Distribution of option payoffs
Long position in call option (Long Gamma)
Long positions in options (calls or puts) have positive gamma and hence have a
long right tail.
The distribution of long option positions has positive skew.
Short position in call option (Short Gamma)
Short positions in options (calls or puts) have negative gamma and hence have a
long left tail.
The distribution of returns for short options positions has negative skew.
Measuring risk from options positions:
The iid normal model for the returns of the underlying for “short” horizons, h:
x  ln  t  h
 St
x ~ N ( ,  2 )
 0
Value at Risk given confidence level, 1-p and horizon h:
VaR in return form:
x*      h
VaR in levels simple (dS):
dS *      h  S t
or continuous
dS *  S t   e x  1
  pth - quantile from standard normal
  annualized volatility ln(St+h/St)
h  risk measurement horizon in years.
f t  f ( S t , rt , rt* ,  t , K ,  )
: ft = value of derivative written on S
St = price of underlying at time t
rt = annual continuously compounded risk free rate from t till T
r*t = annual continuously compounded yield on underlying
t = annualized standard deviation of logarithmic change in underlying’s price
K = strike price
 = T – t = time to expiration
df 
2 f
 dS  12
 (dS ) 2 
 dr 
 dr * 
 d 
 d  Rn
S 2
r *
Two approximations to VaR for option positions based on simple transformations of dS
Linear approximation – Delta approximation:
VaR(df ) 
 VaR(dS )    VaR(dS )
Quadratic approximation – Delta-Gamma approximation:
1 2 f
 VaR(dS )  
 VaR(dS )2
2 S 2
VaR(df )    VaR(dS )     VaR(dS )2
VaR(df ) 
Where  is the absolute value of the position’s delta and  is positive for long
option positions and of negative sign for short option positions.
There are two approaches commonly used to improve on the methods based on simple
transformations of dS.
Monte Carlo simulation – very accurate but computationally expensive. Many trials
required to accurately estimate a quantile in the tail of the predictive distribution.
Analytical approximation of the position’s change in value – The Johnson transformation
and the Cornish-Fisher transformation are examples of this method.
See: “Delta-Gamma Four Ways,” by Jorge Mina and Andrew Ulmer, Rismetrics Group,
LLC, August 31, 1999 for a numerical evaluation of alternative approximation methods.
To analytically approximate characteristics (quantiles) of the distribution of a position’s
value change also starts from the Taylor series approximation
Given the assumed distribution for changes in the risk factor(s) affecting the position’s
value, the moments of the position’s change in value distribution are developed.
These moments can be used for Monte-Carlo simulation of the relevant quantile of the
position’s change in value distribution.
Or analytical approximations can be implemented to construct the relevant quantile of the
position’s change in value distribution.
To contrast simple and analytical methodologies, the Delta, Delta-Gamma and CornishFisher analytical approximation are applied to evaluate the risk exposure from a long
position in a call option.
S = price of underlying asset
V = value of a position that contains options written on S.
q = vector of quantities of the position
V = f(S, , T, r, K, q)
Step 1: Use a Taylor series expansion to describe the changes in position value in terms
of the price of the underlying asset.
dV  VS dS  12  V2 dS 2  ....... +additional terms in expansion
The delta-gamma approach uses only terms of second order or lower.
Substituting for dS = S*(dS/S)=S*X and the option’s delta and gamma:
X = logarithmic change in price of underlying assumed normal distributed.
dV  qSX  1 qGS 2 X 2 or Y = aX + bX2
a= qS
b= 1 qGS 2
The next step is to develop the moments of the position’s change in value from this
expression. Handwritten notes using standardized moments of a normal distribution.
sb = volatility of X
first = b*sb^2;
second = a^2*sb^2 + 3*b^2*sb^4;
third = 9*a^2*b*sb^4 + 15*b^3*sb^6;
fourth = 3*a^4*sb^4 + 90*a^2*b^2*sb^6 + 105*b^4*sb^8;
The next step is to develop the mean, standard deviation, skewness, excess kurtosis, etc.
of the change in position value from the moments of the distribution of change in position
skew=(third-3*second.*uv + 2*uv.^3)./((varv.^0.5).^3)
kurtosis=(fourth-5*third.*uv + 3*second.*uv.^2 + uv.^4)./((varv.^0.5).^4)-3
The next step is to apply the Cornish-Fisher transformation to obtain quantiles (0.05) of
the distribution of the position’s change in value. (Johnson and Kotz 1970, Distributions
in Statistics: Continuous Univariate Distributions pg. 33-35)
wq=z + (1/6)*((z^2)-1)*skew +(1/24)*((z^3)-3*z)*kurtosis…
Hull (2002) Options, Futures and Other Derivatives, 5th edition, pg. 370 illustrates a
correction for skewness only:
wq2 = z + (1/6)*((z^2)-1)*skew;
After obtaining the “adjusted” quantiles, state VAR in terms of the adjusted quantiles and
the standard deviation of the distribution of change in position value.
VARS=-(wq2.*varv.^0.5 + uv);
VARSK=-(wq.*varv.^0.5 + uv);
% correcting for skewness of dV
% correcting for skewness and kurtosis of dV

Notes 12