7) and (14.8).The PVBP vector A can be computed by making a one basis pointshift in each of the input rates while keeping the rest of the rates unchanged, and calculating the change in price of the portfolio, i.e.,(14.9)
Once the PVBP vector A is obtained, the portfolio can be hedged byusing a set of par inflation swaps.The basic principle for hedging is that we can offset the portfolio'srisk exposure by selling an equivalent position of par inflation swaps.The PVBP vector 6 of a par swap of maturity ti is given by(14.10)k=6 month
and the portfolio is delta hedged if we sellA.2 . =L ( j =1,2, ..., n )
' Jj
(14.11)
units of each of the par swaps of maturity ti.TIPS, Treasuries, htures contracts on TIPS, and other structured inflation linked instruments can also be used to hedge inflation swaps bymatching up the PVBP exposures. For example the delta hedging of inflation swaps by using TIPS and Treasuries transfers the risk exposure toreal interest rate levels into risk exposures to TIPShnflation swap ratespreads which are presumably much less volatile than real rates.For large interest rate movements (> 10 basis points), the deltahedge may not be adequate and we will have to make r corrections.Other concerns are that the volatilities at different parts of the real ratecurve can be quite different and the portfolio and the equivalent hedgingposition may have different time decay rates. Under these circumstances0 (Theta) and V (Vega) corrections may be incorporated. These treatments are analogous to the corresponding methods for handling interestrate swaps.
460
14.7
Inflation Linked Instruments Modeling
Chapter 14
Prospect of the Inflation Swap Business
The reference for inflation in the UK is the Retail Price Index (RPI). TheUK government started to issue index-linked Gilts on March 27, 1981.Over the years the total outstanding amount has increased from E800million to E45 billion, 16% of the total Gilt market. The UJS indexlinked Gilts suffered from initial difficulties in tax treatment and largebid-ask spreads due to illiquidity. The Gilt market in general does notpossess a reliable rep0 market. Despite these limitations, there is now anindex-linked Gilt options market and a sizable RPI swap business. TheRPI swap is the UK version of the inflation swap discussed in this chapter. The RPI swap business is offered by structured productslderivativesdesks of financial service firms to institutional clients and the outstanding notional amount of RPI swaps is estimated over E l billion. Thesize of the first U.S. TIPS issuance in January 1997 is 7 billion dollars.The instrument has a coupon of 3.5% and a 10 year term to maturity. Afraction of the issue will be stripped after Jan 29, 1997. Coupon stripsand principal strips will start to trade in the market.The U S . TIPS operation has had a very promising start. On Jan 27,1997, the pre settlement trading of TIPS indicates that the real yield is3.50/48,with a bidlask spread of only 2 basis points and a rep0 rate of
4.5%.In comparison with the UK index-linked Gilt experience, we can seethat the U.S. TIPS market is more liquid, short selling of TIPS is practical, and there will be a full term structure of TIPS real yields because ofthe trading of strips. All these will provide a fertile environment for inflation linked derivatives instruments, such as inflation swaps, to flourish.
Chapter 15
Interest RateProprietaryTradingStrategies
According to the comments of experienced traders, intuition and simplicity count in successful trading. Mathematical model and computer technology turns the horrendous amount of market data into simple and actionable pieces of information upon which human judgment reigns.We have introduced several finance models for interest rate instruments, such as the Exponential Spline Model, TFRM, and PCA. We alsohave experiences of applying these models in the real world. For example, in early 1994, we applied the Exponential Spline Model to derivesmooth, single-piece and accurate yield curves by using U.S. Treasurycoupon notes and bonds as inputs, we started systematically buildingyield curves and swap curves for the international markets. The countries we cover include the U.S., Japan, Canada, Australia, UK, France,Germany, Italy, Spain, the Netherlands, Sweden, Belgium, and Denmark.The accumulation of reliable yield and swap curves for all the countriesmakes it possible for us to systematically study the relationships amongthe yield curves and to better understand the dynamics of interest ratemovements. In 1995, we developed the TFRM for quantifying interestrate level and curve risks and for identifying relative value among yieldcurve sectors. Our analysis results, in the form of reports, are provideddaily to the New YorWLondon sales and trading desks. Salespersons,traders, and clients have continually shared their opinions and experiences with us about applying these analyses in actual transactions. Over
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time we have seen how the results can be best applied to the real worldsituation. In this chapter, we will share our experiences with you abouthow to apply our models to the real world to derive favorable tradingstrategies. Six different trading strategies will be covered.The trading strategies discussed in this chapter should be treated asstatistical arbitrage methods. They are not strategies for making moneywithout risks. Pure arbitrage opportunities are very rare in today’s efficient market environment. The strategies aim at making profits with favorable probabilities. As the proprietary trading desk executes a greatmany risky trades with good winning odds and rather small positions, bythe laws of statistics, the chance of success is greatly enhanced. We callthis the “Casino Trading” method. It should be one of the most important guiding principles for successful proprietary trading business.
15.1
RichlCheap Trade
A trade cannot be done without using specific instruments. Analyzingthe richness/cheapness of each instrument is the first step in trading.Sometimes the ricwcheap discrepancy itself is large enough to justifyexecuting a trade to capture it. This type of trade is said to be arichkheap trade and can be identified by the rich/cheap spread. Thericwcheap spread is defined as the market yield minus the theoreticalyield:ricwcheap spread: S = YMarket- YTheory.
(15.1)
Given the spot curve, we can discount the cashflows of a bond to gettheir present values, and add them up to obtain the theoretical price. Thetheoretical yield can be calculated by performing the regular price-toyield calculation.In practice, we often observe “persistence” in richness or cheapness.Due to liquidity or other market effects, a bond may always trade rich orcheap. A better measure of relative value is the current richkheap spreadminus the historical mean spread. Usually we compute the r i d c h e a pspreads for the 3 month period preceding the trade date, and calculate themean p , the standard deviation 0 ,and the Z-score z. The 2-score z isdefined as:
Sec. 15.1 RicWCheap Trade
463
s-P .Z-score: z = -
(15.2)
0
If we assume that richkheap spread S follows a normal distribution,then a 2-score of 1.6 means that there's a 95% chance that the instrumentis cheap and it will most likely revert to the mean. Table 15.1 showssuch an analysis for the Spanish market.Table 15.1: Spanish Bond RicWCheap Analysis for 4/2/97.Yrs toThe0Sprd- StdZTicker Coupon Maturity MatPrice Yield Yield Sprd Days High Low Mean Mean Dev ScoreSPGB11.450 08/30/98 1.392 107.050 5.961 5.85111.0 62.0 23.1 -2.8 13.2 -2.2 5.9-0.4-0.6SPGB9.900 10/31/98 1.562 105.600 5.989 5.883 10.6 62.0 21.0 1.3 13.4 -2.8 4.40.0-0.1 3.5-2.7-2.8 62.0 4.5 -10.6SPGB7.400 07/30/99 2.307 102.850 6.018 6.0460.70.02.6 3.92.6 62.0 7.4 -9.912.250 03/25/00 2.959 115.850 6.210 6.185SPGB-1.50.4 -2.9 1.9-2.5 62.0 6.7 -3.1SPGB10.900 08/30/03 6.392 120.450 6.825 6.850-1.31.a -2.6 1.9-0.7 62.0 6.4 -1.4SPGB10.500 10/30/03 6.559 118.550 6.872 6.8800.91.8 2.18.000 05/30/04 7.140 105.570 6.972 6.994 -2.2 62.0 1.2 -10.4 -4.0SPGB3.90.70.8 1.34.7 62.0 6.8 1.710.000 02/28/05 7.890 116.900 7.1 17 7.069SPGB2.30.93.8 62.0 5.9 -2.81.5 1.510.150 01/31/06 8.814 118.640 7.207 7.169SPGB1.81.42.3 62.0 3.9 -5.2 -0.22.58.800 04/30/06 9.058 110.100 7.234 7.211SPGB4.5-5.3 62.0 11.9 -5.3-1.87.350 03/31/07 9.975 100.600 7.258 7.311-9.8 5.3SPGB0.6 62.0 0.6 -8.2 -4.14.7 2.51.9SPGB8.200 02/28/09 11.890 106.100 7.402 7.396-1.60.0 62.0 1.6 0.00.8 -0.8 0.5SPGB8.700 02/28/12 14.890 110.720 7.475 7.474
-
According to this analysis, the simplest trade we can execute is longand short a pair of relatively rich and cheap bonds with small maturitydifferences and match up the durations. For example, we could long107MM of SPGB 8.8 4/30/06 and short 100MM of SPGB 7.35 3/31/07.We may expect to pick up roughly 0.618~(2.5+9.8)~ 7 . 6basispoints.'43 According to Bloomberg quotes, the transaction cost is about10 cents per $1 10, or 2 basis points. The trade may have a net profit of 5basis points, or $200,000 per $lOOMM. If we also want to hedge curverisk, we would need to execute a bulletharbell trade, such as long 49MMpar amount of SPGB 8.8 4/30/06, short 1OOMM SPGB 7.35 3/31/07, andlong 47MM SPGB 8.2 2/28/09. We can get the hedge ratios by applyingthe two-factor duration hedge presented in the previous chapter. Thistrading method has been put to extensive empirical tests in the U.S.Treasury and STRIP markets. They worked well if the trades could bedone at the levels specified in this section. In practice, it is not an easytask to execute such trades. Short selling a non-dollar bond can be143 0.618 is the golden mean number. The golden mean reversion corresponds to thehighest probability, even though we may not be sure how the spread is distributed inreality.
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costly.
15.2
RichlCheap Analysis
Richlcheap analysis is one of the most important aspects in making investment and trading decisions. In order to maximize profits, we usuallywant to buy cheap instruments and sell rich ones. Within the limitationof priorities and constraints (such as riskheturn considerations), for making an investment, we would like to choose a relatively cheap class ofinstruments, and then select a cheap instrument in that class. On theother hand, if we want to sell part of our portfolio or short the market, wewould like to sell a rich instrument.In this section, we first show how to apply the two-factor model toidentify ricwcheap yield curve sectors and quantify the relative value ofyield curve sectors in basis points. Then we further explain how practitioners make richlcheap analysis at the individual instrument level.75.2.7
Yield Curve Sector RichlCheap Analysis
The key to the yield curve sector ricwcheap analysis is the fact that theyield curve is mainly two-factor driven, and the correlation among different points on the yield curve is persistent and stable over relativelylong periods of time, such as a few weeks or a few months. Within thecontext of the two-factor risk model (TFRM) presented in the previouschapter, every point of the yield curve is driven by the two factor movements as described by the following equationY ( z )= P L ( z ) f L + PT (r>fT+ C(z>+ 6(z)
7
(15.3)
where z is the time to maturity, Y ( z ) is the yield curve point of maturityz , fL is the level factor, f T is the twist factors, pL(z) and &(z) arethe corresponding correlation curves, C(z) is a constant curve, and thelast term S(z) is the regression error curve which represents randomfluctuations and higher order contributions. Over a historical period, the6(z) curve averages to zero and its standard deviations at different curvesectors indicates how much the yield curve typically departs from thetwo-factor reconstruction.If the correlation pattern is stable, in other words, the correlation coefficient curves pL(z), PT (z) , and C(z) are slowly changing functions
RicWCheup Analysis
Sec. 15.2
465
of time, we can separate the yield curve into two parts: a two-factor project curve f ( ~and) a transient error term S(z) with the former given by(15.4)In this analysis, we consider the two-factor projected yield curvef(r) the fair yield curve (or the equilibrium yield curve) given the factors f L and fT . The error term S(z) absorbs noises and all other higherfrequency, less important factor contributions, and is to be taken as therelative value indicator. As time passes by, the error term oscillatesaround zero, signaling cheapness if it is positive and richness if negative.Over a period of time, say three months, we can treat the error term as awhite noise and compute its high, low, mean and standard deviation, andcompute the Z-score for the current day,(15.5)where o(z) is the standard deviation of S(z). The Z-score is the number of standard deviations from the mean and has probabilistic interpretation. For example, if the noise is normally distributed, a Z-score of -1.6means that there is a 95% chance the yield curve point to be rich (showing a yield less than the fair value).I
I
I
I
I
................................................................................
-10
0
I
I
I
I
I
5
10
15
20
25
-2000
30
Years to maturity
Figure 15.1. Yield curve richhheap sector analysis.
Figure 15.1 illustrates the yield curve sector ricldcheap analysis ofthe US. Treasury par coupon curve for trade date March 11, 1998. The
466
Interest Rate Proprietary Trading Strategies
Chapter 15
solid curve represents the difference between the actual curve and thetwo-factor projected curve in basis points, the vertical bars show thestandard deviation over a period of 3 months, and the dashed line represents the relative value in dollars amounts, which is equal to the relativevalue in basis points times the corresponding modified duration.The effectiveness of this analysis really depends on how stable thecorrelation patterns are, and how fast does a significantly rich or cheapsector reverts to the mean. Over the period from 1994 to 1998, we havesuccessfully applied the yield curve sector analyses to the U.S., Germany, UK, Spain, France, Italy, Australia, and Japan Government markets. Empirical evidence reveals that, the optimal period for calculatingthe regression coefficient curves is about 3 to 6 months, and a rich(cheap) sector with an absolute value of Z score greater than 2 is rarelyobserved, and it usually reverts to zero in a few days or a few weekswithout exception.With the help of this analysis, a trader or an investor can easily picka rich or cheap sector of the curve when implementing his trading or investment strategies. A report in the form of Figure 15.1 is a powerfultool, and is very well liked by traders, sales forces, and clients.15.2.2 RichICheap Analysis for Notes and BondsTo execute a trade or make an investment, one always has to use actualinstruments; either it is a bond, a futures contract, a traded index, or aderivatives instrument. A richkheap analysis at the individual instrument level is an indispensable step in making a trading or investmentdecision intelligently. As a general rule, one would like to buy cheapinstruments, and sell rich instruments.Even though the modeling and pricing of different kinds of instruments can be quite different, there is a generic way of making richkheapanalysis for individual instruments. To start with, we need to have amodel for valuing the instrument under analysis, and the model can provide a theoretical price for the instrument. We usually define therichkheap spread for a bond as the market yield minus the theoreticalyield, that is,Rich/Cheap Spread RC= Market Value -Theoretical Value,
and use it as a measure of an instrument’s richness or cheapness.
(15.6)
Sec. 15.2 RicWCheap Analysis
467
One would naively think that the richkheap spread reveals the instrument’s attractiveness, so that one would buy the instrument if one canbuy the instrument from the market at a price lower than the model price,or vice versa. In reality this rarely works. First of all, the model mayintroduce a systematic error in the pricing of the instrument. Secondly,the market does not need to agree with the model for a number of technical reasons. For example, supply-demand relationship, financing costs,and liquidity considerations may be important factors in determining theinstrument’s price. Usually the richkheap spread exhibits persistence.An instrument will always trade rich or cheap to the model. A moresuitable measure of an instrument’s attractiveness is the relative valuegiven by
RV=RC-RC,
(15.7)
where RC is the average of RC . Similar to Eq. (15.2), we define theZ-score as:
z=-RV
(15.8)
0
where 0 is the standard deviation of the richkheap spread. If we believe that RV follows a Normal distribution, we can assign probabilitymeaning to the attractiveness of an instrument according to the Z-score.The richkheap analysis for notes and bonds is straightforward. Wefirst apply the orthonormal Exponential Spline Model to create the spotcurve using all the notes and bonds as input, then use the spot curve todiscount the cashflows of a bond and add them up to obtain the theoretical price. Lastly, we perform the conventional price-to-yield calculationto compute the theoretical yield of the bond. See Figure 15.2.
468
Interest Rate Proprietary Trading StrategiesUS Treasury Notes &Bonds Historical Rich Cheap AnalysisCpn Ma(
Yield Spread: Act - Theo (BP)
P* Yld MDur
5.875 u m / o 4 101.1765.638 4.943 I . .7.875 I 1/15/04 I 11.824 5.718 5.142
..
I. . ....
Chapter 15
Trade: 03/10/98RC Hi Lo Mn
RV Sld 2
.I -5.0 -4.3 -7.4 -5.8 0.8 0.9 0.90.6 0.6 -1.6 -0.3 0.9 0.5 1.8-1.9 0.5 -2.3 - o . n - i . i u . 7 - 1 . ~
CIRCLE Historical Mean
SQUARE: CurrentSpread
B A R HighlLow
Figure 15.2. An Example of RicWCheap Analysis.
15.3
BondlSwap Trade
Taking a position in bond and an opposite position in swap is a way tocapture value from swap spread movements. Over the years we havesystematically studied and recorded the swap spreads of most of thebonds of all the markets. The conclusion we can draw is that, swapspreads always vary within a certain range so that it is possible to execute trades when they reach extremes.The swap spread (also called asset swap spread) for a bond is definedas the swap implied yield minus the market yield:(15.9)We use the swap zero curve to calculate the present value of thecashflows of a bond and call it the swap implied price. Then we performthe regular price-to-yield calculation to obtain the bonds swap curve implied yield. The statistics of swap spreads, such as highs, lows, means,standard deviations, and Z scores for all the markets are calculated on adaily basis and are stored in a database. A glance of the swap spreads forthe Spanish market on April 17th, 1997 is shown in Table 15.2.
469
Sec. 15.4 Curvature Trade
Table 15.2: Spanish Bond/Swap Spread Analysis for 4/17/97.Y n toTicker Coupon Malurlly MatSPGB10.250 11/30/98 1.60SPGB8.300 12/15/98 1.64SPGB10,100 02/28/01 3.85SPGB8.400 04/30/01 4.02SPGB 11,300 01/15/02 4.73SPGB7.900 02/28/02 4.85SPGB 10.300 06/15/02 5.14SPGB 10.900 08/30/03 6.35SPGB8.000 05/30/04 7.10SPGB 10.150 01/31/06 8.77SPGB8.800 04/30/06 9.02SPGB8.700 02/28/12 14.85
Price Yield107.00 5.52104.22 5.51114.40 5.80108.90 5.85121.10 6.02107.70 6.01118.05 6.09123.30 6.32108.50 6.46122.05 6.73113.35 8.77114.60 7.07
Swap Imp SpreadSpread- Sld(BP) Days High Low Mean Mean Dev ZSCOreYield-7.9 4.4-1.85.6714.559 31.6 12.7 22.4-6.7 4.1-1.616.25 9 2 9 . 3 1 3 . 2 22.95.67-0.816.259 23.3 11.0 18.0-1.8 2.35.97-1.018.759 25.8 13.5 21.0-2.3 2.36.03-2.010.359 18.5 10.3 14.3-4.0 2.06.120.84.40.26.1614.95 9 2 0 . 8 0.8 14.1-2.3 2.2-1.06.2111.859 18.9 9.1 14.10.22.50.16.4210.359 18.1 5.2 10.1-0.32.0-0.16.5812.059 17.9 8.0 12.31.6-0.70.859 7.1 0.22.0-1.26.74-0.3-0.72.16.803.359 11.4 0.74.00.64.10.16.038.35 9 1 7 . 0 0.37.7
According to the analysis, a simple trade would be short the 5-yearswap against long the SPGB 11.30 01/15/02, because the swap spread isat the low end with a Z score of -1.98. We may expect to capture 4 basis points before transaction costs. The trade should be hedged againstinterest rate level movements. The hedge ratio can be determined bymatching the price movement per one basis point shift of the Treasuryand the swap curves.
15.4
Curvature Trade
From empirical studies of yield curve movements, we found that, for theU.S., Japan and the European countries we covered, more than 92% ofvariances can be explained by using a level and a twist two-factor model.The most important application of the TFRM is hedging. For everyinstrument, we provide a pair of level and twist durations as the measureof its interest rate level and curve exposures. Table 15.3 is extractedfrom the TRFM report of the German market on April 21, 1997. Thisreport is generated daily for all the markets we cover.
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Table 15.3: German Market Two-Factor Report for 3/25/97.Yrs toFullMod Level Twist R/C 2- Sector R/C Sector Z.Ticker Coupon Maturity MatPrice Yield PriceDurDurDur Score Indexcbp) ScoreDBR7.000 04/20/99 2.055 106.290 3.760 112.901 1.804 0.664 0.645-0.8-0.1-0.1OBL1116.125 05/20/99 2.137 104.650 3.813 109.924 1.902 0.734 0.6721.4-0.2-0.2TOBL6.375 07/01/99 2.252 105.260 3.881 110.024 2.005 0.831 0.7050.6-0.3-0.4DBR7.125 12/20/99 2.723 107.660 4.084 109.639 2.436 1.289 0.8090.7-0.9-1.0DBRUF8.750 07/20/00 3.304 113.320 4.326 119.396 2.764 1.881 0.890-0.4-1.6-1.5DBR8.500 08/21/00 3.392 112.740 4.364 117.911 2.854 1.980 0.902-0.7-1 .8-1.5OBL1205.000 08/20/01 4.389 101.350 4.646 104.406 3.764 3.132 1.0020.6-2.9-1.9DBR8.250 09/20/01 4.474 113.710 4.762 118.064 3.645 3.027 0.977-2.9-1.o-1.9DBRUF8.000 01/21/02 4.81 1 113.020 4.889 114.553 3.973 3.299 0.985-3.21.3-2.0THA7.750 10/01/02 5.504 112.490 5.081 116.343 4.367 3.747 0.9891.a-3.4-1.9THA6.625 07/09/03 6.274 106.950 5.288 111.753 4.887 4.216 0.961-1.1-3.3-1.7THA6.750 05113/04 7.118 107.360 5.471 113.304 5.332 4.608 0.869-1.7-2.8-1.37.50