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Option Hedging Examples. 2-factor Hedging. Assume the IBM stock position from before. 100 shares of IBM covered by 1.32 call options. Remember slippage with only delta hedge (1.3% Stock Price change met with only .04% change in portfolio). Eliminate Slippage. Delta – Gamma hedge

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2 factor hedging
2-factor Hedging
  • Assume the IBM stock position from before.
  • 100 shares of IBM covered by 1.32 call options.
  • Remember slippage with only delta hedge (1.3% Stock Price change met with only .04% change in portfolio)
eliminate slippage
Eliminate Slippage
  • Delta – Gamma hedge
  • Stock: Delta = 1, Gamma = 0
  • Call Option: Delta = .7580, Gamma = .02944
  • Need additional option:
  • IBM 6-mo., X=80 call
  • Delta = .4035 Gamma = .03651
simultaneous equations
Simultaneous Equations
  • In general:

S Ns + C1 NC1 + C2 NC2 = 0

 ( S) Ns +  (C1) NC1 +  (C2) NC2 = 0,

where: S = 1, C1 = C1 , C2 = C2 ,

 ( S) = 0 ,  ( C1) = C1,  ( C2) = C2

  • Point is to solve for NC1 and NC2.
fill in and plug chug
Fill-In and Plug&Chug

1 Ns + 0.758 NC1 + .4035 NC2 = 0

0 Ns + 0.02944 NC1 + 0.03651 NC2 = 0

  • If we deal with Ns = 1, then

NC1 = -2.311 and

NC2 = +1.864

delta gamma hedge
Delta-Gamma Hedge
  • Thus, to hedge a long position in 100 shares of IBM at $75, and also insure the hedge will not detriorate,

Sell 2.311 IBM 6 mo. X=70 calls &

Buy 1.864 IBM 6 mo. X=80 calls

starting position
Starting Position

Long IBM (100 shares @ $75) 7500.00

Short X=70 calls (2.311@ $8.015) -1852.66

Long X=80 calls (1.864@ $2.829) 527.23

Total Cost of Position 6174.57

ibm 74
IBM = 74
  • Long IBM (100 shares @ $74) 7400.00
  • Short X=70 calls (2.311@$7.272) -1680.93
  • Long X=80 calls (1.864@$2.443) 455.41
  • Total Value of Position 6174.47
  • A change of $0.10 or 0.0017%
  • (Delta-only, change = $2.00 or 0.03%)