Decomposing and valuing callable convertible bonds: a new method based on exotic options

Transcription

1 MPRA Munch Personal RePEc Archve Decomposng and valung callable convertble bonds: a new method based on exotc optons Q-Yuan Zhou and Chong-eng Wu and Yun eng Natonal Scence und of Chna 6. ebruary 7 Onlne at MPRA Paper No. 74, posted 3. March 8 9:9 UTC

2 Decomposng and Valung Callable Convertble Bonds: A New Method Based on Exotc Optons * Q-yuan Zhou, Chong-feng Wu, Yun eng nancal Engneerng Research Center Shangha Jao Tong Unversty, 5, Shangha Cty, Chna Abstract: In the framework of Black-Scholes-Merton opton prcng models, by employng exotc optons nstead of plan optons or warrants, ths paper presents an equvalent decomposton method for usual Callable Convertble Bonds (CCB). urthermore, the analytc valuaton formulae for CCB are worked out by usng the analytc formulae for those smpler securtes decomposed from CCB. Moreover, ths method s valdated by comparng wth Monte Carlo smulaton. Besdes, the effects of call clauses, coupon clauses and soft call condton clauses are analyzed respectvely. These gve lots of new nsghts nto the valuaton and analyss of CCB and much help to hedge ther rsks. Key words: Callable convertble bonds; Equvalent decomposton; Up-and-out calls; Amercan bnary calls; Dervatve prcng * Ths paper had been accepted by the 7 th Asa-Pacfc utures Research Symposum (APRS) and pcked out for a presentaton. We greatly apprecated very useful comments on the 7 th APRS by the dscussant Kn Lam from Hong Kong Baptst Unversty. undng for ths work was provded by the Natonal Scence und of Chna. Besdes, we would lke to acknowledge helpful comments by Guofu Zhou n Washngton Unversty and Zongwu Ca n Unversty of North Carolna at Charlotte. Of course, all errors reman our own. Correspondng author, e-mal: qyuanzhousun@63.net. Professor and the vce dean of Anta College of Economcs and Management, n Shangha Jao Tong Unversty Assocate professor n Shangha Jao Tong Unversty

3 . Introducton Convertble bonds have been playng a major role n the fnancng of companes because of ther appealng hybrd feature that provdes nvestors wth both the downsde protecton of ordnary bonds and the upsde return of equtes. In practce, there are multfarous convertble bonds wth dversfed addtonal clauses, such as call clauses, put clauses, reset clauses, screw clauses and negatve pledge clauses and so on. Although convertble bonds n the developed dervatve markets such as Amercan dervatve market are generally very complex, those n the developng dervatves markets such as Chnese dervatve market are relatve smple. Anyway, callable convertble bonds are the most popular. There are many lteratures on the valuaton of the callable convertble bonds. The Black-Scholes-Merton opton prcng theory has become the defntve theoretc foundaton for valung the convertble bonds snce the poneer paper by Ingersoll (977a). or the frst tme, he obtaned the analytc formulae for the callable convertble bonds by employng the theoretcally reasonable one-factor (.e. frm value) no-arbtrage model. rom then on, the theoretcal equlbrum prce of the callable convertble bond s defned as the one that offers no arbtrage opportunty to ether the holders or the ssuers, on the assumpton that at each pont n tme the ssuers execute the optmal call polcy that maxmzes the common shareholder s wealth (.e. mnmzes the value of ths convertble bond) and that the holders execute the optmal converson strateges that maxmze the value of ths convertble bond. The vast majorty of subsequent research has focused on ether extendng

4 Ingersoll s work to more complcated convertble bonds, or further relaxng hs deal condtons. The two-factor (.e. frm value and nterest rate) no-arbtrage model was presented frstly by Brennan and Schwartz (98) and then developed further by Buchan (997), Carayannopoulos (996) and Lvov et al. (4). Although these models based on frm value are theoretcally appealng, they are mpractcal because they nvolve some unobservable parameters (notably, the volatlty of the frm value). The more practcal one-factor (.e. stock prce) no-arbtrage model was gven for the frst tme by McConnell and Schwartz (986). However, n order to capture default rsk of convertble bonds, ther model had to adopt the credt spread approach that would necessarly result nto the theoretc nconsstence because a convertble bond as a knd of hybrd dervatves conssts of a debt part that s subject to default rsk and an equty part that s not. Ths theoretc nconsstence was reduced greatly by Goldman Sachs (994) and Tsverots and ernandes (998). Subsequently, the more reasonable two-factor (.e. stock prce and nterest rate) no-arbtrage model was proposed frstly by Cheung and Nelken (994) and developed further by ntroducng more reasonable nterest rate models (Ho and Pfeffer, 996; Ygtbasoglu, ). Recently, the reduced-form approach has been adopted to consder default rsk of the convertble bonds (Davs and Lschka, 999; Takahash et al. ; Ayache, orsyth and Vetzal, 3; Ygtbasoglu and Alexander, 4, Lao and Huang, 6). To sum up, wth the development of these models, the prcng results have become more and more reasonable and accurate, and the mean of predcton errors can be less than 5% (Barone-Ades, Bermudez and Hatgoanndes, 3). 3

5 However, these models above could not provde the nvestors wth enough help to deeply understand the value components of the callable convertble bonds and the effect of every knd of typcal clauses, and to convenently replcate them so that ther rsk can be effectvely hedged. urthermore, solvng these models generally has to adopt ntrcate numercal procedures that are very dffcult for nvestors, especally n developng dervatve markets. Obvously, those problems wll be solved easly as long as we are able to completely decompose the callable convertble bonds nto smple tradable securtes n the actual market. Snce 96s, researchers have attempted to reasonably decompose the convertble bonds nto smple tradable securtes. Baumol, Malkel and Quandt (966) proposed that a non-callable convertble bond could be regarded ether as ts correspondng ordnary bond (wth the same prncpal and coupons and maturty) wth a detachable call opton struck at the value of ths ordnary bond, or as stocks plus a put opton struck at the value of ths ordnary bond, whch s greater. However, n lght of later research, ther concluson s demonstrably ncorrect. Ingersoll (977a), under hs deal condtons, proved that a non-callable convertble bond had the same value as ts correspondng ordnary bond plus an attached call warrant, and obtaned ts analytc valuaton formula. Nyborg (996) extended hs decomposton by allowng the underlyng stock to pay dvdends and the captal structure to be more complex. However, both Ingersoll and Nyborg vewed the convertble bonds as contngent clams on the frm value. Ths makes parameter estmaton very dffcult snce not all of frm assets are tradable. Connolly (998, chapter 8) vewed them as 4

6 dervatves on the underlyng stock prce, and completely decomposed a non-callable convertble bond nto ts correspondng ordnary bond and European call warrants. Hs decomposton s relatve reasonable n prncpal. However, n the exstng lteratures, untl now s there no method to completely decompose the callable convertble bonds nto smple securtes tradng n the actual market. To all appearances, one callable convertble bond can be drectly decomposed nto three smpler securtes: one ordnary bond, one call opton (.e. the holders convertble opton) and another call opton (.e. the ssuers callable opton). However, ths drect decomposton s not vald because of the unneglgble nteracton between the exercsng of the embedded call opton. As a result, the dfference between the value of ths callable convertble bond and that of the portfolo of these three securtes can not be gnored (Ingersoll, 977a; Ho and Pfeffer, 996). Ingersoll (977a) proved that a callable convertble dscount bond had the same value as ts correspondng ordnary dscount bond plus an attached stock call warrant mnus an addtonal thrd term representng the cost of gvng the callable opton to the ssuers. However, hs model s mpractcal because he vewed the callable convertble dscount bonds as contngent clams on the frm value. Ho and Pfeffer (996) consdered the callable convertble bonds as dervatves on the underlyng stock prce and presented that the value of one callable convertble bond was equal to ts nvestment value (.e. the value of ts correspondng ordnary bond) plus ts embedded warrant value mnus ts forced converson value. However, they only demonstrated the mportance of ts forced converson value and dd not work out ts 5

7 analytc valuaton formula. In a word, none of these exstng decompostons above s good enough to fully llustrate the value components of the callable convertble bonds and to convenently replcate them so that ther rsk can be effectvely hedged. As a matter of fact, due to the nteractons between the embedded convertble opton and the embedded callable opton, one callable convertble bond s equvalent to ts correspondng ordnary bond (wth the same prncpal and coupons and maturty) plus an embedded pecular path-dependent exotc opton, whose exercse prce and exercse tme are ndetermnate. Thus, nevtably, f a callable convertble bond s decomposed wth only non-path-dependent plan optons or warrants, there must be some unregular resdual (e.g. the addtonal thrd term and the forced converson value mentoned above). In ths paper, n the framework of Black-Scholes-Merton opton prcng models, accordng as the rsk-neutral valuaton prncple, by employng smple exotc optons nstead of plan optons or warrants, an equvalent decomposton method s presented for the Callable Convertble Bonds (CCB) defned n Subsecton 3.. Usng ths method, one callable convertble dscount bond can be completely decomposed nto ts correspondng ordnary dscount bond and three knds of smple exotc optons: regular Amercan bnary calls wth an mmedately-made fxed payment, regular up-and-out calls and regular Amercan bnary calls wth a fxed payment deferred untl maturty. Smlarly, one coupon-bearng callable convertble bond can be completely decomposed nto ts correspondng ordnary bond and fve knds of smple 6

8 exotc optons. Intutvely and exactly, ths method shows us the value components of CCB. Obvously t s very helpful to convenently replcate CCB and effectvely hedge ther rsks. urthermore, the analytc valuaton formulae for CCB are worked out by makng full use of the exstng analytc valuaton formulae for these smple securtes decomposed from CCB. At the same tme, these analytc formulae for CCB are valdated by comparng wth Monte Carlo smulaton. Wthout doubt, these formulae can produce prcng results and correspondng Greeks more convenently and quckly, because they need not to consume huge computatonal resources necessary for numercal procedures. Besdes, they can be used to analyze the effects of call clauses, coupon clauses and soft call condton clauses respectvely. These obvously gve a lot of new nsghts nto the valuaton and analyss of CCB. The remander of ths paper s organzed as follows. In the next secton, the assumptons and the ratonale needed n ths paper are explcated n detal. In Secton 3, we present an equvalent decomposton method for CCB. In Secton 4, the analytc valuaton formulae are worked out. Subsequently, Secton 5 valdates these formulae by comparng wth Monte Carlo smulaton. In Secton 6, we further analyze n detal the effect of every knd of typcal clauses respectvely. Secton 7 concludes the paper.. Valuaton framework.. Assumptons (a) The framework of Black-Scholes-Merton opton prcng models s adopted. 7

9 It s well-known that ths framework s very rgorous and has been relaxed gradually n order to value stock optons more exactly. However, ths framework has stll often been adopted n order to obtan analytc valuaton formulae for those complex dervatve securtes. As we know, n the Black-Scholes-Merton framework, captal market s both perfect and effcent; the term structure of the rsk-free rate of nterest s flat; there s no rskless arbtrage opportunty; and the underlyng stock prce follows the dffuson process below. ds = μ Sdτ + σ SdW () where the varable W follows a standard Wener process under the probablty measure Ρ ; μ and σ are the expected rate of return and volatlty of the underlyng stock prce respectvely. Let r denote the contnuous rsk-free nterest rate and assume that r s constant **. Ths assumpton s relatvely reasonable snce both Brennan and Schwarz (98) and Carayannopoulos (996) concluded that, for the reasonable range of parameters, the addton of an nterest rate factor dd not sgnfcantly mprove the model s accuracy. (b) All nvestors prefer more wealth to less. That s to say, the holders of the convertble bonds always seek to maxmze the prce of the convertble bonds; the ssuers of the convertble bonds, as the deputes of the shareholders, act at all tmes to maxmze the shareholders wealth,.e. the underlyng stock prce. (c) Both the holders and the ssuers behave wth symmetrc market ratonalty. ** Snce Black and Scholes (973) are only nterested n the underlyng asset prce at maturty, they can allow r to be known functons of tme. However, CCB and exotc optons nvolved n ths paper depend n complex ways on the tme path of the varable r. Smply, we assume here that r s constant through tme. 8

10 Ths mples that both the holders and the ssuers are completely ratonal and one part can expect the optmal behavors of the other. The same assumpton was adopted n many lteratures such as Ingersoll (977a) and Barone-Ades and Bermudez and Hatgoanndes (3). (d) The potental dluton, whch results from the possble converson n the future, has already been reflected n the current underlyng stock prce. That s to say, the convertble bonds can be valued wthout correcton for dluton by usng the volatlty of the quoted share (Connolly, 998)... The ratonale Accordng as the rsk-neutral valuaton prncple, n the rsk-neutral world, the expected return on all securtes s the rsk-free nterest rate and the present value of any payoff can be obtaned by dscountng ts expected value at the rsk-free nterest rate (Cox and Ross, 976). Although the rsk-neutral world s merely an artfcal devce for prcng dervatve securtes n the framework of the Black-Scholes-Merton opton models, the valuaton formulae obtaned n the rsk-neutral world are vald n all worlds. When we move from a rsk-neutral world to a rsk-averse world, two thngs happen. The expected growth rate n the stock prce changes and the dscount rate that must be used for any payoff from the dervatves changes. It happens that these two changes always offset each other exactly (Hull,, chapter ). As seen n Harrson and Kreps (979), n the rsk-neutral world, the dffuson process that the underlyng stock prce follows becomes 9

11 ds = rsdτ + σsdw () where the varable W follows another standard Wener process under the rsk-neutral probablty measure Ρ, whch s equvalent to the probablty measure Ρ. Obvously, n the rsk-neutral world, the expected return rate becomes the rsk-free nterest rate, but the expected volatlty has no change. 3. Decomposng the callable convertble bond 3.. Defnton In ths paper, we focus on the usual Callable Convertble Bond (CCB) whose converson feature and call feature are defned as follows. More specfcally, (d) they enttle the holders to convert them nto common shares at the predetermned converson prce at any tme n the future; (d) they enttle the ssuers to call them back at the predetermned call prce at any tme n the future; (d3) they have no call notce perod (ths lmt s relatve reasonable because the effect of the call notce perod s relatve lttle); (d4) both the converson prce and the call prce are constant; (d5) they have the usual screw clauses,.e. upon converson the holders can not receve accrued nterests any longer; (d6) they have no put clauses and reset clauses and other non-standard clauses. In Subsecton 6.4, we wll dscuss further when they have the soft call condton clauses. Although CCB wth these clauses are relatve smple, ther value components are very smlar wth those of more complex convertble bonds wth varous flavor and forms. Therefore, f we completely decompose ths knd of CCB nto smple securtes

12 tradng n the actual market, we wll better understand the value components of CCB and better replcate them, even the more complex convertble bonds. Consder one CCB defned above. or convenence, we denote ts face value by B, converson prce by P, call prce by B, remanng tme to maturty by T. c Then, ts converson rato,.e. the number of shares of the underlyng common stocks nto whch t can be converted, s ( ) B P. Wthout loss of generalty, assume that t stll has N tmes payments of nomnal coupons from now to maturty. Let (,, ) τ = N denote correspondngly the tme span from now to the th ex-coupon date. Obvously, τ N = T. Let ( =,, ) C N and tme R ( =,, N ) denote respectvely the coupon amount and the coupon rate at τ. In ths way, obvously C =. And let ( ; ) B R Pv T C denote the present value of all comng nomnal coupons from now to maturty and v( T; C ) denote the future value of them at maturty. Let Pv( τ ; C ) denote the present value of all comng nomnal coupons from now to the tme τ announce a call on ther own ntatve and v( τ ; C, ) at whch the ssuers wll denote the future value of them at tme τ. Besdes, let S, S τ and S T denote the underlyng stock prce respectvely at current tme zero, at any future tme τ and at maturty T, where < τ T. Let CCB( S, T; C) denote ts theoretcal value at current tme zero and B ( TC ; ) denote the theoretcal value at current tme zero of ts correspondng ordnary bond (wth the same prncpal and coupons and maturty),.e. the so-called nvestment value.

13 3.. Constrant Condtons Based on the assumpton (d) above, the converson of CCB would not result n the mmedate reducton of the underlyng stock prce snce the underlyng stock prce has already reflected the potental dluton. Thus, ts converson value at any tme τ wll be exactly equal to ( B ) P S τ. rom McConnell and Schwartz (986), ts theoretcal value must be at least as great as ts converson value and otherwse a rskless arbtrage opportunty exsts. In addton, ts so-called nvestment value can provde t wth the downsde protecton at any tme. Hence, the theoretcal value of CCB at any tme n the future before the call announcement and maturty must satsfy ( τ ) ( τ ) ( ) CCB Sτ, T ; C max B Sτ, T ; C, B P S τ (3) ollowng McConnell and Schwartz (986) and Barone-Ades, Bermudez and Hatgoanndes (3), due to the callable opton, ts theoretcal value wll not be possble to exceed the predetermned call prce. ( τ ) CCB Sτ, T ; C Bc (4) Puttng (3) and (4) together, we can obtan ( τ ) ( ) ( τ ) max, ;,, ; B Sτ T C B P Sτ CCB Sτ T C Bc (5) If a call were to be announced at tme τ pror to maturty, snce no call notce perod (see Subsecton 3.), the holders would have to choose mmedately the more attractve of the two optons: acceptng the call prce B c n cash or obtanng the converson value ( ) B P S τ, where S τ denote the underlyng stock prce at τ. ( τ τ ) ( ) CCB S, T ; C = max B P S, B at call (6) τ c If no call were to be announced pror to maturty, accordng to the optmal

14 converson strateges gven n the next subsecton, CCB would be held untl maturty. At maturty, the holders can accept the balloon payment or convert to obtan the converson value, whch s greater. Due to the usual screw clauses, the balloon payment s B +. Therefore, the fnal condton s ( ) ( ) CCB ST,; C = max B P ST, B + C N (7) 3.3. Optmal converson strateges ( B ) The holders are enttled to convert one unt of CCB at any tme n the future nto P unts of shares of the underlyng common stock. Based on the assumpton (b) above, optmal converson strateges of the holders are those strateges that maxmze the theoretcal value of CCB. Theorem : Gven the assumptons n the subsecton., t s optmal for the holders never to voluntarly convert the callable convertble bond defned n the subsecton 3. except at maturty or the call announcement. The proof of ths theorem sees Appendx A. In fact, ths theorem s smlar wth Ingersoll s Theorem II (Ingersoll, 977a) that a callable convertble securty wll never be exercsed except at maturty or call. The only dfference s that he vewed CCB as the contngent clams on the frm value, but we vew CCB as dervatves on the underlyng stock prce. Pror to maturty, f a call were to be announced, from (6) the holders must choose mmedately between acceptng the call prce n cash and convertng. Based on the assumpton (c) above, the holders can expect the optmal call polcy of the ssuers. 3

15 rom Theorem n the next subsecton, t s optmal for the ssuers to announce a call as soon as the underlyng stock prce reaches S = ( Bc / B) value reaches the call prce, ( ) τ c τ P,.e. the converson B P S = B. Therefore, upon the call announcement, the holders would be ndfferent between acceptng the call prce n cash and convertng. If no call were to be announced pror to maturty, CCB would be held untl maturty. At maturty, from the fnal condton (7), t s self-evdent that the holders should voluntarly convert f the converson value ( B ) P S s greater than the balloon payment B +,.e. the underlyng stock prce at maturty S T s greater T than the adjusted converson prce ( B ) P +, and otherwse clam the balloon payment Optmal call polces The ssuers are enttled to call CCB back at the predetermned call prce at any tme n the future. Based on the assumpton (b), optmal call polces of the ssuers are those polces that maxmze the underlyng stock prce or, what s the same thng, mnmze the theoretcal value of CCB. Theorem : Gven the assumptons n the subsecton., t s optmal for the ssuers to announce to call back the callable convertble bond defned n the = / P. subsecton 3. as soon as the underlyng stock prce reaches S ( Bc B) The proof of ths theorem sees Appendx B. In fact, ths theorem s smlar wth Ingersoll s Theorem IV (Ingersoll, 977a). Upon the call announcement, the holders τ 4

16 wll be n the same way ndfferent between acceptng the call prce n cash and convertng, though he vewed CCB as the contngent clams on the frm value and we vew CCB as dervatves on the underlyng stock prce,. In practce, however, the call polces executed by the ssuers are not consstent wth these theoretcal works. The ssuers generally delay announcng a call untl the converson value s substantally hgher than the call prce (Ingersoll, 977b; Constantndes and Grundy, 987). Some reasons are demonstrated by Jalan and Barone-Ades (995) and Ederngton, Caton and Campbell (997) and so on. In order to consder ths nconsstency, followng Barone-Ades, Bermudez and Hatgoanndes (3), the restrcton condton (4) can be modfed as: ( τ ) CCB S, T ; C kbc τ (8) where k s a convenently-chosen factor bgger than one. In the same way, we can obtan that t s optmal for the ssuers to announce a call as soon as the underlyng ( c ) stock prce reaches Sˆ = k B / B P. τ 3.5. The equvalent decomposton Concerned wth the endng of CCB, based on the assumptons n the subsecton. and the optmal converson strateges n the subsecton 3.3 and the optmal call polcy n the subsecton 3.4, there exst only three possble cases. or convenence, let ( ) P = S = B / B P. τ c In the frst case, the underlyng stock prce wll reach P pror to maturty, and then the ssuers wll announce a call at once on ther own ntatve. At that tme, the 5

17 holders wll be ndfferent between acceptng the call prce n cash and convertng. In the second case, the underlyng stock prce wll not reach P pror to maturty but at maturty wll exceed the adjusted converson prce ( B ) P +, and then CCB wll be voluntarly converted at maturty by the holders on ther own ntatve. In the thrd case, the underlyng stock prce wll nether reach P pror to maturty nor at maturty exceed the adjusted converson prce, and then CCB wll be redeemed at maturty by the ssuers. As a matter of fact, snce the crtcal stock prce P can be regarded as the barrer of a regular Amercan bnary call wth an mmedately-made fxed payment, the payoff feature of CCB n the frst case s smlar wth that to ths regular Amercan bnary call. urthermore, snce the crtcal stock prce P and the adjusted converson prce ( B ) P + can be regarded respectvely as the barrer and the exercse prce of a regular up-and-out call, the payoff feature of CCB n the second case s smlar wth that to ths regular up-and-out call. Therefore, frstly we can try to separate ths Amercan bnary call and regular up-and-out call from CCB respectvely. nally, CCB can be completely decomposed nto ts correspondng ordnary bond and fve knds of smple exotc optons through four steps as follows. At the frst step, off one unt of CCB, we strp ( B / ) Amercan bnary calls, denoted as ABC ( S, T; P P, P ) P unts of long regular, whose fxed payment ( P P ) s made mmedately when the underlyng stock prce reaches the barrer P for the frst tme. At the second step, from the rest, we separate ( B / ) P unts of long regular 6

18 (, ; P, P ) up-and-out calls, denoted as UOC S T ( B ) +, whose barrer s also P and whose exercse prce s the adjusted converson prce ( B ) P +. After two steps above, the resdual can be completely decomposed nto three smpler securtes. One s a short non-regular Amercan bnary call, denoted as d ABC ( S, T; B + v( T; C ), P ), wth a tme-varyng payment B + v( T; C) deferred untl maturty when the underlyng stock prce reaches the barrer P for the frst tme. Another s a long non-regular Amercan bnary call, denoted as (, ; ( ; ), τ ) ABC S T B v C P +, wth an mmedately-made ndetermnate payment B + v( τ ; C) when the underlyng stock prce reaches the barrer P for the frst tme. And the thrd one s ts correspondng ordnary bond B ( S, T; C ). In order to better demonstrate the value components of CCB, we contnue the ABC ( S, T; B + v T; C, P ) can be further completely d fourth step. In bref, ( ) decomposed nto one regular Amercan bnary call wth a fxed payment B d deferred untl maturty, denoted as ( ) ABC S, ;, T B P, and one non-regular Amercan bnary call wth a tme-varyng payment v( T; C ) deferred untl maturty, ABC ( S, T; v T; C, P ). ( τ ) d denoted as ( ) (, ; ;, ) ABC S T B + v C P can be further completely decomposed nto one regular Amercan bnary call wth an mmedately-made fxed payment B, denoted as ( ) ABC S, ;, T B P, and one non-regular Amercan bnary call wth an mmedately-made ndetermnate payment v (, ; ;, ) ( τ ; C), denoted as ( ) ABC S T v τ C P. Theorem 3: Gven the assumptons n the subsecton., one unt of the 7

19 callable convertble bond defned n the subsecton 3. has the same value at any tme as the portfolo consstng of ( B / P unts of long regular Amercan bnary calls ) ABC ( S, T; P P, P ), ( B / P ) unts of long regular up-and-out calls (, ; ( ), B UOC S T d ABC ( S, T; B, P ) ) ) ) + P P), one unt of short regular Amercan bnary call (, ; ( ; ), d ABC S T v T C P ABC ( S, T; B, P (, ; ( ; ), ABC S T v C P, one unt of short non-regular Amercan bnary call, one unt of long regular Amercan bnary call, one unt of long non-regular Amercan bnary call B S, T; C. Ths τ, and ts correspondng ordnary bond ( ) can be shown as the followng equaton. (, ; ) CCB S T C ( ) ( ) ( ) ( ) ( + B ) = + B / P ABC S, T; P P, P B / P UOC S, T; P, P d (, ;, ) (, ;, ) + ABC S T B P ABC S T B P d (, ; ( τ ; ), ) (, ; ( ; ), ) + ABC S T v C P ABC S T v T C P (, ; ) + B S T C (9) The proof of ths theorem s proved n Appendx C. Obvously the equaton (9) demonstrates fully the value components of CCB. It s worth notng that ( ) d ABC ( S, T; v( T; C ), P ) and ABC S, ; T v( τ ; C) are non-regular Amercan bnary calls. ortunately, both of them result only from coupon payments and the holders take the short poston n the former and the long poston n the latter. It turns out that ther total contrbuton to the value of CCB s relatvely small, especally at near maturty and low current stock prce. In fact, ABC ( S, ;, T B P ) and ( B / P ) unts of (, ;, ) ABC S T P P P may be merged nto ABC ( S, T; ( B P / P ), P ), whose fxed payment ( B P / P ) 8

20 s made mmedately. Then, the equaton (9) becomes (, ; ) ( ( ) ) ( ) ( + B ) CCB S T C ( ) ( τ ) = ABC S, T; B P / P, P + B / P UOC S, T; P, P ( ) ( ) (, ; ( ; ), ) (, ; ) d ABC S, T; B, P + ABC S, T; v ; C, P d ABC S T v T C P B S T C + () Ths equaton mples that CCB can be completely replcated wth only fve knds of exotc optons and ts correspondng ordnary bond. Let ( C = =,, N ), then CCB retrogresses to the callable convertble dscount bond. Accordngly, the equaton () becomes (, ;) = (, ;( / ), ) + ( / ) (, ;, ) d ABC ( S, T; B, P ) + DB ( S, T;) CCDB S T ABC S T B P P P B P UOC S T P P () Ths equaton mples that the callable convertble dscount bond can be completely replcated wth only three regular exotc optons and ts correspondng ordnary ( ) dscount bond, DB S, T;. Let Bc +, then P +, the callable opton wll never be exercsed. Then, CCB retrogresses to the non-callable convertble bond. Accordngly, the equaton () becomes ( ) ( ( ) ( ) ( B ) CB S, T; C = B / P W S, T; + P + B S, T; C) () ( + P) where, ; ( W S T B ) ( B ) P denotes a European call warrant wth the exercse prce + and the remanng tme to maturty T. Ths equaton mples that the non-callable convertble bonds can be completely replcated wth European call warrants and ts correspondng ordnary bond. In essence, ths equaton s the same as the one derved from the bnomal tree method by Connolly (998, Chapter 8). 9

21 4. Aanalytc valuaton formulae or regular Amercan bnary calls and up-and-out calls mentoned above, ther analytc formulae have already been obtaned n the Black-Scholes-Merton framework by Rubnsten and Rener (99a and 99b). or the non-regular Amercan bnary call ( τ ) ABC ( S, T ; v ; C, P ), ts analytc formula has been derved n Appendx D. In short, the analytc formulae for these securtes decomposed from CCB can be drectly expressed below. ( ) ( ) ( ) ( ) ( ) ( ) ( ) μ+ μ, ;, / σ μ μ / σ = + ( ) ABC S T P P P P P P S N a P S N a (, ;( + B ) P, P ) rt ( ) ( B ) ( σ ) rt SN ( d) ( B ) Pe N( d σ T) UOC S T = SN d + Pe N d T + + S P S N d3 + Pe P S N d3 T ˆ μ/ σ C / N rt μ σ ( / ) ( ) ( B ) ( / ) ( + σ ) ˆ μ/ σ C / N rt μ σ ( / ) ( ) ( B ) ( / ) ( + σ ) S P S N d4 + Pe P S N d4 T (3) (4) rt μ/ σ (, ;, ) = ( / ) ( ) + ( ) d ABC S T B P Be P S N a3 N a4 ( ) ( ) ( ) ( ) ( ) ( ) μ+ μ, ;, / σ μ μ / σ = + ( ) ABC S T B P B P S N a P S N a μ/ σ (, ; ( ; ), ) = ( ; ) ( / ) ( 3) + ( 4) d ABC S T v T C P Pv T C P S N a N a (, ; ( τ ; ), ) ABC S T v C P μ/ σ N ( P/ S) N( a3) N( a4) + rτ = BRe μ/ σ = ( P / S) N( a5) + N( a6) ( ) N = (5) (6) (7) (8) rt r B S, T; C = B e + Ce τ (9) where, μ = r σ, ˆ μ, = r + σ, μ = ( μ + rσ )

22 ( ) ( T ), a = ln ( P / S) μt / ( σ ) a= ln P/ S +μt / σ ( ( N ) ) μ ( σ ) d = ln S ˆ / + C B P + T / ( ( N ) ) μ ( σ ) d3 = ln P ˆ / S + C B P + T / T, T, d = ln ( S / P) +μˆ T / ( σ ) T, T, d4 = ln ( P / S) +μˆ T / ( σ ) T, ln ( / ) / ( T ), a4 = ln ( P / S) μt / ( σ ) a = P S +μt σ 3 ln ( / ) / ( ), a6 = ln ( P / S) μτ / ( σ τ ) a = P S + μτ σ τ 5 T, and ( ) N x s the cumulatve probablty dstrbuton functon for a varable x that s normally dstrbuted wth a mean of zero and a standard devaton of.. By substtutng the equatons (3) through (9) nto the equaton (), the analytc formula for CCB can be obtaned easly. Despte the seemngly complex form, ths formula s theoretcally rgorous. Moreover, ts dervaton requres only the same precondtons about captal markets as the Black-Scholes opton prcng formulae. Besdes, t needs to estmate only σ. In practce, wdespread use of ths formula can be expected owng to ts several obvous advantages below. rst, t can be used to quckly estmate the value of CCB wthout consumng huge computaton resource always requred by numercal procedures. Second, base on t, the mportant Greeks (such as delta and gamma) for rsk management can be drectly calculated. Thrd, t may be used for senstvty analyss that can gve much help to desgn CCB. our, t may also help nvestors seze possble rskless arbtrage opportuntes between CCB and ts duplcate portfolo mentoned n Theorem 3.

23 5. Comparson To assess the valdty of the equvalent decomposton above, we have compared the prcng results from our analytc formula wth those from Monte Carlo smulaton (Boyle, Broade and Glasserman 997), whch has been wdely consdered as an essental method n the prcng of daly montored dervatve securtes. In ths paper, Monte Carlo prces are computed by usng, smulaton paths on assumpton that there are 5 closng prces per year,.e. Δ t = / 5. Moreover, the antthetc varable technque for varance reducton s adopted. Snce our analytc formula s obtaned n the contnuous context, ts prcng results for the daly montored CCB consequentally ncludes contnuty errors. In order to remove the contnuty errors, we have adopted the contnuty correcton by Broade, Glasserman and Kou (997). Specfcally, the orgnal barrer adjusted to be P exp ( βσ Δ t ), where β.586. P should be Wthout loss of generalty, consder a numercal example of the daly montored CCB: $, B = R.4(,, N) = =, P = $, B = $, r =.3, σ =.3. Snce both the current underlyng stock prce and the remanng tme to maturty are state varables, comparsons are made n the followng two dfferent cases. In the frst case, we set the remanng tme to maturty to be fve years and the current stock prce to be varable wthn the reasonable range from $3 to $, whch s equally dvded nto 5 ntervals,.e. Δ S = ( 3) / 5 = $.8. In the second case, we set the current stock prce to be $ (at the money) and the remanng tme to maturty to be varable wthn the range from zero to fve years, whch s equally c

24 dvded nto 5 ntervals too,.e. Δ τ = 5/5=.. As llustrated n g. and, the prcng results from our analytc formula wth the contnuty correcton (denoted as Soluton wth correcton ) are extremely close to those from Monte Carlo smulaton (denoted as Smulaton ). The mean of percentage errors relatve to the results from smulaton s only.3% and the largest does not exceed.8% n magntude. Moreover, wth the number of smulaton paths ncreasng, the percentage errors become smaller. Hence, our analytc formula s ndeed vald. 8 6 Smulaton Soluton wthout correcton Soluton wth correcton CCB(S,T;C) S g. Comparson when the current stock prce s varable 3

25 8 6 CCB(S,T;C) Smulaton Soluton wthout correcton Soluton wth correcton g. Comparson when the remanng tme to maturty s varable T To llumnate the effect of contnuty errors, the prcng results from our analytc formula wthout the contnuty correcton (denoted as Soluton wthout correcton ) are also llustrated n g. and. By comparson, t can be concluded that the uncorrected results are always greater than the corrected ones. Moreover, the closer the current stock prce s to the barrer P, the larger ther dfferences are. The mean of the percentage errors s.6% and the largest reaches.38%. Hence, t s better to adopt the contnuty correcton when our analytc formula s appled to the dscretely montored CCB. 6. Analyzng the callable convertble bond 6.. Theoretcal value and state varables On the assumptons stated n the subsecton., the theoretcal value of CCB depends on two state varables: ts remanng tme to maturty and the current 4

26 underlyng stock prce. By employng the same numercal example n the secton 5, the three-dmensonal graph (see g. 3) has been plotted to demonstrate the relatonshps between ts theoretcal value and two state varables. g. 3 shows clearly that ts value ncreases wth the current underlyng stock prce. g. 3 also shows that ts value rupture downsde shortly after the ex-coupon dates and ncreases gradually wth the remanng tme to maturty decreasng durng the perods between two conjont coupon dates except the last. In the same way, based on the formulae from (3) to (9), the three-dmensonal graphs can be plotted easly to demonstrate the relatonshps between the value of each component of CCB and two state varables. 5 CCB(S,T;C) S 6 4 T g. 3 Relatonshps between CCB ( S ), T; C and two state varables 6.. The effect of coupon clauses Wthout doubt, coupon payments must add the theoretcal value of CCB. However, the added value by coupon payments s always less than the present value 5

27 of all comng nomnal coupons because of two reasons below. rst, f CCB were to be called back pror to maturty, the nomnal coupons hereafter would not be pad any more. Second, f t were to be voluntarly converted at maturty, due to the screw clauses the last nomnal coupon would not be pad. In prncpal, the added value by coupon payments obvously should be the dfference between CCB ( S, T; C ) and CCB( S, T ;). In terms of the equatons (9) and (), t can be expressed as (, ) = (, ; ) (, ;) CCBCoupon S T CCB S T C CCB S T d ( ; ) { (, ; ( ; ), ) (, ; ( τ ) )} ;, ( B / P) UOC ( S, T; P, P) UOC S, T; ( B ) P, P = Pv T C ABC S T v T C P ABC S T v C P () { ( ) } + By employng the same example above, ts three-dmensonal graph (see g. 4) has been plotted too. g. 4 shows clearly that t decreases wth the current stock prce ncreasng. Moreover, the curves of the relatonshp between t and the remanng tme to maturty look saw-toothed. 5 CCBCoupon(S,T) S 6 4 T g. 4 Relatonshps between CCBCoupon ( S, T ) and state varables To further demonstrate the effect of coupon clauses, we have desgned another 6

28 ndcator that s the rato of the added value by coupon payments to the present value of all comng nomnal coupons. It can be expressed as (, ; ) (, ) / ( ; ) Rato S T C = CCBCoupon S T Pv T C () Smlarly, we plot ts three-dmensonal graph (see g. 5). g. 5 clearly shows that t decreases from to wth the current stock prce ncreasng. Ths s because the hgher the current stock prce s, the more possble t s for the ssuers to call CCB back pror to maturty. In addton, t ncreases gradually wth the remanng tme to maturty decreasng durng the perods between two conjont coupon dates except the last, but ruptures downsde shortly after the coupon dates, especally near at-the-money. Rato(S,T) T 8 S 6 g. 5 Relatonshps between ( ) Rato S, ; T C and state varables 6.3. The effect of call clauses Snce the only dfference between CCB and ts correspondng non-callable 7

29 convertble bond rests wth call clauses, the effect of call clauses can be obtaned by subtractng the value of the former from that of the latter. In terms of the equatons () and (), ts analytc formula can be derved below. (, ; ) = (, ; ) (, ; ) Call S T C CB S T C CCB S T C ( ) ( ( ) ) ( ) ( ) ( / ), ;( B ), ; /, d ( ) ( + B ) + ( d ( τ ) ) + ( ) = B P W S T + P ABC S T B P P P B / P UOC S, T; P, P ABC S, T; B, P ( ) ABC S, T; v C,, P ABC S, T; v C, T, P () Its three-dmensonal graph has also been plotted (see g. 6) by employng the same example above. g. 6 clearly shows that t ncreases wth the current stock prce and/or the remanng tme to maturty CCBCall(S,T) S 6 4 T g. 6 Relatonshps between CCBCall ( S ), T and state varables 6.4. The effect of soft call condton clauses Commonly, CCB are ssued wth soft call condton clauses that restrct the ssuers to exercse the callable opton. In ths secton, we analyze the effect of the soft 8

30 call condton clauses where the ssuers may call CCB back only f the underlyng stock trades for no less than a predetermned trgger prce (denoted as P ). Based on Theorem, P must be greater than the crtcal stock prce P = S = ( B / B ) P,.e. τ c ( / ) ( / ) B P P > B P P = B, or else the ssuers wll not be restrcted at all by the c soft call condton clauses to exercse the callable opton. Obvously, the soft call condton clauses beneft the holders. Based on the analyss n the subsecton 3.3 and 3.4, snce P > P = S τ, t s optmal for the ssuers to announce a call mmedately as soon as the underlyng stock prce reaches the trgger prce P ; and then the holders must choose convertng at once snce at that tme ( / ) ( / ) B P P > B P P = B. Except at the call c announcement, the soft call condton clauses have no effect on the converson optmal strateges n the subsecton 3.3. Therefore, wth the same proof as the equaton (), the analytc valuaton formula for CCB wth the soft call condton clauses can be expressed as. (, ; ) ( ( ) ) ( ) ( + B ) d ABC ( S, T; B, P) + ABC ( S, T; v( τ ; C), P) d ABC ( S, T; v( T; C), P) + B( S, T; C) CCB S T P ( ) = ABC S, T; B P / P, P + B / P UOC S, T; P, P (3) In ths way, the effect of the soft call condton clauses can be expressed as (, ;, ) (, ; ) (, ; ) CCBSoft S T P P = CCB S T P CCB S T P (4) Its three-dmensonal graph has also been plotted (see g. 7) by usng the same example above and settng P = $3. g. 7 clearly shows that t ncreases wth the current stock prce and/or the remanng tme to maturty. 9

31 CCBSoft(S,T) S 6 4 T g. 7 Relatonshps between (, ;, ) CCBSoft S T P P and state varables 7. Concluson Ths paper presents an equvalent decomposton method for the callable convertble bonds (CCB) defned n Subsecton 3., on the assumpton that they are dervatves on ther underlyng stock prces accordng to Brennan and Schwarz (98) and Carayannopoulos (996). Usng ths method, the callable convertble dscount bond can be completely replcated wth ts correspondng ordnary dscount bond and three knds of regular exotc optons; the coupon-bearng callable convertble bond can be completely replcated wth ts correspondng ordnary bond and fve knds of exotc optons. These are very helpful to understand the value components of varous callable convertble bonds and to replcate them and hedge ther rsks. urthermore, the analytc valuaton formulae for CCB have been obtaned and valdated by comparng wth Monte Carlo smulaton. These formulae can save huge computatonal resources requred by numercal procedures. Moreover, although these 3

32 formulae seem complcated, both the requred assumptons about captal market and parameter estmatons are the same as the Black-Scholes opton prcng formulae. Therefore, wdespread use of these formulae n practce would be expected, especally n the developng dervatves markets such as Chnese market. In addton, we analyze n detal respectvely the effects of coupon clauses, call clauses and soft call condton clauses on the theoretc value of CCB. These gve a lot of new nsghts nto the analyss of varous callable convertble bonds. A useful drecton for further research s to analyze the mpacts of other clauses such as put clauses or other factors such as default rsk and dvdends, whch have not been consdered n ths paper. Appendx A Proof: Consder two nvestment portfolos: Portfolo I conssts of only one unt of CCB; Portfolo II conssts of ( B ) P unts of shares of the underlyng stocks. Snce no dvdend has been assumed n Subsecton., Portfolo II always conssts of ( B ) P unts of shares of the underlyng stock. If no call were to be announced pror to maturty, from the nequalty (3), pror to maturty Portfolo I would be worth at least as great as Portfolo II, even f there s no coupon. At maturty, n terms of the equalty (7), the payoffs to these two portfolos are compared n Table. Table shows clearly that Portfolo I s generally worth more than Portfolo II unless not only the holders voluntarly convert at maturty but 3

33 also there s no coupon, n whch case they have the same value. Table. Demonstraton that at maturty the payoff to Portfolo I wll be Portfolo at least as great as that to Portfolo II. Current value S T Stock prce at maturty < ( + B ) P ST ( + B ) P I CCB ( S, T; C ) B ( ; ) + v T C ( ) ( ; ) B P S + v T C C T N II ( B P) S ( B P) S ( B ) T P S T Relatonshp between termnal values of Portfolo I and II V I > VI I V I V II If a call were to be announced pror to maturty, assumng at that tme the underlyng stock prce s S τ, from the equalty (6) Portfolo I would be worth ( ) max B P Sτ, B c. The payoffs to these two portfolos at the call announcement are compared n Table. Table shows that Portfolo I wll never be worth less than Portfolo II and n some cases wll be worth more, even f there s no coupon. Table. Demonstraton that at the call announcement the payoff to Portfolo I wll never be less than that to Portfolo II. Portfolo Current value I ( ), ; stock prce at the call announcement ( ) B P S B B P S < B τ c ( ) CCB S T C ( ) ( B ) P Sτ + v τ ; C Bc + v( τ ; C) II ( B P) S ( B P) S τ ( B ) τ P S τ c Relatonshp between the values of Portfolo I and II V I VI I V I > V II 3

34 To sum up, both condtons for domnance defned by Merton (973) exst. Hence, unless the current value of Portfolo I exceeds the current value of Portfolo II, CCB S, ; T C > B P S, the former wll domnate the latter. Obvously, CCB.e. ( ) ( ) should never be voluntarly converted except at maturty or the call announcement. Appendx B Proof: Suppose that ths theorem s not the case. rom the nequalty (5), both the declne of nterest rates and the rse of the underlyng stock prce can ncrease the lower lmt of CCB. However, snce the flat term structure has been assumed n Subsecton., only the latter s relevant here. In terms of the nequalty (5), t s very clear that the lower lmt wll approach the upper lmt wth the underlyng stock prce ncreasng. Therefore, the optmal call polcy must yeld a crtcal stock prce S τ so that t s optmal for the ssuers to announce a call as soon as the underlyng stock prce reaches S τ. rom the nequalty (5) agan, we can be sure S ( Bc / B) P. Assume that t s optmal for the ssuers to announce a call as soon as the S B / B underlyng stock prce reaches ( c ) τ < τ P. Let τ denote the tme at whch the underlyng stock prce reaches S τ for the frst tme. Accordng to ths assumed optmal call polcy, f τ < T, the ssuers wll mmedately announce a call at tme τ. S B / B rom the equalty (6) together wth ( c ) τ < accept the call prce n cash when the ssuers announce a call. P, the holders must choose to 33

35 ( τ τ ) ( ) CCB S, T ; C = max B P S, B τ c = B c (B) On the other hand, assume that the ssuers do not follow the assumed optmal call polcy and wll announce a call as soon as the underlyng stock prce reaches ( c / ) B B P. Let ˆ τ denote the tme at whch the underlyng stock prce reaches ( c / ) B B P for the frst tme. Due to ( c / ) B B P > S τ, we must get τ < ˆ τ. rom the nequty (5), we can obtan at tme τ ( τ ) ( ) τ ( τ ) max B T ; C, B P S CCB S, T τ; C Bc (B) rom Barone-Ades, Bermudez and Hatgoanndes (3), the equalty ( ) τ ( τ, τ ; ) = = c s vald only when S = τ ( Bc / B) B P S CCB S T C B < τ ( c / S B B P. However, )P. Hence, f the ssuers announce a call as soon as the underlyng stock prce reaches S = ( Bc / B) τ ( τ τ ) P, we can obtan CCB S, T ; C < Bc (B3) rom (B) and (B3), we can know that the assumed optmal call polcy can not result n the mnmum prce for CCB, so t s not optmal. Hence, t must be optmal for the ssuers to call CCB back as soon as the underlyng stock prce reaches ( / ) P. S = B B τ c Appendx C Let U denote the set of the paths where the underlyng stock prce wll reach the crtcal value P from below pror to maturty. Let V denote the set of the paths where the underlyng stock prce at maturty wll exceed ( B ) P +. In ths way, the 34

36 set U can be expressed as U = { τ T } where τ denotes the frst tme at whch the underlyng stock prce reaches the crtcal value V { ST ( B ) P } { τ, ST ( B ) } { τ, T ( B ) } the set V as P from below pror to maturty, = > +, the ntersecton UV as UV = > T > + P and the ntersecton UV as UV = > T S + P. In terms of the descrpton descrbed n Subsecton 3.5, the frst, second and thrd case of the endng of CCB respectvely corresponds to the set U, UV and UV. Let ( A ) denote the ndcator functon of the set A. Then, t s easy to get ( ) ( ) ( ) E U + UV + UV = (C) Based on these, n the rsk-neutral world, the payoffs to the correspondng ordnary bond and exotc optons decomposed from CCB can be expressed respectvely as follows. rt ( ) ( ) B S, ; ; T C = B e + Pv T C (C) ( ) rτ * e P P, τ < T Ρ rτ ABC ( S, T; P P, P) = = ( P P) E e ( U ), otherwse C N rt (, ;( + B ), ) = ( ),, ( ) T + B τ > T > + B, τ > T, ST ( + B ) rt Ρ ( = e E S ) T + B P UV (C3), τ T UOC S T P P e S P T S P (C4) P ( ) ( ) rτ, * e B τ T Ρ rτ ABC ( S, T; B, P) = = BE e ( U ) (C5), otherwse rt d e B, τ T rt Ρ ABC ( S, T; B, P) = = Be E ( U ), otherwse (C6) 35

37 where ( τ ) rτ e v ; C, τ T Ρ ABC ( S, T; v( τ ; C), P) = = E Pv( τ ; C) ( U ) (C7), otherwse ( ) rt d e v T; C, τ T Ρ ABC ( S, T; v( T; C), P) = = Pv( T; C) E ( U ), otherwse (C8) N r ; Ce τ = ( ) Pv T C = (C9) N ; ( ) rt ( ) rt τ ( ; ) v T C = C e = e Pv T C = (C) k r ( ; ) k Pv τ C = C e τ τ τ < τ = k+ (C) ( ) v τ C e Pv τ C C e τ τ τ τ τ ( ) ( ) ; = k rτ r ; = k < k+ (C) = Obvously, the payoffs to CCB n the rsk-neutral world are a lot more complex than these exotc optons above. If the frst case of ts endng happens, ts present rτ value can be expressed as e ( B / P) P + v( τ ; C) rt ts present value s ( / ) ( ; ). If the second case happens, e B P ST + v T C C N. If the thrd case happens, ts present value s ( ; ) rt e B + v T C. So the total payoffs to CCB can be expressed as follows. 36

38 (, ; ) CCB S T C ( / ) + ( τ ; ) f τ ( / ) T ( ; ) N f τ, T ( + B ) + ( ; ) f τ >, T ( + B ) Ρ rτ Ρ rτ v ( τ ; C ) e ( U ) rt Ρ e E { ( B / P ) ST v( T; C) ( UV) } rt Ρ e E { B v( T; C) ( UV) } rτ e B P P v C T = + > > rt e B P S v T C C T S P rt e B v T C T S P ( / ) ( ) = B P PE e U + E (C3) Substtutng the equatons (C) through (C8) nto the equaton (C3) yelds (, ; ) CCB S T C Ρ rτ Ρ rτ Ρ ( B / P)( P P) E e ( U) B E e ( U) E Pv( τ ; C) ( U) = + + rt ( ) ( B ) rt ( ) ( ) ( ) ( ) Ρ Ρ + B / P e E ST + P UV + B + v T; C e E UV rt Ρ ( ; ) ( ) + B + v T C e E UV ( B / P ) ABC ( S, T; P P, P ) ABC ( S, T; B, P ) = + ( ) ( ( τ ) ) ( ) ( B ) + ABC S, T; v ; C, P + B / P UOC S, T; + P, P Ρ ( ; ) ( ) rt Be + PvTC E U Ρ ( ; ) ( ) ( ) ( ) rt + Be + PvTC E U + UV + UV ( ) ( ) ( ) ( ) ( B ) = B / P ABC S, T; P P, P + B / P UOC S, T; + P, P ( ) + ABC S, T; B, P ABC S, T; B, P d ( ) d (, ; ( τ ; ), ) (, ; ( ; ), ) + ABC S T v C P ABC S T v T C P (, ; ) + BS TC (C4) Therefore, Theorem 3 holds n the rsk-neutral world. Accordng to the rsk-neutral valuaton prncpal, Theorem 3 stll holds even f the assumpton of the rsk-neutral world s relaxed. Appendx D rom Rubnsten and Rener (99b), we can get the expresson of 37

39 Ρ E ( τ τ < τ j ) (, ; * ( ; ), ) ABC S T v C P. Based on ths expresson, the analytc formula for τ can be derved below. (, ; ( τ ; ), ) ABC S T v C P = ( ) < k Ρ rτ r( τ τ ) E e Ce τ T τk τ τk+ = ( τ τ τ ) ( ) ( τ τ τ ) = Ce E < + Ce + Ce E < rτ Ρ rτ rτ Ρ 3 N N rτ Ρ rτ Ρ Ce E ( N T) Ce E = = ( ) + τ τ < + τ = T N rτ Ρ { Ce E ( τ τ τn) } = < = N rτ { Ce Pr ( τ τn) Pr ( τ τ) } = < < < < = + μ/ σ N ( P / S) N( a3) N( a4) + rτ = BRe μ/ σ = ( P / S) N( a5) + N( a6) (D) References Ayache, E., orsyth, P.A., Vetzal, K.R., all 3. The valuaton of convertble bond wth default rsk. Journal of Dervatves (), 9-9. Barone-Ades, G., Bermudez, A., Hatgoanndes, J., 3. Two-factor convertble bonds valuaton usng the method of characterstcs/fnte elements. Journal of Economc Dynamcs & Control 7, Baumol, W.J., Malkel, B.G., Quandt, R.E., 966. The valuaton of convertble securtes. Quarterly Journal of Economcs 8, Bermudez,A., Webber, N., 3. An asset based model of defaultable convertble bonds wth endogensed recovery. Workng paper. Cass Busness School of Cty Unversty. 38