C074 論文題目： Pricing Foreign-Currency Convertible Bonds And Their Asset Swaps With Interest Rate,

Equity, FX, And Credit Risk

參與類組： 財務管理組

Abstract

This paper is the first article to price foreign-currency (or inflation-indexed) convertible bonds and their asset swaps subject to interest rate, equity, exchange rate, and credit risk. We also provide the suitable swap rate for asset swap and prove that the value of a foreign-currency convertible bond is less than (equal to) the value of a synthesis straight bond plus the value of a call option on foreign-currency convertible bond while the foreign-currency convertible bond is (not) embedded with the call or put provisions prior to the maturity date of foreign-currency convertible bond asset swap. In practice, the value of a call option on foreign-currency convertible bond is equal to the value of a foreign-currency convertible bond minus the value of a synthetic straight bond. It is correct only if there is not call or put provisions prior to maturity date of asset swap. From numerical analysis, we also find out the properties of foreign-currency convertible bonds, synthesis straight bonds, call options on foreign-currency convertible bonds, and the swap rates. Taking the five year-to-maturity foreign-currency convertible bond issued by Tom.com Ltd. as an example, we provide the theoretical values of the foreign-currency convertible bond, call option on foreign-currency convertible bond and the appropriate swap rate .The empirical results indicate that the numerical value is closed to the market price. Hence, our pricing model is useful for market practitioners.

Keywords: Foreign-currency Convertible Bonds, Foreign-currency Convertible Bond Asset Swap, Stochastic interest rate, Credit risk, Foreign-currency Convertible Bond Parity

摘要

在考慮無風險利率、有風險利率、股價及匯率等隨機因子下，本文首次在文獻上提出海外可轉換公司債(或通貨膨脹指數連結之可轉換公司債)及其資產交換之訂價。本文亦證明在資產交換期間內，若海外可轉換公司債具有買回權或賣回權，海外可轉換公司債價值必小於等於合成的純粹債券與海外可轉換公司債選擇權

價值之加總；反之，海外可轉換公司債的價值等於合成純粹債券加上海外可轉換公司債選擇權價值。在實

務上，海外可轉換公司債資產交換的到期日為海外可轉換公司債下一個賣回日，海外可轉換公司債選擇權

評價則直接由海外可轉換公司債價格與其合成純粹債券價格相減求得。因此，實務上作法必須在資產交換

期間，海外可轉換公司債沒有買回權情況下才正確，否則將低估海外可轉換公司債選擇權價值。 由數值分析得知，當海外可轉換公司債本身無買回權與賣回權時，海外可轉換公司債的價值與匯率變

動率的波動度及債券利息呈現正向關係。在合成純粹債券方面，其價值與債券利息呈現正向關係，和違約

機率呈現負向關係，但不受匯率變動率之波動度影響。此外，海外可轉換公司債選擇權與匯率波動度和違

約機率呈現正向關係，但與債券利息呈現負向關係。因此，當違約機率升高時，海外可轉換公司債將呈現 U型反轉，顯示先降後昇現象。若海外可轉換公司債本身具有買回權或賣回權，則海外可轉換公司債、合成

之純粹債券和海外可轉換公司債選擇權結果與上述情況一致。在最適交換利率部分，若在資產交換期間，

海外可轉換公司債無買回權與賣回權，最適交換利率與債券利息呈現負相關，與違約機率呈現正相關，而

與股價報酬率波動度無關，代表海外可轉換公司債資產交換可規避股價變動風險。 最後，以香港 Tom.com控股公司發行之五年期零息海外可轉換公司債為例，計算其理論價格、海外可

轉換公司債選擇權價值及其合理之交換利率。實證結果發現理論價格與市場價格接近，驗證模型之正確性。

此外，切割期數增加至 500 期，發現理論價格、合成的純粹債券、海外可轉換公司債選擇權價值及其合理之交換利率呈現收斂現象，此驗證模型的準確性。

關鍵字：海外可轉換公司債、海外可轉換公司債資產交換、隨機利率、信用風險、海外可轉換公司債等價

公式

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Ⅰ. INTRODUCTION Convertible bonds (hereafter CBs) are hybrid financial instruments entitle investors the rights to forgo coupon and principle, and to convert into a pre-specified numbers of shares of common stocks instead. In recent years, the commercial growth of CBs is impressive. The worldwide CBs market is exceed of US$ 400 billion1.

Due to financial innovation, the convertible structure offers an ideal starting point for creating new features tailored to suit issuers and different investor groups, including foreign-currency CBs (henceforth FCCBs) and inflation-indexed CBs whose synthesis straight bonds link to a foreign exchange rate or a general price index, respectively. FCCBs or inflation-indexed CBs market has been expanding rapidly.

Since CB is a mixture of corporate bond and equity nature, this property renders it sensitive to equity, interest rate, and credit risk. Besides, the special cases of exchange rate risk arise for those FCCBs which the payment denominated in foreign currency but convert into domestic equity. As shown by Moody’s report, the expected default probabilities for CB issuers are higher than those without CBs in their capital structures. Consequently, credit risk is a crucial factor in pricing CBs and should not be neglected.

Another emerging market for convertibles is CB strip market that has grown dramatically since the mid 1990s and at least 30% of FCCBs have been stripped in 2001. As we aware that FCCB asset swap dealer strips a FCCB into credit component and equity component to suit different investor groups. Credit component is viewed as FCCB asset swap. The basic FCCB asset swap transaction involves: 1) One party (credit investor) purchase a FCCB and write a American-style call on FCCB to another party (equity investor), give the latter the right to purchase FCCB at strike price equal to synthetic straight bond value. The maturity date of call on FCCB is the same as it of FCCB asset swap. Hence, credit investor holds a synthesis straight bond in FCCB asset swap. Credit investors are usually the fixed-income investors such as commercial banks, retired funds, and insurance corporations while most of equity investors are convertible arbitrage funds. 2) Interest rate swap: If default doesn’t occur, issuer doesn’t call FCCB back, or call on FCCB aren’t early exercised, the FCCB asset swap dealers will pay the credit investor fixed (or floating) swap rate in return for cash flows from FCCB. Otherwise, the asset swap transaction is early terminated2. Hence, the credit investor owns a synthetic straight bond without equity exposure but faces credit risk. On the other hand, the equity investor owns a call option on FCCB that is free of the credit exposure of outright FCCB investment, while still benefit from the upward movement of stock prices.

Ⅱ. LITERATURE REVIEW The modern literature for pricing CBs begins with Ingersoll [1977] and Brennan and Schwartz [1977]. They extend Black and Scholes’ [1973] and Merton’s [1973] model to provide analytical solution and numerical technique using finite differences method for pricing CBs as contingent claims on the firms as a whole. However, the main drawback of structural models such as Ingersoll [1977] needs to estimate unobservable parameters of total value and volatility of the firm’s assets.

Different from the setup of Ingersoll [1977], Mcconnell and Schwartz [1986] use a contingent claims pricing model for valuing Liquid Yield Option Note (LYON) with finite differences method and utilize the model to analyze a specified LYON issue. They assume the value of LYON depends upon the issuer’s stock price and avoids estimating the unobservable parameters such as the volatility of the firm’s value.

Nevertheless, the above pricing models don’t consider default risk and preclude the possibility of bankruptcy, and it may overestimate the CBs values. As a result, considering default risk, Nyborg [1996] evaluates CBs values by using total values of firms as stochastic variables and accounting for the debt obligations of issuers in defining random behavior of the firm value. However, their model also involves many parameters to estimate. Tsiveriotis and Fernandes [1998] view a CB as a contingent claim on the underlying equity and divide it into two parts by different credit levels. One is cash-only part of CB, which is subject to credit risk, and the other part is the CB related to payment in equity, which is not. This 1 Yigitbasioglu, A. (2001), ISMA Centre Discussion Papers In Finance, 2001-14, November, 2001 2 If default occurs, the credit investors will still hold the FCCBs, and the asset swap transaction is early terminated. Otherwise,

while the issuer redeems the FCCBs or the equity investor early exercises, the asset swap dealer immediately pays the accrued interest and nominal principal of the asset swap to credit investors in return for the synthesis straight bonds will also cause the asset swap transaction is early terminated.

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leads a pair of coupled partial differential equations that can be solved to value CBs. However, Ayache, Forsyth, and Vetzal [2003] demonstrate that the widely used CB model of Tsiveriotis and Fernandes [1998] is internally inconsistent. They provide a CB model subject to credit risk using an approach based on the numerical solution of linear complementarity problems.

Recently, an alternative for considering credit risk is the setup of hazard function. The probability of default in the next time partition is determined by a specified hazard rate. When default occurs, some portion of the bond is assumed to be recovered. For example, following Duffie and Singleton [1999], Takahashi, Kobayashi, and Nakagawa [2001] propose a reduced-form model to price convertible debts with credit risk. They model the hazard rate as a decreasing function of stock price and assume the pre-default process of the stock price follows a diffusion process.

Hung and Wang [2002] combine Jarrow and Turnbull [1995] model involving the default-free and the risky discount rate processes with the tree for stock prices to value CBs that may be default. However, their model ignores the default hazard rate and hence underestimates the upward probability of stock price as well as the CB values. Meanwhile, their model also assumes that the risk-free interest rate is the same after the default occurs. However, since that the default event is uncorrelated with the behavior of default-free interest rate, the default-free interest rate is still stochastic after the default event occurs.

For pricing FCCBs and inflation-indexed CBs, Yigitbasioglu [2001] presents a two factor model to deal with FX sensitive cross-currency convertibles by using Crank-Nicholson technique. Landskroner and Raviv [2002] consider two sources of uncertainty allowing both the underlying stock and the CPI (or exchange rate) to be stochastic and incorporate credit risk by using a Rubinstein [1994] three-dimensional binomial tree. However, the risky interest rate is deterministic in their model.

As well as we know, our paper is the first article to explore the pricing model for valuing the values of FCCBs, call options on FCCBs and FCCB asset swaps. We provide a three-dimensional lattice method for valuing callable and puttable FCCBs (or inflation-indexed CBs), call on FCCB, and FCCB asset swap for which the equity price, exchange rate (or CPI), default-free, and risky interest rates are stochastic.

If there are (not) call and put provisions prior to the maturity date of FCCB asset swap, we prove that the value of a synthetic straight bond plus the value of a call option on FCCB are higher than (equal to) the value of a FCCB. In practice, the FCCB strip is to make the value of synthetic straight bond plus the value of a call on FCCB equal to the value of FCCB. Consequently, it is correct only if there is no call or put provision prior to maturity date of FCCB asset swap.

From numerical analysis, we find that the values of synthesis straight bonds are decreasing function of the hazard rate, and increasing function of coupon rate, and are unrelated with the volatility of stock return or exchange rate. The value of a call option on FCCB is increasing function of hazard rate and volatility of stock return or exchange rate. The relationship of FCCB values and hazard rate is a U-shaped curve. The suitable swap rate is positive relationship with hazard rate and unrelated with volatility of stock return. This proves that the credit investor is free of equity exposure, nevertheless, they require higher swap rate if they face the higher degree of credit risk.

Taking the FCCBs issued by Tom.com Ltd. in Hong Kong as an example, we show that the numerical results are convergence and also provide the fair prices of the FCCB, straight bond, a call on FCCB and the appropriate swap rate. Consequently, our pricing model is suitable for market practitioners.

Ⅲ. PRICING MODEL First, we describe setups of cross-currency economy. Then, modifying Hung and Wang’s [2002] model and incorporating exchange rate, and we develop a valuation algorithm for FCCBs.

The Economy We assume there are two countries, α and β, in cross-currency economy. The firm in country β issues a CB denominated in currency of country α. Uncertainty of trading market in country α is described by filtered probability space ))(,,,(

*

0* T

ttFPF =Ω , where *P is spot martingale measure under discounted values of underlying assets in country α.

Under *P , exchange rate process )(tQ , which is used to convert the payoffs of country β into currency of country α, is assumed as follows:

3

)()()()( * tdWdtrr

tQtdQ

QQσβα +−= (1)

where *QW stands for one-dimensional standard Brownian motion. αr and βr are stochastic risk-free interest

rates in country α and β, respectively. Qσ is the volatility of return of exchange rate. Furthermore, we assume that the dynamics of issuer’s stock price )(tS β is:

( ) )()()](1)[()()( * tdNtdWdttNtqr

tStdS

SSQSSQ −+−+−−= σλµσσραβ

β

(2) where ( )τtItN ≥≡)( , ( )⋅I is the indicator function, τ is exponentially distributed over ),0[ ∞ with parameter λ , and )(tμ is Poisson bankruptcy process under *P in country α. )(tλµ represents hazard rate. q is continuous dividend payout rate. Sσ is volatility of issuer’s stock return in country β. SQρ is instantaneous correlation coefficient between issuer’s stock return and exchange rate and satisfies:

dttdWtdWE SQQS ρ=)]()([ ** (3) From country α’ s investors point of view, when they convert a FCCB into issuer’s stocks, they

obtain the amount equal to stock price multiplied by the concurrent exchange rate. Hence, we assume that stock price demonstrated in currency of country α as )()()(* tQtStS ββ ≡ . By Ito’s Lemma, we have3:

( ) )()()](1)[()()( *

3*

*

tdNtdWdttNtqrtStdS

−+−+−= σλµαβ

β (4)

where 222 2 QQSSQS σσσρσσ ++≡ . Let us consider a finite collection of date jT , nj ,...,1= , where *

1 ...0 TTTT n ≤=<<< , tjT j ∆×= , and nTt /=∆ . From (4), we can see that )(* tSβ is a lognormal distribution and can be rewritten as follows:

)()()()(~

11 −−

≡jj

jjT TQTS

TQTSj

β

βξ

njforTif

TifdssTTqrTWTW

j

j

T

Tjjjjj

j

,...,1,0

)())(21()]()([exp

11

21

*3

*3

=≥

=

<

+−−−+− ∫

−−−

τ

τλµσσ α

(5)

Hence, if underlying stock doesn’t default, jTξ

~ is a lognormal distribution.

Furthermore, let ),( Ttpα and ),( Ttvα denote time t price of default-free and risky zero coupon bond maturing at time T in country α and issued by FCCB issuer4, respectively. If default occurs, FCCB investor in country α would receive not the whole face value, but rather a fraction of face value. This fraction is usually defined as the recovery rateδ . As described in Jarrow and Turnbull [1995], we have:

×

−+×

−−= ∫∫ 1)(exp)(exp1),(),(

T

t

T

tdssdssTtpTtv λµδλµαα (6)

Lattice Methods for Default-free and Risky Interest Rates We adapt Jarrow and Turnbull’s [1995] model for default-free and risky interest rates, where pseudo-probability π is determined for default-free interest rate process and is set to be 0.5. Recovery rateδ is given exogenously. Besides, a FCCB has positive default probabilities in each period. Given that

3 Similar to Yigitbasioglu [2001], we also use the domestic price of foreign equity to reduce the required state variables.

However, we use a reduced-form approach and consider the hazard rate process in the setup of equity price. 4 If FCCB issuer doesn’t issue zero-coupon bonds in units of currency of country α, we can use the risky zero coupon bonds in

the same credit class as a proxy.

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recovery rateδ , we can use market data to compute 1−jµλ , where 1−jµλ is default probability prior to one period for time jT , nj ...,,1= . To demonstrate pricing procedure of Jarrow and Turnbull [1995], we take a two-period trading economy as example. The default-free interest rate process is consistent with Black, Derman, and Toy [1990]. For computing default probability at each period, at time 1, we have:

]1)1([),0(),0( 0011 ×−+××= µµαα λδλTpTv (7) Using the market data of ),0( 1Tpα and ),0( 1Tvα with δ , we can derive 0µλ . Similarly, at time 2, we have:

]1)1([)1(),0(),0( 110022 ×−+××−+××= µµµµαα λδλλδλTpTv (8) Using values of ),0( 2Tpα , ),0( 2Tvα with δ and 0µλ , we can obtain 1µλ . According to this procedure, we can calculate 1−jµλ at each time jT recursively.

It is worth to note that if 0=t and 2TT = , we can rewrite (6) as follow:

×

−−= ∫ δλµαα

1

022 )(exp1),0(),0(T

dssTpTv

×

−+×

−−×

−+ ∫∫∫ 1)(exp)(exp1)(exp 2

1

2

1

1

0

T

T

T

T

Tdssdssdss λµδλµλµ (9)

By comparing (8) with (9), we obtain that

−−= ∫

−−

j

j

T

Tjμ dssλμλ1

)(exp11 , for nj ...,,1= (10)

Hung and Wang [2002] combine Jarrow and Turnbull’s [1995] model involving default-free and risky discount rate processes with the tree for stock prices to value CBs that may be defaulted. According to their model, if default occurs, the stock price jump to zero and the default-free interest rate is also unchanged in the successive periods. However, the stochastic behavior of default-free interest rate should be independent with default event, and hence we can’t ignore the pseudo-probability for upward or downward movements of default-free interest rate. Hence, we modify Hung and Wang’s [2002] method by revising the movements of default-free interest rate after default occurs and shown in Exhibit 1.

Lattice Method for Stock Price in Unit of Another Currency To construct a tree for stock price demonstrated in currency of country α, we use a discrete process

jTξ to

approximatejTξ

~ in (5). AssumingjTξ can only change for three possible values, 0 with probability )( jPn , u

with probability )( jPu , or d with probability )( jPd , for nj ,...,1= . The setup of lattice method for )(*jTS β

is shown in the following theorem. Theorem 1. Under the setup of (1) and (2), the possible values of )(*

jTS β , nj ,...,1= , are 0 with

probability )( jPn , uTS j )( 1*

−β with probability )( jPu , or dTS j )( 1*

−β with probability )( jPd , where u, d, )( jPu , )( jPd , and )( jPn are defined as :

)exp(1nT

du σ== , 1)( −= jn jP µλ ,

αµλ rju pjP )1()( 1−−= , )1)(1()( 1 αµλ rjd pjP −−= − (11)

where nTqr

Tnp j

r σσ

σλ αµ

α 25.0

2)1ln(

21 2

1 −−+

−−= − (12)

We prove Theorem 1 in Appendix A. From (12), since that αr

p is a function of αr , we can see that the stock price and the risk-free interest rate in country α is correlated.

In Hung and Wang’s [2002] pricing CBs model, the upward probability of stock price is as follows:

ααα σ

λ

σσ µα

rj

rHWr p

Tnp

nTqr

p ≤−

+=−−

+= −

2)1ln(

25.0

21 1

2

(13)

Prior to bankruptcy, (13) doesn’t consider hazard rate and hence underestimate both upward probabilities

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of stock price and CB values. For example, as shown in numerical example of Hung and Wang’s [2002] model, since that continuum value 86.15 is smaller than conversion value 90, the optimal decision is to convert CB. However, using Theorem 1 and default-free interest rate is still stochastic even though default occurs, the continuum value should be 91.1148 and hence the optimal decision for CB investors is to hold the CB. As a result, we should apply (12) to compute the prices of CBs subject to credit risk. We illustrate the possible paths and corresponding probabilities for *

βS in Exhibit 2.

Pricing Method for Zero-Coupon FCCBs After that, we can incorporate these lattice methods of default-free and risky interest rates and stock price demonstrated in currency of country α into our pricing method for T-year maturity FCCB with call and put provisions. We define that )(trP

α is the probability that stock price will go up when risk-free rate at

time t is αr . Exhibit 3 indicates an intuitive four-period pricing method for FCCB. There are three main cases as one move through the tree. First, if FCCB doesn’t default such as both node A and node C, we have six branches in the next period, which αr goes up or down and *

βS goes up, down, or jumps to 0. If default occurs, the tree enters into second case such as node B and node D. The stock prices will

jump to zero and hence the equity component of FCCB is zero and the debt part can receive only the frictions of par value. As a result, the value of FCCB will be the product of recovery rate and par value. However, unlike the setup of Hung and Wang’s [2002] model, after default occurs, the default-free interest rate process still fluctuates. Hence, at node B and node D, we have two branches at each period, which αr goes up or down and *

βS still jumps to 0. Therefore, the tree of default-free interest rate is recombined after default event. For the terminal

node such as node E, the default-free interest rate as well as the stock price doesn’t fluctuate again. The third case such as node F is a special condition that uuur )3(α is the discount rate between 3=t and

4=t . Thus, the node F has three branches, which αr is fixed and *βS goes up, down, or jumps to 0.

After the construction of the tree, we should backwardly decide the payoff at each node from the terminal nodes. If default does not occur, the total value of the FCCB at each node can be stated as:

Max [ Min (Continuum Value, Call Price), Put Price, Conversion Value] (14) If default occurs, the stock price will jump to zero. FCCB investors will not exercise the conversion

right to get the zero-value underlying stock. Because of bankruptcy, the put provision of FCCB is also invalid. Hence, the total value of FCCB at this node is equal to the present value of recovery rate multiplied by cash flows (including successive coupon and principal payments) from FCCB. Finally, once rolling back through the tree, the total value at time 0T is the theoretical value of FCCB with credit risk.

Refined Lattice Model for Zero-Coupon FCCBs However, if default don’t occur, there are 4 j nodes for period j, and explode when n is too large in Exhibit 3. To overcome this drawback, in Exhibit 4, we provide a refined three-dimensional lattice method for pricing FCCBs. Each node in Exhibit 3 is all included in Exhibit 4. The main advantage of refined pricing method is reducing numbers of nodes from 4 j to ( j+1)2 and hence is timesaving and useful for market practitioner. For example, CPU time for five year in 120 time steps is around 6 seconds.

Ⅳ. EXTENSIONS In this section, we extend the FCCB pricing model as follows:

Coupon-Bearing FCCBs and Synthesis Straight Bonds

We can price the coupon-bearing FCCBs by slightly changing pricing model mentioned above. If default occurs, the call, put, and conversion provisions are invalid and hence the total value of FCCB at this node is equal to the present value of recovery rate multiplied by the accrued interests and the principal payment in the subsequent periods. Otherwise, at the dates of coupon payments the total value of the FCCB at each node is determined as follows:

Max [ Min (Continuum Value + Coupon Payment, Call Price), Put Price, Conversion Value] (15)

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Finally, once having rolled back through the tree, the total value at time 0T is the fair price of coupon-bearing FCCB subject to credit risk.

For pricing synthesis straight bond, since it is the same as FCCB without conversion provision, by using the same FCCB model and assuming the conversion ratio equal to zero, we can obtain the value of synthesis straight bond.

Call on FCCB In the transaction of FCCB asset swap, the credit investors buy FCCB from asset swap dealers and simultaneously sell a call on FCCB to equity investors. Hence, the credit investor holds a synthesis straight bond and equity investor holds a call on FCCB. It gives the equity investors the right to buy FCCB at a strike price equal to the value of synthesis straight bond prior to the maturity of asset swap. For pricing call on FCCB, if default occurs, it is zero since that the value of a FCCB is equal to the value of a synthesis straight bond and simultaneously the FCCB asset swap is early terminated. If default does not occur, for ease of the explanation, we denote that:

mT : The maturity date of FCCB asset swap (or call on FCCB).

iFCCB : The value of a FCCB at time iT , ni ,...,0= , where TTT nm =≤ .

iSB : The value of a synthesis straight bond at time iT , ni ,...,0= .

icontOption _ : The continuum value of a call on FCCB option at time iT , mi ,...,0= .

iOption : The value of a call on FCCB at time iT , mi ,...,0= . Hence, at time mT , the value of a call option on FCCB is as follows:

mOption = Max [( mFCCB – mSB ), 0 ] (16) Otherwise, the option value in each node is set as:

iOption = Max [( iFCCB – iSB ), icontOption _ ], 1,...,0 −= mi (17) where icontOption _ is the discounted values of 1+iOption in the (i+1)th period. Once rolling back through the tree, the value at time 0T is the theoretical price of a call on FCCB subject to credit risk.

It is worth to note that the FCCB asset swap dealer usually set the maturity date of FCCB asset swap (or call on FCCB) equal to the closeness put date of FCCB. They also make the value of a FCCB be equal to the value of a call on FCCB plus the value of a synthesis straight bond. However, it is not correct if there is call or put provisions prior to maturity date of FCCB asset swap. We demonstrate the relationship between FCCB, synthesis straight bond, and call option on FCCB in the following theorem. Theorem 2. With call and put provisions prior to the maturity date of FCCB asset swap, the value of a FCCB is less than the value of a call on FCCB plus the value of a synthesis straight bond and states as:

iii OptionSBFCCB +≤ , mi ,...,0= (18) Otherwise, we have the FCCB Parity as follows:

iii OptionSBFCCB += , mi ,...,0= (19) We prove Theorem 2 in Appendix B.

As a result, unless there is call and put provisions prior to the maturity date of FCCB asset swap, equality, iii SBFCCBOption −= holds. In practice, the maturity date of a call on FCCB is set to the nearness put date of FCCB, while issuer promises not to exercise the call right or there is no call provisions prior to the maturity date, the pricing mechanism of FCCB strip in practice work well.

Swap Rate for FCCB Asset Swap For pricing FCCB asset swap, our goal is to find a suitable swap rate such that FCCB asset swap is a zero sum game for both credit investors and FCCB asset swap dealer at the issue date. We denote M is the nominal principal of FCCB asset swap. For time iT , mi ,...,1= , if default occurs, the asset swap is early terminated and credit investors must hold synthesis straight bond which is already defaulting. Credit investors have a loss equal to the cash inflows (including the accrued interest payment plus M, or equivalently, [1+swap rate]×M ) minus default value of a synthesis coupon-bearing straight bond. If

7

default doesn’t occur, we have the following cases. First, if issuer doesn’t call FCCB back and call on FCCB isn’t early exercised, the payoff of credit investors is equal to the accrued interest payment (swap rate multiplied by M) from FCCB asset swap dealer plus continuum value of FCCB asset swap minus coupon payment of a FCCB. Second, while issuer call FCCB back, equity investors early exercise, or FCCB asset swap is matured, the payoff of credit investors equal the cash flows from asset swap dealer( [1+swap rate]×M ) minus the value of a synthesis coupon-bearing straight bond.

At issuance day 0T , credit investors pay M and get synthesis coupon- bearing straight bond. Consequently, the gain or loss at issue date 0T is equal to the value of a synthesis straight bond minus M and plus continuum value of FCCB asset swap. Suitable swap rate for FCCB asset swap is determined by the criterion that the gain or loss at time 0T is equal to zero.

Inflation-indexed CB Similar to a FCCB, the coupons and principal payments of inflation-indexed CBs are linked to the change of consumer-price-index (CPI). We denote CPI at time t as tC , and assume issue data of inflation-indexed CB is at time 0T , and redefine tCCtQ 0)( ≡ . Then, using a cross-currency trading economy analogy, real prices correspond to prices in country α and nominal prices correspond to prices in country β. αr is the real risk-free interest rate. *

βS is the real price of underlying stock. Since that the nominal price is equal to real price at time 0, following the same valuation algorithm for pricing FCCBs, once we roll back through the tree, the total value at the node 0T is the fair price of an inflation-indexed CB subject to credit risk.

Ⅴ. NUMERCIAL ANALYSIS First, we discuss the characteristics of FCCBs, synthesis straight bonds, and call on FCCB under the following situations: (1) FCCBs without call and put provisions. (2) Without call or put provisions prior to maturity date of FCCB asset swap. (3) With call or put provisions prior to maturity date of FCCB asset swap. Then, we show the relationship of FCCBs, synthesis straight bonds, and call on FCCBs coincides with Theorem 2. Finally, we also investigate the characteristics of suitable swap rate in FCCB asset swap.

In the following section, we assume a German corporation issues a five-year maturity FCCB with face value US$ 100. We also divide 5 year into 120 periods, it means semi-week per time step. The conversion ratio is 3 shares. The initial stock price is EUR 25.4. Initial exchange rate is US$ 1 = EUR$ 0.8155. Hence, the stock price is US$ 31.1465 (equal to 25.4/0.8155). The volatility of the stock return and the instantaneous correlation coefficient of stock return and exchange rate are 20% and 15%, respectively. We also assume that the time-t price function of default-free zero coupon bond in America is

)]1(*025.0015.0exp[),0( −−−= ttpα , where 1≥t . The volatility of the default-free interest rate in America is equal to 1% and the maturity date of FCCB asset swap (call option on FCCB) is at the third year.

Numerical Result Using our pricing method for these coupon-bearing FCCBs, we report the numerical results of FCCBs, synthesis straight bonds, and call on FCCBs by varying levels of coupon rate, hazard rate, and volatility of exchange rate (or equivalently, volatility of stock return) in Exhibit 5A, 5B, and 5C, individually.

In Exhibit 5A, FCCBs are increasing function of volatility of exchange (stock return) and coupon rate. Similarly, synthesis straight bonds are increasing function of coupon rate, and are decreasing function of hazard rate. However, since that the synthesis straight bonds are debt part of FCCBs, their values are irrelevant with the volatility of exchange rate (stock return).

The values of three-year maturity call options on FCCBs have a positive relationship with volatility of exchange rate (stock return). The influence of coupon rate is also clear. If coupon rate increases, and hence values of synthesis straight bonds (strike price) increase, values of call options on FCCBs decrease. Since that FCCBs Parity is valid, the tradeoff of influence of hazard rate for synthesis straight bonds and call options on FCCBs results in the U-shaped relationship of hazard rate and FCCBs values.

In Exhibit 5B, we assume that FCCBs are embedded with call and put provisions. The call price at the 4th year is at par, and the put prices at the 3rd and 5thyear are US$ 107 and US$ 110, respectively. The

8

properties are similar to the case 1. However, due to the call and put provisions, the value of call option on FCCB is increasing function of coupon rate. Since that the call or put provisions will influence the strike price of call options on FCCBs, it is reasonable for this result. Similarly, since that there is no call and put provisions prior to maturity date of FCCB asset swap, the FCCB parity also holds even though the FCCBs are embedded with call and put provisions.

The properties of Exhibit 5C are the same as the above case. However, the only difference is the FCCB parity is invalid and (18) holds. The numerical results verify Theorem 2. In practice, given the market prices of FCCB and the corresponding straight bond, the FCCB asset swap dealers use the FCCB parity to price the values of call options on FCCBs. In most case, the maturity date of FCCB asset swap is the nearness put date. However, it is also embedded with the call provision prior to the maturity date of FCCB asset swap and hence the FCCB parity fails. The miss-pricing algorithm results in the low values of call options on FCCBs and benefit the equity investors such as the hedge fund.

Swap Rate for FCCB Asset Swap We examine the suitable swap rates of the above three cases for FCCB asset swaps in Exhibit 6 by different levels of coupon rate, hazard rate, and volatility of stock return (exchange rate). In the first and second cases, there is no call and put provisions prior to maturity date of FCCB asset swap, as coupon rate increases, the present value of synthesis straight bond increases and hence the initial cash outflow paid by credit investor decreases. As a result, it is clear that the swap rate is decreasing function of coupon rate. The swap rate increase for higher hazard rate due to higher default probability. It is deserved to be mentioned that volatility of stock return is irrelevance for the suitable swap rate. Hence, credit investors only face credit risk and are free of equity exposure. For the third case, the influence for hazard rate and volatility of stock return are also the same. However, the effect of coupon rate is uncertain.

It is worth to note that all of the above numerical results can be explained as a German CB with principals linked to CPI in Germany. The initial CPI is set as 0.8155, the real prices of default-free and risky zero-coupon bonds correspond to the prices of default-free and risky zero coupon bonds in America. Hence, using the same pricing algorithm, we can obtain the theoretical values of inflation-indexed CBs.

Ⅵ. EMPIRICAL RESULTS In 2003, Tom.com Ltd., the greater china media group in Hong Kong, issued a five-year zero-coupon FCCB. We use our pricing model to provide realistic computations for this FCCB. On the issue day, initial stock price and exchange rate were HK$ 2.55 and US$ 1.00 = HK$ 7.77. The volatilities of return of stock and exchange rate are 31.8597% and 1.3585%, respectively. The correlation coefficient between stock return and exchange rate was 0.190352. The volatility of interest rate is 0.8686%. The put price was US$ 102.31 on the third anniversary and the redemption price was US$ 103.86 at the maturity date. Prior to maturity, the investors may convert FCCB into 2352.941 shares of stock per US$ 1000 nominal.

Here, we provide the values of FCCB, straight bond, call on FCCB, and suitable swap rate with different time steps and shown in Exhibit 7A, 7B, 7C, and 7D, individually. These values are convergent when time step exceeds 90 periods. Hence, we divide five years into 120 periods and the theoretical value at the issue day is US$103.0114. When the issue price is US$ 100, we find the implied volatility of stock return equal to 26.127%. We also assume that the first put date is the maturity date of FCCB asset swap, and the value of synthesis straight bond on the issue day is US$ 80.0042. Given implied volatility and synthesis straight bond, call on FCCB is US$ 19.9958 (100 – 80.0042). Because of high credit risk exposure, the required swap rate for FCCB asset swap of Tom.com Ltd. is estimated as 15.77%.

Ⅶ. CONCLUSION AND CONTRIBUTION This article provides a new pricing method for valuing FCCBs, call options on FCCBs, and FCCB asset swap under the consideration of credit risk. We incorporate default-free and risky interest rates, exchange rate, and foreign stock price into one tree. Hung and Wang’s [2002] model for pricing CBs is the special case when the volatility of exchange is zero and exchange rate equals one. However, prior to bankruptcy, Hung and Wang’s [2002] model ignores that the expected stock return should be adjusted for the hazard function, and hence undervalue the upward probability of stock price and the values of FCCBs. After default occurs, their model also ignores that the default-free interest rate possibly goes up or down.

9

This paper is the first article to price foreign-currency (or inflation-indexed) convertible bonds and their asset swaps under the consideration of risk-free and risky interest rates, stock price, and exchange rate. We also compute the suitable swap rate for asset swap and prove that the value of a FCCB is less than (equal to) the value of a synthesis straight bond plus the value of a call option on FCCB while the FCCB is (not) embedded with the call or put provisions prior to the maturity date of FCCB asset swap. From numerical analysis, we also discover the properties of FCCBs, synthesis straight bonds, call options on FCCBs, and the suitable swap rates. Taking the FCCB issued by Tom.com Ltd. as an example, we provide the fair prices of FCCB, a call on FCCB, and the appropriate swap rates and show that our pricing model is convergence. As a result, our model can price not only FCCBs (or CBs if exchange rate and volatility of exchange rate equal to 1 and 0, respectively) but also other cross-currency credit derivatives, as long as they depend on the stochastic default-free and risky interest rates, the exchange rate, and the foreign stock price. Hence, our pricing model is useful for market practitioners.

Exhibit 1 Four-Period Risky Interest Rate Tree Method

not default

not default

default

uudr )3(α

not default

ur )1(α

)0(αr

0µπλ

0)1( µλπ−

)1( 0µλπ −

)1)(1( 0µλπ −−

)1( 2µλπ −

)1)(1( 2µλπ −−

2)1( µλπ−

2µπλ

π

π−1

π

π

π

π

π

π

π−1

π−1

π−1

π−1

π−1

π−1

1µπλ

1)1( µλπ−

)1( 1µλπ −

)1)(1( 1µλπ −−

31 µλ−

3µλ

default

uuur )3(α

uudr )3(α

uuur )3(α

defaultuuur )3(α

default

uudr )3(α

default

uddr )3(α

defaultuuur )3(α

default

uudr )3(α

default

uddr )3(α

default

dddr )3(α

default

uur )2(α

default

udr )2(α

defaultddr )2(α

defaultuur )2(α

defaultudr )2(α

not default

not default

udr )2(α

uur )2(α

default

ur )1(α

default

dr )1(α

not default

dr )1(α

δ

1

δ

131 µλ−

3µλ

δ

δ

δ

δ

δ

δ

δ

δ

δ

10

Exhibit 2 Possible Values of *

βS

)0(*βS uS )1(*

β

dS )1(*β

0)1(* =βS

*)1( 0β

µλ SP−

0µλ

)1)(1( *0β

µλ SP−−

Exhibit 3 Intuitive Pricing Model for FCCBs (or Inflation-indexed CBs)

0)2(

)2(* =β

α

S

r uu0)3(

)3(* =β

α

S

r uuu 0)4(* =βS

0)4(* =βS

uuuuS )4(*β

uuudS )4(*β

0)3(

)3(* =β

α

S

r uud

0)3(

)3(* =β

α

S

r udd

0)3(

)3(* =β

α

S

r ddd

0)3(

)3(* =β

α

S

r uuu

0)3(

)3(* =β

α

S

r uuu

0)3(

)3(* =β

α

S

r uud

0)3(

)3(* =β

α

S

r uud

0)3(

)3(* =β

α

S

r udd

0)2(

)2(* =β

α

S

r ud

0)2(

)2(* =β

α

S

r dd

0)2(

)2(* =β

α

S

r uu

0)2(

)2(* =β

α

S

r ud

uuu

uuu

S

r

)3(

)3(*β

α

uud

uuu

S

r

)3(

)3(*β

α

uuu

uud

S

r

)3(

)3(*β

α

uud

uud

S

r

)3(

)3(*β

α

uu

uu

S

r

)2(

)2(*β

α

ud

uu

S

r

)2(

)2(*β

α

uu

ud

S

r

)2(

)2(*β

α

ud

ud

S

r

)2(

)2(*β

α

0)1(

)1(* =β

α

S

r u

0)1(

)1(* =β

α

S

r d

u

u

S

r

)1(

)1(*β

α

d

u

Sr

)1(

)1(*β

α

u

d

S

r

)1(

)1(*β

α

d

d

S

r

)1(

)1(*β

α

)0(

)0(*β

α

S

r

0µπλ

0)1( µλπ−

)1( 0)0( µλπα

−rP

)1)(1( 0)0( µλπα

−− rP

)1()1( 0)0( µλπα

−− rP

)1)(1)(1( 0)0( µλπα

−−− rP

)1)(1)(1( 2)2( µλπα

−−−uurP

)1()1( 2)2( µλπα

−−uurP

)1)(1( 2)2( µλπα

−−uurP

)1( 2)2( µλπα

−uurP

2)1( µλπ−

2µπλ

π

π−1

π

π

π

π

π

π

π−1

π−1

π−1

π−1

π−1

π−1

1µπλ

1)1( µλπ−

)1( 1)1( µλπα

−urP

)1)(1( 1)1( µλπα

−−urP

)1()1( 1)1( µλπα

−−urP

)1)(1)(1( 1)1( µλπα

−−−urP

)1( 3)3( µλα

−uuurP

)1)(1( 3)3( µλα

−−uuurP

3µλ

0)4(* =βS 0)4(* =βS

0)4(* =βS

0)4(* =βS

0)4(* =βS

0)4(* =βS

0)4(* =βS

0)4(* =βS

0)4(* =βS

0)4(* =βS

11

Exhibit 4 Refined Lattice Model for Zero-Coupon FCCBs

)0(

)0(*β

α

S

r

ur )1(α

uS )1(*β

dS )1(*β

dr )1(α

P1 P3P2

uur )2(α udr )2(α

uuS )2(*β

udS )2(*β

ddr )2(α

ddS )2(*β

uuur )3(α

uuuS )3(*β

uudr )3(α uddr )3(α dddr )3(α

uudS )3(*β

uddS )3(*β

dddS )3(*β

0)3(

)3(* =β

α

S

r uuu0)4(* =βS

0µπλ

0)1( µλπ−

π

π−10)4(* =βS

0)4(* =βS

0)4(* =βS

0)3()3(

* =β

α

Sr uud

0)3()3(

* =β

α

Sr udd

0)3()3(

* =β

α

Sr ddd

0)2(

)2(* =β

α

Sr uu

0)2(

)2(* =β

α

S

r ud

0)2()2(

* =β

α

Sr dd

0)1()1(

* =β

α

Sr u

0)1(

)1(* =β

α

Sr d

π

π

π

π

π−1

π−1

π−1

π−1

0)4(* =βS

0)4(* =βS

0)4(* =βS

0)3(

)3(* =β

α

S

r uuu

0)3()3(

* =β

α

Sr udd

0)3(

)3(* =β

α

Sr uud

0)2(

)2(* =β

α

S

r uu

0)2(

)2(* =β

α

S

r udπ

π

π−1

π−1

1µπλ

1)1( µλπ−

0)4(* =βS

0)4(* =βS0)3(

)3(* =β

α

Sr uud

0)3(

)3(* =β

α

S

r uuu

2)1( µλπ−

2µπλ

3µλ0)4(* =βS

P4

)1( 0)0( µλπα

−rP)1)(1( 0)0( µλπ

α−− rP

)1()1( 0)0( µλπα

−− rP)1)(1)(1( 0)0( µλπ

α−−− rP

P1 =

)1( 1)1( µλπα

−urP

)1)(1( 1)1( µλπα

−−urP

)1()1( 1)1( µλπα

−−urP

)1)(1)(1( 1)1( µλπα

−−−urP)1( 2)2( µλπ

α−

uurP)1)(1( 2)2( µλπ

α−−

uurP)1()1( 2)2( µλπ

α−−

uurP)1)(1)(1( 2)2( µλπ

α−−−

uurP

P7P8

P5P6

P12P11

P10P9

P8 =P7 =P6 =P5 =P4 =P3 =P2 =

P12 =P11 =P10 =P9 =

t = 0

t = 1

t = 2

t = 3

P13P14

P14 =P13 = )1( 3)3( µλ

α−

uuurP)1)(1( 3)3( µλ

α−−

uuurP

t = 4

uuuuS )4(*β

uuudS )4(*β

uuddS )4(*β

udddS )4(*β

ddddS )4(*β

12

Exhibit 5A Ten-Year Maturity FCCB without Call and Put Provisions

FCCB Straight Bond Call on FCCB Qσ

Qσ Qσ

Hazard Rate

)(tλµ

Coupon Rate

20% 30% 40% 20% 30% 40% 20% 30% 40% 0% 115.62 121.76 128.17 89.14 26.48 32.63 39.03 0 5% 137.29 143.65 150.25 112.58 24.71 31.07 37.68 0% 114.24 117.97 122.22 69.43 44.81 48.54 52.80 0.1 5% 133.30 137.32 141.83 90.34 42.96 46.98 51.49

0% 116.21 118.70 121.85 62.71 53.51 56.00 59.14 0.15 5% 134.44 137.17 140.54 82.62 51.82 54.55 57.92 0% 120.98 121.82 123.24 53.39 67.59 68.43 69.85 0.25 5% 138.09 139.03 140.60 71.71 66.38 67.33 68.89

Exhibit 5B Ten-Year Maturity FCCB without Call and Put Provisions Prior to FCCB Asset Swap

FCCB Straight Bond Call on FCCB Qσ

Qσ Qσ

Hazard Rate

)(tλµ

Coupon Rate

20% 30% 40% 20% 30% 40% 20% 30% 40% 0% 118.29 124.06 130.10 100.27 18.02 23.79 29.83 0 5% 130.37 136.09 142.09 110.00 20.37 26.09 32.10 0% 115.06 119.44 124.26 84.40 30.66 35.04 39.86 0.1 5% 128.78 132.90 137.54 95.42 33.36 37.49 42.12 0% 115.65 119.06 123.08 78.08 37.57 40.98 45.00 0.15 5% 129.91 133.07 136.82 89.58 40.32 43.49 47.24 0% 118.72 120.39 122.79 67.96 50.76 52.43 54.83 0.25 5% 133.53 135.01 137.18 80.18 53.35 54.83 57.00

Exhibit 5C Ten-Year Maturity FCCB with Call and Put Provisions Prior to FCCB Asset Swap

FCCB Straight Bond Call on FCCB Qσ

Qσ Qσ

Hazard Rate

)(tλµ

Coupon Rate

20% 30% 40% 20% 30% 40% 20% 30% 40% 0% 110.96 115.19 119.71 96.08 18.28 23.96 29.95 0 5% 115.89 120.12 124.63 101.00 20.37 26.09 32.10 0% 108.80 112.39 116.30 84.40 30.66 35.04 39.86 0.1 5% 115.36 118.98 122.91 92.21 33.36 37.49 42.12 0% 108.79 111.84 115.29 78.08 37.57 40.98 45.00 0.15 5% 116.13 119.27 122.80 88.43 40.32 43.49 47.24 0% 110.50 112.43 114.88 67.96 50.76 52.43 54.83 0.25 5% 118.92 120.99 123.59 80.18 53.35 54.83 57.00

Exhibit 6 Suitable Swap Rate for Three-Year Maturity FCCB Asset Swap

)(tλµ Call and Put Provisions Coupon Rate Sσ 0 0.01 0.1 0.15 0.2 0.25

20% 0% 40% 0.0218 0.0341 0.1506 0.2204 0.2942 0.3721

20%

Without Call and Put Provisions 5% 40% 0.0218 0.0320 0.1283 0.1859 0.2466 0.3105

20% 0% 40% 0.0218 0.0341 0.1505 0.2204 0.2942 0.3721

20%

Put: 3rd and 5th year at 107% and 110%

Call: 4th year at 100% 5% 40% 0.0218 0.0320 0.1283 0.1858 0.2465 0.3104

20% 0% 40% 0.0364 0.0465 0.1505 0.2204 0.2942 0.3722

20%

Put: 3rd and 5th year at 107% and 110%

Call: 2nd and 4th year at 100% 5%

40% 0.0530 0.0613 0.1418 0.1913 0.2465 0.3104

13

Exhibit 7A The FCCB Values with Different Time Steps

Exhibit 7B Straight Bond Values with Different Time Steps

Exhibit 7C Call on FCCB with Different Time Steps

Exhibit 7D Suitable Swap Rates with Different Time Steps

REFERENCE 1. Ayache, E., Forsyth, P.A., and Vetzal, K.R. “Valuation of Convertible Bonds with Credit Risk.”

Journal of Derivatives, 11 (2003), pp. 9-29. 2. Black, F., and Scholes, M. “The Pricing of Options and Corporate liabilities.” Journal of Political

Economy, 81 (1973), pp. 637-654. 3. Brennan, M.J., and Schwartz, E.S. “Convertible Bonds: Valuation and Optimal Strategies for Call

and Conversion.” Journal of Finance, 32 (1977), pp. 1699-1715. 4. Black, F., Derman, E., and Toy. W. “A One-Factor Model of Interest Rates and Its Application to

Treasury Bond Options.” Financial Analysts Journal, Vol. 46, No. 1. (1990), pp. 33-39. 5. Duffie, D., and Singleton, K.J. “Modeling Term Structures of Defaultable Bonds.” Reviews of

Financial Studies, 12 (1999), pp. 687-720. 6. Huag, M.-W., and Wang, J.-Y. “Pricing Convertible Bonds Subject to Default Risk.” Journal of

Derivatives, 10 (2002), pp. 75-87. 7. Ingersoll, J., Jr. “A Contingent-Claims Valuation of Convertible Securities.” Journal of Financial

Economics, 4 (1977a), pp. 289-322. 8. Ingersoll, J., Jr. “An Examination of Corporate Call Policies on Convertible Securities.” Journal of

Finance, 32 (1977b), pp. 463-478. 9. Jarrow, R.A., and Turnbull, S.M. “Pricing Derivatives on Financial Securities Subject to Credit

Risk.” Journal of Finance, 50 (1995), pp. 53-85. 10. Landskroner, Y., and Raviv, A. “Pricing Inflation-Indexed and Foreign-Currency Linked Convertible

Bonds with Credit Risk.” Working Paper, Hebrew University Business School, (2002). 11. McConnell, J.J., and Schwartz, E.S. “Taming LYONS.” Journal of Finance, 41 (1986), pp. 561-576. 12. Merton, R.C. “The Theory of Rational Option Pricing.” Bell Journal of Economics and Management

Science, 4 (1973), pp. 141-183. 13. Nyborg, K.G. “The Use and Pricing of Convertible Bonds.” Applied Mathematical Finance, 3 (1996),

pp. 167-190.

14

14. Rubinstein, M. “Implied Binominal Trees.” Journal of Finance, 49 (1994), pp. 771-818. 15. Takahashi, A., Kobayashi, T. and Nakagawa, N. “Pricing Convertible Bonds with Default Risk.”

Journal of Fixed Income, 11 (2001), pp. 20-29. 16. Tsiveriotis, K., and Fernandes, C. “Valuing Convertible Bonds with Credit Risk.” Journal of Fixed

Income, 8 (1998), pp. 95-102. 17. Yigitbasioglu, A.B. “Pricing Convertible Bonds with Interest Rate, Equity, Credit and FX Risk.”

Discussion Paper 2001-14, ISMA Center, University of Reading, 2001. Available at : www.ismacentre.rdg.ac.uk.

APPENDIX A Proof of Theorem 1 By using equation (5) and (10), we match the first two moments as follows:

djPujPE duT jln)(ln)()]([ln +=ξ

+−−−= ∫∫

−−

j

j

j

j

T

T

T

Tdss

nTqrdss

11)()

21())(exp( 2 λµσλµ

−−−−−= −− )1ln()

21()1( 1

21 jj n

Tqr µµ λσλ

(A.1)

[ ]21

2222 )]([ln)1())(ln())(ln()]([lnjj TjμduT ξE

nTλσdjPujPξE +−=+= − (A.2)

Defining that 1)( −= jn jP µλ , αµλ rju pjP )1()( 1−−= , )1)(1()( 1 αµλ rjd pjP −−= − , and the equal jump size

condition: du lnln −= , we have:

)exp(nTu σ= , )exp(

nTd σ−= ,

nTqr

Tnp j

r σσ

σ

λµ

α 25.0

2)1ln(

21 2

1 −−+

−−= − (A.3)

This completes the proof of Theorem 1.

APPENDIX B Proof of Theorem 2 Case 1: With the call and put provision prior to maturity date of a FCCB asset swap First, if default occurs, the FCCB asset swap is early terminated and the equity investors should not exercise since that the value of a FCCB and stock price are equal to the value of straight bond and zero, respectively. We have: iii SBFCCBOption −== 0 (B.1)

If default does not occur and iii contOptionSBFCCB _≥− , we have

iiiiii SBFCCBcontOptionSBFCCBMaxOption −=−= )_,( (B.2) Therefore, the equity investors should early exercise. If default does not occur and iii contOptionSBFCCB _<− , we obtain

iiii SBFCCBcontOptionOption −>= _ (B.3) The equity investors should hold the call on FCCB. Briefly, we have

iii SBFCCBOption −≥ , mi ,...,0= (B.4) This completes the proof of (18).

15

Case 2: Without the call and put provision prior to maturity date of a FCCB asset swap Similarly, if default occurs, (B.1) holds again. If default does not occur, we can use the mathematical induction to prove that (19) holds. We denote that:

jir , : The default-free interest rate between time iT and 1+iT for the thj node.

jiP , : The probability for the thj node at time iT .

jicontFCCB ,_ : The continuum value of a FCCB for the thj node at time iT .

jiconvFCCB ,_ : The conversion value of a FCCB for the thj node at time iT .

jicontSB ,_ : The continuum value of a synthesis straight bond for the thj node at time iT .

More specificity, we also define that the subscript i, j indicate the thj node at time iT . At the maturity date of call on FCCB (or FCCB asset swap), the value of call on FCCB at thy node at time mT is equal to

ymymymymym SBFCCBSBFCCBMaxOption ,,,,, )0,( −=−= (B.5) where we use the fact that the value of a FCCB is not less than the value of a synthesis straight bond. Let (19) holds for all nodes at time kT , i.e., jkjkjk SBFCCBOption ,,, −= . At time 1−kT , since that each node has six branches if default does not occur, we have

∑ ×−= −−j

jkjkxkxk OptionPrcontOption ))(exp(_ ,,,1,1 , 6,...,1=j (B.6)

The value of a call on FCCB at thx node at time 1−kT is equal to )_,( ,1,1,1,1 xkxkxkxk contOptionSBFCCBMaxOption −−−− −=

]))(exp(,[ ,,,1,1,1 ∑ ×−−= −−−j

jkjkxkxkxk OptionPrSBFCCBMax

−×−−= ∑−−− ])()[exp(, ,,,,1,1,1

jjkjkjkxkxkxk SBFCCBPrSBFCCBMax (B.7)

Because of no call and provisions prior to the maturity date of call on FCCB, we have xkxk contSBSB ,1,1 _ −− = (B.8)

First, if xkxk convFCCBcontFCCB ,1,1 __ −− ≥ , we obtain

xkxkxkxk contSBcontFCCBSBFCCB ,1,1,1,1 __ −−−− −=−

xkj

jkjkj

jkjkxk contOptionSBPFCCBPr ,1,,,,,1 _])[exp( −− =×−×−= ∑∑ (B.9)

or equivalently xkxkxkxk contOptionSBFCCBOption ,1,1,1,1 _ −−−− =−= (B.10)

If xkxk convFCCBcontFCCB ,1,1 __ −− < , we have

xkxkxkxk contSBconvFCCBSBFCCB ,1,1,1,1 __ −−−− −=−

xkxkxk contOptioncontSBcontFCCB ,1,1,1 ___ −−− =−≥ (B.11) As a result, it is clear that

)_,( ,1,1,1,1 xkxkxkxk contOptionSBFCCBMaxOption −−−− −= xkxk SBFCCB ,1,1 −− −= (B.12) By mathematical induction, this completes the proof of (19).