About SDE of Geometric Brownian Motion

Question

It's known that most of the financial assets are subject to Geometric Brownian Motion, which satisfies the following equations:

$\frac{dS}{S}=\mu dt + \sigma dX$ (1)

$S_t = S_0 e^{(\mu + \frac{1}{2} \sigma^2)t + X_t}$ (2)

Here my questions are:

In practice, when we use this SDE to simulate asset price path, i.e. the price movement of a stock index, what parameters should I use for $\mu$ and $\sigma$.

Does the $\mu$ stand for annualized mean log return of the stock index? But in equation(1), the $\mu$ doesn't appear as the exponent of e. Although, in equation(2) $\mu$ is really the exponent of e.

Does the $\sigma$ stand for the volatility of stock price, or volatility of logarithm of price, or volatility of log returns?

Here the volatility is an annualized or not?

The volatility is a rolling volatility or realized volatility?

thanks for your attention!

Answer 1

Equation (2) is wrong, it is is $(\mu - \sigma^2/2)t$ and $\sigma X_t$. Then estimating $\mu$ is not just taking the mean of log returns. Using $\log(S_{n+1}) - \log(S_{n}) \sim N(\mu - \sigma^2/2, \sigma^2)$, you estimate $\sigma$ with the sample standard deviation of log returns, say $\hat \sigma$, and then you get an estimate for $\mu$ with "sample mean of log returns+ ${\hat \sigma}^2/2$." And actually GBM is not a very good model for most stocks, it is too simple.

Answer 2

It's all in return units. Assume, for example, that from 1950 to 2017 annualized average returns on the S&P 500 are about 11.5% (.115) and standard deviation/vol is about 17% (0.17). That is you $\mu$ and $\sigma$.