Paper review: regimeswitching models with applications in finance

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Paper review regime-switching models with
applications in finance

• Matthew Couch
• March 5, 2009

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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)

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A Markov Model For Switching Regressions

• Stephen M. Goldfeld and Richard E. Quandt (1973)

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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
  

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A Markov Model For Switching Regressions

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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.

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A Markov Model For Switching Regressions
The ? Method
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A Markov Model For Switching Regressions
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A Markov Model For Switching Regressions

• 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.

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A Markov Model For Switching Regressions
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A Markov Model For Switching Regressions
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A Markov Model For Switching Regressions
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A Markov Model For Switching Regressions
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A Markov Model For Switching Regressions
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A New Approach to the Economic Analysis of
Nonstationary Time Series and the Business Cycle

• James D. Hamilton (1989)

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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)

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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.

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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

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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.

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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.

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

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

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

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

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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.

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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 (O,F,P) with P the
  real world probability measure
• Time index set

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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

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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

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

• Let and
• be standard Brownian Motions with respect to
• the filtration
• and

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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

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

• Bond price dynamics
• Appreciation rate
• Long term volatility rate

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

• Let ß 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

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

• Define the regime switching Esscher Transform By
• Thus the RadonNikodym derivative of the regime
  switching Esscher transform is given by

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

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

• Thus the RadonNikodym derivative is given by

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

• Applying Girsanovs theorem we obtain the
  following expression for the dynamics of S and d
  the risk neutral measure Q

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

• The conditional price of the variance swap P(X)
  is given by

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

• Hedging
• The Vega of the variance swap is given by

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

• The Vega of the volatility swap is given by

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

• Monte Carlo Experiment
• Two N2 regimes assumed corresponding to states
  of the econimy
• Regime 1 good state, regime 2 bad state
• 10,000 simulation runs
• 20 time steps

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

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

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

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