Empirically validating GBM assumptions

Question

I'm trying to get a better grasp on the suitability of Geometric Brownian Motion as a model for asset prices, and I've stumbled across a hurdle which I can't seem to get over. Assuming GBM, the asset price is given by

$ S_t = S_0 \exp\{ (\mu - \frac{1}{2}\sigma^2)t + \sigma W_t \} $,

which implies that daily log-returns are normally distributed:

$ \log(\frac{S_{t+1}}{S_t}) \sim \mathcal{N}(\mu - \frac{1}{2}\sigma^2, \sigma^2) $.

Now my problem is: how would you go about verifying this with empirical data? If I compute the sample mean of log-returns with real data, then I have to already assume the model in order to extract the drift $ \mu $ by subtracting half the variance. Is there a way to get the drift term directly from the data, without assuming the model? Or is the drift simply a theoretical quantity postulated in the SDE

$ dS_t = S_t (\mu dt + \sigma W_t) $

which defines GBM? I.e. does the drift correspond to any computable statistical quantity?

EDIT: here is my attempt to gain clarity by simulation:

T = 1
N = 1000
S0 = 1
mu = -0.00028117*252
sigma = 0.008723*252
seed = 1
f, g = f_g_black_scholes(lamda = mu, mu = sigma)
numpaths = 10000
paths = pd.DataFrame()
for k in range(1, numpaths):
    t, S, W = euler_maruyama(seed=seed+k, X0=X0, T=T, N=N, f=f, g=g)
    S = pd.DataFrame(S)
    paths = pd.concat([paths, S], axis = 1)

final_price = paths.iloc[-1, :]
sigma_sim = np.std(np.log(final_price))
mu_sim = np.mean(np.log(final_price)) + 0.5*sigma_sim**2

The drift and volatility I estimated from S&P 500 data, and I made use of this implementation of the Euler-Maruyama solver to numerically generate sample paths. The results I obtain are:

mu = -0.070855
mu_sim = -0.010407
sigma = 2.198196
sigma_sim = 2.221657

As you can see there is still a significant discrepancy between the input drift and the result of the simulation. How should I make sense of this?

Answer 1

All you can do is fit the values of the parameters of a model you have already assumed.

Now my problem is: how would you go about verifying this with empirical data?

As Kermittfrog mentioned, after fitting a particular model, you can use test to check whether your initial assumptions make sense or not.

If I compute the sample mean of log-returns with real data, then I have to already assume the model in order to extract the drift $\mu$ by subtracting half the variance. Is there a way to get the drift term directly from the data, without assuming the model? Or is the drift simply a theoretical quantity postulated in the SDE

Parameters are quantities associated with a model in this case. For example consider normal and lognormal distributions, in both you have a $\mu$ and a $\sigma$, but depending on the model you assume you'll get different quantities from your fit, because this parameters mean different things (no matter on the label or greek letter you use to describe them).