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Mean-Reverting Models in Financial and Energy Markets. Anatoliy Swishchuk Mathematical and Computational Finance Laboratory, Department of Mathematics and Statistics, U of C 5 th North-South Dialog, Edmonton, AB, April 30, 2005. Outline.

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Mean reverting models in financial and energy markets l.jpg

Mean-Reverting Models in Financial and Energy Markets

Anatoliy Swishchuk

Mathematical and Computational Finance Laboratory,

Department of Mathematics and Statistics, U of C

5th North-South Dialog,

Edmonton, AB, April 30, 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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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 a=365, 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-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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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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Stock ( (Volatility)a security representing partial ownership of a company)

Bonds (bank accounts)

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

Swaps

Security-a piece of paper representing a promise

Basic Securities

Derivative Securities


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Volatility swaps (Volatility) are forward contracts on future realized stock volatility

Variance swaps are forward contract on future realized stock variance

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


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Realized Continuous Variance and Volatility (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 (Volatility)

A Variance Swap is a forward contract on realized variance.

Its payoff at expiration is equal to (Kvaris the delivery price for variance and N is the notional amount in $ per annualized variance point)


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Volatility Swaps (Volatility)

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 (Volatility)SwapWork?


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Example: Payoff for Volatility and Variance Swaps (Volatility)

For Volatility Swap:

a) volatility increased to 21%:

Strike price Kvol =18% ; Realized Volatility=21%;

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 for (Volatility)Stochastic 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 E{V},

where

and


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Calculation E[V] (Volatility)


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Valuing of Volatility Swap (Volatility)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 E{V} (as inthe case of variance swap), but also Var{V}.


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Calculation of Var[V] (continuation) (Volatility)

After calculations:

Finally we obtain:


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Numerical Example 1: (Volatility)S&P60 Canada Index


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Numerical Example: S&P60 Canada Index (Volatility)

  • We apply the obtained analytical solutions to price a swap on the volatility of the S&P60 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 (Volatility)

Logarithmic returns are used in practice to define discrete sampled variance and volatility

Logarithmic Returns:

where



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Histograms of Log. Returns (1997-2002)for S&P60 Canada Index



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Realized Continuous Variance for (1997-2002)Stochastic 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 GARCH Model)

Main Features of this Model

  • Continuous-time analogue of GARCH(1,1)

  • Mean-reversion

  • Does not contain another Wiener process

  • Complete market

  • Incorporates theexpectation of log-return


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Valuing of Variance Swap for GARCH Model)Stochastic 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 E{V},

where

and



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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 Delay

We need to find EP*[Var(S)]:





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Numerical Example 3: S&P500 Canada Index(1990-1993)






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Explicit Option Pricing Formula for European Call Option under Physical Measure(assumption: W(phi_t^-1)-Gaussian?)


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Parameters: under Physical Measure


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Mean-Reverting Risk-Neutral Asset under Physical MeasureModel (MRRNAM)


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Transformations: under Physical Measure


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Explicit Solution for MRRNAM under Physical Measure



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Numerical Example: AECO Natural Gas Index under Risk-Neutral Measure(1 May 1998-30 April 1999)(Bos, Ware, Pavlov: Quantitative Finance, 2002)


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Variance for New Process W(phi_t^-1) under Risk-Neutral Measure


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Mean-Value for MRRNAM under Risk-Neutral Measure


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Mean-Value for MRRNAM under Risk-Neutral Measure


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Volatility for MRRNAM under Risk-Neutral Measure


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Price C(T) of European Call Option (S=1) under Risk-Neutral Measure(Sonny Kushwaha, Haskayne School of Business, U of C, (my student, AMAT371))


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European Call Option Price for MRM under Risk-Neutral Measure(Sonny Kushwaha, Haskayne School of Business, U of C, (my student, AMAT371))


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L. Bos, T. Ware (U of C) and Pavlov (U of Auckland, NZ) under Risk-Neutral Measure(Quantitative Finance, V. 2 (2002), 337-345)


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Comparison under Risk-Neutral Measure(approximation vs. explicit formula)


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Conclusions under Risk-Neutral Measure

  • 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 (for Heston model + models with delay)

  • We can price options for an asset in energy markets

  • Drawbacks: 1) one-factor models (L is a constant)

    2) W(phi_t^-1)-Gaussian process

  • Future work: 1) consider two-factor models: S (t) and L (t) (L->L (t)) (possibly with jumps) (analytical approach)

    2) 1) with probabilistic approach

    3) to study the process W(\phi_t^-1)


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Drawback of One-Factor Mean-Reverting Models under Risk-Neutral Measure

  • 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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Future work I. under Risk-Neutral Measure(Joint Working Paper with T. Ware: Analytical Approach (Integro - PDE), Whittaker functions)


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Future Work II under Risk-Neutral Measure(Probabilistic Approach: Change of Time Method).


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Acknowledgement under Risk-Neutral Measure

  • I’d like to thank very much to Robert Elliott, Tony Ware, Len Bos, Gordon Sick, 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.

  • I’d also like to thank very much to PIMS for partial support of this talk


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Thank you for your attention! under Risk-Neutral Measure


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