Paper review regimeswitching models with applications in finance

Paper review: regimeswitching models with applications in finance Matthew Couch March 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) 2

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

A Markov Model For Switching Regressions One of the earliest papers with regime switches based on a Markov Chain Applied to a regression model for housing markets 4

A Markov Model For Switching Regressions 5

A Markov Model For Switching Regressions We 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. 6

A Markov Model For Switching Regressions The λ Method 7

A Markov Model For Switching Regressions 8

A Markov Model For Switching Regressions The essence of the λ-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. 9

A Markov Model For Switching Regressions 10

A Markov Model For Switching Regressions 11

A Markov Model For Switching Regressions 12

A Markov Model For Switching Regressions 13

A Markov Model For Switching Regressions 14

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

A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle Early financial/economic application of Regime switching (modeling GNP) Helped popularize the Regime Switching Models, (often cited in current papers) 16

A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle This 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. 17

A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle Gross 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 18

A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle The 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. 19

A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle The 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. 20

A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle A Markov Model of Trend: 21

A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle Stochastic 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. 22

A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle With and 23

A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle CONCLUSIONS This 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 Statesbased 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. 24

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 Heston’s SV model depend on the states of a continuoustime 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. 26

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching The Model Two primary securities: A risk free bond B and a risky asset S A complete probability space (Ω, F, P) with P the real world probability measure Time index set 27

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching 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 28

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching 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 29

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching Let and be standard Brownian Motions with respect to the filtration and 30

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching We assume that X is independent of W Let be the instantaneous market rate of interest of B depend on X, that is 31

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching Bond price dynamics : Appreciation rate: Long term volatility rate: 32

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching Let β and γ 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: 33

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching Then, we can write the dynamics of price process and the short-term volatility process of the risky as 34

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching 35

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching 36

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching Define the regime switching Esscher Transform By Thus the Radon–Nikodym derivative of the regime switching Esscher transform is given by 37

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching Then satifies The martingale condition 38

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching Thus the Radon–Nikodym derivative is given by 39

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching Applying Girsanov’s theorem we obtain the following expression for the dynamics of S and δ the risk neutral measure Q 40

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching 41

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching Variance Swaps A 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. 42

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching 43

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching The conditional price of the variance swap P(X) is given by 44

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching 45

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching 46

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching Volatility Swaps 47

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching 48

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching 49

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching 50

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching Hedging The Vega of the variance swap is given by: 51

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching The Vega of the volatility swap is given by: 52

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching Monte Carlo Experiment: Two N=2 regimes assumed corresponding to states of the econimy Regime 1: “good” state, regime 2: “bad” state 10, 000 simulation runs 20 time steps 53

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching 54

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching We 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. 55

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching 56

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching 57

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching Further Research: For further investigation, it is of interest to explore and develop some criteria to determine the number of states of the Markov chain in our framework which will incorporate important features of the volatility dynamics for different types of underlying financial instruments, such as commodities, currencies and fixed income securities. It would also be interesting to explore the applications of our model to price various volatility derivative products, such as options on volatilities and VIX futures, which are a listed contract on the Chicago Board Options Exchange. It is also of practical interest to investigate the calibration and estimation techniques of our model to volatility index options. Empirical studies comparing the performance of models on volatility swaps are interesting topics to be investigated further. 58

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