Figure 11.5: The Gamma of an Option
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The Option Gamma
37
• When option is at-the-money, the gamma is relatively large
and when option is deep out-of-the-money or deep in-themoney gamma is relatively small
• This is because the nonlinearity of the option price is highest
when the option is close to at-the-money
• Deep in-the-money call option prices move virtually onefor-one with the price of the underlying asset because the
options will almost surely be exercised
• Deep out-of-the-money options will almost surely not be
exercised, and they are therefore worthless regardless of
changes in the underlying asset price.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
The Option Gamma
38
• For these options, the linear delta-based model can be
highly misleading
• Finally, we note that gamma can be computed using
binomial trees as well
• The formula used for gamma in the tree is simply
• and it is based on the change in the delta from point B to C
in the tree divided by the average change in the stock price
when going from points B and C
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Portfolio Risk Using Gamma
39
• In the previous delta-based model, when considering a
portfolio consisting of options on one underlying asset, we
have
• where denotes the weighted sum of the deltas on all the
individual options in the portfolio
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
40
Table 11.2: Gamma of American Put Option
Market Variables
St=
1000
D
Annual r =
0.05
1528.47
Contract Terms
0.00
X=
1100
T=
0.25
B
1236.31
Parameters
-0.19
Annual Vol=
0.6
tree steps =
2
dt=
0.125
53.48
A
E
1000.00
u=
1.236311
1000.00
d=
0.808858
-0.56
RNP =
0.461832
180.25
100.00
0.001855
Stock is black
American Put Delta is Green
American Put Price is Red
American Put Gamma is Blue
C
808.86
-1.00
291.14
F
654.25
445.75
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
Portfolio Risk Using Gamma
41
When incorporating the second derivative, gamma, we
instead rely on the quadratic approximation
• where the portfolio and are calculated as
• where again mj denotes the number of option contract j in
the portfolio
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen