ASM Quiz 15.2

[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 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?

[jason.]

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 is then is . Since the result follows.

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.

[Potato2007]

Jason, I agree with you, but how would you explain the part in red in my question 1?

[jason.]

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 to be. In the text, everywhere in this section that you see replace it by so that the formula reads is . Now set 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 . For a stock, a special case of geometric Brownian motion, the drift is where is the expected return on the stock and is the continuous dividend rate. The author of that text should not have used to refer to the drift and to refer to the expected return on the stock.

[Potato2007]

Thank you! You will pass the exam.