A Note on Risky Bond Valuation C. H. Hui Banking Poliy Depatment Hong Kong Monetay Authoity 0th Floo,, Gaden Road, Hong Kong Email: Cho-Hoi_Hui@hkma.gov.hk C. F. Lo Physis Depatment The Chinese Univesity of Hong Kong hatin, Hong Kong E-mail: flo@phy.uhk.edu.hk Abstat This pape develops a opoate bond valuation model that inopoates a default baie with dynamis depending on stohasti inteest ates and vaiane of the opoate bond funtion. ine the volatility of the fim value affets the level of leveage ove time though the vaiane of the opoate bond funtion, moe ealisti default senaios an be put into the valuation model. When the fim value touhes the baie, bondholdes eeive an exogenously speified numbe of iskless bonds. We deive a losed-fom solution of the opoate bond pie as a funtion of fim value and a shot-tem inteest ate, with time-dependent model paametes govening the dynamis of the fim value and inteest ate. The numeial esults show that the dynamis of the baie has mateial impat on the tem stutues of edit speads. This model povides new insight fo futue eseah on isky opoate bonds analysis and modelling edit isk.
I. INTRODUCTION In piing opoate bonds, Blak and Cox (976) assume a bankupty-tiggeing level fo the opoate assets wheeby default an ou at any time. Longstaff and hwatz (995) extend Blak-Cox model to allow inteest ates to follow the Onstein-Uhlenbek poess. Upon bankupty tiggeed by touhing the baie, bondholdes eeive an exogenously given numbe of iskless bonds. Following Longstaff-hwatz s model, Biys and de Vaenne (997) and höbel (999) develop piing models to define the bankupty-tiggeing baie as a fixed quantity disounted at the iskless ate up to the matuity date of the isky opoate bond. As a esult, the model is haateised by a baie following the stohastiity of the inteest ates. It is obvious to obseve that the baie goes downwads as the time to matuity of the opoate bond ineases. ine the baie denotes the theshold level at whih bankupty ous, highe fim value volatility should imply a highe level of leveage ove time and thus highe pobability of default. The main objetive of this pape is to develop a opoate bond valuation model in whih the bankupty-tiggeing baie is defined as a difted fim value level govened by stohasti isk-fee inteest ates and instantaneous vaiane of the opoate bond value. Though the instantaneous vaiane of the opoate bond value, the fim value volatility is inopoated into the baie dynamis. Thee is an additional fee paamete β to speify the ontibution of the instantaneous vaiane of the opoate bond to the ate of the dift of the baie. We deive a losed-fom solution of the bond pie as a funtion of fim volatility, oelation, dift and mean-level of the inteest ate. In the following setion we develop the piing model of disount opoate bonds of edit speads. In the last setion we shall summaise ou investigation. II. In the valuation of opoate bonds, we assume a ontinuos-time famewok. The dynamis of the shot-tem inteest ate Vasiek (977):
( t) [ θ ( t) ] dt σ ( t dz d κ + ) () whee the shot-tem inteest ate is mean-eveting to long-un mean θ(t) at speed κ(t), and σ (t) is the volatility of. The fim value is assumed to follow a lognomal diffusion poess: ( t) dt σ ( t) dz d µ + () whee µ(t) and σ (t) ae the dift and volatility of the fim value espetively. The Wiene poesses dz and dz ae oelated with dz dz ρdt () and the oelation oeffiient ρ is also assumed to be time dependent. We let the pie of a opoate bond be P(,, t). Using Ito s lemma and the standad no-abitage aguments, the patial diffeential equation govening the bond is P σ t P + σ P + ρσ P [ κ ( t) θ ( t) κ ( t) λ] P P P σ + + (4) whee λ is the maket pie of inteest ate isk. The value of the opoate bond is obtained by solving equation (4) subjet to the final payoff ondition and the bounday ondition imposed by the default baie. In ode to inopoate the dynamis of the fim value into the dynamis of the default baie, we popose the baie H(, t) to have a difted dynamis with the fom: (, t) Q(, t) exp[ β ( t) ] H o (5) whee o is the pe-defined asset value of the baie, Q(, t) is the iskless bond funtion aoding to the Vasiek model with time-dependent paametes, (t) is defined as ( t) dτ σ ( τ ) + ρ( τ ) σ ( τ ) σ ( τ ) ( τ ) + σ ( τ ) ( τ ) t 0 ( ) ( ) ( ) t t τ t exp dτκ τ dτ exp dτ ' κ τ 0 0 0 ' (6) and β is a eal numbe paamete to adjust the ate of the dift. It is noted when the paamete β is put to be zeo, the baie follows the dynamis of a iskless bond, i.e. eoveing Biys-de Vaenne s and höbel s models. The funtion (t) is the
integated instantaneous vaiane of the opoate bond funtion ove the life of the opoate bond, and the funtion ( t) ( t) σ is the instantaneous vaiane of a iskless disount bond pie of the Vasiek model with time to matuity t. The poess of the baie an theefoe be inteupted as a mean dift (adjusted by β) aising fom the dynamis of and P(,, t). The fim value volatility σ (t) is inopoated into the baie dynamis though (t). Fo a positive β, (t) offsets the deeasing effet of the iskless bond value with time to matuity. It makes the deease in the baie level with the time to matuity at a slowe ate. It means that given an initial o as the pe-defined default level, when the vaiane of the opoate bond value is high, the pobability of default to ou ineases with the value β. When the fim value beahes the baie H(, t), bankupty ous befoe matuity t 0. The payoffs to bondholdes ae speified by ( t) P H,, t) α o Q, t > 0 ; α (7) ( Fo β 0, the payoffs to bondholdes at the baie should be always less than the fim value sine (t) is positive definite. On the othe hand, if the fim value has neve beahed the baie, then the payoffs to bondholdes at the bond matuity ae: P (,, t 0) F F P (,, t 0) α < F ; α (8) The solution of equation (4) subjet to equation (7) and (8) is P FQ α α lq + α lq αl + N q [ ] l [ N( d ) N( d )] α N( d + ) N ( d + ) ( β + ) β β ( β + ) Q e [ N( d ) N( d )] β Q l Q β q ( ) ( ) 4 [ ( ) β β e N d + N d4 + ] β β β β ( β ) ( d ) ( q α lq ) Q e N ( d )} + α Q whee l / F is the asset-to-liability atio, q / o is an ealy default atio, and o (9) Campbell (986) shows that a onstant λ an be justified in a maket equilibium with log-utility investos. λ is absobed into the tem κ(t)θ(t) in the following alulation. It an be shown by ompleting squae of (t). If the payoff is defined as [ ] ( H,, t) α o Q(, t) exp β ( t) P, it is less than the fim value at the default baie fo all β. Howeve in this pape, we onside the ase of β 0 to be moe ealisti. The detailed deivation is available upon equest. 4
d d lnl + lnq lnl + ln q lnq d ln q + lnq ( β + ) ln q lnq ( β + ) The edit spead C s of a disount opoate bond pie P(,, T) with time to matuity T and fae value F is given as d 4 (,, T ) ( T ) P C s (,, T ) ln (0) T FQ, The tem stutues of edit speads fo a fim with l.5 and q.78 ae illustated in Figue using diffeent β fom 0 to.5. Othe paametes used in the alulations ae σ 0., σ 0.0, ρ 0.5, 4%, θ 6%, κ 0. and α α 0.8. The edit speads inease with positive β. The levels of the default baie with diffeent β imply diffeent ealy default isk. At the long end, the diffeene between the edit speads fo β 0 and β.5 is about 0bp whih is signifiant ompaed with the edit spead of 4bp fo β 0. The numeial esults show simila tem stutues obtained in pevious studies, whih math the empiial evidene 4. The esults also show that the vaiane of the opoate bond whih is inopoated into the default baie s dynamis has mateial impat on the default pobability. III. UMMARY This pape develops a opoate bond valuation model that inopoates a default baie with dynamis depending on stohasti inteest ates and the vaiane of the opoate bond funtion. ine the volatility of the fim value affets the level of the default baie ove time though the vaiane of the opoate bond funtion, moe ealisti default senaios an be put into the valuation model. When the fim value touhes the baie, bondholdes eeive an exogenously speified numbe of iskless bonds. We deive a losed-fom solution of the opoate bond pie as a funtion of fim value and a shot-tem inteest ate, with time-dependent model paametes govening the dynamis of the fim value and inteest ate. The numeial esults show that the difted default baie has mateial impat on the tem stutues of edit speads. 4 ee Ogden (987), and aig and Waga (989). 5
ACKNOWLEDGEMENT This wok is patially suppoted by the Diet Gant fo Reseah fom the Reseah Gants Counil of the Hong Kong Govenment. The onlusions heein do not epesent the views of the Hong Kong Monetay Authoity. REFERENCE Blak, F. and Cox, J. (976) Valuing Copoate euities: ome Effets of Bond Indentue Povisions, Jounal of Finane, 5, -4. Biys, E. and de Vaenne, F. (997) Valuing Risky Fixed Rate Debt: An Extension, Jounal of Finanial and Quantitative Analysis,, 0-48. Campbell J. Y. (986) A Defene of Taditional Hypotheses about the Tem tutue of Inteest Rates, Jounal of Finane, 4, 8-9. Longstaff, F. and hwatz, E. (995) A imple Appoah to Valuing Risky Fixed and Floating Rate Debt, Jounal of Finane, 50, 789-89. Meton, R. C. (974) On the Piing of Copoate Debt: The Risk tutue of Inteest Rates, Jounal of Finane,, 449-470. Ogden, J. P. (987) "Deteminants of the Ratings and Yields on Copoate Bonds: Tests of the Contingent-Claims Model." Jounal of Finanial Reseah, 0, 9-9. aig, O., and A. Waga. (989) "ome Empiial Estimates of the Risk tutue of Inteest Rates." Jounal of Finane, 44, 5-60. höbel, F. (999) A Note on the Valuation of Risky Copoate Bonds, OR pektum,, 5-47. Vasiek, O. A. (977) An Equilibium Chaateisation of the Tem tutue, Jounal of Finanial Eonomis, 5, 77-88. 6
Cedit pead Cs 0.90% 0.80% 0.70% 0.60% 0.50% 0.40% 0.0% 0.0% 0.0% 0.00% 0 5 0 5 0 Time to Matuity (yea) β0.0 β0.5 β.0 β.5 Figue. Cedit spead as a funtion of time to matuity with l.5, q.78 and diffeent β. The paametes used ae σ 0., 4%, σ 0.0, θ 6%, κ 0., ρ 0.5 and α α 0.8. 7