108156.pdf

Solutions 295

8.14Consider the distribution function

F(x)=P∗{W(t)<x}=P∗

{

V(t)<x+

(

m−r+^12 σ^2

)t

σ

}

=

∫x+(m−r+ (^12) σ (^2) )σt
−∞
√^1
2 π
e−
y 22
dy,
whereV(t)=W(t)+
(
m−r+^12 σ^2
)t
σis normally distributed underP∗.As
a result, the density ofW(t) underP∗is
dF(x)
dx =
√^1
2 π
e−
(^12) (x+(m−r+ (^12) σ (^2) )σt)^2
.
8.15By put-call parity, fort=0
PE(0) =CE(0)−S(0) +Xe−rT
=S(0)(N(d 1 )−1)−Xe−rT(N(d 2 )−1)
=−S(0)N(−d 1 )+Xe−rTN(−d 2 ).
Now, by substitutingtfor 0 andT−tforT, we obtain the Black–Scholes
formula forPE(t).

Chapter 9

9.1By put-call parity (7.1)

d

dSP

E(S)=d

dSC

E(S)−1=N(d 1 )−1=−(1−N(d 1 )) =−N(−d 1 ),

whered 1 is given by (8.9). The delta of a put option is negative, consistently

with the fact that the value of a put option decreases as the price of the

underlying asset increases.

9.2We maximise 581. 96 ×S− 30 , 779. 62 − 1 , 000 ×CE(S, 3651 ), whereSstands for

the stock price after one day, andCE(S, t) is the price of a call at timet, one

day in our case, with 89 days to maturity, and whereσ= 30% andr=8%,

as before.Equating the derivative with respect toSto zero, we infer that the

delta of the option after one day should be the same as the delta on day zero,d

dSC

E(S, 1

365 )=0.58196. This gives the following condition for the stock price

(after inverting the normal distribution function):

ln 60 S+(r+^12 σ^2 )× 36589

σ

√ 89

365

=

ln^6060 +(r+^12 σ^2 )× 36590

σ

√ 90

365

.

The result isS∼= 60 .0104 dollars.

9.3The premium for a single put is 0.031648 dollars (from the Black–Scholes

formula), so the bank will receive 1, 582 .40 dollars by writing and selling 50, 000

puts. The delta of a single put is− 0 .355300, so the delta-hedging portfolio

requires shorting 17, 765 .00 shares, which will raise 32, 332 .29 dollars. This

gives a total of 33, 914 .69 dollars received to be invested at 5%. The value of

the delta neutral portfolio consisting of the shored stock, invested cash and

sold options will be− 32 , 332 .29 + 33, 914. 69 − 1 , 582 .40 = 0.00 dollars.