Methods for Constructing a Yield Curve

Transcription

1 Methods for Constructng a Yeld Curve Patrck S. Hagan Chef Investment Offce, JP Morgan Wood Street London, EC2V 7AN, England, e-mal: patrck.s.hagan@jpmorgan.com Graeme West Fnancal Modellng Agency, 9 Frst Ave East, Parktown North, 293, South Afrca e-mal: graeme@fnmod.co.za, Abstract In ths paper we survey a wde selecton of the nterpolaton algorthms that are n use n fnancal markets for constructon of curves such as forward curves, bass curves, and most mportantly, yeld curves. In the case of yeld curves we also revew the ssue of bootstrappng and dscuss how the nterpolaton algorthm should be ntmately connected to the bootstrap tself. As we wll see, many methods commonly n use suffer from problems: they post unreasonable expectons, or are not even necessarly arbtrage free. Moreover, many methods result n materal varaton n large sectons of the curve when only one nput s perturbed (the method s not local). In Hagan and West [26] we ntroduced two new nterpolaton methods the monotone convex method and the mnmal method. In ths paper we wll revew the monotone convex method and hghlght why ths method has a very hgh pedgree n terms of the constructon qualty crtera that one should be nterested n. Keywords yeld curve, nterpolaton, fxed ncome, dscount factors Basc Yeld Curve Mathematcs Much of what s sad here s a reprse of the excellent ntroducton n [Rebonato, 998,.2]. The term structure of nterest rates s defned as the relatonshp between the yeld-to-maturty on a zero coupon bond and the bond s maturty. If we are gong to prce dervatves whch have been modelled n contnuous-tme off of the curve, t makes sense to commt ourselves to usng contnuously-compounded rates from the outset. Now s denoted tme. The prce of an nstrument whch pays unt of currency at tme t such an nstrument s called a dscount or zero coupon bond s denoted Z(, t). The nverse of ths amount could be denoted C(, t) and called the captalsaton factor: t s the redempton amount earned at tme t from an nvestment at tme of unt of currency n sad zero coupon bonds. The frst and most obvous fact s that Z(, t) s decreasng n t (equvalently, C(, t) s ncreasng). Suppose Z(,t ) < Z(,t 2 ) for some t < t 2. Then the arbtrageur wll buy a zero coupon bond for tme t, and sell one for tme t 2, for an mmedate ncome of Z(,t 2 ) Z(,t ) >. At tme t they wll receve unt of currency from the bond they have bought, whch they could keep under ther bed for all we care untl tme t 2, when they delver n the bond they have sold. What we have sad so far assumes that such bonds do trade, wth suffcent lqudty, and as a contnuum.e. a zero coupon bond exsts for every redempton date t. In fact, such bonds rarely trade n the market. Rather what we need to do s mpute such a contnuum va a process known as bootstrappng. It s more common for the market practtoner to thnk and work n terms of contnuously compounded rates. The tme contnuously compounded rsk free rate for maturty t, denoted r(t), s gven by the relatonshp C(, t) = exp(r(t)t) () Z(, t) = exp( r(t)t) (2) r(t) = ln Z(, t) (3) t In so-called normal markets, yeld curves are upwardly slopng, wth longer term nterest rates beng hgher than short term. A yeld curve whch s downward slopng s called nverted. A yeld curve wth one or more turnng ponts s called mxed. It s often stated that such mxed yeld curves are sgns of market llqudty or nstablty. Ths s not the case. Supply and demand for the nstruments that are used to bootstrap the curve may smply mply such shapes. One can, n a stable market wth reasonable lqudty, observe a consstent mxed shape over long perods of tme. 7 WILMOTT magazne

2 TECHNICAL ARTICLE 2 Z(, t 2 ) f (t) = d ln(z(t)) (7) dt = d r(t)t (8) dt Z(, t ) t t 2 So f (t) = r(t) + r (t)t, so the forward rates wll le above the yeld curve when the yeld curve s normal, and below the yeld curve when t s nverted. By ntegratng, Fgure : The arbtrage argument that shows that Z(, t) must be decreasng. The shape of the graph for Z(, t) does not reflect the shape of the yeld curve n any obvous way. As already mentoned, the dscount factor curve must be monotoncally decreasng whether the yeld curve s normal, mxed or nverted. Nevertheless, many bootstrappng and nterpolaton algorthms for constructng yeld curves mss ths absolutely fundamental pont. Interestngly, there wll be at least one class of yeld curve where the above argument for a decreasng Z functon does not hold true a real (nflaton lnked) curve. Because the actual sze of the cash payments that wll occur are unknown (as they are determned by the evoluton of a prce ndex, whch s unknown) the arbtrage argument presented above does not hold. Thus, for a real curve the Z functon s not necessarly decreasng (and emprcally ths phenomenon does on occason occur).. Forward rates If we can borrow at a known rate at tme to date t, and we can borrow from t to t 2 at a rate known and fxed at, then effectvely we can borrow at a known rate at untl t 2. Clearly Z(, t )Z(; t, t 2 ) = Z(, t 2 ) (4) s the no arbtrage equaton: Z(; t, t 2 ) s the forward dscount factor for the perod from t to t 2 t has to be ths value at tme wth the nformaton avalable at that tme, to ensure no arbtrage. The forward rate governng the perod from t to t 2, denoted f (; t, t 2 ) satsfes exp( f (; t, t 2 )(t 2 t )) = Z(; t, t 2 ) Immedately, we see that forward rates are postve (and ths s equvalent to the dscount functon decreasng). We have ether of f (; t, t 2 ) = ln(z(, t 2)) ln(z(, t )) t 2 t (5) = r 2t 2 r t t 2 t (6) Let the nstantaneous forward rate for a tenor of t be denoted f(t), that s, f (t) = lm f (; t, t + ), for whchever t ths lmt exsts. Clearly then Also r(t)t = t Z(t) = exp r t r t t t = f (s)ds (9) ( t t t ) f (s)ds t t () f (s)ds () whch shows that the average of the nstantaneous forward rate over any of our ntervals [t, t ] s equal to the dscrete forward rate for that nterval. Fnally, t r(t)t = r t + f (s)ds, t [t, t ] (2) t whch s a crucal nterpolaton formula: gven the forward functon we easly fnd the rsk free functon. 2 Interpolaton And Bootstrap Of Yeld Curves Not Two Separate Processes As has been mentoned, many nterpolaton methods for curve constructon are avalable. What needs to be stressed s that n the case of bootstrappng yeld curves, the nterpolaton method s ntmately connected to the bootstrap, as the bootstrap proceeds wth ncomplete nformaton. Ths nformaton s completed (n a non unque way) usng the nterpolaton scheme. In Hagan and West [26] we llustrated ths pont usng swap curves; here we wll make the same ponts focusng on a bond curve. Suppose we have a reasonably small set of bonds that we want to use to bootstrap the yeld curve. (To decde whch bonds to nclude can be a non-trval exercse. Excludng too many runs the rsk of dsposng of market nformaton whch s actually meanngful, on the other hand, ncludng too many could result n a yeld curve whch s mplausble, wth a multtude of turnng ponts, or even a bootstrap algorthm whch fals to converge.) Recall that we nsst that whatever nstruments are ncluded wll be prced perfectly by the curve. Typcally some rates at the short end of the curve wll be known. For example, some zero-coupon bonds mght trade whch gve us exact rates. In some markets, where there s nsuffcent lqudty at the short end, some nter-bank money market rates wll be used. Each bond and the curve must satsfy the followng relatonshp: ^ WILMOTT magazne 7

3 where [A] = n p Z(; t settle, t ) = A s the all-n (drty) prce of the bond; t settle s the date on whch the cash s actually delvered for a purchased bond; p, p,...,p n are the cash flows assocated wth a unt bond (typcally p = e c, p 2 = c for < n and p 2 n = + c where c s the annual 2 coupon and e s the cum-ex swtch); t, t,...,t n are the dates on whch those cash flows occur. On the left s the prce of the bond tradng n the market. On the rght s the prce of the bond as strpped from the yeld curve. We rewrte ths n the computatonally more convenent form n [A ] Z(, t settle ) = p Z(, t ) (3) Suppose for the moment that the rsk free rates (and hence the dscount factors) have been determned at t, t,...,t n. Then we solve Z(, t n ) easly as ] Z(, t n ) = n [[A ] Z(, t settle ) p Z(, t ) p n whch s wrtten n the form of rsk-free rates rather than dscount factors as [ ]] r n = [ln n p n ln [A ] e rsettle tsettle p e rt (4) t n where the t s are now denomnated n years and the relevant day-count conventon s beng adhered to. Of course, n general, we do not know the earler rates, nether exactly (because t s unlkely that any money market nstruments expre exactly at t ) nor even after some nterpolaton (the rates for the smallest few t mght be avalable after nterpolaton, but the later ones not at all). However, as n the case of swap curves, (4) suggests an teratve soluton algorthm: we guess r n, ndeed other expry-date rates for other bonds, and take the rates already known from e.g. the money market, and nsert these rates nto our nterpolaton algorthm. We then determne r settle and r, r,...,r n. Next, we nsert these rates nto the rght-hand sde of (4) and solve for r n. We then take ths new guess for ths bond, and for all the other bonds, and agan apply the nterpolaton algorthm. We terate ths process. Even for farly wld curves (such as can often be the case n South Afrca) ths teraton wll reach a fxed pont wth accuracy of about 8 decmal places n 4 or 5 teratons. Ths then s our yeld curve. 3 How To Compare Yeld Curve Interpolaton Methodologes In general, the nterpolaton problem s as follows: we have some data x as a functon of tme, so we have τ,τ 2,...,τ n and x, x 2,...,x n known. An nterpolaton method s one that constructs a contnuous functon x(t) satsfyng x(τ ) = x for =, 2,...,n. In our settng, the x values = = = mght be rsk free rates, forward rates, or some transformaton of these the log of rates, etc. Of course, many choces of nterpolaton functon are possble accordng to the nature of the problem, one mposes requrements addtonal to contnuty, such as dfferentablty, twce dfferentablty, condtons at the boundary, and so on. The Lagrange polynomal s a polynomal of degree n whch passes through all the ponts, and of course ths functon s smooth. However, t s well known that ths functon s nadequate as an nterpolator, as t demonstrates remarkable oscllatory behavour. The typcal approach s to requre that n each nterval the functon s descrbed by some low dmensonal polynomal, so the requrements of contnuty and dfferentablty reduce to lnear equatons n the coeffcents, whch are solved usng standard lnear algebrac technques. The smplest example are where the polynomals are lnear, and these methods are surveyed n 4. However, these functons clearly wll not be dfferentable. Next, we try quadratcs however here we have a remarkable zg-zag nstablty whch we wll dscuss. So we move on to cubcs or even quartcs they overcome these already-mentoned dffcultes, and we wll see these n 5. All of the nterpolaton methods consdered n Hagan and West [26] appear n the rows of Table. We wll restrct attenton to the case where the number of nputs s reasonably small and so the bootstrappng algorthm s able to prce the nstruments exactly, and we restrct attenton to those methods where the nstruments are ndeed always prced exactly. The crtera to use n judgng a curve constructon and ts nterpolaton method that we wll consder are: (a) In the case of yeld curves, how good do the forward rates look? These are usually taken to be the m or 3m forward rates, but these are vrtually the same as the nstantaneous rates. We wll want to have postvty and contnuty of the forwards. It s requred that forwards be postve to avod arbtrage, whle contnuty s requred as the prcng of nterest senstve nstruments s senstve to the stablty of forward rates. As ponted out n McCulloch and Kochn [2], a dscontnuous forward curve mples ether mplausble expectatons about future short-term nterest rates, or mplausble expectatons about holdng perod returns. Thus, such an nterpolaton method should probably be avoded, especally when prcng dervatves whose value s dependent upon such forward values. Smoothness of the forward s desrable, but ths should not be acheved at the expense of the other crtera mentoned here. (b) How local s the nterpolaton method? If an nput s changed, does the nterpolaton functon only change nearby, wth no or mnor spll-over elsewhere, or can the changes elsewhere be materal? (c) Are the forwards not only contnuous, but also stable? We can quantfy the degree of stablty by lookng for the maxmum bass pont change n the forward curve gven some bass pont change (up or down) n one of the nputs. Many of the smpler methods can have ths quantty determned exactly, for others we can only derve estmates. (d) How local are hedges? Suppose we deal an nterest rate dervatve of a partcular tenor. We assgn a set of admssble hedgng nstruments, for example, n the case of a swap curve, we mght (even should) 72 WILMOTT magazne

4 TECHNICAL ARTICLE 2 decree that the admssble hedgng nstruments are exactly those nstruments that were used to bootstrap the yeld curve. Does most of the delta rsk get assgned to the hedgng nstruments that have maturtes close to the gven tenors, or does a materal amount leak nto other regons of the curve? We wll now survey a handful of these methods, and hghlght the ssues that arse. 4 Lnear Methods 4. Lnear on rates For t < t < t the nterpolaton formula s Usng (8) we get r(t) = t t r + t t r (5) t t t t f (t) = 2t t t t r + t 2t t t r (6) Of course f s undefned at the t, as the functon r(t)t s clearly not dfferentable there. Moreover, n the actual rate nterpolaton formula, by the tme t reaches t, the mport of r t has been reduced to zero that rate has been forgotten. But we clearly see that ths s not the case for the forward, so the left and rght lmts f (t + ) and f (t ) are dfferent the forward jumps. Furthermore, the choce of nterpolaton does not prevent negatve forward rates: suppose we have the (t, r) ponts (y, 8%) and (2y, 5%). Of course, ths s a rather contrved economy: the one year nterest rate s 8% and the one year forward rate n one year s tme s 2%. Nevertheless, t s an arbtrage free economy. But usng lnear nterpolaton the nstantaneous forwards are negatve from about.84 years onwards. 4.2 Lnear on the log of rates Now for t t t the nterpolaton formula s whch as a rate formula s ln(r(t)) = t t ln(r ) + t t ln(r ) t t t t r(t) = r t t t t t t t r t (7) A smple objecton to the above formula s that t does not allow negatve nterest rates. Also, the same argument as before shows that the forward jumps at each node, and smlar expermentaton wll provde an example of a Z functon whch s not decreasng. 4.3 Lnear on dscount factors Now for t t t the nterpolaton formula s whch as a rate formula s r(t) = t Z(t) = t t Z + t t Z t t t t ln [ t t e rt + t ] t r r t t t t t Agan, the forward jumps at each node, and the Z functon may not be decreasng. 4.4 Raw nterpolaton (lnear on the log of dscount factors) Ths method corresponds to pecewse constant forward curves. Ths method s very stable, s trval to mplement, and s usually the startng pont for developng models of the yeld curve. One can often fnd mstakes n fancer methods by comparng the raw method wth the more sophstcated method. By defnton, raw nterpolaton s the method whch has constant nstantaneous forward rates on every nterval t < t < t. From () we see that that constant must be the dscrete forward rate for the nterval, so rt r t f (t) = for t < t < t. Then from (2) we have that t t r(t)t = r t + (t t ) r t r t t t By wrtng the above expresson wth a common denomnator of t t, and smplfyng, we get that the nterpolaton formula on that nterval s (8) r(t)t = t t r t + t t r t (9) t t t t whch explans yet another choce of name for ths method: lnear rt ; the method s lnear nterpolaton on the ponts r t. Snce ± r t s the logarthm of the captalsaton/dscount factors, we see that callng ths method lnear on the log of captalsaton factors or lnear on the log of dscount factors s also merted. Ths raw method s very attractve because wth no effort whatsoever we have guaranteed that all nstantaneous forwards are postve, because every nstantaneous forward s equal to the dscrete forward for the parent nterval. As we have seen, ths s an achevement not to be sneezed at. It s only at the ponts t, t 2,...,t n that the nstantaneous forward s undefned, moreover, the functon jumps at that pont. 4.5 Pecewse lnear forward Havng decded that the raw method s qute attractve, what happens f we try to remedy ts only defect n the most obvous way? What we wll do s that nstead of the forwards beng pecewse constant we wll demand that they be a pecewse contnuous lnear functon. What could be more natural than to smply ask to gently rotate the raw nterpolants so that they are now not only pecewse lnear, but contnuous as well? Unfortunately, ths very plausble requrement gves rse to at least two types of very unpleasant behavour ndeed. Ths s easly understood by means of an example. ^ WILMOTT magazne 73

5 Curve, frst scenaro. Curve, second scenaro Fgure 2: The pecewse lnear forward method. Forward, frst scenaro. Forward, second scenaro Quadratc splnes To complete a quadratc splne of a functon x, we desre coeffcents (a, b, c ) for n. Gven these coeffcents, the functon value at any term τ wll be x(τ ) = a + b (τ τ ) + c (τ τ ) 2 τ τ τ + (2) The constrants wll be that the nterpolatng functon ndeed meets the gven data (and hence s contnuous) and the entre functon s dfferentable. There are thus 3n 4 constrants: n left hand functon values to be satsfed, n rght hand functon values to be satsfed, and n 2 nternal knots where dfferentablty needs to be satsfed. However, there are 3n 3 unknowns. Wth one degree of freedom remanng, t makes sense to requre that the left-hand dervatve at τ n be zero, so that the curve can be extrapolated wth a horzontal asymptote. Suppose we apply ths method to the rates (so x = r ). The forward curves that are produced are very smlar to the pecewse lnear forward curves the curve can have a zg-zag appearance, and ths zg-zag s subject to the same party of nput consderatons as before. So, next we try a cubc splne. 5.2 Cubc splnes Frstly, suppose we have a curve wth nput zero coupon rates at every year node, wth a value of r(t) = 5% for t =, 2,...,5 and r(t) = 6% for t = 6, 7,...,. We must have f (t) = r() for t. In order to assure contnuty, we see then we must have f (t) = r() for every 5. Now, the dscrete forward rate for [5, 6] s %. In order for the average of the pecewse lnear functon f on the nterval [5, 6] to be % we must have that f (6) = 7%. And now n turn, the dscrete forward rate for [6, 7] s 6% and so n order for the average of the pecewse lnear functon f on the nterval [6, 7] to be 6% we must have that f (7) = 5%. Ths zg-zag feature contnues recursvely; see Fgure 2. Note also the mplausble shape of the actual yeld curve tself. Secondly, suppose now we nclude a new node, namely that r(6.5) = 6%. It s farly ntutve that ths mparts lttle new nformaton 2. Nevertheless, the bootstrapped curve changes dramatcally. The party of the zg-zag s reversed. So we see that the localness of the method s exceptonally poor. 5 Splnes The varous lnear methods are the smplest examples of polynomal splnes: a polynomal splne s a functon whch s pecewse n each nterval a polynomal, wth the coeffcents arranged to ensure at least that the splne concdes wth the nput data (and so s contnuous). In the lnear case that s all that one can do the lnear coeffcents are now determned. If the polynomals are of hgher degree, we can use up the degrees of freedom by demandng other propertes, such as dfferentablty, twce dfferentablty, asymptotes at ether end, etc. The frst thng we try s a quadratc splne. Ths tme we desre coeffcents (a, b, c, d ) for n. Gven these coeffcents, the functon value at any term τ wll be x(τ ) = a + b (τ τ ) + c (τ τ ) 2 + d (τ τ ) 3 τ τ τ + (2) As before we have 3n 4 constrants, but ths tme there are 4n 4 unknown coeffcents. There are several possble ways to proceed to fnd another n constrants. Here are the ones that we have seen: x = r. The functon s requred to be twce dfferentable, whch for the same reason as prevously adds another n 2 constrants. For the fnal two constrants, the functon s requred to be lnear at the extremes.e. the second dervatve of the nterpolator at τ and at τ n are zero. Ths s the so-called natural cubc splne. x = r. The functon s agan requred to be twce dfferentable; for the fnal two constrants we have that the functon s lnear on the left and horzontal on the rght. Ths s the so-called fnancal cubc splne Adams [2]. x = r τ. The functon s agan requred to be twce dfferentable; for the fnal two constrants we have that ths functon s lnear on the rght and quadratc on the left. Ths s the quadratc-natural splne proposed n McCulloch and Kochn [2]. x = r. The values of b for < < n are chosen to be the slope at τ of the quadratc that passes through (τ j, r j ) for j =,, +. The value of b s chosen to be the slope at τ of the quadratc that passes through (τ j, r j ) for j =, 2, 3; the value of b n s chosen lkewse. Ths s the Bessel method [de Boor, 978, 2, Chapter IV], although often somewhat rregularly called the Hermte method by software vendors. 74 WILMOTT magazne

6 TECHNICAL ARTICLE 2 x = r τ. Agan, Bessel nterpolaton. Gong one step further, quartc splnes. Accordng to Adams [2] the quartc splne gves the smoothest nterpolator of the forward curve. The splne can proceed on nstantaneous forward rates, ths tme there are 5n 5 unknowns and 3 addtonal condtons at τ or τ n requred. Although one must ask: when does one actually have a set of nstantaneous forwards as nputs for nterpolaton? Alternatvely f we apply (9) then the nputs are rsk free rates, and the splne s of the form r(τ ) = a + b τ + c τ + d τ 2 + e τ 3 + g τ 4, wth 6n 6 unknowns and 4 addtonal condtons requred. x = r. The monotone preservng cubc splne of Hyman [983]. The method specfes the values of b for n, n a way to be dscussed n more detal shortly. Sgnfcant problems can become apparent when usng some of these methods. The splne s supposed to allevate the problem of oscllaton seen when fttng a sngle polynomal to a data set (the Lagrange polynomal), nevertheless, sgnfcant oscllatory behavour can stll be present. Furthermore, the varous types of clampng we see wth some of the methods above (clampng refers to mposng condtons at the boundares τ or τ n ) can compromse localness of the nterpolator, sometmes grossly. In fact, the teratve procedure from 2 often fals to converge for the quartc nterpolaton methods, and we exclude them from further analyss. The method of Hyman s a method whch attempts to address these problems. Ths method s qute dfferent to the others; t s a local method the nterpolatory values are only determned by local behavour, not global behavour. Ths method ensures that n regons of monotoncty of the nputs (so, three successve ncreasng or decreasng values) the nterpolatng functon preserves ths property; smlarly f the data has a mnmum/maxmum then the output nterpolator wll have a mnmum/maxmum at the node. frst thng we do s calculate f d rτ r τ = for n, r τ τ =. (Here we also check that these are all postve, and so conclude that the curve s legal.e. arbtrage free (except n those few cases where forward rates may be negatve). As an nterpolaton algorthm the monotone convex method wll now bootstrap a forward curve, and then f requred recover the contnuum of rsk free rates usng (2). One rather smple observaton s that all of the splne methods we saw n 5 fal n forward extrapolaton beyond the nterval [τ,τ n ]. Clearly f the nterpolaton s on rates then we wll apply horzonal extrapolaton to the rate outsde of that nterval: r(τ ) = r for τ <τ and r(τ ) = r n for τ>τ n. So far so good. What happens to the forward rates? Perhaps surprsngly we cannot apply the same extrapolaton rule to the forwards, n fact, we need to set f (τ ) = r for τ <τ and f (τ ) = r n for τ >τ n consder (8). Ths makes t almost certan that the forward curve has a materal dscontnuty at τ, and probably one at τ n too (the latter wll be less severe as the curve, ether by desgn or by nature, probably has a horzontal asymptote as τ τ n ). In order to avod ths pathology, we now have terms = τ,τ,...,τ n and the generc nterval for consderaton s [τ,τ ]. A short rate (nstantaneous) rate may be provded, f not, the algorthm wll model one. Usually the shortest rate that mght be nput wll be an overnght rate, f t s provded, the algorthm here smply has some overkll there wll be an overnght rate and an nstantaneous short rate but t need not be modfed. f d s the dscrete rate whch belongs to the entre nterval [τ,τ ]; t would be a mstake to model that rate as beng the nstantaneous rate at τ. Rather, we begn by assgnng t to the mdpont of the nterval, and then modellng the nstantaneous rate at τ. as beng on the straght lne that jons the adjacent mdponts. Let ths rate f (τ ) be denoted f. Ths explans (22). In (23) and (24) the values f = f () and f n = f (τ n ) are selected so that f () = = f (τ n ). Thus 6 Monotone Convex Many of the deas of the method of Hyman wll now have a natural development the monotone convex method was developed to resolve the only remanng defcency of Hyman [983]. Very smply, none of the methods mentoned so far are aware that they are tryng to solve a fnancal problem ndeed, the breedng ground for these methods s typcally engneerng or physcs. As such, there s no mechansm whch ensures that the forward rates generated by the method are postve, and some smple expermentaton wll uncover a set of nputs to a yeld curve whch gve some negatve forward rates under all of the methods mentoned here, as seen n Hagan and West [26]. Thus, n ntroducng the monotone convex method, we use the deas of Hyman [983], but explctly ensure that the contnuous forward rates are postve (whenever the dscrete forward rates are themselves postve). The pont of vew taken n the monotone convex method s that the nputs are (or can be manpulated to be) dscrete forwards belongng to ntervals; the nterpolaton s not performed on the nterest rate curve tself. We may have actual dscrete forwards FRA rates. On the other hand f we have nterest rates r, r 2,...,r n for perods τ,τ 2,...,τ n then the f = τ τ τ + τ f d + + τ + τ τ + τ f d, for =, 2,...,n (22) f = f d 2 (f f d ) (23) f n = f d n 2 (f n f d n ) (24) Note that f the dscrete forward rates are postve then so are the f for =, 2,...,n. We now seek an nterpolatory functon f defned on [,τ n ] for f, f,...,f n that satsfes the condtons below (n some sense, they are arranged n decreasng order of necessty). τ () f (t)dt = f d τ τ τ, so the dscrete forward s recovered by the curve, as n (). () f s postve. () f s contnuous. (v) If f d < f d < f d + then f (τ ) s ncreasng on [τ,τ ], and f f d > f d > f d + then f (τ ) s decreasng on [τ,τ ]. Let us frst normalse thngs, so we seek a functon g defned on [, ] such that 3 ^ WILMOTT magazne 75

7 g() g() Fgure 3: The functon g. x = τ = τ x = τ = τ g(x) = f (τ + (τ τ )x) f d. (25) Before proceedng, let us gve a sketch of how we wll proceed. We wll choose g to be pecewse quadratc n such a way that () s satsfed by constructon. Of course, g s contnuous, so () s satsfed. As a quadratc, t s easy to perform an analyss of where the mnmum or maxmum occurs, and we thereby are able to apply some modfcatons to g to ensure that (v) s satsfed, whle ensurng () and () are stll satsfed. Also, we see a posteror that f the values of f had satsfed certan constrants, then () would have been satsfed. So, the algorthm wll be to construct (22), (23) and (24), then modfy the f to satsfy those constrants, then construct the quadratcs, and then modfy those quadratcs. Fnally, ( ) τ τ f (τ ) = g + f d. (26) τ τ Thus, the current choces of f are provsonal; we mght make some adjustments n order to guarantee the postvty of the nterpolatng functon f. Here follow the detals. We have only three peces of nformaton about g: g() = f f d, g() = f f d, and g(x)dx =. We postulate a functonal form g(x) = K + Lx + Mx 2, havng 3 equatons n 3 unknowns we get K g() L = g(), and easly solve to fnd that M 2 3 g(x) = g()[ 4x + 3x 2 ] + g()[ 2x + 3x 2 ] (27) Note that by (22) that (v) s equvalent to requrng that f f < f d < f then f (τ ) s ncreasng on [τ,τ ], whle f f > f d > f then f (τ ) s decreasng on [τ,τ ]. Ths s equvalent to requrng that f g() and g() are of opposte sgn then g s monotone. Now g (x) = g()( 4 + 6x) + g()( 2 + 6x) g () = 4g() 2g() g () = 2g() + 4g() g beng a quadratc t s now easy to determne, smply by nspectng g () and g (), the behavour of g on [, ]. The cases where g () = and g () = are crucal; these correspond to g() = 2g() and g() = 2g() respectvely. These two lnes dvde the g()/g() plane nto eght sectors. We seek to modfy the defnton of g on each sector, takng care that on the boundary of any two sectors, the formulae from those g() () () (v) () C () (v) () A g() = 2g() two sectors actually concde (to preserve contnuty). In actual fact the treatment for every dametrcally opposte par of sectors s the same, so we really have four cases to consder, as follows (refer Fgure 4): () In these sectors g() and g() are of opposte sgns and g () and g () are of the same sgn, so g s monotone, and does not need to be modfed. () In these sectors g() and g() are also of opposte sgn, but g () and g () are of opposte sgn, so g s currently not monotone, but needs to be adjusted to be so. Furthermore, the formula for () and for () need to agree on the boundary A to ensure contnuty. () The stuaton here s the same as n the prevous case. Now the formula for () and for () need to agree on the boundary B to ensure contnuty. (v) In these sectors g() and g() are of the same sgn so at frst t appears that g does not need to be modfed. Unfortunately ths s not the case: modfcaton wll be needed to ensure that the formula for () and (v) agree on C and () and (v) agree on D. The orgn s a specal case: f g () = = g () then g(x) = for all x, and f d = f d = f d d +, and we put f (τ ) = f for τ [τ,τ ]. So we proceed as follows: () As already mentoned g does not need to be modfed. Note that on A we have g(x) = g()( 3x 2 ) and on B we have g(x) = g() ( 3x x2 ). () A smple soluton s to nsert a flat segment, whch changes to a quadratc at exactly the rght moment to ensure that g(x)dx =. So we take { g() for x η g(x) = ( ) 2 g() + (g() g()) x η (28) for η<x η g() η = + 3 g() g() = g() + 2g() g() g() g() = 2g() g() (29) Note that η as g() 2g(), so the nterpolaton formula reduces to g(x) = g()( 3x 2 ) at A, as requred. () Fgure 4: The reformulated possbltes for g. B D 76 WILMOTT magazne

8 TECHNICAL ARTICLE 2 () Here agan we nsert a flat segment. So we take { ( ) 2 g() + (g() g()) η x for < x <η g(x) = η g() for η x < g() η = 3 g() g() Note that η as g() g(), so the nterpolaton formula 2 reduces to g(x) = g()( 3x x2 ) at B, as requred. (v) We want a formula that reduces n form to that defned n () as we approach C, and to that defned n () as we approach D. Ths suggests ( ) 2 A + (g() A) η x for < x <η η g(x) = ( ) 2 (32) A + (g() A) x η for η<x < η (3) (3) where A = when g() = - so the frst lne satsfes ()) and A = when g() = (so the second lne satsfes (). Straghtforward calculus gves and so g(x)dx = 2 3 A + η 3 g() + η 3 g() () Determne the f d from the nput data. (2) Defne f for =,,..., n as n (22), (23) and (24). (3) If f s requred to be everywhere postve, then collar f between and 2f d, for =, 2,...,n collar f between and 2mn(f d, f d + ), and collar f n between and 2fn d. If f s not requred to be everywhere postve, smply omt ths step. (4) Construct g wth regard to whch of the four sectors we are n. (5) Defne f as n (26). (6) If requred recover r as n (2). Integraton formulae are easly establshed as the functons forms of g are straghtforward. Pseudo-code for ths recpe s provded n an Appendx. Workng code for ths nterpolaton scheme s avalable from the second author s webste. 6.2 Ameloraton In Hagan and West [26] an enhancement of ths method s consdered where the curve s amelorated (smoothed). Ths s acheved by makng the nterpolaton method slghtly less local.e. by usng as nputs not only neghbourng nformaton but also nformaton whch s two nodes away. 7 Hedgng We can now ask the queston: how do we use the nstruments whch have been used n our bootstrap to hedge other nstruments? In general A = [ηg() + ( η)g()] 2 A smple choce satsfyng the varous requrements s η = 6. Ensurng postvty g() g() + g() A = g()g() g() + g() Suppose we wsh to guarantee that the nterpolatory functon f s everywhere postve. Clearly from the formula (26) t suffces to ensure that g(x) > f d for x [, ]. Now g() = f f d > f d and g() = f f d > f d snce f, f are postve. Thus the nequalty s satsfed at the endponts of the nterval. Now, n regons (), () and (), g s monotone, so those regons are fne. In regon (v) g s not monotone. g s postve at the endponts and has a mnmum of A (as n (34)) at the x-value η (as n (33)). So, t now suffces to prove that g()g() < f d g()+g(). Ths s the case f f, f < 3f d. To see ths, note that then < g(), g() <2f d and the result follows, snce f < y, z < 2a then y+z = + > + = yz and so > a. yz z y 2a 2a a y+z We choose the slghtly strcter condton f, f < 2f d. Thus, our algorthm s (33) (34) Fgure 5: The g functon as we cross the boundares. From left to rght: boundares A, B, C and D. From top to bottom: approachng the boundary, at the boundary (central), leavng the boundary. Only at the boundary of Cand D are there dscontnutes. ^ WILMOTT magazne 77

9 Hedgng under waves m 3x6 6x9 9x2 2x5 5x8 2y 3y 4y 5y 6y 7y 8y 9y y 5y 2y 25y 3y Hedgng under forward trangles. 2 3m 3x6 6x9 9x2 2x5 5x8 2y 3y 4y 5y 6y 7y 8y 9y y 5y 2y 25y 3y.8 Hedgng under forward boxes m 3x6 6x9 9x2 2x5 5x8 2y 3y 4y 5y 6y 7y 8y 9y y 5y 2y 25y 3y Fgure 6: The obvous superorty of usng forward boxes to determne hedge portfolos: not only s the hedge portfolo smple and ntutve, but the portfolo composton s practcally nvarant under the nterpolaton method. the trader wll have a portfolo of other, more complcated nstruments, and wll want to hedge them aganst yeld curve moves by usng lqudly traded nstruments (whch, n general, should exactly be those nstruments whch were used to bootstrap the orgnal curve). For smplcty, we wll assume that these nstruments are ndeed avalable for hedgng, and the rsky nstrument to be hedged s nothng more complcated that another vanlla swap: for example, one wth term whch s not one of the bootstrap terms, s a forward startng swap, or s a stubbed swap. Suppose ntally that, wth n nstruments beng used n our bootstrap, there are exactly n yeld curve movements that we wsh to hedge aganst. It s easy to see that we can construct a perfect hedge. Frst one calculates the square matrx P where P j s the change n prce of the j th bootstrappng nstrument under the th curve. Next we calculate the change n value of our rsk nstrument under the th curve to form a column vector V. The quantty of the th bootstrappng nstrument requred for the perfect replcaton s the quantty Q where Q s the soluton to the matrx equaton PQ = V. Assumng for the moment that P s nvertble, we fnd the soluton. Of course, n realty, the set of possble yeld curve movements s far, far greater. What one wants to do then s fnd a set of n yeld curve changes whose moves are somehow representatve. Some methods have been suggested as follows: Perturbng the curve: creaton of bumps. In bumpng, we form new curves ndexed by : to create the th curve one bumps up the th nput rate by say bass pont, and bootstraps the curve agan. Perturbng the forward curve wth trangles. One approach s to agan form new curves, agan ndexed by : the th curve has the orgnal forward curve ncremented by a trangle, wth left hand endpont at t, fxed heght say one bass pont and apex at t, and rght hand endpont at t +. (The frst and last trangle wll n fact be rght angled, wth ther apex at the frst and last tme ponts respectvely.) Perturbng the forward curve wth boxes. In boxes: the th curve has the orgnal forward curve ncremented by a rectangle, wth left hand endpont at t and rght hand endpont at t, and fxed heght say one bass pont. Such a perturbaton curve corresponds exactly wth what we get from bumpng, f one of the nputs s a t t FRA rate, we bump ths rate, and we use the raw nterpolaton method. More generally the user mght want to defne generc key terms e.g. w, m, 3m, 6m, y, 2y, etc. and defne trangles or boxes relatve to these 78 WILMOTT magazne

10 TECHNICAL ARTICLE 2 Table : A synopss of the comparson between methods. Yeld curve type Forwards postve? Forward smoothness Method local? Forwards stable? Bump hedges local? Lnear on dscount no not contnuous excellent excellent very good Lnear on rates no not contnuous excellent excellent very good Raw (lnear on log of dscount) yes not contnuous excellent excellent very good Lnear on the log of rates no not contnuous excellent excellent very good Pecewse lnear forward no contnuous poor very poor very poor Quadratc no contnuous poor very poor very poor Natural cubc no smooth poor good poor Hermte/Bessel no smooth very good good poor Fnancal no smooth poor good poor Quadratc natural no smooth poor good poor Hermte/Bessel on rt functon no smooth very good good poor Monotone pecewse cubc no contnuous very good good good Quartc no smooth poor very poor very poor Monotone convex (unamelorated) yes contnuous very good good good Monotone convex (amelorated) yes contnuous good good good Mnmal no contnuous poor good very poor terms - the nputs to the bootstrap do not necessarly correspond to these nodes. In ether case we have (an automated or user defned) set of dates t, t 2,...,t n whch wll be the bass for our waves, where the trangles are defned as above. Some obvous deas whch are just as obvously rejected are to form correspondng perturbatons to the yeld curve tself - such curves wll not be arbtrage free (the derved Z functon wll not be decreasng). As an example, consder a 5m swap, where (for smplcty, and ndeed, n some markets, such as the second author s domestc market) both fxed and floatng payments n swaps occur every 3 months. Thus our swap les between the 4 and 5 year swap, whch let us assume are nputs to the curve. The type of results we get are n Fgure 6. The very plausble and popular bump method performs adequately for many methods, but some methods - for example, the mnmal method - can be rejected out of hand f bumpng s to be used. Furthermore, for all of the cubc splnng methods, there s hedge leakage of varyng degrees. Perturbng wth trangles can be rejected out of hand as a method - ndeed, the pathology that occurs here s akn to the pathology that arses when one uses the pecewse forward lnear method of bootstrap: addng or removng an nput to the bootstrap wll reverse the sgn of the hedge quanttes before the nput n queston. Anyway, to have these magntudes n the hedge portfolo s smply absurd. Perturbng wth boxes s the method of choce, but unfortunately does not enable us to dstngush between the qualty of the dfferent nterpolaton methods. 8 Concluson The comparson of the methods we analyse n Hagan and West [26] appears n Table. It s our opnon that the new method derved n Hagan and West [26], namely monotone convex (n partcular, the unamelorated verson) should be the method of choce for nterpolaton. To the best of our knowledge ths s the only publshed method where smultaneously () all nput nstruments to the bootstrap are exactly reproduced as outputs of the bootstrap, (2) the nstantaneous forward curve s guaranteed to be postve f the nputs allow t (n partcular, the curve s arbtrage free), and (3) the nstantaneous forward curve s typcally contnuous. In addton, as bonuses (4) the method s local.e. changes n nputs at a certan locaton do not affect n any way the value of the curve at other locatons. (5) the forwards are stable.e. as nputs change, the nstantaneous forwards change more or less proportonately. (6) hedges constructed by perturbatons of ths curve are reasonable and stable. In Hagan and West [26] we have revewed many nterpolaton methods avalable and have ntroduced a couple of new methods. In the fnal analyss, the choce of whch method to use wll always be subjectve, and needs to be decded on a case by case bass. But we hope to have provded some warnng flags about many of the methods, and have outlned several qualtatve and quanttatve crtera for makng the selecton on whch method to use. ^ WILMOTT magazne 79

11 FOOTNOTES & REFERENCES. We have r(s)s + C = f (s)ds, so r(t)t = [r(s)s] t = t f (s)ds. 2. It would be wrong to say that there s no new nformaton; that would be the case under lnear nterpolaton of rates, but not necessarly here. 3. Strctly speakng, we are defnng functons g, each correspondng to the nterval [τ,τ ]. As the g are constructed one at a tme, we suppress the subscrpt. [] Ken Adams. Smooth nterpolaton of zero curves. Algo Research Quarterly, 4(/2): 22, 2. [2] Carl de Boor. A Practcal Gude to Splnes: Revsed Edton, volume 27 of Appled Mathematcal Scences. Sprnger-Verlag New York Inc., 978, 2. Pseudo Code For Monotone Convex Interpolaton Frst the estmates for f, f,...,f n. Ths mplementaton assumes that t s requred that the output curve s everywhere postve. Varous arrays [3] Patrck S. Hagan and Graeme West. Interpolaton methods for curve constructon. Appled Mathematcal Fnance, 3 (2):89 29, 26. [4] James M. Hyman. Accurate monotoncty preservng cubc nterpolaton. SIAM Journal on Scentfc and Statstcal Computng, 4(4): , 983. [5] J. Huston McCulloch and Levs A. Kochn. The nflaton premum mplct n the US real and nomnal term structures of nterest rates. Techncal Report 2, Oho State Unversty Economcs Department, 2. URL [6] Rcardo Rebonato. Interest-Rate Opton Models. John Wley and Sons Ltd, second edton, 998. have already been dmensoned, the raw data nputs have already been provded, and t has been specfed wth the boolean varable InputsareForwards where those nputs are rates r, r 2,...,r n or dscrete forwards f d, f 2 d,...,f n d. Further, a collar and mn utlty functons are used (not shown). Of course, collar(a, b, c) = max(a, mn(b, c)). Havng found the estmates for f, f,...,f n, we can fnd the value of f (τ ) for any τ. The key functon here s LastIndex, whch determnes the unque value of for whch τ [τ,τ + ). Extrapolaton s as n the thrd paragraph of 6. 8 WILMOTT magazne

12 TECHNICAL ARTICLE 2 Workng code for ths nterpolaton scheme, wth proper dmensonng of all arrays and code for all the mssng functons, s avalable from the second author s webste on the resources page. W WILMOTT magazne 8