HULL-WHITE ONE FACTOR MODEL: RESULTS AND IMPLEMENTATION

HULL-WHITE ONE FACTOR MODEL: RESULTS AND IMPLEMENTATION QUANTITATIVE RESEARCH Absrac. Deails regarding he implemenaion of he Hull-Whie one facor model are provided. The deails concern he model descripion and parameers, he vanilla insrumens pricing and he Mone Carlo implemenaion. 1. Inroducion This documen provides a brief descripion of he Hull-Whie / exended Vasicek model Hull and Whie [1990] and possible implemenaions. A general overview of he model can be found in Brigo and Mercurio [2006]. When a specific volailiy funcion is required, a piecewise consan volailiy and consan mean reversion is used. The documen provides he resuls necessary for he implemenaion, he heoreical developmens are no provided. The reader is referred o he original papers and books menioned in he ex for he heoreical jusificaions. The noaions of he muli-curves framework are he one of Henrard [2010a]. 2. Ibor and Swaps Le P D, u be he discouning curve. Given he forward curve P j, u, he Ibor fixing in 0 for he period [ 1, 2 ] is 1 I j 0 1, 2 = 1 P j 0, 1 δ P j 0, 2 1 The forward Ibor rae is defined in by 2 F j 0, 1 = 1 δ P j, 0 P j, 1 1 where δ is he accrual fracion beween 0 and 1 in he day coun fracion associaed o he Ibor rae. An swap IRS exchanges a se of n fixed cash-flows agains floaing Ibor raes paid in heir naural day coun convenion on heir naural periods. An IRS is described by a se of fixed coupons or cash flows c i a daes i 1 i ñ. For hose flows, he discouning is used. I also conains a se of floaing coupons over he periods [ i 1, i ] wih i = i 1 + j 1 i n 1. The value of a fixed rae receiver IRS is 3 ñ c i P D, i P P D j, i 1, i P j 1., i Dae: Firs version: 1 July 2011; his version: 9 May 2012. Version 1.1. 1 In pracice, due o weekends and holidays, he periods used for he fixings can be slighly differen from he paymen daes. We will no make ha disincion here. 1

2 QUANTITATIVE RESEARCH Le δ i be he accrual fracion associaed o he fixed coupons paid in i. The level or presen value of a basis poin of he swap is given by PVBP = δ i P D, i. The forward swap rae is given by n P D, i P j, i 1 P S = j, i 1. PVBP We define 4 β j u, u + j = P j, u P j, u + j P D, u + j P D., u Wih ha definiion, a floaing coupon price is P P D j, i 1, i P j 1 = β j i 1, i P D, i 1 P D, i., i Throughou his noe we assume he consan spread hypohesis S0 of Henrard [2010a]. The swap is ofen represened by is cash-flow equivalen i, d i,...,n. The dae 0 is he selemen dae and i,...,m he paymen daes fixed and floaing legs. The cash flow amouns are c 0 = β j 0, 1, d i i = 1,..., n 1 he coupons including he adjusmens for floaing coupons and d n = 1 + cñ he final coupon plus 1 for he noional. Noe ha he swap and Ibor raes can be compued using only he re-based discoun facors in any numeraire as boh involve only raios. 3. Model A erm srucure model describes he behavior of P D, u, he price in of he zero-coupon bond paying 1 in u 0, u T. When he discoun curve P D,. is absoluely coninuous, which is somehing ha is always he case in pracice as he curve is consruced by some kind of inerpolaion, here exiss f, u such ha 5 P D, u = exp u f, sds. The shor rae associaed o he curve is r 0 T wih r = f,. The cash-accoun numeraire is N = exp r s ds. 0 3.1. Shor rae model. A good reference for he descripion of he shor rae approach is [Brigo and Mercurio, 2006, Secion 3.3]. The book descripion refers o he case of he consan volailiy model. The sochasic one facor equaion for he shor rae is, in he cash-accoun numeraire, 6 dr = θ ar d + ηdw. When he numeraire is changed from he cash-accoun o P D, u, he equaion is 7 dr = θ ar ην, u d + ηdw where ν is defined below.

HULL-WHITE: RESULTS AND IMPLEMENTATION 3 3.2. Heah-Jarrow-Moron. The idea of Heah e al. [1992] was o model f wih a sochasic differenial equaion df, u = µ, ud + σ, udw for some suiable µ and σ and deducing he behavior of P D from here. To ensure he arbirage-free propery of he model, a relaionship beween he drif and he volailiy is required. The model echnical deails can be found in he original paper or in he chaper Dynamical erm srucure model of Hun and Kennedy [2004]. To simplify he wriing, he noaion ν, u = u σ, sds is used. The equaions of he model in he cash-accoun numeraire measure associaed o N are df, u = σ, uν, ud + σ, udw. The following separabiliy hypohesis will be used: H: The funcion σ is deerminisic and saisfies σs, = gsh for some posiive funcions g and h. The Hull and Whie [1990] volailiy model saisfies he condiion H wih νs, = 1 exp a sηs/a and σs, = ηs exp a s. The model is analyzed wih a piecewise consan volailiy. By his we mean ha here exiss 0 = τ 0 < τ 1 < < τ n = + such ha ηs = η i for τ i 1 s τ i. 4. Preliminary resuls The forward volailiy of a re-based zero-coupon bond is he posiive number defined by αθ 0, θ 1, u, v 2 = θ1 θ 0 νs, v νs, u 2 ds. The expiry daes are beween some of he daes defining he piecewise consan funcion. The daes are denoed τ p θ 0 < θ 1 τ q. To shoren he noaion an inermediary noaion is used: r p = θ 0 < r l = τ l < r q = θ 1. Wih hose noaions, one has αθ 0, θ 1, u, v = exp au exp av 1 q 1 2a 3 ηl 2 exp2ar l+1 exp2ar l. Lemma 1. Le 0 s u, v. In a HJM one facor model, he price of he zero coupon bond can be wrien in he P D., u numeraire as P D s, v 8 P D s, u = P D, v P D, u exp α, s, u, vx s, 1 2 α2, s, u, v for a sandard normally disribued random variable X s, independen of F. Noe ha, hanks o he separabiliy hypohesis, he variable X s, is he same for all mauriies v. To speed-up Mone-Carlo simulaions on several periods, i is useful o decompose he volailiy ino a mauriy dependen par and an expiry par. Le gs = ηs expas and h = expa. We define Hu = u hd = expau/a. The expiry dependen par is 0 γθ 0, θ 1 = θ1 θ 0 g 2 sds = 1 2a The volailiy is αθ 0, θ 1, u, v = γθ 0, θ 1 Hv Hu. l=p q 1 ηl 2 exp2ar l+1 exp2ar l. l=p

4 QUANTITATIVE RESEARCH as Le s 0 = 0 < s 1 < s i < and Z i = s i s i 1 gsdw s. The sochasic variables can be wrien α0, s i, u, vx 0,si = Hv HuY i where Y i = i j=1 Z j. The random variables Y i are such ha Y i = Y i 1 + Z i wih Z i independen normally disribued wih variance γτ i 1, τ i. 5. Cap/floor The Cap and Floors are a se of caple/floorle. Each caple is a zero-coupon bond on is period. The price of zero-coupon opion is described in [Brigo and Mercurio, 2006, Secion 3.3.2]. I can also be deduced from he coupon bond formula wih n = 1. In his case he exercise boundary κ is explici. In his secion he price is described in he muli-curves framework under he consan spread hypohesis S0. The expiry dae is denoed θ. The sar and end dae of he rae are 0 and 1 and he fixing accrual facor is δ I. The srike is denoed K, he paymen dae is p and he paymen accrual facor is δ p. For sandard caple, he paymen dae is equal o he end of fixing period p = 1. For non-sandard caple in arrear, shor/long enor he daes can be significanly differen. The volailiy used is α i = α0, θ, p, i i = 1, 2. Theorem 1 Explici cap/floor formula in Hull-Whie. In he exended Vasicek model, he price of a cap wih srike K is given a ime 0 by δ p P P D j 0, 0 0, p δ I P j 0, 1 N κ α 0 1 + δ I KN κ α 1 where κ is given by 1 κ = α 1 α 0 ln 1 + δi KP j 0, 1 P j 0, 0 1 2 α2 1 α0 2. The price of a floor is given by δ p P D 0, p 1 + δ I KNκ + α 1 P j 0, 0 δ I P j 0, 1 Nκ + α 0. Noe ha book formulas can be slighly differen as hey usually do no consider he difference beween expiry dae θ and sar dae 0 and he difference beween paymen dae p and end of fixing period 1. 6. European Swapions 6.1. Jamshidian rick or decomposiion. The sandard pricing formula for physical delivery swapion in he model uses he Jamshidian decomposiion proposed in Jamshidian [1989]. The deails are also available in [Brigo and Mercurio, 2006, Secion 3.11.1]. As his is no he mos efficien implemenaion, we don deail i here. 6.2. Explici formula: physical delivery swapion. The formula was iniially proposed in Henrard [2003]. I is adaped here for he muli-curve framework under he deerminisic spread hypohesis S0. Theorem 2 Exac swapion price in Hull-Whie model. Suppose we work in he HJM onefacor model wih a separable volailiy erm saisfying H and in he muli-curves framework wih hypohesis S0. Le θ 0 < < n, c 0 < 0 and c i 0 1 i n. The price of an European receiver swapion wih physical delivery, wih expiry θ on a swap wih cash-flows represenaion i, d i is given a ime by he F -measurable random variable d i P D, i Nκ + α i

HULL-WHITE: RESULTS AND IMPLEMENTATION 5 where κ is he F -measurable random variable defined as he unique soluion of 9 d i P D, i exp 1 2 α i 2 α i κ = 0 and The price of he payer swapion is α i = α, θ, θ, i. d i P D, i N κ α i Noe ha he original proof required ha d i > 0 1 i n while here we have coupon equivalen like 1 β j wih usually β j > 1; some of he coupon equivalen are poenially slighly negaive. 6.3. Approximaed formula: physical delivery swapion. The resuls of his secion are from Henrard [2009]. The forward values of he zero-coupon bonds and swap wihou he iniial noional are given in by P i = P D n, i P D and B = d ip D, i, 0 P D., 0 Those value are he value re-based by he numeraire P., 0 Le ν i = ν, i ν, 0 0 i n. In he maringale probabiliy associaed o he numeraire P D., 0, he re-based prices are maringale ha saisfy he equaions dp i = P i ν i dw 0. The zero-coupon bond prices are exacly log-normal as he volailiy ν i is deerminisic. The re-based swap value saisfy db = c i P i ν i dw 0. Using he noaion α i = c i P i /B and σ = α i ν i, he equaion becomes db = B σ dw 0. This is formally a log-normal equaion bu he σ coefficien is sae dependen. A srike value of he differen parameers is seleced. The swap price is a-he-money a expiry when c i Pθ i = 0 The discouning value of he zero-coupon bond can be approximaed iniial freeze by Pθ i = P D 0, i P D 0, 0 exp τ i X i 1 2 τ i 2 wih τ 2 i = θ 0 ν i 0 2 ds = α 2 0, θ, i, 0 and he X i sandard normally disribued random variables. By choosing arbirarily o have all he sochasic variables X i equal a he srike he one dimensional equaion o solve is c i P0 i exp τ i x 1 2 τ i 2 = 0. Obaining he soluion o he above equaion requires o solve a one dimensional equaion equivalen o he one solved in he swapion price see above. For numerical reasons one may prefers no

6 QUANTITATIVE RESEARCH o have o solve his ype of equaion. The above equaion can be replaced by i firs order approximaion c i P0 i 1 τ i x 1 2 τ i 2, he soluion of which is explici: n x = c ip0 i 1 n 2 c ip0τ i i 2 n c ip0 iτ. i The zero-coupon prices, in he exponenial case and he approximaed firs order case, are given by PK i = P0 i exp τ i x 1 2 τ i 2, respecively PK i = P0 i 1 τ i x 1 2 τ i 2. The raes and bond prices are By defining i 1 1 + δ j L j K = P K i 1 and B K = j=0 c i PK i = K. αk i = c ipk i B K he swapion can be priced wih a opion srike dependen approximaed volailiy 10 σ K = 1 n α0 i + α i 2 Kν i. Noe ha in he muli-facor model, he volailiy σ K is a vecor, as ν 0 and ν K are. Theorem 3 Approximaed swapion price in Hull-Whie model. In he exended Vasicek model, he price, wih iniial freeze and correcor approximaion, of a receiver swapion is given a ime 0 by R 0 = P D 0, 0 B 0 Nκ K + σ K KNκ K where and The price of a payer swapion is κ K = 1 ln B 0 /K 1 σ K 2 σ2 K σ 2 K = θ 0 σ K 2 d. P 0 = P D 0, 0 KN κ K B 0 N κ K σ K 6.4. Approximaed formula: cash-seled swapion. An efficien approximaed formula for cash-seled swapions is proposed in [Henrard, 2010b, Appendix A]. 7. ineres Rae Fuures A general pricing formula for eurodollar fuures in he Gaussian HJM model was proposed in Henrard [2005]. The formula exended a previous resul proposed in Kirikos and Novak [1997]. The exension o he muli-curve framework was proposed in Henrard [2010a] and is reproduced here. The fuures are liquid only for he hree monh Ibor up o wo or hree years. To a lesser exen some one monh fuures are available on he shorer par of he curve.

HULL-WHITE: RESULTS AND IMPLEMENTATION 7 The fuure fixing dae is denoed 0. The fixing is on he Libor rae beween 1 = Spo 0 and 2 = 1 + j. The accrual facor for he period [ 1, 2 ] is δ, he fixing is linked o he yield curve by 1 + δl j 0 = P j 0, 1 P j 0, 2. The fuures price is Φ j. On he fixing dae, he relaion beween he price and he rae is Φ j 0 = 1 L j 0. The fuures margining is done on he fuures price muliplied by he noional and divided by 4. Theorem 4. Le 0 0 1 2. In he HJM one-facor model on he discoun curve under he hypoheses D, L and SI, he price of he fuures fixing on 0 for he period [ 1, 2 ] wih accrual facor δ is given by Φ j = 1 1 P j, 1 11 δ P j, 2 γ 1 where = 1 γf j + 1 1 γ δ 0 γ = exp νs, 2 νs, 2 νs, 1 ds. Proof. Using he generic pricing fuure price process heorem [Hun and Kennedy, 2004, Theorem 12.6], [ ] Φ j = E N 1 L j 0 F. In L j 0, he only non-consan par is he raio of j-discoun facors which is, up o β j 0, he raio of D-discoun facors. Using [Henrard, 2005, Lemma 1] wice, we obain, wih W s he Brownian moion associaed o he N-numeraire, P D 0, 1 P D 0, 2 = P D, 1 P D, 2 exp + 0 1 0 ν 2 s, 1 ν 2 s, 2 ds 2 νs, 1 νs, 2 dw s. Only he second inegral conains a sochasic par. This inegral is normally disribued wih variance 0 νs, 1 νs, 2 2 ds. So he expeced discoun facors raio value is reduced o P D, 1 P D, 2 exp + 1 2 0 0 ν 2 s, 1 ν 2 s, 2 ds νs, 1 νs, 2 2 ds. The coefficien β j 0 is independen of he D-discoun facors raio and a maringale, hence we have he announced resul. 8. Oher insrumens 8.1. Bermudan swapions. The pricing of Bermudan swapions in he one-facor Gaussian HJM model using an ierae numerical inegraion procedure is presened in Henrard [2008b]. 8.2. CMS and CMS cap/floor. The pricing of CMS and CMS cap/floor in he one-facor Gaussian HJM model using approximaions is presened in Henrard [2008a].

8 QUANTITATIVE RESEARCH 9. Mone Carlo Suppose we work in he P D., u numeraire. The numeraire choice should be adaped for each produc see below. Le s i,...,nf wih s 0 = 0 be he fixing and expiry daes used in he pricing. The fixing daes can be relaed o Ibor or swap CMS raes. The daes used in s i for paymens and o compue he fixings are denoed i,j,...,nf ;j=0,...,n R i. Some examples are given below. The paymen daes are i,j wih 0 j n P i and he daes used for he fixing compuaion are i,j wih n P i + 1 j n R i. The re-based discoun facor iniial values are denoed P F 0,i,j = P D 0, i,j P D 0, u. The re-based discoun facors used in he pricing are, for 0 i n F, i j n F and 0 k n R i P F i,j,k = P D s i, j,k P D s i, u. A each fixing daes s i, amouns o be paid in i,j 0 j n P i are C i,j = F P F l,l,k 0 l i;np l+1 k n R l. The amouns will be made explici for each specific produc see below. In general, for each i, here is a limied number of paymen daes one or wo. Also he amoun ofen depend direcly only on Pi,i,k F ; he dependence on he previous raes is indirec rough C i 1,.. In he Mone Carlo approach, he price on one pah of he above opion is compued as P D 0, u Pi,i,jC F i,j.,...,n 0 j n P i 9.1. HJM jump beween fixing daes. The values Pi,i,j F are simulaed from heir iniial values P0,i,j F and repeaed HJM evoluion 8. For he evoluion, one needs o compue he quaniies αs l 1, s l, u, i,j. Those quaniies need o be compued only once and no for each pah. The one sep evoluion is Pl,i,k F = Pl 1,i,k F exp 12 α2 s l 1, s l, u, i,k αs l 1, s l, u, i,k X l. For one pah, he X l 1 l nf are independen sandard normally disribued draws. Even if only he final value Pi,i,k F F is used, i is imporan o esimae Pl,i,k from X l o have a coherence in he pahs. The final value can be compued as Pi,i,k F = P0,i,k F exp 1 i i α 2 s l 1, s l, u, i,k αs l 1, s l, u, i,k X l. 2 l=1 This las expression is faser o implemen han he recursive one above less exponenial and muliplicaion o compue. The simulaion is done wih long jumps, i.e. he seps are from one fixing dae o he nex, wihou exra ime discreizaion. 9.2. HJM one very long jump o each fixing dae. The idea is ha each discouning facor Pi,i,k F is compued direcly from is iniial value and one random variable Y i by Pi,i,k F = P0,i,j F exp H i,j Y i 1 2 H2 i,jγ0, s i wih H i,j = H i,j Hu and Y i = Y i 1 + Z i as described in Secion 4. l=1

HULL-WHITE: RESULTS AND IMPLEMENTATION 9 The variables Y i are no independen; hey can be viewed as equal on some ime inerval wih one of hem coninuing afer. The covariance beween Y i and Y j wih i j is he variance of Y i and is i sk si g 2 sds = g 2 sds = γ0, s i. s k 1 0 k=1 As he variables are no independen anymore, we simulae independen variables and muliply hem by he Cholesky decomposiion of he covariance marix. 9.3. European swapions. For he European swapion, here is only one fixing dae, he expiry dae: s 1 = θ. Le d i, i 0 i n be he cash flow equivalen of he underlying swap as described in Secion 2. Le 1,j = 1,n+1+j = j 0 j n, n R 1 = 2n and n P 1 = n. Wih ha noaion, he payoff of he European swapion is n P 1 d j P1,1,j F or in he Mone-Carlo noaions, n P 1 C 1,j = d j 1 d j P1,1,j F > 0. j=1 j=1 As he cash flows are regularly disribued over he period almos -1 in 0 and 1 + c n in n, he numeraire choice has a very small impac. We sugges o use u = 0. +, 9.4. CMS and CMS cap/floor. The descripion is done for only one paymen. There is only one fixing dae s 1 and one paymen dae 1,0. Le d i, i 0 i n be he cash flow equivalen of he underlying swap as described in Secion 2. Le 1,j+1 = j 0 j n, n R 1 = n + 1 and n P 1 = 1. For a cap, ω = 1 and for a floor ω = 1. Wih ha noaion, he payoff of he CMS cap/floor is in 1,0, C 1,0 = ωs s1 K +. There is only one cashflow, paid in 1,0 ; we sugges o use u = 1,0. 9.5. Rache on Ibor. The rache is described by a se of fixing daes s i 1 i n F. A each fixing dae a Ibor rae is recorded. To each fixing dae a paymen dae i,0 is associaed and he fixing rae depends on wo daes i,j 1 j 2. The Ibor rae I si is given by Equaion 1. To each coupon are associaed hree coefficiens: a memory αi M, a muliplicaive coefficien βi M and an addiive erm γi M. The amoun paid is he coupon muliplied by he accrual fracion and he noional. For a floaing firs coupon, he amoun is C 1,0 = β M 1 I s1 + γ M 1 and for a fixed firs coupon c he amoun is C 1,0 = c. From he second coupon, o each coupon is associaed an upper barrier cap wih memory αi C, muliplicaive coefficien βc i and an addiive erm γi C and a down barrier floor wih memory αi F, muliplicaive coefficien βf i, and addiive erm γi F. The coupon, paid in i,0, is given by C i,0 = max α F i C i 1,0 + β F i I si + γ F i, min α C i C i 1,0 + β C i I si + γ C i, α M i C i 1,0 + β M i I si + γ M i.

10 QUANTITATIVE RESEARCH 10. Implemenaion The one-facor Hull-Whie model parameers wih consan mean reversion and piecewise consan volailiy are available from he objec HullWhieOneFacorPiecewiseConsanParameers. The differen funcions relaed o he model α, γ, H, ec. are available in he class HullWhieOneFacorPiecewiseConsanIneresRaeModel. The pricing of Ibor cap/floor is available in CapFloorIborHullWhieMehod. The pricing of ineres rae fuures is available in IneresRaeFuureHullWhieMehod. The pricing of European physical delivery swapions wih he exac formula is available in SwapionPhysicalFixedIborHullWhieMehod and he approximaed formula in SwapionPhysicalFixedIborHullWhieApproximaionMehod. The pricing is also available wih a numerical inegraion in SwapionPhysicalFixedIborHullWhieNumericalInegraionMehod. This las mehod is more for esing and analysis han for producion; he exac formula is faser and more precise. The pricing of European cash-seled swapions by approximaion is available in SwapionCashFixedIborHullWhieApproximaionMehod. The numerical inegraion approach is available in SwapionCashFixedIborHullWhieNumericalInegraionMehod. The pricing of Bermuda swapions is available in SwapionBermudaFixedIborHullWhieNumericalInegraionMehod The pricing of CMS coupons is available in CouponCMSHullWhieApproximaionMehod; he numerical inegraion approach is available in CouponCMSHullWhieNumericalInegraionMehod. The pricing of CMS cap/floor by approximaion is available in CapFloorCMSHullWhieApproximaionMehod; he numerical inegraion approach is available in CapFloorCMSHullWhieNumericalInegraionMehod. The pricing wih Mone Carlo is available in HullWhieMoneCarloMehod and MoneCarloDiscounFacorCalculaor. References D. Brigo and F. Mercurio. Ineres Rae Models, Theory and Pracice. Springer Finance. Springer, second ediion, 2006. 1, 2, 4 D. Heah, R. Jarrow, and A. Moron. Bond pricing and he erm srucure of ineres raes: a new mehodology for coningen claims valuaion. Economerica, 601:77 105, January 1992. 3 M. Henrard. Explici bond opion and swapion formula in Heah-Jarrow-Moron one-facor model. Inernaional Journal of Theoreical and Applied Finance, 61:57 72, February 2003. 4 M. Henrard. Eurodollar fuures and opions: Convexiy adjusmen in HJM one-facor model. Working paper 682343, SSRN, March 2005. Available a ssrn.com/absrac=682343. 6, 7 M. Henrard. CMS swaps and caps in one-facor Gaussian models. Working Paper 985551, SSRN, February 2008a. URL ssrn.com/absrac=985551. Available a ssrn.com/absrac=985551. 7 M. Henrard. Bermudan swapions in Gaussian HJM one-facor model: Analyical and numerical approaches. Working Paper 1287982, SSRN, 2008b. URL ssrn.com/absrac=1287982. Available a ssrn.com/absrac=1287982. 7 M. Henrard. Efficien swapions price in Hull-Whie one facor model. Quaniaive finance, arxiv, 2009. URL arxiv.org/abs/0901.1776. Available a arxiv.org/abs/0901.1776. 5 M. Henrard. The irony in he derivaives discouning - Par II: he crisis. Wilmo Journal, 26: 301 316, December 2010a. 1, 2, 6 M. Henrard. Cash-seled swapions: How wrong are we? Technical repor, OpenGamma, 2010b. URL ssrn.com/absrac=1703846. Available a ssrn.com/absrac=1703846. 6 J. Hull and A. Whie. Pricing ineres rae derivaives securiies. The Review of Financial Sudies, 3:573 592, 1990. 1, 3 P. J. Hun and J. E. Kennedy. Financial Derivaives in Theory and Pracice. Wiley series in probabiliy and saisics. Wiley, second ediion, 2004. ISBN 0-470-86359-5. 3, 7

HULL-WHITE: RESULTS AND IMPLEMENTATION 11 F. Jamshidian. An exac bond opion formula. The journal of Finance, XLIV1:205 209, march 1989. 4 G. Kirikos and D. Novak. Convexiy conundrums. Risk, pages 60 61, March 1997. 6 Conens 1. Inroducion 1 2. Ibor and Swaps 1 3. Model 2 3.1. Shor rae model 2 3.2. Heah-Jarrow-Moron 3 4. Preliminary resuls 3 5. Cap/floor 4 6. European Swapions 4 6.1. Jamshidian rick or decomposiion 4 6.2. Explici formula: physical delivery swapion 4 6.3. Approximaed formula: physical delivery swapion 5 6.4. Approximaed formula: cash-seled swapion 6 7. ineres Rae Fuures 6 8. Oher insrumens 7 8.1. Bermudan swapions 7 8.2. CMS and CMS cap/floor 7 9. Mone Carlo 8 9.1. HJM jump beween fixing daes 8 9.2. HJM one very long jump o each fixing dae 8 9.3. European swapions 9 9.4. CMS and CMS cap/floor 9 9.5. Rache on Ibor 9 10. Implemenaion 10 References 10 Marc Henrard, Quaniaive Research, OpenGamma E-mail address: marc@opengamma.com