Figure 10.2: Implied BSM Daily Volatility from S&P 500 Index Options with 43, 99, 71 and 162 Days to Maturity (DTM) Quoted on January 06, 2010
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Allowing for Skewness and Kurtosis
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• We now try to make up for mispricing in BSM model
• We again have one day returns defined as
• and
-period returns as
• We now define skewness of the one-day return as
• Skewness is informative about the degree of asymmetry of
the distribution
• A negative skewness arises from large negative returns
being observed more frequently than large positive returns
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Allowing for Skewness and Kurtosis
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• Kurtosis of the one-day return is now defined as
• which is sometimes referred to as excess kurtosis due to
the subtraction by 3
• Kurtosis tells us about the degree of tail fatness in the
distribution of returns
• If large returns are more likely to occur in data than in the
normal distribution, then the kurtosis is positive
• Asset returns typically have positive kurtosis
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Allowing for Skewness and Kurtosis
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• Assuming that returns are independent over time, the skewness at
horizon can be written as a simple function of the daily
skewness,
• and correspondingly for kurtosis
• Notice that both skewness and kurtosis will converge to
zero as the return horizon, ; and thus the maturity of the
option increases
• This corresponds well with implied volatility in Figure
10.2, which displayed a more pronounced smirk pattern for
short term as opposed to long-term options
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Allowing for Skewness and Kurtosis
• We now define the standardized return at the
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-day horizon as:
• So that
• and assume that the standardized returns follow the
distribution given by the Gram-Charlier expansion, which
is written as
• where
j
and D is its jth derivative
is the standard normal density
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Allowing for Skewness and Kurtosis
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• We have
• The Gram-Charlier density function
is an expansion
around the normal density function,
, allowing for a
nonzero skewness,
, and kurtosis
• The Gram-Charlier expansion can approximate a wide
range of densities with nonzero higher moments, and it
collapses to the standard normal density when skewness
and kurtosis are both zero
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Allowing for Skewness and Kurtosis
• To price European options, we can again write the generic riskneutral call pricing formula as
• Thus, we must solve
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Allowing for Skewness and Kurtosis
58
• But we now instead define the standardized risk-neutral
return at horizon as
• and assume it follows Gram-Charlier (GC) distribution
• In this case, the call option price can be derived as
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Allowing for Skewness and Kurtosis
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• where we have substituted in for skewness using
a
and for kurtosis using
• The approximation comes from setting the terms involving σ 3 and
σ 4 to zero, which also enables us to use the definition of d from the
BSM model
• Using this approximation, the GC model is just the simple BSM
model plus additional terms that vanish if there is neither skewness
nor kurtosis in the data
• The GC formula can be extended to allow for a cash flow q in the
same manner as the BSM formula shown earlier
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
Model Implementation
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GC model has 3 unknown parameters: σ , ζ 11 and ζ 21
• They can be estimated using a numerical optimizer minimizing the
mean squared error
• We can calculate the implied BSM volatilities from the GC
model prices by
• Where c-1BSM(∗) is the inverse of the BSM model with
respect to volatility
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Model Implementation
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• But we can also rely on the following approximate formula for
daily implied BSM volatility:
• Notice this is just volatility times an additional term, which
equals one if there is no skewness or kurtosis
• Main advantages of GC option pricing framework are
– It allows for deviations from normality,
– It provides closed-form solutions for option prices
– It is able to capture the systematic patterns in implied
volatility found in observed option data
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Figure 10.3: Implied BSM Volatility from GramCharlier Model Prices
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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