The math there is wrong: 1. log((1-f)r)!= (1-f)r. 2. E(log(a)+lob(b))!=E(log(a))+E(log(b)) 3. They seem to be computing based on arithmetic growth i.e. a 5% increase is .05 instead of 1.05 and 5% drop is -.05 instead of .95. This is wrong and log will blow up on neg numbers.
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Replying to @TrevorVossberg
So what is the optimal amount to invest in a stock with positive variance but zero (risk-adjusted) drift?
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Replying to @s_r_constantin
Hmm I believe you are right here. I seem to have misunderstood the definition of a martingale. I'm not sure how one would have a positive rate of return.
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Replying to @TrevorVossberg @s_r_constantin
Since a martingale's expected value is it's current value
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Replying to @TrevorVossberg
What I’m wondering about specifically is an asset whose value is exp(sigma W_t), where W_t is the Wiener process and sigma is a constant. That is, geometric Brownian motion with no drift.
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Replying to @s_r_constantin
Wikipedia has it at exp(mu*t). https://en.m.wikipedia.org/wiki/Geometric_Brownian_motion#Properties …
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Replying to @TrevorVossberg
yeah, no, that confused me too the first time, that's after a change of variables. the wiki page there says the prob distribution that has mean exp(mu*t) is exp((mu - sigma^2/2)t + sigma W_t). The drift term is *not* the same as the exponent in the mean.
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Replying to @s_r_constantin
ok so there is geometric Brownian motion which is exp((mu - sigma^2/2)t + sigma W_t) and mean exp(mu*t). Mu/sigma are for the Brownian motion. Due to needing to preserve expectation a "negative drift" sigma^2/2t is introduced since exp is biased.
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Replying to @TrevorVossberg @s_r_constantin
Then there is your formula exp(mu*t + sigma W_t) which is geometric Brownian motion(GBM) but with an incorrect mu since it is missing the sigma^2/2 term. If you add back that term to your mu you'll have GBM with mu=mu'+sigma^2/2.
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Replying to @TrevorVossberg @s_r_constantin
So then (mu-r)/sigma^2 becomes (mu+sigma^2/2-r)/sigma^2 becomes 1/2. So you invest half your money, I think.


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Thanks! Yeah, that makes sense but it's kind of a shocking result!
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