Hi I'm currently taking a 4th year Econ course and the topic is computational economics. But the subjects are pretty similar to finance so I'm posting my question here. The question has three parts and I did part a) but have no idea how to do b) and c). Please help and discuss how to do them, thanks.
Here's the question.
Ito's Lemma: Let the price s(t) of a security follow the Ito process ds = (alpha)sdt + (sigma)sdz in
which (alpha) and (sigma) are constants.
(a) Use Ito's Lemma to determine the process followed by y(t) = ln s(t).
(b) What is the probability distribution of y(2) in terms of y(0), (alpha) and (sigma)? (i.e., what is the
type of distribution, its mean and its variance)
(c) If you were given s0 = s(0) and a random draw X from the type of distribution in (b),
but it was standardized to have mean 0 and variance 1, how would you convert it into
a random draw of s(2)? (i.e., of the security price at time 2)
Here is the pdf version: http://www.sfu.ca/~rjones/econ482/homework/hw1a.pdf
For part a) what I got is:
dy/dt = 0
dy/ds = 1/s
(d^2)y/ds^2 = 1/s^2
Use Ito's formula and I got: dy = ((alpha) - ((sigma)^2)/2)dt + (sigma)dz
Hope I got it right.
But for part b) I have no idea what's the meaning of y(2) in terms of y(0) and not sure whether I actually find (alpha) and (sigma) as numbers or as functions.
part c) I just know it has to be a markov process but have no idea how to convert it.
We are using Options, Futures, and other derivatives 7th ed. by John Hull. But I've read ch.12 about Ito's lemma and got nothing helpful.
Please help! Thanks.
Bruzzo
Here's the question.
Ito's Lemma: Let the price s(t) of a security follow the Ito process ds = (alpha)sdt + (sigma)sdz in
which (alpha) and (sigma) are constants.
(a) Use Ito's Lemma to determine the process followed by y(t) = ln s(t).
(b) What is the probability distribution of y(2) in terms of y(0), (alpha) and (sigma)? (i.e., what is the
type of distribution, its mean and its variance)
(c) If you were given s0 = s(0) and a random draw X from the type of distribution in (b),
but it was standardized to have mean 0 and variance 1, how would you convert it into
a random draw of s(2)? (i.e., of the security price at time 2)
Here is the pdf version: http://www.sfu.ca/~rjones/econ482/homework/hw1a.pdf
For part a) what I got is:
dy/dt = 0
dy/ds = 1/s
(d^2)y/ds^2 = 1/s^2
Use Ito's formula and I got: dy = ((alpha) - ((sigma)^2)/2)dt + (sigma)dz
Hope I got it right.
But for part b) I have no idea what's the meaning of y(2) in terms of y(0) and not sure whether I actually find (alpha) and (sigma) as numbers or as functions.
part c) I just know it has to be a markov process but have no idea how to convert it.
We are using Options, Futures, and other derivatives 7th ed. by John Hull. But I've read ch.12 about Ito's lemma and got nothing helpful.
Please help! Thanks.
Bruzzo