Mean-Reverting Models in

Financial and Energy Markets

Anatoliy SwishchukMathematical and

Computational Finance Laboratory,

Department of Mathematics and Statistics, U of C

ColloquiumThursday, March 31, 2005

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

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

The Mean-Reversion Deterministic Process

Mean-Reverting Plot (a=4.6,L=2.5)

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

Mean-Reversion Stochastic Process

Mean-Reverting Models in Financial Markets

• Stock (asset) Prices follow geometric Brownian motion

• The Variance of Stock Price follows Mean-Reverting Models

Mean-Reverting Models in Energy Markets

• Asset Prices follow Mean-Reverting Stochastic Processes

Heston Model for Stock Price and Variance

Model for Stock Price (geometric Brownian motion):

or

follows Cox-Ingersoll-Ross (CIR) process

deterministic interest rate,

Standard Brownian Motion andGeometric Brownian Motion

Standard Brownian motion

Geometric Brownian motion

Heston Model: Variance follows mean-reverting (CIR) process

or

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 pointstowards the current mean value .

Swaps

• Stock (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

Basic Securities Derivative Securities

Security-a piece of paper representing a promise

Variance and Volatility Swaps

• Volatility swaps are forward contracts on future realized stock volatility

• Variance swaps are forward contract on future realized stock variance

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

Realized Continuous Variance and Volatility

Realized (or Observed) Continuous Variance:

Realized Continuous Volatility:

where is a stock volatility, is expiration date or maturity.

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)

Volatility Swaps

A Volatility Swap is a forward contract on realized volatility.

Its payoff at expiration is equal to:

How does the Volatility Swap Work?

Example: Payoff for Volatility and Variance Swaps

Kvar = (18%)^2; N = $50,000/(one volatility point)^2.

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

N =$50,000/(volatility point).

Payment(HF to D)=$50,000(21%-18%)=$150,000.

For Volatility Swap:

For Variance Swap:

Payment(D to HF)=$50,000(18%-12%)=$300,000.

b) volatility decreased to 12%:

a) volatility increased to 21%:

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 E{V},

where and

Calculation E[V]

Valuing of Volatility Swap for Stochastic Volatility

Value of volatility swap:

To calculate volatility swap we need not only E{V} (as in the case of variance swap), but also Var{V}.

We use second order Taylor expansion for square root function.

Calculation of Var[V] (continuation)

After calculations:

Finally we obtain:

Numerical Example:S&P60 Canada Index

Numerical Example: S&P60 Canada Index

• 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)

Logarithmic Returns

Logarithmic Returns:

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

where

Statistics on Log-Returns of S&P60 Canada Index for 5 years

(1997-2002)

Histograms of Log. Returns for S&P60 Canada Index

S&P60 Canada Index Volatility Swap

Realized Continuous Variance for

Stochastic Volatility with Delay

Initial Data

deterministic function

Stock Price

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

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

Valuing of Variance Swap forStochastic Volatility with Delay

Value of Variance Swap (present value):

To calculate variance swap we need only E{V},

where and

where E is an expectation (or mean value), r is interest rate.

Continuous-Time GARCH Model

Deterministic Equation for Expectation of Variance with Delay

There is no explicit solution for this equation besides stationary solution.

Valuing of Variance Swap with Delay in General Case

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

Numerical Example 1: S&P60 Canada Index (1997-2002)

Dependence of Variance Swap with Delay

on Maturity (S&P60 Canada Index)

Variance Swap with Delay (S&P60 Canada Index)

Numerical Example 2: S&P500 (1990-1993)

Dependence of Variance Swap with Delay on Maturity (S&P500)

Variance Swap with Delay (S&P500 Index)

Mean-Reverting Models in Energy Markets

Explicit Solution for MRAM

Explicit Option Pricing Formula for European Call Option under Physical Measure

Parameters:

Mean-Reverting Risk-Neutral Asset Model (MRRNAM)

Transformations:

Explicit Solution for MRRNAM

Explicit Option Pricing Formula for European Call Option under Risk-Neutral Measure

Numerical Example: AECO Natural Gas Index (1 May 1998-30 April 1999)

(Bos, Ware, Pavlov: Quantitative Finance, 2002)

Variance for New Gaussian Process

Mean-Value for MRRNAM

Mean-Value for MRRNAM

Volatility for MRRNAM

Price C(T) of European Call Option (S=1)(Sonny Kushwaha, Haskayne School of Business, U of

C, (my student, AMAT371))

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T

C(T

)

European Call Option Price for MRM (Sonny Kushwaha, Haskayne School of Business, U of

C, (my student, AMAT371))

L. Bos, T. Ware and Pavlov (Put Option)(Quantitative Finance, V. 2 (2002), 337-345)

Comparison (Put vs. Call)

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)

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->L (t)) (possibly with jumps)

Future work I.(Joint Working Paper with T. Ware: Analytical Approach (Integro - PDE),

Whittaker functions)

Future Work II

(Probabilistic Approach: Change of Time Method).

Acknowledgement

• I’d 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.

Thank you for your attention!