Optimal Mortgage Refinancing Based on Monte Carlo Simulation

Transcription

1 IAENG Inernaional Journal of Applied Maheaics, 42:2, IJAM_42_2_6 Opial Morgage Refinancing Based on Mone Carlo Siulaion Jin Zheng, Siwei Gan, Xiaoxia Feng, and Dejun Xie Absrac The pricing of orgages in he conex of sochasic ineres rae plays an iporan role for financial anageen. The conribuing facors ipacing he orgage conrac value have been explored by abundan lieraures. However, he arke players anicipae a syseaic bu low-cos approach o iniize he ne presen value of he payen sreas by aking advanage of refinancing, for insance. This paper focuses on finding a desirable refinancing ie for orgage borrowers o iniize he oal payen in a sochasic ineres rae environen. The underlying ineres rae is assued o follow a sochasic process wih ean-revering propery, he seing of which is broad enough o accoodae a large specru of arke realiies. Two ypes of coonly adoped orgage balance seleen schees are analyzed and copared o ensure he applicabiliy of our sudy. Our nuerical algorih is validaed wih wih varying saplings, leading o several ineresing characerisics peraining o he opial orgage refinancing period. As one of he applicaions, we obain he opial boundary condiions for he value of he orgage conrac for all ie before he expiry of he conrac. Our approach and algorih provide cos effecive and easy o use financial ools for boh insiuional and individual propery invesors. Index Ters orgage refinancing, loan valuaion, financial opiizaion, Mone-Carlo siulaion, sochasic ineres rae odel I. INTRODUCTION As one of he os influenial financial insruens in boh he priary and secondary arke, residenial orgage conrac ypically grans he borrower several opions o faciliae his reacing o he arke oveen, aong which he opions of prepayen and refinancing are of pivoal iporance. Prepayen refers o he behavior ha he borrower chooses o sele all or in par he loan balances even hough he lender s preference ay be o keep receiving he conraced coninuous or periodical insalens. The ain financial reason leading o prepayen is ypically he low invesen reurn ha he borrower ay earn using he oney a hand. The sudies on his aspec have seen iporan developen recenly, especially hose conained in [], [], [], for insance, where he cobinaion of advanced aheaical analysis wih novely nuerical ehods has ade i possible o find very fas and cos effecive soluions o he proble when he underlying ineres rae is assued as a specific bu coonly adoped This work is parially suppored by he Naural Science Foundaion of Jiangsu Province of China. Jin Zheng (Jin.Zheng a liv.ac.uk) is wih he School of Maheaical Sciences, Universiy of Liverpool, UK; and he Deparen of Maheaical Sciences, Xi an Jiaoong Liverpool Universiy, China. Siwei Gan (Siwei.Gan1 a suden.xjlu.edu.cn) and Dejun Xie (Dejun.Xie a xjlu.edu.cn) are wih he Deparen of Maheaical Sciences, Xi an Jiaoong Liverpool Universiy, China. Xiaoxia Feng (x.x.f.9 a su.xju.edu.cn) is wih he Deparen of Maheaics, Xi an Jiaoong Universiy, China. Correspondence auhor eail: Jin.Zheng a liv.ac.uk. ean revering odel. On he oher hand, no all borrowers have sufficien fund o ake alernaive invesen. In fac, a raher ore coon scenario in China s arke is ha ajoriy orgage borrowers ake periodical orgage payen using heir fixed incoe inflow fro oher sources, ypically in he for of salary, for insance. This econoy realiy underscores he iporance of he opion of refinancing in orgage conrac. The ain reason for debors o refinance is o iprove he financial leverage efficiency by obaining an alernaive orgage loan wih a lower ineres rae. Mos of he previous lieraures in his opic are epirical in naural fro he perspecive of opial refinancing differenials, where he opial differenial is defined when he ne presen value of he ineres payen saved reaches he su of refinancing coss (see [] and relevan references conained herein). In his work, we inend o address he proble by siulaing he alernaive ineres rae ha he arke ay offer wih a raher siple assupion on he srucure of he ineres rae process. We exhibi he procedure wih he Vasicek ([]) odel for is racabiliy and ore iporanly, for coparison wih available resuls in relaed lieraures. The vasicek Model has been widely used in financial odeling and financial producs valuaion, including characerizing he price of discoun bond (see []) and residenial orgages (see [], [], [], for insance). Anoher reason for using Vasicek odel o ipleen our algorih is he exisence of convenien paraeer esiaion procedures for he odel, including axiu likelihood ehod or Bayesian based ehod. References of such esiaions can be found in [], for insance. Malab algorihs are proved o be helpful in solving he peraining financial opiizaion probles. We reark ha alhough he algorih is exhibied wih Vasicek odel, he ipleenaion of our approach does no resric he choice of any sochasic odel, as long as such a odel explains he arke rend wih accepable significance. Our curren work considers wo ypes of loan payen in he financial arke, one is aching he principal payen ehod and he oher is aching he payen of principal and ineres. Boh of he wo payen schees are adoped o generae and copue he onhly insallens wihin he whole conracual duraion. To copare, we also consider wo scenarios wih respec o he presen value of he fuure payen flows, one is for he zero discouning rae and he oher is for a posiive discouning rae. In addiion, we would like o ephasize he following wo coonly adoped pracices in orgage indusry. Firs, he debor is allowed o refinance only once afer he conrac is signed bu before he expiry dae. Under his assupion, he debor should (Advance online publicaion: 26 May 212)

2 IAENG Inernaional Journal of Applied Maheaics, 42:2, IJAM_42_2_6 grasp he bes opporuniy of he a lower enough ineres rae o iniize he oal payen. Second, if a raional debor chooses o refinance, he is no longer responsible for he subsequen residuary ineres upon refinancing and he new deb is he ousanding loan balance inheried fro he old deb. This assupion is essenial since oherwise he debor will have no oivaion o refinance. However, in realiy, he arke is no copleely efficien and liquid in he sense ha he debor is ofen required o pay addiional refinancing fee. The ipac of ransacion fee is no considered in he curren work. The res of his paper is organized as follows. We choose a odel o siulae alernaive orgage rae, hen derive he cash flow schees for he wo ypes of loan seleen in Secion 2. In Secion 3, we forulae and presen our algorih for nuerical siulaion. Nuerical experienaion and discussion for finding he desirable refinancing ie for debors are provided in Secion 4. The scenario of posiive discouning facor is included and ipleened in Secion 5. In Secion 6, we provide nuerical experiens for odel calibraion wih varying saplings. The opial refinancing curve as a funcion in ie is defined and presened in Secion 7. We suarize in Secion 8 wih concluding rearks and possible applicaions in relaed fields. II. MODEL DERIVATION AND INTEREST RATE SIMULATION A. Maching The Principal Repayen Mehod Suppose he debor borrows P wih onhly ineres rae r during he ie period [ T ] and repays a he beginning of each onh, where denoes he h onh. According o aching he principal repayen ehod, equals o a cerain porion of principal plus a decreasing value of ineres. = P n (1 1 n )P r (1) where n is he oal nuber of payen ies. The er P n could be explained as a fixed porion of principal, and (1 1 n )P r is an aoun of decreasing ineres due o he reducion of principal every onh. A ie k, he debor prefers o refinance he deb wih anoher lender when a lower ineres rae r k is offered. On he k h onh, he owes he previous bank P k and has paid A k. P k = (1 k 1 n )P k 1 A k = i = P (k 1)(r 1 n k 2 2n r ) (2) i=1 The aoun of oney P () is he new principal he debor borrows fro anoher bank wih he ineres rae r(). This ransacion will las fro ie k o ie T. The oal payen over ie [ T ] could be described as follows: P (T ) = A k n i=k i = P (k 1)(r 1 n k 2 2n r ) P k [1 (n 1)r k ] (3) 2 where n = n k 1 B. Maching he Repayen of Principal and Ineres Mehod The second ehod o repay loan is o ach he repayen of principal and ineres. Assue he debor borrows P wih ineres rae r over ie [ T ] and he aoun of onhly payen is kep he sae. In he beginning of he conrac, he ineres accouns for os of payen due o a large aoun of loan while principal is sall. Le P () denoe he aoun of oney owed a ie and is he onhly payen. { dp () = d r P ()d P () = P (4) The onhly payen, should be: = P r (1 r ) n (1 r ) n 1 where n is he nuber of oal repayen ies. A ie k, he debor owes he P (k) o he previous bank. Again, due o he lower ineres rae r k, he debor would borrow P (k) fro anoher bank o repay he reaining debs P (k). The oal payen over ie [ T ] could be described as follows: where (5) P (T ) = 1 (k 1) 2 (n k 1) (6) 1 = P r (1 r ) k (1 r ) k 1 2 = P kr k (1 r k ) n k (1 r k ) n k 1 P k = r [1 e r (k T ) ] To carry ou nuerical siulaions for boh payen schees, we assue ha he principal P is 1,, he iniial onhly lending rae r is 5 12 %, and he oal payen period, couned in nuber of onhs, is T = 24. III. NUMERICAL SIMULATION The Vasicek shor er ineres rae process is a aheaical odel describing he evoluion of ineres rae (see []). The odel specifies ha he insananeous ineres rae follows he sochasic differenial equaion: (7) dr = k(θ r )d σdw (8) where k is he reversion rae, θ is long er ean ineres rae and σ is he sandard deviaion, all of which are posiive consans. We le r denoe he insananeous spo rae a ie, and W is he sandard Brownian Moion. which yields he explici soluion for equaion (8) (Advance online publicaion: 26 May 212)

3 IAENG Inernaional Journal of Applied Maheaics, 42:2, IJAM_42_2_6 r = e k r θ(1 e k ) σ e k( s) dw s (9) Under he Euler approxiaion, equaion (1) can be rewrien as: r = k(θ r ) σ W (1) Boh equaion (8) and equaion (9) can be used equivalenly o describe he alernaive orgage rae ha a loan borrower ay choose fro he open arke. Bu equaion (1) is ofen ore useful for siulaion purposes. We would like o reark ha alhough Vasicek odel is considered in he curren paper, our ehod is equally applicable o any oher classes of sochasic odels. Then we use siulaed daa o carry ou he experien. The ai of our odel is o obain he bes period o refinance. The bes period in our experien eans he onh during which o refinance yields a lowes oal payen. We siulae boh repayen ehods o obain he frequency disribuions. The ajor seps for he algorihs are as follows: 1. Iniialize r and generae ineres rae r for each onh by Mone Carlo siulaion. By Euler s Approxiaion r j = r j 1 r j 1 = r j 1 k(θ r j 1 ) σdw j 1 2. Updae oal payen by he siulaed ineres rae. Boh of he wo payen schees are experiened in his sudy. 3. Find he period where he oal payen achieves he lowes. The followings Figures 1-6 are he exaple of our siulaions by boh ehods. Figures 1-3 are siulaed by aching he principal repayen ehod, and res Figures 4-6 are siulaed by aching he repayen of principal and ineres. We observe ha for all 6 scenarios, he rend of he plo becoes fla afer, say, years, which eans he oal presen value of payen roughly keeps consan afer cerain years. This is a srong hin ha early sage of he conrac is criical for pruden financial decisions. In he following Secion 6, one will see ha as σ, he soluion curve defining he noralized ne presen value of oal cash payens goes fas o an asypoic value beyond cerain ie. Under his scenario, here is no difference o refinance or no afer such a asypoic ie period. However for any σ >, he sochasic naure of he underlying ineres rae can lead o a very volaile opial decision aking process before, say, < 13 15, which eans adjacen ies for refinancing ay resul in quie divergen ne presen values of fuure payens. IV. NUMERICAL EXPERIMENTATION AND DISCUSSION In his secion, we use siulaed daa o carry ou he experien. The ai of our odel is o obain he bes period o refinance. The bes period in our experien eans he onh during which o refinance yields a lowes oal payen. We siulae boh ehods o obain he frequency disribuions. Fig. 1. Fig. 2. he oal liabiliies he oal liabiliies 1.65 x x A. Maching The Principal Payen Mehod The following Figure 7 provides he inforaion of he frequency disribuion of he bes period hroughou he conraced duraion. The frequency space is 6 onhs. I can be seen ha he frequency arrives he peak a he second half of he firs year. The frequency of following onhs declines over ie. Fro he resuls repored in Table 1, we find ha unil he 5h year, he oal ies o refinance is up o 9252 (he frequency rae is %), which iplies i is beer o Fig. 3. he oal liabiliies 1.66 x (Advance online publicaion: 26 May 212)

4 IAENG Inernaional Journal of Applied Maheaics, 42:2, IJAM_42_2_ x he oal liabiies he frequency Fig. 4. Fig. 7. The frequency disribuion over 24 onhs duraion by 1 ies of siulaions wih aching he principal payen ehod. Fig. 5. he oal liabiies 1.65 x refinance early. We include he ineres rae facor ino our ipleenaion and discussion. As we igh reasonably assue, he bes opporuniy o refinance probably arise when he loan ineres is coparaively low. We define a new variable coun o record he ies ha he bes onh o refinance ( f ) coincide wih he onh where he lowes ineres rae ( r ) occurs. In each siulaion, if he difference beween f and r is less han 3 onhs, we regard he o be a coincidence TABLE I FREQUENCY AND CUMULATIVE FREQUENCY OF THE BEST TIME TO REFINANCE Monhs Frequency Cuulaive Frequency Fig. 6. he oal liabiies 1.75 x and he value of coun increases by 1. for k = 1 : 1 if f r 3 coun = coun 1 The above procedure is circulaed 1 ies and we choose ( r ) in differen ie inervals. The siulaed resuls of he coincidence as easured by he variable coun are shown in Table II. The second colun 1 36 represens he ie inerval fro he 1s onh o he 36h onh of he conrac. Siilarly, 1 6, 1 9 and 1 24 represen he corresponding onh inervals. For insance, he ies ha he opial refinance period locaes in he inerval fro he firs onh o he 9h onh is 6295 in he firs siulaion. The boo row in Table II displays he average value (Advance online publicaion: 26 May 212)

5 IAENG Inernaional Journal of Applied Maheaics, 42:2, IJAM_42_2_6 TABLE II FREQUENCY AND CUMULATIVE FREQUENCY OF THE BEST TIME TO REFINANCE Ties Average of coun. I is observed ha he average percenage value of coun during 1s 24h is only 27.45%, which is he lowes copared o ohers. This resul is no surprising since i has been shown in above ha he possibiliy of refinance is up o 92.52% in he firs five years. As he ie inerval is shorened o, say, he firs hree years or he firs five years, he percenage of coincidence subsanially increases. Fro Table I, we have seen ha he bes refinancing onh is considerably ore possibly locaed in he earlier ie. Bu how early is sill a proble deserving careful analysis. The duraion of he firs 9 onhs apparenly shows he highes possibiliy 99.36%. However, he inerval is so long ha i ay no be an operaive suggesion o debors. In fac, he frequency rae seadily increases afer he 6h onh. On he oher hand, when we inspec he firs 36 onhs duraion, i is noed ha alhough he range becoes sall, he possibiliy ha locaes in his range is sill as high as 73.16%. As for he duraion of he firs 6 onhs, he frequency rae is 92.52%, and he corresponding average percenage of coincidence is he highes aong all hese hree cases. This coparison provides a useful hin on he disribuional paern of he bes refinance period, which, aken in conjuncion wih he observaions of he real arke ineres rae, will faciliae he borrower s financial decisions. We focus on a period of 6 onhs. In he above discussion, we define he coincidence as he difference beween he salles ineres rae onh in a specific period and he bes refinance onh less han hree onhs. As a aer of fac, when we sricly define he coincidence eans he bes refinance onh equal o he local salles ineres rae onh, he consequence reveals he value of coun jus reduces by around 3.3%. In addiion, here is ore likely o refinance afer he salles ineres rae happens han refinance before i. B. Maching The Payen of Principal and Ineres Figure 8 is he frequency disribuion generaed by siulaing 1 ies of aching payen of principal and ineres ehod. I has he siilar bu no idenical properies copared o Figure 7. In his payen schee, he principal balance decreases raher slowly a early sage while in he firs payen schee (aching he payen of principle) ha he principle decreases by an equal aoun each onh. he frequency Fig. 8. The frequency disribuion over 24 onhs duraion by 1 ies of siulaions wih aching he payen of principal and ineres ehod. TABLE III FREQUENCY AND CUMULATIVE FREQUENCY OF THE BEST TIME TO REFINANCE Monhs Frequency Cuulaive Frequency Thus, he less indifference of change of principle leads o he ore divergen disribuion. Again, he ineres rae facor should be involved in our discussion. As enioned above, we use coun o record he ies ha he bes onh o refinance ( ) coincides wih he onh when he salles ineres rae ( r ) occurs. Figure 8 and Table III illusraes he frequency disribuion for a conrac of 2 years. The resuls we have obained are siilar o he previous ehod. The frequency firs increases, reaching he peak during he 13h onh o he 18h onh. Aferwards i decreases gradually, down o afer 11 years. Unil he 7h year, he cuulaive frequency is 9494 in oal, (Advance online publicaion: 26 May 212)

6 IAENG Inernaional Journal of Applied Maheaics, 42:2, IJAM_42_2_6 TABLE IV FREQUENCY AND CUMULATIVE FREQUENCY OF THE BEST TIME TO REFINANCE Ties Average which provides a srong evidence for early refinance. As for coincidence, again, he duraion of 9 onhs has he highes value in hese hree periods. C. Coens on he Resuls In our paper, we wish o deerine which period is a beer choice for debors o refinance. The sudy has found soe iporan properies for refinancing. Firs, he possibiliy of refinancing in he early sage ay surpass 9%, which iplies ha debors should refinance early. Second, he frequency curve arrives is peak a he las half of he firs year. Afer ha, he frequency of refinancing will drop and he coincidence increases a firs and decreases afer is peak value. Finally, a duraion neiher relaively oo long nor oo shor is regarded as a perfec soluion, i.e., a duraion of 9 onhs (7.5 years) is relaively oo long o he whole duraion of 2 years. In consideraion of hese four properies, he debors should refinance in he period of he 1s o he 6h onh when he ineres rae is locally low, for conrac condiions and arke rae oveen specified in his paper. Tha eans in cerain onh when he ineres rae will be expeced o fall down o cerain lower enough level, i is probably he bes ie o refinance. V. OPTIMAL REFINANCING WITH DISCOUNTED PAYMENT In real financial arke, he consideraion of discouned payen wih aching he principal and ineres rae ehod is ore relevan and applicable o os of he indusry praciioners. We assue he debor borrows P wih onhly ineres rae r during he ie period [ T] and pays for each onh. A ie, where he arke ineres reaches r and he debor has he choice of wheher o refinance or no. The onhly payen corresponding o he originally conraced orgage rae is = r P 1 (1 r T ) (11) If he debor does no wan o refinance and holds he curren conrac, he presen value of oal payen fro ie 1 o he frequency Fig. 9. The frequency disribuion over 24 onhs duraion by 1 ies of siulaions wih discouned fuure payen. T should be: M 1 () = = 1 r 1 (1 r 1 )(1 r 2 ) (1 r 1 )(1 r 2 )(1 r 3 ) (1 r 1 )(1 r 2 )(1 r 3 ) (1 r T ) T i j=1 [1 r (12) j] i=1 A ie, if he debor chooses o refinance wih a lower ineres rae r. Afer refinance, he onhly payen becoes 1 insead of, and he discouned oal payen is: M 2 () = 1 r 1 (1 r 1 )(1 r 2 ) (1 r 1 )(1 r 2 )(1 r 3 ) (1 r 1 )(1 r 2 )(1 r 3 ) (1 r ) = 1 1 r 1 1 (1 r 1 )(1 r 2 ) 1 (1 r 1 )(1 r 2 )(1 r 3 ) 1 (1 r 1 )(1 r 2 )(1 r 3 ) (1 r T ) T i j=1 [1 r j] 1 i j=1 [1 r (13) j] i=1 i=1 The aheaical forulaion and he siulaion procedure are siilar o hose discussed in he previous secion, where he discouning facor is zero. The only difference is ha we now discoun fuure cash payen in copuing he presen value. Siilar nuerical experiens are carried ou wih 1 siulaed ineres rajecories. We use he sae esiaed paraeers for Vasicek odel and he sae iniial conrac condiions. Figure 9 describes he frequency disribuion wih discouned fuure payens before he expiraion of he conrac by aching he principal and ineres rae ehod. The frequency reaches is peak in he firs 6 onhs and decline exponenially over he ie. Copared o Figure 8, he (Advance online publicaion: 26 May 212)

7 IAENG Inernaional Journal of Applied Maheaics, 42:2, IJAM_42_2_6 TABLE V FREQUENCY AND CUMULATIVE FREQUENCY OF THE BEST TIME TO REFINANCE Monhs Frequency Cuulaive Frequency frequency disribuion is Figure 9 ends o have a long-ail, which eans he iniu oal payen also be achieved when he ie is close o auriy. The coparison of Table III and Table V shows such a difference for he wo differen discouning scenarios. In Table III, he frequency is afer 143 onhs, which reflecs he fac ha afer 143 onhs, i is no necessary o replace he exising deb obligaion. Neverheless, even near he expiraion dae, he discouned fuure payen schee allows an opial refinance o iniize he oal payen. Thus he difference beween able III and Table V verifies he ie value of oney conserved in he discouned payen schee. A. Analyical Soluions VI. APPLICATIONS The resuls in previous secions show ha he debors should refinance as earlier as possible when he lending rae is relaively low. In his secion, we ry o use analyical approach o find he closed for soluions for soe special cases. On he oher hand, he nuerical resuls are also verified, in par, by he following heoreical analysis on he raion of P (T ) P. To illusrae he idea of our analyical idea, we assue ha he conrac adops he aching he principal payen ehod for he borrower o pay back his deb. Le = k 1, hen equaion (9) yields P (T ) = [r r 1 r P 2 r 1 r 2n 2n (n 1)r 1 ] 2n 2 n 1 r = r 1 r 2 (1 1 2n 2n )(r r 1 ) n 1 r 1 1 (14) 2 We proceed he analysis by idenifying he following wo scenarios. 1) r = θ : When he iniial borrowing rae equals o he long er ean rae, he sochasic process for he arke ineres rae becoes r = e k r θ(1 e k ) σ = r σ e k( s) dw s e k( s) dw s (15) I is inuiive and worhwhile o noe ha he debor is likely o refinance only when he insananeous spo rae is less han he iniial borrowing rae, i.e., only when he sochasic inegral er σ e k( s) dw s resuls in a negaive value. Bu even wih his in ind, he saisically easured iniizer o he sochasic funcion P (T ) P is no iediae since he equaion (17), as a quadraic for in wih sochasic coefficiens, is coposed of ers wih differen signs in differenials in. For insance, one igh wan go o zero on he se of where r > r if only he firs order er of is concerned, bu his ove ay no gran enough ie for r o achieve sufficienly lower level, which is desirable if he second order or zero order er of is concerned. An equilibriu of he opposing facors in (17), as shown by our siulaed resuls in Figure 7 and 8, says ha he bes refinance ie is os likely locaed in he early sage of he conrac for he usual condiions se in his paper. This is rue despie ha he expecaion of P (T ) P is independen of ie. The resul is consisen wih he nuerical resuls conained in he previous secion and offers a saisical explanaion o he opial sraegy ha a borrower should ake o iniize his oal financial cos. 2) r > θ: When he iniial borrowing rae is higher han he long er ean rae, noe ha he sochasic process for he arke ineres rae can be wrien as r = (r θ)e k θ σ e k( s) dw s (16) Figure 1 reveals ha when he value of σ is sall (i.e..1 or less), he siulaed ineres raes are flucuaing around he drif wih very sall deviaions. In his scenario, he general rend of ineres rae drops exponenially o he ean level. Wih he paraeers we choose for he odel, and wih he curren siulaion specificaions, such as he ie sep for he Euler approxiaion and he axiu nuber of siulaed rajecories, conained in his paper, we find ha he sochasic inegral er σ e k( s) dw s is negligible in saisical sense for undersanding he refinancing sraegy. (Advance online publicaion: 26 May 212)

8 IAENG Inernaional Journal of Applied Maheaics, 42:2, IJAM_42_2_6 1 x ineres rae dirf variance 35 x 16 r =.51/12 r 3 =.52/12 r =.53/ Fig. 12. The disribuion of oal payen for r > θ, when θ =.5/12, σ =.3 Fig. 1. drif represens he er (r θ)e k θ, variance represens he er σ e k( s) dw s and ineres raes are he siulaed spo insananeous raes, where r =.12, k=.1 and σ= k=.1 k=.2 k=.3 2 x 16 r =.47/12 r =.48/ r =.49/12 he frequency Fig. 13. The frequency disribuion a differen values of k, when θ =.5/12, σ =.3 and r =.5/ Fig. 11. The disribuion of oal payen for r < θ, when θ =.5/12, σ = k=.1 k=.2 k=.3 6 3) σ = : As presen in he previous secion, he sochasic inegral er is negligible in saisical sense. In his secion we ignore he effec of he sochasic er and consider he ipac of r and θ on oal payen. Fro equaion (14) and (16), we find ha he frequency n1 P (T ) = (r θ)e k r 2 P 2n (1 1 2n )(r (r θ)e k ) n 1 (r θ)e k 1 (17) 2 Consider when r = θ, he equaion(17) becoes P (T ) P () = 2 r 1 and he oal payen is consan over he conracual duraion. I is no essenial for he debor o refinance because r = r a any ie, iplying he arke ineres rae is fixed. Figure 11 and Figure 12 describe he disribuion of oal payen when r r. In Figure 11, when r < θ, due o he ean-revering propery of he Vasicek odel, he ineres rae generaed will increase unil asypoically reaching θ. In his case, he debor will no choose o refinance for he sake of aking advanage of lower ineres rae r. However, when r r, he ineres onoonically decreases o he asypoe. In his case, here exiss a unique opial refinance ie beween = and = T, as shown in Figure 12. Fig. 14. The frequency disribuion a differen values of k, when θ =.5/12, σ =.3 and r =.12/12. B. Variaion Analysis The previous secion nuerically displays siilar resuls of hese wo payen schees. The facors leading o such consequence include, say, he paraeer value and he rend of ineres rae in he conex of Vasicek odel. Here we provide ore nuerical experiens o show how he opial refinancing frequency disribuion will change as he he paraeer value changes. Due o he ean-revering propery, he siulaed ineres rae, r, is expeced o be revering o he ean value θ in he long run. We siulae he process for 1 ies for differen values of paraeers and adop he aching he principal repayen ehod in all siulaions. We also include wo condiions ino siulaion. One is ha he iniial ineres rae equals o long-er ean ineres rae. The oher is an exree condiion ha he iniial ineres rae is greaer han he long-er ean ineres rae. (Advance online publicaion: 26 May 212)

9 IAENG Inernaional Journal of Applied Maheaics, 42:2, IJAM_42_2_ he frequency siga=.1 siga=.2 siga=.3 he boundary condions for he opial refinancing ineres rae k=.1 k=.2 k= Fig. 15. The frequency disribuion a differen values of σ, when θ=.5/12, k=.1 and r =.5/12. Fig. 17. of k. The opial refinancing ineres rae boundary a differen values siga=.1 siga=.2 siga=.3 A he expiraion ie T, M(T ) =, and he ODE offers a unique soluion he frequency Fig. 16. The frequency disribuion a differen values of σ, when θ=.5/12, k=.1 and r =.12/12. 1) Paraeer k: The paraeer k easures how fas he ineres rae process will be driven back o he long er ean under Vasicek odel. Figure 13 and 14 show he fac ha, as he value of k rises, he likelihood of refinancing in he second half of he firs year sees a growh when he iniial lending rae equals o he ean lending rae. In he exree condiion ha he iniial ineres rae is greaer han he iniial lending rae, he increase of he reversion speed leading o relaively early opial refinancing. 2) Paraeer σ: To observe he effec of arke rae volailiy on he refinance frequency disribuion, we change he value of σ while keeping oher paraeers fixed. Figure 6 provides he nuerical oupus when r equals o he long-er ean. In his exaple, changes in he value of σ do no lead o significan changes in he frequency disribuion. When r is relaively higher han he longer ean ineres rae θ, he consequence is ore apparen. Figure 15 shows he nuerical plos for his scenario. An apparen convergen paern can be drawn fro Figure 15, where he bes refinance period converges o around he 25h onh as σ decreases. VII. OPTIMAL REFINANCING BOUNDARY In his secion, we review he proble and our soluions fro sochasic conrol perspecive. Consider a orgage conrac odel wih aching he repayen of principal and ineres ehod. We assue he debor pays each onh wih iniial ineres rae r. Again, he debor is only allowed o refinance once when he or she hinks he ie is reasonable. The ousanding balance M(), can be deerined by he ODE: dm() = d r M()d (18) M() = r (1 e r (T ) ) (19) To sudy he opial refinancing sraegy in general, we se V (r, ) as he presen value, a ie, of a coninuous fuure cashflow of s for s T, subjec o a sochasic ineres rae process of r s, for s T, where r = r. Then V (r, ) = E r [ T s exp( s r u du)ds] (2) According o he properies of orgage conrac, a ie T, he expiraion ie, he value of he conrac should be V (r, T ) = (21) Also, one ay ipose anoher boundary condiion for r as one financial arke requireen: V (, ) =, T (22) Consider he curren sudy where one bu only one refinancing is allowed a soe sage hroughou he duraion of he conrac. Suppose he curren ie is and he corresponding arke ineres rae is r, o find he opial arke ineres r opial, a a specified curren ie, for he borrower o refinance, he proble becoes he following sochasic conrol proble. Find r opial such ha he raional debor prefers o refinance when he arke ineres rae is less or equal o r opial. Since he conracual duraion is 24 onh, we should obain r opial for each onh. Under he assupion ha he conrac is signed a ie, we should obain a curve of r opial as a funcion of. If he arke ineres rae level is equal o or below r opial, he debor can grasp he opporuniy o refinance. Because of he challenge in solving such a sochasic conrol proble analyically, we appeal o he ieraive Mone Carlo siulaion ehod o find is nuerical soluion. For his purpose, we define he opial refinancing ineres rae, r opial, as he ineres rae ha he likelihood o refinance a ie is in he inerval P =[.92,.94]. The inerval is seleced for he following reasonable consideraions. Firs, if he probabiliy is oo low for opial refinancing, he resuls will no be eaningless for orgage borrowers. A raional invesor ay no prefer o refinance if he or she will bear ore risk. Second, if a higher or safer probabiliy is chosen, say 99%, he opial refinancing ineres rae will drop drasically o 1.8% in he firs year and 2% in he second (Advance online publicaion: 26 May 212)

10 IAENG Inernaional Journal of Applied Maheaics, 42:2, IJAM_42_2_6 Fig. 18. of σ. he boundary condions for he opial refinancing ineres rae siga=.3 siga=.2 siga=.1 The opial refinancing ineres rae boundary a differen values year, which is unrealisic in he real financial arke. On balance, i is necessary o siulae opial ineres rae level which converges o a relaively sable value while he he randoness of he ineres odel is aken ino consideraion. Suppose ha he opial refinancing ineres raes a he k h and he (k 1) h year are copued as r k, (r k 1 ), respecively. Le F (x) denoe he probabiliy o refinance when ineres rae is x. The probabiliy is obained wih 5, siulaions. The following procedures are carried ou o obain he iniial guess of r opial for ie a (k 2) year. 1. Given he value of r k and r k 1, we se he upper and lower boundary of he opial refinancing ineres rae for r k 2. The upper bound: u = r k 1 The lower bound: l = 2r k 1 r k 2. Le j = (u l)/2 If F (j) is locaed in he inerval P Then we se r k 2 = j. 3. If F (j) < 9.2 Se he new upper bound u = j Else if F (j) > 9.4 Se he new lower bound l = j Reurn o he Sep 2 unil he value of j converges o he opial ineres rae as defined by he inerval P ; 4. Repea he sep 1 3 o obain he opial ineres rae on specific dae and boundary of he opial refinancing ineres rae. To siplify he proble, he ineres rae in siulaed in he uni of onh. Since F (j) is copued by siulaion, he opial ineres raes obained by he previous procedures are unsable and he boundary line is rough. Therefore, i is necessary o odify and enhance he accuracy a each poin by he following wo seps. 1. Check wheher he value of F (r k ) is locaed in he inerval P for coninuously 3 ies. If i does, he ineres rae r k is considered as he eligible one, else go he nex TABLE VI THE OPTIMAL REFINANCING INTEREST RATE FOR DIFFERENT k Tie k =.1 k =.2 k = sep. 2. If F (r k ) > 9.4 Then r k = r k.1 Else if F (r k ) < 9.2 Then r k = r k.1 Repea he Sep 1-2 o ge an enough accurae value of r k. Figure 17 and 18 provides nuerical plos of r opial using ypical paraeers peraining o he usual orgage conrac under sudy. Fro hese plos, one ay end o conjecure soe analyical feaures of r opial as a funcion in ie, including onooniciy and lower boundedness, for insance. I ay be also worhwhile o copare such an opial refinance boundary wih he opial prepayen boundary conained in [] and []. However, he connecions beween hese wo ypes of boundaries are no iediaely clear. Theoreically, any sochasic conrol probles can be equivalenly forulaed as variaional inequaliies or parial differenial equaion wih free boundaries (see []). The opial poin for an opion holder o exercise he conraced righ usually corresponds o he free boundary for he parial differenial equaion. To successfully forulae he proble ino a parial differenial equaion syse, soluion condiions on he free boundary us be specified. For insance, in [], he following wo free boundary condiions are prescribed in he sudy of he prepayen sraegy under siilar ineres rae process: V r r=r opial = 2 V r 2 r=r opial (23) This igh presens useful hins o he opial refinancing boundary for orgage borrowers. However, furher research needs o be carried ou o idenify he righ boundary condiions for he proble discussed in his work. (Advance online publicaion: 26 May 212)

11 IAENG Inernaional Journal of Applied Maheaics, 42:2, IJAM_42_2_6 TABLE VII THE OPTIMAL REFINANCING INTEREST RATE FOR DIFFERENT σ Tie σ =.1 σ =.2 σ = VIII. CONCLUDING REMARKS This paper focuses on he nuerical siulaion approach for finding he bes refinancing sraegy for orgage borrowers in a sochasic ineres environen. Ineresing properies of he opial refinancing ie, including is relaive closeness o he originaion of he conrac and he saisically lowes poin of he ineres curve, are discovered. In his work, Vasicek Model is applied o siulae he onhly ineres rae and boh aching he principal payen ehod and aching he payen of principal and ineres ehod are considered o generae he oal payen. Resuls fro hese epirical experiens end o sugges relaively early refinancing for boh scenarios under he condiions of he orgage conracs se in he paper, pariculary when he iniial borrowing rae is large copared o he long er ean rae. These findings shed lighs on he very iporan financial queries for any propery invesors. In addiion, since orgage conrac is also a ype of opion, he usefulness of our approach is no liied o he proble a hand. Tradiional analyical echniques for characerizing opion conracs, if possible, usually require aheaically srong and soeies paraeer sensiive properies aached o he forulaion of he proble, such as he convexiy exised in he early exercise boundary of he classic Aerican pu opion (see [], [], for insance). In coparison o such analyical ehods, our approach is robus and easy o ipleen. The algorihs conained in his work can be readily applied o a broad class of probles arising fro financial opiizaion and opion pricing. [3] X. Chen, J. Chada, L. Jiang, and W. Zhang, Convexiy of he Exercise Boundary of he Aerican Pu Opion on a Zero Dividend Asse, Maheaical Finance, vol. 18, pp , 28. [4] J. Dai and P. Huang, Prepayen Characerisics of Morgage and Morgage Pricing, Syses Engineering-Theory Mehdology Applicaion, vol 1, 211 [5] X. Feng and D. Xie, Bayesian Esiaion of CIR Model, Journal of Daa Science, vol. 1, pp , 212. [6] W. Hurliann, Valuaion of Fixed and Variable Rae Morgages: Binoial Tree versus Analyical Approxiaions, Decisions in Econoics and Finance, DOI: 1.17/s z. [7] C. F. Lo, C. S. Lau and C. H. Hui, Valuaion of Fixed Rae Morgages by Moving Boundary Approach, Proceedings of he World Congress on Engineering (WCE), London, vol. II, 29. [8] O. Vasicek, An Equilibriu Characerisaion of he Ter Srucure, Journal of Financial Econoics vol. 18, pp , [9] S. Gan, J. Zheng, X. Feng and D. Xie, When o Refinance Morgage Loans in a Sochasic Ineres Rae Environen, Proceedings of The Inernaional MuliConference of Engineers and Copuer Scieniss 212, IMECS 212, March, 212, Hong Kong, pp [1] D. Xie, An Inegral Equaion Approach o Pricing Fixed Rae Morgages, Far Eas Journal of Applied Maheaics, vol. 35, pp , 29. [11] D. Xie, Fixed Rae Morgages: Valuaion and Closed For Approxiaions, Inernaional Journal of Applied Maheaics, vol. 39 (1), 29. [12] D. Xie, D. Edwards, and G. Schleiniger, An Asypoic Mehod o a Financial Opiizaion Proble. In Advances in Machine Learning and Daa Analysis, vol. 72, pp.79-94, S. Ao, e al., eds.; New York: Springer, 29. [13] D. Xie, D. Edwards, G. Schleiniger, and Q. Zhu, Characerizaion of he Aerican Pu Opion Using Convexiy, Applied Maheaical Finance, vol. 18, pp , 211. REFERENCES [1] Alison. E, A course in Financial Calculus, Cabridge Universiy Press, pp , 22. [2] F. Agarwal, J. Driscoll, and D. Laibson, Opial Morgage Refinancing: A Closed For Soluion, Federal Reserve Board or he Federal Reserve Bank of Chicago, March. 28. (Advance online publicaion: 26 May 212)