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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

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



52



• 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



53



• 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



54



• 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



55



-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



56



• 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



57



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



59



• 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



60



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



61



• 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



62



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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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