Black Scholes Differential Form

Question

My instructor has mostly self contained notes, where our textbook is mostly a reference. She has it written that $$ S_t = S_0e^{(\mu - \frac{\sigma^2}{2})t + \sigma W_t} \iff dS_t = S_t(\mu\, dt + \sigma\, dW_t). $$ I feel that basic differentiation of the exponential e implies that on the right hand side we should have $dS_t = S_t((\mu - \frac{\sigma^2}{2})\,dt + \sigma\, dW_t)$.

I'd appreciate understanding why the $\frac{\sigma^2}{2}$ disappears from the differentiation when this is a basic rule about differentiating the exponential $e$.

$W_t$ follows the normal distribution with mean 0 and variance t. — user7348, Oct 28 '14 at 01:03
You might find answers to this one helpful: http://math.stackexchange.com/questions/894526/in-stochastic-calculus-why-do-we-have-dt2-0-and-other-results — 3x89g2, Oct 28 '14 at 01:04
@Misakov, you're actually correct. I can't believe I missed that. Thanks. If you put that link in a formal answer, I'll upvote you. — user7348, Oct 28 '14 at 01:07
Yeah, that's fine. I don't even understand the whole BS-Formula haha. I assume you are taking some sort of actuarial exam? — 3x89g2, Oct 28 '14 at 01:09
If you want to know why the $\frac{sigma^2}{2}$ disappear, one thing I can assure you is that you cannot use the usual way of taking differentials here. It has something to do with a branch callled stochastic calculus, and a lot of weird things could happen there. Still I think if you go ask your instructor, she should be able to offer some sort of "intuitive explanation" which, probably will not be that rigorous. Cheers. — 3x89g2, Oct 28 '14 at 01:10

score 1 · Accepted Answer · answered Oct 28 '14 at 03:49

The answer comes from Ito Lemma.

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This differs from the regular calculus that there is an additional term on the right hand side. Stochastic calculus is different to regular calculus in that a Brownian Motion can't be differentiated with respect to t.

The additional term explains your observation.

Furthermore, this has nothing to do with Black-Scholes. Black-Scholes is a hedging strategy that replicates an option. Your formula is a GBM, and is lognormal distributed. It just happens that in the BS world, we assume the stock price exhibit log-normal returns.

Black Scholes Differential Form

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