References and Tools for Estimating Geometric Brownian Motion Drift
Summary
The document collects references and software suggestions for estimating the drift parameter of geometric Brownian motion and related stochastic differential equation models. It points readers to a financial econometrics textbook, a study comparing short-term interest-rate models, and R packages for diffusion-model estimation. Later additions mention resources on maximum-likelihood estimation for multivariate diffusion models.
The material is a bibliography rather than a tutorial: it does not derive an estimator, compare its statistical properties, or report empirical results. A separate reply includes a code fragment that computes sample moments from simulated returns and transforms them into volatility and drift-related quantities, but does not explain assumptions or validate the procedure. Readers should consult the cited sources for estimation choices, especially the effects of sampling frequency, noise, and return predictability on drift inference.
Key ideas
- The document directs readers to financial econometrics references on estimating asset-dynamics parameters.
- It recommends R tools designed for stochastic differential equation and diffusion-model estimation.
- The material does not provide a full derivation or comparison of drift estimators.
- A code example appears, but its assumptions and accuracy are not discussed.
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# Estimation of Geometric Brownian Motion drift
# Estimation of Geometric Brownian Motion drift
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 by Matt (score 15, accepted)
https://quant.stackexchange.com/a/2957
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 by steven (score -4)
https://quant.stackexchange.com/a/32864
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
```Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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