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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).

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