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Option Hedging Examples

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

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Eliminate Slippage Delta – Gamma hedge Delta – Gamma hedge Stock: Delta = 1, Gamma = 0 Stock: Delta = 1, Gamma = 0 Call Option: Delta =.7580, Gamma = Call Option: Delta =.7580, Gamma = Need additional option: Need additional option: IBM 6-mo., X=80 call IBM 6-mo., X=80 call Delta =.4035Gamma = Delta =.4035Gamma =.03651

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Simultaneous Equations In general: In general: S N s + C 1 N C1 + C 2 N C2 = 0 S N s + C 1 N C1 + C 2 N C2 = 0 ( S) N s + ( C 1 ) N C1 + ( C 2 ) N C2 = 0, ( S) N s + ( C 1 ) N C1 + ( C 2 ) N C2 = 0, where: S = 1, C 1 = C1, C 2 = C2, ( S) = 0, ( C 1 ) = C1, ( C 2 ) = C2 ( S) = 0, ( C 1 ) = C1, ( C 2 ) = C2 Point is to solve for N C1 and N C2. Point is to solve for N C1 and N C2.

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Fill-In and Plug&Chug 1 N s N C N C2 = 0 1 N s N C N C2 = 0 0 N s N C N C2 = 0 0 N s N C N C2 = 0 If we deal with N s = 1, then If we deal with N s = 1, then N C1 = and N C1 = and N C2 = N C2 =

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Delta-Gamma Hedge Thus, to hedge a long position in 100 shares of IBM at $75, and also insure the hedge will not detriorate, Thus, to hedge a long position in 100 shares of IBM at $75, and also insure the hedge will not detriorate, Sell IBM 6 mo. X=70 calls & Buy IBM 6 mo. X=80 calls

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Starting Position Long IBM (100 $75) Short X=70 calls $8.015) Long X=80 calls $2.829) Total Cost of Position

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IBM = 74 Long IBM (100 $74) Long IBM (100 $74) Short X=70 calls Short X=70 calls Long X=80 calls Long X=80 calls Total Value of Position Total Value of Position A change of $0.10 or % A change of $0.10 or % (Delta-only, change = $2.00 or 0.03%) (Delta-only, change = $2.00 or 0.03%)

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