Mean-Reverting Models in Financial and Energy Markets

1
Mean-Reverting Models in Financial and Energy
Markets

• Anatoliy Swishchuk
• Mathematical and Computational Finance
  Laboratory,
• Department of Mathematics and Statistics, U of C
• Colloquium
• Thursday, March 31, 2005

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Outline

• Mean-Reverting Models (MRM) Deterministic vs.
  Stochastic
• MRM in Finance Variances (Not Asset Prices)
• MRM in Energy Markets Asset Prices
• Some Results Swaps, Swaps with Delay, Option
  Pricing Formula (one-factor models)
• Drawback of One-Factor Models
• Future Work

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Mean-Reversion Effect

• Violin (or Guitar) String Analogy if we pluck
  the violin (or guitar) string, the string will
  revert to its place of equilibrium
• To measure how quickly this reversion back to the
  equilibrium location would happen we had to pluck
  the string
• Similarly, the only way to measure mean reversion
  is when the variances of asset prices in
  financial markets and asset prices in energy
  markets get plucked away from their non-event
  levels and we observe them go back to more or
  less the levels they started from

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The Mean-Reversion Deterministic Process
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Mean-Reverting Plot (a4.6,L2.5)
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Meaning of Mean-Reverting Parameter

• The greater the mean-reverting parameter value,
  a, the greater is the pull back to the
  equilibrium level
• For a daily variable change, the change in time,
  dt, in annualized terms is given by 1/365
• If a365, the mean reversion would act so quickly
  as to bring the variable back to its equilibrium
  within a single day
• The value of 365/a gives us an idea of how
  quickly the variable takes to get back to the
  equilibrium-in days

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Mean-Reversion Stochastic Process
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Mean-Reverting Models in Financial Markets

• Stock (asset) Prices follow geometric Brownian
  motion
• The Variance of Stock Price follows
  Mean-Reverting Models

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Mean-Reverting Models in Energy Markets

• Asset Prices follow Mean-Reverting Stochastic
  Processes

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Heston Model for Stock Price and Variance
Model for Stock Price (geometric Brownian motion)
or
deterministic interest rate,
follows Cox-Ingersoll-Ross (CIR) process
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Standard Brownian Motion andGeometric Brownian
Motion
Standard Brownian motion
Geometric Brownian motion
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Heston Model Variance follows mean-reverting
(CIR) process
or
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Cox-Ingersoll-Ross (CIR) Model for Stochastic
Variance (Volatility)
The model is a mean-reverting process, which
pushes away from zero to keep it positive.
The drift term is a restoring force which always
points towards the current mean value .
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Swaps
Security-a piece of paper representing a promise
Basic Securities
Derivative Securities

• Option (right but not obligation to do something
  in the future)
• Forward contract (an agreement to buy or sell
  something in a future date for a set price
  obligation)
• Swaps-agreements between two counterparts to
  exchange cash flows in the future to a prearrange
  formula obligation

• Stock (a security representing partial ownership
  of a company)
• Bonds (bank accounts)

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Variance and Volatility Swaps
Forward contract-an agreement to buy or sell
something at a future date for a set price
(forward price)
Variance is a measure of the uncertainty of a
stock price.
Volatility (standard deviation) is the square
root of the variance (the amount of noise, risk
or variability in stock price)
Variance(Volatility)2

• Volatility swaps are forward contracts on future
  realized stock volatility

• Variance swaps are forward contract on future
  realized stock variance

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Realized Continuous Variance and Volatility
Realized (or Observed) Continuous Variance
Realized Continuous Volatility
where is a stock volatility,
is expiration date or maturity.
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Variance Swaps
A Variance Swap is a forward contract on realized
variance. Its payoff at expiration is equal to
(Kvar is the delivery price for variance and N is
the notional amount in per annualized variance
point)
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Volatility Swaps
A Volatility Swap is a forward contract on
realized volatility. Its payoff at expiration
is equal to
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How does the Volatility Swap Work?
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Example Payoff for Volatility and Variance Swaps
For Volatility Swap
a) volatility increased to 21
Strike price Kvol 18 Realized Volatility21
N 50,000/(volatility point). Payment(HF
to D)50,000(21-18)150,000.
b) volatility decreased to 12
Payment(D to HF)50,000(18-12)300,000.
For Variance Swap
Kvar (18)2 N 50,000/(one volatility
point)2.
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Valuing of Variance Swap forStochastic Volatility
Value of Variance Swap (present value)
where E is an expectation (or mean value), r is
interest rate.
To calculate variance swap we need only EV,
where
and
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Calculation EV
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Valuing of Volatility Swap for Stochastic
Volatility
Value of volatility swap
We use second order Taylor expansion for square
root function.
To calculate volatility swap we need not only
EV (as in the case of variance swap), but also
VarV.
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Calculation of VarV (continuation)
After calculations
Finally we obtain
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Numerical ExampleSP60 Canada Index
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Numerical Example SP60 Canada Index

• We apply the obtained analytical solutions to
  price a swap on the volatility of the SP60
  Canada Index for five years (January
  1997-February 2002)
• These data were kindly presented to author by
  Raymond Theoret (University of Quebec,
• Montreal, Quebec,Canada) and Pierre Rostan
  (Bank of Montreal, Montreal, Quebec,Canada)

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Logarithmic Returns
Logarithmic returns are used in practice to
define discrete sampled variance and volatility
Logarithmic Returns
where
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Statistics on Log-Returns of SP60 Canada Index
for 5 years (1997-2002)
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Histograms of Log. Returns for SP60 Canada Index
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SP60 Canada Index Volatility Swap
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Realized Continuous Variance forStochastic
Volatility with Delay
Stock Price
Initial Data
deterministic function
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Equation for Stochastic Variance with Delay
(Continuous-Time GARCH Model)
Our (Kazmerchuk, Swishchuk, Wu (2002) The Option
Pricing Formula for Security Markets with
Delayed Response) first attempt was
This is a continuous-time analogue of its
discrete-time GARCH(1,1) model
J.-C. Duan remarked that it is important to
incorporate the expectation of log-return into
the model
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Stochastic Volatility with Delay

• Main Features of this Model
• Continuous-time analogue of GARCH(1,1)
• Mean-reversion
• Does not contain another Wiener process
• Complete market
• Incorporates the expectation of log-return

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Valuing of Variance Swap forStochastic
Volatility with Delay
Value of Variance Swap (present value)
where E is an expectation (or mean value), r is
interest rate.
To calculate variance swap we need only EV,
where
and
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Continuous-Time GARCH Model
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Deterministic Equation for Expectation of
Variance with Delay
There is no explicit solution for this equation
besides stationary solution.
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Valuing of Variance Swap with Delay in General
Case
We need to find EPVar(S)
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Numerical Example 1 SP60 Canada Index
(1997-2002)
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Dependence of Variance Swap with Delay on
Maturity (SP60 Canada Index)
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Variance Swap with Delay (SP60 Canada Index)
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Numerical Example 2 SP500 (1990-1993)
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Dependence of Variance Swap with Delay on
Maturity (SP500)
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Variance Swap with Delay (SP500 Index)
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Mean-Reverting Models in Energy Markets
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Explicit Solution for MRAM
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Explicit Option Pricing Formula for European Call
Option under Physical Measure
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Parameters
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Mean-Reverting Risk-Neutral Asset Model (MRRNAM)
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Transformations
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Explicit Solution for MRRNAM
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Explicit Option Pricing Formula for European Call
Option under Risk-Neutral Measure
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Numerical Example AECO Natural Gas Index (1
May 1998-30 April 1999)(Bos, Ware, Pavlov
Quantitative Finance, 2002)
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Variance for New Gaussian Process
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Mean-Value for MRRNAM
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Mean-Value for MRRNAM
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Volatility for MRRNAM
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Price C(T) of European Call Option (S1)(Sonny
Kushwaha, Haskayne School of Business, U of C,
(my student, AMAT371))
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European Call Option Price for MRM (Sonny
Kushwaha, Haskayne School of Business, U of C,
(my student, AMAT371))
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L. Bos, T. Ware and Pavlov (Put
Option)(Quantitative Finance, V. 2 (2002),
337-345)
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Comparison (Put vs. Call)
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Drawback of One-Factor Mean-Reverting Models

• The long-term mean L remains fixed over time
  needs to be recalibrated on a continuous basis in
  order to ensure that the resulting curves are
  marked to market
• The biggest drawback is in option pricing
  results in a model-implied volatility term
  structure that has the volatilities going to zero
  as expiration time increases (spot volatilities
  have to be increased to non-intuitive levels so
  that the long term options do not lose all the
  volatility value-as in the marketplace they
  certainly do not)

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Conclusions

• Variances of Asset Prices in Financial Markets
  follow Mean-Reverting Models
• Asset Prices in Energy Markets follow
  Mean-Reverting Models
• We can price variance and volatility swaps for an
  asset in financial markets
• We can price options for an asset in energy
  markets
• Drawback one-factor models (L is a constant)
• Future work consider two-factor models S (t)
  and L (t) (L-gtL (t)) (possibly with jumps)

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Future work I.(Joint Working Paper with T.
Ware Analytical Approach (Integro - PDE),
Whittaker functions)
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Future Work II (Probabilistic Approach Change
of Time Method).
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Acknowledgement

• Id like to thank very much to Robert Elliott,
  Gordon Sick, Tony Ware and Graham Weir for
  valuable suggestions and comments, and to all the
  participants of the Lunch at the Lab (weekly
  seminar, usually Each Thursday, at the
  Mathematical and Computational Finance
  Laboratory) for discussion and remarks during all
  my talks in the Lab.

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Thank you for your attention!