# Why is the rate of change of a stock price proportional to the stock price?

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?

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.

• Thanks some interesting points. If the drift is time and price dependent why is it introduced as a constant? Nov 6, 2014 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. Nov 7, 2014 at 7:59

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.

• Do you mean log-normal distributed? Nov 4, 2014 at 0:13
• No. What if I said relative returns? I think of returns as being measured in percent. Nov 4, 2014 at 0:17
• Thanks. Relative returns make total sense. There're a various return types. Nov 4, 2014 at 0:18
• 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. Nov 5, 2014 at 17:41
• 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. Nov 7, 2014 at 0:02