{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

m48-ch20

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

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern