abhishreshthaa
Abhijeet S
Gamma
Gamma (G) is the rate of change of delta (D) with respect to the price of the underlying asset.Gamma addresses delta hedging errors caused by curvature.
A portfolio, hedged with the combined delta and gamma values eliminates the gamma errors incorporated in a delta neutral strategy.
The number of shares (n1) and the number of options (n2) for delta and
gamma hedge are chosen as under
1+n1Delta1+n2Delta2 = 0
n1Gamma1+n2Gamma2=0
Example of Delta and Gamma Hedge
Hedging a short position on a 3 month call option with delta=0.5948 and gamma=0.0026.
Hedge Instruments
Underlying Asset, delta=1, gamma=0 and 1 month call,
delta=0.5551,gamma=0.0046
Choose n1=number of share and n2= number of calls as under
Delta= -0.5948+n1x1+n2x0.5551=0
Gamma=-0.0026+n1x0+n2x0.0046=0
N1= 0.2810……………Long 0.2810 shares
N2 = 0.5652……………Long 0.5652 calls
Delta-Gamma Hedging
Delta, D, can be changed by taking a position in the underlying asset
To adjust gamma, G, it is necessary to take a position in an option or other derivative
Delta-Gamma Hedging Example
A traded call option on the same stock with X = 95 and T = 0.6 (Option 1) will
be used to make the portfolio Delta-Gamma neutral.
Price Delta Gamma
Option to be Hedged 4.2 0.434 0.027
Option 1 6.3 0.531 0.042
Let w1 be the number of options 1 in the portfolio
Gamma neutral implies
-1,0000 x 0.027 + w1 x 0.042 = 0
w1 = 270 / 0.025 = 6,428
Delta Gamma Hedging
Let ws be the number of shares in the portfolio
Delta neutral implies
-10,000 x 0.434 + ws + w1 x 0. 531= 0
ws = -4340 + 6428 x 0. 531 = -927
The Delta-Gamma neutral portfolio is
Short 10,000 calls
Short 3413 Shares
Long 6,428 options 1
Gamma (G) is the rate of change of delta (D) with respect to the price of the underlying asset.Gamma addresses delta hedging errors caused by curvature.
A portfolio, hedged with the combined delta and gamma values eliminates the gamma errors incorporated in a delta neutral strategy.
The number of shares (n1) and the number of options (n2) for delta and
gamma hedge are chosen as under
1+n1Delta1+n2Delta2 = 0
n1Gamma1+n2Gamma2=0
Example of Delta and Gamma Hedge
Hedging a short position on a 3 month call option with delta=0.5948 and gamma=0.0026.
Hedge Instruments
Underlying Asset, delta=1, gamma=0 and 1 month call,
delta=0.5551,gamma=0.0046
Choose n1=number of share and n2= number of calls as under
Delta= -0.5948+n1x1+n2x0.5551=0
Gamma=-0.0026+n1x0+n2x0.0046=0
N1= 0.2810……………Long 0.2810 shares
N2 = 0.5652……………Long 0.5652 calls
Delta-Gamma Hedging
Delta, D, can be changed by taking a position in the underlying asset
To adjust gamma, G, it is necessary to take a position in an option or other derivative
Delta-Gamma Hedging Example
A traded call option on the same stock with X = 95 and T = 0.6 (Option 1) will
be used to make the portfolio Delta-Gamma neutral.
Price Delta Gamma
Option to be Hedged 4.2 0.434 0.027
Option 1 6.3 0.531 0.042
Let w1 be the number of options 1 in the portfolio
Gamma neutral implies
-1,0000 x 0.027 + w1 x 0.042 = 0
w1 = 270 / 0.025 = 6,428
Delta Gamma Hedging
Let ws be the number of shares in the portfolio
Delta neutral implies
-10,000 x 0.434 + ws + w1 x 0. 531= 0
ws = -4340 + 6428 x 0. 531 = -927
The Delta-Gamma neutral portfolio is
Short 10,000 calls
Short 3413 Shares
Long 6,428 options 1