Research ArticlePricing Chinese Convertible Bonds with Default Intensity byMonte Carlo Method

Xin Luo 12 and Jinlin Zhang1

1School of Finance Zhongnan University of Economics and Law Wuhan China2College of Economics and Management Hubei University of Automotive Technology Shiyan China

Correspondence should be addressed to Xin Luo xinluozueleducn

Received 18 December 2018 Accepted 28 March 2019 Published 15 April 2019

Academic Editor Paolo Renna

Copyright copy 2019 Xin Luo and Jinlin Zhang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This article proposes a new way to price Chinese convertible bonds by the Longstaff-Schwartz Least Squares Monte CarlosimulationThe default intensity and the volatility are the two important parameters which are difficultly obtained in the emergingmarket in pricing convertible bonds By developing theMerton theory we find a new effectivemethod to get the theoretical value ofthe two parameters In the pricing method the default risk is described by the default intensity and a default on a bond is triggeredby the bottom Q(T) (default probability) percentile of the simulated stock prices at the maturity date In the present simulation arisk-free interest rate is used to discount the cash flows So the new pricingmodel is considered to tally with the general pricing ruleunder martingale measureThe empirical results of the CEB and the XIG convertible bonds by the proposed method are comparedwith those obtained by the credit spreads method It is also found that the theoretical prices calculated by the method proposed inthe article fit the market prices well especially in the long run tendency

1 Introduction

Convertible bonds are highly hybrid financial derivativeswhich can be converted into the issuerrsquos stock under somespecified conditions They are important refinancing toolsfor listed companies and essential investment targets amonghedge funds and other institutional investors Due to theirflexibility convertible bonds are also well received by per-sonal investors The convertible bond is considerably morecomplex than the warrant not only because it pays a periodiccoupon but also because it involves a dual option on theone hand the bond holder possesses the option to convertthe bond into common stock at his or her discretion andon the other hand the firm possesses the option to call thebond for redemption and the bondholder retaining the rightto convert the bond or to redeem itThis call option is usuallysubject to some kind of restriction The investorrsquos optimalconversion strategy then depends on the firmrsquos call strategyand it appears at first sight that the optimal call strategiesmust depend on the inventorrsquos conversion strategy so thatboth optimal strategies must be solved for simultaneously

Convertible bond valuation has been an important issuein both the academia and the industry Pricing convertiblesecurities was initiated by Ingersoll [1] who supposed thatthe bond value is a function solely of the firm value A and thetime t by It119900 Lemma and the usual arbitrage arguments hegot the Black-Sholes partial differential equation the optimalstrategies for call conversion and put give the boundary con-ditions subject to which the B-S partial differential equationshould be For some special bonds (the discount bond theconsol bond) or in some special cases[1 2] analytic solutionsfor these bond valuation could be obtained with the Black-Sholes-Merton model

Following Ingersollrsquos structural approach Brennan andSchwartz [3 4] were the first to apply the principles ofthe option pricing model to the convertible bond valuationBy optimal strategies for call and conversion they got theboundary conditions of the convertible bond By resortingto the finite differences method Brennan and Schwartzgot the numerical solution of the convertible bond valueThe structural approach of the bond valuation developedby Ingersoll Brennan and Schwartz has two advantages at

HindawiDiscrete Dynamics in Nature and SocietyVolume 2019 Article ID 8610126 8 pageshttpsdoiorg10115520198610126

2 Discrete Dynamics in Nature and Society

least One advantage of the approach is that any contingentclaim whose value can be written as a function solely of themarket value of the firm and the remaining time to maturity120591 V(A 120591) must satisfy the basic B-S partial differentialequation the various convertible securities can be describedby the different boundary conditions So only needing tochange the boundary conditions according to the indentureof the bond we can get the solutions of the value of thevarious bonds through solving the B-S partial differentialequation subject to the boundary conditions determined bythe indenture of the bondAnother advantage of the approachis that it can measure the default risk by supposing that adefault will occur only when the balloon payment cannot bemet by all the assets of the firm ie AltB where B is theface value of the bond plus the accrued interest at maturityThe default criterion can be demonstrated in the boundaryconditions So the solutions of the convertible securitiesvalue include the default risk

The structural approach of the bond valuation has adisadvantage ie it is difficult to get the firm value The firmvalue variable was soon replaced by the stock price for itssimplicity in observation and measurement which is the so-called reduced-form approach developed by McConnell andSchwartz [5] The driving force of the valuation is changedto be the issuing firm stock price which is also assumedto follow a stochastic process This approach precludes thedefault risk because there is no unequivocal way to linkthe stock price to a default event In order to compensatethe overestimate of the value in the reduced-form approachMcConnell and Schwartz used a risky interest rate as thediscount rate This method to measure the default risk iscalled the credit spreads (CS) method

In the emerging Chinarsquos convertible bond market theconvertible bonds are designed to have more complicatedclauses compared with those of the mature markets forexample stipulating the put provision call provision subjectto more restrictions and the reset provisions Because of thedual characters of the convertible bond ie equity and bondand moreover the varieties of terms of Chinese convertiblebonds it is difficult to price the Chinese convertible bondsDefault risk is an important factor in convertible bondpricing With some quasi-default events appearing recentlyin Chinarsquos market more attention to model the default risk ispaid than before in China [6ndash8] Because of the complexitythe convertible bond value can only be solved numerically inmost practical cases Monte Carlo simulation became widelyused in convertible bond pricing [9ndash11]

As is known to all it is important to describe thedefault risk for pricing the convertible bonds On the wholethere exist two methods to measure the default risk Thefirst one is to use the credit spreads (CS) which is firstlyused by McConell and Schwartz [5] to price convertiblebonds Following the work of McConell and Schwartz [5]Tsiveriotis and Fernandes [12] split a convertible bond intotwo components a cash-only part which was discounted byrisky interest rate and an equity part which was discountedby risk-free interest rate The second method is to use thedefault intensity (DI) which was introduced by Duffie andSingleton [13] to price corporate bonds Then Ayache et

al [14] applied the DI method to price convertible bondsthey used the default intensity to represent the risky interestrate the convertible bond value can be gotten by solvingthe B-S partial differential equation in which the default-adjusted interest rate was contained subject to the boundaryconditions determined by the indenture of the convertiblebond The first method is more widely used for its simplicityand convenience than the second one Early works [5 12])priced the default risk of the convertible bond mainly bydiscounting at a higher risk-adjusted rate However as Battenet al [15] pointed out this approach is subject to criticismsince credit risk spreads are neither constant over time norconstant along the yield curve

Park K W et al [16] used the default intensity (DI)obtained from the transition probability matrix for ratedfirms to price the defaultable convertible bond In the maturebonds markets the DI can be obtained from the transitionprobability matrix for rated firms which shows the prob-ability of rating migration between the year and the nextyear Up to now there are no real default events occurringin the emerging bonds market of China so the transitionprobability matrix for rated firms did not show the defaultprobability So Parkrsquos approach is not effective for pricingthe Chinese convertible bonds If the CS method is adoptedto price the default risk it is likely that the researchers areentangled by the otherwise great difference of the CS fordifferent bonds besides the above-mentioned criticism Thevolatility is another difficulty for pricing Chinese convertiblebonds Issuing convertible bonds will affect the volatility ofthe underlying stock return and generally make the volatil-ity of the underlying stock return decrease The volatilitycalculated from the historical stock prices before issuingthe convertible bond cannot be simply used to price theconvertible bond How to estimate the volatility of the stockafter issuing the convertible bond and how to estimate theDI more efficiently at least in theory In the article we willtry to broaden the application of the Merton model [17] tosolve these problems Merton supposed that the firm valueis comprised of the equity and the liability The equity isconsidered as a call option on the firm value struck at thebook value of the liability From the Merton model we canget the cumulative default probability and the volatility of thefirm value after issuing the convertible bonds To a certainextent this volatility can be supposed to be the volatility ofthe underlying stock after issuing the bonds this suppositionshould be reasonable The further explanation will be givenin the bottom of the next section

The advantage of the present convertible bond pricingmethod lies in that the two important parameters the long-run average volatility and the DI can be directly calculatedby the Merton model In our approach we price the defaultrisk by immediately reducing cash flows to a fraction of theface value of the bond when a default event occurs hencethe resulting cash flows are discounted at a risk-free rate Inthis way the general pricing rule under martingale measureis preserved in our pricing model

This article is organized as follows The second sectiondiscusses the convertible bond pricing model in whichwe focus on dealing with the default risk and getting the

Discrete Dynamics in Nature and Society 3

default intensity and the long-term average volatility of theunderlying stock price after issuing convertible bonds Thesimulation design is given in the third section The fourthsection shows the results of the empirical research includingthe comparison of the prices obtained by the DI method andthe CSmethod Conclusions are presented in the last section

2 The Chinese Convertible Bond PricingModel and the Parameters Estimation

The Chinese convertible bonds are embedded with manyoptions such as the conversion option call option putoption and option to lower the conversion price The val-uation of the convertible bonds crucially depends on howthe underlying stock price evolves over time and on theoptimal stopping rules of both the investor and the issuer Itis assumed that the underlying stock price follows Brownianmotion process with a constant volatility

d119878119905 = r119878119905119889119905 + 120590119878119905119889119885 (1)

where 119878119905 is the underlying stock price at the time t r is therisk-free interest rate 120590 is the volatility of the return of thestock and dZ denotes the usual Wiener process

Every moment before the maturity of the convertiblebond investors and issuers will gamble over the benefitInvestors will maximize the value of the convertible bondswhile the issuers will minimize the value of the convertiblebonds from exercising the call option Hence at any timethe value of the convertible bond must not be less than itsconversion value

V (119878119905 119905) ge 119898119878119905 (2)

where V(119878119905 119905) is the value of the convertible bond at thetime t m is the conversion ratio and m119878119905 is the immediateconversion value If (2) does not hold the investor wouldvoluntarily convert bonds into stocks to make an immediateprofit The call provision of the Chinese convertible bondsemphasizes to force the bondholders to convert bonds intostocks immediately for example the call provision of theconvertible bonds issued by China Everbrigh Bank Co Ltd(CEB) ldquoIn 30 consecutive trading days the closing stockprice is not less than 130 of conversion price in 15 tradingdaysThe firm has the option to call the bond at the face valueof the bond plus the current accrued interestrdquo Because thecall price when the call condition is satisfied is much lowerthan the conversion value the firm should call the bonds andif the bond is called the bondholders must elect to convertthe bonds into stocks So if the call condition is satisfied thevalue of the convertible bond is the conversion value

V (119878119905 119905) = max 119870119905 119898119878119905 = 119898119878119905 (3)

where 119870119905 is the call price at the time t The investor will putthe bond if the continuation value is less than the put priceSo the value of the convertible bondmust not be less than theput price if the put condition is satisfied

V (119878119905 119905) ge 119875119905 (4)

Suppose the asset of the firm consists of only the issuingconvertible bonds and its underlying stocks Suppose furtherthat the indenture of the bond issue contains the followingprovisions and restrictions (1) The firm promises to paythe balloon payment B ie the face value of the bond plusthe accrued interest to the bondholders at maturity (2) Inthe event this payment is not met the firm will default onthe bond The firm value A(t) can be expressed as A(t) =S(t) + v(t) where S(t) is the value at time t of the equityV(t) is the value of the bond and all the stock value and thefirm value follow Brownian motion process with the sameWiener process and the instantaneous returns on S andA areperfectly correlated [17 eq (3c)] When A lt B at maturitythe firm will default on the bond In order to consider thedefault risk in the convertible bond pricing we introducethe exogenous variable cumulative default probability Q(T)and default intensity 120582(t) Given the cumulative defaultprobability Q(T) it means that at the maturity date thebottom Q(T) percentile of the simulated firm values is equalto B we can get that the critical firm value 1198601015840119905 ie equalto B be the lowest level below which the issuer will defaulton the bonds Because there is a one-to-one correspondencebetween S(t) and A(t) and the stock value and the firmvalue follow Brownian motion process with the sameWienerprocess [17] we always can find a unique 1198781015840119879 correspondingto 1198601015840119879 by the bottom Q(T) percentile of the simulated stockprices at the maturity date When the stock value is less than1198781015840119879 the firm will default on the bond

In the Merton model the equity is a call option on thefirm value with the strike price equivalent to the liability bookvalue According to the call option model of the no-dividendasset there exits the one-to-one correspondence between 119878119905and the firm value 119860 119905 and 119878119905 is the monotonically increasedfunction of119860 119905 If we let1198601015840119905 be the lowest level belowwhich theissuer will default on the bonds a unique 1198781015840119905 corresponding to1198601015840119905 always can be found at the time t So we can use variables119878119905 and 119860 119905 to describe equivalently the conditional defaultprobability for t isin [1 2 119879]

119875119903 (119878119905 le 1198781015840119905 | (119878119905minus1 gt 119878

1015840119905minus1))

= 119875119903 (119860 119905 le 1198601015840119905 | 119860 119905minus1 gt 119860

1015840119905minus1)

(5)

Equation (5) guarantees that we can determine the defaultcritical value 1198781015840119905 by the same way as determining the defaultcritical value 1198601015840119905 ie 119878

1015840119905 is the bottom 120582 (default intensity)

percentile of the simulated stock pricesAssuming a constant recovery ratio in the default event

the investor will receive a fraction120579 of the face value of thebond when the issuer defaults on the bond at time t

V (119878119905 119905) = 120579119865 for 0 le 119878119905 le 1198781015840119905 (6)

where F is the face value of the bond and 1198781015840119905 is the lowest stockprice level at the time t below which the issuer will default onthe bond

4 Discrete Dynamics in Nature and Society

At the maturity date the payoff to the investor is given bythe following

V (119878119879 119879) = 119898119878119879 for 119898119878119879 ge 119861

= 119861 for 1198981198781015840119879 lt 119898119878119879 lt 119861

= 120579119865 for 0 le 119898119878119879 le 1198981198781015840119879

(7)

where the recovery ratio is assumed to be 0 le 120579 le 1In the credit risk analysis should the risk-neutral default

probability or the physical default probability be adoptedThe answer to the question depends on the purpose of thestudy The discounted present value of the cash flow needsto be calculated in pricing the convertible bonds henceone ineluctably employ the risk-neutral valuation theory Sothe default probability in the risk-neutral world should beadopted to price the credit derivatives In estimation of theexpected shortfall triggered by a default event by scenarioanalysis the physical default probability should be adoptedAccording to the Merton model at the maturity date thestock value can be denoted as

119878119879 = max (119860119879 minus 119861 0) (8)

where 119878119879 B and 119860119879 are the stock value at the maturitydate the balloon payment and the firm asset value at thematurity date respectively We suppose that the stock valueis the function of the firm value A and the remaining timeto maturity 120591 S = S(119860 120591) By the Lt119900 lemma and the usualarbitrage arguments the stock price must satisfy the partialdifferential equation

1212059021198602119878119860119860 + 119903119860119878119860 minus 119903119878 minus 119878120591 = 0 (9)

where 120590 is the volatility of the return of the stock r isthe risk-free rate of interest and 120591 is the remaining time tomaturity date The solution of (9) subject to the boundarycondition (8) is

1198780 = 1198600119873(1198891) minus 119863119890minus119903119879119873(1198892) (10)

where

1198891 =ln (1198600119863) + (119903 + 12059021198602) 119879

120590119860radic119879(11)

and 1198892 = 1198891 minus120590119860radic119879 and 120590119860 is the volatility of the firm valuewhich is assumed to be constant for t isin [1 2 119879]1198600 is thecurrent value of the firm asset From Itorsquos Lemma we get

1205901198780 = 119873 (1198891) 1205901198601198600 (12)

where 120590 is the volatility of the return of the stock beforeissuing the bonds

The volatility of the return of the stock before issuing thebonds can be estimated by the historical stock prices From(10) and (12) we can get 1198600 and 120590119860 then the cumulativedefault probability of the firm can be calculated by N(minus1198892)

The convertible bond is a long-term security so a long-termaverage volatility should be adopted in pricing convertiblebonds The issuing of the convertible bonds will have animpact on the volatility of the underlying stock The data ofthe underlying stock price after issuing the convertible bondsfor pricing the convertible bond is generally insufficient orabsent hence a long-term average volatility of the underlyingstock after issuing the convertible bonds cannot be obtainedfrom the historical data So we hope to find a theoreticalmethod to get the long-term average volatility of the under-lying stock after issuing the convertible bonds The Chineselisted firm can prevail upon the investors to convert bonds bylowering the conversion price in Chinese convertible bondmarket nearly all convertible bonds have been converted intostocksThe firmrsquos asset will become only equity So we use 120590119860of the firm value to replace the long-term average volatilityof the underlying stock price after issuing the convertiblebonds

The cumulative default probability Q(119905) can be expressedby the default intensity 120582(119905)

Q (119905) = 1 minus 119890minusint119905

0120582(120591)119889120591 (13)

Assuming the default intensity is a constant we can get

120582 = minus ln (1 minus 119876 (119879))119879

(14)

The cumulative default probability N(minus1198892) is the probabilitythat the firm cannot afford the balloon payment at thematurity If the convertible bond is the only one senior debtin theory the firm may default only on the maturity date Soin the convertible bond pricing we only need to considerdefaults that occur on the maturity date In view of manyfactors that can result in defaults (including various termdebts and adverse emergencies) these factors can result inthe firm default on the bond prior to the maturity dateThese factors also are among the driving forces that makethe firm value change In order to study the affecting factorsof the convertible bond more overall we still use the defaultintensity 120582(119905) to measure default risk prior to the maturitydate in spite of that call in question the rationality of doingso

3 Simulation Design

Most of the Chinese firms stipulate the first put provisionsldquoif the investors do not exercise the put option when theput provision is firstly triggered the investors will not beallowed to exercise the put option in the current year whenthe put provision is triggered againrdquo The firm just needsto change the conversion price when the put provision isfirstly triggered so as to promote the value of the convertiblebond more than the put price The investors will hold theconvertible bonds instead of putting the convertible bondThe adjusted conversion price119883119905 should satisfy

P = 119878119905119888119873(1198891) minus 119865119890minus119903(119879minus119905)119873(1198892) + (119865 + 119868) 119890

minus119903(119879minus119905) (15)

Discrete Dynamics in Nature and Society 5

Table 1 Rules of optimal exercise decision in convertible bonds

Payoff Condition Decision119862119905 119881119905 gt 119862119905 119886119899119889 gt 119898119878119905 callm119878119905 119881119905 gt 119862119905 119886119899119889 119862119905 lt 119898119878119905 forced conversionm119878119905 119881119905 lt 119898119878119905 119886119899119889 119875119905 le 119898119878119905 voluntary conversion0 119881119905 lt 119862119905 119881119905 gt 119898119878119905 119886119899119889 119881119905 gt 119875119905 continuation119875119905 119875119905 gt 119881119905 119886119899119889 119898119878119905 le 119875119905 PutB m119878119905 lt 119861 redemption at maturity120579F St lt S1015840t default

where

119878119905119888 =100119878119905119883119905

1198891 =ln (119878119905119888119865) + (119903 + 12059021198602) (119879 minus 119905)

120590119860radic119879 minus 119905

1198892 = 1198891 minus 120590119860radic119879 minus 119905

(16)

and P and I are the put price and the accrued interest duringT minus t respectively

Including a default decision by the issuer the optimalexercise decision of the investor and the issuer is summarizedin Table 1 In our simulation the convertible bond value isthe expected discounted cash flow which is estimated by theLongstaff-Schwartz (LS) Least Squares Monte Carlo (LSM)simulation method [18] The optimal stopping rule can thenbe determined by the simultaneous optimal exercise strate-gies of both the investor and the issuer For our simulationwe assume the discrete time by the day over the period witha finite time set t isin [0 1 2 119879] where t = 0 for today andt = T for the maturity date The basic framework of pricingChinese convertible bonds by LSM is as follows

(1)Theduration of convertible bonds contains T days andthe convertible bonds can only be exercised at the stipulatedconversion period Depending on the current stock price wecan get theM paths containing T days of the underlying stockprice by Monte Carlo simulation

(2) The defaulted group of the sample paths belongs to abottom 120582 percentile of the realized stock prices at each dayThe issuer may default on the bond any time before or on thematurity and we directly take into account the default risk inthe present valuation This is done by reducing the resultingcash flows immediately to a fraction 120579 of the face value of thebond when a default occurs in the simulated paths By doingso we adhere to the general pricing rule under martingalemeasure

(3)When the put provisions have been triggered the newconversion price is calculated by (15) and the value of theconvertible bond is promoted equivalent to the put price

(4)When the call provision has been triggered nomatterhow big the conversion value is at that day the convertiblebonds will be exercised and terminated

(5) By (7) the cash flows on the maturity date areobtained In the other conversion time node we choosethe in-the-money option paths to estimate the continuation

value by the Longstaff-Schwartz Least Squares simulationwhich is accomplished by regressing the subsequent realizedcash flows on a basis function of constants S and 1198782Comparing the continuation value with the exercise value ifthe continuation value is bigger the optimal stopping valueremains the same otherwise the new stopping time and thenew stopping value are obtained For the out-of-the-moneyoption paths the convertible bonds will not be exercised andone does not need to change the optimal stopping value

(6) The convertible bond can be priced by discountingeach cash flowback to time t=0with the risk-free interest rateand averaging over all paths

4 The Empirical Research

To test the performance of our model two convertible bondsissued by China Everbrigh Bank Co Ltd (CEB) with largemarket shares and Xiamen Itg Group Co Ltd (XIG) arerandomly chosen for the empirical study We adopt the dataof the CEB and XIG convertible bonds at the issuing dateobtained from WIND database The information of the twoconvertible bonds is in Table 2

For comparing with the CS method we compute thetheoretical prices of the two bonds using our simulationmodel and the CS method It is well known to all that theconvertible bond can be split into two components a cash-only part and an equity part for pricing the convertiblebond [12] In fact the credit spreads of AAA-rated firm debtsare only 1 in Chinarsquos convertible market For conveniencein the present calculation with the CS method we use arisky interest rate to discount each cash flow without greatinfluence on the price of the convertible bond In the articlewe use the mean yield of the defaultable enterprise debts inthe same market as the risky interest rate Before pricing wefirst need to estimate the related parameters The long-termmean volatility of the underlying stock the equity value ofthe firm at issuing date and the 6-year risk-free interest rateand the 6-year risky interest rate at the issuing date can beobtained fromWIND database By solving (10) and (12) withoptimization method we get the volatility of the firm assetvalue and the cumulative default probability of the firmThese6 parameter values are given inTable 3Here we have adoptedthe historical stock prices of 2 years for CEB and 5 yearsfor XIG respectively The purpose for which we adopted thedifferent term historical data is to lower the effect of the 2015stock market crash on the pricing of the convertible bondIf a default occurs over the duration of convertible bondswe choose an empirical constant recovery ratio of 20 forenterprise debts [19]

We simulate 50000 paths with 1440 time steps (240 daysper year) before maturity to compute theoretical prices of thebonds The simulated paths are generated by the risk-neutraldynamics of the stock return as (1) We also should note thatthe implied default probability of Merton model is the risk-neutral Hence we can discount the cash flows that the risk-neutral investorwould receive over the period t isin [1 2 T]at a risk-free interest rate r for pricing the convertible bond

As the Chinese convertible bonds are generally set theconversion period which is from the first day after sixmonths

6 Discrete Dynamics in Nature and Society

Table 2 CEB and XIG convertible bonds

CEB convertible bond XIG convertible bondIssue date 2017331 2016118Time horizon 6 6Face value 100 100Coupon () 02 05 10 15 18 20 03 05 09 14 17 20Call value till maturity 105 108The first conversion price 436 903Change of conversion price 201775 adjusted to 426 2016621 adjusted to 893

Reset clause

In 30 consecutive tradingdays the closing stock price is smallerthan 80 of conversion price in 15

trading days

In 30 consecutive tradingdays the closing stock price is smallerthan 90 of conversion price in 15

trading days

Call on conditionIn 30 consecutive trading days the

closing stock price is not less than 130of conversion price in 15 trading days

In 30 consecutive trading days theclosing stock price is not less than 130of conversion price in 15 trading days

Call value Face value plus the accrued interest Face value plus the accrued interest

Put on condition When the use of the capital is changed

(1) When the use of the capital is changed(2) In 30 consecutive trading days theclosing stock price is less than 70 of

conversion pricePut value Face value plus the accrued interest Face value plus the accrued interest

Table 3 Parameters of the CEB and XIG convertible bonds ldquo1198640rdquo (hundred million) stands for the equity value of the firm at issuing dateldquo120590119904rdquo the long-termmean volatility of the underlying stock before issuing date ldquorrdquo the 6-year risk-free interest rate at the issuing date ldquo119903119910rdquo the6-year risky interest rate at the issuing date ldquo120590119860rdquo the volatility of the firm asset value and ldquoQrdquo the cumulative default probability

1198640 120590119878 r 119903119910 120590119860 QCEB 19185 02672 003158 004514 0236 00007XIG 1065 04155 002780 003716 03447 00766

of issuing bonds to the maturity date So in the presentsimulation we assume a default may occur in this periodfor convenience As seen in Table 3 the cumulative defaultprobability of the CEB and XIG is 00007 and 00766 for 6years respectively By (14) we can get the default intensity ofXIG 6times10minus5 per day In the present pricing of the convertiblebonds with DI method we consider that a default may occuronly on thematurity date as well as prior to or on thematuritydate Because the cumulative default probability of CEB istoo small we just calculate the price of the CEB convertiblebond with supposition that a default may occur only on thematurity date

Figures 1 and 2 are the comparison of the theoreticalprices and market prices of the two convertible bonds for150 trading days From Figures 1 and 2 three observationsare noteworthy Firstly the theoretical prices calculated byCS method and DI method with supposition that a defaultmay occur any time prior to or on the maturity date are veryclose for XIG convertible bondThe default probability of theCEB convertible bonds is so low that the calculated prices byassuming a default eventmay occur only on thematurity dateand on any time prior to or on thematurity it should be nearly

0 20 40 60 80 100 120 140 160

120

115

110

105

100

Pric

e (RM

B)

MarketCSDI1

Time (day)

Figure 1 Comparison of themarket prices and the theoretical pricesof CEB convertible bond calculated by CS method and DI methodwith supposition that a default may occur only on the maturity date(DI1)

Discrete Dynamics in Nature and Society 7

0 20 40 60 80 100 120 140 160

Time (day)MarketCS

DI1DI2

135

130

125

120

115

110

105

100

Pric

e (RM

B)

Figure 2 Comparison of the market prices and the theoreticalprices of XIG convertible bond calculated by CS method and DImethod with supposition that a default may occur only on thematurity date (DI1) and prior to or on the maturity (DI2)

the same So the theoretical price of the CEB convertiblebond calculated by CS method is evidently smaller than thatby the DI method This is due to the fact that we adopt themean risky interest rate of the defaultable enterprise debtsin the same market to discount the cash flows In fact theconvertible bond with the large market shares like CEB hasa rather low default probability compared with the meandefault probability of the market So the CS pricing methodgenerally underestimates the price of the convertible bondwith the large market shares like CEB This is a shortcomingof the CS method Secondly with the application of the samedefault probability the price estimated by supposition that adefaultmay occur only on thematurity date is evidently largerthan that by supposition that a default may occur any timeprior to or on the maturity By checking the simulated cashflows we find that an otherwise defaulted firm may continueto survive and pays out a higher sum to the investor before thematurity However the investor loses the chance of a highersum being paid out when a default is allowed to occur inany time Lastly we can see that the tendencies of the marketprices and the theoretical prices calculated with suppositionthat a default may occur only on the maturity date as inFigures 1 and 2 fit well in the long run

We introduce the variable AD to describe the absolutedeviation of the theoretical price from the market pricewhich is given in the following formula

119860119863119905 =10038161003816100381610038161003816119881119905 minus 119881119905

10038161003816100381610038161003816119881119905

(17)

The mean absolute deviation (MAD) is used to describe theintegral result of the theoretical models

MAD = 1150

150

sum119905=1119860119863119905 (18)

Table 4 The mean absolute deviation (MAD) of the theoreticalprices of the CEB and XIG convertible bonds calculated by the CSmethod and the DI method with supposition that a default mayoccur only on thematurity date (DI1) and prior to or on thematurity(DI2)

CS DI1 DI2CEB 297 386XIG 607 358 675

Table 5 The standard deviation (STD) of the theoretical prices ofthe CEB and XIG convertible bonds calculated the CS method andthe DI method with supposition that a default may occur only onthe maturity date (DI1) and prior to or on the maturity (DI2)

CS DI1 DI2CEB 00040 00037XIG 00073 00070 00106

Table 4 reports the MAD of the CEB and XIG convertiblebonds calculated by different methods As can be seen therethe MADs of CEB and XIG convertible bonds calculated byCS and DI are all within 7 The theoretical prices obtainedby CS and DI on the whole can reflect market price TheDI method with supposition that a default may occur onlyon the maturity matches the market best especially in thelong run tendency So we can use our pricing frameworkto forecast market price of convertible bond and makeinvestment decision

The standard errors (SEs) of the calculated prices of theconvertible bonds by the CS method and the DI method atthe first trading day are given in Table 5 The SEs for DImethod with supposition that a default may occur only onthe maturity are little less than those of the CS method price

5 Conclusion

In this article we have presented a new way to price theconvertible bond by LSM inwhich the credit risk ismeasuredby DI The key idea in our model is to get importantparameters such as the long-run average volatility and the DIdirectly fromMertonmodel which avoids the difficulty fromthe data lack of the emergingmarket especially up to now noeffective method estimating the long-run average volatility ofthe underlying stock after issuing convertible bonds In thepresent empirical study the theoretical prices fit the marketvery well So our model including the method to get the DIand the long-run average volatility is proved to be effectiveThe present simulation has shown that the convertible bondprice is quite sensitive to the rule on the default decisiontime allowing an earlier default decision which will lower theconvertible bond price

Data Availability

The data used to support the findings of this study aresupplied by wind database and are also available from thecorresponding author upon request

8 Discrete Dynamics in Nature and Society

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] J E Ingersoll Jr ldquoA contingent-claims valuation of convertiblesecuritiesrdquo Journal of Financial Economics vol 4 no 3 pp 289ndash321 1977

[2] C M Lewis ldquoConvertible debt Valuation and conversion incomplex capital structuresrdquo Journal of Banking amp Finance vol15 no 3 pp 665ndash682 1991

[3] M J Brennan and E S Schwartz ldquoConvertible bonds valuationand optimal strategies for call and conversionrdquo e Journal ofFinance vol 32 no 5 pp 1699ndash1715 1977

[4] M J Brennan and E S Schwartz ldquoAnalyzing convertiblebondsrdquo Journal of Financial and Quantitative Analysis vol 15no 4 pp 907ndash929 1980

[5] J J McConnell and E S Schwartz ldquoLYONTamingrdquoe Journalof Finance vol 41 no 3 pp 561ndash576 1986

[6] C Fan X Luo and Q Wu ldquoStochastic volatility vs jump dif-fusions Evidence from the Chinese convertible bond marketrdquoInternational Review of Economics amp Finance vol 49 pp 1ndash162017

[7] X Hu and H Mao ldquoEmpirical study on the financial char-acteristics of chinese companies issuing convertible bondsrdquoInternational Journal of Business and Management vol 4 no6 pp 59ndash64 2009

[8] L Lu and W Xu ldquoA simple and efficient two-factor willow treemethod for convertible bond pricing with stochastic interestrate and default riskrdquo e Journal of Derivatives vol 25 no 1pp 37ndash54 2017

[9] M Ammann A Kind andCWilde ldquoSimulation-based pricingof convertible bondsrdquo Journal of Empirical Finance vol 15 no2 pp 310ndash331 2008

[10] S Crepey and A Rahal ldquoPricing convertible bonds with callprotectionrdquo e Journal of Computational Finance vol 15 no2 pp 37ndash75 2011

[11] R Schoftner ldquoOn the estimation of credit exposures usingregression-basedMonte Carlo simulationrdquoe Journal of CreditRisk vol 4 no 4 pp 37ndash62 2008

[12] K Tsiveriotis and C Fernandes ldquoValuing convertible bondswith credit riskrdquo e Journal of Fixed Income vol 8 no 2 pp95ndash102 1998

[13] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999

[14] E Ayache P A Forsyth and K R Vetzal ldquoValuation ofconvertible bonds with credit riskrdquo e Journal of Derivativesvol 11 no 1 pp 9ndash29 2003

[15] J A Batten K L-H Khaw andM R Young ldquoConvertible bondpricing modelsrdquo Journal of Economic Surveys vol 28 no 5 pp775ndash803 2014

[16] K Park M Jung and S Lee ldquoPricing a defaultable convertiblebond by simulationrdquo Korean Journal of Financial Studies vol46 no 4 pp 947ndash965 2017

[17] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo e Journal of Finance vol 29 no2 pp 449ndash470 1974

[18] 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

[19] R d Cheng and J R Lu ldquoA monte carlo method of integratedrisk measurement for defaultable zero-coupon bondsrdquo Journalof Management Sciences in China vol 15 no 4 pp 88ndash97 2012

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Discrete Dynamics in Nature and Society 3

default intensity and the long-term average volatility of theunderlying stock price after issuing convertible bonds Thesimulation design is given in the third section The fourthsection shows the results of the empirical research includingthe comparison of the prices obtained by the DI method andthe CSmethod Conclusions are presented in the last section

2 The Chinese Convertible Bond PricingModel and the Parameters Estimation

The Chinese convertible bonds are embedded with manyoptions such as the conversion option call option putoption and option to lower the conversion price The val-uation of the convertible bonds crucially depends on howthe underlying stock price evolves over time and on theoptimal stopping rules of both the investor and the issuer Itis assumed that the underlying stock price follows Brownianmotion process with a constant volatility

d119878119905 = r119878119905119889119905 + 120590119878119905119889119885 (1)

where 119878119905 is the underlying stock price at the time t r is therisk-free interest rate 120590 is the volatility of the return of thestock and dZ denotes the usual Wiener process

Every moment before the maturity of the convertiblebond investors and issuers will gamble over the benefitInvestors will maximize the value of the convertible bondswhile the issuers will minimize the value of the convertiblebonds from exercising the call option Hence at any timethe value of the convertible bond must not be less than itsconversion value

V (119878119905 119905) ge 119898119878119905 (2)

where V(119878119905 119905) is the value of the convertible bond at thetime t m is the conversion ratio and m119878119905 is the immediateconversion value If (2) does not hold the investor wouldvoluntarily convert bonds into stocks to make an immediateprofit The call provision of the Chinese convertible bondsemphasizes to force the bondholders to convert bonds intostocks immediately for example the call provision of theconvertible bonds issued by China Everbrigh Bank Co Ltd(CEB) ldquoIn 30 consecutive trading days the closing stockprice is not less than 130 of conversion price in 15 tradingdaysThe firm has the option to call the bond at the face valueof the bond plus the current accrued interestrdquo Because thecall price when the call condition is satisfied is much lowerthan the conversion value the firm should call the bonds andif the bond is called the bondholders must elect to convertthe bonds into stocks So if the call condition is satisfied thevalue of the convertible bond is the conversion value

V (119878119905 119905) = max 119870119905 119898119878119905 = 119898119878119905 (3)

where 119870119905 is the call price at the time t The investor will putthe bond if the continuation value is less than the put priceSo the value of the convertible bondmust not be less than theput price if the put condition is satisfied

V (119878119905 119905) ge 119875119905 (4)

Suppose the asset of the firm consists of only the issuingconvertible bonds and its underlying stocks Suppose furtherthat the indenture of the bond issue contains the followingprovisions and restrictions (1) The firm promises to paythe balloon payment B ie the face value of the bond plusthe accrued interest to the bondholders at maturity (2) Inthe event this payment is not met the firm will default onthe bond The firm value A(t) can be expressed as A(t) =S(t) + v(t) where S(t) is the value at time t of the equityV(t) is the value of the bond and all the stock value and thefirm value follow Brownian motion process with the sameWiener process and the instantaneous returns on S andA areperfectly correlated [17 eq (3c)] When A lt B at maturitythe firm will default on the bond In order to consider thedefault risk in the convertible bond pricing we introducethe exogenous variable cumulative default probability Q(T)and default intensity 120582(t) Given the cumulative defaultprobability Q(T) it means that at the maturity date thebottom Q(T) percentile of the simulated firm values is equalto B we can get that the critical firm value 1198601015840119905 ie equalto B be the lowest level below which the issuer will defaulton the bonds Because there is a one-to-one correspondencebetween S(t) and A(t) and the stock value and the firmvalue follow Brownian motion process with the sameWienerprocess [17] we always can find a unique 1198781015840119879 correspondingto 1198601015840119879 by the bottom Q(T) percentile of the simulated stockprices at the maturity date When the stock value is less than1198781015840119879 the firm will default on the bond

In the Merton model the equity is a call option on thefirm value with the strike price equivalent to the liability bookvalue According to the call option model of the no-dividendasset there exits the one-to-one correspondence between 119878119905and the firm value 119860 119905 and 119878119905 is the monotonically increasedfunction of119860 119905 If we let1198601015840119905 be the lowest level belowwhich theissuer will default on the bonds a unique 1198781015840119905 corresponding to1198601015840119905 always can be found at the time t So we can use variables119878119905 and 119860 119905 to describe equivalently the conditional defaultprobability for t isin [1 2 119879]

119875119903 (119878119905 le 1198781015840119905 | (119878119905minus1 gt 119878

1015840119905minus1))

= 119875119903 (119860 119905 le 1198601015840119905 | 119860 119905minus1 gt 119860

1015840119905minus1)

(5)

Equation (5) guarantees that we can determine the defaultcritical value 1198781015840119905 by the same way as determining the defaultcritical value 1198601015840119905 ie 119878

1015840119905 is the bottom 120582 (default intensity)

percentile of the simulated stock pricesAssuming a constant recovery ratio in the default event

the investor will receive a fraction120579 of the face value of thebond when the issuer defaults on the bond at time t

V (119878119905 119905) = 120579119865 for 0 le 119878119905 le 1198781015840119905 (6)

where F is the face value of the bond and 1198781015840119905 is the lowest stockprice level at the time t below which the issuer will default onthe bond

4 Discrete Dynamics in Nature and Society

At the maturity date the payoff to the investor is given bythe following

V (119878119879 119879) = 119898119878119879 for 119898119878119879 ge 119861

= 119861 for 1198981198781015840119879 lt 119898119878119879 lt 119861

= 120579119865 for 0 le 119898119878119879 le 1198981198781015840119879

(7)

where the recovery ratio is assumed to be 0 le 120579 le 1In the credit risk analysis should the risk-neutral default

probability or the physical default probability be adoptedThe answer to the question depends on the purpose of thestudy The discounted present value of the cash flow needsto be calculated in pricing the convertible bonds henceone ineluctably employ the risk-neutral valuation theory Sothe default probability in the risk-neutral world should beadopted to price the credit derivatives In estimation of theexpected shortfall triggered by a default event by scenarioanalysis the physical default probability should be adoptedAccording to the Merton model at the maturity date thestock value can be denoted as

119878119879 = max (119860119879 minus 119861 0) (8)

where 119878119879 B and 119860119879 are the stock value at the maturitydate the balloon payment and the firm asset value at thematurity date respectively We suppose that the stock valueis the function of the firm value A and the remaining timeto maturity 120591 S = S(119860 120591) By the Lt119900 lemma and the usualarbitrage arguments the stock price must satisfy the partialdifferential equation

1212059021198602119878119860119860 + 119903119860119878119860 minus 119903119878 minus 119878120591 = 0 (9)

where 120590 is the volatility of the return of the stock r isthe risk-free rate of interest and 120591 is the remaining time tomaturity date The solution of (9) subject to the boundarycondition (8) is

1198780 = 1198600119873(1198891) minus 119863119890minus119903119879119873(1198892) (10)

where

1198891 =ln (1198600119863) + (119903 + 12059021198602) 119879

120590119860radic119879(11)

and 1198892 = 1198891 minus120590119860radic119879 and 120590119860 is the volatility of the firm valuewhich is assumed to be constant for t isin [1 2 119879]1198600 is thecurrent value of the firm asset From Itorsquos Lemma we get

1205901198780 = 119873 (1198891) 1205901198601198600 (12)

where 120590 is the volatility of the return of the stock beforeissuing the bonds

The volatility of the return of the stock before issuing thebonds can be estimated by the historical stock prices From(10) and (12) we can get 1198600 and 120590119860 then the cumulativedefault probability of the firm can be calculated by N(minus1198892)

The convertible bond is a long-term security so a long-termaverage volatility should be adopted in pricing convertiblebonds The issuing of the convertible bonds will have animpact on the volatility of the underlying stock The data ofthe underlying stock price after issuing the convertible bondsfor pricing the convertible bond is generally insufficient orabsent hence a long-term average volatility of the underlyingstock after issuing the convertible bonds cannot be obtainedfrom the historical data So we hope to find a theoreticalmethod to get the long-term average volatility of the under-lying stock after issuing the convertible bonds The Chineselisted firm can prevail upon the investors to convert bonds bylowering the conversion price in Chinese convertible bondmarket nearly all convertible bonds have been converted intostocksThe firmrsquos asset will become only equity So we use 120590119860of the firm value to replace the long-term average volatilityof the underlying stock price after issuing the convertiblebonds

The cumulative default probability Q(119905) can be expressedby the default intensity 120582(119905)

Q (119905) = 1 minus 119890minusint119905

0120582(120591)119889120591 (13)

Assuming the default intensity is a constant we can get

120582 = minus ln (1 minus 119876 (119879))119879

(14)

The cumulative default probability N(minus1198892) is the probabilitythat the firm cannot afford the balloon payment at thematurity If the convertible bond is the only one senior debtin theory the firm may default only on the maturity date Soin the convertible bond pricing we only need to considerdefaults that occur on the maturity date In view of manyfactors that can result in defaults (including various termdebts and adverse emergencies) these factors can result inthe firm default on the bond prior to the maturity dateThese factors also are among the driving forces that makethe firm value change In order to study the affecting factorsof the convertible bond more overall we still use the defaultintensity 120582(119905) to measure default risk prior to the maturitydate in spite of that call in question the rationality of doingso

3 Simulation Design

Most of the Chinese firms stipulate the first put provisionsldquoif the investors do not exercise the put option when theput provision is firstly triggered the investors will not beallowed to exercise the put option in the current year whenthe put provision is triggered againrdquo The firm just needsto change the conversion price when the put provision isfirstly triggered so as to promote the value of the convertiblebond more than the put price The investors will hold theconvertible bonds instead of putting the convertible bondThe adjusted conversion price119883119905 should satisfy

P = 119878119905119888119873(1198891) minus 119865119890minus119903(119879minus119905)119873(1198892) + (119865 + 119868) 119890

minus119903(119879minus119905) (15)

Discrete Dynamics in Nature and Society 5

Table 1 Rules of optimal exercise decision in convertible bonds

Payoff Condition Decision119862119905 119881119905 gt 119862119905 119886119899119889 gt 119898119878119905 callm119878119905 119881119905 gt 119862119905 119886119899119889 119862119905 lt 119898119878119905 forced conversionm119878119905 119881119905 lt 119898119878119905 119886119899119889 119875119905 le 119898119878119905 voluntary conversion0 119881119905 lt 119862119905 119881119905 gt 119898119878119905 119886119899119889 119881119905 gt 119875119905 continuation119875119905 119875119905 gt 119881119905 119886119899119889 119898119878119905 le 119875119905 PutB m119878119905 lt 119861 redemption at maturity120579F St lt S1015840t default

where

119878119905119888 =100119878119905119883119905

1198891 =ln (119878119905119888119865) + (119903 + 12059021198602) (119879 minus 119905)

120590119860radic119879 minus 119905

1198892 = 1198891 minus 120590119860radic119879 minus 119905

(16)

and P and I are the put price and the accrued interest duringT minus t respectively

Including a default decision by the issuer the optimalexercise decision of the investor and the issuer is summarizedin Table 1 In our simulation the convertible bond value isthe expected discounted cash flow which is estimated by theLongstaff-Schwartz (LS) Least Squares Monte Carlo (LSM)simulation method [18] The optimal stopping rule can thenbe determined by the simultaneous optimal exercise strate-gies of both the investor and the issuer For our simulationwe assume the discrete time by the day over the period witha finite time set t isin [0 1 2 119879] where t = 0 for today andt = T for the maturity date The basic framework of pricingChinese convertible bonds by LSM is as follows

(1)Theduration of convertible bonds contains T days andthe convertible bonds can only be exercised at the stipulatedconversion period Depending on the current stock price wecan get theM paths containing T days of the underlying stockprice by Monte Carlo simulation

(2) The defaulted group of the sample paths belongs to abottom 120582 percentile of the realized stock prices at each dayThe issuer may default on the bond any time before or on thematurity and we directly take into account the default risk inthe present valuation This is done by reducing the resultingcash flows immediately to a fraction 120579 of the face value of thebond when a default occurs in the simulated paths By doingso we adhere to the general pricing rule under martingalemeasure

(3)When the put provisions have been triggered the newconversion price is calculated by (15) and the value of theconvertible bond is promoted equivalent to the put price

(4)When the call provision has been triggered nomatterhow big the conversion value is at that day the convertiblebonds will be exercised and terminated

(5) By (7) the cash flows on the maturity date areobtained In the other conversion time node we choosethe in-the-money option paths to estimate the continuation

value by the Longstaff-Schwartz Least Squares simulationwhich is accomplished by regressing the subsequent realizedcash flows on a basis function of constants S and 1198782Comparing the continuation value with the exercise value ifthe continuation value is bigger the optimal stopping valueremains the same otherwise the new stopping time and thenew stopping value are obtained For the out-of-the-moneyoption paths the convertible bonds will not be exercised andone does not need to change the optimal stopping value

(6) The convertible bond can be priced by discountingeach cash flowback to time t=0with the risk-free interest rateand averaging over all paths

4 The Empirical Research

To test the performance of our model two convertible bondsissued by China Everbrigh Bank Co Ltd (CEB) with largemarket shares and Xiamen Itg Group Co Ltd (XIG) arerandomly chosen for the empirical study We adopt the dataof the CEB and XIG convertible bonds at the issuing dateobtained from WIND database The information of the twoconvertible bonds is in Table 2

For comparing with the CS method we compute thetheoretical prices of the two bonds using our simulationmodel and the CS method It is well known to all that theconvertible bond can be split into two components a cash-only part and an equity part for pricing the convertiblebond [12] In fact the credit spreads of AAA-rated firm debtsare only 1 in Chinarsquos convertible market For conveniencein the present calculation with the CS method we use arisky interest rate to discount each cash flow without greatinfluence on the price of the convertible bond In the articlewe use the mean yield of the defaultable enterprise debts inthe same market as the risky interest rate Before pricing wefirst need to estimate the related parameters The long-termmean volatility of the underlying stock the equity value ofthe firm at issuing date and the 6-year risk-free interest rateand the 6-year risky interest rate at the issuing date can beobtained fromWIND database By solving (10) and (12) withoptimization method we get the volatility of the firm assetvalue and the cumulative default probability of the firmThese6 parameter values are given inTable 3Here we have adoptedthe historical stock prices of 2 years for CEB and 5 yearsfor XIG respectively The purpose for which we adopted thedifferent term historical data is to lower the effect of the 2015stock market crash on the pricing of the convertible bondIf a default occurs over the duration of convertible bondswe choose an empirical constant recovery ratio of 20 forenterprise debts [19]

We simulate 50000 paths with 1440 time steps (240 daysper year) before maturity to compute theoretical prices of thebonds The simulated paths are generated by the risk-neutraldynamics of the stock return as (1) We also should note thatthe implied default probability of Merton model is the risk-neutral Hence we can discount the cash flows that the risk-neutral investorwould receive over the period t isin [1 2 T]at a risk-free interest rate r for pricing the convertible bond

As the Chinese convertible bonds are generally set theconversion period which is from the first day after sixmonths

6 Discrete Dynamics in Nature and Society

Table 2 CEB and XIG convertible bonds

CEB convertible bond XIG convertible bondIssue date 2017331 2016118Time horizon 6 6Face value 100 100Coupon () 02 05 10 15 18 20 03 05 09 14 17 20Call value till maturity 105 108The first conversion price 436 903Change of conversion price 201775 adjusted to 426 2016621 adjusted to 893

Reset clause

In 30 consecutive tradingdays the closing stock price is smallerthan 80 of conversion price in 15

trading days

In 30 consecutive tradingdays the closing stock price is smallerthan 90 of conversion price in 15

trading days

Call on conditionIn 30 consecutive trading days the

closing stock price is not less than 130of conversion price in 15 trading days

In 30 consecutive trading days theclosing stock price is not less than 130of conversion price in 15 trading days

Call value Face value plus the accrued interest Face value plus the accrued interest

Put on condition When the use of the capital is changed

(1) When the use of the capital is changed(2) In 30 consecutive trading days theclosing stock price is less than 70 of

conversion pricePut value Face value plus the accrued interest Face value plus the accrued interest

Table 3 Parameters of the CEB and XIG convertible bonds ldquo1198640rdquo (hundred million) stands for the equity value of the firm at issuing dateldquo120590119904rdquo the long-termmean volatility of the underlying stock before issuing date ldquorrdquo the 6-year risk-free interest rate at the issuing date ldquo119903119910rdquo the6-year risky interest rate at the issuing date ldquo120590119860rdquo the volatility of the firm asset value and ldquoQrdquo the cumulative default probability

1198640 120590119878 r 119903119910 120590119860 QCEB 19185 02672 003158 004514 0236 00007XIG 1065 04155 002780 003716 03447 00766

of issuing bonds to the maturity date So in the presentsimulation we assume a default may occur in this periodfor convenience As seen in Table 3 the cumulative defaultprobability of the CEB and XIG is 00007 and 00766 for 6years respectively By (14) we can get the default intensity ofXIG 6times10minus5 per day In the present pricing of the convertiblebonds with DI method we consider that a default may occuronly on thematurity date as well as prior to or on thematuritydate Because the cumulative default probability of CEB istoo small we just calculate the price of the CEB convertiblebond with supposition that a default may occur only on thematurity date

Figures 1 and 2 are the comparison of the theoreticalprices and market prices of the two convertible bonds for150 trading days From Figures 1 and 2 three observationsare noteworthy Firstly the theoretical prices calculated byCS method and DI method with supposition that a defaultmay occur any time prior to or on the maturity date are veryclose for XIG convertible bondThe default probability of theCEB convertible bonds is so low that the calculated prices byassuming a default eventmay occur only on thematurity dateand on any time prior to or on thematurity it should be nearly

0 20 40 60 80 100 120 140 160

120

115

110

105

100

Pric

e (RM

B)

MarketCSDI1

Time (day)

Figure 1 Comparison of themarket prices and the theoretical pricesof CEB convertible bond calculated by CS method and DI methodwith supposition that a default may occur only on the maturity date(DI1)

Discrete Dynamics in Nature and Society 7

0 20 40 60 80 100 120 140 160

Time (day)MarketCS

DI1DI2

135

130

125

120

115

110

105

100

Pric

e (RM

B)

Figure 2 Comparison of the market prices and the theoreticalprices of XIG convertible bond calculated by CS method and DImethod with supposition that a default may occur only on thematurity date (DI1) and prior to or on the maturity (DI2)

the same So the theoretical price of the CEB convertiblebond calculated by CS method is evidently smaller than thatby the DI method This is due to the fact that we adopt themean risky interest rate of the defaultable enterprise debtsin the same market to discount the cash flows In fact theconvertible bond with the large market shares like CEB hasa rather low default probability compared with the meandefault probability of the market So the CS pricing methodgenerally underestimates the price of the convertible bondwith the large market shares like CEB This is a shortcomingof the CS method Secondly with the application of the samedefault probability the price estimated by supposition that adefaultmay occur only on thematurity date is evidently largerthan that by supposition that a default may occur any timeprior to or on the maturity By checking the simulated cashflows we find that an otherwise defaulted firm may continueto survive and pays out a higher sum to the investor before thematurity However the investor loses the chance of a highersum being paid out when a default is allowed to occur inany time Lastly we can see that the tendencies of the marketprices and the theoretical prices calculated with suppositionthat a default may occur only on the maturity date as inFigures 1 and 2 fit well in the long run

We introduce the variable AD to describe the absolutedeviation of the theoretical price from the market pricewhich is given in the following formula

119860119863119905 =10038161003816100381610038161003816119881119905 minus 119881119905

10038161003816100381610038161003816119881119905

(17)

The mean absolute deviation (MAD) is used to describe theintegral result of the theoretical models

MAD = 1150

150

sum119905=1119860119863119905 (18)

Table 4 The mean absolute deviation (MAD) of the theoreticalprices of the CEB and XIG convertible bonds calculated by the CSmethod and the DI method with supposition that a default mayoccur only on thematurity date (DI1) and prior to or on thematurity(DI2)

CS DI1 DI2CEB 297 386XIG 607 358 675

Table 5 The standard deviation (STD) of the theoretical prices ofthe CEB and XIG convertible bonds calculated the CS method andthe DI method with supposition that a default may occur only onthe maturity date (DI1) and prior to or on the maturity (DI2)

CS DI1 DI2CEB 00040 00037XIG 00073 00070 00106

Table 4 reports the MAD of the CEB and XIG convertiblebonds calculated by different methods As can be seen therethe MADs of CEB and XIG convertible bonds calculated byCS and DI are all within 7 The theoretical prices obtainedby CS and DI on the whole can reflect market price TheDI method with supposition that a default may occur onlyon the maturity matches the market best especially in thelong run tendency So we can use our pricing frameworkto forecast market price of convertible bond and makeinvestment decision

The standard errors (SEs) of the calculated prices of theconvertible bonds by the CS method and the DI method atthe first trading day are given in Table 5 The SEs for DImethod with supposition that a default may occur only onthe maturity are little less than those of the CS method price

5 Conclusion

In this article we have presented a new way to price theconvertible bond by LSM inwhich the credit risk ismeasuredby DI The key idea in our model is to get importantparameters such as the long-run average volatility and the DIdirectly fromMertonmodel which avoids the difficulty fromthe data lack of the emergingmarket especially up to now noeffective method estimating the long-run average volatility ofthe underlying stock after issuing convertible bonds In thepresent empirical study the theoretical prices fit the marketvery well So our model including the method to get the DIand the long-run average volatility is proved to be effectiveThe present simulation has shown that the convertible bondprice is quite sensitive to the rule on the default decisiontime allowing an earlier default decision which will lower theconvertible bond price

Data Availability

The data used to support the findings of this study aresupplied by wind database and are also available from thecorresponding author upon request

8 Discrete Dynamics in Nature and Society

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] J E Ingersoll Jr ldquoA contingent-claims valuation of convertiblesecuritiesrdquo Journal of Financial Economics vol 4 no 3 pp 289ndash321 1977

[2] C M Lewis ldquoConvertible debt Valuation and conversion incomplex capital structuresrdquo Journal of Banking amp Finance vol15 no 3 pp 665ndash682 1991

[3] M J Brennan and E S Schwartz ldquoConvertible bonds valuationand optimal strategies for call and conversionrdquo e Journal ofFinance vol 32 no 5 pp 1699ndash1715 1977

[4] M J Brennan and E S Schwartz ldquoAnalyzing convertiblebondsrdquo Journal of Financial and Quantitative Analysis vol 15no 4 pp 907ndash929 1980

[5] J J McConnell and E S Schwartz ldquoLYONTamingrdquoe Journalof Finance vol 41 no 3 pp 561ndash576 1986

[6] C Fan X Luo and Q Wu ldquoStochastic volatility vs jump dif-fusions Evidence from the Chinese convertible bond marketrdquoInternational Review of Economics amp Finance vol 49 pp 1ndash162017

[7] X Hu and H Mao ldquoEmpirical study on the financial char-acteristics of chinese companies issuing convertible bondsrdquoInternational Journal of Business and Management vol 4 no6 pp 59ndash64 2009

[8] L Lu and W Xu ldquoA simple and efficient two-factor willow treemethod for convertible bond pricing with stochastic interestrate and default riskrdquo e Journal of Derivatives vol 25 no 1pp 37ndash54 2017

[9] M Ammann A Kind andCWilde ldquoSimulation-based pricingof convertible bondsrdquo Journal of Empirical Finance vol 15 no2 pp 310ndash331 2008

[10] S Crepey and A Rahal ldquoPricing convertible bonds with callprotectionrdquo e Journal of Computational Finance vol 15 no2 pp 37ndash75 2011

[11] R Schoftner ldquoOn the estimation of credit exposures usingregression-basedMonte Carlo simulationrdquoe Journal of CreditRisk vol 4 no 4 pp 37ndash62 2008

[12] K Tsiveriotis and C Fernandes ldquoValuing convertible bondswith credit riskrdquo e Journal of Fixed Income vol 8 no 2 pp95ndash102 1998

[13] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999

[14] E Ayache P A Forsyth and K R Vetzal ldquoValuation ofconvertible bonds with credit riskrdquo e Journal of Derivativesvol 11 no 1 pp 9ndash29 2003

[15] J A Batten K L-H Khaw andM R Young ldquoConvertible bondpricing modelsrdquo Journal of Economic Surveys vol 28 no 5 pp775ndash803 2014

[16] K Park M Jung and S Lee ldquoPricing a defaultable convertiblebond by simulationrdquo Korean Journal of Financial Studies vol46 no 4 pp 947ndash965 2017

[17] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo e Journal of Finance vol 29 no2 pp 449ndash470 1974

[18] 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

[19] R d Cheng and J R Lu ldquoA monte carlo method of integratedrisk measurement for defaultable zero-coupon bondsrdquo Journalof Management Sciences in China vol 15 no 4 pp 88ndash97 2012

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

4 Discrete Dynamics in Nature and Society

At the maturity date the payoff to the investor is given bythe following

V (119878119879 119879) = 119898119878119879 for 119898119878119879 ge 119861

= 119861 for 1198981198781015840119879 lt 119898119878119879 lt 119861

= 120579119865 for 0 le 119898119878119879 le 1198981198781015840119879

(7)

where the recovery ratio is assumed to be 0 le 120579 le 1In the credit risk analysis should the risk-neutral default

probability or the physical default probability be adoptedThe answer to the question depends on the purpose of thestudy The discounted present value of the cash flow needsto be calculated in pricing the convertible bonds henceone ineluctably employ the risk-neutral valuation theory Sothe default probability in the risk-neutral world should beadopted to price the credit derivatives In estimation of theexpected shortfall triggered by a default event by scenarioanalysis the physical default probability should be adoptedAccording to the Merton model at the maturity date thestock value can be denoted as

119878119879 = max (119860119879 minus 119861 0) (8)

where 119878119879 B and 119860119879 are the stock value at the maturitydate the balloon payment and the firm asset value at thematurity date respectively We suppose that the stock valueis the function of the firm value A and the remaining timeto maturity 120591 S = S(119860 120591) By the Lt119900 lemma and the usualarbitrage arguments the stock price must satisfy the partialdifferential equation

1212059021198602119878119860119860 + 119903119860119878119860 minus 119903119878 minus 119878120591 = 0 (9)

where 120590 is the volatility of the return of the stock r isthe risk-free rate of interest and 120591 is the remaining time tomaturity date The solution of (9) subject to the boundarycondition (8) is

1198780 = 1198600119873(1198891) minus 119863119890minus119903119879119873(1198892) (10)

where

1198891 =ln (1198600119863) + (119903 + 12059021198602) 119879

120590119860radic119879(11)

and 1198892 = 1198891 minus120590119860radic119879 and 120590119860 is the volatility of the firm valuewhich is assumed to be constant for t isin [1 2 119879]1198600 is thecurrent value of the firm asset From Itorsquos Lemma we get

1205901198780 = 119873 (1198891) 1205901198601198600 (12)

where 120590 is the volatility of the return of the stock beforeissuing the bonds

The volatility of the return of the stock before issuing thebonds can be estimated by the historical stock prices From(10) and (12) we can get 1198600 and 120590119860 then the cumulativedefault probability of the firm can be calculated by N(minus1198892)

The convertible bond is a long-term security so a long-termaverage volatility should be adopted in pricing convertiblebonds The issuing of the convertible bonds will have animpact on the volatility of the underlying stock The data ofthe underlying stock price after issuing the convertible bondsfor pricing the convertible bond is generally insufficient orabsent hence a long-term average volatility of the underlyingstock after issuing the convertible bonds cannot be obtainedfrom the historical data So we hope to find a theoreticalmethod to get the long-term average volatility of the under-lying stock after issuing the convertible bonds The Chineselisted firm can prevail upon the investors to convert bonds bylowering the conversion price in Chinese convertible bondmarket nearly all convertible bonds have been converted intostocksThe firmrsquos asset will become only equity So we use 120590119860of the firm value to replace the long-term average volatilityof the underlying stock price after issuing the convertiblebonds

The cumulative default probability Q(119905) can be expressedby the default intensity 120582(119905)

Q (119905) = 1 minus 119890minusint119905

0120582(120591)119889120591 (13)

Assuming the default intensity is a constant we can get

120582 = minus ln (1 minus 119876 (119879))119879

(14)

The cumulative default probability N(minus1198892) is the probabilitythat the firm cannot afford the balloon payment at thematurity If the convertible bond is the only one senior debtin theory the firm may default only on the maturity date Soin the convertible bond pricing we only need to considerdefaults that occur on the maturity date In view of manyfactors that can result in defaults (including various termdebts and adverse emergencies) these factors can result inthe firm default on the bond prior to the maturity dateThese factors also are among the driving forces that makethe firm value change In order to study the affecting factorsof the convertible bond more overall we still use the defaultintensity 120582(119905) to measure default risk prior to the maturitydate in spite of that call in question the rationality of doingso

3 Simulation Design

Most of the Chinese firms stipulate the first put provisionsldquoif the investors do not exercise the put option when theput provision is firstly triggered the investors will not beallowed to exercise the put option in the current year whenthe put provision is triggered againrdquo The firm just needsto change the conversion price when the put provision isfirstly triggered so as to promote the value of the convertiblebond more than the put price The investors will hold theconvertible bonds instead of putting the convertible bondThe adjusted conversion price119883119905 should satisfy

P = 119878119905119888119873(1198891) minus 119865119890minus119903(119879minus119905)119873(1198892) + (119865 + 119868) 119890

minus119903(119879minus119905) (15)

Discrete Dynamics in Nature and Society 5

Table 1 Rules of optimal exercise decision in convertible bonds

Payoff Condition Decision119862119905 119881119905 gt 119862119905 119886119899119889 gt 119898119878119905 callm119878119905 119881119905 gt 119862119905 119886119899119889 119862119905 lt 119898119878119905 forced conversionm119878119905 119881119905 lt 119898119878119905 119886119899119889 119875119905 le 119898119878119905 voluntary conversion0 119881119905 lt 119862119905 119881119905 gt 119898119878119905 119886119899119889 119881119905 gt 119875119905 continuation119875119905 119875119905 gt 119881119905 119886119899119889 119898119878119905 le 119875119905 PutB m119878119905 lt 119861 redemption at maturity120579F St lt S1015840t default

where

119878119905119888 =100119878119905119883119905

1198891 =ln (119878119905119888119865) + (119903 + 12059021198602) (119879 minus 119905)

120590119860radic119879 minus 119905

1198892 = 1198891 minus 120590119860radic119879 minus 119905

(16)

and P and I are the put price and the accrued interest duringT minus t respectively

Including a default decision by the issuer the optimalexercise decision of the investor and the issuer is summarizedin Table 1 In our simulation the convertible bond value isthe expected discounted cash flow which is estimated by theLongstaff-Schwartz (LS) Least Squares Monte Carlo (LSM)simulation method [18] The optimal stopping rule can thenbe determined by the simultaneous optimal exercise strate-gies of both the investor and the issuer For our simulationwe assume the discrete time by the day over the period witha finite time set t isin [0 1 2 119879] where t = 0 for today andt = T for the maturity date The basic framework of pricingChinese convertible bonds by LSM is as follows

(1)Theduration of convertible bonds contains T days andthe convertible bonds can only be exercised at the stipulatedconversion period Depending on the current stock price wecan get theM paths containing T days of the underlying stockprice by Monte Carlo simulation

(2) The defaulted group of the sample paths belongs to abottom 120582 percentile of the realized stock prices at each dayThe issuer may default on the bond any time before or on thematurity and we directly take into account the default risk inthe present valuation This is done by reducing the resultingcash flows immediately to a fraction 120579 of the face value of thebond when a default occurs in the simulated paths By doingso we adhere to the general pricing rule under martingalemeasure

(3)When the put provisions have been triggered the newconversion price is calculated by (15) and the value of theconvertible bond is promoted equivalent to the put price

(4)When the call provision has been triggered nomatterhow big the conversion value is at that day the convertiblebonds will be exercised and terminated

(5) By (7) the cash flows on the maturity date areobtained In the other conversion time node we choosethe in-the-money option paths to estimate the continuation

value by the Longstaff-Schwartz Least Squares simulationwhich is accomplished by regressing the subsequent realizedcash flows on a basis function of constants S and 1198782Comparing the continuation value with the exercise value ifthe continuation value is bigger the optimal stopping valueremains the same otherwise the new stopping time and thenew stopping value are obtained For the out-of-the-moneyoption paths the convertible bonds will not be exercised andone does not need to change the optimal stopping value

(6) The convertible bond can be priced by discountingeach cash flowback to time t=0with the risk-free interest rateand averaging over all paths

4 The Empirical Research

To test the performance of our model two convertible bondsissued by China Everbrigh Bank Co Ltd (CEB) with largemarket shares and Xiamen Itg Group Co Ltd (XIG) arerandomly chosen for the empirical study We adopt the dataof the CEB and XIG convertible bonds at the issuing dateobtained from WIND database The information of the twoconvertible bonds is in Table 2

For comparing with the CS method we compute thetheoretical prices of the two bonds using our simulationmodel and the CS method It is well known to all that theconvertible bond can be split into two components a cash-only part and an equity part for pricing the convertiblebond [12] In fact the credit spreads of AAA-rated firm debtsare only 1 in Chinarsquos convertible market For conveniencein the present calculation with the CS method we use arisky interest rate to discount each cash flow without greatinfluence on the price of the convertible bond In the articlewe use the mean yield of the defaultable enterprise debts inthe same market as the risky interest rate Before pricing wefirst need to estimate the related parameters The long-termmean volatility of the underlying stock the equity value ofthe firm at issuing date and the 6-year risk-free interest rateand the 6-year risky interest rate at the issuing date can beobtained fromWIND database By solving (10) and (12) withoptimization method we get the volatility of the firm assetvalue and the cumulative default probability of the firmThese6 parameter values are given inTable 3Here we have adoptedthe historical stock prices of 2 years for CEB and 5 yearsfor XIG respectively The purpose for which we adopted thedifferent term historical data is to lower the effect of the 2015stock market crash on the pricing of the convertible bondIf a default occurs over the duration of convertible bondswe choose an empirical constant recovery ratio of 20 forenterprise debts [19]

We simulate 50000 paths with 1440 time steps (240 daysper year) before maturity to compute theoretical prices of thebonds The simulated paths are generated by the risk-neutraldynamics of the stock return as (1) We also should note thatthe implied default probability of Merton model is the risk-neutral Hence we can discount the cash flows that the risk-neutral investorwould receive over the period t isin [1 2 T]at a risk-free interest rate r for pricing the convertible bond

As the Chinese convertible bonds are generally set theconversion period which is from the first day after sixmonths

6 Discrete Dynamics in Nature and Society

Table 2 CEB and XIG convertible bonds

CEB convertible bond XIG convertible bondIssue date 2017331 2016118Time horizon 6 6Face value 100 100Coupon () 02 05 10 15 18 20 03 05 09 14 17 20Call value till maturity 105 108The first conversion price 436 903Change of conversion price 201775 adjusted to 426 2016621 adjusted to 893

Reset clause

In 30 consecutive tradingdays the closing stock price is smallerthan 80 of conversion price in 15

trading days

In 30 consecutive tradingdays the closing stock price is smallerthan 90 of conversion price in 15

trading days

Call on conditionIn 30 consecutive trading days the

closing stock price is not less than 130of conversion price in 15 trading days

In 30 consecutive trading days theclosing stock price is not less than 130of conversion price in 15 trading days

Call value Face value plus the accrued interest Face value plus the accrued interest

Put on condition When the use of the capital is changed

(1) When the use of the capital is changed(2) In 30 consecutive trading days theclosing stock price is less than 70 of

conversion pricePut value Face value plus the accrued interest Face value plus the accrued interest

Table 3 Parameters of the CEB and XIG convertible bonds ldquo1198640rdquo (hundred million) stands for the equity value of the firm at issuing dateldquo120590119904rdquo the long-termmean volatility of the underlying stock before issuing date ldquorrdquo the 6-year risk-free interest rate at the issuing date ldquo119903119910rdquo the6-year risky interest rate at the issuing date ldquo120590119860rdquo the volatility of the firm asset value and ldquoQrdquo the cumulative default probability

1198640 120590119878 r 119903119910 120590119860 QCEB 19185 02672 003158 004514 0236 00007XIG 1065 04155 002780 003716 03447 00766

of issuing bonds to the maturity date So in the presentsimulation we assume a default may occur in this periodfor convenience As seen in Table 3 the cumulative defaultprobability of the CEB and XIG is 00007 and 00766 for 6years respectively By (14) we can get the default intensity ofXIG 6times10minus5 per day In the present pricing of the convertiblebonds with DI method we consider that a default may occuronly on thematurity date as well as prior to or on thematuritydate Because the cumulative default probability of CEB istoo small we just calculate the price of the CEB convertiblebond with supposition that a default may occur only on thematurity date

Figures 1 and 2 are the comparison of the theoreticalprices and market prices of the two convertible bonds for150 trading days From Figures 1 and 2 three observationsare noteworthy Firstly the theoretical prices calculated byCS method and DI method with supposition that a defaultmay occur any time prior to or on the maturity date are veryclose for XIG convertible bondThe default probability of theCEB convertible bonds is so low that the calculated prices byassuming a default eventmay occur only on thematurity dateand on any time prior to or on thematurity it should be nearly

0 20 40 60 80 100 120 140 160

120

115

110

105

100

Pric

e (RM

B)

MarketCSDI1

Time (day)

Figure 1 Comparison of themarket prices and the theoretical pricesof CEB convertible bond calculated by CS method and DI methodwith supposition that a default may occur only on the maturity date(DI1)

Discrete Dynamics in Nature and Society 7

0 20 40 60 80 100 120 140 160

Time (day)MarketCS

DI1DI2

135

130

125

120

115

110

105

100

Pric

e (RM

B)

Figure 2 Comparison of the market prices and the theoreticalprices of XIG convertible bond calculated by CS method and DImethod with supposition that a default may occur only on thematurity date (DI1) and prior to or on the maturity (DI2)

the same So the theoretical price of the CEB convertiblebond calculated by CS method is evidently smaller than thatby the DI method This is due to the fact that we adopt themean risky interest rate of the defaultable enterprise debtsin the same market to discount the cash flows In fact theconvertible bond with the large market shares like CEB hasa rather low default probability compared with the meandefault probability of the market So the CS pricing methodgenerally underestimates the price of the convertible bondwith the large market shares like CEB This is a shortcomingof the CS method Secondly with the application of the samedefault probability the price estimated by supposition that adefaultmay occur only on thematurity date is evidently largerthan that by supposition that a default may occur any timeprior to or on the maturity By checking the simulated cashflows we find that an otherwise defaulted firm may continueto survive and pays out a higher sum to the investor before thematurity However the investor loses the chance of a highersum being paid out when a default is allowed to occur inany time Lastly we can see that the tendencies of the marketprices and the theoretical prices calculated with suppositionthat a default may occur only on the maturity date as inFigures 1 and 2 fit well in the long run

We introduce the variable AD to describe the absolutedeviation of the theoretical price from the market pricewhich is given in the following formula

119860119863119905 =10038161003816100381610038161003816119881119905 minus 119881119905

10038161003816100381610038161003816119881119905

(17)

The mean absolute deviation (MAD) is used to describe theintegral result of the theoretical models

MAD = 1150

150

sum119905=1119860119863119905 (18)

Table 4 The mean absolute deviation (MAD) of the theoreticalprices of the CEB and XIG convertible bonds calculated by the CSmethod and the DI method with supposition that a default mayoccur only on thematurity date (DI1) and prior to or on thematurity(DI2)

CS DI1 DI2CEB 297 386XIG 607 358 675

Table 5 The standard deviation (STD) of the theoretical prices ofthe CEB and XIG convertible bonds calculated the CS method andthe DI method with supposition that a default may occur only onthe maturity date (DI1) and prior to or on the maturity (DI2)

CS DI1 DI2CEB 00040 00037XIG 00073 00070 00106

Table 4 reports the MAD of the CEB and XIG convertiblebonds calculated by different methods As can be seen therethe MADs of CEB and XIG convertible bonds calculated byCS and DI are all within 7 The theoretical prices obtainedby CS and DI on the whole can reflect market price TheDI method with supposition that a default may occur onlyon the maturity matches the market best especially in thelong run tendency So we can use our pricing frameworkto forecast market price of convertible bond and makeinvestment decision

The standard errors (SEs) of the calculated prices of theconvertible bonds by the CS method and the DI method atthe first trading day are given in Table 5 The SEs for DImethod with supposition that a default may occur only onthe maturity are little less than those of the CS method price

5 Conclusion

In this article we have presented a new way to price theconvertible bond by LSM inwhich the credit risk ismeasuredby DI The key idea in our model is to get importantparameters such as the long-run average volatility and the DIdirectly fromMertonmodel which avoids the difficulty fromthe data lack of the emergingmarket especially up to now noeffective method estimating the long-run average volatility ofthe underlying stock after issuing convertible bonds In thepresent empirical study the theoretical prices fit the marketvery well So our model including the method to get the DIand the long-run average volatility is proved to be effectiveThe present simulation has shown that the convertible bondprice is quite sensitive to the rule on the default decisiontime allowing an earlier default decision which will lower theconvertible bond price

Data Availability

The data used to support the findings of this study aresupplied by wind database and are also available from thecorresponding author upon request

8 Discrete Dynamics in Nature and Society

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] J E Ingersoll Jr ldquoA contingent-claims valuation of convertiblesecuritiesrdquo Journal of Financial Economics vol 4 no 3 pp 289ndash321 1977

[2] C M Lewis ldquoConvertible debt Valuation and conversion incomplex capital structuresrdquo Journal of Banking amp Finance vol15 no 3 pp 665ndash682 1991

[3] M J Brennan and E S Schwartz ldquoConvertible bonds valuationand optimal strategies for call and conversionrdquo e Journal ofFinance vol 32 no 5 pp 1699ndash1715 1977

[4] M J Brennan and E S Schwartz ldquoAnalyzing convertiblebondsrdquo Journal of Financial and Quantitative Analysis vol 15no 4 pp 907ndash929 1980

[5] J J McConnell and E S Schwartz ldquoLYONTamingrdquoe Journalof Finance vol 41 no 3 pp 561ndash576 1986

[6] C Fan X Luo and Q Wu ldquoStochastic volatility vs jump dif-fusions Evidence from the Chinese convertible bond marketrdquoInternational Review of Economics amp Finance vol 49 pp 1ndash162017

[7] X Hu and H Mao ldquoEmpirical study on the financial char-acteristics of chinese companies issuing convertible bondsrdquoInternational Journal of Business and Management vol 4 no6 pp 59ndash64 2009

[8] L Lu and W Xu ldquoA simple and efficient two-factor willow treemethod for convertible bond pricing with stochastic interestrate and default riskrdquo e Journal of Derivatives vol 25 no 1pp 37ndash54 2017

[9] M Ammann A Kind andCWilde ldquoSimulation-based pricingof convertible bondsrdquo Journal of Empirical Finance vol 15 no2 pp 310ndash331 2008

[10] S Crepey and A Rahal ldquoPricing convertible bonds with callprotectionrdquo e Journal of Computational Finance vol 15 no2 pp 37ndash75 2011

[11] R Schoftner ldquoOn the estimation of credit exposures usingregression-basedMonte Carlo simulationrdquoe Journal of CreditRisk vol 4 no 4 pp 37ndash62 2008

[12] K Tsiveriotis and C Fernandes ldquoValuing convertible bondswith credit riskrdquo e Journal of Fixed Income vol 8 no 2 pp95ndash102 1998

[13] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999

[14] E Ayache P A Forsyth and K R Vetzal ldquoValuation ofconvertible bonds with credit riskrdquo e Journal of Derivativesvol 11 no 1 pp 9ndash29 2003

[15] J A Batten K L-H Khaw andM R Young ldquoConvertible bondpricing modelsrdquo Journal of Economic Surveys vol 28 no 5 pp775ndash803 2014

[16] K Park M Jung and S Lee ldquoPricing a defaultable convertiblebond by simulationrdquo Korean Journal of Financial Studies vol46 no 4 pp 947ndash965 2017

[17] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo e Journal of Finance vol 29 no2 pp 449ndash470 1974

[18] 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

[19] R d Cheng and J R Lu ldquoA monte carlo method of integratedrisk measurement for defaultable zero-coupon bondsrdquo Journalof Management Sciences in China vol 15 no 4 pp 88ndash97 2012

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Discrete Dynamics in Nature and Society 5

Table 1 Rules of optimal exercise decision in convertible bonds

Payoff Condition Decision119862119905 119881119905 gt 119862119905 119886119899119889 gt 119898119878119905 callm119878119905 119881119905 gt 119862119905 119886119899119889 119862119905 lt 119898119878119905 forced conversionm119878119905 119881119905 lt 119898119878119905 119886119899119889 119875119905 le 119898119878119905 voluntary conversion0 119881119905 lt 119862119905 119881119905 gt 119898119878119905 119886119899119889 119881119905 gt 119875119905 continuation119875119905 119875119905 gt 119881119905 119886119899119889 119898119878119905 le 119875119905 PutB m119878119905 lt 119861 redemption at maturity120579F St lt S1015840t default

where

119878119905119888 =100119878119905119883119905

1198891 =ln (119878119905119888119865) + (119903 + 12059021198602) (119879 minus 119905)

120590119860radic119879 minus 119905

1198892 = 1198891 minus 120590119860radic119879 minus 119905

(16)

and P and I are the put price and the accrued interest duringT minus t respectively

Including a default decision by the issuer the optimalexercise decision of the investor and the issuer is summarizedin Table 1 In our simulation the convertible bond value isthe expected discounted cash flow which is estimated by theLongstaff-Schwartz (LS) Least Squares Monte Carlo (LSM)simulation method [18] The optimal stopping rule can thenbe determined by the simultaneous optimal exercise strate-gies of both the investor and the issuer For our simulationwe assume the discrete time by the day over the period witha finite time set t isin [0 1 2 119879] where t = 0 for today andt = T for the maturity date The basic framework of pricingChinese convertible bonds by LSM is as follows

(1)Theduration of convertible bonds contains T days andthe convertible bonds can only be exercised at the stipulatedconversion period Depending on the current stock price wecan get theM paths containing T days of the underlying stockprice by Monte Carlo simulation

(2) The defaulted group of the sample paths belongs to abottom 120582 percentile of the realized stock prices at each dayThe issuer may default on the bond any time before or on thematurity and we directly take into account the default risk inthe present valuation This is done by reducing the resultingcash flows immediately to a fraction 120579 of the face value of thebond when a default occurs in the simulated paths By doingso we adhere to the general pricing rule under martingalemeasure

(3)When the put provisions have been triggered the newconversion price is calculated by (15) and the value of theconvertible bond is promoted equivalent to the put price

(4)When the call provision has been triggered nomatterhow big the conversion value is at that day the convertiblebonds will be exercised and terminated

(5) By (7) the cash flows on the maturity date areobtained In the other conversion time node we choosethe in-the-money option paths to estimate the continuation

value by the Longstaff-Schwartz Least Squares simulationwhich is accomplished by regressing the subsequent realizedcash flows on a basis function of constants S and 1198782Comparing the continuation value with the exercise value ifthe continuation value is bigger the optimal stopping valueremains the same otherwise the new stopping time and thenew stopping value are obtained For the out-of-the-moneyoption paths the convertible bonds will not be exercised andone does not need to change the optimal stopping value

(6) The convertible bond can be priced by discountingeach cash flowback to time t=0with the risk-free interest rateand averaging over all paths

4 The Empirical Research

To test the performance of our model two convertible bondsissued by China Everbrigh Bank Co Ltd (CEB) with largemarket shares and Xiamen Itg Group Co Ltd (XIG) arerandomly chosen for the empirical study We adopt the dataof the CEB and XIG convertible bonds at the issuing dateobtained from WIND database The information of the twoconvertible bonds is in Table 2

For comparing with the CS method we compute thetheoretical prices of the two bonds using our simulationmodel and the CS method It is well known to all that theconvertible bond can be split into two components a cash-only part and an equity part for pricing the convertiblebond [12] In fact the credit spreads of AAA-rated firm debtsare only 1 in Chinarsquos convertible market For conveniencein the present calculation with the CS method we use arisky interest rate to discount each cash flow without greatinfluence on the price of the convertible bond In the articlewe use the mean yield of the defaultable enterprise debts inthe same market as the risky interest rate Before pricing wefirst need to estimate the related parameters The long-termmean volatility of the underlying stock the equity value ofthe firm at issuing date and the 6-year risk-free interest rateand the 6-year risky interest rate at the issuing date can beobtained fromWIND database By solving (10) and (12) withoptimization method we get the volatility of the firm assetvalue and the cumulative default probability of the firmThese6 parameter values are given inTable 3Here we have adoptedthe historical stock prices of 2 years for CEB and 5 yearsfor XIG respectively The purpose for which we adopted thedifferent term historical data is to lower the effect of the 2015stock market crash on the pricing of the convertible bondIf a default occurs over the duration of convertible bondswe choose an empirical constant recovery ratio of 20 forenterprise debts [19]

We simulate 50000 paths with 1440 time steps (240 daysper year) before maturity to compute theoretical prices of thebonds The simulated paths are generated by the risk-neutraldynamics of the stock return as (1) We also should note thatthe implied default probability of Merton model is the risk-neutral Hence we can discount the cash flows that the risk-neutral investorwould receive over the period t isin [1 2 T]at a risk-free interest rate r for pricing the convertible bond

As the Chinese convertible bonds are generally set theconversion period which is from the first day after sixmonths

6 Discrete Dynamics in Nature and Society

Table 2 CEB and XIG convertible bonds

CEB convertible bond XIG convertible bondIssue date 2017331 2016118Time horizon 6 6Face value 100 100Coupon () 02 05 10 15 18 20 03 05 09 14 17 20Call value till maturity 105 108The first conversion price 436 903Change of conversion price 201775 adjusted to 426 2016621 adjusted to 893

Reset clause

In 30 consecutive tradingdays the closing stock price is smallerthan 80 of conversion price in 15

trading days

In 30 consecutive tradingdays the closing stock price is smallerthan 90 of conversion price in 15

trading days

Call on conditionIn 30 consecutive trading days the

closing stock price is not less than 130of conversion price in 15 trading days

In 30 consecutive trading days theclosing stock price is not less than 130of conversion price in 15 trading days

Call value Face value plus the accrued interest Face value plus the accrued interest

Put on condition When the use of the capital is changed

(1) When the use of the capital is changed(2) In 30 consecutive trading days theclosing stock price is less than 70 of

conversion pricePut value Face value plus the accrued interest Face value plus the accrued interest

Table 3 Parameters of the CEB and XIG convertible bonds ldquo1198640rdquo (hundred million) stands for the equity value of the firm at issuing dateldquo120590119904rdquo the long-termmean volatility of the underlying stock before issuing date ldquorrdquo the 6-year risk-free interest rate at the issuing date ldquo119903119910rdquo the6-year risky interest rate at the issuing date ldquo120590119860rdquo the volatility of the firm asset value and ldquoQrdquo the cumulative default probability

1198640 120590119878 r 119903119910 120590119860 QCEB 19185 02672 003158 004514 0236 00007XIG 1065 04155 002780 003716 03447 00766

of issuing bonds to the maturity date So in the presentsimulation we assume a default may occur in this periodfor convenience As seen in Table 3 the cumulative defaultprobability of the CEB and XIG is 00007 and 00766 for 6years respectively By (14) we can get the default intensity ofXIG 6times10minus5 per day In the present pricing of the convertiblebonds with DI method we consider that a default may occuronly on thematurity date as well as prior to or on thematuritydate Because the cumulative default probability of CEB istoo small we just calculate the price of the CEB convertiblebond with supposition that a default may occur only on thematurity date

Figures 1 and 2 are the comparison of the theoreticalprices and market prices of the two convertible bonds for150 trading days From Figures 1 and 2 three observationsare noteworthy Firstly the theoretical prices calculated byCS method and DI method with supposition that a defaultmay occur any time prior to or on the maturity date are veryclose for XIG convertible bondThe default probability of theCEB convertible bonds is so low that the calculated prices byassuming a default eventmay occur only on thematurity dateand on any time prior to or on thematurity it should be nearly

0 20 40 60 80 100 120 140 160

120

115

110

105

100

Pric

e (RM

B)

MarketCSDI1

Time (day)

Figure 1 Comparison of themarket prices and the theoretical pricesof CEB convertible bond calculated by CS method and DI methodwith supposition that a default may occur only on the maturity date(DI1)

Discrete Dynamics in Nature and Society 7

0 20 40 60 80 100 120 140 160

Time (day)MarketCS

DI1DI2

135

130

125

120

115

110

105

100

Pric

e (RM

B)

Figure 2 Comparison of the market prices and the theoreticalprices of XIG convertible bond calculated by CS method and DImethod with supposition that a default may occur only on thematurity date (DI1) and prior to or on the maturity (DI2)

the same So the theoretical price of the CEB convertiblebond calculated by CS method is evidently smaller than thatby the DI method This is due to the fact that we adopt themean risky interest rate of the defaultable enterprise debtsin the same market to discount the cash flows In fact theconvertible bond with the large market shares like CEB hasa rather low default probability compared with the meandefault probability of the market So the CS pricing methodgenerally underestimates the price of the convertible bondwith the large market shares like CEB This is a shortcomingof the CS method Secondly with the application of the samedefault probability the price estimated by supposition that adefaultmay occur only on thematurity date is evidently largerthan that by supposition that a default may occur any timeprior to or on the maturity By checking the simulated cashflows we find that an otherwise defaulted firm may continueto survive and pays out a higher sum to the investor before thematurity However the investor loses the chance of a highersum being paid out when a default is allowed to occur inany time Lastly we can see that the tendencies of the marketprices and the theoretical prices calculated with suppositionthat a default may occur only on the maturity date as inFigures 1 and 2 fit well in the long run

We introduce the variable AD to describe the absolutedeviation of the theoretical price from the market pricewhich is given in the following formula

119860119863119905 =10038161003816100381610038161003816119881119905 minus 119881119905

10038161003816100381610038161003816119881119905

(17)

The mean absolute deviation (MAD) is used to describe theintegral result of the theoretical models

MAD = 1150

150

sum119905=1119860119863119905 (18)

Table 4 The mean absolute deviation (MAD) of the theoreticalprices of the CEB and XIG convertible bonds calculated by the CSmethod and the DI method with supposition that a default mayoccur only on thematurity date (DI1) and prior to or on thematurity(DI2)

CS DI1 DI2CEB 297 386XIG 607 358 675

Table 5 The standard deviation (STD) of the theoretical prices ofthe CEB and XIG convertible bonds calculated the CS method andthe DI method with supposition that a default may occur only onthe maturity date (DI1) and prior to or on the maturity (DI2)

CS DI1 DI2CEB 00040 00037XIG 00073 00070 00106

Table 4 reports the MAD of the CEB and XIG convertiblebonds calculated by different methods As can be seen therethe MADs of CEB and XIG convertible bonds calculated byCS and DI are all within 7 The theoretical prices obtainedby CS and DI on the whole can reflect market price TheDI method with supposition that a default may occur onlyon the maturity matches the market best especially in thelong run tendency So we can use our pricing frameworkto forecast market price of convertible bond and makeinvestment decision

The standard errors (SEs) of the calculated prices of theconvertible bonds by the CS method and the DI method atthe first trading day are given in Table 5 The SEs for DImethod with supposition that a default may occur only onthe maturity are little less than those of the CS method price

5 Conclusion

In this article we have presented a new way to price theconvertible bond by LSM inwhich the credit risk ismeasuredby DI The key idea in our model is to get importantparameters such as the long-run average volatility and the DIdirectly fromMertonmodel which avoids the difficulty fromthe data lack of the emergingmarket especially up to now noeffective method estimating the long-run average volatility ofthe underlying stock after issuing convertible bonds In thepresent empirical study the theoretical prices fit the marketvery well So our model including the method to get the DIand the long-run average volatility is proved to be effectiveThe present simulation has shown that the convertible bondprice is quite sensitive to the rule on the default decisiontime allowing an earlier default decision which will lower theconvertible bond price

Data Availability

The data used to support the findings of this study aresupplied by wind database and are also available from thecorresponding author upon request

8 Discrete Dynamics in Nature and Society

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] J E Ingersoll Jr ldquoA contingent-claims valuation of convertiblesecuritiesrdquo Journal of Financial Economics vol 4 no 3 pp 289ndash321 1977

[2] C M Lewis ldquoConvertible debt Valuation and conversion incomplex capital structuresrdquo Journal of Banking amp Finance vol15 no 3 pp 665ndash682 1991

[3] M J Brennan and E S Schwartz ldquoConvertible bonds valuationand optimal strategies for call and conversionrdquo e Journal ofFinance vol 32 no 5 pp 1699ndash1715 1977

[4] M J Brennan and E S Schwartz ldquoAnalyzing convertiblebondsrdquo Journal of Financial and Quantitative Analysis vol 15no 4 pp 907ndash929 1980

[5] J J McConnell and E S Schwartz ldquoLYONTamingrdquoe Journalof Finance vol 41 no 3 pp 561ndash576 1986

[6] C Fan X Luo and Q Wu ldquoStochastic volatility vs jump dif-fusions Evidence from the Chinese convertible bond marketrdquoInternational Review of Economics amp Finance vol 49 pp 1ndash162017

[7] X Hu and H Mao ldquoEmpirical study on the financial char-acteristics of chinese companies issuing convertible bondsrdquoInternational Journal of Business and Management vol 4 no6 pp 59ndash64 2009

[8] L Lu and W Xu ldquoA simple and efficient two-factor willow treemethod for convertible bond pricing with stochastic interestrate and default riskrdquo e Journal of Derivatives vol 25 no 1pp 37ndash54 2017

[9] M Ammann A Kind andCWilde ldquoSimulation-based pricingof convertible bondsrdquo Journal of Empirical Finance vol 15 no2 pp 310ndash331 2008

[10] S Crepey and A Rahal ldquoPricing convertible bonds with callprotectionrdquo e Journal of Computational Finance vol 15 no2 pp 37ndash75 2011

[11] R Schoftner ldquoOn the estimation of credit exposures usingregression-basedMonte Carlo simulationrdquoe Journal of CreditRisk vol 4 no 4 pp 37ndash62 2008

[12] K Tsiveriotis and C Fernandes ldquoValuing convertible bondswith credit riskrdquo e Journal of Fixed Income vol 8 no 2 pp95ndash102 1998

[13] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999

[14] E Ayache P A Forsyth and K R Vetzal ldquoValuation ofconvertible bonds with credit riskrdquo e Journal of Derivativesvol 11 no 1 pp 9ndash29 2003

[15] J A Batten K L-H Khaw andM R Young ldquoConvertible bondpricing modelsrdquo Journal of Economic Surveys vol 28 no 5 pp775ndash803 2014

[16] K Park M Jung and S Lee ldquoPricing a defaultable convertiblebond by simulationrdquo Korean Journal of Financial Studies vol46 no 4 pp 947ndash965 2017

[17] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo e Journal of Finance vol 29 no2 pp 449ndash470 1974

[18] 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

[19] R d Cheng and J R Lu ldquoA monte carlo method of integratedrisk measurement for defaultable zero-coupon bondsrdquo Journalof Management Sciences in China vol 15 no 4 pp 88ndash97 2012

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Discrete Dynamics in Nature and Society

Table 2 CEB and XIG convertible bonds

CEB convertible bond XIG convertible bondIssue date 2017331 2016118Time horizon 6 6Face value 100 100Coupon () 02 05 10 15 18 20 03 05 09 14 17 20Call value till maturity 105 108The first conversion price 436 903Change of conversion price 201775 adjusted to 426 2016621 adjusted to 893

Reset clause

In 30 consecutive tradingdays the closing stock price is smallerthan 80 of conversion price in 15

trading days

In 30 consecutive tradingdays the closing stock price is smallerthan 90 of conversion price in 15

trading days

Call on conditionIn 30 consecutive trading days the

closing stock price is not less than 130of conversion price in 15 trading days

In 30 consecutive trading days theclosing stock price is not less than 130of conversion price in 15 trading days

Call value Face value plus the accrued interest Face value plus the accrued interest

Put on condition When the use of the capital is changed

(1) When the use of the capital is changed(2) In 30 consecutive trading days theclosing stock price is less than 70 of

conversion pricePut value Face value plus the accrued interest Face value plus the accrued interest

Table 3 Parameters of the CEB and XIG convertible bonds ldquo1198640rdquo (hundred million) stands for the equity value of the firm at issuing dateldquo120590119904rdquo the long-termmean volatility of the underlying stock before issuing date ldquorrdquo the 6-year risk-free interest rate at the issuing date ldquo119903119910rdquo the6-year risky interest rate at the issuing date ldquo120590119860rdquo the volatility of the firm asset value and ldquoQrdquo the cumulative default probability

1198640 120590119878 r 119903119910 120590119860 QCEB 19185 02672 003158 004514 0236 00007XIG 1065 04155 002780 003716 03447 00766

of issuing bonds to the maturity date So in the presentsimulation we assume a default may occur in this periodfor convenience As seen in Table 3 the cumulative defaultprobability of the CEB and XIG is 00007 and 00766 for 6years respectively By (14) we can get the default intensity ofXIG 6times10minus5 per day In the present pricing of the convertiblebonds with DI method we consider that a default may occuronly on thematurity date as well as prior to or on thematuritydate Because the cumulative default probability of CEB istoo small we just calculate the price of the CEB convertiblebond with supposition that a default may occur only on thematurity date

Figures 1 and 2 are the comparison of the theoreticalprices and market prices of the two convertible bonds for150 trading days From Figures 1 and 2 three observationsare noteworthy Firstly the theoretical prices calculated byCS method and DI method with supposition that a defaultmay occur any time prior to or on the maturity date are veryclose for XIG convertible bondThe default probability of theCEB convertible bonds is so low that the calculated prices byassuming a default eventmay occur only on thematurity dateand on any time prior to or on thematurity it should be nearly

0 20 40 60 80 100 120 140 160

120

115

110

105

100

Pric

e (RM

B)

MarketCSDI1

Time (day)

Figure 1 Comparison of themarket prices and the theoretical pricesof CEB convertible bond calculated by CS method and DI methodwith supposition that a default may occur only on the maturity date(DI1)

Discrete Dynamics in Nature and Society 7

0 20 40 60 80 100 120 140 160

Time (day)MarketCS

DI1DI2

135

130

125

120

115

110

105

100

Pric

e (RM

B)

Figure 2 Comparison of the market prices and the theoreticalprices of XIG convertible bond calculated by CS method and DImethod with supposition that a default may occur only on thematurity date (DI1) and prior to or on the maturity (DI2)

the same So the theoretical price of the CEB convertiblebond calculated by CS method is evidently smaller than thatby the DI method This is due to the fact that we adopt themean risky interest rate of the defaultable enterprise debtsin the same market to discount the cash flows In fact theconvertible bond with the large market shares like CEB hasa rather low default probability compared with the meandefault probability of the market So the CS pricing methodgenerally underestimates the price of the convertible bondwith the large market shares like CEB This is a shortcomingof the CS method Secondly with the application of the samedefault probability the price estimated by supposition that adefaultmay occur only on thematurity date is evidently largerthan that by supposition that a default may occur any timeprior to or on the maturity By checking the simulated cashflows we find that an otherwise defaulted firm may continueto survive and pays out a higher sum to the investor before thematurity However the investor loses the chance of a highersum being paid out when a default is allowed to occur inany time Lastly we can see that the tendencies of the marketprices and the theoretical prices calculated with suppositionthat a default may occur only on the maturity date as inFigures 1 and 2 fit well in the long run

We introduce the variable AD to describe the absolutedeviation of the theoretical price from the market pricewhich is given in the following formula

119860119863119905 =10038161003816100381610038161003816119881119905 minus 119881119905

10038161003816100381610038161003816119881119905

(17)

The mean absolute deviation (MAD) is used to describe theintegral result of the theoretical models

MAD = 1150

150

sum119905=1119860119863119905 (18)

Table 4 The mean absolute deviation (MAD) of the theoreticalprices of the CEB and XIG convertible bonds calculated by the CSmethod and the DI method with supposition that a default mayoccur only on thematurity date (DI1) and prior to or on thematurity(DI2)

CS DI1 DI2CEB 297 386XIG 607 358 675

Table 5 The standard deviation (STD) of the theoretical prices ofthe CEB and XIG convertible bonds calculated the CS method andthe DI method with supposition that a default may occur only onthe maturity date (DI1) and prior to or on the maturity (DI2)

CS DI1 DI2CEB 00040 00037XIG 00073 00070 00106

Table 4 reports the MAD of the CEB and XIG convertiblebonds calculated by different methods As can be seen therethe MADs of CEB and XIG convertible bonds calculated byCS and DI are all within 7 The theoretical prices obtainedby CS and DI on the whole can reflect market price TheDI method with supposition that a default may occur onlyon the maturity matches the market best especially in thelong run tendency So we can use our pricing frameworkto forecast market price of convertible bond and makeinvestment decision

The standard errors (SEs) of the calculated prices of theconvertible bonds by the CS method and the DI method atthe first trading day are given in Table 5 The SEs for DImethod with supposition that a default may occur only onthe maturity are little less than those of the CS method price

5 Conclusion

In this article we have presented a new way to price theconvertible bond by LSM inwhich the credit risk ismeasuredby DI The key idea in our model is to get importantparameters such as the long-run average volatility and the DIdirectly fromMertonmodel which avoids the difficulty fromthe data lack of the emergingmarket especially up to now noeffective method estimating the long-run average volatility ofthe underlying stock after issuing convertible bonds In thepresent empirical study the theoretical prices fit the marketvery well So our model including the method to get the DIand the long-run average volatility is proved to be effectiveThe present simulation has shown that the convertible bondprice is quite sensitive to the rule on the default decisiontime allowing an earlier default decision which will lower theconvertible bond price

Data Availability

The data used to support the findings of this study aresupplied by wind database and are also available from thecorresponding author upon request

8 Discrete Dynamics in Nature and Society

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] J E Ingersoll Jr ldquoA contingent-claims valuation of convertiblesecuritiesrdquo Journal of Financial Economics vol 4 no 3 pp 289ndash321 1977

[2] C M Lewis ldquoConvertible debt Valuation and conversion incomplex capital structuresrdquo Journal of Banking amp Finance vol15 no 3 pp 665ndash682 1991

[3] M J Brennan and E S Schwartz ldquoConvertible bonds valuationand optimal strategies for call and conversionrdquo e Journal ofFinance vol 32 no 5 pp 1699ndash1715 1977

[4] M J Brennan and E S Schwartz ldquoAnalyzing convertiblebondsrdquo Journal of Financial and Quantitative Analysis vol 15no 4 pp 907ndash929 1980

[5] J J McConnell and E S Schwartz ldquoLYONTamingrdquoe Journalof Finance vol 41 no 3 pp 561ndash576 1986

[6] C Fan X Luo and Q Wu ldquoStochastic volatility vs jump dif-fusions Evidence from the Chinese convertible bond marketrdquoInternational Review of Economics amp Finance vol 49 pp 1ndash162017

[7] X Hu and H Mao ldquoEmpirical study on the financial char-acteristics of chinese companies issuing convertible bondsrdquoInternational Journal of Business and Management vol 4 no6 pp 59ndash64 2009

[8] L Lu and W Xu ldquoA simple and efficient two-factor willow treemethod for convertible bond pricing with stochastic interestrate and default riskrdquo e Journal of Derivatives vol 25 no 1pp 37ndash54 2017

[9] M Ammann A Kind andCWilde ldquoSimulation-based pricingof convertible bondsrdquo Journal of Empirical Finance vol 15 no2 pp 310ndash331 2008

[10] S Crepey and A Rahal ldquoPricing convertible bonds with callprotectionrdquo e Journal of Computational Finance vol 15 no2 pp 37ndash75 2011

[11] R Schoftner ldquoOn the estimation of credit exposures usingregression-basedMonte Carlo simulationrdquoe Journal of CreditRisk vol 4 no 4 pp 37ndash62 2008

[12] K Tsiveriotis and C Fernandes ldquoValuing convertible bondswith credit riskrdquo e Journal of Fixed Income vol 8 no 2 pp95ndash102 1998

[13] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999

[14] E Ayache P A Forsyth and K R Vetzal ldquoValuation ofconvertible bonds with credit riskrdquo e Journal of Derivativesvol 11 no 1 pp 9ndash29 2003

[15] J A Batten K L-H Khaw andM R Young ldquoConvertible bondpricing modelsrdquo Journal of Economic Surveys vol 28 no 5 pp775ndash803 2014

[16] K Park M Jung and S Lee ldquoPricing a defaultable convertiblebond by simulationrdquo Korean Journal of Financial Studies vol46 no 4 pp 947ndash965 2017

[17] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo e Journal of Finance vol 29 no2 pp 449ndash470 1974

[18] 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

[19] R d Cheng and J R Lu ldquoA monte carlo method of integratedrisk measurement for defaultable zero-coupon bondsrdquo Journalof Management Sciences in China vol 15 no 4 pp 88ndash97 2012

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Discrete Dynamics in Nature and Society 7

0 20 40 60 80 100 120 140 160

Time (day)MarketCS

DI1DI2

135

130

125

120

115

110

105

100

Pric

e (RM

B)

Figure 2 Comparison of the market prices and the theoreticalprices of XIG convertible bond calculated by CS method and DImethod with supposition that a default may occur only on thematurity date (DI1) and prior to or on the maturity (DI2)

the same So the theoretical price of the CEB convertiblebond calculated by CS method is evidently smaller than thatby the DI method This is due to the fact that we adopt themean risky interest rate of the defaultable enterprise debtsin the same market to discount the cash flows In fact theconvertible bond with the large market shares like CEB hasa rather low default probability compared with the meandefault probability of the market So the CS pricing methodgenerally underestimates the price of the convertible bondwith the large market shares like CEB This is a shortcomingof the CS method Secondly with the application of the samedefault probability the price estimated by supposition that adefaultmay occur only on thematurity date is evidently largerthan that by supposition that a default may occur any timeprior to or on the maturity By checking the simulated cashflows we find that an otherwise defaulted firm may continueto survive and pays out a higher sum to the investor before thematurity However the investor loses the chance of a highersum being paid out when a default is allowed to occur inany time Lastly we can see that the tendencies of the marketprices and the theoretical prices calculated with suppositionthat a default may occur only on the maturity date as inFigures 1 and 2 fit well in the long run

We introduce the variable AD to describe the absolutedeviation of the theoretical price from the market pricewhich is given in the following formula

119860119863119905 =10038161003816100381610038161003816119881119905 minus 119881119905

10038161003816100381610038161003816119881119905

(17)

The mean absolute deviation (MAD) is used to describe theintegral result of the theoretical models

MAD = 1150

150

sum119905=1119860119863119905 (18)

Table 4 The mean absolute deviation (MAD) of the theoreticalprices of the CEB and XIG convertible bonds calculated by the CSmethod and the DI method with supposition that a default mayoccur only on thematurity date (DI1) and prior to or on thematurity(DI2)

CS DI1 DI2CEB 297 386XIG 607 358 675

Table 5 The standard deviation (STD) of the theoretical prices ofthe CEB and XIG convertible bonds calculated the CS method andthe DI method with supposition that a default may occur only onthe maturity date (DI1) and prior to or on the maturity (DI2)

CS DI1 DI2CEB 00040 00037XIG 00073 00070 00106

Table 4 reports the MAD of the CEB and XIG convertiblebonds calculated by different methods As can be seen therethe MADs of CEB and XIG convertible bonds calculated byCS and DI are all within 7 The theoretical prices obtainedby CS and DI on the whole can reflect market price TheDI method with supposition that a default may occur onlyon the maturity matches the market best especially in thelong run tendency So we can use our pricing frameworkto forecast market price of convertible bond and makeinvestment decision

The standard errors (SEs) of the calculated prices of theconvertible bonds by the CS method and the DI method atthe first trading day are given in Table 5 The SEs for DImethod with supposition that a default may occur only onthe maturity are little less than those of the CS method price

5 Conclusion

In this article we have presented a new way to price theconvertible bond by LSM inwhich the credit risk ismeasuredby DI The key idea in our model is to get importantparameters such as the long-run average volatility and the DIdirectly fromMertonmodel which avoids the difficulty fromthe data lack of the emergingmarket especially up to now noeffective method estimating the long-run average volatility ofthe underlying stock after issuing convertible bonds In thepresent empirical study the theoretical prices fit the marketvery well So our model including the method to get the DIand the long-run average volatility is proved to be effectiveThe present simulation has shown that the convertible bondprice is quite sensitive to the rule on the default decisiontime allowing an earlier default decision which will lower theconvertible bond price

Data Availability

The data used to support the findings of this study aresupplied by wind database and are also available from thecorresponding author upon request

8 Discrete Dynamics in Nature and Society

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] J E Ingersoll Jr ldquoA contingent-claims valuation of convertiblesecuritiesrdquo Journal of Financial Economics vol 4 no 3 pp 289ndash321 1977

[2] C M Lewis ldquoConvertible debt Valuation and conversion incomplex capital structuresrdquo Journal of Banking amp Finance vol15 no 3 pp 665ndash682 1991

[3] M J Brennan and E S Schwartz ldquoConvertible bonds valuationand optimal strategies for call and conversionrdquo e Journal ofFinance vol 32 no 5 pp 1699ndash1715 1977

[4] M J Brennan and E S Schwartz ldquoAnalyzing convertiblebondsrdquo Journal of Financial and Quantitative Analysis vol 15no 4 pp 907ndash929 1980

[5] J J McConnell and E S Schwartz ldquoLYONTamingrdquoe Journalof Finance vol 41 no 3 pp 561ndash576 1986

[6] C Fan X Luo and Q Wu ldquoStochastic volatility vs jump dif-fusions Evidence from the Chinese convertible bond marketrdquoInternational Review of Economics amp Finance vol 49 pp 1ndash162017

[7] X Hu and H Mao ldquoEmpirical study on the financial char-acteristics of chinese companies issuing convertible bondsrdquoInternational Journal of Business and Management vol 4 no6 pp 59ndash64 2009

[8] L Lu and W Xu ldquoA simple and efficient two-factor willow treemethod for convertible bond pricing with stochastic interestrate and default riskrdquo e Journal of Derivatives vol 25 no 1pp 37ndash54 2017

[9] M Ammann A Kind andCWilde ldquoSimulation-based pricingof convertible bondsrdquo Journal of Empirical Finance vol 15 no2 pp 310ndash331 2008

[10] S Crepey and A Rahal ldquoPricing convertible bonds with callprotectionrdquo e Journal of Computational Finance vol 15 no2 pp 37ndash75 2011

[11] R Schoftner ldquoOn the estimation of credit exposures usingregression-basedMonte Carlo simulationrdquoe Journal of CreditRisk vol 4 no 4 pp 37ndash62 2008

[12] K Tsiveriotis and C Fernandes ldquoValuing convertible bondswith credit riskrdquo e Journal of Fixed Income vol 8 no 2 pp95ndash102 1998

[13] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999

[14] E Ayache P A Forsyth and K R Vetzal ldquoValuation ofconvertible bonds with credit riskrdquo e Journal of Derivativesvol 11 no 1 pp 9ndash29 2003

[15] J A Batten K L-H Khaw andM R Young ldquoConvertible bondpricing modelsrdquo Journal of Economic Surveys vol 28 no 5 pp775ndash803 2014

[16] K Park M Jung and S Lee ldquoPricing a defaultable convertiblebond by simulationrdquo Korean Journal of Financial Studies vol46 no 4 pp 947ndash965 2017

[17] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo e Journal of Finance vol 29 no2 pp 449ndash470 1974

[18] 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

[19] R d Cheng and J R Lu ldquoA monte carlo method of integratedrisk measurement for defaultable zero-coupon bondsrdquo Journalof Management Sciences in China vol 15 no 4 pp 88ndash97 2012

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Discrete Dynamics in Nature and Society

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] J E Ingersoll Jr ldquoA contingent-claims valuation of convertiblesecuritiesrdquo Journal of Financial Economics vol 4 no 3 pp 289ndash321 1977

[2] C M Lewis ldquoConvertible debt Valuation and conversion incomplex capital structuresrdquo Journal of Banking amp Finance vol15 no 3 pp 665ndash682 1991

[3] M J Brennan and E S Schwartz ldquoConvertible bonds valuationand optimal strategies for call and conversionrdquo e Journal ofFinance vol 32 no 5 pp 1699ndash1715 1977

[4] M J Brennan and E S Schwartz ldquoAnalyzing convertiblebondsrdquo Journal of Financial and Quantitative Analysis vol 15no 4 pp 907ndash929 1980

[5] J J McConnell and E S Schwartz ldquoLYONTamingrdquoe Journalof Finance vol 41 no 3 pp 561ndash576 1986

[6] C Fan X Luo and Q Wu ldquoStochastic volatility vs jump dif-fusions Evidence from the Chinese convertible bond marketrdquoInternational Review of Economics amp Finance vol 49 pp 1ndash162017

[7] X Hu and H Mao ldquoEmpirical study on the financial char-acteristics of chinese companies issuing convertible bondsrdquoInternational Journal of Business and Management vol 4 no6 pp 59ndash64 2009

[8] L Lu and W Xu ldquoA simple and efficient two-factor willow treemethod for convertible bond pricing with stochastic interestrate and default riskrdquo e Journal of Derivatives vol 25 no 1pp 37ndash54 2017

[9] M Ammann A Kind andCWilde ldquoSimulation-based pricingof convertible bondsrdquo Journal of Empirical Finance vol 15 no2 pp 310ndash331 2008

[10] S Crepey and A Rahal ldquoPricing convertible bonds with callprotectionrdquo e Journal of Computational Finance vol 15 no2 pp 37ndash75 2011

[11] R Schoftner ldquoOn the estimation of credit exposures usingregression-basedMonte Carlo simulationrdquoe Journal of CreditRisk vol 4 no 4 pp 37ndash62 2008

[12] K Tsiveriotis and C Fernandes ldquoValuing convertible bondswith credit riskrdquo e Journal of Fixed Income vol 8 no 2 pp95ndash102 1998

[13] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999

[14] E Ayache P A Forsyth and K R Vetzal ldquoValuation ofconvertible bonds with credit riskrdquo e Journal of Derivativesvol 11 no 1 pp 9ndash29 2003

[15] J A Batten K L-H Khaw andM R Young ldquoConvertible bondpricing modelsrdquo Journal of Economic Surveys vol 28 no 5 pp775ndash803 2014

[16] K Park M Jung and S Lee ldquoPricing a defaultable convertiblebond by simulationrdquo Korean Journal of Financial Studies vol46 no 4 pp 947ndash965 2017

[17] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo e Journal of Finance vol 29 no2 pp 449ndash470 1974

[18] 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

[19] R d Cheng and J R Lu ldquoA monte carlo method of integratedrisk measurement for defaultable zero-coupon bondsrdquo Journalof Management Sciences in China vol 15 no 4 pp 88ndash97 2012

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom