Empirical Study of Multi-Objective Optimization In The Multi-Currency Hull- White Two Factor Model

Transcription

1 Empirical Sudy of Muli-Objecive Opimizaion In The Muli-Currency Hull- Whie Two Facor Model Prepared by Yujiro Osuka Presened o he Acuaries Insiue ASTIN, AFIR/ERM and IACA Colloquia Augus 205 Sydney This paper has been prepared for he Acuaries Insiue 205 ASTIN, AFIR/ERM and IACA Colloquia. The Insiue s Council wishes i o be undersood ha opinions pu forward herein are no necessarily hose of he Insiue and he Council is no responsible for hose opinions. Yujiro Osuka The Insiue will ensure ha all reproducions of he paper acknowledge he auhor(s) and include he above copyrigh saemen. Insiue of Acuaries of Ausralia ABN Level 2, 50 Carringon Sree, Sydney NSW Ausralia (0) f +6 (0) e acuaries@acuaries.asn.au w

2 ASTIN, AFIR/ERM and IACA Colloquia Augus 205 Sydney Submied 4/205 An Empirical Sudy of Muli-objecive opimizaion in he Mulicurrency Hull-Whie wo Facor model Yujiro Osuka Milliman Inc. Urbanne Kojimachi Building 8F -6-2 Kojimachi, Chiyoda-ku Tokyo, , Japan Absrac The paper concerns calibraion of muli-currency ineres rae models wih muli-objecive opimizaion echniques. Firs, he paper consrucs a muli-objecive opimizaion problem o calibrae ineres rae models in differen currencies o swapions in each currency. Then he paper invesigaes he flucuaion of prices of several ineres rae derivaives including foreign currency denominaed annuiies on he Pareo fronier in he parameer space of he Mulicurrency Hull-Whie wo Facor model. I is found ha his calibraion problem can be solved by a weighed sum mehod and ha i can produce he Pareo fronier by changing he weighs of each objecive funcion. The paper also demonsraes in an empirical way he flucuaion of he derivaive prices. I shows no specific rend wih changing model parameers on he Pareo fronier in objecive funcion space. Keywords: Muli-currency, Hull-Whie, Ineres rae, calibraion, muli objecive, opimizaion, Pareo fronier, Inroducion A se of muli-facor ineres rae models over muli-currencies involve a poenially wide range of correlaion coefficiens describing he mulidimensional Brownian moion. To assume ha he correlaed indices generaed by he model reflec he correlaion envisioned by a user, he mulidimensional Brownian moion mus be calibraed o a cerain correlaion coefficien marix subjec o specified consrains. Generally, in he muli-facor model over muli-currencies he dimension of he Brownian moion is a leas equal o he number of economies imes (number of facors +). I should be emphasized ha he correlaion coefficien for he Brownian moion and for observaion variables are no one-o-one. For example, he year rae or 0 year rae are described by a funcion of he facors in he mulifacor model, and he correlaion of he wo is a funcion of he correlaion coefficiens beween he facors and he ineres rae model parameers. Therefore, in order o consruc a realisic model in which observaion variables have an user defined correlaion marix, calibraion should be performed in a way ha consrain funcions deermine relaions among he correlaion marix of observaion variables, he correlaion marix of Brownian moions, and he ineres rae model parameers. If he arge correlaion marix of observaion variables covers iner-currency, he ineres rae model calibraion for each economy could no be accomplished independenly due o he consrains which conrol he relaions beween he model parameers for each objecive funcion o be opimized. This yields he muli-objecive opimizaion problem. However, here is no enough research on he behavior of derivaive prices derived from he calibraed model in he soluion se for his problem, or on he opimal condiion o erminae he opimizaion. I is worh invesigaing hese characerisics because one should selec only one soluion from he opimal soluion se which is called he Pareo fronier. If he pricing flucuaes so severely ha one canno deermine he price wih cerain level of confidence, i is difficul in pracice o implemen he model. Therefore, his paper describes a muli-objecive opimizaion problem o calibrae he muli-currency Hull-Whie wo facor model. The paper hen shows numerically he behavior of several ypes of derivaives in he Pareo fronier obained as he soluion se for his 205 Yujiro Osuka

3 YUJIRO OTSUKA problem. Finally he paper offers a discussion of opimal condiions o erminae he calibraion from he numerical resul. Noe ha many sudies for he muli-currency ineres rae model have been conduced by many auhors using various assumpions and frameworks. Such examples are Andreasen (995), Frey and Sommer (996), Flesaker and Hughson (996), Rogers (997), Mikkelsen (200), Schlögl (2002), Pelsser (2003), Amin (2003), and Brigo and Mercurio (2006). 2

4 AN EMPIRICAL STUDY OF MULTI-OBJECTIVE OPTIMIZATION IN THE MULTI-CURRENCY HULL-WHITE TWO FACTOR MODEL 2 Index Correlaion in he Muli-currency Hull-Whie wo facor model This chaper aims a expressing a correlaion marix of observable indices in he mulicurrency Hull-Whie wo facor model as a funcion of model parameers including correlaion marix of mulidimensional Brownian moion. This correlaion marix funcion of observable indices will be needed in he nex chaper o consruc consrains on he calibraion parameers for implemenaion of a arge correlaion marix of observable indices. Since he ineres rae model used is a wo facor model, he same number of ineres rae indices per currency can be considered in he calibraion. -year zero rae and -year zero rae are seleced as observable ineres rae indices, and exchange raes beween differen currencies are also seleced as observable index. To simplify he problem, le us consider he case in wo currencies. 2. The Hull-Whie wo Facor Model and he Forward Rae Process The shor rae process in he Hull-Whie wo facor model follows he following sochasic differenial equaion (Hull and Whie, 994). dr() = [θ() + u() ar()]d + σ dw, r(0) = r 0, () where, sochasic mean revering level follows; du() = bu()d + σ 2 dw 2, u(0) = 0, (2) The forward rae a ime beween imes and is f(, T, T 2 ) = logp (, T ) logp (, T 2 ) T 2 T, (3) whose differenial form can be wrien as df(, T, T 2 ) = d + B(a,, T 2 ) B(a,, T ) T 2 T σ dw ) e z(t B(z,, T ) z + B(b,, T 2 ) B(b,, T ) B(a,, T 2 ) + B(a,, T ) σ (T 2 T )(a b) 2 dw 2 (4) (5) 3

5 YUJIRO OTSUKA 2.2 Correlaion Coefficien beween -year Rae and -year Rae in he Same Currency wih observaion inerval Consider R s -year zero rae a ime (= f(,, + R s )) and R L -year zero rae a ime (= f(,, + R L )), heir insananeous covariance can be wrien as df(,, + R S ) df(,, + R L ) B(a,, + R = S ) σ [ R dw + B(b,, + R S) B(a,, + R S ) σ S R S (a b) 2 dw 2 ] B(a,, + R L ) σ [ R dw + B(b,, + R L) B(a,, + R L ) σ L R L (a b) 2 dw 2 ] = B(a,, + R S)B(a,, + R L ) σ 2 R S R d L + {B(b,, + R S) B(a,, + R S )}{B(b,, + R L ) B(a,, + R L )} σ 2 R S R L (a b) 2 2 d σ + σ 2 ρ R S R L (a b) [ B(a,, + R S )B(b,, + R L ) + B(a,, + R L )B(b,, + R S ) 2B(a,, + R S )B(a,, + R L )]d (6) By inegraing he insananeous covariance above along, i seems one can obain covariance in observaion inerval δ. However, afer he observaion inerval δ, he o + δ par of f(,, + R s ) and f(,, + R L ) disappear a ime + δ and heir alernaive pars appear a he ip of each forward rae. To our inuiion, i is naural ha observaion objecive does no change essenially before and afer he inerval. In his paper, he covariance of R s -year zero rae and R L -year zero rae in observaion ime o + δ is defined as follows. +δ df(s, + δ, + δ + R L) (s)df(s, + δ, + δ + R S )(s) = [ + + = +δ {B(a,s,+δ+R S ) B(a,s,+δ)}{B(a,s,+δ+R L ) B(a,s,+δ)} 2 σ R S R L {B(b,s,+δ+R S ) B(b,s,+δ) B(a,s,+δ+R S )+B(a,s,+δ)} {B(b,s,+δ+R L ) B(b,s,+δ) B(a,s,+δ+R L )+B(a,s,+δ)} 2 σ R S R L (a b) 2 2 {B(a,s,+δ+R S ) B(a,s,+δ)} {B(b,s,+δ+R L ) B(b,s,+δ) B(a,s,+δ+R L )+B(a,s,+δ)} +{B(a,s,+δ+R L ) B(a,s,+δ)} {B(b,s,+δ+R S ) B(b,s,+δ) B(a,s,+δ+R S )+B(a,s,+δ)} R S R L (a b) 2a 3 R S R L ( σ 2 + σ 2 2 (a b) 2σ σ 2 ρ 2 (a b)) { e ars }{ e arl }{ e 2aδ } σ b 3 R S R L (a b) 2 { e brs }{ e brl }{ e 2bδ } 2 σ σ 2 ρ ] ds + (a b)σ σ 2 ρ σ 2 ab(a + b)r S R L (a b) 2 [{ e ars }{ e brl } + { e arl }{ e brs }] { e (a+b)δ } (7) 4

6 AN EMPIRICAL STUDY OF MULTI-OBJECTIVE OPTIMIZATION IN THE MULTI-CURRENCY HULL-WHITE TWO FACTOR MODEL Thus correlaion coefficien beween R s -year rae and R L -year rae for inerval o + δ is given by +δ ρ SL = df(s, + δ, + δ + R L )(s) df(s, + δ, + δ + R S )(s) +δ +δ df(s, + δ, + δ + R S )(s) df(s, + δ, + δ + R S )(s) df(s, + δ, + δ + R L)(s) df(s, + δ, + δ + R L )(s) (8) 2.3 Correlaion Coefficien beween -year Rae and he differen currency s - year Rae Secion 2.2 derives a correlaion coefficien beween R s -year rae and R L -year rae in he same currency. This secion will provide one where R s -year rae and R L -year rae belonging o differen currencies. The below gives covariance beween domesic R s -year rae and foreign R L -year rae Cholesky Decomposiion Le us assume model parameers for he foreign currency by (a F, b F, σ F, σ 2 F, ρ F ), he differenial form of he forward rae a ime beween imes T and T 2 by df F (, T, T 2 ), he facor Brownian moion by (dw F, dw 2 F ), and he correlaion wih hose for domesic facors by dw dw F = γ, dw dw 2 F = γ 2, dw 2 dw F = γ 2, dw 2 dw 2 F = γ 22. Then he correlaion marix for hese are wrien as ρ γ γ 2 ρ γ 2 γ 22 γ γ 2 ρ F γ 2 γ 22 ρ F Assuming here is he Cholesky decomposiion for he above as c c 2 c 3 c 4 C = 0 c 22 c 23 c c 33 c c 44 Then one can wrie facor Brownian moion for wo currencies by using independen Brownian moion as dw dw dw 2 F dw F = C T dw 2 dw 3 dw 2 dw 4 (9) (0) () 5

7 YUJIRO OTSUKA Therefore, he covariance of domesic R s -year zero rae and foreignr L -year zero rae in observaion ime o + δ is +δ df(s, + δ, + δ + R S )(s)df F (s, + δ, + δ + R L )(s) +δ = [{ B(a,s,+δ+R S ) B(a,s,+δ) σ R c dw (s) S + B(b,s,+δ+R S ) B(b,s,+δ) B(a,s,+δ+R S )+B(a,s,+δ) σ R S (a b) 2(c 2 dw (s) + c 22 dw 2 (s))} B(a F,s,+δ+R L) B(a F,s,+δ) F { σ R L ( c 3 dw (s) + c 23 dw 2 (s) + c 33 dw 3 (s)) + B (b F,s,+δ+R L) B(b F,s,+δ) B(a F,s,+δ+R L)+B(a F,s,+δ) F σ R L (a F b F ) 2 ( c 4 dw (s) + c 24 dw 2 (s) + c 34 dw 3 (s) + c 44 dw 4 (s))}] = (a b) (a F b F )c c 3 σ σ F (a b)c c 4 σ σ 2 F (a F b F )(c 2 c 3 +c 22 c 23 )σ 2 σ F +(c2 c 4 +c 22 c 24 )σ 2 σ 2 F aa F R S R L (a b)(a F b F )(a+a F ) (2) { e ars }{ e af R L }{ e (a+af )δ } + (a b)c c 4 σ σ 2 F (c 2 c 4 + c 22 c 24 )σ 2 σ 2 F ab F R S R L (a b)(a F b F )(a + b F ) { e ars }{ e bf R L }{ e (a+bf )δ } + (af b F )(c 2 c 3 + c 22 c 23 )σ 2 σ F (c 2 c 4 + c 22 c 24 )σ 2 σ 2 F a F br S R L (a b)(a F b F )(a F + b) { e brs }{ e af R L }{ e (af +b)δ } F (c 2 c 4 + c 22 c 24 )σ 2 σ + 2 bb F R S R L (a b)(a F b F )(b + b F ) { e brs }{ e bf R L }{ e (b+bf )δ } Thus correlaion coefficien beween R s -year rae and R L -year rae for inerval δ is given by +δ ρ SFL = df(s, + δ, + δ + R S )(s) df F (s, + δ, + δ + R L )(s) +δ +δ df(s, + δ, + δ + R S)(s) df(s, + δ, + δ + R S )(s) df F (s, + δ, + δ + R L ) (s)df F (s, + δ, + δ + R L )(s) (3) 6

8 AN EMPIRICAL STUDY OF MULTI-OBJECTIVE OPTIMIZATION IN THE MULTI-CURRENCY HULL-WHITE TWO FACTOR MODEL 2.4 Correlaion Coefficien beween -year Rae and Lognormal Process A muli-currency model needs a currency exchange rae model in i. In his paper a simple lognormal process is assumed as a currency exchange rae process. This secion will provide correlaion coefficien beween R s -year rae and logarihm of currency exchange rae S(). Lognormal process for currency exchange rae S() can be wrien as ds() = μs()d + σ FX S()dW FX (4) where expeced rae of reurn, volailiy, and dw FX he Winner process. The variance of log (S()) in is +δ d{log (S(s))}d{log(S(s))} = σ 2 FX δ (5) Assuming correlaion coefficiens wih ineres rae facors dw FX dw = ξ, and dw FX dw 2 = ξ 2, he covariance beween R s -year rae and logarihm of currency exchange rae S() in ime o + δ is +δ df(s, + δ, + δ + R S ) d {log(s(s))} +δ B(a, s, + δ + R = S ) B(a, s, + δ) ξ [ R σ S + B(b, s, + δ + R S ) B(b, s, + δ) B(a, s, + δ + R S ) + B(a, s, + δ) ξ R S (a b) 2 σ = (a b)ξ σ σ FX ξ 2 σ 2 σ FX a 2 R S (a b) + ξ 2σ 2 σ FX b 2 R S (a b) { e brs }{ e bδ } { e ars }{ e aδ } Therefore, he correlaion coefficien beween R s -year rae and logarihm of currency exchange rae S() in ime o + δ is +δ ρ ES = df(s,, + δ + R S ) (s)d{log(s(s))} +δ df(s, + δ, + δ + R S )(s) df(s, + δ, + δ + R S )(s) +δ d{log (S(s))}d{log(S(s))} Noe ha variance of R s -year rae is provided in secion 2.2. In he same way, he correlaion coefficien beween R L -year rae and logarihm of currency exchange rae S() in ime o + δ is +δ ρ EL = df(s, + δ, + δ + R L )(s) d{log (S(s))} +δ df(s, + δ, + δ + R L)(s) df(s, + δ, + δ + R L )(s) +δ d{log (S(s))}d{log(S(s))} (7) (6) (8) 7

9 YUJIRO OTSUKA 3 Opimizaion Problem This chaper presens an opimizaion problem o implemen he muli-currency Hull-Whie wo facor model wih arge correlaion marix of observable indices. I is ofen he case ha an objecive funcion is defined as he sum of errors beween model prices and marke prices for some ineres rae derivaives. I is also ofen he case ha an opimizaion algorihm is run for each currency independenly. However, as seen in chaper 2, he correlaion marix of observable indices is a funcion of model parameers which are he members of calibraion parameers hemselves. This means he Hull-Whie parameers in currency 2 are uniquely calculaed from oher parameers wih given arge correlaions of observable indices. Since hose parameers for he Hull- Whie parameers in currency 2 are no necessarily opimal for he objecive funcion in currency 2, here is a radeoff relaionship beween he Hull-Whie parameers for currency and 2. This finally yields he muli-objecive opimizaion problem. 3. Overview of Consrains The figure shows overview of consrains for he opimizaion problem. Targe Model correlaion Brownian moion = ρ Targe ρ Ceq(a, b, σ, σ 2, ρ, a F, b F, σ F, σ 2 F, ρ F, γ, ξ) ρ Figure Overview of Consrains The lef marix represens he arge correlaion. The cener one represens he model correlaion derived from formulas described in chaper 2 whose variables are ineres rae model parameers and correlaion coefficiens among he Brownian moions. These wo marixes are conneced by an equaliy sign, which will be equaliy consrains in he calibraion. Noe ha values of arge correlaion are illusraive. 3.2 Deails of Equaliy Consrains on Correlaion Coefficiens 3.2. Consrains on Ineres Rae Correlaion in he Same Currency Denoe he Hull-Whie model parameers in he currency by A = (a, b, σ, σ 2, ρ), and hose in he currency 2 by A F = (a F, b F, σ F, σ 2 F, ρ F ). The correlaion beween R s -year zero rae and R L -year zero rae shown by formula (8) is a funcion of he Hull-Whie model parameers and observaion inerval δ. For convenience, le us denoe he correlaion beween R s -year zero rae and R L -year zero rae in he currency by ceq (A, δ), and ha in he currency 2 by ceq 2(A F, δ). Tha is ceq (A, δ) = ρ SL (Currency ) (9) ceq 2(A F, δ) = ρ SL (Currency 2) (20) Then consrains on ineres rae correlaion in he same currency can be se by Targe ceq (A, δ) = ρ 2 ceq 2(A F Targe, δ) = ρ 34 (2) (22) 8

10 AN EMPIRICAL STUDY OF MULTI-OBJECTIVE OPTIMIZATION IN THE MULTI-CURRENCY HULL-WHITE TWO FACTOR MODEL Targe where ρ ij is he elemen of arge correlaion marix described in he figure Ineres Rae Correlaion Iner-currency In he same way, he correlaion beween R s -year zero rae and R L -year zero rae in he differen currency is shown by formula (3) is a funcion of he Hull-Whie model parameers in he wo currencies, correlaion coefficiens among he Brownian moions and observaion inerval δ. Assuming dw dw 2 F dw F ρ γ γ 2 (dw dw 2 dw F F dw 2 ) = γ 2 γ 22 ρ 2 d = Pd (23) dw 2, le us denoe he correlaion coefficiens for he combinaion of R s -year rae and R L -year rae in he currency, and R s -year rae and R L -year rae in he currency 2 by ceq 3(A, A F, P, S, S, δ) = ρ SFS (R s rae (curr )vs R s rae (curr 2)) (24) ceq 4(A, A F, P, S, L, δ) = ρ SFL (R s rae (curr )vs R L rae (curr 2)) (25) ceq 5(A, A F, P, L, S, δ) = ρ LFS (R L rae (curr )vs R s rae (curr 2)) (26) ceq 6(A, A F, P, L, L, δ) = ρ LFL (R L rae (curr )vs R L rae (curr 2)) (27) Then consrains on ineres rae correlaion iner-currency can be se by ceq 3(A, A F Targe, P, S, S, δ) = ρ 3 ceq 4(A, A F Targe, P, S, L, δ) = ρ 4 ceq 5(A, A F Targe, P, L, S, δ) = ρ 23 ceq 6(A, A F Targe, P, L, L, δ) = ρ 24 (28) (29) (30) (3) Correlaion beween Ineres Rae and Exchange Rae For correlaion beween ineres rae and exchange rae, he model correlaion is given by formula (7) and (8). Assuming correlaions among ineres rae in he wo currencies and exchange rae by dw dw 2 F dw F ξ dw FX = ξ 2 ξ 3 dw 2 ξ 4 d = ξd (32) and denoing ceq 7 (A, ξ, ξ 2, S, δ) = ρ ES (R s rae (curr )vs exchange rae) (33) ceq 8 (A, ξ, ξ 2, L, δ) = ρ EL (R s rae (curr )vs exchange rae) (34) ceq 9(A F, ξ 3, ξ 4, S, δ) = ρ EFS (R L rae (curr 2)vs exchange rae) (35) ceq 0(A F, ξ 3, ξ 4, L, δ) = ρ EFL (R L rae (curr 2)vs exchange rae) (36) Then consrains on he correlaion beween ineres rae and exchange rae can be se by 9

11 YUJIRO OTSUKA Targe ceq 7 (A, ξ, ξ 2, S, δ) = ρ 5 Targe ceq 8 (A, ξ, ξ 2, L, δ) = ρ 25 ceq 9(A F Targe, ξ 3, ξ 4, S, δ) = ρ 35 ceq 0(A F Targe, ξ 3, ξ 4, L, δ) = ρ 45 (37) (38) (39) (40) 3.3 Opimizaion Problem Here finally he opimizaion problem for he muli-currency Hull-Whie wo facor model wih arge correlaion marix of observable indices can be se as min A,A F K = f(a), KF = f(a F ) (4) lb A, A such ha F ub { P s.. ceq(a, A F, S, L, P, δ) ρ Targe = 0, Eigen(P) 0 (42) where lb and ub are appropriae lower and upper bounds respecively. The second row in consrain is ranslaed as follows; for a se of A, A F in a cerain rial in he opimizaion, here exis correlaion coefficiens among he 5 dimensional Brownian moion represened by posiive definie marix P which saisfies ha model correlaions for observable indices equal o arge correlaions as described in secion 3.2. For objecive funcions K and K F, appropriae error funcions shall be se such as sum of errors beween ineres rae derivaives of model price and marke price. 0

12 AN EMPIRICAL STUDY OF MULTI-OBJECTIVE OPTIMIZATION IN THE MULTI-CURRENCY HULL-WHITE TWO FACTOR MODEL 4 Mehod of he Empirical Sudy Chaper 3 provided an opimizaion problem o esimae all model parameers in he mulicurrency Hull-Whie wo facor model. This chaper shows deailed mehods for he empirical sudy performed in he paper. 4. Calibraion Targe and Capial Marke Daa 4.. Swap Rae and Swapion EUR and GBP are seleced as model currencies in his sudy. Marke daa as of he December-end 203 is seleced for he case sudy. The swap rae and swapions in each currency are seleced as ineres rae base and a calibraion arge. The able below shows swapion quoes as of he valuaion dae. Table. EUR Swapion Quoes for Differen Tenor and Term Opion Term Swap Tenor % 37.88% 29.28% 23.7% 2.44% 22.80% % 27.89% 25.02% 22.80% 23.07% 23.33% % 25.0% 23.47% 22.47% 23.20% 23.02% % 23.59% 23.2% 2.57% 2.5% 20.29% % 24.0% 22.84% 2.72% 20.20% 8.43% Table 2. GBP Swapion Quoes for Differen Tenor and Term Opion Term Swap Tenor % 30.20% 23.90% 9.52% 7.76% 8.03% % 23.2% 9.73% 7.28% 6.52% 6.29% % 9.94% 7.96% 6.30% 5.44% 5.2% % 8.79% 7.34% 5.98% 4.88% 4.32% % 8.08% 6.75% 4.88% 4.2% 3.49% 4..2 Correlaion Targe for Observable Indices The observaion inerval wih which correlaion coefficien is calculaed is assumed monhly. year zero rae and 0 year zero rae are seleced as observable ineres rae indices. And he arge correlaion for he observable indices are se appropriaely by he following able. Table 3. Correlaion Targe for Observable Indices EUR Y EUR 0Y GBP Y GBP 0Y GBPEUR EUR Y 00% 60% 30% 40% 20% EUR 0Y 60% 00% 40% 75% 40% GBP Y 30% 40% 00% 40% 55% GBP 0Y 40% 75% 40% 00% 60% GBPEUR 20% 40% 55% 60% 00%

13 YUJIRO OTSUKA Objecive Funcion To perform muli-objecive opimizaion for objecive funcions Kand K F which represen sum of errors beween swapion prices in he model and marke, a weighed sum mehod is seleced. The weighed sum mehod convers a muli-objecive problem o a single-objecive problem by replacing weighed sum of objecive funcions. In general, i is proved ha he opimal soluion of he weighed sum mehod falls in he Pareo fronier if each objecive funcion is convex (Mieinen, 999). Though i is no proved Kand K F are convex, i is ineresing o invesigae wheher his mehod can be used in his specific problem. The global objecive funcion is hen se as where K K Global = pk + ( p)k F (43) = { Swapion imodel(currency ) (A) Swapioni Marke(Currency ) } 2 wi i (44) K F = { Swapion Model(Currency 2) Marke(Currency 2) 2 i F Swapioni } wi i (45) and p, w i, and w i F are appropriae weigh consan. In his sudy, swapion price is denominaed as swapion volailiy in black model, and w i, w i F are all se as 00%. 4.2 Tes Insrumens Using he calibraed muli-currency model, pricing of wo ypes of derivaive is performed. The aim is o invesigae flucuaion of he derivaive pricing in he Pareo fronier of model parameer Differenial Swap For one good example o sudy he muli-currency model, a differenial swap, also known as a quano swap, is seleced. A differenial swap is a ypical derivaive using muli-currency ineres rae model. Here a brief explanaion of a differenial swap is given. Company A pays o company B in a currency (EUR) an amoun expressed by he semiannual compounded LIBOR rae associaed wih currency 2 (GBP) a every paymen dae occurring every half year. On he same paymen daes, B pays o A he semiannual compounded LIBOR rae (+spread) associaed wih currency. From B side, he economic resul is equal o he combinaion of a bond issued in currency, invesmen on currency 2 currency, and exchange forward on every cash flow of he invesmens. The pricing es is performed on he following conrac; mauriy 0 years, noional in EUR, spread % Foreign Currency Denominaed Single Premium Deferred Annuiy Anoher ineresing example is foreign currency denominaed annuiy wih single premium. For some counries in a low ineres rae environmen, i is one soluion o earn high income o inves on foreign currency asses. And o avoid currency exchange risk, he annuiy paymen is denominaed in he foreign currency. The lapse rae is hough o be associaed wih ineres rae level of domesic currency and exchange rae because he policyholder is in currency. Mainenance expense is incurred in currency. As a resul, o evaluae ime value of financial opion and guaranee, muli-currency ineres rae model is needed. 2

14 AN EMPIRICAL STUDY OF MULTI-OBJECTIVE OPTIMIZATION IN THE MULTI-CURRENCY HULL-WHITE TWO FACTOR MODEL The pricing es is performed on he following conrac; domesic currency is EUR, invesmen currency GBP, deferred period 0 years, single premium and deah benefi 0,000,000 GBP, MVA margin 0.45%, expense loading.65%, issue age 57, lapse rae 2.5% if surrender benefi afer MVA > single premium else.25%, surrender charge 7%, 6%, 5%, 4%, 3%, 2%, %, 0% for issue year, 2, 3, 4, 5, 6, 7, 8+ respecively, mainenance expense 30,000 per policy. 3

15 YUJIRO OTSUKA 5 Resul 5. Calibraion Resul By changing weigh value p in he global objecive funcion, he Pareo fronier has been obained as 3 ses of opimal soluions Calibraion Resul f f Figure 2 The Pareo Fronier as a Resul of Muli-objecive Opimizaion The horizonal axis shows values of K, and he verical axis shows K F. Each do is he opimal soluion for weighed sum of K and K F by changing he weigh value. 5.2 Differenial Swap Resul Differenial swap pricing for each opimal se of model parameers given in he prior secion is shown in he figure Differenial Swap RunNo Figure 3 Differenial Swap Pricing Resul for each Opimal Each plo corresponds 3 opimal soluions given in he prior secion. The number order coincides he numbering from lef side of figure 2. 4

16 AN EMPIRICAL STUDY OF MULTI-OBJECTIVE OPTIMIZATION IN THE MULTI-CURRENCY HULL-WHITE TWO FACTOR MODEL 5.3 Foreign Currency Denominaed Single Premium Deferred Annuiy Time value of financial opion and guaranee (TVFOG) of foreign currency denominaed single premium deferred annuiy for each opimal se of model parameers given in he prior secion is shown in he figure TVFOG RunNo Figure 4 TVFOG of Foreign Denominaed Deferred Annuiy wih Single Premium Each plo corresponds 3 opimal soluion given in he prior secion. The number order coincides he numbering from lef side of figure 2. 5

17 YUJIRO OTSUKA 6 Discussions 6. Consrains Effec on he Calibraion The soluion for he muli-objecive opimizaion problem esablished for he muli-currency Hull-Whie wo facor model has been obained as he Pareo fronier shown in he figure 2. Since he opimizaion problem has 0 of equaliy consrains, i was one of he subjecs ha how sricly he domain space of each objecive funcion is consrained?. The feasible region is obviously consrained by 0 of equaliy consrains, hus i is naural ha each objecive funcion could no be less han ha of independen calibraion in each currency. However, i is sill imporan o see how deep he opimizaion funcion can be lower in he problem in pracice. As i is seen in figure 2 ha gaps of boh sides of 3 plos in which ever axis, are insignifican. In his sudy, he gap was 2.62E-06 and 2.276E-06 for currency and currency 2 respecively. I means objecive funcion for currency does no change significanly when weighing he oher side of objecive funcion (currency 2) and vice versa. Consider he number of effecive dimensions in he opimizaion problem, i is roughly esimaed 8 dimensions as he number of variables is 8 agains he 0 equaliy consrains. Though he number of consrains is 0, here migh be sill enough size of he feasible region o minimize he boh objecive funcions which resuls in he small gaps of each objecive funcions in he Prao fronier. Therefore, he following hypohesis is possible. I is imporan o see he number of effecive dimensions which is he dimensions of feasible region afer subracing he effec of consrains for considering sufficiency of he feasible region, and he consrains do no limi oo srongly he region if appropriae number of effecive dimensions remains. 6.2 Derivaive Flucuaion in he Pareo Fronier Anoher subjec on he sudy is wha range he derivaive price flucuaes according o he model parameer in he Pareo fronier. In he figure 3 and 4, each do corresponds hose in figure 2 (he Pareo fronier plos) in he order lef o righ. Changing he weigh value p in he global objecive funcion, i is seen ha wo ypes of derivaive prices change wih no specific characerisic. And he price gap wihin his range seems immaerial for boh pracically. Therefore, he following hypohesis is also possible. Given he small range of Pareo fronier, he model price flucuaion of ineres rae derivaives shows no specific paern, and he price gap is no significan. 6.3 Compuaional Error in Sochasic Inegral In pracical implemenaion, here is a problem of conradicion ha he inegral of insananeous covariance of wo sochasic processes is no equal o he covariance of discreizaion of he wo variables, as is well known in he heory of sochasic differenial equaions. Depending on which view is adoped, he calculaed resul of correlaion beween he wo indices is differen. Therefore, one has wo opions when implemening consrains on he correlaion marix among observable indices in he calibraion. This paper inroduced he consrains on he correlaions from he former view for simpliciy, bu i migh be more accurae using he covariance of discreizaion he wo variables when calculaing he covariance of hose insead of using he inegral of he insananeous covariance. 6

18 AN EMPIRICAL STUDY OF MULTI-OBJECTIVE OPTIMIZATION IN THE MULTI-CURRENCY HULL-WHITE TWO FACTOR MODEL References Hull, J., Whie, A. (994) Numerical Procedures for Implemening Term Srucure Models II: Two-Facor Models. The Journal of Derivaives 2, Brigo, D. Mercurio, F. (2006) Ineres Rae Models: Theory and Pracice. Second Ediion. Springer, Berlin Heidelberg New York. fp://fp.freesofware.com/pub/ex/can/macros/conrib/suppored/nabib, 997. Andreasen, J.F. (995) Pricing by Arbirage in an Inernaional Economy. Research in Inernaional Business and Finance 2, 936. Frey, R., and Sommer, D. (996) A Sysemaic Approach o Pricing and Hedging Inernaional Derivaives wih Ineres Rae Risk. Applied Mahemaical Finance 3(4), Flesaker, B., Hughson, L. (996) Posiive Ineres. Risk 9, Rogers, L.L.G. (997) The Poenial Approach o he Term Srucure of Ineres Raes and Foreign Exchange Raes. Mahemaical Finance 7, Mikkelsen, P. (200) Cross-Currency LIBOR Marke Models. CAF working paper 85. Schlögl, E. (2002) A Mulicurrency Exension of he Lognormal Ineres Rae Marke Models. Finance and Sochasics 6(2), Pelsser, A. (2003) Mahemaical Foundaion of Convexiy Correlaion. Quaniaive Finance 3(), Amin, A. (2003) Muli-Facor Cross Currency LIBOR Marke Models: Implemenaion, Calibraion and Examples. Available online a: hp:// Mieinen, K.M. (999) Nonlinear Muliobjecive Opimizaion, Kluwer academic, Dordrech. 7