فیلترهای معامله بهینه تحت فرصت های تجاری زودگذر: تصویر سازی تئوری و تجربی

If transitory profitable trading opportunities exist, transaction filters mitigate trading costs. We use a dynamic programming framework to design an optimal filter that maximizes after-cost expected returns. The filter size depends crucially on the degree of persistence of trading opportunities, transaction cost, and standard deviation of shocks. For daily dollar–yen exchange trading, the optimal filter can be economically significantly different from a naïve filter equal to the transaction cost. The candidate trading strategies generate positive returns that disappear after transaction costs. However, when the optimal filter is used, returns after costs remain positive and higher than for naïve filters

It is inarguable that opportunities for above-normal returns are available to market participants at some level. These opportunities may be exploitable for instance at an intra-daily frequency as a reward for information acquisition when markets are efficient, or at a lower frequency to market timers when markets are inefficient. By nature these profit opportunities are predicable but transitory, and transaction costs may be a major impediment in exploiting them.1 This paper explores the optimal trading strategy when transitory opportunities exist and transactions are costly. The model we present is applicable to the arbitrage of microstructure inefficiencies that require frequent and timely transactions, which may be largely riskless. An example is uncovered interest speculation in currency markets where a trader takes either one side of the market or the reverse. Alternatively, a trader arbitrages differences between an asset's return and that of one of its derivatives: going long on the arbitrage position or reversing the position and going short, as is the case in covered interest arbitrage. Alexander (1961) and Fama and Blume (1966) introduced “filter rules” according to which traders buy (sell) an asset only if its current price exceeds (is below) the previous local minimum (maximum) by more than X (more than Y) percent, where X and Y are parameters of the rule, commonly set equal and chosen in the range of 0.5–5% (e.g., Sweeney, 1986). The parameters X (and Y if different) determine a “band of inactivity” that prompts one to trade once a realization exceeds the local minimum or is below the local maximum by a certain percentage. A larger band of inactivity (larger X) filters out more trades, thus reducing transactions costs.2 The general idea of filters, in filter rules, as well as other trading rules, is that if the trade indicator is weak the expected return from the transaction may not compensate for the transaction cost. Lehmann (1990) provides an interesting alternative filter by varying portfolio weights according to the strength of the return indicators—in trading smaller quantities of the assets with the weaker trade indicators, transaction costs are automatically reduced relative to the payoff. Knez and Ready (1996) and Cooper (1999) explore different filters and find that the after-transaction-cost returns indeed improve compared to trading strategies with zero filter. The problem with the filter approach is that there is no way of knowing a priori which filter band is reasonable because the buy/sell signal and the transaction cost are not in the same units—the filter is the percentage by which the effective signal exceeds the signal at which a change in position first appears profitable before transaction costs, but this percentage bears no relation to the percentage return expected. This also implies that there is no discipline against data mining for researchers: many filters with different bands can be tried to fabricate positive net strategy returns. While Lehmann's (1990) approach provides more discipline as it specifies a unique strategy, the filter it implies is not generally optimal. The purpose of this paper is to design an optimal filter that a priori maximizes the expected return net of transaction cost. To accomplish this we employ a “parametric” approach (e.g., Balvers et al., 2000) that allows the trading signal and the transaction cost to be in the same units. In effect we convert a filter into returns space and then are able to derive the filter's optimal band. The optimal filter depends on the exact balance between maintaining the most profitable transactions and minimizing the transactions costs. The optimal filter (band) can be no larger than the transaction cost (plus interest). This is clear because there is no reason to exclude trades that have an immediate expected return larger than the transaction cost. In general, the optimal filter is significantly smaller than the transaction cost. This occurs when the expected return is persistent: even if the immediate return from switching is less than the transaction cost, the persistence of the expected return makes it likely that an additional return is foregone in future periods by not switching. Roughly, the filter must depend on the transaction cost, as well as a factor related to the probability that a switch occurs. Our model characterizes the determinants of the filter in general and provides an exact solution for the filter when zero-investment returns have a uniform distribution. In exploring the effect of transaction costs when returns are predictable, this paper has the same objective as Balduzzi and Lynch (1999), Lynch and Balduzzi (2000), and Lynch and Tan (2009).3 The focus of these authors, however, differs significantly from ours in that they consider the utility effects and portfolio rebalancing decisions, respectively, in a life cycle portfolio choice framework. They simulate the welfare cost and portfolio rebalancing decisions given a trader's constant relative risk aversion utility function, but they do not provide analytical solutions and it is difficult to use their approach to quantify the optimal trading strategies for particular applications. Our approach, in contrast, provides specific theoretical results yielding insights into the factors affecting optimal trading strategies. Moreover, our results can be applied based on observable market characteristics that do not depend on subjective utility function specifications. In contrast to Balduzzi and Lynch (1999), Lynch and Balduzzi (2000), and Lynch and Tan (2009), we sidestep the controversial issue of risk in the theory. This simplifies our analysis considerably and is reasonable in a variety of circumstances. First, we can think of the raw returns as systematic-risk-adjusted returns, with whichever risk model is considered appropriate. The systematic risk adjustment is sufficient to account for all risk as long as trading occurs at the margins of an otherwise well-diversified portfolio. Second, in particular at intra-daily frequencies, traders may create arbitrage positions so that risk is irrelevant. Third, in many applications risk considerations are perceived as secondary compared to the gains in expected return; if risk adjustments are relatively small so that the optimal trading rules are approximately correct then risk corrections can be safely applied to ex post returns. Our optimal filter will produce higher expected returns than the naïve strategies of either using no filter or using a transaction-cost-sized filter. We apply the optimal filter to a natural case for our model: daily foreign exchange trading in the yen/dollar market. As is well-known (e.g., Cornell and Dietrich, 1978; Sweeney, 1986; LeBaron, 1998; Gencay, 1999; Qi and Wu, 2006), simple moving-average trading rules improve forecasts of exchange rates and generate positive expected returns (with or without risk adjustment) in the foreign exchange market. However, for daily trading, returns net of transaction costs are negative or insignificant if no filter is applied (Neely and Weller, 2003).4 We find that for the optimal filter, the net returns are still significantly positive and higher than those when the filter is set equal to the transaction cost. Further, the optimal filter derived from the theory given a uniform distribution and two optimal filters derived numerically under normality and bootstrapping assumptions all generate similar results that are relatively close to the ex post maximizing filter for actual data. These results are important as they suggest an approach for employing trading strategies with filters to deal with transaction costs, without inviting data mining. The results also hint that in some cases the conclusion that abnormal profits disappear after accounting for transaction costs may be worth revisiting. Section 2 develops the theoretical model and provides a general characterization of the optimal filter for an ARMA(1,1) returns process with general shocks, as well as a specific formula for the case when the shocks follow the uniform distribution. In Section 3, we apply the model to uncovered currency speculation. We show first that the moving-average strategy popular in currency trading can be related to our ARMA(1,1) specification. We then use the first one-third of our sample to develop estimates of the returns process, which we employ to calculate the optimal filter for an ARMA(1,1), an AR(1), and two representative MA returns processes. The optimal filter is obtained from the theoretical model for the uniform distribution but also numerically for the normal distribution and the bootstrapping distribution. In Section 4, we conduct the out-of-sample test with the final two-thirds of the sample to compare mean returns from a switching strategy before and after transaction costs. The switching strategies are conducted under a variety of filters, including the optimal ones, for each of the ARMA(1,1), AR(1), and MA returns cases. Section 5 concludes the paper.

If transitory profitable trading opportunities exist, transaction filters are used in practice to mitigate trading costs; but the filter size is difficult to determine a priori. This paper uses a dynamic programming framework to design a filter that is optimal in the sense of maximizing expected returns after transaction costs. The optimal filter size depends negatively on the degree of persistence of the profitable trading opportunities, positively on transaction costs, and positively on the standard deviation of shocks. We apply our theoretical results to foreign exchange trading by parameterizing the moving average strategy often employed in foreign exchange markets. The parameterization implies the same decisions as the moving average rule in the absence of transaction costs, but has the advantage of translating the buy/sell signal into the same units as the transaction costs so that the optimal filter can be calculated. Application to daily dollar–yen trading demonstrates that the optimal filter is not solely of academic interest but may differ to an economically significant extent from a naïve filter equal to the transaction cost. This depends importantly on the time series process that we assume for the exchange rate dynamics. In particular, we find that for an AR(1) process, the optimal filter is close to the naïve transaction cost filter, but for the ARMA(1,1) process, the optimal filter is only around 30% of the naïve transaction cost filter, and for the more stable MA processes, the optimal filter is smaller still as a fraction of the transaction cost. Impressively, the ex ante optimal filters under the assumptions of uniform, normal, and bootstrap distributions are all very close to one another and all are quite close to the ex post after-cost return-maximizing level. We confirm that simple daily moving average foreign exchange trading generates positive returns that disappear after accounting for transaction costs. However, when the optimal filter is used, returns after transaction costs remain positive and are higher than for naïve filters. This result has implications beyond foreign exchange markets. It cautions against dismissing abnormal returns as due to transactions costs, merely because the after-cost return is negative or insignificant. For instance, Lesmond et al. (2004) argue convincingly that momentum profits disappear when actual transaction costs are properly considered, even after accounting for the proportion of securities held over in each period. But their after-cost returns are akin to those for our suboptimal zero filter strategy. It would be interesting to see what outcome would arise if an optimal filter were used. In our sample, the trading strategy returns, gross of transaction costs, are significantly positive, but no longer significant after transaction costs are subtracted. But if we optimally eliminate trades that do not make up for their transaction cost, then the after-cost profits are only slightly lower than the gross profits from unrestricted trading and are statistically significant. They are also economically significant, around 0.5% per month after transaction costs, which raises the issue of market efficiency. The profits are of similar magnitude as the momentum profits after transaction costs and may in fact be closely related to the momentum phenomenon. However, given the lower total variance of the trading strategy returns,23 it is even more difficult here, compared to the momentum case for equity returns, to argue that an unobserved systematic risk is responsible. So the “anomaly” may be exploitable and, in the absence of a risk explanation, could suggest market inefficiency. Apart from the practical advantages of using the optimal filter, there is also a methodological advantage: in studies attempting to calculate abnormal returns from particular trading strategies in which transaction costs are important, there is no guideline as to what filter to use in dealing with transaction costs. Lesmond et al. (2004, p. 370) note: “Although we observe that trading costs are of similar magnitude to the relative strength returns for the specific strategies we consider, there is an infinite number of momentum-oriented strategies to evaluate, so we cannot reject the existence of trading profits for all strategies”. Rather than allowing the data-mining problem that is likely to arise when a variety of filter sizes are applied, our approach here provides a unique filter in Eq. (13) that can be unambiguously obtained in advance from observable variables.