108156.pdf

296 Mathematics for Finance

One day later the shorted shares will be worth 17, 765 × 1 .81 = 32, 154. 64

dollars, whereas the cash investment will grow to 33, 914 .69e^0.^05 /^365 ∼= 33 , 919. 34

dollars. The put price will increase to 0.035182 dollars, so the price of 50, 000

puts will be 1, 759 .11 dollars. The value of the delta neutral portfolio will be

− 32 , 154 .64 + 33, 919. 34 − 1 , 759. 11 ∼= 5 .59 dollars.

9.4The price of a single put after one day will now be 0.038885 dollars, the 50, 000

options sold will therefore be worth 1944.26 dollars, the stock and cash deposit

positions remaining as in Solution 9.3. The delta neutral portfolio will bring

a loss of 179.56 dollars.

9.5If the stock price does not change,S(t)=S(0) =S, then the value of the

portfolio after timetwill be given by

V(t)=SN(d 1 )−Xerte−rTN(d 2 )−CE(S, t),

whereCE(S, t) is given by the Black–Scholes formula andd 1 ,d 2 by (8.9). Then

d

dtV(t)

∣∣

∣∣

t=0

=−rXe−rTN(d 2 )−ddtCE(S, t)

∣∣

∣∣

t=0

=−rXe−rTN(d 2 )−thetaCE

= σS

2

√

2 πT

e−d

(^21) / 2
,
which is positive.
9.6Using put-call parity and the Greek parameters for a call, we can find those
for a put:
deltaPE=N(d 1 )−1=deltaCE−1=−N(−d 1 ),
gammaPE= gammaCE,
thetaPE=− Sσ
2
√
2 πT
e−
d^21
(^2) +rXe−rTN(−d 2 ),
vegaPE=vegaCE,
rhoPE=−TXe−rTN(−d 2 ).
(The Greek parameters are computed at timet= 0.) These equalities can also
be verified directly by differentiating the Black–Scholes formula for the put
price.
9.7The rho of the original option is 7.5878, the delta of the additional option is
0 .4104 and the rho is 7.1844. The delta-rho neutral portfolio requires buying
approximately 148.48 shares of stock and 1, 056 .14 additional options, while
borrowing $7, 693 .22. The position after one day is presented in the following
table, in which we also recall the results of the delta hedge:
S( 3651 ) delta-rho delta
r=8% r=9% r= 15% r=9%
58. 00 − 7. 30 − 9. 65 − 26. 14 − 133. 72
58. 50 − 2. 71 − 4. 63 − 17. 95 − 97. 22
59. 00 0. 18 − 1. 23 − 10. 93 − 72. 19
59. 50 1. 59 0. 77 − 4. 85 − 58. 50
60. 00 1. 76 1. 60 0. 52 − 55. 96
60. 50 0. 92 1. 50 5. 45 − 64. 38
61. 00 − 0. 68 0. 72 10. 16 − 83. 51
61. 50 − 2. 78 − 0. 47 14. 90 − 113. 07
62. 00 − 5. 13 − 1. 84 19. 91 − 152. 78