Interest Rates
Frederick Sturm, first vice president at Fuji Securities, offers some firehose strategies to combat massive basis point moves in Treasuries.
Quick and Dirty Treasury Hedges
In some trading scenarios, speed is more important than precision. Many players with heavy exposure to U.S. Treasury securities, such as hedge fund managers and corporates, have known this for years.
Any trader with three hours of time and a calculator can finetune a hedge against short and intermediateterm Treasury exposure with the TreasuryEurodollar (TED) spread. But what if heavily exposed investors suddenly find themselves facing a massive price movement? When this happens, the TED spread usually widens, making traditional hedging strategies much less effective. Traders who take hours—or even minutes—to calculate the optimal hedge are at a disadvantage in times of crisis.
Fortunately, there are rules of thumb, utilizing Eurodollar bundles and packs, that allow traders to manufacture quick and dirty hedges to protect their positions. These strategies won't win many style points, but they can prevent free falls in the wake of enormous market moves.
What's in a bundle?
The Chicago Mercantile Exchange introduced Eurodollar bundles in September 1994. A bundle is the simultaneous sale or purchase of one each of a series of consecutive Eurodollar contracts. The first contract in any bundle is generally the first quarterly contract in the Eurodollar strip. (Approximately one month before the expiration of the spot month Eurodollar contract, a second set of bundles, using the next quarterly contract as the starting point, trade simultaneously with the established bundles.) The exception to this convention is the fiveyear forward bundle, which covers years five through 10 of the Eurodollar futures strip. For example, on November 20, the first contract in the fiveyear forward bundle was the December 2002 (the 21st contract in the strip), and September 2007 (the 40th contract) the last. The terminal contract depends on the bundle's term to maturity. The CME offers one, two, three, four, five, seven and 10year terms to maturity.
Making prices in bundles
For any bundle, the price is quoted in terms of the net change during the current trading session vs. the previous trading day's settlement level. At the end of the trading day, the bundle's price quotation reflects the simple average of the net price changes of each of the bundle's constituent contracts.
Here's an example. Assume that all of the nearest 21 contracts have enjoyed a threetick increase in price since yesterday's settlement; at the same time the prices of each of the next seven contracts have posted net gains of four ticks. Under these conditions, the implied fairvalue price quotation for the sevenyear bundle would be: [(21*3)+(7*4)] / 28 = 3.25 ticks.
This raises a critical point. Unlike Eurodollar futures prices, which are quoted in increments of either 1 basis point or, in the case of the first four quarterly and two serial month contracts, .5 basis point, bundle prices are quoted in increments of .25 basis point.
For an Eurodollar futures contract, the dollar value of a 1basispoint move (DV01) is always $25. By contrast, for Eurodollar bundles the DV01 will always be a multiple of $25, depending on the number of contracts in the bundle. These differences are summarized in the table on Page 51.
Unbundling after the trade
After a buyer and a seller have agreed on the price and quantity of a bundle, they must assign mutually agreeable prices to each of the bundle's constituent contracts. In principle, the transactors may set these component prices arbitrarily, subject to one restriction: The price of at least one constituent Eurodollar contract must lie within that contract's trading range for the day, assuming that at least one of the Eurodollar contracts in the bundle has established a trading range. CME regulations are designed this way to ensure that bundle prices will remain tethered to the price action of the underlining individual Eurodollar contracts.
In the vast majority of cases, traders make use of a computerized system, located on the CME trading floor, that automatically assigns individual prices to the contracts in a bundle. This system was designed by the CME to simplify the administrative aspects of the bundle trade.
The pricing algorithm used by the CME is based on the following principle: To the extent that adjustments are necessary to bring the average price of the bundle's components into conformity with the bundle's own price, these price adjustments should begin with the most deferred Eurodollar contract in the bundle and should work forward to the nearest Eurodollar contract.
By construction, bundles are wellsuited to traders and investors who deal in LIBORbased floatingrate products. Obvious examples of users include investment banks that routinely carry syndication inventories of floatingrate notes, corporate treasuries that issue floating rate debt and commercial bankers who wish to hedge the risk exposure entailed in periodic loanrollover agreements.
The most avid followers of bundles, however, are likely to be those market participants who deal in longdated TED spreads. Such trades entail the purchase (or sale) of a Treasury security and the simultaneous sale (or purchase) of a strip of Eurodollar futures contracts with a comparable notional term to maturity. A frequently encountered version features a long position in the twoyear Treasury note and a short position in some combination of the nearest seven or eight Eurodollar contracts.
In this context, Eurodollar bundles serve two useful ends. First, they establish a readily available, widely acceptable and easily interpretable benchmark by which the performance of any other TED trade can be judged.
Second, and more important, they facilitate cleaner, more rapid execution of the Eurodollar leg of the trade. Instead of being forced to construct lengthy and idiosyncratic Eurodollar strip positions contract by contract (always a risky proposition, especially in a fastmoving market), futures brokers have the capability of executing one trade on behalf of the client that automatically establishes price, quantity and the allocation of the bundle's component prices.
What's in a pack?
A Eurodollar pack is the simultaneous purchase or sale of an equally weighted, consecutive series of four Eurodollar futures, quoted on an averagenetchange basis from the previous day's close. Because this quoting method is similar to that for Eurodollar bundles, packs offer an alternative method of executing a strip trade. All four contract months in the pack are executed in a single transaction, eliminating the inconvenience of partial fills, particularly in the deferred contract months. Just like individual Eurodollar futures contracts, packs are designated by a color code that corresponds to their positions on the yield curve. There are generally nine different packs trading at any given time: red, green, blue, gold, purple, orange, pink, silver and copper, corresponding to Eurodollar futures years two though 10, respectively. Packs are subject to the same tradingrange constraints as bundles.

Win: 
Place: 
Show: 
Treasury Issue 
Best Quick Hedge 
Second Best 
Third Best 
Onthe Run 2Year 
2Yr Bundle 
3Yr Bundle 
Red 1 
Onthe Run 3Year 
3Yr Bundle 
4Yr Bundle 
2Yr Bundle 
Current 4Year 
4Yr Bundle 
5Yr Bundle 
3Yr Bundle 
Onthe Run 5Year 
4Yr Bundle 
5Yr Bundle 
3Yr Bundle 
Current 7Year 
5Yr Bundle 
7Yr Bundle 
4Yr Bundle 
Onthe Run 10Year 
7Yr Bundle 
5Yr Bundle 
Blue 1 
Treasury Issue 
Best Individual ED Contract 
Best ED Pack 
Best ED Bundle 
Onthe Run 2Year 
Red 1 
White 
2Year 
Onthe Run 3Year 
Red 2 
Red 
3Year 
Current 4Year 
Green 1 
Green 
4Year 
Onthe Run 5Year 
Green 1 
Green 
4Year 
Current 7Year 
Green 4 
Green 
5Year 
Onthe Run 10Year 
Blue 1 
Blue 
7Year 
Some rules of thumb for traders faced with massive price movements. 
Quick and dirty hedging
Suppose you are caught with an unhedged Treasury note position in a fastmoving market. Smashing the firealarm box is clearly preferable to slowly aiming for exactitude. Which spot on the Eurodollar futures strip should you grab first?
The table above answers this question for Treasury notes at several ontherun maturities, both in terms of overall goodness of performance and by transactional device (that is, individual Eurodollar contracts vs. packs or bundles).
How quick and dirty works: Some examples
On July 31, 1997, 6 percent of July 31, 2002 was the ontherun fiveyear note. The dollar value of a 1basispoint change in its yield (DV01) was $42,780 per $100 million face value. The most effective quickdirty hedge was the fouryear Eurodollar bundle, for which DV01 is $400 (16 contracts x $25 per contract). Thus, the appropriate hedge ratio was 107 ($42,780 / $400) fouryear bundles.
Suppose we buy $100 million face value of the note and sell 107 bundles at close on August 1. The price of the note has fallen 24/32nds, giving us a loss of $750,000 (apart from the one day of coupon accrual that we earn). At the same time, the price of the fouryear Eurodollar bundle has fallen 16.5 ticks, benefiting our short position by $706,200 (107 x 16.5 x $400). Thus, our net loss with a fouryear bundle hedge is $43,800.
Using the same reasoning and similar arithmetic, we find the second best onthefly hedge tactic is to sell 86 fiveyear bundles, and third best is to sell 143 threeyear bundles. If for whatever reason we wish to hedge with a stack of one single Eurodollar contract, then the results in the lower table on Page 51 say our best choice is to sell 1,711 ($42,780 / $25) of Green1. With the fiveyear bundles as a hedge, our net loss would be $29,750; with the threeyear bundle as a hedge, our net loss would be $52,875; and by stackhedging with Green1, we could transform our $750,000 loss on the Treasury note into a net gain of $62,725.
Some rules of thumb
For notes in the twoyear to fouryear maturity sector, the best quickdirty hedge on the average is the Eurodollar bundle nearest to the note's term to maturity.
 For fiveyear to 10year maturities, the best quickdirty hedge is the Eurodollar bundle nearest to the note's duration.
 In most cases, the secondbest alternative is to grab into the middle of the Eurodollar strip for a fouryear or fiveyear bundle.
 Though stack hedging with individual Eurodollar contracts is never the first choice, it ranks among the top three for twoyear notes (Red1) and 10year notes (Blue1).
Note that whenever the yield curve shifts, with the long end of the curve pivoting around the short end, longerdated Eurodollar hedges should outperform their shorterdated counterparts. The examples on Page 51 implicitly illustrate this point. Although the fouryear Eurodollar bundle provides the best protection on average for the Treasury fiveyear note, both the fiveyear bundle and the Green1 stack hedge gave superior performance on August 1, 1997. Why? Because the yield curve steepened sharply that day, with longterm rates rising more than shortterm rates.
Conversely, whenever the yield curve shifts, with the short end of the curve pivoting around the long end, shorterdated Eurodollar hedges should outperform their longerdated counterparts. In such circumstances, the holder of a fouryear Treasury note, for example, will find that the threeyear Eurodollar bundle (normally the thirdbest alternative) is likely to give better hedge protection than the fouryear Eurodollar bundle (normally the first choice).
These strategies won't win many style points, but they can prevent free falls in the wake of enormous market moves. 
One might assume that quick and dirty Eurodollar hedges work best in the twoyear maturity sector, if only because that's where many term TED traders focus their activity. This assumption is incorrect: in terms of price change correlation, Eurodollar futures provide noticeably greater protection for threeyear to 10year maturities, and especially for fiveyear notes.
Caveat emptor
The tactics described here are recommended for their singlestroke transactional convenience, for their bluntedged effectiveness, and because they permit Treasury note hedgers to exploit the depth and liquidity of the CME Eurodollar futures pit. Use them because they work, but never forget that they are suitable only as emergency first aid for extremely short holding periods. Indeed, given their shallow roots in finance theory, they are wholly inappropriate for sharppencil strategic trades (unlike the carefully weighted strips of Eurodollar contracts that typically appear in term TED trades). In particular they take no account of the dynamics of the bank credit spread (the spread between Treasury yields and LIBOR); their tacit assumption is that LIBOR rates will move tick for tick with Treasury yields.
Nor do they take explicit account of the relationship between forward rates and spot yields. In effect, the trader who hedges, for example, a Treasury fiveyear note with a single stacked Eurodollar contract is trading the spread between a fiveyear Treasury spot yield and a forward 90day bank rate. More generally speaking, he or she is trading a spot longterm segment of the term structure against a remote shortterm segment.
These trades also fail to compensate explicitly for the nonconvexity bias that is wellknown to be embedded in Eurodollar futures prices, especially those with a long term to expiry. Traders who opt for quick and dirty hedging of Treasuries beyond five years to maturity will do well to remember that they stand unprotected against this aspect of their risk exposure.
This article was adapted from "CME Open Interests: What's New in CME Interest Rates? (Bundles, Packs and Stubs),” published by the CME, and "Picking Your Spots on the Eurodollar Strip: A Graphical Guide to Quick and Dirty Hedging for Treasury Notes,” published by Fuji Securities. Peter Barker, director of interest rate product marketing at the CME, contributed to this article.
Ask Dr. Risk
William Margrabe, president of the William Margrabe Group, discusses Monte Carlo technique costless collars and remedial education.
How to Catch an Unfaithful Model in the Act
Dear Dr. Risk:
We're making a presentation to a bank next week about model risk, in hopes of winning a contract to validate its models. How can one measure and manage the model risk derived from using the wrong models or using models incorrectly?
Ernie Young
Dear Mr. Young:
Thanks for sending in a sophisticated question about one of my favorite topics—model risk, the risk that you'll lose a whole lot of money, your job, even your selfrespect, because some pretty model that means a lot to you, one that has been with you for a long time or one you just met, is being unfaithful…to reality. This is an equal opportunity problem. Women worry about it as much as men. No ethnic group is free from this worry. I can't tell you how many times my secretary has interrupted my philosophical inquiries because a client with this sort of problem is on the phone.
Yours is too big a topic for this spot. But I can lay the philosophical foundation and talk about some practical aspects. The main thing to remember is that testing financial models is like testing scientific theories. A scientific theory is someone's guess about the way the world works. A good theoretical scientist exposes his theory to rejection. A pseudoscientist or quack won't do that. Other theoreticians, as well as empirical scientists, reject scientific theories. According to Karl Popper's school of the philosophy of science, one never validates a scientific theory—one can only hope to reject it.
Similarly, a pricing model is someone's guess about the way a financial market works. A good model builder exposes his or her model to rejection. A quack will use subterfuges or political clout to avoid such a challenge. Other model builders, traders, professional testers and the market reject pricing models. You can never validate a model—but you can always reject it, if you push it far enough. How far do you have to push? That's the question.
How does one reject a theory? In the words of Malcolm X, "By any means necessary!” Paul Feyerabend, a philosopher of science, said basically the same thing in his book Against Method. A tester tries to be ingenious and use any trick available to find out where a model fails—that is, where it is unfaithful to reality. This requires an understanding of how the markets work, how the model is supposed to work and how traders work. One of my most useful approaches is to see if the modeler has made any of the hundreds of errors that I have made—and caught—during 25 years of building financial models. Another trick is to look for mistakes by others that I have caught in two decades of checking somebody else's financial models, as a reviewer for a scholarly journal, an employee of a derivatives dealer and an independent consultant.
Every model has its limits. If you push the model too far, it will bite you, even if it takes five years and extreme conditions for the weakness to show itself. The trader's job is to find the limits, not get fooled and make a lot of money. My job—any tester's job—is to find the limits before the trader or the market do, and help the client avoid losing a fortune to either.
How to get luckier with Monte Carlo
Dear Dr. Risk:
We use a Monte Carlo model to price some of our exotic, multivariate European payoff functions? Even with 60,000 paths, the remaining randomness is, frankly, disappointing. Do you have any ideas that might fix things up?
Al Grimaldi
Email your questions 
and/or comments to Dr. Risk at doctorrisk@aol.com. He will acknowledge each message and answer the most interesting questions here in the future. William Margrabe is a risk management consultant in the New York area. His web address is http://www.margrabe.com. 
Dear Mr. Grimaldi:
Your question came at just the right time. My presentation in Boston at the International Association of Financial Engineers meeting last September focused on this topic. The control variate approach may allow you reduce the standard error around your estimate of the product's value. Here's how it works:
1. You price the target payoff, Max(x1, ..., x6), with an MC model, and get its estimated price, MC{target}.
2. You price a related, control payoff, Max(x1,x2)—for which you know the exact price, V{control}—and get an estimated price, MC{control}.
3. You regress the target payoff against the control payoff, within the Monte Carlo model, and estimate the slope coefficient, b.
4. Finally, you compute the control variate estimate of the target payoff, CV{target}, by adjusting the Monte Carlo estimate of the value of the target payoff up or down to reflect error in the estimate of the control payoff's value and the slope coefficient.
That is, based on the assumption that
MC{target}  Value{target} = a + b * [MC{control}  Value{control}],
you can compute your control variate estimate,
Value{target} = MC{target} a  b * [MC{control}  Value{control}].
People usually assume that a = 0.
For example, suppose that
 Your MC estimate of the value of receiving the maximum of six prices was 116.89.
 Your MC estimate of the value of receiving the sum of six prices is 577.15.
 The known market value of receiving the sum of six prices is 579.03, the present value of the forward value of the portfolio.
 The regression coefficient of the maximum of six vs. the sum of six is 0.2.
What's your control variate estimate of the value of receiving the maximum of six prices?
CV Estimate = 116.89  0  0.2 *[577.15  579.03] = 117.27
You might think of it this way:
 Your MC estimate of the value of the sum of the six prices is low by 1.88.
 The expected error of the max of the six prices is 20 percent of the expected error of the sum.
 So, the conditional expected error of the max is 20 percent of 1.88, or 0.38.
 Subtract that from 116.89—add 0.38 to it—and you get 117.27.
You may find it advantageous to use more than one control payoff and multivariate regression.
When IRS eyes are smiling
Dear Dr. Risk:
I bought 1,000 shares of Fannie Mae in the 1980s at $15. It is now at about $60, after splitting threeforone and fourforone, which means the price would have been about $720 without the splits. That means I've got nearly three quarters of a million at risk in that stock. I can take the heat, but the little lady doesn't like the way the market's been jumping around lately. She wants me to sell it, so we don't end up poor if the stock market tanks. I don't want to sell it and pay the prohibitive capital gains tax. I can't sell short against the box. I tried to get a derivatives dealer to do an equity swap with me, but she just laughed at me and said I was too small. Can I somehow solve my problem with derivatives?
Fred Mack
Dear Mr. Mack:
I see the fix you're in, feel your pain and wish I had that problem, instead of you.
When I provided analytic support to an equity derivatives sales desk, I had a difficult time seeing the appeal of a "costless collar” to customers. The strategy can come close to taking the customers out of the underlying market, and is extremely expensive when you consider the inherent bidask spread. Of course, that's part of the appeal to the salesman. From the customer's point of view, the name "worthless collar” makes about as much sense.
According to a recent article in the Wall Street Journal, the costless collar is the poor man's (that is, individual's) equity swap. If you sell atthemoney calls against Fannie Mae and spend the premium on slightly inthemoney puts, your Fannie Mae position will be (almost) flat, just as if you had sold short against the box (which the IRS frowns on) or had equityswapped away your stock returns (which that dealer laughed at). If the IRS smiles on this trade, you won't incur an immediate tax liability.
This solution isn't permanent, isn't cheap and carries a risk. When those options expire, you're back in the same frying pan. Of course, you can repeat the trade until—in decreasing order of probability—you die and the basis on that stock steps up (probability = 1), the IRS prohibits the strategy as "abusive” or a Libertarian Congress repeals the federal income tax laws (0 < probability < epsilon: a long shot). Unfortunately, the bidask spreads on those options will eat you up, unless you die soon.
Many of my employer's customers were taxexempt institutions. Why did the costless collar appeal to some of them? Maybe they really did want some exposure, but didn't want to see the tails of the distribution. That can make sense for a professional manager who has scored big for the year, doesn't want to see that gain go away before the end of his performance period and doesn't dare tell his customers that he's decided to take a vacation from the market.
One lingering question concerns legal risk: The newpaper piece concluded with a key question, namely, "How far apart must those strike prices be to keep IRS eyes smiling?” You'll have to ask your tax attorney or accountant about that. Dr. Risk can't say.
Learning the basics
Dear Dr. Risk
I would love to be able to read, do all the exercises and "take apart” the most complicated books (such as Duffie's Dynamic Asset Pricing Theory) and articles on derivatives mathematics. Sometimes I get lost in the notation, however, and find that the text is difficult to understand. Perhaps I'm paying the price of being trained as a computer scientist and not a mathematician.
Do you know of a text, series of lecture notes, or even a summer school or other resource that could help me to bridge the gap to work at this level? Perhaps a text that I could work through that would equip me better to deal with the task of completing the books that elude my grasp? To date, I have happily read and understood books to the level of Rebonato's Interest Rate Models, Hull's text, Shimko's Continuous Time Finance, Rennie & Baxter's Financial Calculus, Salih Nefci's Mathematics of Financial Derivatives, and Willmot's Option Pricing. But I am still left perplexed by the more complicated works.
Ashwin
Dear Ashwin:
I understand your desire to understand the foundations of derivatives pricing, as well as your frustration at times. Once, eager to learn about diffusion processes, I spent six weeks studying Chapter 15 of Karlin and Taylor's A Second Course in Stochastic Processes. Afterward, drained by the experience, I contemplated giving up the study of financial theory. Then, as I leafed through the volume's preface I stumbled across these words: "We strongly recommend devoting a semester to diffusion processes (Chapter 15).” At that moment I had a revelation that changed my life: so that's why they put the preface at the beginning! Maybe I ought at least to sample the appetizer before I start devouring the meat of the book. But I digress. I have four reactions for you.
First, take five minutes to congratulate yourself for coming a long way. If you understand the books you mentioned, I would think you could solve nearly all practical problems, and even make contributions to theory.
Second, a number of mathematics departments offer courses in financial theory. You can find them on the Internet. I would bet that one near you in the United Kingdom meets your requirements. I have considered offering such a course over a semester, in conjunction with a university or as an inhouse program, as an alternative to the course I usually deliver at the level of Hull's Options, Futures, and Other Derivatives. If enough people in the New York area express interest, I'll offer such an advanced course in midtown or lower Manhattan.
Third, if you decide to do it yourself, prepare for months of concentrated study. One way to go is to study the references in the notes to Duffie's chapters. He doesn't skimp on ideas for further study. Alternatively, gaining a deep understanding of relatively simple derivatives models in discrete time, plus the ability to transfer that understanding to a context of continuoustime stochastic processes, would be one way to achieve your goal. Here's some more flesh on that skeleton:
1. Since you are a computer scientist, I would imagine that derivatives theory in discrete time and space would be relatively simple for you.
2. Nevertheless, you might gain a deeper understanding at the level of measure theory from the classic paper by J.M. Harrison and D.M. Kreps, "Martingales and Arbitrage in Multiperiod Securities Markets,” Journal of Economic Theory, Vol. 20 (1979), pp. 381–408.
3. While Harrison and Kreps discuss elements of measure theory at a useful level, I needed to study some elementary topology to follow them. Unfortunately, I didn't find a text that was elementary enough and focused enough for our purposes, although nearly everyone mentions J.L. Kelley's classic, General Topology.
4. For a broader and deeper understanding of the intersection of probability and measure—recall the prominence of "equivalent martingale measure” and "riskneutral probability” in derivatives theory—you might look at Billingsley's Probability and Measure or R.B. Ash's Real Analysis and Probability.
5. In order to transfer that deep understanding of discrete models to continuous models, you might try to understand key theorems—such as the central limit theorem and Donsker's theorem—in Billingsley's Convergence of Probability Measures. As the title suggests, Billingsley discusses what happens to a discrete probability measure (think of a binomial distribution for probability and ArrowDebreu prices) as the number of points in the sample space approaches infinity and the space itself approaches a continuum.
Finally, if you are willing to follow such a course of study, don't mention it in a job interview, unless it is for an academic position. It shows an interest in understanding things that outweighs an interest in building things. While I believe that such understanding can be useful at times, most employers of derivatives quants want employees with a bias toward action, not contemplation. So, by all means make the study, but let potential employers see only the results.
Let me know how your quest for knowledge turns out.