Creating a zero coupon curve by bootstrapping with cubic splines.

Transcription

1 MMA 708 Analytcal Fnance II Creatng a zero coupon curve by bootstrappng wth cubc splnes. erg Gryshkevych Professor: Jan R. M. Röman Dvson of Appled Mathematcs chool of Educaton, Culture and Communcaton Mälardalen unversty Bo 883, E Västerås, weden

2 Abstract In ths report we calculate spot rate curve by bootstrappng and nterpolaton wth cubc splnes. For ths task, Ms Ecel applcaton s developed and presented n later secton.

3 Contents. Introducton Problem descrpton Bootstrappng Cubc splnes nterpolaton Computatonal algorthm Applcaton overvew...8 Concluson...0 References... 3

4 . Introducton The concept behnd zero coupon prcng s the evaluaton of all ndvdual cash flows as f they were zero coupon bonds. The evaluaton s made usng a spot rate curve whch accurately descrbes current market condtons. The prcng of lqud, standardzed nstruments are qute smple the current market prce s used. The zero coupon prcng methodology becomes mportant when prcng OTC nstruments, for whch no market prces are avalable. It s also needed for prcng standardzed nstruments, whch do not have relable market prces. In ths case, zero coupon prcng wll be used to prce these nstruments consstently alongsde the lqud nstruments. Ths s a knd of relatve prcng []. Durng such prcng approach, bootstrappng & nterpolaton method s used for obtanng spot rate curve. In ths report we are buldng spot rate curve usng market data and method of nterpolaton wth cubc splnes Also, a general Ms Ecel applcaton, whch performs all calculatons, s presented. 4

5 2. Problem descrpton 2. Bootstrappng Bootstrappng s a procedure used to calculate the zero coupon curve from the gven market data. Because zero coupon bonds offered by the market are not avalable for every tme perod, the bootstrappng method s used to fll n the mssng fgures n order to derve the zero coupon curve. The bootstrap method uses nterpolaton to determne the spot rates for zero coupon securtes wth varous maturtes. o t makes sense to construct a curve of zero coupon nstruments from whch we can prce any yeld, whether forward or spot, wthout the need of more eternal nformaton. As was mentoned above, the bootstrap method uses some nterpolaton method. It depends on nterpolaton method, whch values wll we get between gven spot rates. Interpolaton methods vary sgnfcantly and, n fact, t s the topc for a separate tetbook. In ths paper, nterpolaton wth cubc splnes s selected as the nterpolaton method for bootstrappng, whch we wll hghlght n more detals. More on nterpolaton methods as well, as development of new technque can be found n [2]. 5

6 2.2 Cubc splnes nterpolaton. plne nterpolaton s a form of nterpolaton where the nterpolant s a specal type of pecewse polynomal called a splne. In case of thrd order polynomals, such splnes are called cubc. For a data set { } of n ponts, we can construct a cubc splne wth n pecewse cubc polynomals between the data ponts. ( ) ( ) [ ] ( ) [ ] ( ) [ ] n n n,,, K If () represents the splne functon nterpolatng functon f, t s requred: The nterpolaton property ( )f( ) for all. Actually, n our case, nterpolated functon s a set of dscrete numbers. Twce contnuously dfferentable: ( ) ( ) ( ) ( ) ( ) ( ) n,, K The assumed form for the cubc polynomal curve ft for each segment s: ( ) ( ) ( ) ( ) d c b a 3 2 Usng ths representaton and requred propertes, the cubc splne propertes are found by solvng the system of equatons. Fnaly, a,b,c, and d values correspond to the polynomal defnton for each segment. ( ) h and y d h c b h h h y y a More on cubc spllne nterpolaton can be read n [] and [2]. 6

7 2.3 Computatonal algorthm Our applcaton s bult accordng to the followng algorthm. tart Input: set of bonds Form array of spot rates from ntally gven zero coupon bonds. Buld a zero coupon curve by nterpolaton wth cubc splnes. Obtan a spot rate from net coupon bearng bond. Add obtaned spot rate to correspondng array. More coupon bearng bonds? Yes No Buld a zero coupon curve by nterpolaton wth cubc splnes. Output: zero coupon curve. End 7

8 3. Applcaton overvew Applcaton s bult n M Ecel usng VBA programmng language. It contans two spreadsheets one for nput data and computatons, the second one fro dsplayng a dagram. As was mentoned n the prevous secton, algorthm begns wth nput data. Input data contans bond names, nformaton about coupons and ther frequences, market prces and tme propertes. By default t s set to annually. If there was a transacton, the last prce s taken to account, otherwse average of bd and ask s consdered as the prce of the asset. Bonds are quoted n clean prces, so n order to go from clean prce to the drty one, we add to the gven clean prce accrued, snce last coupon payment, nterest rate. Nomnal amount, by default, s set to 00. It means, that f ths feld s left empty, value of 00 s assgned to the nomnal amount property of the bond. Tme to maturty s calculated wth bult n ecel functon Yearfrac() usng gven bonds epraton date and today s one. The applcaton understands fve types of day count conventons: U 30/360 Actual/actual Actual/360 Actual/365 European 30/360 By default, day count s set to U 30/360. Fgure 3. By pressng Run button, computatonal algorthm, that was overvewed n secton 2.3 s launched. 8

9 As the result, n output there s a set of spot rates, recovered from the ntal market data. By pressng button Dagram, we gve nstructon to dsplay a dagram on the separate spreadsheet. Red dots on the plot correspond to ntal ponts, from the column pot rates. Blue curve s obtaned by cubc nterpolaton. 6,0000% 4,0000% 2,0000% 0,0000% 8,0000% 6,0000% 4,0000% 2,0000% 0,0000% Fgure 3.2 0,89 0,39 0,593 0,795 0,997,98,400,602,804 2,006 2,208 2,40 2,62 2,84 3,06 3,28 3,420 3,622 3,824 4,025 4,227 4,429 4,63 4,833 5,035 5,237 5,439 5,64 5,843 6,045 6,247 6,449 6,65 6,852 Checkbo Plot forward rate controls dsplayng of the forward rate curve, obtaned from the spot rate curve. If checked, forward rate curve s dsplayed as well. After bootstrappng & nterpolaton process s fnshed, one can start prcng bonds. ecton Prce a bond contans felds of nput nformaton. After fllng correspondng felds and pressng button Prces theoretcal drty prce s gven. Fgure 3.3 9

10 Concluson Zero coupon prcng s well developed technque of relatve prcng. pot rates for correspondng cash flows are taken from the spot rate curve Ths curve s bult usng spot rates, that are calculated usng market data. Then the correspondng set of values of spot rates s nterpolated. o, n such way, spot rates curve s obtaned. There s a bg number of nterpolaton methods avalable, modern technques are dscussed n [2]. In ths report we have used cubc splnes as nterpolaton method. Developed Ms Ecel applcaton can be used for zero curve constructng and further valuaton of bonds. 0

11 References [] Jan R. M. Röman, Lecture notes n Analytcal Fnance II, 200 [2] Patrck. Hagan, Graeme West, Interpolaton methods for curve constructon, Appled Mathematcal Fnance, Vol. 3, No. 2, 89 29, June 2006 [3] Espen Gaarder Haug, The complete gude to Opton Prcng Formulas, 2nd ed., Mc Graw Hll, New York, 2007 [4] Charles O Nell, Cubc splne nterpolaton, 2002.