量化投资之一：配对交易（英文期刊）
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The investment strategy we aim at implementing is a market neutral long/short strategy. This implies that we will try to find shares with similar betas, where we believe one stock will outperform the other one in the short term. By simultaneously taking both a long and short position the beta of the pair equals zero and the performance generated equals alpha.Needless to mention, is that the hard part of this strategy is to find market neutral positions that will deviate in returns. To do this we can use a statistical tool developed by Schroder Salomon Smith Barney (SSSB). The starting point of this strategy is that stocks that have historically had the same trading patterns (i.e. constant price ratio) will have so in the future as well. If there is a deviation from the historical mean, this creates a trading opportunity, which can be exploited. Gains are earned when the price relationship is restored. The historical calculations of betas and the millions of tests executed are done by SSSB, but it is our job as portfolio managers to interpret the signals and execute the trades.Summary: find two stocks prices of which have historically moved together,
 mean reversion in the ratio of the prices, correlation is not key
 Gains earned when the historical price relationship is restored
 Free resources invested in riskfree interest rate
2.2  Testing for the mean reversion 

The challenge in this strategy is identifying stocks that tend to move together and therefore make potential pairs. Our aim is to identify pairs of stocks with meanreverting relative prices. To find out if two stocks are meanreverting the test conducted is the DickeyFuller test of the log ratio of the pair. In theA DickeyFuller test for determining stationarity in the logratio y_{t}=logA_{t}log B_{t} of share prices A and BD y_{t} = ? + g y_{t1} + e_{t} (17)
In other words, we are regressing D y_{t} on lagged values of y_{t}.the null hypothesis is that g=0, which means that the process is not mean reverting.If the null hypothesis can be rejected on the 99% confidence level the price ratio is following a weak stationary process and is thereby meanreverting. Research has shown that if the confidence level is relaxed, the pairs do not meanrevert good enough to generate satisfactory returns. This implies that a very large number of regressions will be run to identify the pairs. If you have 200 stocks, you will have to run 19 900 regressions, which makes this quite computerpower and time consuming.Schroder Salomon Smith Barney provide such calculationBy conducting this procedure, a large number of pairs will be generated. The problem is that all of them do not have the same or similar betas, which makes it difficult for us to stay market neutral. Therefore a trading rule is introduced regarding the spread of betas within a pair. The beta spread must be no larger than 0.2, in order for a trade to be executed. The betas are measured on a twoyear rolling window on daily data.We now have meanreverting pairs with a limited beta spread, but to further eliminate the risk we also want to stay sector neutral. This implies that we only want to open a position in a pair that is within the same sector. Due to the different volatility within different sectors, we expect sectors showing high volatility to produce very few pairs, while sectors with low volatility to generate more pairs. Another factor influencing the number of pairs generated is the homogeneity of the sector. A sector like Commercial services is expected to generate very few pairs, but Financials on the other hand should give many trading opportunities. The reason why, is that companies within the Financial sector have more homogenous operations and earnings.The screening process described gives us a large set of pairs that are both market and sector neutral, which can be used to take positions. This should not be done randomly, since timing is an important issue. We will therefore introduce several trading execution rules.All the calculations described above will be updated on a daily basis. However, we will not have to do this ourselves, but we will be provided with updated numbers every day, showing pairs that are likely mean revert within the next couple of weeks. In order to execute the strategy we need a couple of trading rules to follow, i.e. to clarify when to open and when to close a trade. Our basic rule will be to open a position when the ratio of two share prices hits the 2 rolling standard deviation and close it when the ratio returns to the mean. However, we do not want to open a position in a pair with a spread that is wide and getting wider. This can partly be avoided by the following procedure: We actually want to open a position when the price ratio deviates with more than two standard deviations from the 130 days rolling mean. The position is not opened when the ratio breaks the twostandarddeviations limit for the first time, but rather when it crosses it to revert to the mean again. You can say that we have an open position when the pair is on its way back again (see the picture below).
Figure 1: Pairs Trading rules
summary: Open position when the ratio hits the 2 standard deviation band for two consecutive times
 Close position when the ratio hits the mean
Furthermore, there will be some additional rules to prevent us from loosing too much money on one single trade. If the ratio develops in an unfavourable way, we will use a stoploss and close the position as we have lost 20% of the initial size of the position. Finally, we will never keep a position for more that 50 days. On average, the mean reversion will occur in approximately 35 days , and there is no reason to wait for a pair to revert fully, if there is very little return to be earned. The potential return to be earned must always be higher than the return earned on the benchmark or in the fixed income market. The maximum holding period of a position is therefore set to 50 days. This should be enough time for the pairs to revert, but also a short enough time not to loose time value.The rules described are totally based on statistics and predetermined numbers. In addition, there is a possibility for us to make our own decisions. If we for example are aware of fundamentals that are not taken into account in the calculations and that indicates that there will be no mean reversion for a specific pairs, we can of course avoid investing in such pairs.From the rules it can be concluded that we will open our last position no later than 50 days before the trading game ends. The last 50 days we will spend trying to close the trades at the most optimal points of time.Summary: Stop loss at 20% of position value
 Beta spread <0.2
 Sector neutrality
 Maximum holding period < 50 trading days
 10 equally weighted positions
As already mentioned, through this strategy we do almost totally avoid the systematic market risk. The reason there is still some market risk exposure, is that a minor beta spread is allowed for. In order to find a sufficient number of pairs, we have to accept this beta spread, but the spread is so small that in practise the market risk we are exposed to is ignorable. Also the industry risk is eliminated, since we are only investing in pairs belonging to the same industry.The main risk we are being exposed to is then the risk of stock specific events, that is the risk of fundamental changes implying that the prices may never mean revert again, or at least not within 50 days. In order to control for this risk we use the rules of stoploss and maximum holding period. This risk is further reduced through diversification, which is obtained by simultaneously investing in several pairs. Initially we plan to open approximately 10 different positions. Finally, we do face the risk that the trading game does not last long enough. It might be the case that our strategy is successful in the long run, but that a few short run failures will ruin our overall excess return possibilities.2.7  General Discussion on pairs trading 

There are generally two types of pairs trading: statistical arbitrage convergence/divergence trades, and fundamentallydriven valuation trades. In the former, the driving force for the trade is a aberration in the longterm spread between the two securities, and to realize the meanreversion back to the norm, you short one and go long the other. The trick is creating a program to find the pairs, and for the relationship to hold.The other form of pairs trading would be more fundamentallydriven variation, which is the purvey of most marketneutral hedge funds: in essence they short the most overvalued stock(s) and go long the undervalued stock(s). It's normal to "pair" up stocks by having the same number per sector on the long and short side, although the traditional "pairs" aren't used anymore. Pairs trading had originally been the domain of BD's in the late 70's, early 80's before it dissipated somewhat due to the bull market (who would want to be marketneutral in a rampant bull market), and the impossibility of assigning "pairs" due to the morphing of traditional sectors and constituents. Most people don't perform traditional "pairs trading" anymore (i.e. the selection of two similar, but mispriced, stocks from the same industry/sector), but perform a variation. Goetzmann et al wrote a paper on it a few years back, but at the last firm I worked at, the research analyst "poohpaahed" it because he couldn't get the same results: he thinks Goetzmann [7] either waived commissions, or worse, totally ignored slippage (i.e. always took the best price, not the realistically one). Here's the paper : source: forum http://www.wilmott.comsome quotations from this paper: "take a long?short position when they diverge." A test requires that both of these steps must be parameterized in some way. How do you identify "stocks that move together?" Need they be in the same industry? Should they only be liquid stocks? How far do they have to diverge before a position is put on? When is a position unwound? We have made some straightforward choices about each of these questions. We put positions on at a two?standard deviation spread, which might not always cover transactions costs even when stock prices converge. Although it is tempting to try potentially more profitable schemes, the danger in data?snooping refinements outweigh the potential insights gained about the higher profits that could result from learning through testing. As it stands now, data?snooping is a serious concern in our study. Pairs trading is closely related to a widely studied subject in the academic literature ?? mean reversion in stock prices. 2 We consider the possibility that we have simply reformulated a test of the previously documented tendency of stocks to revert towards their mean at certain horizons. To address this issue, we develop a bootstrapping test based upon random pair choice. If pairs?trading profits were simply due to mean?reversion, then we should find that randomly chosen pairs generate profits, i.e. that buying losers and selling winners in general makes money. This simple contrarian strategy is unprofitable over the period that we study, suggesting that mean reversion is not the whole story. Although the effect we document is not merely an extension of previously known anomalies, it is still not immune to the data?snooping argument. Indeed we have explicitly "snooped" the data to the extent that we are testing a strategy we know to have been actively exploited by risk? arbitrageurs. As a consequence we cannot be sure that past trading profits under our simple strategies will continue in the future. This potential critique has another side, however. The fact that pairs trading is already well?known risk?arbitrage strategy means that we can simply test current practice rather than develop our filter?rule ad hoc.
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