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Figure 11.5: The Gamma of an Option

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



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Figure 11.5: The Gamma of an Option

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