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

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Chapter 20 Brownian Motion and Itô’s Lemma Question 20.1. If y = ln (S) then S = e y and dy = α(S,t) S σ(S,t) 2 2 S 2 dt + σ(S,t) S dZ t , a) dy = α e y σ 2 2 e 2 y dt + σ e y dZ t . b) dy = λa e y λ σ 2 2 e 2 y dt + σ e y dZ t . c) dy = α σ 2 2 dt + σdZ t . Question 20.2. If y = S 2 then S = y and dy = ( 2 Sα (S, t) + σ (S, t) 2 ) dt + 2 Sσ (S, t) dZ t where α (S, t) is the drift of S and σ (S, t) is the volatility of S . For the three specifications: a) dy = ( 2 α y + σ 2 ) dt + 2 yσdZ t . b) dy = 2 ( a y ) + σ 2 dt + 2 yσdZ t (1) = 2 λa y 2 λy + σ 2 dt + 2 yσdZ t . (2) c) dy = ( 2 α + σ 2 ) ydt + 2 σydZ t . Question 20.3. If y = 1 /S then S = 1 /y and dy = ( S 2 α (S, t) + S 3 σ (S, t) 2 ) dt S 2 σ (S, t) dZ t , a) dy = ( αy 2 + σ 2 y 3 ) dt σy 2 dZ t . b) dy = ( λ ( ay 2 y ) + σ 2 y 3 ) dt σy 2 dZ t . c) dy = ( α + σ 2 ) ydt σydZ t . 257
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Part 5 Advanced Pricing Theory Question 20.4. If y = S then S = y 2 and dy = 1 2 S 1 / 2 α (S, t) 1 8 S 3 / 2 σ (S, t) 2 dt + 1 2 S 1 / 2 σ (S, t) dZ t (3) = 1 2 y α (S, t) 1 8 y 3 σ (S, t) 2 dt + 1 2 y σ (S, t) dZ t (4) a) dy = α 2 y σ 2 8 y 3 dt + σ 2 y dZ t . b) dy = λa 2 y λ 2 y 2 σ 2 8 y 3 dt + σ 2 y dZ t . c) dy = α 2 σ 2 8 ydt + σ 2 ydZ t . Question 20.5. Let y = S 2 Q 0 . 5 , then dy y = 2 S δ S ) + α Q δ Q 2 + σ 2 S σ 2 Q 8 + ρσ S σ Q dt + 2 σ S dZ S + Q dZ Q . (5) Question 20.6. If y = ln (SQ) = ln (S) + ln (Q) then dy = d ln (S) + d ln (Q) (6) = α S δ S σ 2 S / 2 + α Q δ Q σ 2 Q / 2 dt + σ S dZ S + σ Q dZ Q . (7) Question 20.7. With δ = 0, the prepaid forward price for S a 1 is F P 0 , 1 ( S a ) = S a 0 exp (a 1 ) r + 1 2 a (a 1 ) σ 2 . (8) a) If a = 2, F P 0 , 1 ( S 2 ) = 100 2 exp ( . 06 + . 4 2 ) = 12461 . b) If a = . 5, F P 0 , 1 ( S . 5 ) = 10 exp . 03 . 4 2 8 = 9 . 5123. c) If a = − 2, F P 0 , 1 ( S 2 ) = 100 2 exp ( . 18 + 3 ( . 4 2 )) = 1 . 349 9 × 10 4 . 258
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Chapter 20 Brownian Motion and Itô’s Lemma Question 20.8. Since the process y = S a Q b follows geometric Brownian motion, i.e. dy = α y ydt + σ y ydZ y the price of the claims will be e r E (y 1 ) = y 0 e ( α y r ) . We use Ito’s lemma, as in equation (20.38), with δ = 0 and α S = α Q = r to arrive at the drift α y = ar + br + 1 2 a (a 1 ) σ 2 S + 1 2 b (b 1 ) σ 2 Q + abρσ S σ Q
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