**Maximizing Returns: Forex Trading And Mining Strategies In Ontario** – Optimizing the parameters of a trading strategy using backtesting has a major problem: there are usually not enough historical trades to achieve statistical significance. Whichever optimal parameter is found is likely to suffer from data espionage bias, and there may be nothing optimal about it in the out-of-sample period. That is why parameter optimization of trading strategies often does not add value. On the other hand, optimizing the parameters of a time series model (such as a maximum likelihood fit to an autoregressive or GARCH model) is more robust, since the input data is prices, not transactions, and we have many prices. Fortunately, it turns out that there are clever ways to take advantage of the ease of optimizing time series models to optimize the parameters of a trading strategy.

An elegant way to optimize a trading strategy is to use the methods of stochastic optimal control theory; elegant, that is, if you are mathematically sophisticated and able to solve the Hamilton-Jacobi-Bellman (HJB) equation analytically (see Cartea et al.). Even then, this will only work when the underlying time series is well known, such as the process Ornstein-Uhlenbeck (OU) continuum underlying all mean reversion price series. This OU process is clearly represented by a stochastic differential equation. Furthermore, the HJB equations can usually only be solved exactly if the objective function has a simple form, such as a linear function. If your price series is clearly represented by an OU process and your goal is profit maximization, which happens to be a linear function of the price series, then stochastic optimal control theory will give you the analytically optimal trading strategy: with entry exact and output thresholds given as functions of organizational unit process parameters. It is no longer necessary to find optimal thresholds by trial and error during a tedious backtesting process, a process that invites overfitting a small number of operations. As we indicated above, the parameters of the OU process can be adjusted quite robustly to prices, and indeed there is a maximum likelihood analytical solution for this adjustment given in Leung et al. Alabama.

## Maximizing Returns: Forex Trading And Mining Strategies In Ontario

In many optimization problems, when there is no optimal analytical solution, simulations are often used. Examples of such methods include simulated annealing and Markov Chain Monte Carlo (MCMC). We’ll do the same here: if we couldn’t find an analytical solution for our optimal trading strategy, but were able to fit our underlying price series fairly well to a standard discrete time series model like ARMA, then we can simply simulate many instances of the price series. underlying prices. We will backtest our trading strategy at each instance of the simulated price series and find the best trading parameters that most frequently generate the highest Sharpe ratio. This process is much more robust than backtesting the real time series, because there is only one real price series, but we can simulate as many price series (all following the same ARMA process) as we want. That means we can simulate as many trades as we want and get optimal trading parameters with as high a precision as we want. This is almost as good as an analytical solution.

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Here is a somewhat trivial example of this procedure. We want to find an optimal strategy that trades AUDCAD every hour. First, we fit an AR(1)+GARCH(1, 1) model to the data using

Average prices. The maximum likelihood fit is performed using a one-year rolling window of historical prices, and the model is refitted each month. We use MATLAB’s Econometric Toolbox for this fit. Once the sequence of monthly models is found, we can use them to predict both the log mean price at the end of the hourly bars and the expected variance of the log returns. Therefore, a simple trading strategy can be tried: if the expected log return on the next bar is greater than K times the expected volatility (square root of the variance) of the log returns, buy AUDCAD and hold it for one bar, and vice versa for short positions. But what is the optimal K?

Following the procedure described above, each time we fit a new AR(1)+GARCH(1, 1) model, we use this to

The prices of the registers for the hourly bars of the next month. In fact, we simulate this 1,000 times, generating 1,000 time series, each with the same number of hourly bars in a month. We then simply iterate over all reasonable values of K and remember which K generates the highest Sharpe ratio for each simulated time series. We choose the K that most often results in the best Sharpe ratio among the 1,000 simulated time series (i.e., we choose the

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Of the distribution of optimal K throughout the simulated series). This is the sequence of K (one for each month) that we use for our final backtest. Below is a sample distribution of K optimals for a particular month and the corresponding distribution of Sharpe ratios:

Interestingly, the mode of the optimal K is 0 for any month. This certainly makes for a simple trading strategy: simply buy whenever the expected log return is positive, and vice versa for short positions. The CAGR is approximately 4.5% assuming zero transaction costs and mid-price executions. Here is the cumulative yield curve:

You may exclaim, “This can’t be optimal, because I can trade AUDCAD hourly bars with much better returns and Sharpe ratio!” Of course, optimal in this case only means optimal within a certain universe of strategies, and assuming an underlying price series model AR(1)+GARCH(1, 1). Our universe of strategies is fairly simplistic: simply buy or sell based on whether the expected return exceeds a multiple of the expected volatility. But this procedure can be extended to any price series model you assume and any universe of strategies you can think of. In all cases, it greatly reduces the possibility of overfitting.

We invented this procedure for our own use a few months ago, borrowing similar ideas from Dr. Ng’s computational research in condensed matter physics systems (see Ng

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Here or here). But later we discovered that a similar procedure had already been described in a paper by Carr et al.

Ernie is a leading quantitative hedge fund manager and quantitative finance author. He has previously applied his expertise in machine learning IBM T.J. Watson Research Center’s Human Language Technologies group, Morgan Stanley’s Artificial Intelligence and Data Mining Group, and Credit Suisse’s Horizon Trading Group.

Ray Ng is a Quantitative Strategist at QTS. He received his doctorate. in theoretical condensed matter physics from McMaster University.

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