108156.pdf

266 Mathematics for Finance

1.9We need to find an investment intoxshares andybonds replicating the put

option, that is, such thatxS(1) +yA(1) =P(1),no matter whether the stock

price goes up or down. This leads to the system of equations

{

x120 +y110 = 0,

x80 +y110 = 20.

The solution isx=−^12 andy= 116. To replicate the put option we need to

take a short position of^12 a share in stock and to buy 116 of a bond. The value

of this investment at time 0 is

xS(0) +yA(0) =−^12 ×100 + 116 × 100 ∼= 4. 55

dollars. By a similar argument as in Proposition 1.3, it follows thatxS(0) +

yA(0) =P(0), or else an arbitrage opportunity would arise. Therefore, the

price of the put must beP(0)∼= 4 .55 dollars.

1.10The investor will buy^500100 = 5 shares and 13500. 6364 ∼= 36 .6667 options. Her final

wealth will then be 5×S(1) + 36. 6667 ×C(1), that is, 5×120 + 36. 6667 × 20 ∼=

1 , 333 .33 dollars if the price of stock goes up to $120, or 5×80 + 36. 6667 × 0 ∼=

400 .00 dollars if it drops to $80.

1.11a) Ifp=0.25, then the standard deviation of the return is about 52% when

no option is purchased and about 26% with the option.

b) Ifp=0.5, then the standard deviation of the return is about 60% and 30%,

respectively.

c) Forp=0.75 the standard deviation of the return is about 52% and 26%,

respectively.

1.12The standard deviation of a random variable taking valuesaandbwith

probabilitiespand 1−p, respectively, is|a−b|

√

p(1−p). If no option is in-

volved, then the return on stock will be 60% or−60%,depending on whether

stock goes up or down. In this case|a−b|=|60%−(−60%)|= 120%. If

one option is purchased, then the return on the investment will be 35% or

−25%, and|a−b|=|35%−(−25%)|= 60%. Clearly, the standard deviation

|a−b|

√

p(1−p) will be reduced by a half, no matter whatpis.

Chapter 2

2.1The ratersatisfies

(

1+ 36561 ×r

)

× 9 ,000 = 9, 020.

This givesr∼= 0 .0133, that is, about 1.33%. The return on this investment

will be

K(0, 36561 )=^9 ,^0209 ,− 0009 ,^000 ∼= 0. 0022 ,

that is, about 0.22%.

2.2Denote the amount to be paid today byP. Then the return will be

1 , 000 −P

P =0.^02.

The solution isP∼= 980 .39 dollars.