a detail on modelling

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a detail on modelling

Postby fundamentalGuy » Wed Apr 04, 2012 3:56 pm

Although this posting is long, I hope it is obvious that this question is not asked without proper consideration. I apologize for any typographical errors.

A posting by "mj" on the Wilmott forum in October 2009 advised the following to an individual trying to understand (or get comfortable with) the -0.5 \sigma^2 term in the solution for geometric brownian motion.
(quote from post)
"try evolving the following three processes in a computer using lots of short steps
dS_{t} = \mu S_t dt + \sigma S_t dW_{t}
d \log S_{t} = \mu dt + \sigma dW_{t}
d \log S_{t} = (\mu-0.5\sigma^2) dt + \sigma dW_{t}
Compare the resulting paths for S_t. And in particular compute the expectation of S_t for all three.
This is the most convincing way to see that the term really needs to be there."

I have done so with the following updating (in order)
(1) S_{t+1} = S_t ( 1 + \mu \delta_t + \sigma \delta_t^0.5 \epsilon_t )
(2) S_{t+1} = \exp ( \ln S_t + \mu \delta_t + \sigma \delta_t^0.5 \epsilon_t )
(3) S_{t+1} = \exp [ \ln S_t + (\mu - 0.5 \sigma^2)\delta_t + \sigma \delta_t^0.5 \epsilon_t ]
with \epsilon_t \sim \mathcal{N} (0, 1)
and acknowledge that (3) approximates (1), including the sample expectation of S_t.

My question: "Is this the correct framework?" Should (1) be used or is
(1a) S_{t+1} = S_t \exp (\mu \delta_t + \sigma \delta_t^0.5 \epsilon_t)
a better representation of reality? Even if \delta_t is small, should (1) or (1a) be used to simulate the evolution of an asset price? Which is the 'truth model'?

In a discrete-time framework, (1a) seems more realistic in that \delta_t can be chosen based solely on what 'fits' \epsilon 'best' whereas the multi-step evolution of every term in (1) is affected by the choice of \delta_t. Of course, this has implications for continuous time, but to date, each person I've asked seems to give answers reverse-engineered from his own knowledge base (maybe "memorization base").

Thank you in advance to anyone who responds.
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Re: a detail on modelling

Postby mj » Sat Apr 07, 2012 9:37 am

3) is the exact solution so should be used where possible.
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