0 11/11/04 7.61 6 111.61 0 5.579 114.506 5.698 4.803 0.795DBR2.1-2.2-1.17.375 01/03/05 7.762 110.840 5.608 112.622 5.846 4.903 0.768DBR0.1-2.1-1.1DBR6.250 04/26/06 9.071 103.220 5.782 109,019 6.470 6.041 0.491-0.7-0.7-0.7DBR6.000 01/04/07 9.764 100.990 5.860 102.423 7.156 6.714 0.3551.6-0.2-0.2nFm5.625 09/20/16 19.474 90.300 6.514 93.269 10.843 10.138 0.6041.3-2.4-0.9-0.6DBR6.250 01/04/24 26.764 95.100 6.643 96.593 12.319 10.660 0.667-0.3-1.5
In order to hedge a portfolio or a trading position, we only need tosolve two linear simultaneous equations to match the overall level andtwist exposures. Sometimes the investor or trader may have a view onthe curve movements (steepening or flattening), but is uncertain aboutthe levels. We can then neutralize the level exposure by matching thelevel factor durations, and assume a desired net curve risk. According toour empirical studies, it is very important to apply two factor hedgingmethods in the European markets where the correlation between yieldlevels at different maturities are rather weak. The level factor usuallyexplains between 60% to 85% of yield curve movements. It is crucial tobring in the second factor to increase the coverage to 95%, so that a trading position is reasonably well hedged. With the help of two-factorhedges, it becomes possible to control the risk well so that a trader cantake positions in bonds with very different maturity terms to capture themarket anomalies. However, in the common practice of using durations,it is very risky, say in the German market, to long the one year bill, shortthe five year, and long the 10 year.Another important application of the TFRM in trading is the twofactor yield curve rich/cheap sector analysis presented in Section 15.2.1.
15.5
Spread Trade
A spread trade is a directional bet on the movements of the yield curveslope. Under most circumstances, the movement of yield curve spread
Sec. 15.5 Spread Trade
471
(say the 10/5 spread) is just as unpredictable as the level of interest rates.However, the range of yield curve spread movements is usually morecertain, and the spread often oscillates more frequently than the level.The highs and troughs are often reached within weeks, so that they constitute an excellent vehicle for trading. In an environment of nontrending spreads, it is possible to execute trades at spread extreme andbet on their reversion to mean. A spread trade is risky and the potentialprofit is also large, so that transaction costs are relatively less important,a crucial prerequisite for successful proprietary trading.Table 15.4: Par Coupon Curve 10/5 Spread Analysis for 4/14/97.curent
culerltDays Hgh Low Mean SdDw Mean ZsCae Rior 3Rior Trend t-Test21.7 62 31.3 21.7 27.529-5.8 -20 -20 -1.6Flatten -1.2
29.799.5101.128.981.979.756.6s3.256.592758.9
63 34.9 226 29.363 117.6 88.0 101.463 110.2 88.8 100.163 58.8 128 33.063 104.3 64.8 80.962 lCe9 77.6 93.362 90.1 56.0 78.963 107.7 827 94.462 55.5 30.8 43.263 106.2 83.5 94.361 98.9 26.5 621
-
3.27.25.513.211.06.79.97.35.75.618.0
0.4-1.91 .o-4.11 .o-13.6-224-1.2123-1.6-32
0.1 0.1-0.3 -0.30.2 0.2-0.3 -0.30.1 0.0-20 -22-23 -24-0.2 -0.222 1.6-0.3 -0.4-0.2 -0.4
-0.1 None0.1 Flatten0.0 None-0.2Flatten-0.1 I\bne-22Flatten-20Flatten-0.2 Flatten
0.8 f\bne-0.6 None-0.4Flatten
-
-0.5-1.3-0.6-1.3-0.9-1.6-2.6-1.20.2-0.5-1.1
Table 15.4 is the Global Market Yield Curve Spread Analysis Reportfor April 14th, 1997. The t-Test is the slope of trend divided by the estimated (1 standard deviation) error of the slope of trend. When the tTest is low, it indicates no trend, so that Z-score analysis of the spreadmay be meaningful and one can execute a trade if the spread is at an extreme (say z = 2). From the table, we see that the Australia market isnon-trending, and the 10/5 spread is at an extremely high with z = 2.1 .We suggest to long the 10 year, and short the 5 year. Another consideration underlying the trade idea is that, usually the slope of the Australiancurve is positively correlated with the other European curves' slopes. Asindicated by the table, the European curves are in a flattening environment, so that it is reasonable to bet on the flattening of the Australiancurve. Since the current spread minus the mean spread is 12 basis points,we may expect to capture 0.618 x 12 -. 7.4 basis points.
472
15.6
Interest Rate Proprietary Trading Strategies
Chapter 15
Box Trade
An often used and rather risky trade is the box trade, which is implemented to capture the anomalies between the yield curve slopes of twocountries. The prerequisite is that the slopes of the two countries are correlated in the long run. The simplest box trade is a spread position in onecountry against a spread position in another country so that the overallposition is hedged against level movements in both countries.
15.7
OAT Floater Trade
Floaters represent a significant portion of the Italian and French markets.The most interesting floaters are those that are indexed to long term ratesbut pay coupons quarterly or semiannually. The important feature of thefloaters is that they have very short interest rate level durations, butrather high exposure to curve risk. This is just the opposite of regularfixed-coupon bonds. In a floater trade, we can easily control the levelrisk, hedge the curve risk, and concentrate on capturing discrepanciesbetween the price movements of the floaters. In this section we use theFrench OAT Floaters as an example to demonstrate how we could profitfrom floater trading.Why OAT Floaters? FRTR 10/25/06 and 1/25/09 floaters representnew financial inventions in the French market. They pay quarterly coupons indexed to TEClO which is the 10-year constant maturity yield index. FRTR l0/25/06 started to trade in April 96 and FRTR 1/25/09 onlyappeared in Jan 1997. Because they are relatively more difficult to priceand the balance of demand and supply has not yet been fully established,we expect to see wide price fluctuations. This may lead to profitabletrading opportunities, Both floaters have very small risk exposure to interest rate level shifts, and the level durations are about 0.25 year. Onthe other hand they both have considerable exposure to curve twists.However, the price movement per basis point change of the 10/2 spreadcan be calculated by using our floater pricing model. This leads to thefollowing excellent trade idea: Construct a hypothetical portfolio of thetwo floaters with no curve exposure and execute the trade when we observe price extreme. The portfolio will have negligible level risk and isnon-trending. The recommended actions are:Construct portfolio:P = 1 (FRTR 10/25/06)-0.86 (FRTR 1/25/09).
Sec. 15.9 A Generic Convergence Trading Strategy
4 73
Call the OAT floater traders to get executable bid-ask prices for bothfloaters every day.Compute the portfolio P and record the history.Execute the trade when price fluctuation is 1.5 standard deviationsaway and when the magnitude of expected convergence can well compensate transaction costs.Unwind the trade when the profit exceeds some thresholds.Such a floater pair trade has minimal exposure to interest rate levelsand is hedged against curve movements. It is trying to take advantage ofthe random fluctuations between the price movements of the floaters, andis considered a low risk trade.
15.8
CashlFutures Trade
The two-factor framework makes it convenient to hedge notes, bonds,futures and Euro-dollar futures contracts on a common platform, so thatit opens up the possibility for constructing trading positions among theseinstruments that are hedged against two-factor risks and can capturemarket mispricings.Empirical studies and actual trading experiences have shown thatmispricings within the U.S. notes and bonds market are typically at theorder of $10,000 for a $100 million position. A search among all theliquid notes and bonds sometimes reveals mispricings of $50,000 to$100,000 per $100 million. Transaction costs range from half a tick (thatis about $16,000 per $100 million) to a few ticks. The profit margins aresmall. The situation is similar within the futures market. With fewerinstruments and larger transaction costs, profitable convergence tradingopportunities within the futures market is rare.The prospect is much brighter if one trades across the cash, futures,and Euro futures markets. For two-factor hedged cash and futures trades,the transaction costs stay at about $50,000 per $100 million, but the expected profits increase to about $50,000 to $300,000 per $100 million.
15.9
A Generic Convergence Trading Strategy
The trading strategies discussed in the previous sections rely on eitherour ability to correctly model the instruments or our views on the unhedged components of market movements. Sometimes it is rather diffi-
4 74
Interest Rate Proprietary Trading Strategies
Chapter 15
cult to accurately model certain instruments for a number of reasons. Forexample, a new type of debt instruments may have just appeared in themarket. Though they seem to trade according to a rather strong correlation pattern in connection with other interest rate instruments, but nosimple mathematical models can explain their behavior. Under such circumstances, if our goal is to generate profits without taking directionalrisks, we are left with no choice but to construct correlation based hedgesand statistically arbitrage the market using these instruments.Let us consider the price histories of N instruments that are reasonably strongly correlated. The length of the history is T days. Suppose wehave observed N price time series, S,(j) for i=l,2,...,N andt = 1,2,.-., T . Our goal is to construct a linear combination of the N instruments such that the variance of the trade over the observed history isminimized. In the real world, it is difficult to simultaneously executetrades in more than three instruments, and only a linear combination ofseveral instruments is practical. For convenience, we fix the trading position (or number of units or shares) in the first instrument to be one, andput on the following constant share trade,
p = S(') + x(2)s(2) + ... + x ( N ) S ( N )
(15.10)
where di)is the position in instrument S(j). We also assume that theexpected trading horizon is rather short, such as a few days to a fewweeks, and every position is financed by paying or receiving a fixed termrep0 rate. Letdenote the carry of taking a position in one unit ofinstrument S(')for one day, then daily carry of the trading position isgiven by
c = c(1)+ x(2)c(2) + ...+ x ( N ) C ( N )
(15.11)
With these assumptions, the cost of carry for putting on the trade islinear in time. We now require that this N-instrument trade is constructed to hedge away the market risks associated with the first ( N - 1)driving factors of the market, so that the overall trading position minusthe linear carry is only a noise signal with respect to the ( N - 1) -factorhedge. The trading strategy can be summarized as follows:(1). Construct an (N-1) empirical linear hedge by minimizing the variance:
475
Sec. 15.9 A Generic Convergence Trading Strategy
c(
v = T ,$I) + X ( 2 ) S ( 2 ) + ...+ x ( ~ )t , # -~ kt) - c )2I
I
t=1
(15.12)
This minimization problem is equivalent to solving the following(N + 1) simultaneous linear equations with the (N+l) unknownsk,c,X(2),X(3),...,X(N):
for i=2,3,...,N
(2). Compute statistics such as the standard deviation:(15.14)
the expected convergence:E = S;l) + ~ ( ~ ) s+,..(.+~x) ( ~ ) $ -~ kt) - c ,
(15.15)
and today’s Z-score:E
2=-.0-
(15.16)
k > c and zlz*,
(15.17)
We execute the trade if
where z*is a threshold Z-score value (usually greater than one), or implement the reverse trade ifk l c and z 2 z * ,
(15.18)
476
Interest Rate Proprietary Trading Strategies
Chapter 15
subject to the condition that the expected profit is significantly largerthan the transaction cost:
IEl>f x d ,
(15.19)
where d is the two-way transaction cost for putting on and getting off thetrade, and f is a numerical factor that is greater than one.(3) Exit the trade when E reverses to zero. If the correlation pattern isreliable and will not be broken, we should not cut loss when the tradegoes against us, but instead double up the position.The condition that the slope of the empirical trend should be largerthan or equal to the daily carry in Eq. (15.17) ensures that the trade doesnot have an average disadvantage over the passage of time. The key ofthis trading strategy is to eliminate market risks and focus on the noisepart of the price signals so as to buy low and sell high.In reality a proprietary trader constructs such trades by exhaustingany three combinations from a pool of liquid and strongly correlated instruments, tests for the conditions, puts on the trade if satisfied, andmaintains the positions until they are taken off. For bond and futurestrades, dynamically re-balancing the positions is cumbersome and costly,therefore not recommended.
15.10 Other Factors Related to Trading StrategyIn this section we discuss several factors that may affect the tradingstrategies provided in previous sections.15.10.7 Transaction CostThe typical bid-ask spreads for executing a 10 year maturity bond tradein the global markets are shown in Table 15.5. In order to comprehendhow much the transaction cost can really take away from an apparentlyprofitable trade, we will analyze a richkheap trade in the U.S. market aswell as a curvature trade in the German market.Table 15.5: Typical bid-ask Spreads for 10 Year Bond Trades.BidlAsk US Canada UK France Germany Italy Spain Japan Dutch Australia Sweden DenmarkYield (bp) 1Price($) 0.03
1.5
2
0.05
0.06
20.1
20.08
2
4
0.08 0.06
2
0.3
20.1
2
0.08
30.15
20.1
For the U.S. market, the transaction cost is the smallest, but the mar-
Sec. 15.10 Other Factors Related to Trading Strategy
477
ket discrepancies and opportunities are also tiny. From individualrichkheap considerations, sometimes we can find opportunities of about2 basis points in the 10 year sector. Suppose we have identified a pair ofneighboring instruments in the 10 year sector with a combined Z-score of1.6 and expected convergence of 2 basis points. The standard deviationof the richkheap spread is 2/1.6=1.25 basis points. In the perfectworld with no transaction costs, we expect to capture 2 basis points witha 95% probability, and it is very attractive. However, when we count the1 basis point bid-ask spread in yields, the situation becomes that we expect to capture 1 basis point with a 74% winning probability (corresponding to a Z-score of U1.6 = 0.625). The trade is not very appealingnow if we consider the difficulty of execution and the fact that the spreadmay not be normally distributed and may even be drifting. Next let’sconsider the German curvature trade discussed in Section 15.4. OnMarch 25th 1997, the 5-year sector is found to be rich relative to the 2year and 10-year sectors, and we expect to capture 6 basis points with aZ-score of 2. The standard deviation of the par coupon curve fluctuationis 3 basis points. However, the two-factor hedge is not perfect, and thefluctuation of the beta curves can add about 2 basis points to the standarddeviation. Without transaction costs, the trade expects to capture 6 basispoints with a 90% winning probability. It is a great trade. When weconsider bid-ask spread, the 8 cents per 100 dollar transaction cost translates into about 2 basis points in yield for the five year sector. This notonly reduces the winning probability to about 78%, but also diminishesthe expected profitability to 4 basis points. If we further take into account the difficulties with short selling the German bond and the risks torep0 rates, the trade becomes barely worth executing.The above two opportunities are actually rarely seen in the market,and they are low risk trades. Based on our experience, level and curvehedged low risk statistical arbitrage trading opportunities rarely occur,mainly because transaction costs take away most of the juices.
15.10.2 Higher Risk and Highly Profitable TradesProprietary trading groups should look for more risky trades with highprofithsk ratios where transaction cost is of minimum concern. Spreadtrades and box trades are such examples. Bond/Swap spread trades arealso suitable for proprietary trading. They are more risky than level andcurve hedged trades, but they are much less risky than spread or boxtrades. The potential profit is high in the non-dollar bond/swap markets.
478
Interest Rate Proprietary Trading Strategies
Chapter 15
New products in the financial market often bring forth profitable tradingopportunities.75.70.3 Bet Big When All Components Line Up
It is possible that on some days, we find very rich and cheap sectors ofthe yield curve, and in the rich and cheap sectors we have identified veryrich and very cheap individual instruments. Meanwhile the rep0 rates isalso favorable, and the trade also agrees with our economic intuition. Ifthe after-transaction-cost winning probability is higher than 95%, bet onethird or half of the risk capital.75.70.4 Human Judgment
All the trading and hedging methods suggested in this chapter are basedon observations of the market behavior in the past. The price movementsare after all the results of collective human decisions. What happened inthe past may not always hold true in the future. The mathematical models and computer programs only help the trader make decisions. However, models and strategies are crucial to successful proprietary trading,because under today's market conditions, at any moment, there are vastamounts of information coming at electronic speed. The human brain isinefficient at analyzing lots of raw data, no matter how experienced orsharp the person is. The mathematical models and computer programsshould process the data and elevate them to a highly relevant level uponwhich human judgment can function most efficiently. Every now andthen, after some activities take place, the computer should scrutinize themarket to see if anything fits into our trading strategies, and then bringup a list of possible trades. The trading strategy is just a tool and the final decision is up to the human trader.
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