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  Technical Indicators in Automated Trading

Deploying Advanced Mathematical Strategies for Automated FOREX Trading

By Andrew Luciani

The foreign exchange market is unique because of its huge trading volume, leading to high liquidity, continuous operation … the low margins of relative profit … the use of leverage to enhance profit margins with respect to account size".

The word 'profit' is used twice in the WiKi article, but your equity and the time you spend in (real) trading the FOREX can be highly uncorrelated. The Adaptative Market Hypotesis implicates (Prof. Andrew Lo, 2004):

1. To the extent that a relation between risk and reward exists, it is unlikely to be stable over time.
2. Investment strategies perform well in certain environments and poorly in other ones. This includes quantitatively-, fundamentally and technically-based methods.
3. Survival is the only objective that matters while profit and utility maximization are secondary relevant aspects.
4. Innovation is the key to survival because as risk/reward relation varies through time, the best way of achieving a consistent level of expected returns is to adapt to changing market conditions.

5. Roughly speaking, the FOREX is the hardest market, quite random, highly noisy, especially on intraday basis. On the EUR / USD daily log returns the null hypothesis of random walk cannot be rejected, therefore this market is considered weak-form efficient: a single-series quantitative approach drives straight to over-fitting. Macro environment series (e.g. LIBOR rates, Consumer Confidence, CPI, etc.) have to be accounted in a multi-market, multi-dimensional model in order to extract hyper-volumes (i.e. patterns) wherein a possible dynamic is probable with a statistical significance.

   It sounds complex and it is indeed complex. It requires knowledge of advanced math, non-parametric distance-based or kernel regression, neural networks. Noise modeling and filtering are critical steps in designing a robust mathematical strategy. Physics and mechanics can help. A simple example: an ideal mass-spring-damper system (or a modern shock absorber) with mass m, spring constant k and viscous damper of damping coefficient c. The differential equation is Eq. 1.

   

   (Eq. 1)

   Replace x with the difference between a noisy series (e.g. share or currency prices, TED spread, etc.) and the filter output. The critical damped filter (i.e. It converges to zer0 faster than any other, and without oscillating.

   ζ = 1, Eq. 2) works better than SMA and EMA, and can also be applied in confidence bands modeling. Setting an over-shooting of 2% or 5% (filter under dumped) can reduce the lag, and can help in short timeframes (daily to intraday).

   
   

   (Eq. 2)

   In advanced model k=f(x) or k=f(stdev(x)) or m=f(bar volume). Few lines of code and MATLAB outs numerator and denominator of the transfer function (in the example code a basic system with 2% overshooting):

   period=T;
   overshooting=0.02;
   mm=abs(log(overshooting)/pi);
   zeta=sqrt(mm^2/(mm^2+1));
   teta=acos(zeta);
   omega=(pi-teta)/(period*sqrt(1-zeta^2));
   sys2ord=tf([omega^2],
   [1 2*zeta*omega omega^2]);
   sysdis=c2d(sys2ord,1);
   [num,den]=tfdata(tf(sysdis),'v')

   The second-degree denominator also implies a fast calculation. The design of an advanced parametric adaptive filter requires a robust regression in order to avoid over fitting. Example: let's find the linear function f such that

   k = f(stdev(x)) = a+b*stdev(x)

   maximizes the filter response with respect to an objective function (e.g. profit, Sharpe, lag, etc.). Now split your domain, for example 10 FOREX series with a 20-year-historical-daily data, in small 'reasonable' sub-sets, let's say 200 sub-sets of a year of historical data. Then for each sub-set find the value of k that maximizes your objective function (with an exhaustive search or a Montecarlo method), and calculate the value of the stdev of prices in that sub-set. Create a matrix M with the pairs 3Cstdevi, ki3E, i=1..200 and finally get the two values of a and b with a single line of MATLAB code:

   [a b] = robustfit(M);

   Always test your results in out-of-sample data.

   Please note: this is not the definitive way to trade the FOREX, these are just suggestions in how to build robust toolboxes of analysis. While the above damper filter can be useful in building proxies or in identifying trends, the calculation of the instant frequency and the phase of a series can help in dealing with cycles.

   Again MATLAB:
   f = [0.3 0.7];
   m = [ 1 1];
   n=6;
   b = firgr(n,f,m,'hilbert');

   Then with the simple Hilbert (Fig. 1) filter we can calculate in-phase and quadrature components and finally the phase of the signal/series, with a small group-delay, i.e. lag. An example with CHF / JPY weekly data is in Fig. 2.

   

   Fig. 1 - Hilbert filter, Magnitude and Phase res.

   

   Fig. 2 - CHF / JPY weekly data and phase, May 2010

   While the phase can be useful in identification of overbought and oversold region for prices, or confidence regions for a macroeconomic series, it is clearly not a good idea to use it as a stand-alone support tool for your trading decisions.

   'When' to trade, i.e. market timing, is a key factor in a good investment strategy, but also the choice of 'what' to trade is mainly relevant. Focusing on a single market can be unsafe. Alpha and Beta analysis can help in filtering the domain of securities and in identifying the best opportunities. Few definitions can help the reading. The Alpha is a risk-adjusted measure of the so-called active return on an investment. The return of a benchmark (i.e. a stock index value, the dollar index, the free-risk rate, etc) is subtracted from the return of the investment in order to consider relative performance. The Beta of a stock or portfolio is a number describing the relation of its returns with that of the financial market (or the benchmark) as a whole. An asset with a beta of 0 means that its price is not at all correlated with the market. A positive beta means that the asset generally follows the market. A negative beta shows that the asset inversely follows the market; the asset generally decreases in value if the market goes up and vice versa. Last definition: momentum is the empirically observed tendency for rising asset prices (stocks) to raise further. This is true in economic expansion, while in recession the momentum has to be sold. The same kind of 'inertia' can also be found in currencies, at large timeframes, weekly to monthly. Translating these words into a math language means the selection of securities with a specific (smoothed) value of Alpha and Beta. The chart in Figure 3 gives a graphical idea of this approach: the normalized prices of almost all of Dow Jones Industrial index components are plotted on the chart. For a long/short hedge fund, i.e. a hedge fund that buy long equities that are expected to increase in value and short sell equities that are expected to decrease in value, the stocks that offer the best long/short opportunities, with a statistical significance, are the red ones. A similar approach can be used with currencies, at large timeframes, weekly to monthly. In economic expansion, the stocks plotted in red in the upper part of the chart are the stocks with a high Alpha and are expected to present a positive Alpha also in the near future. The stocks with a low Alpha, the red ones in the bottom part of the chart, are expected to recover faster than other stocks in an overall economic expansion (in condition that they are not removed from Index components): this is well known as the mean-revert effect (in US markets). During a recession the best short opportunities in US markets arise from the stocks with a high Alpha. In other countries there is still a down-trend inertia, so the best short set-ups arise from low-Alpha securities. However, it is not a good idea to use Alpha/Beta filtering as a stand-alone support tool for your trading decisions.

   

   Fig. 3 - DJI components, May 2008 - May 2010

   

   Fig. 4 - Libor rates, 1997 - 2008

   Last but not least, some considerations in international interest rates should be taken into consideration. In Fig. 4, the chart presents LIBOR 3-month rates from 1997 to June 2008, (just before the subprime mortgage crisis). LIBOR, or London Interbank Offered Rate, are daily reference rates based on the interest at which banks borrow unsecured funds from other banks. Identifying currencies with highest and lowest LIBOR, can help in the selection of crosses with the best trading opportunities, i.e. volatility, because of the Currency Carry Trade.

   In the Currency Carry Trade investors borrow low-yielding currencies and lend (invest in) high-yielding currencies: borrow cheap (better if overnight) and lend expensive (better if in the long-term). In the chart the currency related to the lowest LIBOR rate is JPY, while the currency related to the highest LIBOR rate is (approximately) the GBP from 1997 to 2001, then the AUD from 2001 to the present.

   The crash of the GBP / JPY cross in Oct-1998 and the crash of the AUD / JPY cross in Oct-2008 can also be related to these considerations. Please consider all the material presented in this paper just as a suggestion in how to build robust toolboxes of analysis.