Optimizing trading strategies without overfitting By Ernest Chan and Ray Ng

Here is a somewhat trivial example of this procedure. We want to find an optimal strategy that trades  AUDCAD on an hourly basis. First, we fit a AR(1)+GARCH(1,1) model to the data using log midprices. The maximum likelihood fit is done using a one-year moving window of historical prices, and the model is refitted every month. We use MATLAB's Econometrics Toolbox for this fit. Once the sequence of monthly models are found, we can use them to predict both the log midprice at the end of the hourly bars, as well as the expected variance of log returns. So a simple trading strategy can be tested: if the expected log return in the next bar is higher than K times the expected volatility (square root of variance) of log returns, buy AUDCAD and hold for one bar, and vice versa for shorts. But what is the optimal K?

Following the procedure outlined above, each time after we fitted a new AR(1)+GARCH(1, 1) model, we use this to simulate the log prices for the next month's worth of hourly bars. In fact, we simulate this 1,000 times, generating 1,000 time series, each with the same number of hourly bars in a month. Then we simply iterate through all reasonable value of K and remember which K generates the highest Sharpe ratio for each simulated time series. We pick the K that most often results in the best Sharpe ratio among the 1,000 simulated time series (i.e. we pick the mode of the distribution of optimal K's across the simulated series). This is the sequence of K's (one for each month) that we use for our final backtest. Below is a sample distribution of optimal K's for a particular month, and the corresponding distribution of Sharpe ratios: