THEORETICAL AND NUMERICAL VALUATION OF CALLABLE BONDS Dejun Xie, University of Delaware

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1 The Inernaional Journal of Business and Finance Research Volume 3 Number 009 THEORETICAL AND NUMERICAL VALUATION OF CALLABLE BONDS Dejun Xie, Universiy of Delaware ABSTRACT This paper sudies he value of a callable bond and he bond issuer s opimal financial decision regarding wheher o coninue he invesmen on he marke or call he bond. Assume he marke invesmen reurn follows a sochasic model, he value of conrac is formulaed as a parial differenial equaion sysem embedded wih a free boundary, defining he level of marke reurn rae a which i is opimal for he issuer o call he bond. A fundamenal soluion of he parial differenial equaion is derived, and used o formulae he value of he bond. A bisecion scheme is implemened o solve he problem numerically. JEL: D4, D46 G14, G15 KEYWORDS: Callable Bonds, opimal financial decision, sochasic model INTRODUCTION A callable bond is a bond ha can be redeemed by he bond issuer prior o is mauriy under cerain condiions by paying off all he borrowing balance o he bond holders. Naurally he issuer shall inves he raised capial ino marke, expecing he reurn on he invesmen be higher han he amoun he has o pay o he bond holder. A company will raionally choose o call a bond if i is paying a higher coupon while earning a lower reurn from marke. Here we consider a callable bond wih a duraion T > 0 and a fixed coupon rae c > 0, where he issuer is allowed o call (redeem he bond) premaurely by paying off he borrowing a any ime d < T. We assume he bond is fully amorized, i.e., he issuer pays an equal coupon paymen m> 0 for each uni ime period hroughou he duraion of he bond. This is in conras o oher ypes of bonds, say, zero-coupon bond, for insance, where he issuer does no pay back any amoun of borrowing unil he conrac expires. The amorized callable bond is popular in pracice mainly because paymens are disribued ino equal insallmens over he duraion of he bond, which makes i possible for he issuer o make relaively smooh financial planning and avoid paymen shock. Denoe by P(d) he ousanding borrowing balance owed by he bond issuer o he bond holder a ime d. If he coupon rae is zero, hen P(d) will linearly decrease in ime from d=0 o d = T. However, his is unlikely he case in real economy since i would imply ha he bond issuer uses for free he money borrowed from he bond buyers. The usual siuaion is c> 0, which means ha bond issuer pays more, in sum of all he coupons, han is iniial borrowing P(0). For example, company A raised $1,000,000 as of oday by issuing a 0-year callable bond wih a fixed coupon rae c = 5%. Then he company needs o pay a coupon of m = $/year o he bond holders. We would like o remark ha in pracice, a corporae bond someimes only specifies m, P(0), and T, leaving c as an embedded implici erm of he conrac. For his reason, c is someimes referred as he implied coupon rae or implied ineres rae. Now suppose he bond issuer has raised P(0) dollars of capial a d = 0 and for each uni ime period, he pays a coupon of m dollars back o he bond holder, and if he decides o call a ime d, he mus pay off he borrowing balance of P(d) dollars. Then he valuaion of bond is of ineress o boh he bond issuer and he holder. On one hand, he holder or a hird pary invesor may wan o know he fair value of such a bond. For invesmen banks or securiy companies which hold large porfolio of such bonds, he value of hese bonds may have significan impac on heir credi raings and asse performance. And in siuaions like 71

2 D. Xie The Inernaional Journal of Business and Finance Research Vol. 3 No. 009 company merge or liquidaion, he valuaion of hese bonds ofen becomes necessary, as required by sauory accouning principles. On he oher hand, from he bond issuer s poin of view, he expecs o wisely use he capial such ha he invesmen reurn is higher, on average, han he bond coupons he has o pay. In realiy, he issuer may have many ways o use he capial. Here for simpliciy, we consider his overall reurn from all possible marke invesmen, he reurn rae of which follows cerain sochasic process. The opimal decision a any given ime, from he bond issuer s poin of view, depends on how much reurn can be earned if an equal amoun of capial P(d) be invesed in he marke. Inuiively, he issuer should no choose o call unless he overall invesmen reurn rae is expeced o be very low for cerain amoun of ime. Hence, he issuer mus, a every momen while he conrac is in effec, monior he marke invesmen reurn and decide wheher i is of his bes ineres o call immediaely. The paper is organized as follows. In Secion we provide a review of relaed lieraure. In Secion 3 we inroduce he model of he underlying rae of reurn for he opion and he parial differenial equaion governing he value of he bond. In secion 3, we formulae he inegral equaion represenaion of he value of he bond wih a free boundary incorporaed. In secion 4, we apply he Fourier ransform o derive he Green s funcion used in our inegral formulaion of he soluion. In secion 5, we implemen a bisecion scheme o solve he problem ieraively and presen some specific numerical examples. In secion 6, we presen some concluding remarks on our approach and possible fuure direcions. LITERATURE REVIEW A seminal work of bond valuaion can be raced back o Meron (1974), where he equiy of a firm is reaed as an opion on is asses, and he value of a pure discoun bond of a firm is analyzed under he assumpion ha he asse value of he firm follows geomeric Brownian moion. Because of he imporan role played by callable bonds in real economy, relaed problems have been sudied by considerable lieraures. Black and Cox (1976) exend he heoreical framework of Meron (1974) o he siuaion where he value of a firm follows a diffusion process wih insananeous variance proporional o he square of he value. Geske (1977) provides a compound opion approach for valuing deb of a firm where he deb is resriced o be a single issue of discree coupon deb and he firm is assumed dividend free. Wih hese assumpions, Geske shows ha he value of he corporae deb can be compued as he difference beween he oal value of he firm and he value of he equiy. These earlier works share a common limiaion by assuming marke ineres rae as consan. In an aemp o address he ineres rae effec on he value of bonds as well as oher ineres rae opions, Hull and Whie (1990) propose o value hese opions using a class of marke daa consisen spo rae models. Specifically, hey presen wo one-sae variable models and show ha he differences beween he European syle bond prices compued for hese wo models are small under cerain assumpions. In he sequenial, Hull and Whie (1993) propose a general procedure o consruc ineres rae models for fiing a given se of iniial marke daa. They also provide a compuing framework involving he consrucion of a rinomial ree for he shor rae, working back hrough which he value of bond can be obained. Aiming o generalize he framework for erm srucure models, Heah, Jarrow and Moron (199) direcly impose a sochasic srucure on he evoluion of forward ineres rae curve. The model, based on he heory of arbirage asse pricing, is proposed for valuing he ineres rae opions, including corporae bonds, when lacking he informaion of marke price of risk. One shorcoming of his model is ha i poses more compuaional difficulies in implemenaion. In recen developmen, reduced form mehod is inroduced o sudy he bond valuaion problems where when defaul is allowed. According o Duffie and Singleon (1999), a defaulable bond can be valued by discouning is paymen sreams a a defaul adjused ineres rae hrough a risk neural measure. We would like o remark ha he problems of pricing defauable or callable bonds, like many oher opion valuaion problems, ypically do no have closed form soluions, hus a grea deal of effors have been made 7

3 o find numerical soluions as well as approximaed analyical soluions. For insance, a numerical parial differenial equaion scheme is implemened by Brennan and Schwarz (1977), a binomial laice model was applied by Li e al. (1995), a Mone Carlo simulaion approach is proposed by Kind and Wilde (003), and a finie difference mehod is recenly ried by Breon and Ben-Ameur (005). A quasi-closed form approximaion formula is obained by Tourruco e al. (007) for a zero coupon bond under he Black- Karasinski model. One noices (see Buler (1995), for insance) ha usual numerical echniques such as finie difference or binomial mehod ypically provide poor accuracy and sabiliy, which are mainly caused by he complexiy of he free boundary condiions. Mone Carlo simulaion is easy o implemen, bu is no quie efficien for pricing callable bonds wih is drawback of low convergence. More deailed criiques of hese hese mehods can be found in Jiang (005). To recify such numerical shorcomings, an inegral equaion approach has been recenly proposed by Chen and Chadam (007) and Xie (008) for American opion pricing and relaed problems. The main idea is o formulae he value of he opion under review as an appropriae inegral form, on basis of which one can derive an efficien algorihm o solve he problem ieraively. The same idea is applied in his work. We firs derive an inegral represenaion of he bond value in erms of marke reurn rae x a ime, hen use i o design a bisecion algorihm o ieraively solve for he opimal early call boundary. Then he bond value is recovered by numerical inegral algorihm. THE MODEL AND THE PROBLEM FORMULATION In his work we assume ha he reurn rae ha he bond issuer can earn from marke invesmen follows he Vasicek (1977) model, where he marke reurn rae r is reaed as a Markov process, governed by he sochasic differenial equaion dr = k( θ r ) d + σdw (1) where W is he sandard Brownian process. The Vasicek model is composed of one deerminisic erm and one random erm. The deerminisic erm (also he drif erm ) is chosen o produce he so called meanrevering propery. And he random erm is o model he volailiy caused by (possibly infinie) unpredicable facors. Using inegraing facor mehod, as explained in Jiang (005), for insance, one can solve he sochasic differenial equaion and ge he explici (sochasic) soluion for he reurn rae a any ime > d. k( d) k( d u) = θ + ( d θ) + σ d u r e r e dw We are ineresed in he value of he callable bond a any given ime and he corresponding reurn rae. For mahemaical convenience, insead of using real ime d, we inroduce := T -d. Financially is he ime o expiry of he bond conrac (hereafer simply referred as he ime ). Le V(x, ) be he bond value a ime and he corresponding reurn rae x. Sandard heory of mahemaical ells us ha V(x, ), when he bond is no opimal o call, mus saisfy he Black-Sholes ype parial differenial equaion (hereafer referred as PDE ) σ - - k( x) + xv = m. () V V V θ x x When i is opimal o call a cerain ime, V(x, ) becomes P(). Le h() be he unknown opimal reurn 73

4 D. Xie The Inernaional Journal of Business and Finance Research Vol. 3 No. 009 level for he issuer o exercise he call, hen we have V (x, ) = P() for x <= h(), (3) and his auomaically gives V(x, 0) = 0. (4) I is also apparen ha V x (x, ) = 0 for x < h(). (5) Since V(x, ) is assumed a leas wice differeniable in x, we know ha V x (x,) = 0 for x = h(). (6) By a rivial free of arbirage argumen, one can show he h(0) = c, (7) i.e., he opimal call boundary mus sars a c as we look backward from he expiry dae. Mahemaically ()-(7) consiue a free boundary problem we need o solve, where he free boundary x = h(), defining he opimal level of reurn rae a which he bond is o be called, separaes he righ half -x plane ino wo regions. For he coninuaion region where x > h(), he bond conrac is in effec and he value of he bond is governed by (). For he early exercise region where x < h(), he bond is called and he bond holder ges back he borrowing balance of P(). Because i is he issuer, raher han he holder, has he choice o ac in response o he marke, he value of he conrac is always less or equal han he loan balance. Thus he free boundary is where he value of he bond firs reaches value of P(), i.e., i is opimal for he issuer o call he bond if and only if ha value of he bond reaches P() for he firs ime. To solve he free boundary problem ()-(7), we firs derive he fundamenal soluion of (), hen use i o formulae he soluion V(x, ). MATHEMATICAL MOTIVATION When handling a free boundary PDE sysem, i is ofen emping o invesigae if we can find he soluion for he homogeneous PDE wihou boundary condiions. If his can be done, hen he soluion o he sysem can be formulaed in erms of inegral equaions wih boundary condiions incorporaed. Wihou loss of generaliy, we assume m = 1. Define σ V V L( V ): = + k( θ x) xv, (8) x r We see ha, according o (.5) and (.6), V LV ( ) = F ( x, ), x, > 0 (9) Where 1, for x > h( ), > 0 F(,) x = x x c (1 ) e +, for x h ( ), > 0 c c 74

5 If we can find he soluion, say G(r,y,,τ) o he PDE defined in (8), hen, by he Green s ideniy, we would be able o wrie he soluion as V( x, ) F( y, τ) G( x, y,, τ) d ydτ. 0 = (10) And hen he following manipulaions can be made o find he inegral ideniies on which a numerical ieraion scheme can be designed. h( τ ) y y cτ V = ( + (1 )) e G(, r y,, τ) dydτ + G(, r y,, τ) dydτ 0 c c 0 h( τ ) y y c ( (1 )) e τ = + G ( r, y,, τ) dyd τ 0 c c y cτ G( r, y,, τ)(1 )( e 1) dydτ 0 h( τ ) c Denoe y y cτ I = ( + (1 )) e G( x, y,, τ ) dy c c We are going o show ha I = P(). To his end we do no evaluae I direcly. Insead, we inegrae he differenial form of G wih respec o y over he whole x-space. Le s = τ. Because G, by he naure of being he fundamenal soluion o he PDE in (8), saisfies σ Gs Gyy + [ k( θ y) G] y + yg = 0 (11) and G decays exponenially in y, as will become apparen afer we derive is explici formula, we can apply inegral by pars o inegrae (11) wih respec o y in he whole x-space, hus have d ds Then R Gdy = R ygdy cτ cτ 1 e I = e Gdy + ygdy R c cτ cτ 1 e d = e Gdy Gdy R c ds 1 d cτ = {(1 e ) Gdy} c ds R 75

6 D. Xie The Inernaional Journal of Business and Finance Research Vol. 3 No. 009 Now, using he fac ha Gdy = 1, τ =, we ge R 1 d cτ 1 c Idτ = {(1 e ) Gdy} (1 e ) P( ). 0 = 0 c ds R c Le y cτ U ( x, ) = G( r, y,, τ)(1 )( e 1) dydτ, (1) 0 h( τ ) c we ge V(x,) = P() - U(x, ) (13) And now i is sraighforward o ranslae he boundary condiions (3-6) of V(x, ) ino boundary condiions of U(x, ), i.e., when x = h(), U(x, ) = 0, U x (x,) = 0. (14) (15) So far he analysis is carried ou as if he soluion were known. The following secion is dedicaed o finding he fundamenal soluion G using he Fourier Transform mehod. DERIVATION OF THE FUNDAMENTAL SOLUTION Le G be he fundamenal soluion associaed wih (8). For every (x, ) fixed, Define he Fourier ransform in he x variable by: iλr FGr [ (, y,, τ)] = Gr (, y,, τ) e dr= Gˆ ( λ, y,, τ), we have Gˆˆˆ σ G G + k( θ y) + ( k yg ) ˆ = 0, for τ <, y τ y y ˆ (,,, ) i y G λ y = e λ (16) We posulae ha admis a soluion of he form ˆ (,,, ) A(,, τλ) + yb(,, τλ) e. Gr y = (17) wih A(, T, λ) and B(,T, λ) o be deermined shorly. Subsiuing (17) back ino he parial differenial equaion in (16), we ge 76

7 σ A + B kθ B+ k = 0 B + kb 1= 0 A (,, λ) = 0 B (,, λ) = iλ (18) Using he mehod of separaion of variables o solve he second differenial equaion in (18), we ge 1 1 B ( iλ) e k k k ( τ ) = + (19) Subsiue (19) ino he firs differenial equaion in (18), we have σ A = kθ Bds + B ds kds τ + τ τ 1 ( ) 1 k s σ 1 k( s) 1 = kθ [( iλ) e + ] ds + [( i ) e ] ds k( ) τ k k λ + + τ τ k k k( s) σ k( s) = θ [(1 + iλk) e 1] ds + [(1 + iλk) e 1] ds + k( τ) τ k τ Nex, we wan o collec erms for A + yb according o he powers of λ. For his purpose, we wrie k( s) [(1 λ ) 1] τ σ k( s) k( τ ) iλk e ds k τ y iλ e τ A + yb = θ + i k e ds [(1 ) 1] ( ) [( ) ] k k k = αλ + αiλ+ αλ 1 0 A edious bu sraighforward compuaion will lead o exac expressions of α 0, α 1,and α as funcions in y, and τ. Bu he inegral form expressions of α 0, α 1,and α are good enough for compuaional purposes. Now we can apply he inverse of Fourier ransform o derive he desired he soluion 1 ˆ iλr Gr (, y,, τ) = Gr (, y,, τ) e dλ π 1 αλ α1iλ αλ 0 iλr = e + + e dλ π = 1 e 4πα α ( α1+ r) 0 4α I is easy o verify ha G(x, y,, τ) is exponenially decaying in y for x,, τ fixed, and he spaial inegral 77

8 D. Xie The Inernaional Journal of Business and Finance Research Vol. 3 No. 009 wih respec o y is always equal o 1 for fixed au a. Now he expression for U r (r, ) involves a double inegral over an infinie domain, someimes i is beneficial o reduce he double inegral o a single one. This is because ha he inside inegral of U can be evaluaed by, say, change of variables. Since he compuaion is edious and such a furher simplificaion is no necessary in finding numerical soluions, we omi hese furher compuaions. NUMERICAL EXAMPLE AND DISCUSSION Here we presen some numerical examples obained from our mehod. Our numerical algorihm is based on he free boundary condiion defined in (14), where he inegral represenaion of U(x, ) is given by (1). Recall ha h() is defined as he level of reurn rae x a which V(x, ) reaches P() for he firs ime, or equivalenly, U(x, ) decreases o 0 for he firs ime. Sar from h(0) = c and U(x, 0) = 0, for each > 0 fixed, we apply bisecion scheme o find h(), hen use he inegral represenaion (1) o recover U(x, ), and use (13) o recover V(x, ). To implemen he bisecion scheme, we sar wih he wo iniial guesses of h(). If hey are oo high compared o he rue value of h(), we somehow shif downward he nex guess; if oo low, we somehow shif upward he nex guess; if one is above and he oher is below he rue value, we ake he average o updae he guess. For a given bond wih duraion T and coupon rae c, assuming he model parameers are known, we pariion he ime [0, T] ino N evenly spaced subinervals wih d = T/N. Define = ( 0, 1,..., N ), a ime vecor wih n = nd, n = 0,...N. For a prescribed error olerance level, say, Tole=10-8, we implemen our algorihm as follows. (1) h( 0 ) = c, which is known from (7). () Suppose we have found h( 1 ), h( ),...,h( n-1 ), we apply a bisecion scheme o find h( n ): (a) Take reasonable wo iniial guesses of h( n ), say h 1 ( n ) and h ( n ). We assume h 1 ( n ) > h ( n ); if no, simply swich index 1 and. (b) Apply numerical inegraion mehod, say Simpson quadraure, for insance, o evaluae he inegral in (1), wih h(τ) being inerpolaed by ( 0,h( 0 )), ( 1, h( 1 )),..., ( n, h i ( n )), and ge U(h i ( n ), n ) accordingly for i = 1,, respecively. (c) If U(h 1 ( n ), n ) >0 and U(h ( n ), n ) >0, hen we se h 3 ( n ) = h ( n ) - [h 1 ( n ) - h ( n )]. (d) If U(h 1 ( n ), n ) <0 and U(h ( n ), n ) <0, hen we se h 3 ( n ) = h 1 ( n ) + [h 1 ( n ) - h ( n )]; h ( n ) = h 1 ( n ). (e) If U(h 1 ( n ), n ) >0 and U(h ( n ), n ) <0, hen we se h 3 ( n ) =[ h 1 ( n ) + h ( n )]/; 78

9 (i) If U(h 3 ( n ), n ) > 0, se h ( n ) = h ( n ). (ii) Oherwise, se h ( n ) = h 1 ( n ). (f) If U(h 3 ( n ), n ) <Tole, ieraion ends. If no, use h ( n ) and h 3 ( n ) as wo updaed iniial guesses, repea seps (a) hrough (f) o find h 4 ( n ). Repea such an ieraion unil an index k is reached such ha U(h k ( n ), k ) < Tole. (3) Once h( n ) s have been found for n = 0, 1,..., N, V(x, n ) can be recovered by (13) for arbirary given x using numerical inegraion quadraures. For he ieraion o converge faser, i is beer o sar wih one iniial guess above and he oher below he rue value of h( n ). Our numerical experimens show ha he choices of c and -c are good enough for mos cases. Also o increase he accuracy of he numerical soluion, one can increase N, he number of grids for pariioning he ime inerval [0, T]. For ypical parameers wih T < 30, our numerical simulaions show ha N = 4096 is large enough for achieving a soluion wih relaive error less han 10-6, where relaive error is defined as he difference of numerical values of h(t) s achieved wih differen N s. As an example, consider a 5-year bond as of oday wih coupon paymen m = 1 (dollars per year), and coupon rae c = And he corresponding borrowing balance is P() = (dollars). Assume he parameer values appearing in (.1) are 0 = 0.04, k = 0., a = 0.01, we implemen our numerical mehod and ge he bond value as of oday V as a funcion of curren marke reurn rae x. The c = 0.08 column in Table 1 is he oupu of V (in dollars) of such a bond for x = 0.01,0.0,..., 0.1. Indeed one can see ha V increases as x decreases, when oher variables and parameers fixed. To make opimal financial decision, he bond issuer needs o apply he algorihm o compue V and compare i wih P. If he he curren marke reurn rae x = 0.10, say, hen V = (dollars). Since V(0.10, 5) = < = P(5), he issuer should no call he bond. Financially i means ha cos of fund raising hrough issuance of bond is inexpensive enough ha he issuer can benefi by invesing P(5) amoun of capial o earn a relaively higher marke reurn. On he oher hand, if x = 0.07, hen V(0.07, 5) = P(5) = 4.110, and he issuer should call he bond. Financially i means ha he cos of fund raising hrough issuance of bond is so expensive ha i is a wise decision for he issuer o close he deal. And as ime changes, he issuer shall obain he updaed reurn rae x, redo he compuaion, and make updaed comparison. For similar bonds wih coupon rae c = 0.0,0.04,0.06, one can similarly compue V for x = 0.01,0.0,..., 0.1. The resuls are presened in Table 1. Keeping oher variables and parameers unchanged, one can run he program again for = 10 and = 0, for insance, and he oupus are abulaed in Table and 3. CONCLUSION AND DISCUSSION Assuming he reurn rae of marke invesmen follows he Vasicek model, we formulaed and numerically solved a callable bond valuaion problem. An exac soluion of he governing PDE is obained and used o derive he represenaion of he conrac value. A bisecion algorihm is implemened and validaed o solve he problem numerically. Numerical simulaions show ha our algorihm is fas and sable. 79

10 D. Xie The Inernaional Journal of Business and Finance Research Vol. 3 No. 009 Table 1: Numerical Compuaion of Bond Price V(x,) x c = 0.0 c = 0.04 c = 0.06 c = Table 1 presens a numerical compuaion of bond price V(x,) a differen marke ineres rae x and coupon rae c. Here he duraion of bond is =5, and he values for he model parameers are θ =0.04, k=0., σ = 0.01 Table : Numerical Compuaion of Bond Price V(x,), =10 x c = 0.0 c = 0.04 c = 0.06 c = Table presens a numerical compuaion of bond price V(x,) a differen marke ineres rae x and coupon rae c. Here he duraion of bond is =10, and he values for he model parameers are θ =0.04, k=0., σ = 0.01 In addiion o is mahemaical robusness, he algorihm can be a useful ool for porfolio managemen purposes. Praciioners who wish o creae maximum yield using borrowed capial should be able o apply our program o monior he marke condiion and decide when i is opimal o liquidae is invesmen. While he mehod is designed for valuing he callable bond wih marke ineres following he Vasicek model, we feel i can be exended o similar problems where marke ineres follows oher mean-revering models. One limiaion of our curren work is he assumpion ha he issuer mus sele he balance in whole amoun if he decides o call he bond. As a fuure research direcion, we would like o sudy he opimal prepaymen sraegy for he issuer if parial paymens are allowed. 80

11 Table 3: Numerical Compuaion of Bond Price V(x,), =0 x c = 0.0 c = 0.04 c = 0.06 c = Table 3 presens a numerical compuaion of bond price V(x,) a differen marke ineres rae x and coupon rae c. Here he duraion of bond is =0, and he values for he model parameers are θ =0.04, k=0., σ = 0.01 REFERENCES Buler, H.J. (1995), Evaluaion of callable bonds: Finie difference mehods, sabiliy and accuracy, Economic Journal, vol. 105, Duffie, D. & Singleon, K.J. (1999), Modeling erm srucures of defaulable bonds, Review of Financial Sudies, vol. 1, Brennan, M. J. & Schwarz, E. S. (1977), Converible Bonds: Valuaion and Opimal Sraegies for Call and Conversion The Journal of Finance, vol. 3, Chen X. & Chadam, J. (007), Mahemaical analysis of an American pu opion, SIAM J. Mah. Anal., vol. 38, Xie, D. (008), A PDE approximaion approach for he opimal early exercise boundary of American pu opion, Far Eas Journal of Applied Mahemaics, vol. 33, Hull, J. C. & Whie, A. (1990), Pricing ineres rae derivaive securiies, Review of Financial Sudies, vol. 3, Brennan, M. & Schwarz, E. (1977), Savings bonds, reracable bonds, and callable bonds, Journal of Financial Economics, vol. 5, Vasicek, O. A. (1977), An equilibrium characerizaion of he erm srucure, J. Fin. Econ, vol. 5, Kind, A. & Wilde, C. (003), A Simulaion-Based Pricing Mehod for Converible Bonds, Derivaives Research, New York Universiy Faculy Digial Archive, S-DRP Tourruco, F., Hagan, P. S. & Schleiniger, G. F. (007), Approximae Formulas for Zero-coupon Bonds, Applied Mahemaical Finance, vol. 14,

12 D. Xie The Inernaional Journal of Business and Finance Research Vol. 3 No. 009 Meron, R. (1974), On he pricing of corporae deb: he risk of srucure of ineres raes, Journal of Finance, vol. 9, Breon, M. and Ben-Ameur H. (005), A Finie Elemen Mehod for Two Facor Converible Bonds, Numerical Mehods in Finance, Springer US, Kau, J. B. & Keenan, D.C. (1995), An Overview of he Opion-Theoreic Pricing of Morgages, J. of Housing Res., vol. 6, Jiang, L. (005), Mahemaical Modeling and Mehods of Opion Pricing, World Scienific Publishing Company Li, A., Richken, P & Subranmanian, S. (1995), Laice Models for Pricing American Ineres Rae Claims, Journal of Finance, vol. 50, Black, F. and Cox, J. C. (1976), Valuing Corporae Securiies: Some Effecs of Bond Indenure Provisions, Journal of Finance, vol. 31, Geske, R. (1977), The Valuaion of Corporae Liabiliies as Compound Opions, Journal of Financial and Quaniaive Economics, vol. 1, Hull, J. and Whie, A. (1993), One Facor Ineres Rae Models and he Valuaion of Ineres Rae Derivaive Securiies, Journal of Financial and Quaniaive Analysis, vol. 8, Heah D., Jarrow, R. and Moron, A. (199), Bond Pricing and he Term Srucure of Ineres Raes: A New Mehodology for Coningen Claims Valuaion, Economerica, vol. 60, BIOGRAPHY Dejun Xie is a UNIDEL Researcher in he Deparmen of Mahemaical Science a he Universiy of Delaware. He holds a Ph.D. in Applied Mahemaics from he Universiy of Pisburgh. His curren research ineress include quaniaive economics and mahemaical finance. 8