Why Stock Price Models Scale Changes with Price
Summary
The document explains why the Black–Scholes model writes a stock’s price change as a drift proportional to the current price, alongside a random component. Its central idea is to model relative returns: if returns are treated as having a stable distribution, then the corresponding dollar change grows with the price level. The answer contrasts daily price changes, described as skewed, with daily returns, which it says are closer to normally distributed.
It also addresses drift estimation and option pricing. The responses argue that estimating a reliable drift is difficult and may require a time- and price-dependent model. In the Black–Scholes delta-hedging argument, the stock’s drift cancels along with the random exposure, so the drift need not be estimated to price the option under that framework. The discussion is conceptual and brief; it does not provide data, a statistical estimation procedure, or a proof that relative returns must follow a particular distribution.
Key ideas
- Modeling relative returns leads to price changes that scale with the current stock price.
- Daily price changes may be more skewed than daily returns.
- The drift can be difficult to estimate reliably and may vary over time and price.
- In the Black–Scholes delta-hedging argument, the drift cancels from the option-pricing derivation.
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Full text
# Why is the rate of change of a stock price proportional to the stock price? # 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? ## Answer by Ulysses (score 3) https://quant.stackexchange.com/a/15343 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. ## Answer by Degustaf (score 2) https://quant.stackexchange.com/a/15284 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.
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