Tải bản đầy đủ - 0 (trang)
Figure 11.5: The Gamma of an Option

# Figure 11.5: The Gamma of an Option

Tải bản đầy đủ - 0trang

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

• 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

• 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

### Tài liệu bạn tìm kiếm đã sẵn sàng tải về

Figure 11.5: The Gamma of an Option

Tải bản đầy đủ ngay(0 tr)

×