Curiosities on the Monotone Preserving Cubic Spline

Transcription

1 PRE-PRINT UD Curostes on the Monotone Preservng Cubc Splne Mchele Mao and Jorrt-Jaap de Jong ugly Ducklng B.V. Burgemeester le Fevre de Montgnylaan 30, 3055LG Rotterdam, the Netherlands December 9, 2014 Abstract In ths paper we descrbe some new features of the monotone-preservng cubc splnes and the Hyman s monotoncty constrant, that s mplemented nto varous splne nterpolaton methods to ensure monotoncty. We fnd that, whle the Hyman constrant s n general useful to enforce monotoncty, t can be safely omtted when the monotone-preservng cubc splne s consdered. We also fnd that, when computng senstvtes, consstency requres makng some specfc assumptons about how to deal wth non-dfferentable locatons, that become relevant for specal values of the parameter space. Keywords: Yeld curve, fxed-ncome, nterpolaton, Hyman, monotone preservng cubc splnes. m.mao@uglyducklng.nl j.jong@uglyducklng.nl 1

2 Contents 1 Introducton 2 2 Set up and notaton Monotone preservng cubc splnes The man results The proof of the man results The Frtsch-Butland prescrpton The theorem Summary and conclusons 8 References 8 A Appendx: VBA Code 9 1 Introducton In ths paper we revew and streamlne the content of our prevous paper [1], where we consdered one-dmensonal nterpolaton methods that preserve the trend of the nput data ponts. Useful revews of varous nterpolaton technques are e.g. [2, 3, 4, 5]. We wll mostly use [4] 1. One key ssue s to be able to construct the nterpolatng functon n such a way that t follows the trend of the data ponts. Ths property s usually referred to as monotoncty-preservng and t helps to remove spurous effects, such as oscllatng behavor. One of the most famous monotoncty-preservng algorthm was ntroduced by Hyman [8] and can be mplemented n prncple on almost any cubc-splne-lke nterpolaton method. In fact, a large class of cubc splnes s defned by specfyng the vector of dervatves at the nput ponts and the Hyman algorthm works exactly on these dervatves. In ths paper we wll focus on one partcular splne only, namely the so-called monotone preservng splne. It s defned as that splne where Hyman s adjustment s enforced on top of the Frtsch-Butland prescrpton [3, 9, 10]. In many practcal applcatons, we are nterested n the behavor of the specfc nterpolaton method under small changes of the nput data. Contnuty of the nterpolatng functon s vewed n terms of small changes n the x-values, whle stablty s related to changes n the y-values of the nput data. Typcally, contnuty s guaranteed, so we wll not dscuss t here. Instead, we wll focus on stablty, whch s expressed by the dervatve: f(x) (1) Examples where such a quantty s relevant can be found n many places. For nstance, n many fnancal applcaton (e.g. portfolo replcaton and hedgng) one needs to use 1 The papers [4, 6] have become qute famous wthn the mathematcal fnance communty, snce ther author have ntroduced a new nterpolaton method whch s deeply ntertwned wth nterest rates, forward rates, and dscount factors. As noted e.g. n [7], ths method sometmes produces dscontnuous forward rates. 2

3 the so-called PV01 (present value of one bass pont 2 ) or DV01 (dollar value of one bass pont) [11]. In these cases, the nput y-values are the nterest rates r gven for specfc maturtes t (the x-values), and the assocated contnuous curve r(t) s known as the yeld curve. In ths example, the dervatve (1) becomes: r(t) r j. The man result of ths note s that Hyman s constrant does not contrbute to any calculaton n the monotone-preservng cubc splne framework. In secton 2, we defne our set up and fx our notaton, whch follows [4]. Secton 3 contans the man result. We show explctly how Hyman s monotoncty constrant s redundant n the monotonepreservng cubc splne method. Moreover, we wll see that an addtonal assumpton about how to deal wth non-dfferentable locatons must be ncluded n order to get the correct answer for the senstvtes. Secton 4 contans a summary of the work done and some conclusons. In Appendx A we show the Excel/VBA code used for testng purposes. 2 Set up and notaton In ths secton we revew the monotone preservng cubc splnes. The man reference are [4, 6]. We start wth a mesh of data ponts {t 1, t 2,..., t n } (we wll thnk of the x-values as tmes) and correspondng values {f 1, f 2,..., f n } for a generc but unknown functon f(t). Cubc splnes are genercally defned by pece-wse cubc polynomal that pass through consecutve ponts: f(t) = a + b (t t ) + c (t t ) 2 + d (t t ) 3, (2) wth t [t, t +1 ] and = 1,..., n. We wll use the followng defntons: h = t +1 t (3) m = f +1 f h, (4) wth = 1,..., n 1. The coeffcents a, b, c, and d, depends on the detals of the method, and are related to the values of f(t) and ts dervatves at the node ponts. In general, a = f(t ) f, b = f (t ), etc. (5) where the prme denotes the dervatve of the nterpolatng functon f(t) w.r.t. ts argument t. Moreover, gven a and b, we can express c and d as follows: c = 3m b +1 2b h (6) d = b +1 + b 2m h 2 We can use (2) to compute the dervatve f(t) 2 1 bass pont = 0.01%.. (7) = a + b (t t ) + c (t t ) 2 + d (t t ) 3. (8) 3

4 We have a = δ j (9) m = 1 ( ) δ j +1 h δj (10) c = 1 ( 3 m b +1 2 b ) (11) h d = 1 ( b+1 h 2 + b 2 m ), (12) whch depends on the matrx elements b. Here δ j s the Kronecker delta, whch s equal to one f = j and zero otherwse. Due to our prevous formulas, once the dervatves at the ponts, or equvalently the b coeffcents, are specfed, everythng else s fxed. In partcular, f we are nterested n computng f(t), then all the work wll be n the calculaton of the dervatves of b w.r.t. f j. Ths calculaton s however trcky f we use monotone preservng splnes (or any other method whch enforces monotoncty n a smlar fashon), where the b s are non-dfferentable functons of the f j s (whch nvolve the mn and max functons). Let us consder ths case n more detal. 2.1 Monotone preservng cubc splnes Let us start by recallng the formulas for the b s n the monotone preservng cubc splne method as defned n the Hagan-West paper [4]. Frst of all, at the boundares: b 1 = 0, b n = 0. (13) For the nternal data, f the curve s not monotone at t,.e. m 1 m 0, then b = 0 (f m 1 m 0), (14) so that t wll have a turnng pont there. Instead, f the trend s monotone at,.e. m 1 m > 0, one defnes β = 3m 1 m max(m 1, m ) + 2 mn(m 1, m ) (15) and b = { mn (max(0, β ), 3 mn(m 1, m )) f m 1, m > 0 max (mn(0, β ), 3 max(m 1, m )) f m 1, m < 0 The former choce s made when the curve s ncreasng (postve slopes), the latter when decreasng (negatve slopes). Equaton (16) represents the monotoncty constrant ntroduced by Hyman [8] and based on the Frtsch-Butland algorthm [3, 9, 10]. 3 The man results In the case of non-monotonc trend the b coeffcents are zero, and hence also ther dervatves. So we wll focus on case of monotonc trend from now on. The man result wll (16) 4

5 be that wthn the framework of the monotone preservng splne the Hyman adjustment (16) can be omtted, snce b = β, (17) f the trend s monotonc at the node. As a trval consequence of (17), one has b = β. (18) As a sde result, one fnds [1] that, when computng the dervatve above, specal care s needed to handle the non-dfferentable max and mn functons. In fact, f m = m 1, the soluton s to average the dervatves of m and m 1 : max(m 1, m ) = 1 ( m 2 mn(m 1, m ) = 1 ( m 2 + m 1 + m 1 ) ) (19a) (19b) and smlarly for the dervatves of the β s. Ths was confrmed n [1] by performng numercal checks. Whle (17) s lmted to the monotone-preservng algorthm, (19) s always vald. The remanng of ths secton s dedcated to provng (17). 3.1 The proof of the man results Here we recall the relevant formulas that have been worked out n detal n [1] The Frtsch-Butland prescrpton If we denote the denomnator of the β s wth L max(m 1, m ) + 2 mn(m 1, m ), then the dervatves of (15) wll be gven by: β f where β = 3 ( ) f L 2 α m2 β m2 1, h 1 h α = { { 2 f m 1 > m 1 f m 1 > m and β = 1 f m 1 < m 2 f m 1 < m or explctly β f 1 where β = 3 f L 2 2m 2 h 1 m2 1 h f m 1 > m m 2 h 1 2m2 1 h f m 1 < m. β f 1 = k k = 3 L 2 m 2 h 1, { 2 f m 1 > m 1 f m 1 < m, (20) 5

6 or explctly β f +1 where or explctly β = 3 f 1 L 2 m 2 { 2 f m 1 > m h 1 1 f m 1 < m. β 3 = k f +1 L 2 m2 1, h k = { 1 f m 1 > m 2 f m 1 < m, β = 3 { f +1 L 2 m2 1 1 f m 1 > m h 2 f m 1 < m. (21) (22) β β = 0, f j 1,, + 1. (23) As already mentoned, f m 1 = m, then β wll be the average of the answers correspondng to the two optons m 1 < m and m 1 > m. 3.2 The theorem In ths secton we wll prove that n the locally-monotonc case one has: b = β. (24) Let us consder the case of local monotoncty at the node and let us defne the followng four regons n the parameter space that are relevant for the Hyman adjustment: ) β > 0 and β < 3 mn(m 1, m ) ) β > 0 and β > 3 mn(m 1, m ) ) β < 0 and β > 3 max(m 1, m ) v) β < 0 and β < 3 max(m 1, m ) The frst two regon refer to the case of ncreasng trend (m, m 1, b > 0), whle the remanng two are for the case of decreasng trend (m, m 1, b < 0). Theorem 1. Gven β as defned n (15), the followng statements hold both true f m 1, m > 0, then β < 3 mn(m 1, m ); f m 1, m < 0, then β > 3 max(m 1, m ). 6

7 Proof. The proof s straghtforward. Consder the monotoncally ncreasng case frst, m 1, m > 0. By workng out the nequalty β < 3 mn(m 1, m ) and usng the postvty of the denomnator n (15), we end up wth: 0 < 2 (mn(m 1, m )) 2. (25) whch s always satsfed. Smlarly, for the monotoncally decreasng case m 1, m < 0, by workng out the nequalty β > 3 max(m 1, m ) and usng the negatvty of the denomnator n (15), we end up wth: whch s always satsfed. 0 < (max(m 1, m )) 2 + m 1 m (26) Theorem 2 (Man Result). In the monotone-preservng cubc splne nterpolaton method that uses Hyman monotoncty constrant (15)-(16), f the data trend s locally monotonc at node, then we have b = β (27) else b = 0. (28) Proof. Ths follows from the calculaton: { mn (max(0, β ), 3 mn(m b = 1, m )) f m 1, m > 0 max (mn(0, β ), 3 max(m 1, m )) f m 1, m < 0 { mn (β, 3 mn(m = 1, m )) f m 1, m > 0 max (β, 3 max(m 1, m )) f m 1, m < 0 { β f m = 1, m > 0 β f m 1, m < 0 (29) (30) (31) = β. (32) In gong from (29) to (30) we have used the fact that β s postve (negatve) f the trend s ncreasng (decreasng), whle from (30) to (31) we have used Theorem 1. Hence, n the case of local monotoncty we are left wth b = β. (33) Therefore, the whole Hyman constrant (16) s completely superfluous n the monotonepreservng splne and the dentty between the dervatves follows trvally: where the r.h.s. s gven by formulas (20)-(23). b = β, (34) 7

8 As a general remark, the Hyman constrant (16) s trval when the β s are defned by the Frtsch-Butland algorthm [10] as n (15). Ths s what happens for example n the notaton of Hagan and West [4]. The reason for ths dentty s the fact that the β s as defned by the Frtsch-Butland algorthm already guarantee that the resultng curve wll be monotonc. However, one can stll mplement the Hyman constrant (16) to ensure monotoncty n a non-trval way when dfferent frst dervatves β at the node ponts are chosen, and n that case the denttes (33) and (34) wll not hold anymore. 4 Summary and conclusons In ths paper we have consder monotone preservng nterpolaton methods and shown that Hyman s monotoncty constrant does not contrbute wthn the framework of the monotone-preservng cubc splne. However ths s genercally not true anymore when the Hyman constrant s appled on any other splne n order to construct a monotonc curve. In fact, n ths case Theorem 1 does not need to hold and consequently regons ) and v) wll n general contrbute. In ths case one wll need to use to complete formulas for all the four regons. Ths result s mportant for practcal as well as conceptual reasons. Moreover ths quantty s mportant n many areas (e.g. n fnance for prcng and for rsk management of nterest rate dervatves). Acknowledgements We would lke to thank Patrck Hagan for dscussons, the ABN AMRO Model Valdaton team for provdng us wth feedback and Dmytro Makogon n partcular for hs useful comments and remarks. We are also grateful wth the Lawrence Lvermore Natonal Laboratory for sharng wth us copes of the ther reports. References [1] M. Mao, J. de Jong (2014), On the Non-dfferentablty of Hyman s Monotoncty Constrant, pre-prnt nr UD PRE-PRINT-UD pdf. [2] W. H. Press, S. A. Teukolsky, W. T. Vetterlng, B. P. Flannery (2007), Numercal Recpes: The Art of Scentfc Computng, Cambrdge Unversty Press, Thrd Edton, 1256 pp., ISBN-10: [3] F. N. Frtsch, R. E. Carlson (1980), Monotone Pecewse Cubc Interpolaton, SIAM J. Numer. Anal., 17 (1980) [4] P. S. Hagan, G. West (2006), Interpolaton Methods for Curve Constructon, Appled Mathematcal Fnance, 13: 2, , [5] T. M. Lehmann, C. Gonner, K. Sptzer (1999), Survey: Interpolaton Methodsn Medcal Image Processng, IEEE Transactons on Medcal Imagng, Vol. 18, No. 11,

9 url=http%3a%2f%2feeexplore.eee.org%2fxpls%2fabs_all.jsp%3farnumber% 3D [6] P. S. Hagan, G. West (2008), Methods for Constructng a Yeld Curve, Wlmott Magazne, [7] P. F. du Preez, E. Mare (2013), Interpolatng Yeld Curve Data n a Manner that Ensures Postve and Contnuous Forward Curves, SAJEMS NS 16 (2013) No 4: [8] J. M. Hyman (1983), Accurate Monotoncty Preservng Cubc Interpolaton, SIAM J. Sc. Stat. Comput. Vol. 4, No. 4, http: // preservng_cubc_nterpolaton/fle/d912f50c0a8c9585cd.pdf. [9] J. Butland (1980), A Method of Interpolatng Reasonable-Shaped Curves Through Any Data, Proceedngs of Computer Graphcs 80, Onlne Publcatons Ltd, [10] F. N. Frtsch, J. Butland (1980), An Improved Monotone Pecewse Cubc Interpolaton Algorthm, Lawrence Lvermore Natonal Laboratory preprnt UCRL [11] M. Mao, J. de Jong (2014), Geometry of Interest Rate Rsk, pre-prnt nr UD , A Appendx: VBA Code In ths appendx we show the Excel/VBA code that can be used to test our results. Here we wll only reproduce the most mportant methods. The complete code, together wth a workng spreadsheet and examples, can be found at the lnk nl/lbrary_fles/pre-print-ud _companion-spreadsheet.xlsm. The frst step s to compute the b coeffcents. Ths calculaton can be done usng equatons (13)-(16) and s mplemented nto the B Coeffcents functon. Wthn ths functon one can choose whether to swtch on or off the Hyman constrant (16), enclosed n the HymanConstrant method. If the constrant s turned off, then only the Frtsch- Butland prescrpton s used wth b = β (15). The boolean parameter shymanenforced s used for ths purpose and s specfed n the ntalzaton method. Prvate Functon HymanConstrant(ndex As Integer, b() As Double) As Double Dm max As Double, mn As Double max = WorksheetFuncton.max(m(ndex), m(ndex - 1)) mn = WorksheetFuncton.mn(m(ndex - 1), m(ndex)) If m(ndex) > 0 Then HymanConstrant = WorksheetFuncton.mn(WorksheetFuncton.max(0, b(ndex)), 3 * mn) If m(ndex) < 0 Then HymanConstrant = WorksheetFuncton.max(WorksheetFuncton.mn(0, b(ndex)), 3 * max) End Functon 9

10 Prvate Functon B_coeffcents() As Double() Dm As Integer, n As Integer Dm max As Double, mn As Double If numberofnodes > 2 Then n = numberofnodes - 1 Dm b() As Double ReDm b(0 To n) b(0) = 0# b(n) = 0# For = 1 To n - 1 If m( - 1) * m() <= 0 Then b() = 0 Else max = WorksheetFuncton.max(m(), m( - 1)) mn = WorksheetFuncton.mn(m( - 1), m()) b() = 3 * m( - 1) * m() / (max + 2 * mn) If shymanenforced Then b() = HymanConstrant(, b) Next B_coeffcents = b End Functon A crucal ngredent of the code s the FndLowerIndexWthBnarySearch method, whch performs a search usng the bnary algorthm and returns the ndex of the nterval that contans x, gven any nput value x,.e. x [x, x +1 ). For x values that fall outsde the nput range, we use flat extrapolaton. We can then compute all the necessary quanttes, such as the length of the relevant nterval (3) wth method h(), ts slope (4) wth method m(), and the remanng nterpolaton coeffcents (6) and (7) wth methods c() and d() respectvely. Fnally we apply the Interpolate method that computes the nterpolated functon for any nput value. Publc Functon Interpolate(xInput As Double) As Double Dm youtput As Double, ndex As Integer If numberofnodes < 3 Then MsgBox "At least 3 ponts needed for Monotone Preservng" Ext Functon If xinput >= xvalues(numberofnodes - 1) Then youtput = yvalues(numberofnodes - 1) ElseIf xinput < xvalues(0) Then youtput = yvalues(0) Else ndex = FndLowerIndexWthBnarySearch(xInput) youtput = yvalues(ndex) + b(ndex) * (xinput - xvalues(ndex)) + C_coeffcents(ndex) * (xinput - xvalues(ndex))^2 + D_coeffcents(ndex) * (xinput - xvalues(ndex))^3 Interpolate = youtput End Functon 10