This preview shows page 1. Sign up to view the full content.
Unformatted text preview: +r d
5/4 1/2
1
=
=
ud
2 1/2
2 and by (6.12) the option prices follow the recursion:
Vn (a) = .4[Vn+1 (aH ) + Vn+1 (aT )] (6.17) Example 6.2. Callback options. In this option you can buy the stock at
time 3 at its current price and then sell it at the highest price seen in the past
for a proﬁt of
V3 = max Sm
0m3 S3 Our goal is to compute the value Vn (a) and the replicating strategy n (a) for
this option in the binomial model given in (6.16) with S0 = 4. Here the numbers
above the nodes are the stock price, while those below are the values of Vn (a)
and n (a). Starting at the right edge, S3 (HT T ) = 2 but the maximum in the
past is 8 = S1 (H ) so V3 (HT T ) = 8 2 = 6. 188 CHAPTER 6. MATHEMATICAL FINANCE 32
⇠
16 ⇠⇠⇠⇠ 0
⇠
XX
XX
XX8
3.2
8
.25 8
H
H
8
2.24 HH
⇠⇠0
H 4 ⇠ ⇠⇠
.0666
H⇠X
X
XX X 2
2.4
X
6
1 4
1.376@
@
.1733 @ @ 8
⇠⇠0
4 ⇠ ⇠⇠
⇠X
XX
XX 2
0.8
X
2
.333 @2
@
H
2
1.2 HH
H
⇠⇠2
.466
HH⇠⇠⇠⇠
1
XX
XX
XX.5
2.2
3.5
1 On the tree, stock prices are above the nodes and option prices below. To
explain the computation of the option price note that by (6.17). V2 (HH ) = 0.4(V3 (HHH ) + V3 (HHT )) = 0.4(0 + 8) = 3.2
V2 (HT ) = 0.4(V3 (HT H ) + V3 (HT T )) = 0.4(0 + 6) = 2.4
V1 (H ) = 0.4(V2 (HH ) + V2 (HT )) = 0.4(3.2 + 2.4) = 2.24 If one only wants the option price then Theorem 6.5 which says that V0 =
E ⇤ (VN /(1 + r)N ) is much quicker: V0 = (4/5)3 · 1
· [0 + 8 + 0 + 6 + 0 + 2 + 2 + 2 + 3.5] = 1.376
8 Example 6.3. Put option. We wil...
View
Full
Document
 Spring '10
 DURRETT
 The Land

Click to edit the document details