108156.pdf

294 Mathematics for Finance

The corresponding European and American put prices will be

n 01 2

1. 68

1. 68

2. 53

2. 80 <

/^33 ..^3636

PE(n)

PA(n)

3. 42

3. 69

\^33 ..^6666

4. 33

4. 60 <

5. 07

5. 07

At time 1 the payoff of the American put option in both the up and down

states will be higher than the value of holding the option to expiry, so the

option should be exercised in these states (indicated by bold figures).

8.13Takebsuch thatS(0)eσb+ru−^12 σ^2 u=aand putV(t)=W(t)+

(

m−r+^12 σ^2

)t

σ

for anyt≥0, which is a Wiener process underP∗.Inparticular,V(u)isnor-

mally distributed underP∗with mean 0 and varianceu. The right-hand side

of (8.8) is therefore equal to

E∗

(

e−ruS(u)1S(u)<a

)

=S(0)E∗

(

eσV(u)−

(^12) σ (^2) u
(^1) V(u)<b
)
=S(0)
∫b
−∞
eσx−
(^12) σ (^2) u√ 1
2 πt
e−
x 2 u^2
dx
=S(0)
∫b
−∞
√^1
2 πt
e−
(x− 2 σuu)^2
dx.
Now observe that, sinceV(t) is a Wiener process underP∗, the random vari-
ablesV(u)andV(t)−V(u) are independent and normally distributed with
mean 0 and varianceuandt−u, respectively. As a result, their joint density
is 21 πte−
2(ty−^2 u)−x 2 u^2

. This enables us to compute the left-hand side of (8.8),

E∗

(

e−rtS(t)1S(u)<a

)

=S(0)E∗

(

eσV(t)−^12 σ

(^2) u
(^1) V(u)<b
)
=S(0)E∗
(
eσ(V(t)−V(u))+σV(u)−^12 σ^2 u (^1) V(u)<b
)
=S(0)
∫b
−∞
(∫∞
−∞
eσy+σx−
(^12) σ (^2) t 1
2 πte
−2(ty−^2 u)−x 22 udy
)
dx
=S(0)
∫b
−∞
(∫∞
−∞
1
2 πte
−(y−2(σt(−t−uu)))^2 −(x− 2 σuu)^2 dy
)
dx
=S(0)
∫b
−∞
√^1
2 πt
e−
(x− 2 σuu)^2
dx
We can now see that the two sides of (8.8) are equal to one another.