Alpha Trading Part 6

Markets Close at Different Times.

For some of the examples in this chapter, the futures markets used in a pair may not close at exactly the same time. These are mainly in the section that looks at inflation-related markets, gold, crude oil, and the EURUSD. The biggest impact on the results is that there may be some additional trades due to price moves that occur after one market is closed. However, when we compared the periods when the S&P and EuroStoxx did not close at exactly the same time with the later period when they did, we could not see a significant difference in the number of trades. Of course, when actually trading these pairs, entries and exits to both legs must be done simultaneously, when both markets are trading.

MECHANICS OF A PAIRS TRADE IN FUTURES.

The mechanics of a futures trade are slightly different from the trades using stocks that we covered in the previous chapter. The two exceptional advantages are: 1. There is no distinction between entering a long trade and a short sale (simply called a "short" in futures). When you place an order to buy, you don't tell the broker that you're setting a new position or covering a previous short. If you are previously long 10 contracts and you place an order to sell 10, you end up flat. If you are long 5 contracts and sell 10, you end up short 5. There are no uptick rules and no likelihood that shorts would be restricted in any way. You don't need to borrow stock to sell short; it's a normal process when you trade futures. Selling short in futures simply means that you believe prices will go down. The person who takes the other side of that trade expects prices to go up. You can both be right if your time frames are different; prices could go down in the short term, then up in the long term.

2. You can leverage the trades without borrowing funds. In fact, you earn interest on most of the money on deposit in your account. That can offset the commission costs. In trading futures, you only need to put up margin, but this is not the same as margin for stocks. It is simply a good faith deposit. For most futures trades, that deposit amounts to about 10% of the contract value. If you are a commercial trader-that is, if you are trading gold and your business is jewelry-then your margin might be as low as 5%. For example, if gold is trading at $1,100/ounce and the contract is 100 ounces, then one contract is worth $110,000. The typical margin would be about $10,000, but the exact number can be found by going to the COMEX or Chicago Mercantile Exchange (CME) web site or checking with your broker.

Putting up $10,000 to trade a gold contract, with gold at $1,100 an ounce, is the same as buying a.s.sets of $110,000 at a leverage of 11:1. In reality, the leverage is lower because the brokerage firm will require more in the event your trade goes the wrong way. In futures, you are required to restore margin to the full amount ($10,000 in our example) once your balance falls below 75% of the initial margin. To avoid continuous problems, a broker may want you to deposit $20,000 or $30,000 just to trade one contract. That means they won't have to chase clients who lose all their investment and more. In futures, you are responsible for your losses, even if they exceed the amount on deposit in your account. A typical investment manager will keep leverage to 4:1. Then, for one gold contract, they will have $40,000 on deposit, earning either money market interest or 3-month T-bill rates on up to 90% of the balance.

Example of Trading Signals for Futures.

As an example of a pairs trade in futures, we use a heating oilnatural gas pair. Table 4.1 shows two trades, both profitable, but with large equity swings. In this example we use a 14-day calculation period and enter a new trade when the momentum difference exceeds 50 and an exit when the momentum falls to 10 for shorts and rises to 10 for longs.

TABLE 4.1 Two sample trades for the pair natural gas and heating oil.

In the first trade, lasting only two days, the momentum difference on 6/9/2009 is 58.37, below the 50 entry threshold. This is the difference between the natural gas momentum of 29.55 and the heating oil momentum of 87.93, indicating that natural gas was moderately oversold while heating oil was somewhat overbought, as given by the 14-day calculation period. The entry prices are the closing prices for that day, 2.359 and 15.517. Remember that these are back-adjusted futures, so these prices are not the actual prices that occurred on that day.

On the following day, the natural gas momentum rises slightly, and the heating oil momentum drops considerably, resulting in a net momentum difference of 0.53. An exit is triggered because the momentum difference is above the threshold of 10. Most of the profit is due to heating oil, for a net gain of $32,040.

The position size for this trade was determined from the volatility calculation, which is the 14-day average true range. We use the same period as the stochastic calculation for consistency and in order not to introduce an additional parameter. At the time of the entry signal, June 9, 2006, natural gas had a average daily dollar range of $2,837 and heating oil a range of $2,424. With a lower volatility, heating oil will get more contracts than natural gas, in this case 12 to 10.

The second trade on June 14, 2006, is much more volatile. Natural gas is now overbought with a stochastic of 82.80, and heating oil is oversold at 14.52. The volatility is nearly the same as three days prior, so we buy 10 natural and sell 12 heat. However, on the first day, natural gas jumps 3.85% while heat only gains 0.02%, for a net loss of $62,641. After that, the two markets work their way back in the right direction for a net gain on the trade of $43,919 when the momentum difference goes negative. This is a trade that shows how difficult it is to use a stop-loss order, which would have exited the trade at the worst time, locking in the large loss. Remember that the natural profile of performance is to have a few large losses and more smaller profits.

INFLATION SCARES.

The first decade of the 21st century included the end of the tech bubble that sent NASDAQ to nearly 5000, the subsequent drop of 75%, the attack of September 11, 2001, real estate hyperinflation leading to the subprime collapse, a run in crude oil to $150/barrel and related panic over impending inflation, a worldwide government takeover of financial inst.i.tutions, record unemployment, record corporate bonuses, and a smooth rally in the equity markets in 2009 in antic.i.p.ation of more stable times to come. We live in interesting times.

News of these events can be confusing. Too many of us watch the financial news programs, which, because of the need to make everything sound urgent, tend to make minor events into something newsworthy and of great importance to the viewers. Without this high-profile coverage, most market fluctuations would most likely be little blips of noise.

The year with the most extreme price moves was 2008. In Figure 4.1, we can compare the bull (and bear) markets in oil with other key commodities. From late 2007, there is a clear upturn in gold, wheat, and crude oil, with a smooth decline in the Dollar Index. Although the rally in prices is somewhat out of phase, the decline that began after crude peaked was remarkably uniform. Was it a reaction to a false alarm in inflation? Not likely. Again, we would argue that it was mostly caused by investors pulling money out of every market at the same time. Commodity index funds had gained in popularity during the bull market in wheat, and everyone had been concerned that the price of bread was going to double or triple. With wheat dropping, followed by a collapse in the crude market, the dollar gaining strength, and gold falling from its highs of $1,000/ounce, investors saw their profits disappearing. Once the liquidation started, it became contagious.

FIGURE 4.1 Inflation-related commodities: the dollar index, gold, crude oil, copper, and wheat. The relations.h.i.+ps are strongest when the economy is under stress.

In this chapter, we'll look at the periods where these markets all seem to have moved together and see if there were profits to be made-at a reasonable risk. But first we'll focus on the energy markets, which have gained tremendous interest during the past two years.

Setting Up the Trade.

For pairs trading using stocks, we started with a basic unit size of 100 shares for leg 1, the first stock of the pair. The size of the second leg was larger or smaller, depending on its volatility, in order to make the risk of both legs equal. For futures, the starting size will be 10 contracts for the first leg, and the second will then be adjusted in the same way as with stocks. Using 10 contracts does not give us the accuracy of 100 for sizing the second leg, but 10 contracts of futures can be a large investment and is enough to show the effect of adjusting position size to volatility.

Margin and Leverage.

Consider that each contract of crude oil is 1,000 barrels (bbl), currently selling at about $75/bbl. The contract size is then $75,000, the minimum trading unit. Of course, you don't need to invest $75,000 to buy one contract of crude; you only need to put up a margin deposit of about $7,500, or 10% of the contract value.

If you buy a contract and the price of crude goes up $1, you gain $1,000. Your deposit is $7,500, giving you a return of 13.3%. That can easily happen in one day. On the other hand, if the price drops by $1 you lose 13.3%. If the price rises by $10/bbl to $85 (not out of the question), you have earned 133%. But if it drops by $10 (equally possible), you have lost your $7,500, plus you must pay the broker the remaining $2,500 within 24 hours. If you want to continue to trade you must deposit more money.

In reality, it doesn't quite happen that way. Based on the volatility of the market, your deposit should be high enough to cover most normal moves, so doubling or losing all your money in one day should not happen. On the other hand, you may be required to deposit $25,000 and only trade one crude contract. That protects the broker from the possibility of uncovered losses. If you start to lose so much that your deposit is eroding, the broker will ask you to restore your account to its original value or will liquidate your positions for you. There is no such thing as losing more than you have and then borrowing to cover your losses-only the big banks can do that.

TRADING ENERGY PAIRS.

For simplicity, we will trade energy pairs using only the four primary U.S. energy markets, crude oil (symbol CL, also known as WTI for West Texas Intermediate), its products, heating oil (HO) and reformulated blendstock for oxygenate blending (RBOB or RB), gasoline, and natural gas (NG). The gasoline contract, which used to be called unleaded gas changes often because of environmental regulations, so it's best to keep watching the volume for any noticeable, permanent declines. Low volume will tell you that there is another, more active contract.

As mentioned before, the crude contract is 1,000 barrels, heating oil is 42,000 gallons, unleaded gas is 42,000 gallons, and natural gas is 10,000 million British thermal units (Btus). The contracts are sized this way because of commercial needs. An oil refiner typically buys three contracts of crude and produces two contracts of gasoline and one of distillate (heating oil). The crack spread, which gets favorable margin from the exchange, is a spread that buys crude and sells RBOB and heating oil in the ratio 3:2:1, although this is not exact. If producing margins are too low, the refiner has the option of doing a reverse crack, in which it sells crude and buys the products. In our trading, we will only spread pairs and decide the relative position sizes using volatility. It is generally unproductive to trade the crack and try to compete with commercials at their own game.

If the price of crude oil moves from $75 to $76, the profit or loss from trading each contract is $1,000. If heating oil moves from $2.25/gallon to $2.26/gallon, the profit or loss is $420, and the same for unleaded gas. If natural gas rises from $6.500 mmBtu to $6.600 mmBtu, the gain or loss is $1,000. Therefore, the minimum move (smallest decimal move) for crude is worth $10, for heating oil and unleaded gas $4.20, and for natural gas $10.

Trading Hours.

For us to be able to test these pairs correctly, the data must be posted at the same time. In the energy markets, that is no problem because all four futures markets have a pit session that opens at 9:00 A.M. and closes at 2:30 P.M., Eastern time. It is unclear whether the pit session refers to open outcry trading or if it is just the primary electronic session. If it is not all electronic, it will be soon. There is also an electronic session for all markets that opens at 6:00 P.M. (18:00) and closes the next day at 5:15 P.M. (17:15).

The London Intercontinental Exchange (ICE) also trades similar products: Brent crude oil, a close proxy to our WTI, and gas oil (similar to our heating oil). The ICE is only electronic, with hours from 20:00 to 18:00 the following day (22 continuous hours), Eastern time. Although the closing prices are only 45 minutes later than the U.S. electronic counterpart, it would be necessary to capture prices at exactly the same time in the U.S. and London to have accurate data to trade pairs with one leg on each side of the pond.

As we will see from looking at the equity index and interest rate market pairs that span the U.S. and Europe, there is greater opportunity trading markets that respond to the same fundamentals but are applied to different economies. It is likely that energy pairs that have one U.S. leg and one London leg will be more profitable than the combinations of U.S. markets tested here, but that will be an opportunity that readers must pursue on their own. There is no viable arbitrage between the U.S. and European energy markets, but global macro events seem to keep both markets moving in the same way.

More Fundamental Background.

There are some fundamentals that we should discuss ahead of seeing results, as a way of setting expectations. It was already mentioned that heating oil and natural gas have overlapping markets, home heating, but most users cannot change from one to another because of equipment. Some commercial plants have that ability to use both sources, but it is not common. Heating oil is also diesel fuel, used for trucks and an increasing number of cars, especially in Europe. However, there is no practical arbitrage for heating oil. That is, the price of heating oil in Europe, called gas oil, would need to be much higher than the price in the U.S. to make loading and s.h.i.+pping from the U.S. to Europe, or the other direction, a profitable transaction.

Heating oil and gasoline are the refined products of crude oil and are produced in different percentages as the seasons change in North America. The production of heating oil is increased beginning in the early summer to build inventories, and gasoline production is increased in April in antic.i.p.ation of the driving season, which officially starts on Memorial Day (the end of May) in the U.S. Prices of gasoline tend to increase during the high-demand summer season, and heating oil can increase at the beginning of winter, or anytime between December and March, in expectation of any sustained cold spell that would affect demand.

Crude oil does not have any clear seasonal pattern, being the netting of the products, gasoline and heating oil, which are both seasonal. It does make extreme moves due to geopolitical risk, such as attacks on the pipelines in Iraq or Nigeria or labor strikes in Venezuela. Crude prices are also manipulated by OPEC, which sets production limits for its partic.i.p.ating members. This has had mixed success over the years, but they keep trying. In the final a.n.a.lysis, any country in need of money will try to sell more oil.

Crude oil also follows the change in the U.S. dollar because it is quoted in dollars worldwide. For example, if the dollar declines against the euro from 1.35 to 1.40, about 3.7%, we should see a similar rise in the price of crude oil, everything else being equal. Oil maintains what we would call a global value. That holds true for commodities as well. In Figure 4.2, the vertical price scales have been extended to fill the chart, giving both crude and the EURUSD the same relative volatility and showing remarkable similarity. Although crude moved much further measured as a percentage, the overlapping patterns show an undeniable relations.h.i.+p since March 2007. Oddly enough, it appears that the EURUSD started up in the first quarter of 2007 before it was joined by crude. This looks like an opportunity, but the daily correlation is only .33, measured beginning in 2007.

FIGURE 4.2 Crude oil and the EURUSD show a remarkable correlation from 2007.

Despite the different fundamentals, energy prices move together. When crude oil spikes up during the winter, heating oil is the primary focus, but gasoline prices must be dragged higher as well. In the spring and early summer, it is the opposite, with gasoline leading the way. You can't expect a big move in any of the energy markets without all the others showing some proportional reaction. It's the lead and lag in these moves that provides trading opportunities.

REVISITING MOMENTUM WITH ENERGY MARKETS.

Energy markets have been the focus of rising costs for the past two years, topping with crude oil at $150/bbl and taking the products, gasoline and heating oil, along with it. Natural gas moved higher then lower, along with crude, as did all sources of energy (even firewood), although very few users are able to use heating oil and natural gas interchangeably. The equipment needed to use one or the other is not readily interchangeable. Figure 4.3 shows a chart of crude oil, heating oil, and natural gas from 1990 through January 2010. Heating oil tracks crude because it is derived from crude, and natural gas starts to shadow the movements of crude in 1996, but only when the moves are larger.

FIGURE 4.3 Energy prices, 1990January 2010, log scale. There is a similarity in natural gas, but there are also large differences.

Opportunity and Risk The interesting part about natural gas and heating oil is that the traders create strong correlations during periods of stress. During a crisis, it is always money that moves the market, not common sense. Although the big picture of natural gas and heating oil price moves, Figure 4.3 shows a general, smooth downtrend of natural gas prices, compared with a general rise in heating oil prices into 2008; during the times when crude price moved up faster than normal, natural gas followed. The most extreme example was the peak of 2008 shown in Figure 4.4.

FIGURE 4.4 NYMEX energy prices, August 31, 2007October 31, 2008. Natural gas tracks crude oil and heating oil during times of stress. At the peak of crude oil prices, in mid-2008, natural gas tracked crude oil as though it were a product.

During the 14 months that included the peak crude oil price on July 14, 2008, natural gas moved in step with crude oil and especially heating oil. Once prices came back down in late 2008, the similarity of price movement disappeared. Figure 4.5 shows the prices of heating oil and natural gas from March 2009 through the end of 2009. You can find short bursts of similarity in the prices, but overall the picture is one of heating oil rising and natural gas falling. These differences can offer both opportunity and risk.

FIGURE 4.5 Heating oil and natural gas, from March 2009 through January 2010, following the peak in crude oil. Arbitrage would have provided more risk than reward.

The correlations during the three periods may give a more exact picture. Table 4.2 shows the three intervals: 1. All data from 1990 through 2009.

2. Peak crude oil price move, September 2007 through October 2008.

3. After-the-peak price moves, March 2008 through December 2009.

In the top panel of Table 4.2, we see the normal, long-term relations.h.i.+p of the three markets. Unleaded gas is omitted because it has a very similar relations.h.i.+p to both crude and natural gas, as heating oil. The left side of the table used the price differences to find the correlations, and the right side used the percentage changes. Normally we would want to use the percentage changes, but most futures long-term data is continuously back-adjusted from individual futures contracts. In the worst cases, where there is a significant price gap between the contract that is expiring and the next contract to be used, the aggregate back adjusting can cause the oldest prices to become negative, or near zero. For those markets, percentages don't work. For commodity prices such as energy and metals, cash prices offer a good alternative; however, using differences will give similar results if you stay aware of the data problems.

TABLE 4.2 Cross-correlations of crude oil, heating oil, and natural gas during the three intervals: all data from 1990 through 2009 (top panel); the peak crude period, September 2007 through October 2008 (center panel); and the after-the-peak period, March 2009 through December 2009 (bottom panel).

Heating oil and unleaded gas are always highly correlated to crude prices, which is to be expected because they are both products of crude. During the three periods shown in Table 4.2, the correlation of heat to crude stays over .90. It is not higher due to the seasonality of the products. During the spring and summer months, the product prices are driven by unleaded gasoline, and heating oil becomes a secondary market, causing a less direct relations.h.i.+p.

A long-term correlation of 0.351 for crude and natural gas shows some correlation, but it is not clear whether they would be profitable for pairs trading; the risk may be too high. However, during the peak period for crude prices, the correlation with natural gas jumped to 0.557 for crude and 0.536 for heating oil, making it a likely candidate. After the peak, those correlations dropped to below the long-term values, so this opportunity would have increased in risk.

In this chapter, we will look at pairs trading in futures markets that have a long-term history of being fundamentally related, as well as more limited, shorter-term opportunities. Sometimes the best trades are those that spring from special situations.

Trading Energy Pairs.

We've looked at the changing correlations in the most active energy futures, and now it's time to see if they can be traded profitably as pairs. The four futures contracts are crude (CL), heating oil (HO), unleaded gasoline (RB), and natural gas (NG). Both Brent and gas oil, traded on the IPE or ICE, are also candidates but won't be considered here. The data will be the past 10 years, beginning January 2000.

We can first set our expectations by looking at the correlations of the six pairs, shown in Table 4.3. It is immediately clear that the pairs made up of crude and its products have a very high correlation, and those using natural gas are much lower. Because this period included the very volatile run-up in crude to $150, the current correlations with natural gas are toward the high part of the range seen in Table 4.2. The two groups provide very different expectations. The high correlations make it certain that divergences in the momentum will produce a trade that quickly corrects, but the absolute price differences may be too small to be profitable after we subtract our cost of $25 per contract for each round-turn. Because of the lower correlations, natural gas pairs are likely to be more profitable but with much higher risk.

TABLE 4.3 Correlations of pairs, 10 years beginning 2000.

Leg 1 Leg 2 Corr.

Crude Unleaded 0.837 Crude Heating 0.883 Unleaded Heating 0.812 Crude Natural gas 0.391 Unleaded Natural gas 0.368 Natural gas Heating 0.411.

Average Crude & products 0.844.

Average Natural gas 0.390.

The rules for trading will be exactly the same as in Chapter 3, except that we charge a round-turn cost of $25 per contract. We calculate the stochastic momentum values of the two futures markets separately over the same time period, subtract the two values, and test that difference against an entry threshold. When we tested stocks using calculation ranges from 14 to 4, we found that the smaller momentum periods emphasized market noise, which is to our advantage, and increased the number of trades. The entry threshold was from 60 to 40, but the lower value of 40 generated the most trades. The entry thresholds also affect the size of the individual trade profits (and risk), which is the negative part of the trade-off. Getting enough profits per trade was the biggest problem.

TABLE 4.4 Results of energy pairs varying the calculation period and entry threshold. The stochastic momentum calculation period is on the left and the entry threshold along the top axis.

In Table 4.4, the results of these tests are shown for all six pairs, separated into those without natural gas (part a) and those with natural gas (part b). The test period was January 2000 through March 2010. It is immediately evident that the pairs made up of crude and its products all fail to generate profits after costs. On the other hand, the natural gas pairs were all successful, net of costs, for all combinations of calculation periods and entry thresholds except the fastest, 4 days. We can understand that the shortest calculation periods result in the fastest trades and the smallest returns per trade. In this case, 4 days was too fast.

The natural gas pairs show a consistent pattern of improvement as values move from the bottom right to the top left, which is also from faster to slower, peaking at a calculation period of 10 days. Keeping it in perspective, the range of periods, from 4 to 14, is considered a fast range and not suitable for trend following.

It is often the case that the best ratio is among the fastest trading models, and we find that the profits per contract are marginally large enough to offset costs. That doesn't seem to be the situation here, but then the successful pairs are those with natural gas, and the correlations are fairly low. We can expect these markets to move apart often but come back together because the energy complex has underlying similarities and will react to much of the market news in the same way.

It's likely that some traders will choose a different combination of parameters. They might try to maximize the number of trades or the profits per contract. They may want the combination that is out of the market most often, as a way of avoiding price shocks. And we didn't test combinations of exit thresholds, where a short trade is entered at a momentum difference of 60 and exited at 10 rather than 0. That would increase the number of trades, increase the percentage of profitable trades, but decrease the size of the per contract returns. There is no best choice, only trade-offs and trader preference. Those options are left to the reader.

TABLE 4.5 Individual energy pairs' performance using momentum period 10, entry threshold 60, from January 2000.

In this example, the best choice will be the parameters that produced the highest ratio, a calculation period of 10 and an entry threshold of 60. Table 4.5 shows some of the statistics for that choice. There are only 10 trades per year, but the profits per contract are sufficiently large to give us confidence that we can absorb any surprising slippage. The reasonably equal distribution of profits per contract between leg 1 and leg 2 is also good, showing that both markets are moving. Annualized returns averaged only 6.7% but will be higher when this is mixed in a portfolio and diversification reduces the risk. These values are all based on an annualized volatility (standard deviation) of 12%, as are all results shown in this book.

For completeness, we also show the results of the crude-product pairs in Table 4.6. First, the number of trades is very small because the correlations are high. It is rare that the two closely related markets diverge enough to generate a trading signal. The only pair that netted a profit was unleadedheating oil, which have opposite seasonality and are most likely to move apart. The ratio of 0.129 shows that the performance was very erratic, and you would need to take nearly 8 times the risk for each unit of reward. That's not an attractive profile.

TABLE 4.6 Results of crude-product pairs with a period of 10 and entry threshold of 60.

A MINIPORTFOLIO OF NATURAL GAS PAIRS.

Selecting the three natural gas pairs, we can combine the profit and loss streams into a portfolio to see how much diversification we gain. This will be the same process that we use for any group of pairs in the same sector. We can later combine multiple sectors for further diversification. As we will see with futures, we have the ability to choose our level of risk and vary our leverage. It is one of the great advantages of trading futures.

The process begins with either the three streams of daily profits and losses or the c.u.mulative profits shown in Figure 4.6. You start with the c.u.mulative profits, then simply subtract each day's value from the previous day to get the change in profits. As you can see in the chart, heating oil appears to be more volatile than either crude or unleaded gas. The process of adjusting to a target volatility will equalize the volatility of the returns of each of the three pairs.

FIGURE 4.6 c.u.mulative profit and loss streams for the three natural gas pairs, from 2000.

Step 1: Calculation of the Daily Profit or Loss Each day it is necessary to calculate the daily profit or loss, less the round-turn commission when the trade is closed out. Table 4.7 shows the data needed to calculate the profits and losses, along with the momentum difference to confirm that an entry or exit has occurred.

TABLE 4.7 Calculating the daily profits and losses for the crudenatural gas pair.

The first trade is entered at the close on February 9, 2000, selling 10 contracts of crude at 63.11 and buying 7 natural gas at 20.317 based on current volatility. Note that these prices are back-adjusted futures but work accurately for these examples. On the next day, February 10, 2000, the price of crude rises by $0.66, and natural gas also rises by $0.052. The conversion factor for crude is $1,000 for a 1.00 move and $10,000 for a 1.00 move in natural gas, then 10 contracts of crude generate a loss of 10 $0.66 $1,000 = $6,600, and 7 contracts of natural gas generate a gain of 7 $0.052 $10,000 = $3,640. That gives a net loss for the day, as well as a c.u.mulative loss, of $2,960.

On the next day, February 11, 2000, crude gains $0.01 and natural gas drops $0.022, both losses for the pairs trade, netting a daily loss of $1,640 and a c.u.mulative loss of $4,600. The calculations continue in the same way, but with prices going in a favorable direction until the trade is closed out on February 17, 2000. On that day, the daily profit or loss is reduced by an additional $25 for each contract, or $250 for crude and $175 for natural gas, giving the final trade a profit of $765.

There are no trades for the next three days; therefore, the daily profit/loss column shows zeros.

The next trade is entered on February 23, 2000, and is also short crude and long natural gas. This time, the volatility of natural gas has dropped, and the position sizes are equal. If we follow the same calculations, we show a large gain on the final day of the trade, February 28, 2000, resulting in a c.u.mulative profit.

The Net PL column, second from the right, will be used to continue the portfolio construction.

Step 2: Align the Profit/Loss Streams There are three daily profit/loss streams that must be aligned by date. For markets that all trade on the same exchange in the same country, this is not usually a problem. Pasting them into Excel should be all that's necessary. However, dealing with pairs that have one leg in the U.S. and one leg in Europe requires an additional step.