MeanReversion: Ornstein-Uhlenbeck fit and optimal trading bands - libreria per MetaTrader 5

MeanReversion.mqh models a spread that comes back to its mean (the residual of a cointegrated pair, a basis, a z-score) as an Ornstein-Uhlenbeck process, and finds where to enter and exit. The fit is the exact maximum likelihood on equally spaced bars, with standard errors, half-life and long-run standard deviation. The trading levels are those of Bertram (2010): for a given cost per round trip, the entry and exit that give the highest expected return per unit of time, with the expected time of a trade cycle. No DLL.

Usage

#include <MeanReversion\MeanReversion.mqh>

vector spread;                             // for example log(y) - beta log(x), in %
// ... fill spread ...
COrnsteinUhlenbeck ou;
if(ou.Fit(spread))                         // exact maximum likelihood, one bar = 1
  {
   double half = ou.HalfLife();            // bars
   double z = ou.ZScore(spread[spread.Size() - 1]);
   double a, m, rate;
   ou.OptimalSymmetric(0.05, a, m, rate);  // cost 0.05 per round trip: enter at mean + a, exit at mean + m
   double bars = ou.ExpectedCycle(a, m);   // expected bars from one entry to the next
  }

What is in it

• Fit(x, dt): kappa (speed of return), theta (mean), sigma, from the exact transition density of the process (it is a regression of each value on the one before, read in the right units); standard errors from the Hessian of the exact likelihood. LogLikelihoodAt() evaluates it for any parameters.
• ExpectedCycle(a, m): expected time from the entry level a to the exit level m and back to a, pi / kappa (erfi(m sqrt(kappa) / sigma) - erfi(a sqrt(kappa) / sigma)), levels measured from the mean.
• ReturnPerTime(a, m, c) = (m - a - c) / ExpectedCycle(a, m); OptimalSymmetric(c) maximizes it with m = -a, Optimal(c) without that constraint (the optimum is symmetric, as Bertram shows; the function lets you check it).
• Erfi: the imaginary error function, by its series.

In units of the long-run standard deviation the best levels depend only on the cost: with no cost, trade at the mean; a cost of 0.1 sd gives levels at about +/-0.54 sd, 0.5 sd at +/-0.96 sd, 1 sd at +/-1.26 sd.

Example

The script SpreadBands.mq5 builds the spread of two symbols (log prices, hedge ratio by least squares), fits the process, takes the cost of a round trip from the bid-ask spreads of both legs (plus a commission you set), and prints the optimal levels, the expected number of trades and return per year for 1, 2 and 5 times that cost, and the z-score now (picture 1). It can write the spread and the levels to a CSV file (picture 2).

Checked

• the fit on 3 simulated processes (one with dt = 1/252) and a real spread against scipy maximizing the exact likelihood: parameters within 4e-7, the same likelihood, standard errors within 1e-5 of a numerical Hessian (statsmodels);
• erfi against scipy on [-6, 6]: within 8e-16; cycle lengths within 1e-15; optimal levels against scipy's bounded search within 5e-8 sd, the free optimum within 3e-8 sd;
• the cycle formula itself against a simulation of the process (60 million steps, about 5 000 and 1 800 cycles): within 2.2% and 0.9%, the size of the simulation error.

Files

MeanReversion.mqh goes to MQL5\Include\MeanReversion, SpreadBands.mq5 to MQL5\Scripts\MeanReversion.