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#1
04-19-2007, 10:59 PM
 Potato2007 Member Join Date: Apr 2007 Posts: 34
ASM Quiz 15.2

1. For Geometric Brownian motion, (20.12) in page 655 of textbook gives
Ln[X(t)]~N{ln[X(0)]+(α - 0.5σ^2)t, σ^2 t}, while ASM gives
Ln[S(t)/S(0)]~N{(α - δ - 0.5σ^2)t, σ^2 t}.
Are they equivalent?

2. For ASM Quiz 15.2, shouldn't the mean be equal to (0.1-0.05^2)*5, instead of 0.1*5 as in the manual?

Last edited by Potato2007; 04-19-2007 at 11:24 PM..
#2
04-19-2007, 11:18 PM
 jason. Member Join Date: Feb 2007 Studying for CFA II Favorite beer: Samuel Smith Old Brewery Pale Ale Posts: 888

Quote:
 Originally Posted by Potato2007 1. For Geometric Brownian motion, (20.12) in page 655 of textbook gives Ln[X(t)]~N{ln[X(0)]+(α - 0.5σ^2)t, σ^2 t}, while ASM manual gives Ln[S(t)/S(0)]~N{(α - δ - 0.5σ^2)t, σ^2 t}. Are they equivalent?
Yes, because if $\ln X(t)$ is $N\bigl(\ln X(0) + (\alpha - 0.5\sigma^2)t, \sigma^2t\bigr)$ then $\ln X(t) - \ln X(0)$ is $N\bigl( (\alpha - 0.5\sigma^2) t, \sigma^2 t\bigr)$. Since $\ln X(t) - \ln X(0) = \ln \bigl(X(t)/X(0)\bigr)$ the result follows.

Quote:
 2. For ASM Quiz 15.2, shouldn't the mean be equal to (0.1-0.05^2)*5, instead of 0.1*5 given in the manual?
Yes. At least the author was consistent in making this error throughout that chapter.
#3
04-19-2007, 11:26 PM
 Potato2007 Member Join Date: Apr 2007 Posts: 34

Jason, I agree with you, but how would you explain the part in red in my question 1?
#4
04-19-2007, 11:42 PM
 jason. Member Join Date: Feb 2007 Studying for CFA II Favorite beer: Samuel Smith Old Brewery Pale Ale Posts: 888

Quote:
 Originally Posted by Potato2007 Jason, I agree with you, but how would you explain the part in red in my question 1?
Ah! I apologize for overlooking that when I read your post. Both are correct, it just depends on what we consider $\alpha$ to be. In the text, everywhere in this section that you see $\alpha$ replace it by $\beta$ so that the formula reads $\ln X(t)$ is $N\bigl(\ln X(0) + (\beta- 0.5\sigma^2)t, \sigma^2t\bigr)$. Now set $\beta = \alpha - \delta$ and we obtain the ASM Manual's formula for the distribution of the natural logarithm of stock prices.

The point is that on page 655 of the text, the author is discussing general geometric Brownian motion with drift $\alpha$. For a stock, a special case of geometric Brownian motion, the drift is $\alpha - \delta$ where $\alpha$ is the expected return on the stock and $\delta$ is the continuous dividend rate. The author of that text should not have used $\alpha$ to refer to the drift and to refer to the expected return on the stock.
#5
04-19-2007, 11:59 PM
 Potato2007 Member Join Date: Apr 2007 Posts: 34

Thank you! You will pass the exam.

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