200 Mathematics for Finance
day. The values of the portfolio are given below (for comparison we also recall
the values of the delta neutral portfolio):
S( 3651 ) delta-gamma delta
58. 00 − 2. 04 − 71. 35
58. 50 0. 30 − 31. 56
59. 00 1. 07 − 3. 26
59. 50 0. 81 13. 69
60. 00 0. 02 19. 45
60. 50 − 0. 79 14. 22
61. 00 − 1. 11 − 1. 77
61. 50 − 0. 49 − 28. 24
62. 00 1. 52 − 64. 93We can see that we are practically safe within the given range of stock prices.
For larger changes we are also in a better position as compared with delta
hedging:
S( 3651 ) delta-gamma delta
50 − 614. 08 − 2 , 233. 19
55 − 78. 22 − 554. 65
60 0. 02 19. 45
65 63 , 13 − 481. 60
70 440. 81 − 1 , 765. 15
As predicted, a delta-gamma neutral portfolio offers better protection against
stock price changes than a delta neutral one.
Delta-Vega Hedging.Next we shall hedge against an increase in volatility,
while retaining cover against small changes in the stock price. This will be
achieved by constructing a delta-vega neutral portfolio containing, as before,
an additional option. The conditions imposed are
deltaV=x− 1 ,000 deltaCE+̂zdeltaĈE=0,
vegaV=− 1 ,000 vegaCE+̂zvegaĈE=0.They lead to the system of equations
x− 581 .957 + 0. 312373 ̂z=0,
− 1 , 1634 .305 + 8. 610681 ̂z=0,with an approximate solutionx∼= 159 .89,̂z∼= 1 , 351. 15 .The corresponding
money position isy∼=− 7 , 311. 12.