Estimation of Geometric Brownian Motion drift

Question

One can find many papers about estimators of the historical volatility of a geometric Brownian motion (GBM). I'm interested in the estimation of the drift of such a process. Any link on this topic would be very helpful.

Answer 1

One reference is "The Econometrics of Financial Markets" by John Y. Campbell, Andrew W. Lo, & A. Craig MacKinlay -- https://press.princeton.edu/titles/5904.html. In particular:

 9.3.1 Parameter Estimation of Asset Price Dynamics 356
 9.3.4 The Effects of Asset Return Predictability 369

You might also take a look at Chan (1992) "An Empirical Comparison of Alternative Models of the Short-Term Interest Rate" which discusses parameter estimation of several models including the GBM: http://rady.ucsd.edu/faculty/directory/valkanov/classes/mfe/docs/Longstaff_JoF_1992.pdf

There are also rather nice packages for R, 'sde' and 'yuima', which allow you (among many other things) to estimate the parameters of the SDE models. Take a look at the slides "Statistical data analysis of financial time series and option pricing in R" -- http://past.rinfinance.com/agenda/2011/StefanoIacus.pdf -- in particular, you may find the "Estimation of Financial Models" part quite useful.

Edit (2018): Today I'd also take a look at https://yuima-project.com/papers/ and https://yuima-project.com/books/ as well as "MLEMVD: A R Package for Maximum Likelihood Estimation of Multivariate Diffusion Models": https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2944341, http://past.rinfinance.com/agenda/2017/talk/MatthewDixon.pdf, https://channel9.msdn.com/Events/RFinance/RFinance-2017/MLEMVD-A-R-Package-for-Maximum-Likelihood-Estimation-of-Multivariate-Diffusion-Models.

Answer 2

hope the following codes help you

Z = normrnd(0.00112, 0.01525, 15000, 52);
R = Z';
m = sum(R)/52;
p = m';
for k = 1:15000;
    for j = 1:52;
        D(k,j) = (Z(k,j)-p(k,1)).^2;
    end;
end;
V = sum(D')/52;
V = V';
t = 1/52;
S = sqrt(V/t)
A = 0.5 * S.^2 + (1/t)*p