(A PDF version of this strategy is attached at the bottom of this page.)

Market Background
Evaluating the Implied Volatilities
Structuring the Trade
Assessing Possible Results
A Word of Caution

Options on Chicago Board of Trade swap futures create opportunities to structure trades based on your volatility outlook.

Suppose you think the market is mispricing the implied volatility of options on 10-year swap futures relative to the implied volatility of options on 5-year swap futures. One way to trade this situation is with option straddles at the 5-year and 10-year maturities.

Straddles are volatility strategies. You buy a straddle when you don’t know whether the next move will be up or down but you expect either an increase in volatility or a large move in either direction—that is, when you do not have a view on direction, but you do have a view on volatility. You sell a straddle when you expect volatility to decline or for there to be relatively little price action.

Market Background

Consider market conditions 39 days before expiration of options on CBOT swap futures. The 5-year swap rate is trading at 3.55%, and the 10-year swap rate is 4.56%. These swap rates imply the 111-04 and 111-15 CBOT swap futures prices shown in Exhibit 1a.

Your sense of the volatility characteristics of these markets might lead you to make some assumptions about the implied volatilities the market is using to
price these options:

1. 5-year implied volatility is fairly priced at 4.7% and will change little during the option holding period.

2. 10-year implied volatility, at 5.0%, is significantly lower than your estimate and should revert close to its 8.0% long-term historical mean.

A colleague might well ask how you know this, for an important early step in setting up any option trade is
to evaluate the market estimate of implied volatility relative to what you know about historical volatility.

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Evaluating the Implied Volatilities

Volatility cones can be useful tools for this because they provide convenient graphic summaries of the term structure and range of the volatility of an underlying market. Exhibit 2 summarizes a study of historical price volatility of 5-year CBOT swap futures using hypothetical data for the period from November 2000 to October 2002.

The heaviest line shows the median historical volatility to be approximately 4.9% per year, regardless of time until option expiration. The uppermost and bottommost lines define the range of volatility observations during this two-year period. For example, for intervals lasting 210 days, historical volatilities ranged from a high of 6.09% to a low of 3.42%. However, for intervals lasting 21 days, the range widened to over 10% on the high end and 2% on the low end. All of this squares with the common wisdom that volatility increases dramatically when options approach expiration.

Market participants in the LIBOR world, especially those in bank treasuries, tend to focus on yield volatility. As a result, relating an option on CBOT swap futures to an OTC swaption or interest rate cap or to a Eurodollar option requires a comparison on the basis of yield volatility. In contrast, relating a swap option to a Treasury option or an OTC mortgage option requires a comparison on the basis of price volatility. Exhibit 3 provides a parallel study of the yield volatility of
CBOT 5-year swap futures, again using hypothetical historical data.

You can see that this graphic closely resembles the one of price volatility, with only the vertical scale being different. This allows you to make a rough and ready evaluation of the volatility situation no matter whether you start from yield volatilities or price volatilities.

Similarly, Exhibits 4 and 5 display parallel graphic summaries of the price and yield volatility term structures of 10-year CBOT swap futures.

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Structuring the Trade

Given these market conditions and your volatility assumptions, you can use options on CBOT swap futures to sell a 5-year straddle and buy a 10-year straddle. Exhibit 6 summarizes the components of
the transaction and their option “greeks.”

The choice of the 111 strike price puts both straddles nearly at the money. The general trading goal is that if the 5-year volatility does essentially nothing, as you predict, then you will keep all or most of the initial premium that you collect. If the 10-year volatility rises to a level close to its long-term median, as you predict, the long 10-year straddle will generate a gain.

However, to isolate the volatility factor, you need to ratio the trade so that both straddle positions will respond equally to equal changes in implied volatility across the yield curve.

Vega is the option risk parameter that relates option price change to changes in volatility. The 8.9678 vega of the 10-year 111 call, on the table of initial market conditions, predicts that a one percentage point increase in volatility will increase the option price almost 9 option points (that is, 9/64 of a futures price point).

The last line of the initial market conditions table shows the vegas for the two straddles. Notice that the price of the 5-year straddle will respond slightly more to a given volatility increase or decrease than the price of the 10-year straddle will.

To neutralize this difference in responsiveness to volatility change, you can divide the 5-year straddle vega by the 10-year straddle vega.

18.4628/17.9356 = 1.0294

This tells you that a position short 1,000 5-year 111 straddles and long 1,029 10-year straddles will be close to vega neutral. If both implied volatilities change the same amount, the trade should generate essentially no result.

A further refinement concerns directionality. Notice that the 5-year 111 straddle delta is 0.0644. This is the sum of the put and call deltas.

Call delta 0.5315 - Put delta 0.4671= Straddle delta 0.0644

This means that the 5-year straddle has some directional exposure. Indeed, a 1,000-straddle position will have a delta of 64.4 (0.0644 * 1,000 = 64.4). Because futures contracts have a delta of 1.0, by definition, you can eliminate most of this directionality by buying 64 5-year CBOT swap futures contracts for every 1,000 of the 111 option straddles that you sell.

The 0.2095 delta of the 10-year straddle indicates that this position has even more directional exposure than the 5-year straddle. To eliminate most of the directionality, you can sell 209 10-year CBOT swap futures for every 1,000 of the 111 option straddles that you buy. Note that since vega neutrality dictates that you will buy 1,029 10-year straddles instead of 1,000, you will sell 216 (209 x (1,029/1,000)) 10-year swap futures instead of 209 to establish delta neutrality in the 10-year leg of the transaction.

This aggregate position—short 1,000 5-year straddles, long 1,029 10-year straddles, long 64 5-year CBOT swap futures, short 216 10-year CBOT swap futures—should respond only if the 10-year volatility increases relative to the 5-year volatility.

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Assessing Possible Results

This trade can produce a variety of results, depending on what happens in the market. Changes in the prices of the underlying CBOT swap futures should have almost no effect, because of the use of the futures contracts to establish initial delta neutrality. Given the vega weighting, parallel volatility moves should also produce essentially no result. If the 5-year volatility increases relative to the 10-year volatility, this trade can generate a loss. But, in any situation where the 10-year volatility increases relative to the 5-year volatility, this trade should generate gains.

Consider three of the possible market scenarios. In the first, suppose that, during a 20-day holding period, both the 5-year and 10-year CBOT swap futures prices rise 16 ticks but both volatilities hold constant. In this case, the trade will perform as shown in Exhibit 7.

Notice that the 5-year straddle and the 5-year futures position both generate gains, but the 10-year straddle and the 10-year futures position both suffer losses. These are normal results. Short straddles gain when volatility changes little or not at all. Long straddles depend on strong volatility increases for success. When volatility does not increase, as in this case, they lose.

At first glance, the $11,859 net result seems out of line with the expectation that a vega-neutral and delta-neutral position should produce essentially no result when volatility does not change. Two factors account for the drift from a zero result. This trade is in place for 20 days during a period when options experience extreme time decay. While that accounts for most of this result, there is also rounding error at work here. This trade is neither perfectly delta neutral, even on day one, nor is it perfectly vega neutral. Further, without rebalancing along the way, it would experience drift along both the delta and vega parameters. More importantly, the Scenario 1 result pales to insignificance in comparison with the result of Scenario 2.

Scenario 2 incorporates the expectations and assumptions outlined at the outset. The two futures prices change as they do in Scenario 1, and the 5-year volatility holds constant. However, the 10-year volatility reverts to its long-term median level which is 8% with 19 days to expiration. Exhibit 8 shows how this trade can be expected to perform given these conditions.

While the 5-year straddle and futures perform exactly as in Scenario 1, the strong increase in implied volatility causes the 10-year straddle to generate a positive result. With gains at both maturities, the trade generates $529,359 of revenue.

You won’t call the market correctly every time, of course. Yet with this trade, you don’t have to be precisely right in your expectations to profit. It can generate positive results as long as 10-year volatility rises relative to 5-year volatility. Exhibit 9 illustrates.

In this case, both futures perform as in the other two cases. Also, the 10-year volatility rises from 4.7% to 8.0%, as in Scenario 2. What is different is that, contrary to your expectations, 5-year volatility rises from 4.7% to 6.1%. You can see, on the Exhibit 2 volatility cone, that 6.1% is right at the 75th percentile line. As a result, the 5-year straddle generates a gain that is only 16% of the 5-year straddle gain shown in Scenarios 1 and 2. Notice that the trade still books sizable proceeds despite this violation of your volatility expectations.

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A Word of Caution

These three scenarios illustrate that this trade is subject to a modicum of slippage due to rounding error and time decay. You should also remember that these are view-driven trades. Here, you have a view on volatility rather than on interest rate or price direction. It is still a view, and you can be wrong and suffer a loss.

Yet these three scenarios illustrate another truth about the value of trading options on CBOT swap futures. Spread trades like this one, if properly constructed, give you more ways to be right. Indeed, as Scenario 3 illustrates, you can be partly wrong and still do relatively well.

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For More Information

To learn more about options on CBOT swap futures, visit www.cbot.com or call 312-341-7955.