His book is a careful, detailed exposition of the scientific method applied to strategy development. For serious retail traders, I know of no other book that provides this range of examples and level of detail. His discussions of how regime changes affect strategies, and of risk management, are invaluable bonuses. Chan has created a practical guide to algorithmic tradingstrategies that can be readily implemented by both retail andinstitutional traders alike.

More than an academic treatise onfinancial theory, Algorithmic Trading is an accessible resourcethat blends some of the most useful financial research done in thelast few decades with valuable insights Dr.

Chan has gained fromactually exploiting some of those theories in live trading. Engaging and informative, Algorithmic Trading skillfully covers awide array of strategies. Broadly divided into the mean-revertingand momentum camps, it lays out standard techniques for tradingeach category of strategies and, equally important, the fundamentalreasons why a strategy should work.

The emphasis throughout is onsimple and linear strategies, as an antidote to the over-fittingand data-snooping biases that often plague complex strategies. This book stays true to that view by using a level ofmathematics that allows for a more precise discussion of theconcepts involved in financial markets.

While Algorithmic Tradingcontains an abundance of strategies that will be attractive to bothindependent and institutional traders, it is not a step-by-stepguide to implementing them. Personalized experience. Get started with a FREE account. Published in: Lifestyle.

Full Name Comment goes here. Are you sure you want to Yes No. No problem. Erica Bryant Hi there! I just wanted to share a list of sites that helped me a lot during my studies Often, setting the look- back to equal a small multiple of the half-life is close to optimal, and doing so will allow us to avoid brute-force optimization of a free parameter based on the performance of a trading strategy.

CAD is not stationary with at least 90 percent probability. To determine whether USD. CAD is a good candidate for mean reversion trading, we will now determine its half-life of mean reversion. The regression function ols as well as the function lag are both part of the jplv7 package. This code fragment is part of stationaryTests. Depending on your trading horizon, this may or may not be too long. But at least we know what look-back to use and what holding period to expect.

The look-back for the moving average and standard deviation can be set to equal the half-life. We see in Example 2. You might wonder why it is necessary to use a moving average or standard deviation for a mean-reverting strategy at all. Though we usually assume the mean of a price series to be fixed, in practice it may change slowly due to changes in the economy or corporate management.

As for the standard deviation, recall that Equation 2. So it is appropriate to use moving average and standard deviation to allow ourselves to adapt to an ever-evolving mean and standard de- viation, and also to capture profit more quickly. CAD equal to the negative normalized deviation from its moving average. X is nothing but the quote USD. X, so in this case the linear mean reversion is equivalent to setting the market value of the portfolio to be the negative of the Z-Score of USD.

The functions movingAvg and movingStd can be downloaded from my website. As with most example strategies in this book, we do not include transaction costs. There is a more practical version of this mean-reverting strategy in Chapter 5. Since the goal for traders is ultimately to determine whether the ex- pected return or Sharpe ratio of a mean-reverting trading strategy is good enough, why do we bother to go through the stationarity tests ADF or Vari- ance Ratio and the calculation of half-life at all?

Furthermore, the outcome of a backtest is dependent on the specifics of a trading strategy, with a specific set of trading parameters. However, given a price series that passed the stationarity statistical tests, or at least one with a short enough half-life, we can be assured that we can eventually find a profitable trading strategy, maybe just not the one that we have backtested.

The most common com- bination is that of two price series: We long one asset and simultaneously short another asset, with an appropriate allocation of capital to each asset. But the concept of cointegration easily extends to three or more assets. The for- mer is suitable only for a pair of price series, while the latter is applicable to any number of series.

Cointegrated Augmented Dickey-Fuller Test An inquisitive reader may ask: Why do we need any new tests for the sta- tionarity of the portfolio price series, when we already have the trusty ADF and Variance Ratio tests for stationarity? The answer is that given a number of price series, we do not know a priori what hedge ratios we should use to combine them to form a stationary portfolio. The hedge ratio of a particu- lar asset is the number of units of that asset we should be long or short in a portfolio.

If the asset is a stock, then the number of units corresponds to the number of shares. A negative hedge ratio indicates we should be short that asset. But pursuing this line of thought further, what if we first determine the optimal hedge ratio by running a linear regression fit between two price series, use this hedge ratio to form a portfolio, and then finally run a stationarity test on this portfolio price series? This is essentially what Engle and Granger did.

For our convenience, the spatial-econometrics. For example, both Canadian and Australian economies are commodity based, so they seem likely to cointegrate. The program cointegrationTest. From Figure 2. We use the cadf function of the jplv7 package for our test. Other than an extra input for the second price series, the inputs are the same as the adf function. Using English and Greek capital letters to represent vectors and matrices respec- tively, we can rewrite Equation 2.

The number of independent portfolios that can be formed by various linear combinations of the cointegrating price series is equal to r. One test produces the so-called trace statistic, and other produces the eigen statistic.

A good exposition can be found in Sorensen, As a useful by-product, the eigenvectors found can be used as our hedge ratios for the individual price series to form a stationary port- folio.

We will see how many cointegrating relations can be found from these three price se- ries. We also use the eigenvectors to form a stationary portfolio, and find out its half-life for mean reversion. There are three inputs to the johansen function of the jplv7 package: y, p, and k. The input k is the number of lags, which we again set to 1. This code fragment is part of cointegrationTests. What does it mean to have two cointegrating relations when 56 we have only two price series?

Actually, no. Remember when we discussed the CADF test, we pointed out that it is order dependent. If we switched the role of the EWA from the independent to dependent variable, we may get a different conclusion. These two different hedge ratios, which are not necessarily reciprocal of each other, allow us to form two independent stationary portfolios. With the Johansen test, we do not need to run the regression two times to get those portfolios: Running it once will generate all the independent cointegrating relations that exist.

The Johansen test, in other words, is independent of the order of the price series. Assuming that its price series is contained in an array z, we will run the Johansen test on all three price series to find out how many cointegrating relationships we can get out of this trio.

The eigenvalues and eigenvectors are contained in the arrays 57 results. Naturally, we pick this eigenvector to form our stationary portfolio the eigenvector determines the shares of each ETF , and we can find its half-life by the same method as before when we were dealing with a stationary price series.

CAD, so we expect a mean reversion trading strategy to work better for this triplet. We can now confidently proceed to backtest our simple linear mean-reverting strategy on this portfolio. The idea is the same as before when we own a number of units in USD. CAD proportional to their negative normalized de- viation from its moving average i. A unit portfolio is one with shares determined by the Johansen eigenvector. When a unit portfolio has only a long and a short position in two instruments, it is usually called a spread.

We express this in more mathematical form in Chapter 3. This linear mean-reverting strategy is obviously not a practical strategy, at least in its simplest version, as we do not know the maximum capital required Example 2.

The multiple is a negative number if we wish to short the unit portfolio. All other variables are as previously calculated. The positions is a Tx3 array representing the position market value of each ETF in the portfolio we have invested in.

Despite such impracticalities, the importance of backtesting a mean-reverting price series with this simple linear strategy is that it shows we can extract profits with- out any data-snooping bias, as the strategy has no parameters to optimize.

Remember that even the look-back is set equal to the half-life, a quantity that depends on the properties of the price series itself, not our specific trading strategy. Also, as the strategy continuously enters and exits positions, it is likely to have more statistical significance than any other trading strategies that have more complicated and selective entry and exit rules. We can pick and choose from a great variety of cointegrating stocks and ETFs to create our own stationary, mean-reverting portofolio.

Besides the plethora of choices, there is often a good fundamental story behind a mean-reverting pair. Even when a cointegrating pair falls apart stops cointegrating , we can of- ten still understand the reason. For example, as we explain in Chapter 4, the reason GDX and GLD fell apart around the early part of was high energy prices, which caused mining gold to be abnormally expensive.

We hope that with understanding comes remedy. This availability of fundamen- tal reasoning is in contrast to many momentum strategies whose only justi- fication is that there are investors who are slower than we are in reacting to the news. More bluntly, we must believe there are greater fools out there. But those fools do eventually catch up to us, and the momentum strategy in question may just stop working without explanation one day.

Another advantage of mean-reverting strategies is that they span a great variety of time scales. At one extreme, market-making strategies rely on prices that mean-revert in a matter of seconds. The short end of the time scale is particularly beneficial to traders like ourselves, since a short time scale means a higher number of trades per year, which in turn translates to higher statistical confidence and higher Sharpe ratio for our backtest and live trad- ing, and ultimately higher compounded return of our strategy.

Unfortunately, it is because of the seemingly high consistency of mean- reverting strategy that may lead to its eventual downfall. As Michael Dever pointed out, this high consistency often lulls traders into over- confidence and overleverage as a result Dever, Think Long Term Capital Management.

When a mean-reverting strategy suddenly breaks down, perhaps because of a fundamental reason that is discernible only in hindsight, it often occurs when we are trading it at maximum leverage after an unbroken string of successes.

So the rare loss is often very painful and sometimes catastrophic. Hence, risk management for mean reverting is particularly important, and particularly difficult since the usual stop losses cannot be logically deployed. In Chapter 8, I discuss why this is the case, as well as techniques for risk management that are suitable for mean- reverting strategies. This price series may be the market value of a single asset, 63 though it is rare that such stationary assets exist, or it may be the market value of a portfolio of cointegrating assets, such as the familiar long-short stock pair.

Seasonal mean reversion means that a price series will mean-revert only during specific periods of the day or under specific conditions. Conversely, not all stationary series will lead to great profits—not if their half-life for mean reversion is 10 years long. It is not a very practical strategy due to the constant infinitesimal rebalancing and the demand of unlimited buying power.

Finally, we highlight the danger data errors pose to mean-reverting strategies. In presenting the backtests of any strategy in this book, we do not in- clude transaction costs.

We sometimes even commit a more egregious er- ror of introducing look-ahead bias by using the same data for parameter optimization such as finding the best hedge ratio and for backtest. These are all pitfalls that we warned about in Chapter 1. The only excuse for doing this is that it makes the presentation and source codes simpler to understand.

I urge readers to undertake the arduous task of cleaning up such pitfalls when implementing their own backtests of these prototype strategies.

So we can interpret q as the market value of a portfolio of assets with prices y1, y2, …, yn and with constant capital weights h1, h2, …, hn, together with a cash component implicitly included, and this market val- ue will form a stationary time series. This cash does not show up in Equation 3.

So to keep the market value of this portfolio stationary but not constant! The upshot of all these is that mean reversion trading using price spreads is simpler than using log price spreads, but both can be theo- retically justified if both price and log price series are cointegrating.

If we look at Equation 3. In other words, if your two assets are not really cointegrating but you believe their spread is still mean reverting on a short time frame, then using ratio as an indicator may work better than either price spreads or log price spreads. This is the same idea as using moving average and standard deviation in our linear mean-reverting strategy. There is another good reason to use ratio when a pair is not truly coin- tegrating. For such pairs, we often need to use a dynamically changing hedge ratio to construct the spread.

But we can dispense with this trouble if we use the ratio as a signal in this situation. But does a ratio work bet- ter than an adaptive hedge ratio with price or log price spreads? You will find, in that example at least, price spreads with an adaptive hedge ratio work much better than ratio. An interesting special case is currency trading.

If we trade the currency pair EUR. We already demonstrated a simple mean-reverting strategy on trading such currency pairs in Example 2. CAD using ratio as the signal. But about those pairs that have no ready-made cross rates on many brokerages or exchanges, such as MXN.

Should we use the ratio USD. MXN instead? Again, because MXN. NOK may be more effective. MXN instead. Trading USD. Trading MXN. So the two methods are not identical. Example 3. But we try this strategy on the price spread, log price spread, and ratio for comparison. Some traders believe that when oil prices go up, so do gold prices. The logic is that high oil price drives up inflation, and gold prices are positively correlated with inflation. We will gloss over the difference between spot oil prices versus oil futures, which actually constitute USO.

We will come back to this difference in Chapter 5. Nevertheless, we will see if there is enough short-term mean reversion to make a mean- reverting strategy profitable. We will first try the price spread as the signal. The method we used to calculate the hedge ratio is linear regression, using the ols function from the jplv7 package as before. You can, of course, use the first eigenvector from the Johansen test instead.

We will now see if we can create a profitable linear mean reversion strategy. Once again, the number of units shares of the unit portfolio we should own is set to be the negative Z-Score, and the Tx2 positions array represents the market value in dollars of each of the constituent ETFs we should be invested in.

Next, we will see if using log prices will make any difference. Latest commit c Oct 24, History.

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