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When deriving the Black Scholes equation, it is usually stated "we assume the change in the stock price is":

$dS=\mu S(t) dt + $random term

My question is why is the change in the stock price always proportional to the stock price (ignoring the random term for now)? Is it simply because the stock pays dividends which are proprtional to the stock price (in which case $\mu$ must be related to dividends). How do you find what $\mu$ is for a given stock or option? Is $\mu$ always positive?

kotozna
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As Degustaf mentioned above, one of the keys to the exponential dynamics is just the fact that you first model (relative/log) returns and then describe the dynamics of the stock price itself. I am not sure whether the arbitrage argument of experquisite is realistic, though.

Regarding the estimation of the drift: if you know the drift, you can just trade based on it, or at least hedge options much more profitably. For sure, a lot of people would be interested in that knoweldge, and you can expect that people looked into that. You can start with this thread. My guess is that drift is much harder to estimate in a robust way, even harder than volatility, so that's why there is not many methods that would tell you have to compute a reliable estimate of the drift, not to say that the drift is likely to be time- and price-dependent, so you have to estimate a function rather than a single value.

Ulysses
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  • Thanks some interesting points. If the drift is time and price dependent why is it introduced as a constant? – kotozna Nov 06 '14 at 21:20
  • @kotozna: it is only introduced as a constant in BS model, but there the vol is constant as well, and perhaps you are aware of the stochastic vol models that have potential to perform better at least in pricing exotic options. For similar reasons, there are a lot of models with non-constant drift. I don't know a special-named models for stocks, but see e.g. here. For interest rates there are a lot of models with non-constant drift, e.g. Vasicek one. – Ulysses Nov 07 '14 at 07:59
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This is based on observations of historical data. If you looked at a histogram of daily changes, you would notice that the distribution is heavily skewed. Whereas if you looked at a histogram of daily returns, you would see that it is much closer to normally distributed.

As for how to find $\mu$, you don't. The beauty of the Black-Scholes model is that when the option is delta hedged to remove the random term, the $\mu$'s all cancel out as well.

Degustaf
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  • Do you mean log-normal distributed? – SmallChess Nov 04 '14 at 00:13
  • No. What if I said relative returns? I think of returns as being measured in percent. – Degustaf Nov 04 '14 at 00:17
  • Thanks. Relative returns make total sense. There're a various return types. – SmallChess Nov 04 '14 at 00:18
  • Thanks for your response but I am not sure it answers the question. It is interesting that it is experimental fact that stocks prices behave in this way, but that doesn't explain why. And maybe you don't need to find $\mu$ in the Black-Scholes model, but I am still interested in how you would find it. Considering it is one of the most fundamental parameters in finance I'm surprised there is no way to at least estimate it or say how it is related eg. stock dividends or some other aspect of stocks. – kotozna Nov 04 '14 at 19:55
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    Here's a simple reason: stocks can be split, arbitrarily, so if drift wasn't relative to stock price, there would be arbitrage in stock splits. – experquisite Nov 05 '14 at 17:41
  • @experquisite That's a good point. But is it the reason? I mean if a stock pays dividends that's an entirely different reason why a stock price would be proportional to itself...which one is right? – kotozna Nov 06 '14 at 20:53
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    Well there are lots of reasons, but no true reason other than that is what the model is. It's a limited liability corporation with a specific equity value and an arbitrary share count. Arguably the purpose of a corporation is to produce some return-on-equity through internal reinvestment, and that reinvestment implies compounding and exponential dynamics. – experquisite Nov 07 '14 at 00:02