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Unformatted text preview: world or physical measure rather than the risk 193 6.4. CAPITAL ASSET PRICING MODEL neutral probabilities p⇤ = q ⇤ = 1/2 which are a ﬁction used to compute option
prices.
Our wealth at time 1 satisﬁes
5
+ (W0
4
5
W1 (T ) = 2 0 + (W0
4 W1 (H ) = 8 Writing 1 = 1 (H ), = 2 5
W0 + 3
4
5
4 0 ) = W0 + 3
4
4 0 1 (T ), and 0 = 0) 0 = 0
0 our wealth at time 2 is 5
15
25
+ (W1 (H ) 8 1 ) = 6 1 +
W0
0+
4
4
16
5
15
25
W2 (HT ) = 4 1 + (W1 (H ) 8 1 ) = 6 1 +
W0
0+
4
4
16
5
3
15
25
W2 (T H ) = 4 2 + (W1 (H ) 2 2 ) = 2
W0
0+
4
2
4
16
5
3
15
25
W2 (T T ) = 2 + (W1 (H ) 2 2 ) =
W0
2
0+
4
2
4
16 W2 (HH ) = 16 1 Let y0 , y1 , y2 , and y3 be our wealth at time 2 under outcomes HH , HT , T H ,
and T T . To see the correspondence think of binary digits H = 0 and T = 1.
With this notation we want to maximize
4
2
2
1
ln y0 + ln y1 + ln y2 + ln y3
9
9
9
9 V = E ln W2 = Di↵erentiating and using the formulas for the yi we have
✓
◆
@V
15 4 1
21
21
11
=
·
+·
·
·
@0
4 9 y0
9 y1
9 y2
9 y3
✓
◆
@V
41
21
=6
·
·
@1
9 y0
9 y1
✓
◆
@V
321
11
=
·
·
@2
2...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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