Figure 10.3: Implied BSM Volatility from Gram-Charlier Model Prices
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Allowing for Dynamic Volatility
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• The GC model is able to capture the strike price structure but not
the maturity structure in observed options prices
• We now consider option pricing allowing for the underlying asset
returns to follow a GARCH process
• The GARCH option pricing model assumes that the expected
return on the underlying asset is equal to the risk-free rate, rf , plus
a premium for volatility risk, λ , as well as a normalization term
• The observed daily return is then equal to the expected return plus
a noise term
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
Allowing for Dynamic Volatility
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The noise term is conditionally normally distributed with mean zero
and variance following a GARCH(1,1) process with leverage
• By letting past return feed into variance in a magnitude depending
on the sign of the return, the leverage effect creates an asymmetry in
distribution of returns
• This asymmetry is important for capturing the skewness implied in
observed option prices
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Allowing for Dynamic Volatility
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• Specifically, we can write the return process as
• Notice that the expected value and variance of tomorrow’s
return conditional on all the information available at time t
are
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Allowing for Dynamic Volatility
• For a generic normally distributed variable
and therefore we get
66
we have
• where we have used
• This expected return equation highlights the role of λ as the
price of volatility risk
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
Allowing for Dynamic Volatility
We can again solve for the option price using the risk-neutral
expectation as in
• Under risk neutrality, we must have that
• so that expected rate of return on risky asset equals riskfree rate and conditional variance under risk neutrality is
same as the one under original process
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
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Allowing for Dynamic Volatility
Consider the following process:
• Here we can check that the conditional mean equals
• which satisfies the first condition
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Allowing for Dynamic Volatility
• Furthermore, the conditional variance under the risk-neutral process equals
• where last equality comes from tomorrow’s variance
being known at the end of today in GARCH model
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
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Allowing for Dynamic Volatility
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The conclusion is that the conditions for a risk-neutral process are
met
• An advantage of the GARCH option pricing approach introduced
here is its flexibility
• The previous analysis could easily be redone for any of GARCH
variance models introduced in Chapter 4
• More important, it is able to fit observed option prices quite well
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Model Implementation
• While we have found a way to price the European option under
risk neutrality, we do not have a closed-form solution available.
• Instead, we have to use simulation to calculate price
• The simulation can be done as follows: First notice that
we can get rid of a parameter by writing
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Model Implementation
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• For a given conditional variance σ 2t+1 , and parameters
ω, α, β, λ *, we can use Monte Carlo simulation to create future
hypothetical paths of the asset returns
• We can illustrate the simulation of hypothetical daily returns from
day t+1 to the maturity on day
as
• where the
are obtained from a N(0,1) random number
generator and where MC is the number of simulated return
paths
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Model Implementation
73
• We need to calculate the expectation term E*t[*] in the option
pricing formula using the risk-neutral process, thus, we calculate
the simulated risk-neutral return in period t+j for simulation path i
as
• and the variance is updated by
• the simulation paths in the first period all start out
from the same σ2t+1; therefore, for all i, we have
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen