PRICING CONVERTIBLE BONDS WITH KNOWN INTEREST RATE. Jong Heon Kim

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1 Kngweon-Kyungki Mth. Jour , No. 2, pp PRICING CONVERTIBLE BONDS WITH KNOWN INTEREST RATE Jong Heon Kim Abstrct. In this pper, using the Blck-Scholes nlysis, we will derive the prtil differentil eqution of convertible bonds with both non-stochstic nd stochstic interest rte. We lso find numericl solutions of convertible bonds eqution with known interest rte using the finite element method. 1. Introduction Interest rte derivtives re instruments whose pyoffs re dependent in some wy on the level of interest rtes. In lst two decdes, the vlue of trding in interest rte derivtives in both the over-counter nd exchnge-trded mrkets incresed very quickly. Mny new products were developed to meet prticulr needs of end users. A key chllenge for derivtives trders is to find good, robust procedures for pricing nd hedging these products. Wrrnts re long-term cll options issued by firm tht give the holder the right to purchse the firm s common stock t predetermined pricethe exercise price, on or before n expirtion dte. Convertible bonds re hybrid instruments, hving chrcteristics of both debt nd equity. Like stright bonds, convertible bonds re entitled to receive coupons nd principl pyments. However, convertible bondholder hs the option to forgo these rights by converting their bonds into stock t prespecified rte. In its simplest form, convertible bond cn be decomposed into stright bonds nd wrrnt. The theory of option nd wrrnt pricing hs only of lte been plced on sound theoreticl bsis in context of security mrket equilibrium[2, 7]; closed form expressions hve been derived by Blck-Scholes[2] nd Received October 11, Mthemtics Subject Clssifiction: 74S05. Key words nd phrses: convertible bonds, two-fctor model, finite element method.

2 186 Jong Heon Kim Merton[8] for the vlue of n option when the underlying stock pys no dividend or the option is protected ginst dividends, nd when the stock pys continuous dividend which is proportionl to the mrket vlue of the stock. Cox nd Ross[5] hs extended this option pricing model to tke ccount of jumps in security returns, nd the bsic option pricing model hs been shown to obtin under certin ssumptions, even in the bsence of continuous trding opportunities[9]. More recently, Schwrtz developed lgorithms to solve the relevnt dynmic progrmming problem when the stock does py dividends nd the option is not protected ginst dividend pyments, so tht the possibility of exercise prior to mturity must be considered for n Americn type option. Merton[8] hs considered the relted problem of vluing cllble wrrnts on nondividend pying stocks : cllble wrrnts differ form convertible bonds in hving no coupon pyments. In recent yers there hve been number of ttempts to extend the models so tht they involve two or more fctors. A number of reserchers hve investigted the properties of two-fctor equilibrium models. Brennn nd Schwrtz[3, 4] developed model where the process for the short rte reverts to long rte. Another two-fctor model, proposed by Longstff nd Schwrtz[7], strts with generl equilibrium model of the economy nd derives term structure model where there is stochstic voltility. In section 2, using the Blck-Scholes nlysis, we derive the prtil differentil eqution of convertible bonds under the ssumption of the known interest rte. In section 3, we lso derive convertible bonds eqution with stochstic interest rte. In section 4, we find the numericl solutions of the convertible bond with known interest rte using the finite element method[1, 6]. 2. Convertible bonds with known interest rte In this section, we mke preprtions for the vlution of convertible bonds with ssumption of known interest rte, these bonds re very similr to Americn vnill option. We illustrte the ides with constnt interest rte nd, t the section 3, we briefly bring together convertible bonds nd stochstic interest rtes in two-fctor model. A convertible bond hs mny of the sme chrcteristics s n ordinry bond but with the dditionl feture tht the bond my, t ny time of the owner s

3 Pricing convertible bonds with known interest rte 187 choosing, be exchnged for specified sset. This exchnge is clled conversion. The convertible bond on n underlying sset with price S returns Z, sy, t time T unless t some previous time the owner hs converted the bond into mconversion rtio of the underlying sset. The bond my lso py coupon to the holder. Since the bond price depends on the vlue of tht sset we hve V V S, t the contrct vlue now depends on n sset price nd the time to mturity. Repeting the Blck-Scholes nlysis, with portfolio Π consisting of one convertible bond nd ssets under ds µsdt + σsdx dv Π V S µs V + V t σ2 S 2 2 V dt + σs V 2 dx where dx is the Wiener process nd µ is the drift rte nd σ is the voltility for the stock, respectively. Let we include coupon pyment of KS, tdt on the bond nd dividend pyment of DS, tdt on the sset, then we find tht the chnge in the vlue of the portfolio is We cn choose then dπ dv ds + KS, t DS, tdt µs V + V t σ2 S 2 2 V dt + σs V 2 dx µsdt σsdx + KS, t DS, tdt. V V dπ t σ2 S 2 2 V + KS, t DS, t dt. 2 The return on this risk-free portfolio is equl tht from bnk deposit nd so dπ rπdt V t σ2 S 2 2 V + rs DS, t V rv + KS, t 0 2

4 188 Jong Heon Kim where r is the interest rte. This prtil differentil eqution is recognized s the bsic Blck-Scholes eqution but with the ddition of the coupon pyment term. The finl condition is V S, T Z. Reclling tht the bond my be converted into m ssets we hve the constrint V S, t ms. In ddition to this constrint, we require the continuity of V nd V /. Thus the convertible bond is similr to n Americn option problem. It is interesting to note tht the finl dt itself does not stisfy the pricing constrint. Thus, lthough the vlue t mturity my be Z the vlue just before is mxms, Z. Boundry conditions re nd V S, t ms s S V 0, t Z exp{ rt t} this lst condition ssumes tht it is not optiml to exercise when S 0. Theorem 1. If the vrible S stisfies ds σsdx + µsdt where dx is Wiener process nd µ nd σ re constnt. Then the function V S, t stisfies following prtil differentil eqution V t σ2 S 2 2 V + rs DS, t V rv + KS, t 0 2 provided Z if t T mx{ms, Z} if 0 < t < T V S, t ms if S Z exp{ rt t} if S 0 V S, t ms.

5 Pricing convertible bonds with known interest rte Convertible bonds with stochstic interest rte When interest rtes re stochstic, convertible bond hs vlue of the form V V S, r, t with dependence on T suppressed. The vlue of the convertible bond is now function of both S, r nd t. We ssume tht the sset price is governed by the stndrd model 1 ds µsdt + σsdx 1 nd the interest rte by 2 dr ur, tdt + ωr, tdx 2. Since we re only modelling the convertible bond, nd do not intend finding explicit solutions, we llow u nd ω to be ny functions of r nd t. Observe tht in 1 nd 2 the Wiener processes hve been given subscripts. This is becuse we re llowing S nd r to be governed by two different rndom vribles ; this is two-fctor model. Thus, lthough dx 1 nd dx 2 re both drwn from norml distributions with zero men nd vrince dt, they re not necessrily the sme rndom vrible. They re, however, correlted by E[dX 1 dx 2 ] ρdt with 1 ρs, r, t 1. In order to mnipulte V S, r, t we need to know how Itô s lemm pplies to functions of two rndom vribles. As might be expected, the usul Tylor series expnsion together with few rules of thumb results in the correct expression for the smll chnge in ny function of both S nd r. Remrk. The Wiener processes of 1 nd 2 hve the following properties dx 2 1 dt dx 2 2 dt dx 1 dx 2 ρdt. Proof. By the Winner process δx ε δt

6 190 Jong Heon Kim where ε is drwn from stndrdized norml distribution. E[δX 2 ] δte[ε 2 ] δt V r[ε] + E[ε] 2 δt V r[δx 2 ] V r[ε 2 δt] δt 2 V r[ε 2 ] tends to zero s δt 0. Thus δx 2 is non-stochstic nd dx 2 1 dt. And E[dX 1 dx 2 ] ρdt V r[dx 1 dx 2 ] E[dX 1 dx 2 2 ] E[dX 1 dx 2 ] 2 δt 2 ρ 2 δt 2 1 ρδt 2 V r[dx 1 dx 2 ] tends to zero s δt 0. Thus dx 1 dx 2 lso non-stochstic nd dx 1 dx 2 ρdt. Applying Tylor s theorem to V S + ds, r + dr, t + dt we find tht 1 V S + ds, r + dr, t + dt ds i! + dr r + dt i V S, r, t t To leding order, nd dv V V ds + r i0 dr + V t dt 2 V 2 ds2 + 2 V r dsdr ds 2 σ 2 S 2 dx 2 1 σ 2 S 2 dt dr 2 ω 2 dx 2 2 ω 2 dt dsdr σsωdx 1 dx 2 ρσsωdt. 2 V r 2 dr2 + Thus Itô s lemm for the two rndom vribles governed by 1 nd 2 becomes dv V V V ds + dr + r t dt σ2 S 2 2 V dt + ρσsω 2 V 2 r dt V 2 ω2 r dt. 2

7 Pricing convertible bonds with known interest rte 191 Now we come to the pricing of the convertible bond. Let us construct portfolio consisting of one bond with mturity T 1, 2 bonds with mturity dte T 2 nd of the underlying sset. Thus Π V 1 2 V 2 S V1 dπ t V 2 V1 dt + t 2 V1 + r V 2 2 r 2 V 1 + ρσsω r 2 V 2 2 r We cn choose so we find nd V 2 ds dr σ2 S 2 2 V 1 2 V 2 2 dt 2 2 dt V 1 2 ω2 r 2 2 V 1 V V 1 r V 2 2 r 0 2 V 1/ r V 2 / r V 1 V 1/ r V 2 V 2 / r 2 V 2 r 2 elimintes risk from the portfolio. Now the portfolio is risk-free, dπ rπdt dt. dπ r V 1 2 V 2 S dt rv 1 rs V 1 dt 2 V1 t V 2 2 t 2 V 1 + ρσsω r 2 V 2 2 r rv 2 rs V 2 dt σ2 S 2 2 V 1 dt 2 V 2 2 dt 2 2 dt V 1 2 ω2 r V 2 dt. r 2

8 192 Jong Heon Kim Gthering together ll V 1 terms on the left-hnd side nd ll V 2 terms on the right-hnd side we find tht V1 t σ2 S 2 2 V 1 + ρσsω 2 V 1 2 r V 1 2 ω2 r rv rs V 1 V2 2 t σ2 S 2 2 V 2 + ρσsω 2 V 2 2 r V 2 2 ω2 r rv rs V 2 Let then I V 1 t σ2 S 2 2 V 1 + ρσsω 2 V 1 2 r V 1 2 ω2 r rv rs V 1 I b V 2 t σ2 S 2 2 V 2 + ρσsω 2 V 2 2 r V 2 2 ω2 r rv rs V 2 I V 1 / r I V 1/ r V 2 / r I b I b V 2 / r. This is one eqution in two unknowns. However, the left-hnd side is function of T 1 nd the right-hnd side is function of T 2. The only wy for this to be possible is for both side to be independent of the mturity dte. Thus, dropping the subscript from V, V t σ2 S 2 2 V +ρσsω 2 V 2 r V V 2 ω2 rv +rs r2 V r for some function S, r, t. It is convenient to write S, r, t ωr, tλs, r, t ur, t for given ωr, t nonzero nd ur, t, this is lwys possible. function λs, r, t is the mrket price of risk. V t +1 2 σ2 S 2 2 V +ρσsω 2 V 2 r +1 2 V V 2 ω2 +rs r2 +u ωλ V r S, r, t The rv 0. This is the convertible bond pricing eqution. Note tht it contins the known interest rte problem u 0 ω : Blck-Scholes eqution. / 0 : The simple bond problem zero-coupon bond.

9 Pricing convertible bonds with known interest rte 193 More generlly, when the underlying sset pys dividends nd the bond pys coupon we hve V t σ2 S 2 2 V + ρσsω 2 V 2 r V 2 ω2 r 2 + rs D V + u ωλ V rv + K 0. r Since this is diffusion eqution with two spce-like stte vribles S nd r tht is, there re double derivtives of V with respect to ech of S nd r, s well s cross-term-we need to impose boundry conditions on the edge of the S, r spce. In other words, we must prescribe V 0, r, t nd V, r, t for ll t, V S,, t for ll S nd t nd second boundry condition on fixed r boundry, gin for ll S nd r. Some of these boundry conditions re very obvious nd others re result of insisting tht V remin finite. For exmple, for convertible bond with on cll feture we hve V S, r, t ms s S V 0, r, t is given by the solution of the simple bond problem no convertibility nd stochstic interest rtes. V S, r, t 0 s r nd the lst boundry condition, to be pplied on the lower r boundry, is equivlent to finiteness of V. Theorem 2. If the vrible S nd r stisfy the eqution 1 nd 2 respectively, then the function V S, r, t stisfies following eqution provided V S, r, t V t σ2 S 2 2 V 2 + ρσsω 2 V r ω2 2 V r 2 + rs D V V S, r, t ms. + u ωλ V r rv + K 0 bond conditions if S 0 Americn cll option conditions if r 0 ms if S mx{ms, bond price} if 0 < t < T 0 if r

10 194 Jong Heon Kim 4. Numericl solutions for convertible bonds In this section we find the numericl solutions of convertible bonds under the ssumption of the known interest rte using the finite element method. Suppose the convertible bond pricing eqution is s following 3 V S, t t σ2 S 2 2 V S, t V S, t + r d S rv S, t + K 0 2 where r, ddividend yield nd K re constnts nd Boundry conditions re with constrint S b, 0 t T. V S, T mx {ms, Z} V 0, t Z exp { r T t} V b, t m V S, t ms. Now we represent the differentil term of time to difference of time, then the eqution 3 becomes 1 t + r V S, t n σ2 S 2 V S, t n 1 r dsv S, t n 1 1 t V S, t n K 0 where t t i t i 1 0 t 0 < t 1 < < t n T nd Define V S, t V S, t, 2 V S, t 2 us : V S, t n 1 u S : V S, t n V S, t.

11 Pricing convertible bonds with known interest rte 195 then the eqution 3 becomes 1 4 t + r us 1 2 σ2 S 2 u S r dsu S 1 u S K 0. t Since t is not vrible from now on, u is function with only vrible S. Suppose ΨS C 1 [, b] is test function with compct supprt on [, b], we multiple the test function to both side of the eqution 4 nd integrte 1 b t + r usψsds 1 b 2 σ2 S 2 u SΨSdS 5 r d K b b Su SΨSdS 1 t ΨSdS 0. b u S ΨSdS Becuse eqution 5 is zero for ll test function ΨS, eqution 3 nd 5 hve the sme solution us. We find the solution us of the eqution 3 by solving the integrl eqution 5 with proper bsis functions nd test functions. First of ll, we define two spces { } 2 P f S f S c k S k, c k R, S b k0 Q {g S P g 0 g b, g i 1} where i + ih i 0, 1, 2,, N nd h b /2k N 2k. We cn choose the test function Ψ i S nd the bsis function Ψ j S on Q s following Cse I. i is even Ψ i 2 S 1 2h 2 S i 1 S i when Ψ i 1 S 1 h 2 S i 2 S i Ψ i l S 1 2h 2 S i 2 S i 1 i 2 S i

12 196 Jong Heon Kim when when Cse II. i is odd Ψ i r S 1 2h 2 S i+1 S i+2 Ψ i+1 S 1 h 2 S i S i+2 Ψ i+2 S 1 2h 2 S i S i+1 i S i+2 Ψ i 1 S 1 2h 2 S i S i+1 Ψ i 0 S 1 h 2 S i 1 S i+1 Ψ i+1 S 1 2h 2 S i 1 S i i 1 S i+1. Finding nlytic solutions of eqution 3 is very difficult, so we look for the numericl solutions. Suppose the pproximted solution of u S is u h S, us u h S N j0 j Ψ j S But we know the solution when j 0, N u h S j Ψ j S j1 Our gol is to find the coefficient j. To simplify the nottion, we define f, g h : b f h S g S ds Now we clculte ech terms of eqution 5 using chosen test nd bsis functions. u, Ψ i h N 1 j1 j b Ψ j SΨ i SdS

13 CseI. i is even Pricing convertible bonds with known interest rte 197 i ji 2 i+2 + ji i j Ψ j SΨ i SdS i 2 j i+2 i Ψ j SΨ i SdS h 15 αn 1 i 2 + 2h 15 αn 1 i 1 + 8h 15 αn 1 i + 2h 15 αn 1 i+1 h 15 αn 1 i+2 CseII. i is odd i+1 ji 1 j i+1 i 1 Ψ j SΨ i SdS 2h 15 αn 1 i h 15 αn 1 i + 2h 15 αn 1 i+1 CseIII. i is N N jn 2 N j Ψ j SΨ i SdS N 2 h 15 αn 1 N 2 + 2h 15 αn 1 N 1 + 4h 15 αn 1 N. S 2 u, Ψ i h [ u SS 2 Ψ i S ] b N 1 j1 j b Ψ js { 2SΨ i S + S 2 Ψ is } ds

14 198 Jong Heon Kim CseI. i is even i ji 2 i+2 ji 1 30h h 1 15h h 1 30h CseII. i is odd i+1 ji h 8 15h h i j Ψ js { 2SΨ i S + S 2 Ψ is } ds i 2 j CseIII. i is N N mb 2 i+2 i Ψ js { 2SΨ i S + S 2 Ψ is } ds ih + h i 2 i ih + h i 2 i ih + h i 2 i ih + h i 2 i ih + h i 2 i+2 i+1 j Ψ js { 2SΨ i S + S 2 Ψ is } ds i ih + h i 2 i ih + h i 2 i ih + h i 2 i+1 jn 2 mb h h 1 30h N j Ψ js { 2SΨ N S + S 2 Ψ NS } ds N Nh + h N 2 N Nh + h N 2 N Nh + h N 2 N

15 Pricing convertible bonds with known interest rte 199 Su, Ψ i h N 1 j1 j b SΨ jsψ i SdS CseI. i is even i ji 2 i ji i j SΨ jsψ i SdS i 2 j i+2 i SΨ jsψ i SdS 5 + h 4 + 5i αn 1 i h 2 + 5i αn 1 i h 15 αn 1 i h 2 + 5i αn 1 i h 4 + 5i αn 1 i+2 30 CseII. i is odd i+1 ji 1 j i+1 i 1 SΨ jsψ i SdS h 2 + 5i αn 1 i h 15 αn 1 i h 2 + 5i αn 1 i+1

16 200 Jong Heon Kim CseIII. i is N N jn 2 N j SΨ jsψ i SdS N 2 CseI. i is even h 4 + 5N αn 1 N h 2 + 5N αn 1 N h N αn 1 1, Ψ i h 2h 3 i+2 i 2 b Ψ i SdS Ψ i SdS N. CseII. i is odd i+1 i 1 Ψ i SdS 4h 3 CseIII. i is N N N 2 Ψ i SdS h 3. For ll cses i, the eqution 5 becomes 1 t + r u, Ψ i h 1 2 σ2 S 2 u, Ψ i h r d Su, Ψ i h 6 1 t u, Ψ i h K 1, Ψ i h 0. In Tble is shown tht the vlue of convertible bonds with Z 1, n 1, r 0.1, σ 0.25 nd with one yer before mturity. In both cses there re no coupon pyments. We cn find tht the price of convertible

17 Pricing convertible bonds with known interest rte 201 bonds with no dividendd is higher thn the price with d It cn be shown tht n increse in d mkes erly exercise more likely. Stock Price CB Priced 0.00 CB Priced References [1] O. Axelsson nd V. A. Brker, Finite element solution of boundry vlue problems, Acdemic Press [2] F. Blck nd M. S. Scholes, The pricing of options nd corporte libilities, Journl of Politicl Economy, , [3] M. J. Brennn nd E. S. Schwrtz, A continuous time pproch to pricing bonds, Journl of Bnking nd Finnce, , [4] M. J. Brennn nd E. S. Schwrtz, An equilibrium model of bond pricing nd test of mrket efficiency, Journl of Finncil nd Quntittive Anlysis, 21, , [5] J. C. Cox nd S. A. Ross, The vlution of options for lterntive stochstic processes, Journl of Finncil Economics, [6] C. Johnson, Numericl solution of prtil differentil equtions by the finite element method, Cmbridge Univ. Press [7] F. A. Longstff nd E. S. Schwrtz, Interest rte voltility nd the term structure: A two fctor generl equilibrium model, Journl of Finnce, 47, , [8] R. C. Merton, The theory of rtionl option pricing, Bell Journl of Economics nd Mngement science, , Number 1. [9] M. E. Rubinstein, The vlution of uncertin income strems nd the pricing of options, Bell Journl of Economics, 7, 1976,

18 202 Jong Heon Kim Deprtment of Applied Mthemtics Kumoh Ntionl University of Technology Kumi , Kore E-mil:

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