Research Article Pricing Chinese Convertible Bonds with ...
6
Research Article Pricing Chinese Convertible Bonds with Dynamic Credit Risk Ping Li and Jing Song School of Economics and Management, Beihang University, Beijing 100191, China Correspondence should be addressed to Ping Li; [email protected] Received 14 March 2014; Accepted 23 May 2014; Published 9 June 2014 Academic Editor: Chuangxia Huang Copyright © 2014 P. Li and J. Song. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. To price convertible bonds more precisely, least squares Monte Carlo (LSM) method is used in this paper for its advantage in handling the dependence of derivatives on the path, and dynamic credit risk is used to replace the fixed one to make the value of convertible bonds reflect the real credit risk. In the empirical study, we price convertible bonds based on static credit risk and dynamic credit risk, respectively. Empirical results indicate that the ICBC convertible bond has been overpriced, resulting from the underestimation of credit risk. In addition, when there is an issue of dividend, the conversion price will change in China's convertible bonds, while it does not change in the international convertible bonds. So we also empirically study the difference between the convertible bond's prices by assuming whether the conversion price changes or not. 1. Introduction Convertible bond is an innovative and complex financial instrument which can be converted to the issuer’s stock at some specified circumstance. It is hard to be valued because of its characters of both equity and bond, in addition to its vari- eties of terms. China’s convertible bond market is an emerging market, so it is important to value the product considering some changes compared with the existed valuation methods. eoretical research on convertible bonds was initiated by Ingersoll [1], who applied the Black-Sholes-Merton model of pricing options. Following his work, Brennan and Schwartz [2] firstly used corporate value as the basic variable to price convertible bonds. However, corporate value was soon replaced by stock price for its simplicity of observation and measurement, which was first introduced by McConnell and Schwartz [3]. Brennan and Schwartz [4] considered stochastic interest rate firstly and constructed a two-factor pricing model for convertible bonds. en, stochastic credit risk was introduced by Davis and Lischka [5]. From the above tendency we can see that researchers gradually added factors to increase the accuracy of valuation. To solve the more and more complex models, Monte Carlo simulation became widely used. Longstaff and Schwartz [6] firstly introduced the least squares Monte Carlo (LSM) to price American options. Following them, Moreno and Navas [7] studied the robustness of LSM method for pricing American options; Stentoſt [8] studied the convergence of the LSM approach. Due to its significant advantage in pricing derivatives, LSM method was soon applied to the valuation of convertible bonds, such as the work of Cr´ epey and Rahal [9] and Ammann et al. [10]. Credit risk is an important factor in convertible bonds valuation and is paid more attention than before in China. Currently, there are two methods to measure the credit risk. e first one is credit spread, which is firstly used by McConnell and Schwartz [3] to value convertible bonds. Following their work, Tsiveriotis and Fernandes [11] split convertible bond into two components: a cash-only part, which was discounted by risky interest rate, and an equity part, which was discounted by risk-free interest rate. e second method is to use default density, which was used by Duffie and Singleton [12] to price corporate bonds. en, Ayache et al. [13] applied default density to convertible bonds, deriving a partial differential equation for valuation. Comparing the two methods, the first one is more widely used for its simplicity and convenience. However,the estimation of the spread is essential. To get more accurate price, researchers began to employ dynamic credit risk. Davis and Lischka [5] supposed that hazard rate obeys a Brownian motion, with the Vasicek model of interest rate, and then established a two and a half model. But they did not verify the effect of the model using real data. Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2014, Article ID 492134, 5 pages http://dx.doi.org/10.1155/2014/492134
Transcript of Research Article Pricing Chinese Convertible Bonds with ...
Research ArticlePricing Chinese Convertible Bonds with Dynamic Credit Risk
Ping Li and Jing Song
School of Economics and Management Beihang University Beijing 100191 China
Correspondence should be addressed to Ping Li lipingxx126com
Received 14 March 2014 Accepted 23 May 2014 Published 9 June 2014
Academic Editor Chuangxia Huang
Copyright copy 2014 P Li and J Song This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
To price convertible bonds more precisely least squares Monte Carlo (LSM) method is used in this paper for its advantage inhandling the dependence of derivatives on the path and dynamic credit risk is used to replace the fixed one to make the valueof convertible bonds reflect the real credit risk In the empirical study we price convertible bonds based on static credit risk anddynamic credit risk respectively Empirical results indicate that the ICBC convertible bond has been overpriced resulting fromthe underestimation of credit risk In addition when there is an issue of dividend the conversion price will change in Chinasconvertible bonds while it does not change in the international convertible bonds So we also empirically study the differencebetween the convertible bonds prices by assuming whether the conversion price changes or not
1 Introduction
Convertible bond is an innovative and complex financialinstrument which can be converted to the issuerrsquos stock atsome specified circumstance It is hard to be valued because ofits characters of both equity and bond in addition to its vari-eties of termsChinarsquos convertible bondmarket is an emergingmarket so it is important to value the product consideringsome changes compared with the existed valuation methods
Theoretical research on convertible bondswas initiated byIngersoll [1] who applied the Black-Sholes-Merton model ofpricing options Following his work Brennan and Schwartz[2] firstly used corporate value as the basic variable toprice convertible bonds However corporate value was soonreplaced by stock price for its simplicity of observation andmeasurement which was first introduced by McConnelland Schwartz [3] Brennan and Schwartz [4] consideredstochastic interest rate firstly and constructed a two-factorpricing model for convertible bonds Then stochastic creditrisk was introduced byDavis and Lischka [5] From the abovetendency we can see that researchers gradually added factorsto increase the accuracy of valuation To solve the moreand more complex models Monte Carlo simulation becamewidely used Longstaff and Schwartz [6] firstly introducedthe least squares Monte Carlo (LSM) to price Americanoptions Following them Moreno and Navas [7] studied therobustness of LSM method for pricing American options
Stentoft [8] studied the convergence of the LSM approachDue to its significant advantage in pricing derivatives LSMmethod was soon applied to the valuation of convertiblebonds such as the work of Crepey and Rahal [9] andAmmann et al [10]
Credit risk is an important factor in convertible bondsvaluation and is paid more attention than before in ChinaCurrently there are two methods to measure the creditrisk The first one is credit spread which is firstly used byMcConnell and Schwartz [3] to value convertible bondsFollowing their work Tsiveriotis and Fernandes [11] splitconvertible bond into two components a cash-only partwhich was discounted by risky interest rate and an equitypart which was discounted by risk-free interest rate Thesecond method is to use default density which was used byDuffie and Singleton [12] to price corporate bonds ThenAyache et al [13] applied default density to convertiblebonds deriving a partial differential equation for valuationComparing the twomethods the first one ismorewidely usedfor its simplicity and convenience Howeverthe estimation ofthe spread is essential
To get more accurate price researchers began to employdynamic credit risk Davis and Lischka [5] supposed thathazard rate obeys a Brownianmotion with theVasicekmodelof interest rate and then established a two and a half modelBut they did not verify the effect of the model using realdata
Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 492134 5 pageshttpdxdoiorg1011552014492134
2 Discrete Dynamics in Nature and Society
Currently there is not any model applying dynamiccredit risk to price convertible bonds In addition whenthere is a distribution of dividend the conversion price willchange in Chinarsquos market while it does not change in theinternationalmarket In this paperwewill study the pricing ofChinarsquos convertible bonds using dynamic credit risk and thenempirically study the difference between the prices obtainedby assuming whether the conversion price changes or not
The rest of this paper is organized as follows Section 2gives the basic framework of convertible bond pricing byleast squares Monte Carlo simulation Section 3 derives theequation of dynamic credit spread Section 4 is the empiricalpart including the effect of dynamic credit spread and thecomparison of price obtained by assuming whether theconversion price changes or not Finally Section 5 concludesthe paper
2 Basic Framework of Convertible BondPricing by LSM
Besides the common debt convertible bonds are embeddedwith many options such as conversion option call optionput option and option to lower the conversion price So weshould compare the value of these options comprehensivelywhen pricing the convertible bonds At the expiration thefinal boundary can be 119881 = max(119899119878
119879 119862119879)
Every moment before the maturity of the convertiblebond investors and issuers will gamble over the benefitInvestors will maximize the value of convertible bonds whilethe issuer will minimize the value of convertible bonds fromexercising the call option To make it clear we use Table 1 toshow the rules of option exercise
The basic framework of convertible bonds pricing by LSMwhen the credit risk is static is as follows
(1)Considering the stock volatility is one of the importantparameters affecting stock path we assume that the stockprice follows the stochastic volatility model presented byHeston [14]
119889119878119905= 119878119905(119903119891minus 119902) 119889119905 + 119878
119905radicV119905119889119882
119889V119905= 120581 (V minus V
119905) 119889119905 + radicV119905120590V119889119882V
(1)
where 119903119891is the risk-free interest rate 119902 is the dividend yield
of the underlying stock V119905is the stochastic volatility of 119878
119905
and is modeled by the second equation of (1) 120581 is the meanreversion coefficient of V
119905 V is the long-term mean reversion
level of V119905 and 120590V is the volatility of V
119905 119882 and 119882V are both
Wiener processesWe split the duration of convertible bonds into119873 sections
on average and assume that the convertible bonds can onlybe exercised at these discrete times We can generate randomnumbers by Monte Carlo simulation and then get 119873
119906paths
from the Heston model Then the stock price matrix isobtained
(2)Applying the optimal stopping theory we compare thevalue of continuation and immediate exercise If the latter oneis bigger then we get the stopping time denoted by 119869 andstopping value denoted by119872119873119906
119894
(3) We assume (Ω 119865 119875) to be the complete probabilityspace and assume [0 119879] to be the finite time horizon R =
(R)119894=0119873
is defined to be the augmented filtration generatedby the actual market performance andR
0sub R1sub sdot sdot sdot sub R
119873
The state variable 119878 is the simulated stock price from whichwe calculate the cash flow 119862(120596 119895) on the path 120596 at time 119895 andthe temporary optimal stopping value119872119873119906
119894Then we take the
expectation of the cash flows 119862(120596 119895) discounted by the riskyinterest rate 119903(120596 119904) and get the value of continuation 119884(120596 119905)as follows
119884 (120596 119905) = 119864119876[
[
119873
sum
119895=119905+1
exp(minusint119895
119905
119903 (120596 119904) 119889119904)119862 (120596 119895) | R]
]
(2)
(4)The least square regression process is as follows Weput the value of continuation to be the dependent variable 119884and the underlying stock price to be the independent variable119883 and Laguerre polynomials are chosen to be the basisfunction to make the least square regression The procedureto get the estimated value of 119884 can be described by thefollowing equations
1198840 (119883) = exp (minus119883
2)
1198841 (119883) = exp(minus119883
2) (1 minus 119883)
1198842 (119883) = exp(minus119883
2)(1 minus 2119883 +
1198832
2)
(120596 119905) =
2
sum
119895=0
119886119895119884119895 (119883)
(3)
(5)Compare (120596 119905)with the conversion value call valueand the put value on the basis of the exercise rules ofconvertible bonds If (120596 119905) is bigger then the optimalstopping value remains the same if (120596 119905) is smaller thenwe get the new stopping time 119905 and the new stopping value119872119873119906
119894(6) The convertible bond can be valued by discounting
each119872119873119906119894
back to time 119905 = 0 with the risky interest rate andaveraging over all paths
119881 =1
119873119906
119873119906
sum
119895=1
exp(minusint119895
0
119903 (120596 119904) 119889119904)119872119895 (4)
It is worth noticing that not all values of continuationare estimated by the Laguerre polynomials To increase theestimated accuracy the following three conditions must beexcluded
(a) When the call provisions have been triggered nomatter how big the value of continuation is theconvertible bonds will be exercised and terminatedso the value of continuation does not need to beestimated
Discrete Dynamics in Nature and Society 3
Table 1 Rules of option exercise in convertible bonds
Case Payoff Rules Exerciserestriction
Conversion 119899119905119878119905
119899119905119878119905gt 119865(120596 119905)
and 119875119905le 119899119905119878119905
119905 isin Ωconv119905 isin Ωput cap Ωconv
Call 119875119905
119875119905gt 119865(120596 119905)
and 119899119905119878119905le 119875119905
119905 isin Ωput119905 isin Ωput cap Ωconv
Put 119862119905
119865(120596 119905) gt 119862119905
and 119862119905gt 119899119905119878119905
119905 isin Ωcall119905 isin Ωcall cap Ωconv
Forced conversion 119899119905119878119905
119865(120596 119905) gt 119862119905
and 119862119905lt 119899119905119878119905
119905 isin Ωcall119905 isin Ωcall cap Ωconv
(b) When the put provisions have been triggered andthe call value is smaller than the discounted value ofthe minimum payoff 119881min until the maturity the callaction will not be exercised definitely So the value ofcontinuation does not need to be estimated
(c) When the call provisions and the put provisions havenot been triggered if the conversion value is smallerthan the discounted value of the minimum payoff119881min until the maturity the conversion action willnot be exercised definitely So there is no need toestimate the value of continuation where 119881min is thediscounted value of the call value until maturity
3 The Dynamic Credit Risk Model
Currently there are two main methods to describe thedynamic credit risk The first one is the two and a half modelof Davis and Lischka [5] who assumed that the hazard rateobeys a Brownian motion The second one is of Huang et al[15] who assume the credit spread to be linked with the assetvalue of the company
In this paper we describe the credit risk to be the dynamiccredit spread linked with the stock price because the creditrisk is mostly affected by the stock price Moreover theconversion price can be the benchmark of the stock price andthen we can derive the equation of dynamic credit spread 119903
119888
as follows
119903119888= 119903119898119888(119883
119878)
120578
(5)
where 119903119898119888
denotes the average credit spread119883 is the conver-sion price and 119878 is the stock priceThe equation indicates thatwhen 119878 gt 119883 the credit spread will decrease when 119878 lt 119883 thecredit spread will increase 120578 denotes the adjustment speedFinally the risky interest rate 119903(120596 119905) in the last section can bewritten to be
119903 (120596 119905) = 119903119891 + 119903119898119888(119883
119878)
120578
(6)
4 Empirical Research
To test the performance of our model we consider ICBCconvertible bond one of the largest convertible bonds issuedin China to do the empirical study We adopt 146 weekly
Table 2 ICBC convertible bond
Convertible bond ICBC convertible bondIssue date 2010831Time horizon 6Face value 100Coupon () 05 07 09 11 14 18Call value till maturity 105The first conversion price 42
Change of conversion price
20101126 adjusted to 41620101227 adjusted to 4152011615 adjusted to 3972012614 adjusted to 3772013626 adjusted to 353
Reset clause
In 30 consecutive trading daysthe closing stock price is smallerthan 80 of conversion price in
15 trading days
Call on condition
In 30 consecutive trading daysthe closing stock price is biggerthan 130 of conversion price in
15 trading days
Call value Face value plus the accruedinterest
Put on condition When the use of capital ischanged
Put value Face value plus the accruedinterest
data from August 12 2011 to January 3 2014 obtained fromCSMAR The information of ICBC convertible bond is inTable 2
Before pricing we first need to estimate the relatedparametersWe get the term structure of risk-free interest ratebased on cubic polynomial spline function using 15 treasurybonds traded in Shanghai Exchange By minimizing the sumof squared errors between themarket price and the simulatedprice of the underlying stock we get the three parametersin Hestonrsquos stochastic volatility model 120590V = 040 120581 = 13and V = 004 We employ the estimation result obtained byZheng and Lin [16] to be Chinarsquos average credit spread thatis 119903119898119888= 098
4 Discrete Dynamics in Nature and Society
0 50 100 15095
100
105
110
115
120
125
Time (week)
Pric
e (RM
B)
Market pricesTheoretical prices
Figure 1 Comparison of the theoretical and market prices of ICBCconvertible bond
0 50 100 1500
002
004
006
008
01
012
MA
D
Time (week)
Figure 2 The relative deviation (AD) of ICBC convertible bond
41 Pricing Results with Static Credit Risk We first assumethat the credit risk is static and get 5000 paths of the stockprice by LSM Then we get the theoretical prices of ICBCconvertible bond using our pricing framework Figure 1 is thecomparison of the theoretical prices and market prices
From Figure 1 we can see that the tendencies of thetwo lines fit well in the long run so we can use this priceframework to forecast market price of convertible bond andmake investment decision On the other hand the marketprice is a little higher than the theoretical price so ICBCconvertible bond is a little overpriced
We also define the variable AD to be the absolutedeviation of the theoretical price from the market price asfollows
AD119894=
10038161003816100381610038161003816119881119894minus 119881119894
10038161003816100381610038161003816
119881119894
(7)
0 50 100 15095
100
105
110
115
120
125
Time (week)
Pric
e (RM
B)
Market pricePrices with static credit riskPrices with dynamic credit risk
Figure 3 Comparison of theoretical prices with dynamic and staticcredit risk
Then we get the figure of AD by Matlab and depict it inFigure 2
The following mean absolute deviation MAD is definedto describe the integral result of our model
MAD =1
146
146
sum
119894=1
AD119894 (8)
Through calculation the mean absolute deviation ofICBC convertible MAD is 23 within 5 which demon-strates that the theoretical price obtained from ourmodel canreflect market price
42 Pricing Results Using Dynamic Credit Risk In this sec-tion we give the pricing results of ICBC convertible bondusing the dynamic credit risk modeled by (6) and with 120578 = 2We compare the market prices with the prices obtained fromstatic credit risk and dynamic credit risk in Figure 3
We can see from Figure 3 that the market price of theICBC convertible bond is higher than the prices obtainedfrom both the static and dynamic credit risk models whichmeans that the ICBC convertible bond is overestimatedresulting from the underestimation of credit riskWe can alsosee that the dynamic credit risk can reflect the real credit risksincewhen the price of convertible bond goes downwards theprice obtained from dynamic credit risk is lower than thoseobtained from static credit risk This implies that we shoulduse dynamic credit risk to price convertible bonds
43 Effect of Conversion Price In the above empirical studywe price the ICBC convertible bond based on changingconversion price (CP) resulting from two issues of dividendThough most Chinese companies change conversion pricewhen issuing dividend international companiesrsquo conversionprice does not change So in this section we also give the
Discrete Dynamics in Nature and Society 5
0 44 96 15090
95
100
105
110
115
120
125
Time (week)
Pric
e (RM
B)
Market pricesPrices with unchanging CPPrices with changing CP
Figure 4 Pricing results without changing the conversion price
prices of ICBC convertible bond assuming that the conver-sion price does not change The pricing results together withthose obtained from changing conversion price are shown inFigure 4
Figure 4 indicates that compared with the prices withchanging CP obtained in Section 41 the obtained prices withunchanging CP have two jumps at time 119905 = 44 and 119905 =96 just after the CP changed This demonstrates that whenthe company issues dividend the effect of unchanging CPis significant This can be explained by the fact that whenthere is an issue of dividend investors of Chinarsquos convertiblebonds will have an expectation of price decline which willaffect the payoff of convertible bonds eventually ThereforeChinese companies of convertible bonds are suggested tochange CP when issuing dividend That is we cannot copythe international experience of unchanging CP
5 Conclusion
This paper studies the pricing of convertible bonds withdynamic credit risk using least squares Monte Carlo methodWe employ the dynamic credit spread changing with stockprice In empirical study our model is proved to be effectiveand the comparison test demonstrates that the dynamiccredit risk is important in convertible bond pricing Theprice obtained from dynamic credit risk can reflect thereal credit risk Thus the potential risk resulting from theoverestimation of convertible bonds cannot be neglected bythe investors In addition we also study the empirical effectof changing the conversion price when the issuer distributesdividend Consequently the unchanged conversion price willlead to an unreasonable price So Chinarsquos market is notmature enough to keep the conversion price constant just likeinternational markets
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China (nos 71271015 and 70971006)
References
[1] J E Ingersoll Jr ldquoA contingent-claims valuation of convertiblesecuritiesrdquo Journal of Financial Economics vol 4 no 3 pp 289ndash321 1977
[2] M J Brennan and E S Schwartz ldquoConvertible bonds valuationand optimal strategies for call and conversionrdquo The Journal ofFinance vol 32 no 5 pp 1699ndash1715 1977
[3] J J McConnell and E S Schwartz ldquoLYON tamingrdquoThe Journalof Finance vol 41 no 3 pp 561ndash576 1986
[4] M J Brennan and E S Schwartz ldquoAnalyzing convertiblebondsrdquo Journal of Financial and Quantitative Analysis vol 15pp 907ndash929 1980
[5] M Davis and F R Lischka ldquoConvertible bonds with marketrisk and credit riskrdquoWorking Paper Tokyo-Mitsubishi Interna-tional 1999
[6] F A Longstaff and E S Schwartz ldquoValuing American optionsby simulation a simple least-squares approachrdquo Review ofFinancial Studies vol 14 no 1 pp 113ndash147 2001
[7] M Moreno and J F Navas ldquoOn the robustness of least-squaresMonte Carlo (LSM) for pricing American derivativesrdquo Reviewof Derivatives Research vol 6 no 2 pp 107ndash128 2003
[8] L Stentoft ldquoConvergence of the least squares Monte Carloapproach to American option valuationrdquoManagement Sciencevol 50 no 9 pp 1193ndash1203 2004
[9] S Crepey and A Rahal ldquoPricing convertible bonds with callprotectionrdquo Journal of Computational Finance vol 15 no 2 pp37ndash75 2011
[10] M Ammann A Kind and CWilde ldquoSimulation-based pricingof convertible bondsrdquo Journal of Empirical Finance vol 15 no2 pp 310ndash331 2008
[11] K Tsiveriotis and C Fernandes ldquoValuing convertible bondswith credit riskrdquo The Journal of Fixed Income vol 8 no 2 pp95ndash102 1998
[12] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999
[13] E Ayache P A Forsyth and K R Vetzal ldquoValuation ofconvertible bonds with credit riskrdquo The Journal of Derivativesvol 11 no 1 pp 9ndash29 2003
[14] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[15] J G Huang S C Yang and X Feng ldquoValuation of convertiblebond with dynamic credit riskrdquo Mathematical Statistics andManagement vol 27 no 6 pp 1108ndash1116 2008
[16] Z L Zheng andH Lin ldquoEstimation of the default risk premiumin Chinardquo Securities Market Herald vol 13 no 6 pp 41ndash442003
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2 Discrete Dynamics in Nature and Society
Currently there is not any model applying dynamiccredit risk to price convertible bonds In addition whenthere is a distribution of dividend the conversion price willchange in Chinarsquos market while it does not change in theinternationalmarket In this paperwewill study the pricing ofChinarsquos convertible bonds using dynamic credit risk and thenempirically study the difference between the prices obtainedby assuming whether the conversion price changes or not
The rest of this paper is organized as follows Section 2gives the basic framework of convertible bond pricing byleast squares Monte Carlo simulation Section 3 derives theequation of dynamic credit spread Section 4 is the empiricalpart including the effect of dynamic credit spread and thecomparison of price obtained by assuming whether theconversion price changes or not Finally Section 5 concludesthe paper
2 Basic Framework of Convertible BondPricing by LSM
Besides the common debt convertible bonds are embeddedwith many options such as conversion option call optionput option and option to lower the conversion price So weshould compare the value of these options comprehensivelywhen pricing the convertible bonds At the expiration thefinal boundary can be 119881 = max(119899119878
119879 119862119879)
Every moment before the maturity of the convertiblebond investors and issuers will gamble over the benefitInvestors will maximize the value of convertible bonds whilethe issuer will minimize the value of convertible bonds fromexercising the call option To make it clear we use Table 1 toshow the rules of option exercise
The basic framework of convertible bonds pricing by LSMwhen the credit risk is static is as follows
(1)Considering the stock volatility is one of the importantparameters affecting stock path we assume that the stockprice follows the stochastic volatility model presented byHeston [14]
119889119878119905= 119878119905(119903119891minus 119902) 119889119905 + 119878
119905radicV119905119889119882
119889V119905= 120581 (V minus V
119905) 119889119905 + radicV119905120590V119889119882V
(1)
where 119903119891is the risk-free interest rate 119902 is the dividend yield
of the underlying stock V119905is the stochastic volatility of 119878
119905
and is modeled by the second equation of (1) 120581 is the meanreversion coefficient of V
119905 V is the long-term mean reversion
level of V119905 and 120590V is the volatility of V
119905 119882 and 119882V are both
Wiener processesWe split the duration of convertible bonds into119873 sections
on average and assume that the convertible bonds can onlybe exercised at these discrete times We can generate randomnumbers by Monte Carlo simulation and then get 119873
119906paths
from the Heston model Then the stock price matrix isobtained
(2)Applying the optimal stopping theory we compare thevalue of continuation and immediate exercise If the latter oneis bigger then we get the stopping time denoted by 119869 andstopping value denoted by119872119873119906
119894
(3) We assume (Ω 119865 119875) to be the complete probabilityspace and assume [0 119879] to be the finite time horizon R =
(R)119894=0119873
is defined to be the augmented filtration generatedby the actual market performance andR
0sub R1sub sdot sdot sdot sub R
119873
The state variable 119878 is the simulated stock price from whichwe calculate the cash flow 119862(120596 119895) on the path 120596 at time 119895 andthe temporary optimal stopping value119872119873119906
119894Then we take the
expectation of the cash flows 119862(120596 119895) discounted by the riskyinterest rate 119903(120596 119904) and get the value of continuation 119884(120596 119905)as follows
119884 (120596 119905) = 119864119876[
[
119873
sum
119895=119905+1
exp(minusint119895
119905
119903 (120596 119904) 119889119904)119862 (120596 119895) | R]
]
(2)
(4)The least square regression process is as follows Weput the value of continuation to be the dependent variable 119884and the underlying stock price to be the independent variable119883 and Laguerre polynomials are chosen to be the basisfunction to make the least square regression The procedureto get the estimated value of 119884 can be described by thefollowing equations
1198840 (119883) = exp (minus119883
2)
1198841 (119883) = exp(minus119883
2) (1 minus 119883)
1198842 (119883) = exp(minus119883
2)(1 minus 2119883 +
1198832
2)
(120596 119905) =
2
sum
119895=0
119886119895119884119895 (119883)
(3)
(5)Compare (120596 119905)with the conversion value call valueand the put value on the basis of the exercise rules ofconvertible bonds If (120596 119905) is bigger then the optimalstopping value remains the same if (120596 119905) is smaller thenwe get the new stopping time 119905 and the new stopping value119872119873119906
119894(6) The convertible bond can be valued by discounting
each119872119873119906119894
back to time 119905 = 0 with the risky interest rate andaveraging over all paths
119881 =1
119873119906
119873119906
sum
119895=1
exp(minusint119895
0
119903 (120596 119904) 119889119904)119872119895 (4)
It is worth noticing that not all values of continuationare estimated by the Laguerre polynomials To increase theestimated accuracy the following three conditions must beexcluded
(a) When the call provisions have been triggered nomatter how big the value of continuation is theconvertible bonds will be exercised and terminatedso the value of continuation does not need to beestimated
Discrete Dynamics in Nature and Society 3
Table 1 Rules of option exercise in convertible bonds
Case Payoff Rules Exerciserestriction
Conversion 119899119905119878119905
119899119905119878119905gt 119865(120596 119905)
and 119875119905le 119899119905119878119905
119905 isin Ωconv119905 isin Ωput cap Ωconv
Call 119875119905
119875119905gt 119865(120596 119905)
and 119899119905119878119905le 119875119905
119905 isin Ωput119905 isin Ωput cap Ωconv
Put 119862119905
119865(120596 119905) gt 119862119905
and 119862119905gt 119899119905119878119905
119905 isin Ωcall119905 isin Ωcall cap Ωconv
Forced conversion 119899119905119878119905
119865(120596 119905) gt 119862119905
and 119862119905lt 119899119905119878119905
119905 isin Ωcall119905 isin Ωcall cap Ωconv
(b) When the put provisions have been triggered andthe call value is smaller than the discounted value ofthe minimum payoff 119881min until the maturity the callaction will not be exercised definitely So the value ofcontinuation does not need to be estimated
(c) When the call provisions and the put provisions havenot been triggered if the conversion value is smallerthan the discounted value of the minimum payoff119881min until the maturity the conversion action willnot be exercised definitely So there is no need toestimate the value of continuation where 119881min is thediscounted value of the call value until maturity
3 The Dynamic Credit Risk Model
Currently there are two main methods to describe thedynamic credit risk The first one is the two and a half modelof Davis and Lischka [5] who assumed that the hazard rateobeys a Brownian motion The second one is of Huang et al[15] who assume the credit spread to be linked with the assetvalue of the company
In this paper we describe the credit risk to be the dynamiccredit spread linked with the stock price because the creditrisk is mostly affected by the stock price Moreover theconversion price can be the benchmark of the stock price andthen we can derive the equation of dynamic credit spread 119903
119888
as follows
119903119888= 119903119898119888(119883
119878)
120578
(5)
where 119903119898119888
denotes the average credit spread119883 is the conver-sion price and 119878 is the stock priceThe equation indicates thatwhen 119878 gt 119883 the credit spread will decrease when 119878 lt 119883 thecredit spread will increase 120578 denotes the adjustment speedFinally the risky interest rate 119903(120596 119905) in the last section can bewritten to be
119903 (120596 119905) = 119903119891 + 119903119898119888(119883
119878)
120578
(6)
4 Empirical Research
To test the performance of our model we consider ICBCconvertible bond one of the largest convertible bonds issuedin China to do the empirical study We adopt 146 weekly
Table 2 ICBC convertible bond
Convertible bond ICBC convertible bondIssue date 2010831Time horizon 6Face value 100Coupon () 05 07 09 11 14 18Call value till maturity 105The first conversion price 42
Change of conversion price
20101126 adjusted to 41620101227 adjusted to 4152011615 adjusted to 3972012614 adjusted to 3772013626 adjusted to 353
Reset clause
In 30 consecutive trading daysthe closing stock price is smallerthan 80 of conversion price in
15 trading days
Call on condition
In 30 consecutive trading daysthe closing stock price is biggerthan 130 of conversion price in
15 trading days
Call value Face value plus the accruedinterest
Put on condition When the use of capital ischanged
Put value Face value plus the accruedinterest
data from August 12 2011 to January 3 2014 obtained fromCSMAR The information of ICBC convertible bond is inTable 2
Before pricing we first need to estimate the relatedparametersWe get the term structure of risk-free interest ratebased on cubic polynomial spline function using 15 treasurybonds traded in Shanghai Exchange By minimizing the sumof squared errors between themarket price and the simulatedprice of the underlying stock we get the three parametersin Hestonrsquos stochastic volatility model 120590V = 040 120581 = 13and V = 004 We employ the estimation result obtained byZheng and Lin [16] to be Chinarsquos average credit spread thatis 119903119898119888= 098
4 Discrete Dynamics in Nature and Society
0 50 100 15095
100
105
110
115
120
125
Time (week)
Pric
e (RM
B)
Market pricesTheoretical prices
Figure 1 Comparison of the theoretical and market prices of ICBCconvertible bond
0 50 100 1500
002
004
006
008
01
012
MA
D
Time (week)
Figure 2 The relative deviation (AD) of ICBC convertible bond
41 Pricing Results with Static Credit Risk We first assumethat the credit risk is static and get 5000 paths of the stockprice by LSM Then we get the theoretical prices of ICBCconvertible bond using our pricing framework Figure 1 is thecomparison of the theoretical prices and market prices
From Figure 1 we can see that the tendencies of thetwo lines fit well in the long run so we can use this priceframework to forecast market price of convertible bond andmake investment decision On the other hand the marketprice is a little higher than the theoretical price so ICBCconvertible bond is a little overpriced
We also define the variable AD to be the absolutedeviation of the theoretical price from the market price asfollows
AD119894=
10038161003816100381610038161003816119881119894minus 119881119894
10038161003816100381610038161003816
119881119894
(7)
0 50 100 15095
100
105
110
115
120
125
Time (week)
Pric
e (RM
B)
Market pricePrices with static credit riskPrices with dynamic credit risk
Figure 3 Comparison of theoretical prices with dynamic and staticcredit risk
Then we get the figure of AD by Matlab and depict it inFigure 2
The following mean absolute deviation MAD is definedto describe the integral result of our model
MAD =1
146
146
sum
119894=1
AD119894 (8)
Through calculation the mean absolute deviation ofICBC convertible MAD is 23 within 5 which demon-strates that the theoretical price obtained from ourmodel canreflect market price
42 Pricing Results Using Dynamic Credit Risk In this sec-tion we give the pricing results of ICBC convertible bondusing the dynamic credit risk modeled by (6) and with 120578 = 2We compare the market prices with the prices obtained fromstatic credit risk and dynamic credit risk in Figure 3
We can see from Figure 3 that the market price of theICBC convertible bond is higher than the prices obtainedfrom both the static and dynamic credit risk models whichmeans that the ICBC convertible bond is overestimatedresulting from the underestimation of credit riskWe can alsosee that the dynamic credit risk can reflect the real credit risksincewhen the price of convertible bond goes downwards theprice obtained from dynamic credit risk is lower than thoseobtained from static credit risk This implies that we shoulduse dynamic credit risk to price convertible bonds
43 Effect of Conversion Price In the above empirical studywe price the ICBC convertible bond based on changingconversion price (CP) resulting from two issues of dividendThough most Chinese companies change conversion pricewhen issuing dividend international companiesrsquo conversionprice does not change So in this section we also give the
Discrete Dynamics in Nature and Society 5
0 44 96 15090
95
100
105
110
115
120
125
Time (week)
Pric
e (RM
B)
Market pricesPrices with unchanging CPPrices with changing CP
Figure 4 Pricing results without changing the conversion price
prices of ICBC convertible bond assuming that the conver-sion price does not change The pricing results together withthose obtained from changing conversion price are shown inFigure 4
Figure 4 indicates that compared with the prices withchanging CP obtained in Section 41 the obtained prices withunchanging CP have two jumps at time 119905 = 44 and 119905 =96 just after the CP changed This demonstrates that whenthe company issues dividend the effect of unchanging CPis significant This can be explained by the fact that whenthere is an issue of dividend investors of Chinarsquos convertiblebonds will have an expectation of price decline which willaffect the payoff of convertible bonds eventually ThereforeChinese companies of convertible bonds are suggested tochange CP when issuing dividend That is we cannot copythe international experience of unchanging CP
5 Conclusion
This paper studies the pricing of convertible bonds withdynamic credit risk using least squares Monte Carlo methodWe employ the dynamic credit spread changing with stockprice In empirical study our model is proved to be effectiveand the comparison test demonstrates that the dynamiccredit risk is important in convertible bond pricing Theprice obtained from dynamic credit risk can reflect thereal credit risk Thus the potential risk resulting from theoverestimation of convertible bonds cannot be neglected bythe investors In addition we also study the empirical effectof changing the conversion price when the issuer distributesdividend Consequently the unchanged conversion price willlead to an unreasonable price So Chinarsquos market is notmature enough to keep the conversion price constant just likeinternational markets
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China (nos 71271015 and 70971006)
References
[1] J E Ingersoll Jr ldquoA contingent-claims valuation of convertiblesecuritiesrdquo Journal of Financial Economics vol 4 no 3 pp 289ndash321 1977
[2] M J Brennan and E S Schwartz ldquoConvertible bonds valuationand optimal strategies for call and conversionrdquo The Journal ofFinance vol 32 no 5 pp 1699ndash1715 1977
[3] J J McConnell and E S Schwartz ldquoLYON tamingrdquoThe Journalof Finance vol 41 no 3 pp 561ndash576 1986
[4] M J Brennan and E S Schwartz ldquoAnalyzing convertiblebondsrdquo Journal of Financial and Quantitative Analysis vol 15pp 907ndash929 1980
[5] M Davis and F R Lischka ldquoConvertible bonds with marketrisk and credit riskrdquoWorking Paper Tokyo-Mitsubishi Interna-tional 1999
[6] F A Longstaff and E S Schwartz ldquoValuing American optionsby simulation a simple least-squares approachrdquo Review ofFinancial Studies vol 14 no 1 pp 113ndash147 2001
[7] M Moreno and J F Navas ldquoOn the robustness of least-squaresMonte Carlo (LSM) for pricing American derivativesrdquo Reviewof Derivatives Research vol 6 no 2 pp 107ndash128 2003
[8] L Stentoft ldquoConvergence of the least squares Monte Carloapproach to American option valuationrdquoManagement Sciencevol 50 no 9 pp 1193ndash1203 2004
[9] S Crepey and A Rahal ldquoPricing convertible bonds with callprotectionrdquo Journal of Computational Finance vol 15 no 2 pp37ndash75 2011
[10] M Ammann A Kind and CWilde ldquoSimulation-based pricingof convertible bondsrdquo Journal of Empirical Finance vol 15 no2 pp 310ndash331 2008
[11] K Tsiveriotis and C Fernandes ldquoValuing convertible bondswith credit riskrdquo The Journal of Fixed Income vol 8 no 2 pp95ndash102 1998
[12] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999
[13] E Ayache P A Forsyth and K R Vetzal ldquoValuation ofconvertible bonds with credit riskrdquo The Journal of Derivativesvol 11 no 1 pp 9ndash29 2003
[14] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[15] J G Huang S C Yang and X Feng ldquoValuation of convertiblebond with dynamic credit riskrdquo Mathematical Statistics andManagement vol 27 no 6 pp 1108ndash1116 2008
[16] Z L Zheng andH Lin ldquoEstimation of the default risk premiumin Chinardquo Securities Market Herald vol 13 no 6 pp 41ndash442003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 3
Table 1 Rules of option exercise in convertible bonds
Case Payoff Rules Exerciserestriction
Conversion 119899119905119878119905
119899119905119878119905gt 119865(120596 119905)
and 119875119905le 119899119905119878119905
119905 isin Ωconv119905 isin Ωput cap Ωconv
Call 119875119905
119875119905gt 119865(120596 119905)
and 119899119905119878119905le 119875119905
119905 isin Ωput119905 isin Ωput cap Ωconv
Put 119862119905
119865(120596 119905) gt 119862119905
and 119862119905gt 119899119905119878119905
119905 isin Ωcall119905 isin Ωcall cap Ωconv
Forced conversion 119899119905119878119905
119865(120596 119905) gt 119862119905
and 119862119905lt 119899119905119878119905
119905 isin Ωcall119905 isin Ωcall cap Ωconv
(b) When the put provisions have been triggered andthe call value is smaller than the discounted value ofthe minimum payoff 119881min until the maturity the callaction will not be exercised definitely So the value ofcontinuation does not need to be estimated
(c) When the call provisions and the put provisions havenot been triggered if the conversion value is smallerthan the discounted value of the minimum payoff119881min until the maturity the conversion action willnot be exercised definitely So there is no need toestimate the value of continuation where 119881min is thediscounted value of the call value until maturity
3 The Dynamic Credit Risk Model
Currently there are two main methods to describe thedynamic credit risk The first one is the two and a half modelof Davis and Lischka [5] who assumed that the hazard rateobeys a Brownian motion The second one is of Huang et al[15] who assume the credit spread to be linked with the assetvalue of the company
In this paper we describe the credit risk to be the dynamiccredit spread linked with the stock price because the creditrisk is mostly affected by the stock price Moreover theconversion price can be the benchmark of the stock price andthen we can derive the equation of dynamic credit spread 119903
119888
as follows
119903119888= 119903119898119888(119883
119878)
120578
(5)
where 119903119898119888
denotes the average credit spread119883 is the conver-sion price and 119878 is the stock priceThe equation indicates thatwhen 119878 gt 119883 the credit spread will decrease when 119878 lt 119883 thecredit spread will increase 120578 denotes the adjustment speedFinally the risky interest rate 119903(120596 119905) in the last section can bewritten to be
119903 (120596 119905) = 119903119891 + 119903119898119888(119883
119878)
120578
(6)
4 Empirical Research
To test the performance of our model we consider ICBCconvertible bond one of the largest convertible bonds issuedin China to do the empirical study We adopt 146 weekly
Table 2 ICBC convertible bond
Convertible bond ICBC convertible bondIssue date 2010831Time horizon 6Face value 100Coupon () 05 07 09 11 14 18Call value till maturity 105The first conversion price 42
Change of conversion price
20101126 adjusted to 41620101227 adjusted to 4152011615 adjusted to 3972012614 adjusted to 3772013626 adjusted to 353
Reset clause
In 30 consecutive trading daysthe closing stock price is smallerthan 80 of conversion price in
15 trading days
Call on condition
In 30 consecutive trading daysthe closing stock price is biggerthan 130 of conversion price in
15 trading days
Call value Face value plus the accruedinterest
Put on condition When the use of capital ischanged
Put value Face value plus the accruedinterest
data from August 12 2011 to January 3 2014 obtained fromCSMAR The information of ICBC convertible bond is inTable 2
Before pricing we first need to estimate the relatedparametersWe get the term structure of risk-free interest ratebased on cubic polynomial spline function using 15 treasurybonds traded in Shanghai Exchange By minimizing the sumof squared errors between themarket price and the simulatedprice of the underlying stock we get the three parametersin Hestonrsquos stochastic volatility model 120590V = 040 120581 = 13and V = 004 We employ the estimation result obtained byZheng and Lin [16] to be Chinarsquos average credit spread thatis 119903119898119888= 098
4 Discrete Dynamics in Nature and Society
0 50 100 15095
100
105
110
115
120
125
Time (week)
Pric
e (RM
B)
Market pricesTheoretical prices
Figure 1 Comparison of the theoretical and market prices of ICBCconvertible bond
0 50 100 1500
002
004
006
008
01
012
MA
D
Time (week)
Figure 2 The relative deviation (AD) of ICBC convertible bond
41 Pricing Results with Static Credit Risk We first assumethat the credit risk is static and get 5000 paths of the stockprice by LSM Then we get the theoretical prices of ICBCconvertible bond using our pricing framework Figure 1 is thecomparison of the theoretical prices and market prices
From Figure 1 we can see that the tendencies of thetwo lines fit well in the long run so we can use this priceframework to forecast market price of convertible bond andmake investment decision On the other hand the marketprice is a little higher than the theoretical price so ICBCconvertible bond is a little overpriced
We also define the variable AD to be the absolutedeviation of the theoretical price from the market price asfollows
AD119894=
10038161003816100381610038161003816119881119894minus 119881119894
10038161003816100381610038161003816
119881119894
(7)
0 50 100 15095
100
105
110
115
120
125
Time (week)
Pric
e (RM
B)
Market pricePrices with static credit riskPrices with dynamic credit risk
Figure 3 Comparison of theoretical prices with dynamic and staticcredit risk
Then we get the figure of AD by Matlab and depict it inFigure 2
The following mean absolute deviation MAD is definedto describe the integral result of our model
MAD =1
146
146
sum
119894=1
AD119894 (8)
Through calculation the mean absolute deviation ofICBC convertible MAD is 23 within 5 which demon-strates that the theoretical price obtained from ourmodel canreflect market price
42 Pricing Results Using Dynamic Credit Risk In this sec-tion we give the pricing results of ICBC convertible bondusing the dynamic credit risk modeled by (6) and with 120578 = 2We compare the market prices with the prices obtained fromstatic credit risk and dynamic credit risk in Figure 3
We can see from Figure 3 that the market price of theICBC convertible bond is higher than the prices obtainedfrom both the static and dynamic credit risk models whichmeans that the ICBC convertible bond is overestimatedresulting from the underestimation of credit riskWe can alsosee that the dynamic credit risk can reflect the real credit risksincewhen the price of convertible bond goes downwards theprice obtained from dynamic credit risk is lower than thoseobtained from static credit risk This implies that we shoulduse dynamic credit risk to price convertible bonds
43 Effect of Conversion Price In the above empirical studywe price the ICBC convertible bond based on changingconversion price (CP) resulting from two issues of dividendThough most Chinese companies change conversion pricewhen issuing dividend international companiesrsquo conversionprice does not change So in this section we also give the
Discrete Dynamics in Nature and Society 5
0 44 96 15090
95
100
105
110
115
120
125
Time (week)
Pric
e (RM
B)
Market pricesPrices with unchanging CPPrices with changing CP
Figure 4 Pricing results without changing the conversion price
prices of ICBC convertible bond assuming that the conver-sion price does not change The pricing results together withthose obtained from changing conversion price are shown inFigure 4
Figure 4 indicates that compared with the prices withchanging CP obtained in Section 41 the obtained prices withunchanging CP have two jumps at time 119905 = 44 and 119905 =96 just after the CP changed This demonstrates that whenthe company issues dividend the effect of unchanging CPis significant This can be explained by the fact that whenthere is an issue of dividend investors of Chinarsquos convertiblebonds will have an expectation of price decline which willaffect the payoff of convertible bonds eventually ThereforeChinese companies of convertible bonds are suggested tochange CP when issuing dividend That is we cannot copythe international experience of unchanging CP
5 Conclusion
This paper studies the pricing of convertible bonds withdynamic credit risk using least squares Monte Carlo methodWe employ the dynamic credit spread changing with stockprice In empirical study our model is proved to be effectiveand the comparison test demonstrates that the dynamiccredit risk is important in convertible bond pricing Theprice obtained from dynamic credit risk can reflect thereal credit risk Thus the potential risk resulting from theoverestimation of convertible bonds cannot be neglected bythe investors In addition we also study the empirical effectof changing the conversion price when the issuer distributesdividend Consequently the unchanged conversion price willlead to an unreasonable price So Chinarsquos market is notmature enough to keep the conversion price constant just likeinternational markets
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China (nos 71271015 and 70971006)
References
[1] J E Ingersoll Jr ldquoA contingent-claims valuation of convertiblesecuritiesrdquo Journal of Financial Economics vol 4 no 3 pp 289ndash321 1977
[2] M J Brennan and E S Schwartz ldquoConvertible bonds valuationand optimal strategies for call and conversionrdquo The Journal ofFinance vol 32 no 5 pp 1699ndash1715 1977
[3] J J McConnell and E S Schwartz ldquoLYON tamingrdquoThe Journalof Finance vol 41 no 3 pp 561ndash576 1986
[4] M J Brennan and E S Schwartz ldquoAnalyzing convertiblebondsrdquo Journal of Financial and Quantitative Analysis vol 15pp 907ndash929 1980
[5] M Davis and F R Lischka ldquoConvertible bonds with marketrisk and credit riskrdquoWorking Paper Tokyo-Mitsubishi Interna-tional 1999
[6] F A Longstaff and E S Schwartz ldquoValuing American optionsby simulation a simple least-squares approachrdquo Review ofFinancial Studies vol 14 no 1 pp 113ndash147 2001
[7] M Moreno and J F Navas ldquoOn the robustness of least-squaresMonte Carlo (LSM) for pricing American derivativesrdquo Reviewof Derivatives Research vol 6 no 2 pp 107ndash128 2003
[8] L Stentoft ldquoConvergence of the least squares Monte Carloapproach to American option valuationrdquoManagement Sciencevol 50 no 9 pp 1193ndash1203 2004
[9] S Crepey and A Rahal ldquoPricing convertible bonds with callprotectionrdquo Journal of Computational Finance vol 15 no 2 pp37ndash75 2011
[10] M Ammann A Kind and CWilde ldquoSimulation-based pricingof convertible bondsrdquo Journal of Empirical Finance vol 15 no2 pp 310ndash331 2008
[11] K Tsiveriotis and C Fernandes ldquoValuing convertible bondswith credit riskrdquo The Journal of Fixed Income vol 8 no 2 pp95ndash102 1998
[12] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999
[13] E Ayache P A Forsyth and K R Vetzal ldquoValuation ofconvertible bonds with credit riskrdquo The Journal of Derivativesvol 11 no 1 pp 9ndash29 2003
[14] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[15] J G Huang S C Yang and X Feng ldquoValuation of convertiblebond with dynamic credit riskrdquo Mathematical Statistics andManagement vol 27 no 6 pp 1108ndash1116 2008
[16] Z L Zheng andH Lin ldquoEstimation of the default risk premiumin Chinardquo Securities Market Herald vol 13 no 6 pp 41ndash442003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Discrete Dynamics in Nature and Society
0 50 100 15095
100
105
110
115
120
125
Time (week)
Pric
e (RM
B)
Market pricesTheoretical prices
Figure 1 Comparison of the theoretical and market prices of ICBCconvertible bond
0 50 100 1500
002
004
006
008
01
012
MA
D
Time (week)
Figure 2 The relative deviation (AD) of ICBC convertible bond
41 Pricing Results with Static Credit Risk We first assumethat the credit risk is static and get 5000 paths of the stockprice by LSM Then we get the theoretical prices of ICBCconvertible bond using our pricing framework Figure 1 is thecomparison of the theoretical prices and market prices
From Figure 1 we can see that the tendencies of thetwo lines fit well in the long run so we can use this priceframework to forecast market price of convertible bond andmake investment decision On the other hand the marketprice is a little higher than the theoretical price so ICBCconvertible bond is a little overpriced
We also define the variable AD to be the absolutedeviation of the theoretical price from the market price asfollows
AD119894=
10038161003816100381610038161003816119881119894minus 119881119894
10038161003816100381610038161003816
119881119894
(7)
0 50 100 15095
100
105
110
115
120
125
Time (week)
Pric
e (RM
B)
Market pricePrices with static credit riskPrices with dynamic credit risk
Figure 3 Comparison of theoretical prices with dynamic and staticcredit risk
Then we get the figure of AD by Matlab and depict it inFigure 2
The following mean absolute deviation MAD is definedto describe the integral result of our model
MAD =1
146
146
sum
119894=1
AD119894 (8)
Through calculation the mean absolute deviation ofICBC convertible MAD is 23 within 5 which demon-strates that the theoretical price obtained from ourmodel canreflect market price
42 Pricing Results Using Dynamic Credit Risk In this sec-tion we give the pricing results of ICBC convertible bondusing the dynamic credit risk modeled by (6) and with 120578 = 2We compare the market prices with the prices obtained fromstatic credit risk and dynamic credit risk in Figure 3
We can see from Figure 3 that the market price of theICBC convertible bond is higher than the prices obtainedfrom both the static and dynamic credit risk models whichmeans that the ICBC convertible bond is overestimatedresulting from the underestimation of credit riskWe can alsosee that the dynamic credit risk can reflect the real credit risksincewhen the price of convertible bond goes downwards theprice obtained from dynamic credit risk is lower than thoseobtained from static credit risk This implies that we shoulduse dynamic credit risk to price convertible bonds
43 Effect of Conversion Price In the above empirical studywe price the ICBC convertible bond based on changingconversion price (CP) resulting from two issues of dividendThough most Chinese companies change conversion pricewhen issuing dividend international companiesrsquo conversionprice does not change So in this section we also give the
Discrete Dynamics in Nature and Society 5
0 44 96 15090
95
100
105
110
115
120
125
Time (week)
Pric
e (RM
B)
Market pricesPrices with unchanging CPPrices with changing CP
Figure 4 Pricing results without changing the conversion price
prices of ICBC convertible bond assuming that the conver-sion price does not change The pricing results together withthose obtained from changing conversion price are shown inFigure 4
Figure 4 indicates that compared with the prices withchanging CP obtained in Section 41 the obtained prices withunchanging CP have two jumps at time 119905 = 44 and 119905 =96 just after the CP changed This demonstrates that whenthe company issues dividend the effect of unchanging CPis significant This can be explained by the fact that whenthere is an issue of dividend investors of Chinarsquos convertiblebonds will have an expectation of price decline which willaffect the payoff of convertible bonds eventually ThereforeChinese companies of convertible bonds are suggested tochange CP when issuing dividend That is we cannot copythe international experience of unchanging CP
5 Conclusion
This paper studies the pricing of convertible bonds withdynamic credit risk using least squares Monte Carlo methodWe employ the dynamic credit spread changing with stockprice In empirical study our model is proved to be effectiveand the comparison test demonstrates that the dynamiccredit risk is important in convertible bond pricing Theprice obtained from dynamic credit risk can reflect thereal credit risk Thus the potential risk resulting from theoverestimation of convertible bonds cannot be neglected bythe investors In addition we also study the empirical effectof changing the conversion price when the issuer distributesdividend Consequently the unchanged conversion price willlead to an unreasonable price So Chinarsquos market is notmature enough to keep the conversion price constant just likeinternational markets
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China (nos 71271015 and 70971006)
References
[1] J E Ingersoll Jr ldquoA contingent-claims valuation of convertiblesecuritiesrdquo Journal of Financial Economics vol 4 no 3 pp 289ndash321 1977
[2] M J Brennan and E S Schwartz ldquoConvertible bonds valuationand optimal strategies for call and conversionrdquo The Journal ofFinance vol 32 no 5 pp 1699ndash1715 1977
[3] J J McConnell and E S Schwartz ldquoLYON tamingrdquoThe Journalof Finance vol 41 no 3 pp 561ndash576 1986
[4] M J Brennan and E S Schwartz ldquoAnalyzing convertiblebondsrdquo Journal of Financial and Quantitative Analysis vol 15pp 907ndash929 1980
[5] M Davis and F R Lischka ldquoConvertible bonds with marketrisk and credit riskrdquoWorking Paper Tokyo-Mitsubishi Interna-tional 1999
[6] F A Longstaff and E S Schwartz ldquoValuing American optionsby simulation a simple least-squares approachrdquo Review ofFinancial Studies vol 14 no 1 pp 113ndash147 2001
[7] M Moreno and J F Navas ldquoOn the robustness of least-squaresMonte Carlo (LSM) for pricing American derivativesrdquo Reviewof Derivatives Research vol 6 no 2 pp 107ndash128 2003
[8] L Stentoft ldquoConvergence of the least squares Monte Carloapproach to American option valuationrdquoManagement Sciencevol 50 no 9 pp 1193ndash1203 2004
[9] S Crepey and A Rahal ldquoPricing convertible bonds with callprotectionrdquo Journal of Computational Finance vol 15 no 2 pp37ndash75 2011
[10] M Ammann A Kind and CWilde ldquoSimulation-based pricingof convertible bondsrdquo Journal of Empirical Finance vol 15 no2 pp 310ndash331 2008
[11] K Tsiveriotis and C Fernandes ldquoValuing convertible bondswith credit riskrdquo The Journal of Fixed Income vol 8 no 2 pp95ndash102 1998
[12] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999
[13] E Ayache P A Forsyth and K R Vetzal ldquoValuation ofconvertible bonds with credit riskrdquo The Journal of Derivativesvol 11 no 1 pp 9ndash29 2003
[14] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[15] J G Huang S C Yang and X Feng ldquoValuation of convertiblebond with dynamic credit riskrdquo Mathematical Statistics andManagement vol 27 no 6 pp 1108ndash1116 2008
[16] Z L Zheng andH Lin ldquoEstimation of the default risk premiumin Chinardquo Securities Market Herald vol 13 no 6 pp 41ndash442003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 5
0 44 96 15090
95
100
105
110
115
120
125
Time (week)
Pric
e (RM
B)
Market pricesPrices with unchanging CPPrices with changing CP
Figure 4 Pricing results without changing the conversion price
prices of ICBC convertible bond assuming that the conver-sion price does not change The pricing results together withthose obtained from changing conversion price are shown inFigure 4
Figure 4 indicates that compared with the prices withchanging CP obtained in Section 41 the obtained prices withunchanging CP have two jumps at time 119905 = 44 and 119905 =96 just after the CP changed This demonstrates that whenthe company issues dividend the effect of unchanging CPis significant This can be explained by the fact that whenthere is an issue of dividend investors of Chinarsquos convertiblebonds will have an expectation of price decline which willaffect the payoff of convertible bonds eventually ThereforeChinese companies of convertible bonds are suggested tochange CP when issuing dividend That is we cannot copythe international experience of unchanging CP
5 Conclusion
This paper studies the pricing of convertible bonds withdynamic credit risk using least squares Monte Carlo methodWe employ the dynamic credit spread changing with stockprice In empirical study our model is proved to be effectiveand the comparison test demonstrates that the dynamiccredit risk is important in convertible bond pricing Theprice obtained from dynamic credit risk can reflect thereal credit risk Thus the potential risk resulting from theoverestimation of convertible bonds cannot be neglected bythe investors In addition we also study the empirical effectof changing the conversion price when the issuer distributesdividend Consequently the unchanged conversion price willlead to an unreasonable price So Chinarsquos market is notmature enough to keep the conversion price constant just likeinternational markets
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China (nos 71271015 and 70971006)
References
[1] J E Ingersoll Jr ldquoA contingent-claims valuation of convertiblesecuritiesrdquo Journal of Financial Economics vol 4 no 3 pp 289ndash321 1977
[2] M J Brennan and E S Schwartz ldquoConvertible bonds valuationand optimal strategies for call and conversionrdquo The Journal ofFinance vol 32 no 5 pp 1699ndash1715 1977
[3] J J McConnell and E S Schwartz ldquoLYON tamingrdquoThe Journalof Finance vol 41 no 3 pp 561ndash576 1986
[4] M J Brennan and E S Schwartz ldquoAnalyzing convertiblebondsrdquo Journal of Financial and Quantitative Analysis vol 15pp 907ndash929 1980
[5] M Davis and F R Lischka ldquoConvertible bonds with marketrisk and credit riskrdquoWorking Paper Tokyo-Mitsubishi Interna-tional 1999
[6] F A Longstaff and E S Schwartz ldquoValuing American optionsby simulation a simple least-squares approachrdquo Review ofFinancial Studies vol 14 no 1 pp 113ndash147 2001
[7] M Moreno and J F Navas ldquoOn the robustness of least-squaresMonte Carlo (LSM) for pricing American derivativesrdquo Reviewof Derivatives Research vol 6 no 2 pp 107ndash128 2003
[8] L Stentoft ldquoConvergence of the least squares Monte Carloapproach to American option valuationrdquoManagement Sciencevol 50 no 9 pp 1193ndash1203 2004
[9] S Crepey and A Rahal ldquoPricing convertible bonds with callprotectionrdquo Journal of Computational Finance vol 15 no 2 pp37ndash75 2011
[10] M Ammann A Kind and CWilde ldquoSimulation-based pricingof convertible bondsrdquo Journal of Empirical Finance vol 15 no2 pp 310ndash331 2008
[11] K Tsiveriotis and C Fernandes ldquoValuing convertible bondswith credit riskrdquo The Journal of Fixed Income vol 8 no 2 pp95ndash102 1998
[12] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999
[13] E Ayache P A Forsyth and K R Vetzal ldquoValuation ofconvertible bonds with credit riskrdquo The Journal of Derivativesvol 11 no 1 pp 9ndash29 2003
[14] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[15] J G Huang S C Yang and X Feng ldquoValuation of convertiblebond with dynamic credit riskrdquo Mathematical Statistics andManagement vol 27 no 6 pp 1108ndash1116 2008
[16] Z L Zheng andH Lin ldquoEstimation of the default risk premiumin Chinardquo Securities Market Herald vol 13 no 6 pp 41ndash442003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
