m48-ch20 - Chapter 20 Brownian Motion and Its Lemma...

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Unformatted text preview: Chapter 20 Brownian Motion and Its Lemma Question 20.1. If y = ln (S) then S = e y and dy = (S,t) S (S,t) 2 2 S 2 dt + (S,t) S dZ t , a) dy = e y 2 2 e 2 y dt + e y dZ t . b) dy = a e y 2 2 e 2 y dt + e y dZ t . c) dy = 2 2 dt + dZ t . Question 20.2. If y = S 2 then S = y and dy = ( 2 S (S, t) + (S, t) 2 ) dt + 2 S (S, t) dZ t where (S, t) is the drift of S and (S, t) is the volatility of S . For the three specications: a) dy = ( 2 y + 2 ) dt + 2 ydZ t . b) dy = 2 y ( a y ) + 2 dt + 2 ydZ t (1) = 2 a y 2 y + 2 dt + 2 ydZ t . (2) c) dy = ( 2 + 2 ) ydt + 2 ydZ t . Question 20.3. If y = 1 /S then S = 1 /y and dy = ( S 2 (S, t) + S 3 (S, t) 2 ) dt S 2 (S, t) dZ t , a) dy = ( y 2 + 2 y 3 ) dt y 2 dZ t . b) dy = ( ( ay 2 y ) + 2 y 3 ) dt y 2 dZ t . c) dy = ( + 2 ) ydt ydZ t . 257 Part 5 Advanced Pricing Theory Question 20.4. If y = S then S = y 2 and dy = 1 2 S 1 / 2 (S, t) 1 8 S 3 / 2 (S, t) 2 dt + 1 2 S 1 / 2 (S, t) dZ t (3) = 1 2 y (S, t) 1 8 y 3 (S, t) 2 dt + 1 2 y (S, t) dZ t (4) a) dy = 2 y 2 8 y 3 dt + 2 y dZ t . b) dy = a 2 y 2 y 2 2 8 y 3 dt + 2 y dZ t . c) dy = 2 2 8 ydt + 2 ydZ t . Question 20.5. Let y = S 2 Q . 5 , then dy y = 2 ( S S ) + Q Q 2 + 2 S 2 Q 8 + S Q ! dt + 2 S dZ S + b Q dZ Q . (5) Question 20.6. If y = ln (SQ) = ln (S) + ln (Q) then dy = d ln (S) + d ln (Q) (6) = S S 2 S / 2 + Q Q 2 Q / 2 dt + S dZ S + Q dZ Q . (7) Question 20.7. With = 0, the prepaid forward price for S a 1 is F P , 1 ( S a ) = S a exp (a 1 ) r + 1 2 a (a 1 ) 2 . (8) a) If a = 2, F P , 1 ( S 2 ) = 100 2 exp ( . 06 + . 4 2 ) = 12461 . b) If a = . 5, F P , 1 ( S . 5 ) = 10 exp . 03 . 4 2 8 = 9 . 5123. c) If a = 2, F P , 1 ( S 2 ) = 100 2 exp ( . 18 + 3 ( . 4 2 )) = 1 . 349 9 10 4 . 258 Chapter 20 Brownian Motion and Its Lemma Question 20.8. Since the process y = S a Q b follows geometric Brownian motion, i.e. dy = y ydt + y ydZ y the price of the claims will be e r E (y 1 ) = y e ( y r ) . We use Itos lemma, as in equation (20.38), with = 0 and S = Q = r to arrive at the drift y = ar + br + 1 2 a (a...
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This note was uploaded on 12/06/2011 for the course ECON 101 taught by Professor Adam during the Spring '06 term at Neumann.

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m48-ch20 - Chapter 20 Brownian Motion and Its Lemma...

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