In-Arrears Interest Rate Derivatives under the 3/2 Model

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1 Modern Economy, 5, 6, Publshed Onlne June 5 n ScRes. hp:// hp://d.do.org/.46/me In-Arrears Ineres Rae Dervaves under he / Model Joanna Goard School of Mahemacs and Appled Sascs, Unversy of Wollongong, Wollongong, Ausrala Emal: joanna@uow.edu.au Receved 5 May 5; acceped 5 June 5; publshed 8 June 5 Copyrgh 5 by auhor and Scenfc Research Publshng Inc. Ths work s lcensed under he Creave Commons Arbuon Inernaonal Lcense (CC BY). hp://creavecommons.org/lcenses/by/4./ Absrac Le symmery mehods are used o fnd a closed form soluon for n-arrears swaps under he / model dr = r ( A( ) α r ) d + cr dzˆ. As well, appromae soluons are found for shor-enor n- arrears caples and floorles under he same neres rae model. Comparsons are made of he appromae opon values wh hose obaned wh a compuaonally-nensve numercal scheme. The appromae prcng s found o be subsanally fas and easy o mplemen, whle he relave errors wh respec o he rue prces are very small. Keywords In-Arrears Swaps, Ineres Rae Opons, / Model. Inroducon Ineres rae dervaves are conracs whose value depends n some way on he level of neres raes. Swap conracs have esed snce he early 98s and snce hen here has been sgnfcan growh n erms of volume and dversy of conracs. In general an neres rae swap s an agreemen beween wo companes, whereupon one company agrees o pay cash flows equal o he neres on a predeermned fed rae on a noonal prncpal, X, a regular se mes T, T, (e.g. every 6 monhs) over he lengh of he conrac me and n reurn receves neres a a floang rae (usually he LIBOR rae) on he same noonal amoun on he same se paymen perods whn he conrac me. In he plan vanlla neres rae swap, he floang rae s he rae ha prevals a he prevous paymen dae, or n he case of he frs paymen, he rae a he openng of he conrac. Wh LIBORn-arrears swaps, he floang rae pad on a paymen dae equals he rae observed on he paymen dae self. Hence he floang leg canno be valued as he sum of forward LIBORs. Caples and floorles are he neres rae counerpars of European call and pu opons. Smlar o vanlla How o ce hs paper: Goard, J. (5) In-Arrears Ineres Rae Dervaves under he / Model. Modern Economy, 6, hp://d.do.org/.46/me

2 swaps, he payoff of vanlla caples and floorles s based on he neres rae a he prevous paymen dae whereas he payoff of n-arrears caples and floorles s based on he LIBOR raes a he acual me of he paymen. Hence n-arrears dervaves are no as sraghforward o prce as her vanlla counerpars. As saed by Chen and Sandmann [], ypcally when no assumpons are made abou he erm srucure of neres raes, s no possble o prce hese n-arrears producs; and even when erm srucures are assumed, s ofen no possble o fnd closed form soluons for he producs. For hs reason convey adjusmens (or convey correcons) are ofen used by praconers as a rule of humb n he valuaon of n-arrears erm srucure producs. In [], Maller and Alobad assume ha rsk-neural neres raes follow he Co-Ingersoll-Ross (CIR) model of he form ˆ dr = b ar d + cr dz where a, b and c are consans and Ẑ s a Wener process under a rsk-neural probably measure. By usng a Green s funcon approach hey manage o derve an analycal epresson for n-arrears swaps. I has been shown (see e.g. []) ha when he shor-erm neres rae, r, follows a sochasc dfferenal equaon of he form γ dr = b ar d + cr dz where a, b, γ and c are consans and dz s an ncremen n a Wener process under a real probably measure ; he value of γ s very mporan n dfferenang beween he dfferen models ables o adequaely capure he dynamcs of neres raes. In parcular he unconsraned esmae of γ by Chan e al. [] was.5. Ths was agreed upon by Campbell e al. [4] who showed ha he heeroskedascy of he shor rae was markedly reduced as γ ncreased from o.5. In [5], Ahn and Gao showed ha he neres rae model dr = b ar d + cr dz () ouperformed many of he popular neres rae models ncludng he Vascek and CIR models. The nonlnear drf n () mples a subsanal nonlnear mean-reverng behavour when he neres rae s above s long-run mean. Hence afer a large neres rae rse, he neres rae can poenally quckly decrease, whle afer a low neres rae perod, can be slow o ncrease. I has also been shown ha wh a >, r wll always reman posve. Ths model was furher mproved (see e.g. [6] [7]) ( ) dr = r B ar d + cr dz () o nclude a me-dependen long-run arge allowng yeld-curves o be fed. In [6], a soluon was found for he prce of bonds wh maury T, under he assumpon ha he rsk-neural process for r has a smlar form o (), namely ( α ) ˆ dr = r A r d + cr dz () where Ẑ s a Wener process under an equvalen rsk-neural measure. The soluon gven s α β k Γ k k c rτ β + + c α α β B( rt, ; ) = ( rτ ) e M k,k, c α c c crτ ( k ) Γ + + c where (4a) α α c c c k =, (4b) 78

3 R τ where (,, ) dfferenal Equaon (PDE) for V( r,) B( rt,; ) A d = e, (4c) = ( + β R ( ) d ) R, (4d) β =, (4e) T R d M ab s he Kummer-M funcon (see e.g. [8]). Ths was found by solvng he governng paral = namely V cr V V + + r ( A αr) rv = r r lm r, =. In Secon of hs paper, we eend he resuls n [6] by fndng an eac soluon for n-arrears swaps under he assumpon ha rsk-neural neres raes follow he me-dependen / model (). As s ypcal wh swap prcng, we dvde he swap no a seres of forward rae agreemens (FRAs) and prce each of hese ndvdually. The value of he swap s he sum of he ndvdual FRAs. Ths was also he approach of Maller and Alobad []. Then n Secon 4 we derve analyc appromaons for caples and floorles based on he me-dependen neres rae model () and compare her values wh hose obaned usng an accurae (bu compuaonally nensve) numercal scheme. Frsly however, we brefly summarse Le s classcal symmeres mehod whch s used o solve he PDEs n hs paper. subjec o he fnal condon V( rt, ) = and boundary condons V (, ) =, V( r). Le s Classcal Symmeres Mehod In essence, he classcal mehod for fndng symmery reducons of a second-order PDE n one dependen varable V and wo ndependen varables (r, ) ( rvv V V V ),,,,,, =, (6) r rr r s o fnd a one-parameer Le group of ransformaons n nfnesmal form (,, ) ( ε ) * r = r+ ερ rv + O (7a) (,, ) * = τ + εψ rv + O ε (7b) (,, ) ( ε ) * V = V+ εv rv + O (7c) whch leaves (6) nvaran. The coeffcens ρψ, and V of he nfnesmal symmery are ofen referred o as he nfnesmals. Ths nvarance requremen s deermned by where =, (8) = = ρ( rv,, ) + ψ ( rv,, ) + V( rv,, ) (9) r V are vecor felds ha span he assocaed Le algebra, and are called he nfnesmal generaors of he ransformaon (7a-c), and s he second eenson (or second prolongaon) of, eended o he second je space, co-ordnased by rvv,,, r, V, Vrr, (see Chaper n he book of Bluman and Kume [9]). Then for known funcons ρψ,,v, nvaran soluons V correspondng o (7a-c) sasfy he nvaran surface condon (ISC) (5) 79

4 V V Ω= ρ( rv,, ) + ψ ( rv,, ) V( rv,, ) =, r whch when solved as a frs-order PDE by he mehod of characerscs, yelds he funconal form of he smlary soluon n erms of an arbrary funcon,.e. where ( φ ) V= q r,, z, z= z r,, and where φ s an arbrary funcon of nvaran z for he symmery. Subsung hs funconal form no (6) produces an ordnary dfferenal Equaon whch one solves for he funcon φ ( z). Furher, for a fnal-value problem wh he fnal condon V( rt, ) = j( r), hen we need a lnear combnaon of generaors such ha condon () s sasfed a T V = j r.e. =, ( rt j( r) ) Vr ( rt) ψ ( rt j( r) ) V ( rt) V( rt j( r) ) ρ,,, +,,, =,,. () V( rt, ) can be found from he fnal condon. As well, for evoluon equaons V(, ) r from he governng PDE (see [] for deals).. LIBOR-n-Arrears Swaps () rt can be found In hs secon we derve he analyc soluon for he prce of n-arrears swaps based on he rsk-neural neres rae model (). The value gven here s from he perspecve of he recever.e. he nvesor who receves he fed rae r and pays he floang rae. As n [], we assume ha he acual floang rae s he spo rae r. The value o he payer.e. he nvesor who pays he fed and receves he floang, s smply he negave of he value gven here. Theorem. The value of an n-arrears swap wh noonal value and fed rae r o a recever wh paymen mes T every half year, when he neres rae follows he rsk-neural process () s gven by and (, ) V r (,; ) W( rt,; ) rb o rt = () n where B rt, ; s gven 4a-e and α β p Γ p rr( T ) p c rτ β + c α α β W( rt, ; ) = ( rτ ) e M p, p, + + R c α c c crτ ( p ) Γ + c ( αc ) + ( αc ) + 8c ( + α) where p = = k + R, τ, β, k are gven n (4b-e). Proof. The value of an FRA o he nvesor who receves he fed rae r a T = sasfes (5) subjec o r r r V( rt, ) =, V (, ) =, lm r V( r, ) =. As (5) s lnear and homogeneous, and he soluon o he bond under () s already known (gven by Equaon (4a-e)), we need only fnd he soluon o (5) subjec o V( rt, ) = r, V (, ) =, lm r V( r, ) =. We denoe hs soluon as W( r, ). Wh he help of he package Dmsym [], we fnd ha PDE (5) has he symmery wh generaor where τa A g τ τ = W + τ τ r (4) rc rc rc W r () 7

5 τ + g = D R d D, ( R d ) D R d D = + D + α R R R + c 4 and where, D, D, D4 R =. In order for he fnal condon o sasfy () and he boundary condons o be nvaran we choose D are arbrary consans and τ ( T ) R d g d e A R d and = =, T R R d nong ha τ ( T ) =. Solvng he ISC () correspondng o (4), hen upon smplfcaon, he funconal form of he soluon can be wren as r W( r, ) = φ( z), z= rτ. (5) R Subsuon of hs funconal form no (5) gves ha φ ( z) needs o sasfy cz subjec o φ = RT, and φ ( z) Solvng (6) for φ ( z) we ge where β c z ( z) ( z) ( c ) z φ + φ α + β α + φ = lm = and where z β =. T R d p α α β α α β φ ( z) = z e AM p +, p +, BU p, p, c c cz c c cz α +α p + + p =, and A and B are consans. Takng no consderaon he boundary c c α p Γ p β + c A= RT. c α Γ p + c W r, = W rt,; as gven n (). condons we ake B = and Undong he change of varables we ge he soluon o 4. Asympoc Soluon for Caples Agan assumng equdsan paymen mes T, T,, Tn = T, n a vanlla cap he conrac holder receves λ X ma r L ( T ) K c a me T +, =,,,, N where λ = T+ T, X s he noonal amoun, rl( T ) s he LIBOR rae a me T and K c s he fed cap rae. In a vanlla floor wh fed floor rae K f he paymen sze s λ X ma K rl( T) a T +, =,,,, N wh he frs paymen me a T and las one a T N +. Hence he paymen sze a me T + s based on he LIBOR rae a me T. In conras wh n-arrears caps and floors he conrac holder receves λ X ma r L ( T ) K c and λ X ma K f rl T respecvely a mes T, =,,, N so ha he paymen sze a me T s acually based on he LIBOR rae a me T. Each of he ndvdual cashflows n a cap are called caples and he ndvdual cashflows n a floor are called floorles. Hence caps and floors are sums of he ndvdual caples and floorles respecvely. Furher, he value of a floor can be found from he cap-floor pary, namely floor = cap swap (see e.g. []). The mos common valuaon of neres rae caples s va he Black-76 model []. Under hs model he under- (6) 7

6 lyng neres rae s assumed o follow a log-normal dsrbuon, whch s no n agreemen wh emprcal fndngs. In hs secon we look a appromang he value of caples and floorles for shor mes o epry, based on he rsk-neural neres rae model (). We noe ha caples and floorles characerscally have shor enor, especally when he assocaed caps and floors have maures of abou one year. For smplcy we le λ X = bu he soluons derved can smply be mulpled by λ X. We sar wh he caple. Wh T, from Equaon (5) he value of an n-arrears caple V( r, ) wh fed cap rae K and epry T sasfes V cr V V = + r ( H ( τ) αr) rv τ r r =. To fnd an appromaon o (7) for small τ we follow he mehod oulned by Howson [4]. We le where < and assume he soluon can be epanded as a seres subjec o V ( r,) = ma ( r K,) where H( τ) AT ( τ) Subsung (8) no (7) we ge Upon equang coeffcens of dons, we ge ha where H H ( τ ) ( ) = (7) V r, = V r,. (8) V V cr V r ( H ( ) αr) + + rv =. (9) τ r r and, and wh consderaon of correspondng boundary and nal con- ( ( α )) r K + r H + K r + ; r K K V( r, ) + V( r, ) = ; r K K, =. However, he above soluon s no dfferenable a r = K and as we epec large Gamma V..e near r = K, hs ouer soluon s no vald n he vcny of r = K. For he nner soluon where r r s near K, he second-order dervave wh respec o r needs o be ncluded n he dfferenal sysem. We n- r K roduce he nner varable = K Ths leads o he equaon.e and rescale V o V = KQ. [ ] α Q c = K K Q H H Q K Q rq K where H = H, We now epand Q ck = ε + ε + ε + Q + εq H + εh+ ( ) + ε Q( H + εh+ ) αqk ε + ε + ε rεq H = H, o be solved subjec o Q(,) = ma (,), lm Q(, ) =, Q H K O lm, + ( α ) + (, τ ) (, ) =. () () Q = Q () 7

7 and subsue hs form no (). Equang erms of O we ge J. Goard Q ck Q = () subjec o Q (,) = ma (,), lm Q (, ) =, lm Q (, ). PDE () adms a s-dmensonal fne Le group of ransformaons (see e.g. [9]). Wh consderaon of he nal and boundary condons, we use he symmery wh generaor = + + Q. Ths leads o Q an nvaran soluon of he form Q = φ ( z) where z =. Subsuon of hs nvaran form no () yelds he reduced equaon ck whch needs o be solved subjec o lm z φ = z, lmz φ. ck z z z Hence we ge ha φ ( z) = ep erfc + π ck ck / Now collecng erms of φ + zφ φ = (4) and so c K Q (, ) = ep erfc. + π c K c K Q sasfes O ε we ge ha Q Q, Q Q ck ck = + + [ H α K] subjec o Q (,) =, lm Q(, ) = ( H α K), Q s [ H K] Q ( ) erfc (5) (6) lm, =. The soluon o hs problem c K α, = ep. + 8 π c K (7) c K The wo-erm nner epanson can hen be found by Q(, ) + Q(, ). We hen mach he nner and ouer soluons o ge a soluon ha s unformly vald by calculang ouer + nner common where common s ha par of he soluon ha s common o boh. In hs case as ε he nner soluon s he same as he ouer soluon and so he ouer epanson s n fac he common epanson. Ths means ha he nner epanson s unformly vald. In erms of he orgnal varables our appromae soluon for a caple wh fed cap rae K and shor me o epry T based on he rsk-neural neres rae () s hen ( H α K) r K K r K r Vc ( r, τ) = erfc + τk erfc K ckτ K ckτ ( r K) ( r K) ep + c τk r + ck τ ep. 8 π ckτ 8 π ckτ In a smlar way we ge an appromae soluon for he value of a floorle wh epry T and fed floor rae K as ( H α K) K r r K r K V f ( r, τ) = erfc τk erfc K ckτ K ckτ ( r K) ( r K) ep + c τk r + ck τ ep. 8 π ckτ 8 π ckτ (8) (9) We use he resuls [4] ha () f u = u and v = v + u hen a parcular soluon s v = u and () f v = v + u hen a parcular soluon s v = u + u. u = u and 7

8 To es our appromae soluons we numercally solve PDE (7) usng he mahemacs sofware package MAPLE [5] (whch uses a cenered mplc fne-dfference scheme) wh sep szes of 4 and use hs as a proy for he rue soluon. We noe ha obanng such numercal values s very labour-nensve and compuaonally-nensve whereas our appromae values are fas and easy-o-mplemen. Frsly, o es he accuracy of he fne-dfference scheme, we used he mehod o numercally solve for n-arrears swap values.e. Equaon (5) subjec o he condons for he swap as gven n Secon, and compared hese values o he eac soluon (). Usng he parameer values α =, r =.5, a =.55 and wh c = we found ha for r values from.45 o.65, wh, absolue errors were of he order of ; for 6 absolue errors were of he order of 6 and for 4 absolue errors dd no eceed 5. Usng he same parameer values as our esng procedure,.e. α =, K =.5, H =.55 and wh c =, Pappro Prue we compued sgned percenage errors for caple values.e. %, usng he appromae soluon P (8) and he pary value.e floorle + swap value usng he eac value of he swap gven by Equaon () mulpled by, and he appromae floorle value gven n Equaon (9). The resuls are lsed n Table. Smlarly n Table are he percenage errors of he appromae floorle soluons gven by (9) and he pary values found from caple-swap. From Table and Table can be seen ha as epeced, n general, he shorer mes o epry yeld he more accurae resuls. In parcular we noe Equaon (8) yelds values for caples ha are a-he-money (ATM) or n-he-money (ITM) ha are above our eac values, wh relave errors <.% for, <.5% for and <.% for /4. For caples ha are ATM or ITM, he pary formula yelds he more accurae resuls where he value s domnaed by he eac swap value. ATM opons are slghly overprced and mosly ITM opons are underprced slghly. Table. Sgned percenage errors of caple appromaons wh α =, K =.5, H =.55, c =. rue 6 4 r Equaon (8) Pary Value Equaon (8) Pary Value Equaon (8) Pary Value.45.77% 54.9%.45% 7.%.64%.7%.5.657%.588%.%.9%.6%.9% %.4%.7%.6%.6%.7%.6.7%.%.9%.76%.85%.7%.65.7% < 4 %.%.8%.9%.88% Table. Sgned percenage errors of floorle appromaons wh α =, K =.5, H =.55, c =. 6 4 r Equaon (9) Pary Value Equaon (9) Pary Value Equaon (9) Pary Value.5.544% < 4 %.%.78%.57%.5%.4.64%.9%.7%.76%.7%.46% %.49%.9%.%.9%.67%.5.59%.68%.5%.8%.67%.%.55.69% 7.6%.66%.9%.866%.% 74

9 For he values of r where caples are ou-of-he money (OTM), he floorle values have small percenage errors bu hese errors are large n comparson o he correspondng caple values hus producng very large pary percenage errors. Equaon (9) yelds values for ATM and ITM floorles wh relave errors <.65% for, <.% for and <% for 4. For floorles ha are ITM, he pary formula yelds he more accurae resuls where he value s domnaed by he eac swap value. For he values of r where floorles are ou-of-he money (OTM), he caple values have small percenage errors bu hese are large n comparson o correspondng floorle values producng very large pary percenage errors. The resuls sugges ha pary values would be bes o prce ATM and ITM caples and ITM floorles; her values mosly slghly underprcng compared o he eac soluon, whle Equaon (8) could be used o prce OTM caples and Equaon (9) used o prce ATM and OTM floorles; her values producng small percenage errors especally (and surprsngly) for he larger values of 6 and 4. I should also be noed ha smlar percenage error resuls were found usng oher volaly coeffcen values c. 5. Dscusson The form of an neres rae model s crucal n he subsequen modellng of neres rae producs and he accuracy of her valuaons. I has been shown emprcally by a number of auhors ha he / model () ouperforms many of he popular neres rae models, such as he Vascek and CIR models, n s ably o capure he acual behavour of he neres rae. Includng a free funcon of me n he drf furher enhances he model s ably o capure he neres rae dynamcs. In hs paper we have assumed he rsk-neural neres rae model () and eended he resuls n [6] by fndng an eac soluon for he value of n-arrears swaps and appromae values for caples and floorles wh shor mes o epry. As noed prevously, caples and floorles have characerscally shor enors especally when he maury of he cap/floor o whch hey belong, s abou one year. The appromae opon values have been shown o produce small percenage errors and n parcular he pary values caple = floorle + swap and floorle = caple swap produce bes resuls for ATM and ITM caples and ITM floorles. References [] Chen, A. and Sandmann, K. (9) In Arrear Term Srucure Producs: No Arbrage Prcng Bounds and he Convey Adjusmens. hp:// [] Maller, R. and Alobad, G. (4) Ineres Rae Swaps under CIR. Journal of Compuaonal and Appled Mahemacs, 64-65, hp://d.do.org/.6/s77-47()49-4 [] Chan, K., Karoly, A., Longsaff, F. and Sanders, A. (99) Emprcal Comparson of Alernae Models of he Shor- Term Ineres Rae. Journal of Fnance, 47, 9-7. hp://d.do.org/./j b4. [4] Campbell, J.Y., Lo, A.W. and MacKnlay, A.C. (996) The Economercs of Fnancal Markes. Prnceon Unversy Press, Prnceon. [5] Ahn, D. and Gao, B. (999) A Paramerc Nonlnear Model of Term Srucure Dynamcs. Revew of Fnancal Sudes,, hp://d.do.org/.9/rfs/.4.7 [6] Goard, J. () New Soluons o he Bond-Prcng Equaon va Le s Classcal Mehod. Mahemacal and Compuer Modellng,, 99-. hp://d.do.org/.6/s ()6-9 [7] Goard, J.M. and Hansen, N. (4) Comparson of he Performance of a Tme-Dependen Shor-Ineres Rae Model wh Tme-Dependen Models. Appled Mahemacal Fnance,, hp://d.do.org/.8/ [8] Abramowz, M. and Segun, I.A. (965) Handbook of Mahemacal Funcons. Dover Publcaons, New York. [9] Bluman, G.W. and Kume, S. (989) Symmeres and Dfferenal Equaons. Sprnger-Verlag, New York. hp://d.do.org/.7/ [] Goard, J.M. () Nonnvaran Boundary Condons. Applcable Analyss, 8, hp://d.do.org/.8/68969 [] Wlmo, P. (997) Dervaves: The Theory and Pracce of Fnancal Engneerng. John Wley and Sons, New York. 75

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