1. Option Hedging Examples

2. 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)

3. Eliminate Slippage Delta – Gamma hedge Delta – Gamma hedge Stock: Delta = 1, Gamma = 0 Stock: Delta = 1, Gamma = 0 Call Option: Delta =.7580, Gamma =.02944 Call Option: Delta =.7580, Gamma =.02944 Need additional option: Need additional option: IBM 6-mo., X=80 call IBM 6-mo., X=80 call Delta =.4035Gamma =.03651 Delta =.4035Gamma =.03651

4. 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.

5. Fill-In and Plug&Chug 1 N s + 0.758 N C1 +.4035 N C2 = 0 1 N s + 0.758 N C1 +.4035 N C2 = 0 0 N s + 0.02944 N C1 + 0.03651 N C2 = 0 0 N s + 0.02944 N C1 + 0.03651 N C2 = 0 If we deal with N s = 1, then If we deal with N s = 1, then N C1 = -2.311 and N C1 = -2.311 and N C2 = +1.864 N C2 = +1.864

6. 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 2.311 IBM 6 mo. X=70 calls & Buy 1.864 IBM 6 mo. X=80 calls

7. 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 Position6174.57

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