Interest Rate Barrer Optons Prcng Yang Tsung-Mu Department of Fnance Natonal Tawan Unversty
Contents 1 Introducton 1.1 Settng the Ground 1.2 Survey of Lterature 1.3 Thess Structure 2 Prelmnares 2.1 General Framework 2.2 The Standard Market Models 2.2.1 Black s Model 2.2.2 Bond Optons 2.2.3 Interest Rate Caps 2.2.4 European Swap Optons 2.2.5 Generalzatons 2.3 Hull-Whte Model 2.3.1 Model Formulaton 2.3.2 Prcng Bond Optons wthn the Hull-Whte Framework 2.3.3 Calbratng the Hull-Whte Model 3 Cheuk and Vorst s Method 3.1 Sngle-Barrer Swaptons 3.1.1 Defntons 3.1.2 Tme-Dependent Barrer 3.1.3 Intal Hull-Whte Tree 3.1.4 Adjusted Hull-Whte Tree 3.1.5 Calculatng Prces 3.1.6 Numercal Results 3.2 Sngle-Barrer Bond Optons 3.2.1 Defntons 3.2.2 Tme-Dependent Barrer 3.2.3 Numercal Results 4 Extendng Cheuk and Vorst s Method 4.1 Double-Barrer Swaptons 4.1.1 Movng Barrers 4.1.2 Numercal Results 2
5 Conclusons Bblography Appendx A Calculatng Tme-Dependent Level θ () t 3
Abstract Cheuk and Vorst s method [1996a] can be appled to prce barrer optons usng one-factor nterest rate models when recombnng trees are avalable. For the Hull-Whte model, barrers on bonds or swap rates are transformed to tme-dependent barrers on the short rate and we use a tme-dependent shft to poston the tree optmally wth respect to the barrer. Comparson wth barrer optons on bonds or swaps when the observaton frequency s dscrete confrms that the method s faster than the Monte Carlo method. Unlke other methods whch are only applcable n the contnuously observed case, the lattce methods can be used n both the contnuously and dscretely observed cases. We llustrate the methodology by applyng t to value sngle-barrer swapton and sngle-barrer bond optons. Moreover, we extend Cheuk and Vorst s dea [1996b] to double-barrer swapton prcng. 4
Chapter 1 Introducton 1.1 Settng the Ground The fnancal world has wtnessed an explosve growth n the tradng of dervatve securtes snce the openng of the frst optons exchange n Chcago n 1973. The growth n dervatves markets has not only been that of volume but also of complexty. Many of these more complex dervatve contracts are not exchange traded, but are traded over-the-counter. Over-the-counter contracts provde talor made products to reduce fnancal rsks for clents. Interest rate dervatves are nstruments whose payoffs are dependent n some way on the level of nterest rates. In the 1980s and 1990s, the volume of tradng n nterest rate dervatves n both the over-the-counter and exchange-traded markets ncreased very quckly. Interest rate dervatves have become most popular products n all dervatves markets. Exhbts 1 and 3 show the amount of nterest rate dervatves n the OTC and exchange-traded markets (also plotted n Exhbts 2 and 4). Moreover, many new products have been developed to meet partcular needs of end-users. A key challenge for dervatves traders s to fnd better and more robust procedures for prcng and hedgng these contracts. 5
EXHIBIT 1 OTC DERIVATIVES: AMOUNT OUTSTANDING Global market, by nstruments, n bllon of US dollars 1998 1999 2000 2001 2002 2003 Interest rate dervatves 50015 60091 64668 77568 101658 141991 Forward rate agreements 5756 6775 6423 7737 8792 10769 Interest rate swaps 36262 43936 48768 58897 79120 111209 Interest optons 7997 9380 9476 10933 13746 20012 Other dervatves 30303 28111 30531 33610 40021 55186 Source: Bank of Internatonal Settlements. EXHIBIT 2 OTC DERIVATIVES GRAPH 6
EXHIBIT 3 EXCHANGE-TRADED DERIVATIVES: AMOUNTS OUTSTANDING Notonal prncpal n bllons of US dollars 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 Interest rate dervatves 7304 8431 8618 9257 11227 12655 11681 12642 21758 21711 33933 Interest rate futures 4943 5808 5876 5979 7587 8031 7925 7908 9265 9951 13132 Interest rate optons 2361 2623 2742 3278 3640 4624 3756 4734 12493 11760 20801 Other dervatves 453 467 664 761 1180 1280 1909 1616 2002 2099 2818 Source: Bank of Internatonal Settlements. EXHIBIT 4 EXCHANGE-TRADED DERIVATIVES GRAPH Interest rate dervatves are more dffcult to value than equty and foregn 7
exchange dervatves. Reasons nclude 1. The behavor of nterest rate s harder to capture than that of a stock prce or an exchange rate, so many models are ntroduced to approach t. 2. For the valuaton of many products, t s necessary to develop a model descrbng the behavor of the entre zero-coupon yeld curve, but t s not easy to depct the curve exactly. 3. The volatltes of dfferent ponts on the yeld curve are dfferent. 4. Interest rates are consdered as varables for dscountng as well as for defnng the payoff form the dervatve. In ths thess we focus on nterest rate optons wth barrers. Barrer optons are optons where the payoff depends on whether the underlyng asset s prce/level reaches a certan threshold durng a certan perod of tme. A number of dfferent types of barrer optons are regularly traded n the over-the counter market. They are attractve to some market partcpants because they are less expensve than the standard optons. These barrer optons can be classfed as ether knock-out or knock-n types. A knock-out opton ceases to exst when the underlyng asset s prce hts a certan barrer; a knock-n opton comes nto exstence only when the underlyng asset s prce or level hts a barrer. Interest rate barrer optons usually nvolve one or two tme-dependent boundares affectng the opton prces, and exact closed-form solutons are not avalable for most nterest rate models. It s well-known that for barrer optons, the postons of nodes n the tree wth respect to the barrer value are crtcal. Cheuk and Vorst [1996b] proposed a trnomal tree model for barrer optons. It uses a tme-dependent shft to poston the tree optmally wth respect to the barrer. The model they constructed s flexble and can be used to prce optons wth tme-varyng barrer structures such as nterest rate 8
dervatves. For now, we wll focus our attenton on down-and-out swaptons and down-and -out bond optons. The sngle-barrer down-and-out swapton and the sngle-barrer down-and-out bond optons are prced usng the methodology of Cheuk and Vorst [1996a]. The double-barrer down-and-out swapton s prced by extendng the dea of Cheuk and Vorst [1996a]. Other nterest rate barrer optons can be prced usng smlar deas. 1.2 Survey of Lterature Valuaton approaches have largely focused on equty barrer optons, where n certan nstances analytcal expressons may be avalable. Works on nterest rate barrer optons prcng are relatvely rare. Accurate and effcent valuaton technques are requred snce barrer optons have become very popular n recent years as useful hedgng nstruments for rsk management strateges. Several researches address the prcng of nterest rate barrer optons. Cheuk and Vorst [1996a] extended Rtchken s method by ntroducng a tme-dependent shft n trnomal lattce. The Hull-Whte model s selected and sngle-barrer swaptons are prced n both the contnuously and dscretely observed cases. Kuan and Webber [2003] use one-factor nterest rate models ncludng the Hull-Whte model and the swap market model to value barrer knock-n bond optons and barrer knock-n swaptons. A numercal ntegraton method s used to prce nterest barrer optons when the transton dstrbuton functon of underlyng rate s known but explct prcng formulas are not avalable. Although the convergence s fast, the drawback s that the valuaton only can be appled to the contnuously observed case. Monte Carlo 9
smulaton s known for ts hgh flexblty. However, n the case of barrer opton t produces based results for optons, whch depend on the contnuously montored sample path of some stochastc varable. Barone-Ades and Sorwar [2003] prce contnuously observed barrer bond optons n the corrected Monte-Carlo smulaton of the CKLS dffuson process: ( θ ) γ dr = ar dt + σr dz (1.1) Usng the results of Bald et al [1999], they set up a Monte-Carlo scheme to value nterest rate barrer optons whch takes nto account the possblty of breachng the barrer between successve ntervals of tme. It has enough flexblty to prce all knds of contnuously observed nterest rate barrer optons. Ths procedure then provdes an almost unbased Monte Carlo estmator. However, the speed of convergence s stll very slow and need many paths of the underlyng nterest rate process to obtan accurate results. 1.3 Thess Structure The remander of ths thess wll be organzed as follows. Chapter 2 revews the Black s model prcng technologes and explans how to calbrate Hull-Whte model wth the Black s model consstently. Chapter 3 covers the prcng of sngle-barrer swaptons and bond optons. The prcng of double-barrer swaptons s presented n Chapter 4. Chapter 5 concludes. 10
Chapter 2 Prelmnares Frst, defne the followng varables: T 0 : Current tme, T : Tme to maturty of the opton, P( t, T ) : Prce at tme t of a zero-coupon bond payng $1 at tmet, y( t, T): the (T t)-perod nterest rate (annualzed) at tme t. 2.1 General Framework Vanlla European contngent clams such as caps, floors, bond optons, and swaptons are prced correctly usng the smple model developed by Black [1973]. Ths model makes several smplfyng assumptons whch allow closed-form valuaton formulae to be possble. Svoboda [2004] referred the class of vanlla contngent clams as frst-generaton products. Second- and thrd-generaton dervatves, such as path-dependent and barrer optons, provde exposure to the relatve levels and correlated movements of varous portons of the yeld curve, and the market prces of these frst-generaton nstruments are taken as gven. Ths does not necessarly mply a belef n the ntrnsc correctness of the Black model. Dstrbutonal assumptons whch are not ncluded n the Black model, such as mean revertng and skewness, are ncorporated by adjustng the mpled volatlty nput. More sophstcated models allow the prcng of nstruments dependent on the changng level and slope of the yeld curve. A crucal factor s that these models must prce the exotc dervatves n a manner that s consstent wth the prcng of vanlla nstruments. When assessng the correctness of any more sophstcated model, ts 11
ablty to reproduce the Black prces of vanlla nstruments s vtal. Svoboda [2004] remarked that t s not a model s a pror assumptons but rather the correctness of ts hedgng performance that plays a pvotal role n ts market acceptance. The calbraton of a model s an ntegral part of ts specfcaton. So the usefulness of a model cannot be assessed wthout consderng the relablty and robustness of any parameter estmaton scheme. 2.2 The Standard Market Models 2.2.1 Black s Model We wll show how the Black s model for valung vanlla European nterest rate optons s derved. The market prce wll be gven, and the parameters of more complex models can be valued by mnmzng devatons from market prces. Black s model s used to value optons on foregn exchanges, optons on ndces, and optons on future contracts. Traders have become very comfortable wth both the lognormal assumpton that underles the model and the volatlty measure that descrbes uncertanty. It s no surprse that there have been attempts to extend the model so that t covers nterest rate dervatves. In the followng we wll dscuss three of the most popular nterest rate dervatves (bond optons, nterest rate caps, and swap optons) and descrbe how the lognormal assumpton underlyng the Black-Scholes model can be used to value these nstruments. Now we use Black s model to prce European optons. Consder a European call opton g on a varable whose value s V. Defne T : Tme to maturty of the opton, F : Forward prce of V for a contract wth maturty T, F 0 : Value of F at tme T 0, K : Strke prce of the opton, 12
V T : Value of V at tme T, σ : Volatlty of F. Black s model calculates the expected payoff from the opton assumng: 1. VT has a lognormal dstrbuton wth the standard devaton of lnv T equal to σ T. 2. The expected value of s F. VT 0 3. The world s forward rsk neutral wth respect to P( t, T ) ;.e., martngale n the world. g t (, ) P t T s a So the value of the opton at tme T s max ( V K,0) and the lognormal T assumpton mples that the expected payoff s ( ) ( ) ( ) E V N d KN d, (2.1) T 1 2 where E ( V T ) s the expected value of V T and Because g t (, ) P t T d 1 2 1 ( ) 2 EVT ln σ T K + 2 =. σ T d = d σ T s a martngale wth respect to P( t, T ), t follows that g 0 g T (, = ET 0 ) (, ET g PT T = PTT) Besdes, we are assumng that E ( VT ) F0 ( ) T. (2.2) = and the valdty of dscountng at the rsk-free rate, so the value of the opton s gven by (2.1) and (2.2) (, ) ( ) ( ) g0 = P T0 T F0N d1 KN d2, (2.3) where 2 ln F0 T K + σ 2 d1 = σ T d = d σ T 2 1. 13
2.2.2 Bond Optons A bond opton s an opton to long or short a partcular bond by a certan date for a partcular prce. Consder a zero-coupon bond PT (, 0 T ) and we assume the world s forward rsk-neutral wth respect to a zero-coupon bond maturng at tmet ( T T ). So the prce of a call opton wth strke prce K and maturty T (years) on a bond (, 0 ) PT T s ( ) ( ) * 0 T ( ) c=p T,T E max P T, T K,0, (2.4) where E T denotes expected value n a world that s forward rsk neutral wth respect to a zero-coupon bond maturng at tme T. It mples that T ( (, )) 0 E P T T = F, where F 0 s the forward prce of PTT (, ) at tme T 0. Assumng the bond prce s lognormal wth the standard devaton of the logarthm of the bond prce equal toσ T, the equaton (2.3) becomes: ( )[ ] c= PT, T FNd ( ) KNd ( ), (2.5) 0 0 1 2 where 2 [ ] σ ln F0 / K + T / 2 d= 1 σ T d = d σ T. 2 1 Ths reduces to Black s model. We have shown that we can use today s T -year maturty nterest rate for dscountng provded that we also set the expected bond prce equal to the forward bond prce. 2.2.3 Interest Rate Caps A popular nterest rate opton offered by fnancal nsttutons n the over-the-counter market s the nterest rate cap. Consder a cap wth a total lfe of T, 14
a prncpal of C, and a cap rate of K. Suppose that the reset dates are T1, T2,..., T n and defne T + = T. Defne yt (, T + ) as the smply compounded nterest rate for n 1 k k 1 the perod between tmes T and T + 1 observed at tme T (1 k n) : k k k ( k 1 Tk) (, ) 1 PT, yt ( k 1, Tk) =. (2.6) δ PT T k 1 k The caplet correspondng to the rate observed at tme T k provdes a payoff at tme T k + 1 of where T 1 δ k k+ T k ( ( 1) ) Cδ max y T, T + K,0 k k k =. If the rate (, ) volatltyσ, the value of the caplet s where k k k 1, yt T + s assumed to be lognormal wth (, )[ ( ) ( ] caplet = C P T T F N d KN d ), (2.7) δ k 0 k+ 1 k 1 2 2 [ ] σ ln Fk / RK + ktk / 2 d= 1 σ T k k d2 = d1 σ k Tk and F s the forward rate for the perod between tme and. Note that k (, ) k k 1 F k Tk T k + 1 yt T + and are expressed wth a compoundng frequency equal to frequency of resets n these equatons. The cap s a portfolo of n such optons and the formula also reduces to a summaton of Black-lke formulas. 2.2.4 European Swap Optons Swap optons, or swaptons, are optons on swap rates and are a very popular type of nterest rate opton. They gve the holder the rght to enter a certan nterest 15
rate swap that pays a fxed rate, the strke rate, and receves a floatng nterest rate at a certan tme n the future. Consder a swapton whch gves the rght to pay a rate K and receve ( T) ytk 1, k on s swap settled n arrears at tme Tk = T + kδ, k = 1,..., n wth a notonal prncpal C. Suppose that the swap rate for an n-payments swap at the maturty of the swap opton s s T. Assume the relevant swap rate at the maturty of the opton s lognormal. By comparng the cash flows on a swap where the fxed rate s K. The total value of the swapton s where n Cδ P( T0, T )[ s0n( d1) KN( d2) ] = 1 d 1 2 [ ] + σ ln s0 / K T / 2 = σ t d = d σ T 2 1, (2.8) and s s the forward swap rate startng at tme T and wll be ntroduced below. 0 Forward swap rate The forward swap rate ( T n) κ T 0, at tme T can also be determned usng the formula (Brace and Musela 1997) that makes the value of the forward swap zero,.e., κ T ( T, n) 0 (, ) (, + δ ) PT0 T PT0 T n = n δ PT T j= 1 (, + nδ) 0. (2.9) 2.2.5 Generalzatons We have presented three versons of Black s model: one for bond optons, one for caps, and one for swaptons. Each of the models s nternally consstent wth each other, but they are not consstent wth each other. For example, when future bond prces are lognormal, future zero rates and swap rates can not be, and when future zero rates are lognormal, future bond prces and swap rates can not be. 16
Wth the market prce by Black s formula we now proceed to ntroduce how to calbrate more complex models. 2.3 Hull-Whte Model 2.3.1 Model Formulaton We wll use the short rate model to prce barrer optons n the thess. Hull and Whte proposed an extenson to the Vascek model of the one-factor form: ( θ () ) dr = t ar dt + σ dz, (2.10) where θ () t s a tme-dependent reverson level chosen so that the spot yeld rate curve mpled by the model matches the yeld curve observed ntally, a s the speed of mean reverson, and σ s a known constant. It provdes enough degrees of freedom to ft the current nterest rate term structure. The process descrbng the evoluton of the short rate can be deduced from the observed term structure of nterest rates and nterest rate volatltes. We wll value the European call opton on a zero coupon bond and descrbe how to use the nformaton from observed term structure of nterest rate and volatltes. We then go on to make sure that the model s consstent wth market prces by calbratng the Hull-Whte model for constant mean reverson and volatlty parameter. After fnshng all work above the more complcated nterest rate dervatves, barrer optons, wll be nvestgated n the next chapter. 2.3.2 Prcng Bond Optons wthn the Hull-Whte Framework Let P( t, T ) be the tme t prce of a dscount bond maturng at tme T. The bond prce formula s shown n Hull and Whte [1990, 1994a] to be (, ) (, ) (, ) P t T = A t T e BtT r, (2.11) where P( t, T ) s the prce at tme t of a zero-coupon bond maturng at tme T. 17
Furthermore, A( tt, ) and (, ) B tt are functons only of t and T, and r s the short rate at tme t. The functon A( tt, ) s determned from the ntal values of dscount bonds P(0, T) as ( 0, T) ( 0, t) ( ) ( ) 2 ( ) 2 ( 2 at, 0, σ, 1 ) ( a ) P B t T F t B t T e AtT (, ) = exp P 4 (2.12) (, ) B t T at ( t) ( 1 e ) =, (2.13) a F (0, t ) s the nstantaneous forward rate that apples to tme t as observed at tme zero. It can be computed from the ntal prce of dscount bond as F ( 0, t) By Ito s Lemma we have: ( t) log P 0, =. (2.14) t 2 P P 1 P dp = dt + dr + drdr 2 t r 2 r = ( θ() ) + σ + t 1 = () + 2 P dt ABe Br t ar dt dz AB 2 e Br 2 dt 2 2 Pdt t BP θ t ar dt BPσdz B Pσ dt 1 2 σ (2.15) Hence the prce process of the dscount bond s descrbed by the stochastc equaton: The relatve volatlty of 1 2 2 dp = Pt BP ( θ () t ar) + B Pσ dt PBσdz 2. (2.16) P( r,, t T ) s B( t, T) σ. Snce t s ndependent of r, the dstrbuton of the bond prce at any tme * t, condtonal on ts value at an earler tme t, must be lognormally dstrbuted. Consder a European opton on ths dscount bond. Ths opton has the followng characterstcs: K : exercse prce, 18
T : opton expry tme, T : bond maturty tme, : current (valuaton) tme, where T0 0 T T T. Snce PTT (, ) s lognormally dstrbuted and represents the forward bond prce, the Black s formula can be used to prce the dscount bond opton: * ( 0, ) ( ) ( 0, ) ( p ) C = P T T N h KP T T N h σ, (2.17) ( ) PT0, T ln ( KP( T0, T )) 1 where h = + σ σ 2 p and p ( ) 2 2 2 σ at ( T) 2aT ( T0 ) σ p = 1 e 1 e 3 2a 2.3.3 Calbratng the Hull-Whte Model Havng consdered the model formulaton that allows us to ncorporate observed term structure data nto the prcng formula, we explan how actual data are used n the calbraton exercse. Cubc splne nterpolaton An nterpolaton technque must be appled to term structure so that values for any maturty term maybe extracted. Cubc splne nterpolaton was favored for the smoothness of curve t produces. Cubc splne nterpolaton s a type of pecewse polynomal approxmaton that uses a cubc polynomal between successve pars of nodes. At each of nodes across whch the cubc splne s ftted, the followng hold: The values of the ftted splnes equal the values of the orgnal functon at the node ponts. The frst and the second dervatves of the ftted splnes are contnuous. The algorthm s presented n Appendx A.. 19
Usng the market data For each day, the contnuously compounded rate of nterest and hstorcal volatlty are avalable for a dscrete set of node ponts correspondng to terms to maturty, t, = 1,..., N where t 1 1 = year and t 365 N = T. The nterest rate wth term to maturty 1 day and ts correspondng hstorcal volatlty are taken as proxes for the nstantaneous short-term nterest rate ( ) 0 r T and ts volatlty σ ( rt, ). B ( T0, T ) s calculated usng BT ( 0, T) term structure of nterest rates ( ) 0, r 0 at ( 1 e ) =. Applyng the ntal a yt t, we determne the tme dscount T 0 bond prces as ( ) PT0, T e ( ) yt0, TT =. Apply B ( T0, t ) and PT ( 0, t ) to fnd ( 0, ) Calbraton methodology A T t. Calbraton of the model to the observed market prces nvolves retrevng values of σ and a such that these market prces may be recovered from the model. The reverson speed a and the assocated volatlty parameter σ should gve rse to the smallest prcng error. That s, we want the a and σ such that n = 1 ( p (, ) ) 2 model a σ pmarket s mnmzed for several dfferent maturtes t, = 1,..., n and (2.17) s used to calculate (, ) p a σ. model Tme-dependent mean reverson level θ ( t) As shown by Hull and Whte (1993), the tme-dependent mean reverson level, () t s determned at the ntal tme θ 0 T as 20
θ () (, ) 2 F T0 t σ 2at T0 t = + af( T0, t) + ( 1 e ( ) ). (2.18) t 2a Thereafter, our model can be consstent wth market prces, and exotc optons prcng wll be followed under the calbraton parameters a and σ. 21
Chapter 3 Cheuk and Vorst s Method Frst, defne the followng varables: T 0 : current tme, T : opton maturty, wtn) (, : the spot swap rate at tme t whch makes the value of the swap ( n payments) zero, β : the barrer swap rate fxed at the spot swap rate wt ( 0, n ) mnus the same fxed rate throughout (for example, 25 bass ponts), h : the barrer short rate at tme t s found by settng the value of the swap at the fxed rate β zero, φ : the barrer Δt -perod rate (annualzed) at tme t that corresponds to h, κ T ( tn, ) (, ) : the at-the-money forward swap rate at tme T whch makes the value of forward swap zero. In our tree, we defne three varables at node (, j ) : R j : the Δt -perod nterest rate at tme that s assocated wth node, j, r(, j): the short rate that s assocated wth node (, j ), t ( ) s(, j): the swap rate at tme t that s assocated wth node(, j ). 22
3.1 Sngle-Barrer Swaptons 3.1.1 Defntons Wth a down-and-out swapton the holder can choose to enter nto a swap startng n the swapton s maturty or choose not to exercse the swapton f the swap rate at the swapton s maturty s less than the strke rate. However, f the correspondng swap rate falls below a certan barrer value before the swapton s maturty, the swapton expres worthless. 3.1.2 Tme-Dependent Barrer For the Hull-Whte model, barrers are transformed to smooth tme-dependent barrers on the short rate. We poston the nodes optmally on the tme-varyng barrer that enables us to prce barrer optons effcently. Snce the short rate s the only state varable, the tme-dependent barrer can be transformed to the equvalent tme-dependent barrer on short rate, h. The short rate h can be found through Newton-Raphson teraton or any other teratve scheme to be detaled later. The determnaton of Interest rate swap h can be done before constructng the tree. Consder a payer swap on prncpal C settled n arrears at tmes T T0 jδ, j 1,..., n j = + =. The floatng rate yt (, j 1 Tj ) tmetj 1. receved at tme T j s set at The swap cash flows at tmest j, j = 1,..., n, are C y( T 1, T ) δ (the floatng leg) j j and Ckδ (the fxed leg), where k s the fxed rate determned at. Hence the value T 0 of the swap s (Brace and Musela 1997): 23
n E C P( T0, Tj) ( y( Tj 1, Tj) k) δ j= 1 n ( 0, 0) j ( 0, j) = CP T T C d P T T j= 1 n = C 1 d jp T, T j= 1 ( 0 j). (3.1) where d j = kδ for j = 1,..., n 1 and d = 1+ kδ. n The dentty of (3.1) s explaned as follows. The swap can be vewed as a portfolo of a zero coupon bond and a coupon bearng bond. It receves nterest at a floatng rate on a notonal prncpal C, and the value of the notonal prncpal at tme T0 s C. It pays nterest at a fxed rate on the notonal prncpal C. Its value thus equals n zero coupon bonds wth notonal prncpals Cd for j = 1,..., n maturng j at tme T j. Therefore, the value s C n j= 1 j ( 0, j) d P T T. Swap rate The spot swap rate wt ( 0, n) at tme s that value of the fxed rate whch makes the value of the swap zero,.e., n j= 1 T wt ( n) 0 ( j) ( ) δ = ( ) ( C PT, T wt, n C* PT, T C* PT, T The left expresson s fxed payment value at 0 0 0 0 0 T 0 n ) 0,. (3.2) and the rght expresson s floatng recepton value at T 0. Usng equaton (3.2), we fnd that (, ) wt n 0 j= 1 ( ) 1 PT0, Tn = n δ PT T ( 0, j ). (3.3) Barrer swap rate and barrer short rate Suppose the contractual barrer swap rate β s fxed at the spot swap rate 24
wt ( 0, n) mnus the same fxed rate λ throughout (for example, 25 bass ponts) and the swapton knocks out f the swap rate for T t T ( ) 0 opton maturty s less than or equal to β. So the barrer short rate h can be found by settng the value of the swap at the fxed rate β zero for T 0 t T,.e., we want to make: j= 1 ( nδ ) 1 P t, t + δ n (, + jδ) P t t β = 0. (3.4) We can approxmate the value h through Newton-Raphson teraton gven the bond prce formula ( ) ( ) (, ) Bt T h P t, T = A t, T e. 3.1.3 Intal Hull-Whte Tree Barrer Δt -perod rate Let φ be the Δt -perod barrer nterest rate (annualzed) at tme t that corresponds to the short rate h. We can fnd φ by usng the bond prce formula wth short rate h : ( ) (, ) e Δ = A t t +Δ t e. (3.5) φ t Bt t+δth, So (, +Δ ) log ( (, +Δ )) B t t t h A t t t φ = Δt. (3.6) A swapton s at-the-money when the strke rate equals the forward swap rate at swapton s maturty. We llustrate the results usng the example of a contnuously observed at-the-money knockout swapton on a fve-year swap wth a prncpal of 100 and the fxed leg payng annually. The opton maturty s 2 months from now, and the barrer swap rate s fxed at the spot swap rate at tme T 0 mnus 70 bass ponts. The ( ) 0.18 t zero yeld curve s gven by yt0, t = 0.08 0.05e, the parameter a n the 25
Hull-Whte model equals 0.1, andσ equals 0.015. We buld the Hull-Whte model wth four tme steps, and the results are shown n Exhbt 5. EXHIBIT 5 A TRINOMIAL TREE FOR A BARRIER SWAPTION Forward swap rate : 6.20% Spot swap rate: 6.03% Barrer swap rate: 5.33% (=6.03% 0.70%) Tme t 1 2 3 4 h 2.17% 2.19% 2.21% 2.24% φ 2.21% 2.23% 2.25% 2.27% By the Hull-Whte model, we frst construct a tree that has the form shown n Exhbt 6. We just reveal the frst three out of a total of four steps for brevty. The startng nodes R ( 0,0) and the step sze Δ r can be determned accordng to Hull and Whte (1993): ( 0,0 ) = (, +Δ ) R yt T t 0 0 Δ r = σ 3 a t ( 1 e ) 2 2 2a. The three nodes that can be reached by the branches emanatng from any gven node (, j ) are ( + 1, j+ 1), ( + 1, j), and ( 1, j 1) tree are constructed by: (, ) ( 0,0) R j = R + jδ r. +. The nodes (, ) R j n the 26
EXHIBIT 6 THE INITIAL TREE 4.63% P I O4.10% 3.04% A 3.57% D H N 3.04% 3.04% 3.04% C G M 2.51% B F L 2.21% 2.23% 2.25% E K1.98% 1.45% J Orgnal central nodes Barrer rates Barrer nodes 3.1.4 Adjusted Hull-Whte Tree The second stage n the constructon of the tree nvolves central node adjustments. As remarked above, t s mportant that the barrer Δt -perod rate les on one of the nodes. The best postonng depends on whether observaton of the barrer s contnuous or dscrete. For contnuously observed barrers, t s preferred that the barrer les exactly on one of the nodes. For dscretely observed barrers, t s best to let the barrer fall exactly half way between two nodes. To unfy the treatment for 27
contnuous and dscrete barrer observatons, we ntroduce a new varable barrer s observed contnuously: φ. If the φ = φ. (3.7) If the barrer Δt rate s below the spot Δ t rate at tme T 0 for dscrete observatons: If the barrer observatons: Δt Δr φ = φ. (3.8) 2 rate s above the spot Δ t rate at tme T 0 for dscrete Δr φ = φ +. (3.9) 2 We want to construct a tree that algns φ on a tree node for all 1 n and then adjust the tree accordng to the followng procedure: The node (, j ) wth a value R (, j ) closest to φ sets the value of (, ) R j equal to φ. Let α be the Δt -perod nterest rate at tme t that s assocated wth the adjusted central node at tme t. The frst one equals the current spot yeld rate maturng at t +Δt: 0 α = yt (, T+ Δt). (3.10) 0 0 0 Integers e for 1 ncan be calculated to ensure that barrer rates are an nteger number of steps (wth step sze Δ r ) away from central ratesα : φ α r e = Δ +, (3.11) 2 1 1 where [ x ] s the largest nteger less than or equal to x. The adjusted central rates α can be found wth e thus: 28
α = e Δ r. (3.12) φ The adjusted nterest rate at node (, j ) s then gven by ( ) R j = α + jδ r. (3.13), The fnal result s shown n Exhbt 7. EXHIBIT 7 THE ADJUSTED TREE 4.90% P I 4.37% O 3.11% A 3.84% D H N 3.29% 3.31% 3.11% G M C 2.78% F L B 2.21% 2.23% 2.25% E K 1.72% J Adjusted central nodes Barrer rates Barrer nodes 29
3.1.5 Calculatng Prces Suppose that the tree has already been constructed up to tme T so that t can match the barrer. A value of θ (Δ t) for 0 n must be chosen so that the tree s consstent wth yt ( 0, t + 2 ). The procedure s explaned by Hull and Whte [1993], and we gve the detals n Appendx C. Let r(, j ) be the short rate that s assocated wth node (, j). It can be calculated by bond prce formula (2.11) from R (, j ). Once θ (Δ t) has been determned, the correspondng probabltes to nodes matchng the mean (, ) ( θ ( ) ( )) r j + Δt ar, j Δt and varance (, j ) can be determned by σ aδt ( 1 e ) 2 2 the short rate to the contnuous-tme nterest rate model and the condton that the sum of the probabltes equals 1. By the Lndeberg-Feller theorem (see, for example, Bllngsley (1986)), a tree constructed n the way wll converge to the underlyng contnuous-tme Hull-Whte model. The probabltes are: 2a of 2 V η η Pu (, j) = + + 2 2 2Δr 2Δr 2Δr 2 V η Pm (, j) = 1 2 2 Δr Δr 2 V η η Pd (, j) = + 2 2 2Δr 2Δr 2Δr (3.14) wth ( ( t) ar(, j) ) t 1 η = θ Δ Δ + α α + and σ V = a t ( 1 e ) 2 2 2a. Exhbt 8 llustrates the results calculated form Exhbt 7. 30
EXHIBIT 8 TRINOMIAL TREE FOR EXHIBIT 7 Frst, the short rates r(, j) at nodes (, j ), where = 4, s computed usng the bond prce formula (2.11) wth rates R (, j ). Wth the short rates r(, j) gven, the swap rates s(, j ) at nodes (, ) j, where = 4, are found usng the swap rate formula (3.3) and therefore the payoff V (, j ) at nodes (, ) j, where = 4, at maturty T s determned: 5 V (, j) = C P( T, T + δ ) max ( s(, j) κt ( T0, n ),0) δ. (3.15) = 1 Exhbt 9 shows the calculatons requred to compute the payoff at the opton maturty, two months from now. 31
EXHIBIT 9 OPTION PAYOFF AT TERMINAL NODES (=4) Fnally, Exhbt 10 shows the dscountng of the opton value back through the tree. If the node (, j ) s touched, the value (, ) V j s set to zero f not, V (, j) s calculated usng (, ) ( + 1, + 1 ) + m(, ) ( + 1, ) (, ) ( 1, ) Pu j V j P j V j V (, j) = exp ( R(, j) Δ t + Pd j V + j ). (3.16) 32
EXHIBIT 10 DISCOUNTING THE OPTION PRICE BACK THROUGH THE TREE 3.1.6 Numercal Results We frst consder a sx-month at-the-money payer s swapton wth a notonal value of 100 and a barrer fxed at the spot swap rate mnus 25 bass ponts. The underlyng s a fve-year swap wth fxed payments made annually. The zero yeld ( ) 0.18 curve s gven by, 0.08 0.05 t yt0 t = e, the parameter a n the Hull-Whte model equals 0.1, and σ equals 0.015. Exhbts 11 and 12 gve the prce for such a sngle-barrer opton, when contnuously observed, for dfferent numbers of steps n the trnomal tree. It can be seen that convergence s fast due to our specally constructed tree. Exhbt 11 shows that the method gves very accurate answers wth as lttle as 30 steps, and Exhbt 12 shows how fast and smoothly the method converges. 33
EXHIBIT 11 CONTINUOUSLY OBSERVED SINGLE-BARRIER SWAPTION PRICES Product At-the-money sngle-barrer knockout swapton on fve-year swap wth the prncpal 100 and fxed payments made annually Barrer Spot swap rate mnus 25 bass ponts Opton maturty 0.5 year Steps Prces Tme (sec) 2 1.270025 0.015 5 1.021106 0.031 10 0.986277 0.062 30 0.965275 0.172 100 0.963571 0.547 200 0.963272 1.157 500 0.963417 3.125 1000 0.963320 5.813 Note: Calculatons were made n Dev C++ on Wndow XP system wth Inter(R) Pentum(R) 4 CPU 2.40GHz. 34
EXHIBIT 12 CONVERGENCE OF CONTINUOUSLY OBSERVED SINGLE-BARRIER SWAPTION PRICES More tme steps are needed for a swapton whose barrer s checked at dscrete ntervals. For example, f the barrer s checked daly and two perods between subsequent observatons are allocated, 250 steps are needed for a maturty of sx-months (125 days). The speed of convergence s related to the number of perods between observatons. So the hgher the frequency we observe, the more steps we need to calculate for the same accuracy. Whle 1250 ( 125 10) steps are needed daly montorng, only 60 (6 10) steps are needed for the same accuracy (10 steps between observatons) for monthly montorng. As can be seen from Exhbts 13 and 14, the speed of convergence s very good. The prces generally converge wth only 10 perods between observatons and are very close to Monte Carlo results when our 35
method s appled wth only 50 perods between observatons. EXHIBIT 13 DISCRETELY OBSERVED SINGLE-BARRIER SWAPTION PRICES Number of perods between observatons Monte Carlo Observaton Frequency 2 5 10 20 50 100 Sem-annually (Tme (sec)) Quarterly (Tme (sec)) Monthly (Tme (sec)) Weekly (Tme (sec)) Daly (Tme (sec)) 1.415429 1.471626 1.396712 1.436232 (0.016) (0.031) (0.047) (0.078) 1.465117 1.376347 1.408221 1.404073 (0.016) (0.063) (0.094) (0.204) 1.318001 1.297323 1.292777 1.289523 (0.063) (0.125) (0.266) (0.531) 1.178672 1.160131 1.154721 1.151438 (0.218) (0.562) (1.188) (2.562) 0.704829 0.690894 1.061978 1.059989 (2.226) (3.156) (7.484) (15.383) 1.424825 1.428639 1.42821 (0.219) (0.438) (1141.423) 1.399401 1.398472 1.39813 (0.547) (1.125) (1015.674) 1.287248 1.286702 1.28654 (1.392) (3.047) (798.811) 1.149227 1.148500 1.14856 (7.953) (18.341) (891.615) 1.058798 1.058396 1.0586 (48.312) (108.120) (899.254) Note 1. For the Monte-Carlo method (5000,000 paths and about 200 tme steps), the process s dr ( θ ( t) ar) dt σdz = + and the dscount rate exp( t r ) Δ s calculated usng Smpson s method. Note 2. For example, 250 (125 days 2 ) steps are needed for daly observed swaptons and the number of perods between observatons s 2. 36
EXHIBIT 14 CONVERGENCE OF DISCRETELY OBSERVED SINGLE-BARRIER SWAPTION PRICES 3.2 Sngle-Barrer Bond Optons 3.2.1 Defnton An up-and-out bond opton s one type of knock-out bond opton. It s a standard opton but t ceases to exst f the bond prce reaches a certan barrer level, H. 3.2.2 Tme-Dependent Barrer The lattce method can also be appled to prce sngle-barrer bond optons. Snce the short rate s the only state varable, the tme-dependent barrer can be transformed to an equvalent tme-dependent barrer on short rate h and the 37
determnaton of h can be done before constructng the tree. As shown n Hull and Whte [1990, 1994a]: ( ) ( ) (, ) BtTh P t, T = A t, T e. (3.17) Let the barrer bond prce H be fxed. The barrer short rate h can be calculated for all ( ) 0 t T opton maturty by H ( ) (, ) = A t T e (3.18) Bt Th, so that h ( At ( T) H) B( t, T) log, / =. (3.19) After fndng the barrer short rates, all the remanng procedure follows that for sngle-barrer swaptons earler. 3.2.3 Numercal Results A sx-month barrer bond opton wth a notonal value of 100 and strke prce 0.85 s consdered. We fx the barrer prce at 0.91. The zero yeld curve s gven ( ) 0.18 by, 0.08 0.05 t yt0 t = e, the parameter a n the Hull-Whte model equals 0.1, and σ equals 0.015. Exhbts 15 and 16 gve the prce for such a barrer opton, when contnuously observed, for dfferent numbers of steps n the trnomal tree. It can be seen that convergence s faster than barrer swaptons because barrer swapton wasted some tme n short rates calculaton wth Newton-Raphson teraton. Exhbt 15 shows the method gves very accurate answers wth as lttle as 100 steps and Exhbt 16 shows how fast and smoothly the method converges. 38
EXHIBIT 15 CONTINUOUSLY OBSERVED SINGLE-BARRIER BOND OPTION PRICES Product Sngle knock-out call optons on three-year dscount bond wth prncpal 100 Barrer prce 0.91 Strke prce 0.85 Opton maturty 0.5 year Steps Prces Tme (sec) 2 1.641741 0.000 5 1.912776 0.000 10 1.967791 0.000 30 2.006969 0.000 100 2.020153 0.015 200 2.023302 0.063 500 2.024933 0.36 1000 2.025322 1.531 1500 2.025501 3.359 39
EXHIBIT 16 CONVERGENCE OF CONTINUOUSLY OBSERVED SINGLE-BARRIER BOND OPTION PRICES More tme steps are needed for a bond opton whose barrer s checked at dscrete ntervals. As can be seen from Exhbts 17 and 18, the speed of convergence s also very good. The prces generally converge wth only 10 perods between observatons and are very close to Monte-Carlo results, wth only 50 perods between observatons. 40
EXHIBIT 17 DISCRETELY OBSERVED SINGLE-BARRIER BOND OPTION PRICES Observaton Frequency Semannually (Tme (sec)) Quarterly (Tme (sec)) Monthly (Tme (sec)) Weekly (Tme (sec)) Daly (Tme (sec)) Number of perods between observatons Monte 2 5 10 20 50 100 150 200 Carlo 2.04100 2.15258 2.17812 2.16727 2.17103 2.17033 2.16944 2.16981 2.1698 (0.000) (0.000) (0.000) (0.000) (0.000) (0.016) (0.047) (0.062) (892.344) 2.14913 2.17597 2.16427 2.16761 2.16689 2.16629 2.16628 2.16589 2.1661 (0.000) (0.000) (0.000) (0.000) (0.016) (0.063) (0.140) (0.250) (1093.132) 2.15467 2.14355 2.14117 2.13852 2.13804 2.13763 2.13742 2.13743 2.13777 (0.000) (0.000) (0.000) (0.015) (0.140) (0.578) (1.312) (2.359) (879.516) 2.10666 2.09808 2.09542 2.09413 2.09334 2.09315 2.09310 2.09307 2.09271 (0.016) (0.016) (0.109) (0.422) (2.718) (11.063) (24.39) (43.359) (798.812) 2.06718 2.06298 2.06146 2.06080 2.06037 2.06023 2.06019 2.06016 2.06016 (0.110) (0.609) (2.500) (10.172) (62.437) (250.859) (562.062) (1008.250) (977.311) 41
EXHIBIT 18 CONVERGENCE OF DISCRETELY OBSERVED SINGLE-BARRIER BOND OPTION PRICES 42
Chapter 4 Extendng Cheuk and Vorst s Method 4.1 Double-Barrer Swaptons 4.1.1 Movng Barrers It has shown how a tme-dependent barrer can be matched, and here we explan how a second barrer can be matched by extendng Cheuk and Vorst s method [1996b]. To dstngush the two barrers, we wll refer to an upper and a lower barrer. The adjectves upper and lower descrbe the locatons of the barrers relatve to each other. Double-barrer knock-out swaptons prcng s covered, and the same method s applcable to other types of nterest optons ncludng barrer bond optons and nterest barrer caps, etc. Trnomal models can be constructed so that nodes are always stuated on the two barrers. Our method s to change dr to match two barrers for = 1,..., n. We frst unfy the treatment of contnuous and dscrete observaton barrers and u h d varables and h are ntroduced for = 1,..., n: u h = u h d h = h d for contnuously observed barrers, and: u h d h u dr = h + 2 d dr = h 2 for dscretely observed barrers, where s the upper barrer and s the lower barrer. u h d h 43
Then we defne the dstance as u d M = h h To begn wth, we choose a dr close to 3V that satsfes x dr = M for contnuously observed barrers, and: u d ( x 1) dr = ( h h ) for dscretely observed barrers, where x s an nteger. That s, x s chosen as x M 1 = +. dr 2 When x s known, dr follows from the equaton above. After determnng the dr, we match the lower barrer by shftng the central nodes α we have explaned earler. As the dstance between the two barrers s fxed for = 1,..., n, matchng the lower mples that the upper barrer s matched also, gven dr. All formulas earler can be used by treatng dr as dr. The resultng probabltes also match the mean and varance of the short rate n the contnuous-tme nterest rate model. 44
4.1.2 Numercal Results We consder a sx-month at-the-money payer s swapton wth a notonal value of 100. A lower-barrer s fxed at the spot swap rate mnus 25 bass ponts and a upper-barrer s fxed at the spot rate plus 200 bass ponts. The underlyng s a fve-year swap wth fxed payments made annually. The zero yeld curve s gven ( ) 0.18 by, 0.08 0.05 t yt0 t = e, the parameter a n the Hull-Whte model equals 0.1, and σ equals 0.015. Double-barrer swaptons requre more tme than sngle-barrer opton n prcng because u d h and are calculated smultaneously. Besdes, the speed of h convergence s also slower than sngle-barrer opton because dr changes wth the tme t. But we can stll get very accurate answer wth as lttle as 200 steps. Exhbts 19 and 20 gve the prce for such a barrer opton, when contnuously observed, for dfferent numbers of steps n the trnomal tree. It can be seen that convergence s fast due to our specally constructed tree. Exhbt 19 shows the method gets an accurate answer n a short tme, and Exhbt 20 shows how fast and smoothly the method converges. 45
EXHIBIT 19 CONTINUOUSLY OBSERVED DOUBLE-BARRIER SWAPTIONS PRICES Product At-the-money double-barrer knockout swaptons on fve-year swap wth the prncpal 100 and fxed payments made annually Lower barrer Upper barrer Spot swap rate mnus 25 bass ponts Spot swap rate plus 200 bass ponts Opton maturty 0.5 year Steps Prces Tme (sec) 2 0.495313 0.015 5 0.475678 0.031 10 0.554802 0.062 30 0.569651 0.266 100 0.578415 0.847 200 0.580903 1.782 500 0.582516 4.734 1000 0.582964 10.281 46
EXHIBIT 20 CONVERGENCE OF CONTINUOUSLY OBSERVED DOUBLE-BARRIER SWAPTION PRICES More tme steps are stll needed for a swapton whose barrer s checked at dscrete ntervals. As can be seen from Exhbts 21 and 22, the speed of convergence s very good. The prces generally converge wth only 20 perods between observatons and are very close to Monte-Carlo results when our method s used wth only 50 perods between observatons. 47
EXHIBIT 21 DISCRETELY OBSERVED DOUBLE-BARRIER SWAPTION PRICES Observaton Frequency Sem-annually (Tme (sec)) Quarterly (Tme (sec)) Monthly (Tme (sec)) Weekly (Tme (sec)) Daly (Tme (sec)) Number of perods between observatons 2 5 10 20 50 100 1.090210 1.155344 1.122846 1.121167 (0.016) (0.047) (0.075) (0.188) 1.166872 1.098719 1.091995 1.087980 (0.016) (0.094) (0.172) (0.344) 0.987741 0.963259 0.953694 0.951150 (0.110) (0.266) (0.532) (1.062) 0.818323 0.798535 0.792114 0.788520 (0.432) (1.125) (2.328) (4.732) 0.704829 0.690894 0.686753 0.684701 (2.159) (7.015) (15.487) (36.979) 1.113194 (0.419) 1.115203 (0.875) 1.083037 1.081246 (0.875) (1.766) 0.948474 0.947544 (2.345) (5.127) 0.785881 0.785205 (13.672) (38.86) 0.683409 0.682989 (48.312) (432.563) Monte Carlo 1.112772 (1134.31) 1.080515 (1124.674) 0.946642 (937.118) 0.784679 (1291.615) 0.682474 (1189.254) 48
EXHIBIT 22 CONVERGENCE OF DISCRETELY OBSERVED DOUBLE-BARRIER SWAPTION PRICES 49
Chapter 5 Conclusons We have descrbed computng procedures that mplement the barrer methodology n Cheuk and Vorst [1996a] to value sngle-barrer swaptons under the Hull-Whte nterest rate model. We have also appled the same dea to prce sngle-barrer bond optons. The prces of dscretely observed sngle-barrer swaptons and bond optons are very close to those computed by the Monte-Carlo method, and the rate of convergence of our method s excellent. A second tme-dependent barrer can be accommodated usng the parameter dr and two barrers are completely matched. The prce of dscretely observed double-barrer swaptons computed by our method s close to that derved by the Monte-Carlo method, and the rate of convergence of our method s also excellent. Both contnuously and dscretely observed barrers have been consdered. It s no surprse accuracy s hghly senstve to the number of perods n tree n the contnuously observed case but not n the dscretely observed case. Our results show that the observaton frequency s a very mportant determnant of barrer optons prces. We have presented a novel dea that can deal wth all knds of nterest rate optons wth barrers. The extensve applcablty of our methodology makes t extremely useful n practce. 50
Bblography [1] Cheuk, T.H.F., and T.C.F. Vorst, Breakng Down Barrers, RISK (Aprl 1996a), pp. 64 67. [2] Cheuk, T.H.F., and T.C.F. Vorst, Complex Barrer Optons, Journal of Dervatves (Fall 1996b), pp. 8 22. [3] Hull, J., and A. Whte, Usng Hull-Whte Interest Rate Trees, Journal of Dervatves (Sprng 1996), pp. 26 36. [4] Hull, J., and A. Whte, One-Factor Interest-Rate Models and the Valuaton of Interest-Rate Dervatve Securtes, Journal of Dervatves (June 1993), pp. 235 254. [5] Hull, J., Optons, Futures, and Other Dervatves, Englewood Clff, NJ: Prentce Hall, 2003. [6] Sorwar, G. and Barone-Ades, G., Interest Rate Barrer Optons, Kluwer Appled Optmzaton Seres, (2003). [7] Smona Svoboda, Interest Rate Modelng, London: Antony Rowe, 2004. 51
APPENDIX A Calculatng Tme-Dependent Level θ () t In ths appendx, t s assumed that the tree has been constructed up to tme Δt and t s shown how θ (Δ t) s obtaned. Defne Q(, j ) as the value of a securty that pays off $1 f node (, j) s reached and zero otherwse. It s assumed that Q(, j where ) s calculated as the tree s beng constructed usng the relatonshp: * * * r ( 1, j) t Q(, j) = Q( 1, j ) q( j, j) e Δ, * j ( * * q j, j) s the probablty of movng from node ( 1, j ) to node (, j). The value as seen at node (, j ) of a bond maturng at tme ( 2) (, ) ( 1) () = ( ) rjδt r+ Δt e E e r r, j + Δt s where E s the rsk-neutral expectatons operator and r( ) s the value of nstantaneous rate at tme tme ( + 2) Δt Δt s therefore gven by:. The value at tme zero of a dscount bond maturng at ( 2) ( 2 ) (, ) ( 1) e + y+ Δt = Q(, j) e rjδt E e r+ Δt r() = r(, j). j The value of ( 1) () ( ) r+ Δt E e r = r, j s known analytcally: so that ths leads to ( ) () ( ) = = r 1 t rj, t E e + Δ r r, j e Δ e θ ( ) ( 2 ( Δ t) + ar(, j) + V /2) Δ t 1 V 1 θ t yeld Q j e ( ) = ( + 2) ( + 2) + + log (, ) 2 Δt 2 Δt j ( ) ( ) 2 r, j Δ t+ ar, j Δt 2 P ( j ) ( ) and,, P, j, (, are calculated wth u m P j) θ ( Δ t). Therefore θ ( Δ t) d for 0 n can be determned recursvely wthq(, j) and 52
u ( 1, j) ( ) ( ) P, P 1, j, P 1, j. m d 53