Inernaional Scholarly Research Nework ISRN Probabiliy and Saisics Volume 212, Aricle ID 67367, 16 pages doi:1.542/212/67367 Research Aricle A General Gaussian Ineres Rae Model Consisen wih he Curren Term Srucure Marco Di Francesco Unipol Assicurazioni, via Salingrado 45, Bologna, Ialy Correspondence should be addressed o Marco Di Francesco, marco.difrancesco@unipolassicurazioni.i Received 25 April 212; Acceped 29 July 212 Academic Ediors: J. Abellan and P. E. Jorgensen Copyrigh q 212 Marco Di Francesco. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. We describe an exension of Gaussian ineres rae models sudied in lieraure. In our model, he insananeous spo rae r is he sum of several correlaed sochasic processes plus a deerminisic funcion. We assume ha each of hese processes has a Gaussian disribuion wih ime-dependen volailiy. The deerminisic funcion is given by an exac fiing o observed erm srucure. We es he model hrough various numeric experimens abou he goodness of fi o European swapions prices quoed in he marke. We also show some criical issues on calibraion of he model o he marke daa afer he credi crisis of 27. 1. Inroducion A shor-rae model for he erm srucure of ineres raes is based on he assumpion of a specific dynamics for he insananeous spo-rae process r for he definiion of r, we refer for insance o he monographs 1 3. These models were he firs approach o describe and explain he shape and he moves of he erm srucure of ineres raes. Moreover, hese models are very convenien since he dynamics of he insananeous spo rae drives all he erm srucure, in he sense ha boh raes and prices of bonds are defined as an expecaion of a funcional of r. Indeed, he exisence of a risk-neural measure Q implies ha he price a ime of a coningen claim wih payoff H T a ime T>is given by [ H E e r s ds H T ], 1.1 where E denoes he ime -condiional expecaion under he measure Q. In his paper, we describe all processes under he risk-neural measure Q for he definiion and he exisence of Q, we refer o he monographs 1 4.
2 ISRN Probabiliy and Saisics A full descripion of he main shor-rae models can be found in 1 3, 5. Here, we only recall ha hese models belong o wo general classes: he endogenous ones, in which he observed erm srucure is an oupu, and exogenous model, in which he observed erm srucure is an inpu. A basic sraegy o ransform an endogenous model o an exogenous one is he inclusion of ime-dependen parameers o fi exacly he observed erm srucure. In fac, maching exacly he erm srucure is equivalen o solving a sysem wih an infinie number of equaions, and his is possible only afer inroducing an infinie number of parameers or, equivalenly, a deerminisic funcion of ime. In his paper, we describe a general exogenous model in which he insananeous spo rae r is he sum of several correlaed Gaussian sochasic processes wih ime-dependen volailiy plus a deerminisic funcion which allows an exac fiing o he observed erm srucure. We derive an explici formula for discoun bond, and we recall a formula o approximae a European swapion price proposed by Schrager and Pelsser in 6, ha can be used o calibrae he model o he marke daa. To do ha, we also derive he dynamics of he r under he T forward measure, ha is, he measure associaed wih he zero coupon bond mauring a ime T see 1 4. In lieraure, a model in which he insananeous spo rae is given by he sum of wo correlaed Gaussian processes plus a deerminisic funcion is known as he G2 model. This model wih consan parameers is widely sudied in 2. Moreover, he G2 model is equivalen o he well-known wo-facor Hull-Whie model see for insance 5, 7, 8. We also noe ha one Gaussian process plus a deerminisic funcion is equivalen o he wellknown Hull-Whie model see 5, 7, 9. In his paper, we propose a common noaion for he general case in which here are n correlaed Gaussian processes wih ime-dependen volailiy. In order o use he same noaion as in 2, wewillreferohismodelashegn model. The aim of his paper is no only o provide a common noaion o unify mulivariae Gaussian models wih an exac fi o he observed erm srucure. Wih several numerical examples, we will show ha he G1 and G2 models wih consan parameers worked well before Augus 27, he beginning of he so-called credi-crunch crisis. Then, Augus 27 arrived and one of he several consequences of he liquidiy and credi crisis was an explosion of he swapions volailiy, specially for swapions wih shor enor and mauriy. Since Augus 27, he swapions volailiy surface has become very unsmooh. A he money ATM swapions volailiy surface a 31/12/26 and 31/12/211 are, respecively, shown in Tables 1 and 2 value in percenage. Wih he volailiy as in Table 2,heG1 and G2 models wih consan parameers do no work well, in he sense ha i s impossible o have a good fi o he marke daa. To improve he calibraion o ATM swapions volailiy surface a 31/12/211, we increase he number of facors and ake a funcional specificaion for he volailiy of he processes. We es a calibraion o swapions prices proposed by Schrager and Pelsser in 6. In our experimen, o opimize parameers, we use he differenial evoluion algorihm as implemened in R hrough he package DEopim. In order o make he reader familiar wih he noaions, we sar, in Secion 2, wih one facor model and hen, in Secion 3, we describe he general n facor model. In Secion 4, we show our numerical resuls. Finally, in Secion 5, we summarize our resuls and discuss fuure developmens. As a final remark, we noe ha he Gaussian disribuion of he processes allows a good analyical racabiliy and he consrucion of efficien and fairly fas numerical procedures for pricing any ype of payoff. On he oher hand, he Gaussian disribuion of he processes leads
ISRN Probabiliy and Saisics 3 Table 1: ATM swapions volailiy a 31/12/26. T/Tenor 1 2 5 7 1 1 13.45 13.8 14.5 14.35 13.9 2 14.4 14.75 14.8 14.5 14.1 5 14.7 14.7 14.25 13.9 13.55 7 14.5 14 13.55 13.3 13.1 1 12.95 13 12.8 12.6 12.45 15 12 11.95 12 11.9 11.85 2 11.45 11.5 11.65 11.6 11.65 Table 2: ATM swapions volailiy a 31/12/211. T/Tenor 1 2 5 7 1 1 59.6 49 43.9 4.5 39.2 2 59.4 46.7 38.8 36.5 35.2 5 34.5 31.4 29.3 28.5 28.1 7 29.4 27.9 26.1 25.9 26.6 1 24.7 24.3 24.3 25 26.5 15 25.8 26.4 28 29 3.5 2 31.6 32.1 33.1 33.8 34 o he possibiliy of negaive raes and so he model is hardly applicable o some concree problems. 2. The G1++ Model In his secion, we assume ha he dynamics of he insananeous shor-rae process under he risk-neural measure Q is given by r x f, 2.1 where he process {x : } saisfies he sochasic differenial equaion dx ax d σ dw, x, 2.2 where a is a posiive consan, σ is a deerminisic funcion of ime ha is regular enough o ensure he exisence and uniqueness of a soluion, and W is a sandard Brownian moion. The funcion f is deerminisic and is given by an exac fiing o he erm srucure of discoun facor observed in he marke. Using he same noaion as in 2, we will refer o his model as he G1 model. Inegraing equaion 2.2, we have, for each s, r x s e a s s σ s e a u dw u f. 2.3
4 ISRN Probabiliy and Saisics Therefore, r condiional o F s,heσ-field represening he informaion available in he marke up o ime s see for insance 1, 2, 4 is normally disribued wih mean and variance given by E r F s x s e a s f, VAR r F s s σ 2 s e 2a u du, 2.4 where E and VAR denoe he mean and he variance under he measure Q, respecively. In he case σ σ is a posiive consan funcion, we have VAR r F s σ 2 s e 2a u du σ2 1 e 2a s ). 2.5 2a We denoe by P, T he price a ime of a zero coupon bond mauring a ime T wih uni nominal value. The exisence of he risk-neural measure Q implies ha P, T E [e r s ds F ]. 2.6 The model fis he curren observed erm srucure of he discoun facor if, for each mauriy T, he discoun facor P,T is equal o he one observed in he marke P M,T. To calculae P,T, we have o inegrae he process r over he inerval,t. Noice ha, since he process x is normally disribued condiional on F, hen also x d is iself normally disribued. Indeed, by Fubini s Theorem for sochasic inegral see 3, we have x d σ u e a u dw u d σ u e au e a ddw u u σ u 1 e a T u ) dw u. a 2.7 So, he inegral x d is normally disribued wih mean zero and variance [ ] VAR x d F σ 2 u a 2 1 e a T u ) 2 du V,T. 2.8 In he case σ σ is a posiive consan funcion, we have V,T σ2 T 2 1 e at a 2 a 1 ) e 2aT. 2.9 2a
ISRN Probabiliy and Saisics 5 Finally, recalling ha if Z is normal random variable wih mean μ Z and variance σ 2 Z, hen E e Z e μ Z 1/2 σ 2 Z, we have P,T E [e r d] E [e x f d] e f d E [e x d] e f d e 1/2 V,T. 2.1 Clearly, he model gives an exac fiing o he marke erm srucure of he discoun facors if and only if P M,T P,T 2.11 for every T>. By 2.1, we have P M, e f d e 1/2 V,T. 2.12 So, he model gives an exac fiing o he marke erm srucure of he discoun facors if and only if for every T> e f d P M,T e 1/2 V,T. 2.13 To have an explici expression of f, denoing by f M,T he marke insananeous forward rae, ha is, P M,T e f M, d. 2.14 We can wrie 2.12 e f M, d e f d e 1/2 V,T. 2.15 Then he exponens on boh sides of 2.15 agree, and by differeniaion we ge f T f M,T σ 2 u 1 e a T u ) e a T u du 2.16 a and so, in he case σ σ is a posiive consan funcion, he model perfec fis he marke erm srucure of he discoun facor if and only if for every T> f T f M,T σ2 2a 2 1 e at) 2. 2.17
6 ISRN Probabiliy and Saisics However, o calculae P, T for every <<T, we poin ou ha we dono need o compue he whole f curve. This is very imporan in order o implemen he model. In fac, arguing as in 2.7, we can prove ha x s ds x 1 e a T a σ u 1 e a T u ) dw u. 2.18 a In paricular, he inegral x s ds condiional on F is normally disribued wih mean and variance given by [ ] E x s ds F [ ] VAR x s ds F and, in he case σ σ is a posiive consan funcion x 1 e a T a σ 2 u a 2 1 e a T u ) 2 du V, T 2.19 V, T σ2 T 2 1 e a T 1 ) e 2a T. 2.2 a 2 a 2a So, we have P, T E [e ] r s ds E [e ] T x s f s ds e f s ds E [e ] T x s ds e f s ds e f s ds e 1 e a T /a x 1/2 V,T e ) by 2.13 2.21 P M,T P M, e 1/2 V,T V,T V, e 1 e a T /a x A, T e B,T x, where, A, T P M,T P M, e 1/2 V,T V,T V,, B, T 1 e a T. a 2.22 Expression 2.21 is very imporan in simulaion. We noe ha o generae P, T, we need only he marke discoun curve and he process x. FromP, T, we could derive all he oher raes forward raes, swap raes, ec..
ISRN Probabiliy and Saisics 7 In order o derive an explici formula of he price of an European swapion, we need o change he probabiliy measure as indicaed by Jamshidian in 1 and more generally by German e al. in 11. We denoe by Q T he T-forward measure, given by choosing a zero coupon bond wih mauriy T as numerarie. By he Girsanov s Theorem, he dynamics of he process x under he measure Q T is given by dx ax d σ 2 B, T d σdw T, 2.23 where W T is a sandard Brownian moion under Q T defined by dw T dw σ B, T d. Then, under he measure Q T, he disribuion of he process x is sill Gaussian; in paricular, for every s T, he mean and he variance of he process are given by E T x F s x s e a s M T s,, VAR T x F s s σ 2 u e 2a u du σ s,, 2.24 where E T and VAR T denoe, respecively, he mean and he variance under Q T,and M T, s s σ 2 a u 1 e a T u u e du. a 2.25 In he case σ σ is a posiive consan funcion, we have M T, s σ2 a 2 1 e a s ) σ2 2a 2 e a T e a T 2s ). 2.26 Now, we consider an European call swapion wih mauriy T, srike K, and nominal value N, which gives he holder he righ o ener a ime T ino a swap wih paymen daes { 1,..., k }, < 1,..., k, where he pays he fixed rae K and receives he Libor rae. Jamshidian in 1 showed how a European swapion can be decomposed ino a porfolio of opions on zero coupon bonds, deriving an exac closed formula for any European swapion. In paricular, he price ES,T,K,N a ime of he above swapion is given by ES,T, k,k,n NP,T E T [1 ) ] k c i P T, i F, 2.27 where c i Kτ n and c k 1 Kτ n, where τ i denoes he year fracion from i 1 o i, i 1,...,k. Then, by 2.21 we have ES,T, k,k,n NP,T 1 R k c i A T, i e i x) B T, g x dx, 2.28
8 ISRN Probabiliy and Saisics where g x denoes he densiy funcion of x T condiioned on F under Q T,hais, g x 1 2πσ2,T e x M,T 2 /2σ 2,T. 2.29 We also poin ou ha in he case σ σ is a posiive consan funcion, we have σ 2,T σ 2 /2a 1 e 2aT. Equaion 2.28 may be used in order o calibrae he model o he marke prices of quoed swapions. An analyic formula alernaive o 2.28 is given by ES,T, k,k,n NP,T Φ ) k y 1 c i A T, i e B T,i M,T e B2 T, i σ 2,T /2 Φ y 2 i )), 2.3 where Φ is he sandard normal cumulaive funcion, y 1 and y 2 i are given by y 1 y M,T, σ,t y 2 i y 1 B T, i σ,t, 2.31 where y is he unique soluion of k c i A T, i e B T,i y 1. 2.32 To conclude his secion, we noice ha he G1 model is perfecly analogous o he one proposed by Hull and Whie in 7 and known as he Hull-Whie model. 3. The Gn++ model We now exend he previous one facor model o he mulidimensional case. In his secion, we assume ha he dynamics of he insananeous shor-rae process under he risk-neural measure Q is given by r x i f, 3.1 where, for every i 1,...,n, he process {x i : } saisfies he sochasic differenial equaion dx i a i x i d σ i dw i, x i, 3.2 where a i is a posiive consan, σ i is a deerminisic funcion of ime ha is regular enough o ensure he exisence and uniqueness of a soluion, and W i is a sandard Brownian
ISRN Probabiliy and Saisics 9 moion. We denoe by ρ ij he insananeous correlaion beween W i and W j, for every >, i, j 1,...n, i/ j; wealsoseρ ii 1, for every i 1,...,n. We naurally assume ha 1 ρ ij 1, and he marix ρ ij i,j 1...,n is symmeric and posiive definie. The funcion f is deerminisic and is given by an exac fiing o he erm srucure of discoun facors observed in he marke. Using he same noaion as in 2, we will refer o his model as he Gn model. Inegraing equaion 3.2, we have, for each s, r x i s e ai s s σ i u e a i u dw i u f. 3.3 Therefore, r condiional o F s is normally disribued wih mean and variance given by E r F s x i s e ai s f, VAR r F s i,j 1 s σ i u σ j u ρ ij e a i u e a j u du, 3.4 where E and VAR denoe he mean and he variance under he measure Q, respecively. In he case ha for every i 1,...,n, σ i σ i is a posiive consan funcion, we have VAR r F s σ iσ j ) ρ ij 1 e a i a j s. 3.5 a i a j The model gives an exac fiing o he currenly observed erm srucure of he discoun facors if, for each mauriy T, he discoun facor P,T is equal o he one observed in he marke P M,T. To calculae P,T, we have o inegrae he process r over he inerval,t. Proceeding as in 2.7, we obain ha he inegral n x i d, condiional o F,is normally disribued wih mean zero and variance given by VAR [ ] x i d F ρ ij i,j 1 σ i u σ j u 1 e a i T u a i 1 e aj T u a j du V,T. 3.6 In he case ha, for every i 1,...,n, σ i σ i is a posiive consan funcion, we have V,T i,j 1 σ i σ j ρ ij T 1 e a it a i a j a i 1 e ajt a j 1 ) e ai aj T. 3.7 a i a j So, we have P,T E [e r d] E [e e f d E [e n x f d] n x d] e f d e 1/2 V,T. 3.8
1 ISRN Probabiliy and Saisics Clearly, he model gives an exac fiing o he marke erm srucure of he discoun facors if and only if P M,T P,T, for every T>, ha is, e f d P M,T e 1/2 V,T. 3.9 To have an explici soluion of f, denoing by f M,T he marke insananeous forward rae, we can wrie 3.9 e f M d e f d e 1/2 V,T. 3.1 Then, he exponens on boh sides of 3.1 agree, and by differeniaion we ge f T f M,T ρ ij i,j 1 σ i u σ j u 1 e a i T u ) e a i T u 1 e a j T u 1 ) e a j T u e a j T u 1 ) e a i T u du. a j a i 3.11 In he case, for every i 1,...,n, σ i σ i is a posiive consan funcion, we have f T f M,T 1 2 i,j 1 ρ ij σ i σ j a i a j 1 e a it ) 1 e a jt ). 3.12 However, as in he G1, o calculae P, T for every <<T, we poin ou ha we do no need o compue he whole f curve. This is very imporan in order o implemen he model. In fac, we can prove ha he inegral n x i s ds, condiional o F, is normally disribued wih mean and variance given by [ ] E x s ds F x i 1 e a i T a i [ ] 1 e a i T u 1 e aj T u VAR x s ds F ρ ij σ i u σ j u ρ ij du i,j 1 a i a j 3.13 V, T. In he case, for every i 1,...,n, σ i σ i is a posiive consan funcion, we have V, T i,j 1 σ i σ j ρ ij T 1 e a i T 1 e aj T 1 ) e ai aj T. 3.14 a i a j a i a j a i a j
ISRN Probabiliy and Saisics 11 So, we have P, T E [e ] T r s ds E [e e f s ds E [e n x i s ds ] n x i s f s ds ] e f s ds e f s ds e n 1 e a i T /a i x i 1/2 V,T e ) by 3.9 3.15 P M,T P M, e 1/2 V,T V,T V, e n 1 e a i T /a i x i A, T e n B i,t x i, where A, T P M,T P M, e 1/2 V,T V,T V,, B i, T 1 e a i T a i. 3.16 Expression 3.15 is paricularly imporan in simulaion. We noe ha o generae P, T,we need only he marke discoun curve and he processes x i, i 1,...,n.FromP, T, we could derive all he oher raes forward raes, swap raes, ec.. In order o derive an explici formula of he price of an European swapion, we need o change he probabiliy measure as indicaed by Jamshidian 1 and more generally by German e al. in 11. We denoe by Q T he T-forward measure, given by choosing a zero coupon bond wih mauriy T as numerarie. By he Girsanov s Theorem, for each i 1,...,n, he dynamics of he process x i under he measure Q T, is given by dx i a i x i d σ i n σ j ρ ij B i, T d σ i dw T i, j 1 3.17 where W T i is a sandard Brownian moion under Q T wih dw T i dw j ρ ij. Moreover, he explici soluion of 3.17 is, for s T x i x i s e a i s M T i s, σ i s e a i u dw T i u, 3.18 where M T i s, s σ i u j 1 σ j u a j ρ ij 1 e a i T u ) e a i u du, 3.19
12 ISRN Probabiliy and Saisics and, in he case ha, for every i 1,...,n, σ i σ i is a posiive consan funcion M T i s, σ i j 1 ) σ j 1 e a i s ρ ij e aj T e ajt ai ai aj s. 3.2 a j a i a i a j So, under Q T, he disribuion of r condiional o F s is normal wih mean and variance given by E T r F s x i s e ai s M T i s, f VAR T r F s σ i u ρ ij σ j u e ai u e aj u du. i,j 1 s 3.21 The price ES,T, k,k,n a ime of an European call swapion wih mauriy T, srikek and nominal value N is given by k ES,T, k,k,n NP,T E T 1 c j P ) T, j j 1 F, 3.22 where c j Kτ n and c k 1 Kτ n, where τ j denoes he year fracion from j 1 o j, j 1,...,k. Then, by 3.15 we have ES,T, k,k,n k NP,T 1 c j A ) n T, j e B i T, j x i g x 1,...,x n dx 1 dx n, R n j 1 3.23 where g denoes he densiy funcion of he vecor X n x 1 T,...,x n T condiioned on F under Q T,hais,ishen-dimensional normal densiy funcion wih vecor mean and covariance marix C ij T given by E X n F M 1,T,..., M n,t C ij T i,j 1 σ i u σ j u ρ ij e a i T u e a j T u du 3.24 and, in he case for every i 1,...,n, σ i σ i is a posiive consan funcion C ij T σ iσ j ) ρ ij 1 e a i a j T. 3.25 a i a j Since he numerical soluion of he inegral in 3.23 is compuaionally inefficien for large n, he price of European swapion could be approximaed by he approach proposed by
ISRN Probabiliy and Saisics 13 Schrager and Pelsser in 6. As illusraed in he Appendix of 6, his mehod leads o a very simple analyical pricing formula of a swapion in he framework of Gaussian ineres rae. In he Gn model, in he case ha he srike is in a he money, we have ES,T, k,k,n N VOL 2π k τ i P, i N VOL 2π P k 1, 3.26 where VOL is he approximaed volailiy of he swap rae given by VOL n i,j 1 σ i u σ j u ρ ij A i A j e a i a j u du, 3.27 where, for every i 1,...,n, A i e a it P,T P k 1 e aik P, k P k 1 K k j 1 e a i i τ i P, i P k 1. 3.28 We finally poin ou ha, in he case for every i 1,...,n, σ i σ i is a posiive consan funcion, we have VOL n i,j 1 σ i σ j ρ ij A i A j e a i a j T 1 a i a j. 3.29 4. Numerical Resuls: Calibraion o ATM Swapions Prices We es formula 3.26 in he calibraion o ATM swapions prices a 31/12/26 and a 31/12/211. All marke daa are aken from Bloomberg provider. ATM swapions volailiy surface a he reference daes are, respecively, shown in Tables 1 and 2 value in percenage. In our experimens, o calibrae parameers, we use he differenial evoluion algorihm as implemened in R hrough he package DEopim. We se he defaul characerisics of he opimizaion algorihm. In paricular, we use he second sraegy insead of he classical muaion in he opimizaion procedure; we se a number of maximum populaion generaors allowed equals o 4, a crossover probabiliy equals o.5, and a sep-size equals o.8. The funcion o minimize is he sum of he square percenage difference beween he marke price and he model price of European swapions wih enor and mauriy as in Table 2. 4.1. G1++ wih Piecewise Consan Volailiy We consider he G1 model and we choose a piecewise consan funcional specificaion for he volailiy of he process x : in paricular, he volailiy funcion σ is consan wihin inervals wih endpoins, 1, 3, 2. We deerminae previous inervals afer several experimens: we ry more se of possible inervals, and we find ha he bes se is he previous one, even increasing he number of inervals. A 31/12/26, he bes fi gives
14 ISRN Probabiliy and Saisics Table 3: G1 wih piecewise consan volailiy: he marix of percenage error a 31/12/211. T/Tenor 1 2 5 7 1 1.4139.97.3262.3795.4364 2.12.1352.29.242.6753 5.769.2647.64.41.36 7.72.3631.82.96.142 1.2967.217.1227.845.757 15.1828.1273.33.71.15 2.632.682.269.422.741 Table 4: G2 wih consan parameers: he marix of percenage error a 31/12/211. T/Tenor 1 2 5 7 1 1.1.197.27.7166.173 2.191.3781.2437.549.129 5.775.77.31.336.4999 7.1282.5959.712.33757.2974 1.8195.945.6967.3418.2683 15.66.6257.3658.62.59657 2.2582.1996.532.3179.6581 a mean percenage error of 2.5% on he swapions price surface. Insead of his good resul, a 31/12/211, he calibraed parameers give a mean percenage error of 7.2% on he swapions price surface. The marix of percenage error a 31/12/211 is shown in Table 3. To improve he resul of calibraion, we increase he number of facors. 4.2. G2++ wih Consan Parameers We now consider he G2 model wih consan parameers. A 31/12/26, we have a very good resul: he calibraed parameers give a mean percenage error of 1.6% on he swapions price surface. A 31/12/211, we improve he calibraion resul obained in he previous secion a he same reference dae, bu he insananeous correlaion ρ 12 beween he wo Brownian moions given by he calibraion oupu is near o 1. Anyway, he mean percenage error over he swapions price surface is 5.2% wih he percenage error marix given by Table 4. 4.3. G2++ wih Consan Piecewise Volailiy We consider he G2 model and we choose a piecewise consan funcional specificaion for boh he volailiy of he processes x 1 and x 2 : in paricular, he volailiy funcions σ 1 and σ 2 are consan wihin inervals wih endpoins, 1, 2, 2. As for he G1 model, we arrive o deerminae he previous inervals afer several experimens: we ry more se of possible inervals and we find ha he bes se is he previous one. We make experimens only a 31/12/211, since we obained a nonimproved resul a 31/12/26 in he previous secion. The calibraed parameers give a mean percenage error of 3.4% on he swapions
ISRN Probabiliy and Saisics 15 Table 5: G2 wih piecewise consan volailiy: he marix of percenage error a 31/12/211. T/Tenor 1 2 5 7 1 1.4.67.15.485.4482 2.165.844.1279.3514.3316 5.5272.5583.484.1913.367 7.125.32.73.153.1216 1.868.543.25.235.83 15.694.358.169.19.367 2.276.1.448.51.559 price surface. This is he bes resul ha we obain hrough all our experimens a 31/12/211. The relaive error marix is shown in Table 5. 4.4. G2++ wih a Funcional Insananeous Correlaion Since he calibraed insananeous correlaion ρ 12 in he G2 model wih consan parameers is near o 1, we se a funcional specificaion for ρ 12 ρ 12 ρ 1 ρ ) e λ, 4.1 where λ is a posiive consan and 1 ρ 1. The inerpreaion of hese wo parameers is sraighforward: ρ is he long-erm average insananeous correlaion whereas λ is he rae of convergence of ρ 12 o ρ. The oher parameers are consan funcions. Unforunaely, among he bes calibraed parameers, ha give a mean percenage error of 5.2%, ρ is closer o 1 as aspeced, bu λ is quie large near o 15. This does no improve he calibraion obained wih consan insananeous correlaion. 4.5. G3++ wih Consan Parameers We finally consider he G3 model wih consan parameers. We subsanially do no improve he resul obained wih he G2 model wih consan parameers. In fac, he mean percenage error sill remained near o 5.2%, and he insananeous correlaion ρ 23 beween he second and he hird facor is near o 1. 5. Conclusions In his paper we describe a general exogenous model in which he insananeous spo rae r is he sum of correlaed Gaussian sochasic processes wih ime-dependen volailiy plus a deerminisic funcion given by an exac fiing o he observed erm srucure. We also give a general noaion independen of he number of facors and analogous wih he noaion used in lieraure. In our numerical experimens, we consider an increasing number of facors from one o hree and differen funcional specificaions for he volailiy of facors and one also for he correlaion. To calibrae he model o marke daa, we use an accurae and fas formula o approximae a European swapions price proposed by Schrager and Pelsser in 6. The algorihm used for calibraion is he differenial evoluion as implemened in he R package
16 ISRN Probabiliy and Saisics DEopim, whereas he funcion o minimize is he sum of he square percenage difference beween he marke price and he model price of swapions. A 31/12/26 we obain a good fi o he marke daa. The G2 model wih consan parameers gives a mean percenage error of 1.6% on he swapions price surface. Then, Augus 27 arrived and he world will never be he same. We es he calibraion a 31/12/211. We obain he bes opimizaion wih he wo facors model wih a piecewise consan funcional specificaion for boh he volailiy funcions. We ge a mean percenage error on he swapions price surface of 3.4%. We also noe ha seing all he parameers consan and increasing he number of he facors, we do no improve he performance of he calibraion. Furher invesigaion could be direc o differen direcions: we could se a differen funcional specificaion for he volailiy of he facors, insead of consan or piecewise consan as in his paper; we could use a differen opimizaion echnic o calibrae he parameers insead of differenial evoluion algorihm, or, saring from he oupu of he differenial evoluion algorihm; we could ry o increase he performance of he calibraion using a deerminisic opimizaion algorihm; finally, we could ry o calibrae he parameers o oher marke daa insead of swapions prices. Acknowledgmens The auhor wishes o hank professor Andrea Pascucci and prof. Sergio Polidoro for several helpful conversaions and for some useful remarks ha improved his paper. He also wishes o hank his wife Giulia for some useful remarks ha correc and improve my English language. Finally, he hanks his colleagues Lorenzo and Marina ha allowed him o wrie his paper. References 1 T. Bjork, Arbirage Theory in Coninuous Time, Oxford Universiy Press, 1998. 2 D. Brigo and F. Mercurio, Ineres Rae Models Theory and Pracice, Springer Finance, Springer, Berlin, Germany, 2nd ediion, 26. 3 D. Filipović, Term-Srucure Models, Springer Finance, Springer, Berlin, Germany, 29. 4 A. Pascucci, PDE and maringale mehods in opion pricing, vol. 2 of Bocconi & Springer Series, Springer, Milan, Ialy, 211. 5 J. Hull and A. Whie, Opions, Fuures and Oher Derivaives, Prenice Hall, Upper Saddle River, NJ, USA, 3rd ediion, 1997. 6 D. F. Schrager and A. A. J. Pelsser, Pricing swapions and coupon bond opions in affine erm srucure models, Mahemaical Finance, vol. 16, no. 4, pp. 673 694, 26. 7 J. Hull and A. Whie, Pricing ineres rae derivaive securiies, Review of Financial Sudies, vol. 3, no. 4, pp. 573 592, 199. 8 J. Hull and A. Whie, Numerical procedures for implemening erm srucure models ii: wo-facor models, The Journal of Derivaives, vol. 2, no. 2, pp. 37 48, 1994. 9 J. Hull and A. Whie, Numerical procedures for implemening erm srucure models i: single-facor models, The Journal of Derivaives, vol. 2, no. 1, pp. 7 16, 1994. 1 F. Jamshidian, An exac bond opion formula, The Journal of Finance, vol. 44, no. 1, pp. 25 29, 1989. 11 H. German, N. EL KAROUI, and J. C. Roche, Changes of numeraire, changes of probabiliy measure and pricing of opions, Journal of Applied Probabiliy, vol. 32, pp. 443 458, 1995.