on the observed behavior of the series. An empirical application of this technique to postwar U.S. real GNP suggests that the periodic shift from a positive growth rate to a negative growth rate is a recurrent feature of the U.S. business cycle, and indeed could be used as an objective criterion for defining and measuring economic recessions. The estimated parameter values suggest that a typical economic recession is associated with a 3% permanent drop in the level of GNP.

*A New Approach to the Economic Analysis of Nonstationary Time Series and the Business CycleGross National Product (GNP) is defined as the value of all (final) goods and services produced in a country in one year by the nationals, plus income earned by its citizens abroad, minus income earned by foreigners in the country. At the time of publication of this paper a number of studies had sought to characterize the nature of the long term trend in GNP and its relation to the business cycle. The approaches in these studies were based on the assumption that first differences of the log of GNP follow a linear stationary process; that is, in all of the above studies, optimal forecasts of variables are assumed to be a linear function of their lagged values.This paper, suggests an alternative to the approaches to nonstationarity, exploring the consequences of specifying that first differences of the observed series follow a nonlinear stationary process rather than a linear stationary process

*A New Approach to the Economic Analysis of Nonstationary Time Series and the Business CycleThe nonlinearities with which this paper is concerned arises if the process is subject to discrete shifts in regime-episodes across which the dynamic behavior of the series is markedly different. The basic approach is to use Goldfeld and Quandt's (1973) Markov switching regression to characterize changes in the parameters of an autoregressive process. For example, the economy may either be in a fast growth or slow growth phase, with the switch between the two governed by the outcome of a Markov process.

*A New Approach to the Economic Analysis of Nonstationary Time Series and the Business CycleThe approach taken could also be viewed as a natural extension of Neftci's (1984) analysis of U.S. unemployment data. In Neftci's specification, the economy is said to be in state 1 whenever unemployment is rising and in state 2 whenever unemployment is falling, with transitions between these two states modeled as the outcome of a second-order Markov process. In this paper, by contrast, the unobserved state is only one of many influences governing the dynamic process followed by output, so that even when the economy is in the "fast growth" state, output in principle might be observed to decrease.

*A New Approach to the Economic Analysis of Nonstationary Time Series and the Business CycleA Markov Model of Trend:

*A New Approach to the Economic Analysis of Nonstationary Time Series and the Business CycleStochastic Specification:Several options are available for combining the trend term n t with another stochastic process. Here I discuss the approach that results in the computationally simplest maximum likelihood estimation.

*A New Approach to the Economic Analysis of Nonstationary Time Series and the Business CycleWithand

*A New Approach to the Economic Analysis of Nonstationary Time Series and the Business CycleCONCLUSIONSThis paper explored the possibility that growth rates of real GNP are subject to autocorrelated discrete shifts. Empirical estimation suggested that the business cycle is better characterized by a recurrent pattern of such shifts between a recessionary state and a growth state rather than by positive coefficients at low lags in an autoregressive model. Indeed, statistical estimates of the economy's growth state cohere remarkably well with NBER (The United States-based National Bureau of Economic Research) dating of postwar recessions, and might be used as an alternative objective method for assigning business cycle dates. A move from expansion into recession is associated with a 3% decrease in the present value of future real GNP and similarly portends a 3% drop in the long-run forecast level of GNP.

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching

Robert J. Elliott, Tak Kuen Siu, Leunglung Chan (2005)

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching

Introduction:

A model is developed for pricing volatility derivatives, such as variance swaps and volatility swaps under a continuous-time Markov-modulated version of the stochastic volatility (SV) model developed by Heston. In particular, it is supposed that the parameters of this version of Hestons SV model depend on the states of a continuous-time observable Markov chain process, which can be interpreted as the states of an observable macroeconomic factor. The market considered is incomplete in general, and hence, there is more than one equivalent martingale pricing measure. The regime switching Esscher transform used by Elliott et al. is adopted to determine a martingale pricing measure for the valuation of variance and volatility swaps in this incomplete market.


The Model

Two primary securities: A risk free bond B and a risky asset SA complete probability space (,F,P) with P the real world probability measure Time index set


Consider a finite state continuous time Markov Chain with state space where

The states of the Markov chain process X describe the states of an observable economic indicator


Without loss of generality, identify the state space of the chain with the set of unit vectors in

Let be the generator of X Then X has the following semi martingale representation:

Where is a martingale increment process

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime SwitchingLet andbe standard Brownian Motions with respect to

the filtration and

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime SwitchingWe assume that X is independent of W

Let be the instantaneous market rate of interest of B depend on X, that is

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime SwitchingBond price dynamics :

Appreciation rate:

Long term volatility rate:

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime SwitchingLet and denote the speed of mean reversion and the volatility of volatility respectively. Suppose that the dynamics of the price process and the short-term volatility process of the risky stock are governed by the following equations:

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime SwitchingThen, we can write the dynamics of price process and the short-term volatility process of the risky as

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime SwitchingDefine the regime switching Esscher Transform By

Thus the RadonNikodym derivative of the regime switching Esscher transform is given by

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching Then satifies

The martingale condition

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime SwitchingThus the RadonNikodym derivative is given by

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime SwitchingApplying Girsanovs theorem we obtain the following expression for the dynamics of S and the risk neutral measure Q

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime SwitchingVariance SwapsA variance swap is a forward contract on annualized variance, which is the square of the realized annual volatility.

In practice, variance swaps are written on the realized variance evaluated based on daily closing prices with the integral in above replaced by a discrete sum. Hence, variance swaps with payoffs depending on the realized variance defined above are only approximations to those of the actual contracts.

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime SwitchingThe conditional price of the variance swap P(X) is given by

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime SwitchingVolatility Swaps

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime SwitchingHedging

The Vega of the variance swap is given by:

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime SwitchingThe Vega of the volatility swap is given by:

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime SwitchingMonte Carlo Experiment:Two N=2 regimes assumed corresponding to states of the econimyRegime 1: good state, regime 2: bad state 10,000 simulation runs20 time steps

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime SwitchingWe suppose that we are currently in the good economic state and that the current volatility level is V(0)=0.12. The delivery prices of the variance swap and the volatility swap range from 80% to 125% of the current levels of the variance and the standard deviation of the underlying risky asset, respectively. The time-to-expiry of both the variance swap and the volatility swap is 1 year.

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime SwitchingFurther Research:

For further investigation, it is of interest to explore and develop some criteria todetermine the number of states of the Markov chain in our framework which willincorporate important features of the volatility dynamics for different types ofunderlying financial instruments, such as commodities, currencies and fixed incomesecurities. It would also be interesting to explore the applications of our model toprice various volatility derivative products, such as options on volatilities and VIXfutures, which are a listed contract on the Chicago Board Options Exchange. It isalso of practical interest to investigate the calibration and estimation techniques ofour model to volatility index options. Empirical studies comparing theperformance of models on volatility swaps are interesting topics to be investigatedfurther.

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime SwitchingReferencesBack, K. and Pliska, S. R. (1991) On the fundamental theorem of asset pricing with an infinite state space,Journal of Mathematical Economics, 20, pp. 118.Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics,31, pp. 307327.Brenner, M. and Galai, D. (1989) New financial instruments for hedging changes in volatility, FinancialAnalyst Journal, July/August, pp. 6165.Brenner, M. and Galai, D. (1993) Hedging volatility in foreign currencies, Journal of Derivatives, 1, pp.5359.Brenner, M. et al. (2001) Hedging volatility risk. Working paper, Stern School of Business, New YorkUniversity, USA.Brockhaus, O. and Long, D. (2000) Volatility swaps made simple, Risk, 2(1), pp. 9295.Buffington, J. and Elliott, R. J. (2002a) Regime switching and European options, in: StochasticTheory and Control, Proceedings of a Workshop, Lawrence, Kansas, October (Springer Verlag), pp.7381.Buffington, J. and Elliott, R. J. (2002b) American options with regime switching, International Journal ofTheoretical and Applied Finance, 5, pp. 497514.Carr, P. (2005) Delta hedging with stochastic volatility. Working paper, Quantitative Financial Research,Bloomberg LP, New York.Carr, P. and Madan, D. (1998) Towards a theory of volatility trading, in: R. Jarrow (Ed.) Volatility (RiskBook Publications).Carr, P. et al. (2005) Pricing options on realized variance, Finance and Stochastics, 9(4), pp. 453475.Cox, J., Ingersoll, J. and Ross, S. (1985) A theory of the term structure of interest rates, Econometrica, 53,pp. 385407.

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime SwitchingCox, J. and Ross, S. (1976) The valuation of options for alternative stochastic processes, Journal ofFinancial Economics, 3, pp. 145166.Delbaen, F. and Schachermayer, W. (1994) A general version of fundamental theorem of asset pricing,Mathematische Annalen, 300, pp. 463520.Demeterfi, K. et al. (1999) A guide to volatility and variance swaps, The Journal of Derivatives, 6(4), pp.932.Duffie, D. and Richardson, H. R. (1991) Mean-variance hedging in continuous time, The Annals ofApplied Probability, 1, pp. 115.Dybvig Philip, H. and Ross, S. (1987) Arbitrage, in: J. Eatwell et al. (Eds), The New Palsgrave: ADictionary of Economics, 1, pp. 100106.Elliott, R. J. et al. (1994) Hidden Markov Models: Estimation and Control (New York: Springer-Verlag).Elliott, R. J. et al. (2003) Modelling electricity price risk. Working paper, Haskayne School of Business,University of Calgary.Elliott, R. J. et al. (2005) Option pricing and Esscher transform under regime switching, Annals of Finance,1(4), pp. 423432.Elliott, R. J. and Kopp, E. P. (2004) Mathematics of Financial Markets (New York: Springer-Verlag).Elliott, R. J. and Swishchuk, A. (2004) Pricing options and variance swaps in Brownian and fractionalBrownian market. Working paper, University of Calgary.Engle, R. (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of U.K.inflation, Econometrica, 50, pp. 9871008.

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime SwitchingFo llmer, H. and Schweizer, M. (1991) Hedging of contingent claims under incomplete information, in:M. H. A. Davis & R. J. Elliott (Eds), Applied Stochastic Analysis, Stochastics Monographs, Vol.5, pp.389414 (New York: Gordon and Breach).Grunbuchler, A. and Longstaff, F. (1996) Valuing futures and options on volatility, Journal of Bankingand Finance, 20, pp. 9851001 Harrison, J. M. and Kreps, D. M. (1979) Martingales and arbitrage in multi-period securities markets,Journal of Economic Theory, 20, pp. 381408.Harrison, J. M. and Pliska, S. R. (1981) Martingales and stochastic integrals in the theory of continuoustrading, Stochastic Processes and Their Applications, 11, pp. 215280.Harrison, J. M. and Pliska, S. R. (1983) A stochastic calculus model of continuous trading: completemarkets, Stochastic Processes and Their Applications, 15, pp. 313316.Heston, S. L. (1993) A closed-form solution for options with stochastic with applications to bond andcurrency options, Review of Financial Studies, 6(2), pp. 237343.Heston, S. L. and Nandi, S. (2000) Derivatives on volatility: some simple solution based on observables.Working paper, Federal Reserve Bank of Atlanta.Howison, S. et al. (2004) On the pricings and hedging of volatility derivatives, Applied MathematicalFinance, 11(4), pp. 317346.Javaheri, A. et al. (2002) GARCH and volatility swaps, Wilmott Magazine, January, pp. 117.Matytsin, A. (2000) Mode ling volatility and volatility derivatives. Working paper, Columbia University.Pan, J. (2002) The jump-risk premia implicit in options: evidence from an integrated time-series study,Journal of Financial Economics, 63, pp. 350.

*Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime SwitchingRoss, S. A. (1973) Return, risk and arbitrage. Wharton Discussion Paper published in: Risk and Return inFinance, edited by: I. Friend & J. Bicksler (Eds), pp. 189217 (Cambridge: Ballinger, 1976).Ross, S. A. (1978) A simple approach to the valuation of risky streams, Journal of Business, 3, pp.453476.Ross, S. A. (2005) Neoclassical Finance(Princeton and Oxford: Princeton University Press).Schachermayer, W. (1992) A Hilbert space proof of the fundamental theorem of asset pricing in finitediscrete time, Insurance: Mathematics and Economics, 11, pp. 249257.Schweizer, M. (1991) Option hedging for semimartingales, Stochastic Processes and Their Application, 37,pp. 339363.Schweizer, M. (1992) Mean-variance hedging for general claims, The Annals of Applied Probability, 2, pp.171179.Shreve, S. E. (2004) Stochastic Calculus for Finance(New York: Springer).Swishchuk, A. (2004) Modeling of variance and volatility swaps for financial markets with stochasticvolatilities, Wilmott Magazine, 2, pp. 6472Swishchuk, A. (2005) Modeling and pricing of variance swaps for stochastic volatilities with delay,Wilmott Magazine, Forthcoming.Taylor, S. J. (1986) Modelling Financial Time Series (New York: John Wiley & Sons).Windcliff, H., Forsyth, P. A. and Vetzal, K. R. (2003) Pricing methods and hedging strategies for volatilityderivatives. Working paper, University of Waterloo.

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Paper review: regime- switching models with applications in finance Matthew Couch Matthew Couch March 5, 2009