Download Mean-Reverting Models in Financial and Energy Markets

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
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 MeanReverting Stochastic Processes
Heston Model for Stock Price and
Variance
Model for Stock Price (geometric Brownian motion):
or
deterministic interest rate,
follows Cox-Ingersoll-Ross (CIR) process
Standard Brownian Motion and
Geometric Brownian Motion
Standard Brownian motion
Geometric Brownian motion
Heston Model: Variance follows meanreverting (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 points
towards the current mean value .
Swaps
Security-a piece of paper representing a promise
Basic Securities
• Stock (a security
representing partial
ownership of a
company)
• Bonds (bank
accounts)
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
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
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
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.
Valuing of Variance Swap for
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
Calculation E[V]
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 E{V} (as in the
case of variance swap), but also Var{V}.
Calculation of Var[V]
(continuation)
After calculations:
Finally we obtain:
Numerical Example 1:
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 1997February 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 are used in practice to define discrete
sampled variance and volatility
Logarithmic Returns:
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
Stock Price
Initial Data
deterministic function
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 for
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
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 2: 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 3: 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
(assumption: W(phi_t^-1)-Gaussian?)
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 Process
W(phi_t^-1)
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))
1
0.9
0.8
0.7
C(T)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
T
0.6
0.7
0.8
0.9
1
European Call Option Price for MRM
(Sonny Kushwaha, Haskayne School of Business, U of
C, (my student, AMAT371))
L. Bos, T. Ware (U of C) and Pavlov (U of Auckland, NZ)
(Quantitative Finance, V. 2 (2002), 337-345)
Comparison
(approximation vs. explicit formula)
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 (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)
Drawback of One-Factor MeanReverting 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)
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,
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
Thank you for your
attention!