How to get around flat likelihood function when calibrating GBM parameters?

Question

I want to calibrate jointly the drift mu and volatility sigma of a geometric brownian motion,

$$\log(S_t) = \log(S_{t-1}) + (\mu - 0.5*\sigma^2) \Delta t + \sigma*\sqrt{\Delta t}*Z_t$$

where $Z_t$ is a standard normally distributed random variable, and am testing this by generating data $x = \log(S_t)$ via

x(1) = 0;
for i = 2:N
  x(i) = x(i-1) + (mu-0.5*sigma^2)*Deltat + sigma*sqrt(Deltat)*randn;
end

and my (log-)likelihood function

function LL = LL(x, pars)
  mu    = pars(1);
  sigma = pars(2);
  Nt = size(x,2);
  LL = 0;
  for j = 2:Nt
    LH_j = normpdf(x(j), x(j-1)+(mu-0.5*sigma^2)*Deltat, sigma*sqrt(Deltat));
    LL = LL + log(LH_j);
  end  

which I maximize using fmincon (because sigma is constrained to be positive), with starting values 0.15 and 0.3, true values 0.1 and 0.2, and N = Nt = 1000 or 100000 generated points over one year (i.e. $\Delta t$ = 0.0001 or 0.000001).

Calibrating the volatility alone yields a nice likelihood function with a maximum around the true parameter, but for small Deltat (less than say 0.1) calibrating both mu and sigma persistently shows a (log-)likelihood surface being very flat in mu (at least around the true parameter); I would expect also a maximum there; for a reason I think it should be possible to calibrate a GBM model to a data series of 100 stock prices in a year, making the average of Deltat = 0.01.

Any sharing of experience or help is greatly appreciated (thoughts passing through my mind: the likelihood function is not right / this is a normal behaviour / too few data points / data generation is not correct / ...?).

Answer 1

This is what often happens in optimization problems, i.e. some direction is almost flat. Google 'preconditioning'. Basically the idea is to rescale the variables, so that the Hessian has approx. same order of magnitude values on the diagonals.

Also, that's not a stationary process, so estimation of mu can be difficult.

BTW not sure if it's a very good idea to use the 'normpdf' function here. Instead you should probably just write out the complete loglikelihood expression (not sure if that will help though). It seems about right to me though.

Answer 2

The log likelihood function is indeed rather flat in the $\mu$-direction, for small time horizons (you used $T = 1$ it looks like). As you may have noticed, increasing the number of observations but keeping the time horizon the same DOES NOT IMPROVE the accuracy of the estimate of $\mu$ - this is a bit counterintuitive, if you ask me. But, increasing the time horizon $T$ DOES improve the accuracy of the estimate. To see this, see my question here.

Merton (1980) proposed using a Bayesian technique, in which you first specify a prior distribution for $\mu$ and use Bayes' rule to derive a new log likelihood function. This is nowadays known as "maximum a-posteriori (MAP)" estimation and is one way to try to get better estimate of $\mu$. But the fact remains: MLE estimation of $\mu$ is notoriously inaccurate for short time horizons.