Automated FX Trading System Using ARL

Authors: M. Dempster & V. Leemans
Date: 2004 Website: http://www.cfr.statslab.cam.ac.uk/publications/papers.html

• Adaptative Reinforcement Learning ⇒ ARL
• 3 layers structure:

• Machine Learning Algorithm:
  • Using Recurrent Reinforcement Learning (RRL)

• Risk Management Overlay:
  • Restrain/shutdown trading when in high uncertainty.

• Dynamic Utility Optimization:
  • Render selection of fixed meta-parameters useless.
  • Allow risk-return trade-off control by user.

• Machine Learning layer + Dynamaic optimization layer ⇒ Adaptative Reinforcement Learning.

• Model used: \(F_t = sign\left( \sum\limits_{i=0}^M (w_{i,t} \cdot r_{t-i}) + w_{M+1,t} \cdot F_{t-1} + \nu_t\right)\)

• \(F_t \in \{-1, 1\}\) is the position to take at time t.
• \(w_t\) is the weight vector at time t.
• \(\nu_t\) is the threshold at time t.
• \(r_t = p_t - p_{t-1}\) are the raw return values at time t.

• At evrey trade we buy/sell 1 unit of the currency pair.
• Profit at time T can be calculated as (ignoring interest rates):

\[\begin{align}P_T & = \sum\limits_{t=0}^T R_t \\ R_t & = F_{t-1} \cdot r_t - \delta |F_t - F_{t-1}|\end{align}\]

• Where \(\delta\) is the transaction cost per trade.

• We then define a differential sharpe ratio considering a moving average version of the classical sharpe ratio:

\[\begin{align}\hat{S}(t) & = \frac{A_t}{B_t} \\ A_t & = A_{t-1} + \eta(R_t - A_{t-1}) \\ B_t & = B_{t-1} + \eta (R_t^2 - B_{t-1})\end{align}\]

• We then expand \(\hat{S}(t)\) into a Taylor serie in \(\eta\), and we can consider the first derivative as an instantaneous performance measure:

\[D_t = {\frac{d\hat{S}(t)}{d\eta}}_{|\eta=0} = \frac{B_{t-1}\Delta A_t - \frac 12 A_{t-1} \Delta B_t}{(B_{t-1} - A_{t-1}^2)^{\frac 32}}\]

• We then use a simple gradient descent method to update the weights: \(w_{i,t} = w_{i,t-1} + \rho \Delta w_{i,t}\)

• In case of online learning we can consider only the term that depends on the most recent return \(R_t\), so we get:

\[\Delta w_{i,t} = \frac{dD_t}{dw_i} \approx \frac{dD_t}{dR_t} \left( \frac{dR_t}{dF_t} \cdot \frac{dF_t}{dw_{i,t}} + \frac{dR_t}{dF_{t-1}} \cdot \frac{dF_{t-1}}{dw_{i,t-1}}\right)\]

• Considering that the neural network is recurrent, we can get:

\[\frac{dF_t}{dw_{i,t}} \approx \frac{\partial F_t}{\partial w_{i,t}} + \frac{\partial F_t}{\partial F_{t-1}} \cdot \frac{dF_{t-1}}{dw_{i,t}}\]

• The system is then trained on \(n_e = 10\) epochs on training set of length \(L_{train} = 2000\) ticks [optimal value].
• Then we test it on test set of length \(L_{test} = 500\) ticks [optimal value].

• Extended to take into account other inputs such as signals from 14 technical indicators. ⇒ This didn't improve the results.

• During training phase the transaction cost \(\delta\) is left as a tuning parameter.

• To prevent too big weights, all weights are rescaled by a factor \(f \lt 1\) as soon as a threshold value is hit.

• Improved position updating scheme: recalculating the output \(F_t\) twice:
  1. As before
  2. After the weights are updated ⇒ provide a more accurate value which might be different.

• Build a trailing stop-loss which is always adjusted x points under or above the best price reached during the life of a position.

• If a position is closed because of the stoploss, then a cool-down period [⇒ using 1 minute] is used before trading again.

• This layer can evaluate the strength of the signal received from the NN using the non-thresholded value (inside the sign() function).

• A threshold y can be provided by the optimization layer to only enter a position when we have an higher certainty.

• A maximum draw-down system [parameter z] is implemented to prevent complete failure of the trading system.

• Definition of the risk measure:

\[\Sigma = \frac{\sum_{i=0}^N (R_i)^2 I\{R_i \lt 0\}}{\sum_{i=0}^N (R_i)^2 I\{R_i \gt 0\}}\]

Here \(R_i\) is the raw return at time i: \(R_i = W_i - W_{i-1}\), with \(W_i\) the cumulative profit at time i.

• Then we define the utility function: \(U(\bar{R},\Sigma,\nu) = a \cdot (1 - \nu) \bar{R} - \nu \cdot \Sigma\), with:
  • \(\nu\): Risk aversion parameter.
  • \(\bar{R} = \frac{W_N}{N}\) : average profit per frequency interval.

• The parameter optimization problem becomes:

\[\max_{\delta, \eta, \rho, x, y} U(\bar{R},\Sigma : \delta, \eta, \rho, x, y)\]

⇒ Implemented as a one-at-a-time random search optimization (using 15 random evaluation around current value for each parameter).

• Tested with the EURUSD pair:
  • Frequency of 1 minute from January 200 to January 2002.
  • Spread of 2 pips.
• Trading between 9 am and 5 pm (london time)
• Interdealer platform: EBS / Reuters3000
• Used risk aversion value \(\nu = 0.5\)
• ⇒ Earned 5104 pips over 2 years

• Future work:
  • In [2,7] it has demonstrated that order book or order flow information could enhance the perfs.
  • Risk management layer could control multiple trading systems for different currencies.