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110 The mean utility , can be transformed back to an effective return value by applying Equation 11.17: A/fy/. , = AR(uAt) (11.20) This ARefftAt is the typical, effective return for the horizon At, but it is not yet annualized. As in the case of the Xeg, an annualization is necessary for a comparison between Reg values for different intervals. The annualization factor, is the ratio of the number of Af in a year divided by the number of nonoverlapping /s in the full sample of size T. We have Reff,ann,At = AAt AReff.At (1121) To achieve a measure that simultaneously considers a wide range of horizons, we define Reff as a weighted mean over all the n horizons, p wi Reff,ann.Atj w Reff = v» in (1122) where the weights , are chosen according to the relative importance of the time horizons Atj and may differ for trading models with different trading frequencies. In the case of the trading models described in this book, we have choosen a weighting function Wi = w(Ati) = l (1L23> 2+(log9Tjfe) with the maximum for a Af of 90 days. Both Xeff and Reff are quite natural measures. They treat risk as a discount factor to the value of the investment. In other words, the performance of the model is discounted by the amount of risk that was taken to achieve it. In the Xeg case the risk is treated similarly both for positive or negative outcome, whereas in the case of Reff negative performance is more penalized. 11.4 trading model algorithms We turn now to the description of the techniques used to build the realtime trading models. Trading models have been developed for many decades by a large number of people and applied in all types of financial markets. These models have been designed from a broad class of indicators ranging from classical technical analysis up to chaotic theory. It is just not possible to provide a comprehensive list of the various approaches used in trading system design and the literature in this field is so large that we will leave this exercise to other authors. As in the case of technical analysis, hundreds of articles and books have been written.
Generally, trading systems are built from a few classes of indicators providing specific types of information on the underlying financial time series. For instance, we have these: The trend following indicators, which allow to detect and follow major market trends. The overbought and oversold indicators, which allow to detect important market turning points. The cycle indicators, which try to emphasize periodic market fluctuations. The timing indicators, which provide optimum exit conditions. As an example we will describe one model that we have developed for the FX market and that many large banks have actively used for a decade. As we have pointed out earlier, a trading strategy is built from some indicators and a set of decision rules. Indicators are variables of the trading system algorithm whose values, together with the system rules, determine the trading decision process. In Chapter 3.3 we gave different descriptions of indicators that have been used in conjunction with trading models. 11.4.1 An Example of a Trading Model The realtime trading model (RTT) studied in this section is classified as a onehorizon, highrisk/highreturn model. The RTT is a trendfollowing model and takes positions when an indicator crosses a threshold. The indicator is momentum based, calculated through specially weighted moving averages with repeated application of the exponential moving average operator (see Section 3.3.6). In the case of extreme foreign exchange movements, however, the model adopts an overbought/oversold (contrarian) behavior and recommends taking a position against the current trend. The contrarian strategy is governed by rales that take the recent trading history of the model into account. The RTT model goes neutral only to save profits or when a stoploss is reached. Its profit objective is typically at 3%. When this objective is reached, a gliding stoploss prevents the model from losing a large part of the profit already made by triggering it to go neutral when the market reverses. At any point in time t, the gearing function for the RTT is g,(Ix) = sign(Ix(t)) /(/,(OI) cUit)) where Ix(t)=x, MA(r = 20 days, 4; x) where xt is the logarithmic price at time t and the moving average (MA) of x follows the definition and notation of Equation 3.56 (where the last argument x of
MA indicates the time series to which the MA operator is applied), /(IMOI) = if if I/* (01 I/* (01 if \>x(t)\<a 0.5 0 + 1 if \IAt)\<d 1 if \Ix(t)\ > d and g,. sign(Ix(t)) > 0 and n > P where a < b < d and / is the return of the last deal and P the profit objective. The function, f(\Ix(t)\), measures the size of the signal at time t and the function, c(\Ix\), acts as a contrarian strategy. The model will enter a contrarian position only if it has reached its profit objective with a trend following position. In a typical year, the model will play against the trend two to three times while it deals roughly 60 to 70 times. The hit rate of the contrarian strategy is of about 75%. The parameters a and b depend on the position of the model, a(t) if if 8t8t 0 = 0 and b = 2a. The thresholds are also changed if the model is in a position g, 0 and the volatility of the price has been low, in the following way: a{i) = if if \xe \xP x, I > V x,\ < V where xe is the logarithmic entry price of the last transaction and is a threshold, generally quite low < 0.5%. This means that the model is only allowed to change position if the price has significantly moved from the entry point of the deal. Because Xeg and Reg are implicit functions of the gearing, the optimization of the RTT model is based on the Xeg and Reg performance. The parameters subject to optimization are; r, a, d, and v. There are two other auxiliary parameters, which are the stop loss, S, at which an open position is automatically closed and the profit objective, P. These parameters are only optimized at the end once the others have been found and they are also not allowed to vary all the way because maximum stoploss and maximum gain limits are set by the environment.8 The model is subject to the openclose and holiday closing hours of the Zurich market. 11.4.2 Model Design with Genetic Programming The major problem with trading models is the large amount of time needed to develop and optimize new trading strategies. As we said before, a trading strategy is a small computer program composed of some indicators to forecast price trends combined with a set of rules to determine the trading decision process. 8 For more details on the optimization procedure, see Pictet et al. (1992).
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