AnalyzingConvertibleBonds_1980

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JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSISProceedings IssueVolume XV, No. 4, November 1980

ANALYZING CONVERTIBLE BONDS

Michael J. Brennan and Eduardo S. Schwartz*

The convertible bond is a hybrid security which, while retaining most ofthe characteristics of straight debt, offers, in addition, the upside potentialassociated with the underlying common stock. As a quid pro quo for the upsidepotential the convertible bond is typically subordinated to other corporate debtand carries a lower coupon rate than would an otherwise equivalent straight bond.

The value of a convertible, like that of a straight bond, depends upon thecoupon rate and maturity as well as the risk of default, which in turn reflectsboth the underlying asset risk of the issuing firm and the security provisionsof the bond indenture. Also like a straight bond, the value of the convertibleis influenced by the prevailing level of interest rates. However, the call pro-vision which is standard in most straight bond indentures assumes greater im-portance in the case of the convertible because of its potential use in forcingconversion. Finally, the value of the convertible will depend upon the valueof the conversion privilege: this is a function of the risk and capital struc-ture of the firm, the payout policy of the firm, the call policy to be pursuedby the firm, and the terms on which the bond may be converted into common stockas well as the current stock price.

The equilibrium value of a convertible bond is defined as that value whichoffers the potential of arbitrage profit neither to purchaser nor to short seller,given that the bondholder pursues an optimal strategy with respect to conversionand that the firm pursues an optimal policy with respect to calling the bonds.Given the multifarious factors determining the value of a convertible, the valua-tion problem is extremely complex, so that it is difficult without a formalmodel for the issuing firm to make an assessment of the feasible trade-offs be-tween various characteristics of an issue: the conversion and call features,

Both authors. University of British Columbia. The authors are gratefulto Peter Madderom of the University of British Columbia Computing Centre forextensive programming assistance and to Mark Rubinstein for encouraging us todo this research. The research was supported by a grant from The Berkeley Re-search Program in Finance, University of California, Berkeley, and the paperwas first presented at the Berkeley Program in Finance Silverado Conference inMarch 1980.907


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coupon rate and maturity, etc. The obverse side of this problem is faced by theinvestor choosing between convertibles and common stock.

This paper develops a model of the pricing of convertibles which can be usedas an aid in making the above types of assessme nt. Since both the call and theconversion features are options exercisable by the firm and by the investor r e -spectively, it is not surprising that the appropriate mode of analysis is totreat the convertible bond as a contingent claim and to value it using the option-pricing techniques pioneered by Black-Scholes and Merton.

Convertible bonds have been analyzed previously by Brennan and Schwartz andby Inger soll. This paper differs from that earlier work both in allowing forthe uncertainty inherent in interest rates and in taking acount of the possibilityof senior debt in the firm's capital structure.

I. Call and Conversion StrategiesWe shall consider a firm which has outstanding senior debt, convertible

bonds and common stock. The total market value of the firm, V, is the sum of themarket values of these three securities.

(1) V = N B + N C + N SB C 0where B and N are market value of a straight bond and the number outstandingBrespectively; C and N refer similarly to the convertibles; N Q is the number ofBCshares of common stock before conversion of the bonds; and S is the stockprice before conversion. Similarly, after conversion of the convertible, themarket value of the firm is given by

AC(2) V = N B + (N + AN )SB 0ACwhere AN is the number of shares issued as a result of conversion and S is the

stock price after conversion has taken place.The conversion privilege is conventionally expressed either in terms of the

price at which the bonds are convertible into common stock, the conversion price,or in terms of the number of common shares into which each bond is convertible,the conversion ratio. Assuming that the par value of the convertible is $1,000,these two measures are related by

Conversion Ratio = 1000/Conversion Price.

Writing the conversion ratio as q, it follows that the total number of new

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shares issued on conversion, AN, is

(3) AN = N q

and the fraction of the total shares owned by the holder of each convertiblebond after conversion, Z, is

(4) Z = q/(N + AN) .

In elementary textbooks the conversion value of a bond is defined as theconversion ratio, q, times the preconversion stock price:

Q/->Conversion Value = q x S

The problem with the definition of conversion value is that it is not ingeneral equal to the value of the shares into which each bond is convertible if

ACall the bonds are converted: this value is given instead by q x S . SolvingBC ACequations (1) and (2) for S and S , the two definitions of conversion value1are:BC oConversion Value (1) = q x S = S~[V - N B - N C ] .ACConversion Value (2) = q x S = +

It may be seen that the two definitions are not equivalent, and that Con-version Value (1) depends on the actual value of the convertible bonds; in fact,the two definitions yield the same value only if the bonds are selling at theirconversion value. We shall use Conversion Value (2) because it both representsthe value of the bonds if they are all converted and it expresses the conversionvalue in terms of what is given or known, the values of the firm and the straightdebt, and the conversion ratio.Conversion Strategy

Convertible bondholders will always find it optimal to convert if the valueof the bond falls below its Conversion Value (2). Therefore, the value of theconvertible can never fall below the level so that

This involves the standard assumption that the value of the firm is inde-pendent of the capital structure.909


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(5) C * q S = Z(V - N B) .B

On the other hand it will never be optimal to convert if the value of thebond exceeds its Conversion Value (2) for this involves a sure value loss.Therefore the condition for optimal conversion is that the value of the bond isequal to its conversion value:

(6) C = Z(V - N B) .B

Note that condition (6) does not define the optimal conversion strategyuntil the bond is valued and C is determined. Yet the bond value itself dependson the optimal conversion strategy: therefore the bond value and the optimalconversion strategy must be determined simultaneously.Call Strategy

In determining its optimal call strategy for the convertible bonds themanagement of the firm is assumed to be concerned with maximizing the value ofthe original share s. The valu e of these shares is given from equation (1) by

(7) N QS = V - N B B - N C C

It is clear from this equation that, given the total value of the firm, V,the value of the shares is maximized by pursuing a call strategy which minimizesthe value of the conve rtibles. Such a call strategy is incompatible with allow-ing the convertible to rise in value above the price at which it is currentlycallable, CP. Hence under an optimal call strategy

(8) C S CP.

On the other hand, it will not be optimal to call the convertible when itsvalue is less than the call price, since this would confer a windfall gain onthe convertible holders at the expense of the stockholders. Therefore, under theoptimal strategy the convertible will be called when

(9) C = CP .

Again, equation (9) does not define the optimal call strategy until C isdetermined by solving the valuation problem which itself depends on the optimalcall strategy. It follows that the valuation problem and the optimal call andconversion strategies must be solved for simultaneously.

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CallPrice: CP

Conversion Value:Z(V -

Convertible Bond: Call and Conversion ConditionsFigure 1

Given that the senior debt is perfectly secured so that its value, B, isindependent of the total value of the firm, V, the shaded region of Figure 1represents the combinations of convertible bond and firm values that are con-sistent with the conversion value constraint (5) and the call policy constraint(8). At any time the bond is not convertible or is not callable, the relevantconstraint in Figure 1 is removed. Finally, if the senior debt is not riskless,the conversion value line will no longer be straight but will look like thedotted line shown in the Figure.

II. The ModelThe value of the convertible and the straight debt are assumed to depend

upon the total value of the firm, V, and the current interest rate, r. V af-fects the convertible bond value through its influence on the probability ofdefault (the asset backing of the bonds) and on the conversion value of thebonds; it affects the value of the straight debt only through the probabilityof default. The interest rate affects both bond values since it reflects therate at which certain future returns are discounted. Both the value of the firmand the interest rate are assumed to follow random processes. Over a short in-terval of time the change in the interest rate, Ar, is assumed to be given

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denote partial derivatives with respect to the respective arguments.The convertible value also satisfies:

The Conversio n Condition

(14) C(V, r,t) 5 Z(V - N B ( V , r , t ) ) .a

This is equation (5) rewritt en to show explicitly the dependenc e of bothbond values on V, r, and t.

The Call Condition

(15) C(V,r ,t) CP(t )

where CP(t) is the price at which the converti ble is callable at time t.The Maturity Condition

The convertible is assumed to mature at time T, prior to the matur ity ofthe senior debt. Its value at maturit y is given by

(16) ( = Z(V - NB (V ,r ,T )) if Z(V - N B(V,r,T)) 5 FB B

= F if Z(V - N B(V, r,t) ) S F $ i- (V - N B)C(V,r,T) / B N C B

= N (V " V 0 ) lf F * I (V ~ NB B 0 ) 5 c c= 0 if V < N B .

BThis condition states that the convertible holders receive the conversion

value if it exceeds the par value of the bond; then they receive the par v alueprovided that this does not exceed the value of the firm less the par value ofthe senior debt (B ) . If this condition is not satisfied, the firm goes bank-rupt and the convertible bondholder s are paid after the senior bond hold ers. Thematurity condition is illustrated in Figure 2 on the assumption that the seniordebt is selling at par.The Bankruptcy Co ndition

It is assumed that the convertible bond indenture is written so that thebondholders will receive a fraction k of the par value when the firm goes bank-rupt. This implies that the firm will go bankrupt if its value falls to the sumof the par value of the straight debt and k times the par valu e of the con vert-ibles:

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Z(V - N DB0,

Convertible Value at MaturityFigure 2

(17) C(V,r,t) = kF if V = N B + kN P.B CIII. Solving for the Convertible Bond Value

The convertible bond value is obtained by solving numerically the partialdifferential equation (13) subject to the conditions imposed by conversion (14) ,call (15 ), maturity (16) , and bankruptcy (17) . Note , however, that both theconvers ion and maturity c onditio ns involve the value of the senior debt at thetime of conversion or maturity, B(V,r,t), where t is equal to the time of ma-turity or convers ion. Denote by B*(V,r,t) the value of the senior debt giventhat the converti bles are no longer outstandin g. Then it can be shown by methodssimilar to those described in the Append ix that B*(V,r,t) satisfies the partialdiffere ntial equa tion (13) wit h Q(V,t) replaced by Q*(V,t) where

(18) Q*(V,t) I + D*(V,t) .B

Equat ion (18) gives recogn ition to the fact that whe n the convertibles hav ebeen eliminated from the firm's capital structure by conversion or maturity, theaggregate payout on the firm's securities will no longer include interest on the

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convertibles, and the aggregate dividend payment as a function of firm value willreflect the changed capital structure of the firm.

In addition, the value of the senior debt will satisfy the maturity valuecondition

(19) B*(V,r ,T') = B if B


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the convertible. The dividend payout parameters were based on the assumptionthat the firm's earnings follow a random walk, that the firm's overall cost ofcapital is 10 percent and that both the payout ratio and the corporate tax rateare 50 percent.

The convertible bond is assumed to be a $6m issue with an 8 percent couponand a 10-year maturity. It is callable after 5 years at 108 with the call pricedeclining in equal annual decrements to 100 at maturity. Each $1000 bond isconvertible into 18.52 shares of common stock so that the conversion price is$54.

The stock price at the time the issue is made is taken as $44.02 and theshort-term riskless rate as 15 percent. Given the assumed process for interestrates and the assumption that the pure expectations theory of the term structureholds so that A = 0, this is consistent with a yield to maturity on a default-free bond of 11.46 percent. The convertible bondholders are assumed to recover2/3 of the par value of their investment in the event of bankruptcy.

The differential equation was solved first assuming that the bond was neitherconvertible nor callable. The yield to maturity on this straight corporate bondwas 11.52 percent reflecting a .06 percent default premium over the correspond-ing riskless rate. Finally the convertible bond was valued and each $1000 bondwas found to have a market value at time of issue of $997.

Figure 3 shows the value of each $1000 bond at time of issue as a functionof the firm value and the stock price: the latter is derived by subtracting theaggregate market value of the bonds from the firm value and dividing by the num-ber of shares. The left-hand end of the curve represents bankruptcy when thevalue of the firm is $4m; this is equal to the bankruptcy value of the bonds,$6m x 2/3. The convertible value is a monotonically increasing function of thestock price, asymptotically approaching the conversion value: note that for theparticular parameter values chosen it is not optimal to convert the bond imme-diately for any firm value within the range considered.

Figure 4 shows the bond value as a function of the firm value and the stockprice at the time the bond first becomes callable for the same short-term risk-less rate. Note that the call condition now keeps the bond value below the callprice while the conversion condition keeps it above the conversion value. Forthe interest rate chosen the bond is called optimally when the conversion valueis equal to the call price; for lower interest rates the bond may be calledoptimally at lower firm values (see Figure 8) .

Figure 5 shows the effect of the interest rate on the bond value when thefirm value is $50 at time of issue. As the interest rate rises, the bond valueasymptotically approaches the conversion value: it never reaches it, however,

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TABLE 1PARAMETERS OF NUMERICAL EXAMPLE

1. The Interest Rate ProcessAr = a(p - r) + ra Zr r r

a = .13 per year\i = b% per yeara = 26% per year

2. Firm CharacteristicsAV = [Vyv - Q(V,t)]Q(V,t) = c^V + d 2 +

= .05 = -.12a = 20% per yearp = -0.01

CB = $.48m per year1 million shares of common stock outstanding. No senior debt.

3. Bond CharacteristicsIssue size:Coupon Rate:Conversion Price:Maturity:Callable after 5 yearsRecovery in Bankruptcy:

4. Environmental ParametersStock Price:Short-term Riskless Rate:10-year Riskless Rate:

$6m8%

$5410 years

2/3 of par value.

$44.0215%11.46%

10-year Corporate Bond Rate: 11.52%

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-oco

15.23T40.0 50.0 60.0

34.58 44.02 53.35Bond Value a t Time of Issue

r = 15%Figure 3

71.72100 .0 Firm Value ($ir89.81 Stock Price ($

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coCO

Ca l l P r i ce

00

.0

.02015

.0

.0 840.34 .

073

5044

.0

.3 460.53 .

079

8072

.0

.00100.0 Firm Value ($m)

90.00 Stock Pric e ($)Bond Value at T ime of F i r s t Ca l l

r = .15%Figure 4

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Call Price

Conversion Value

0.0 0.1 0.2 0.3 0.4 Short-Term0.5 Riskless Rate

Bond Value at First Call Date as a Function of theInterest Rate.

V = $50mFigure 6

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2015

i.0.2 3

4034

i.0.5 8

5044

.0

.0 260.53 .

035

80.071.72

1100.0 Firm Value ($ir89.81 Stock Price (i

Bond Value at Time of Issue fo r D iff e re nt Short-Term(r ) and 10 year (1) Riskless In te re st R ates.Figure 7

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since for this firm value the coupon on the bond exceeds the dividend rate onthe shares into which it is convertible. In Figure 6 the same relationship isshown at the time of first call; now the bond value is not only bounded frombelow by the conversion value, but is also bounded from above by the call price.Figures 7 and 8 show how the bond value is related to the firm value for dif-ferent interest rates at time of issue and time of first call respectively. Asone would expect, the bond value is a decreasing function of the interest rate.

V. Sensitivity AnalysisIn addition to providing an estimate of the value of a particular convert-

ible bond, the model may be used to assess the sensitivity of the bond value tochanges in environmental, firm, and bond parameters. The results of some rep-resentative calculations are shown in Table 2: in this table various parametersare changed one at a time.

TABLE 2COMPARATIVE STATICS OF BOND VALUE AT ISSUE (PARAMETER

VALUES OF BASIC EXAMPLE IN PARENTHESES)

Basic ExampleNon-callableNon-callable, non-convertibleStock Price: $48.42 ($44.02)10-year Riskless Rate: 12.5% (11.4%)ay: .22 (.20)d = .055 (.050); d =-.132 (-.12)Coupon: 8.8% (8.0%)Conversion Price: $48.60 ($54.00)Bankruptcy Value: $333 ($666)First Call Date: 6 years (5 years)First Call Price: $1188 ($1080)

First, it may be seen that removal of the corporation's right to call thebond for redemption would increase its value by 3.5 percent; on the other hand,removing the conversion privilege as well, which would make the bond a straightnon-callable bond, would reduce the bonci value by 21.1 percent. Thus, in thiscase the straight bond value at issue is about 79 percent of the equilibriumissue value.

The next three lines of the table show the responsiveness of the bondprice to environmental parameters beyond the control of the firm: of these, the

924

Bond Value997

1032787

1045973

1007987

10281047997

10051005

% Chan-3.5

-21.14.9

-2.31.0

-1.03.25.0

-0.00.80.8


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stock price is the most critical, a 10 percent increase in the stock price rais-ing the bond value by 4.9 percent. As one would expect, the bond value decreasesas the long-term rate of interest increases. In this example the bond valuerises with the risk of the firm, a : it may be noted that the effect of an in-crease in firm risk is to raise both the probability of bankruptcy (which de-creases the bond value) and the value of the conversion option (which raisesthe bond value). In this instance the effect of the latter outweighs the for-mer. An increase in the firm's dividend payout also reduces the bond valuesince this both increases the probability of bankruptcy and reduces the expectedrate of appreciation in the stock price, decreasing the value of the conversionprivilege.

Turning to the characteristics of the bond itself, only the coupon rateand the conversion price have a major effect on the bond value, a 10 percentreduction in the conversion price raising the estimated bond value by 5.0 percent,and a 10 percent increase in the coupon rate raising the value by 3.2 percent.Changes in the security provisions have a negligible effect, reflecting the lowprobability that bankruptcy will occur, and small changes in the call price anddate of first call have only modest effects on the bond value.

One use of the model is to assess the feasible tradeoffs between differentbond characteristics. Thus, assuming that the relationships are approximatelylinear, Table 2 suggests that a 10 percent reduction in the coupon rate could beoffset by a 6 percent reduction in the conversion price (10 percent x 3.2/5.0),or a 31 percent reduction in allowed future dividend payouts, and similar com-parisons may be made between other parameters.

VI. Non-Stochastic Interest RatesThe advantage of the model developed in this paper over the earlier models

of Brennan and Schwartz and Ingersoll is that it allows explicitly for uncer-tainty about future interest rates. On the other hand, this makes the modelmore complex and substantially increases the computational cost of solving thedifferential equation. It is therefore of interest to determine the error thatwould be caused by assuming a single known constant interest rate.

For this purpose, the 10-year riskless rates corresponding to short-termriskless rates of 0, 5, 10, 15 and 20 percent were calculated. The bond wasthen revalued assuming a constant interest rate equal to the 10-year rate.Comparison of the resulting bond value with that generated by the model allow-ing for uncertainty in interest rates yielded the valuation errors reported inTable 3 for different interest rates and firm values. A negative value in thistable implies that the certain interest rate model value exceeds the uncertaininterest rate model value.

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TABLE 3ERRORS IN BOND VALUATION (%) GENERATED BYA MODEL WITH NON-STOCHASTIC INTEREST RATES

Short-term Riskless Rate:10-year Riskless Rate:V ($m)

205080

03.12

3.613.822.15

55.96

-1 . 61-0.42

0.07

1 08.74

-1 . 60-0 . 37

0.00

1 511.46

-1.26-0.40-0.07

2 014.17

0.73-0.63-0.15

The table suggests in a striking manner that for a reasonable range ofinterest rates the errors from the certain interest rate model are likely to beslight, and, therefore, for practical purposes it may be preferable to use thissimpler model for valuing convertible bonds.

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APPENDIXTHE PARTIAL DIFFERENTIAL EQUATION FOR CONVERTIBLE BONDS

The value of a convertible bond is subject to two distinct sources of un-certainty: the uncertainty in the value of the underlying firm val ue, V, andthe uncertainty inherent in the rate of interest. It is necessary for us firstto investigate the interest rate uncertainty.

Taking the limit of equation (10) as the time interval approaches z ero,the process for the interest rat e, r, is given by the stochastic differ entialequation

(Al) dr = ct(y - r)dt + ra dZ .r r r

The value of any default-free discount bond will be a function only of theinterest rate and time. Let G (r,t) (i = 1,2) denote the mark et values of anytwo such bon ds. Then, by Ito's Lem ma, the instantaneo us change in the value ofsuch a bond is given by

(A2)

where, = (G. + G a(y - r) + 1/2G r2a2)/Glb. t r r rr rl

y = G ra /^. r rlConsider a zero net investment portfolio formed by investing an amount

x. (i = 1,2) in bond i and borrow ing an amou nt (x + x ) at the instantaneoi:interest rat e, r. The instantaneous return on this portfo lio is

[x (y - r) + x (y - r)] dt+ (x a + x a )dZ .1 G 1 2 G 2 1 G x 2 G 2 r

If x and x are chose n so that (x a + x a ) = 0, then the ret urn on1 2 1 G 2 Gthe portfolio is non-sto chastic . To avoid the possibil ity of arbitrag e profits

the return must be zero. This will be so if and only if

(A3) y -r y -r1 2G l G 2

where A(r,t) is the same for all bon ds. (A3) says that the reward to variability

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ratio for all default-free bonds is the same. With this result we are now pre-pared to derive the partial differential equation for convertibles (or for thesenior firm debt) by a similar argument.

Taking the limit of equation (11) as the time interval approaches zero,the instantaneous change in the value of the firm is given by

Dropping the superscript i from equation (A2) the instantaneous rate ofreturn on any default-free bond is given by

Finally, given that the only two sources of uncertainty are the value ofthe underlying firm, V, and the interest rate, r, it follows that the value ofa convertible bond may be written as a function of these two variables and time:C(V,r,t). Then using Ito's Lemma, the instantaneous rate of capital gain onthe convertible is given by

wherey C = [Ct + Cr a ( Pr " r) + CV ( UV "

and p is the instantaneous correlation between dZ and dZ .Consider forming a zero net investment portfolio by investing amounts

x , x , x in the convertible, the default-free bond and the firm respectively,and borrowing (x + x + x ) at the instantaneous interest rate r. The instan-C G Vtaneous return on this portfolio is then, using (A4), (A5) and (A6) and takingaccount of the rate of coupon payment on the convertible cF,

(A7) t x n ( ^ + c F / C " r ) + xr{V~ " r ) + X ( u - r ) ] d tC C LJ o V V

[ x c c v v V c + xv av ]d zvx G a G ] d Z r .

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Let

(A8)

Then the instantaneous rate of return on the portfolio is certain, and toavoid the possibility of arbitrage profit it must be equal to zero so that

(A9) xc ( PC + C F / C " r ) + XG C yG " r> + XV ( yV "

Eliminating x , x and x between (A8) and (A9) and using the definitionC V Gu , we obtain

(A10)

+ C a(y - r) + C (rV - Q(V,t)) + cF - rC + CC ra

- -T^-^r - r) = 0. GFinally we may use equation (A3) to write the last term as -C A a which

is then equivalent to equation (13) of the text.

REFERENCES[1] Black, F., and M. Scholes. "The Pricing of Options and Corporate Liabili-

ties." Journal of Political Economy, Vol. 81, Number 3 (May-June 1973),pp. 637-654.

[2] Brennan, M., and E. Schwartz. "Convertible Bonds: Valuation and OptimalStrategies for Call and Conversion." Journal of Finance, Vol. 32, Number5 (December 1977), pp. 1699-1715.[3] Ingersoll, J. "A Contingent-Claims Valuation of Convertible Securities."

Journal of Financial Economics, Vol. 4, Number 3 (May 1977), pp. 289-322.[4] Merton, R. "The Theory of Rational Option Pricing." Bell Journal ofEconomics and Management Science, Vol. 4, Number 1 (Spring 1973), pp. 141-

183.

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