Convertible Bonds Valuation under a Spread-DrivenDefault

Nicolas Benielli1

Crisil Irevna UK LtdSt. Clement’s House, 27-28 Clement’s Lane, London EC4N 7AE,

London, United Kingdom

Abstract

We propose a convertible bond pricing model that may depend on equity and short ratecredit spread, assuming a log-normal dynamics with bounded volatility for the spread anda negative equity-spread correlation. The numerical procedure is a discrete-time approachbased on multinomial trees and a backward induction algorithm. We show the steps forthe construction of the trees, and then we provide some numerical examples of the model.We also show the convergence and a sensitivity analysis.

1Senior Quantitative Analyst, nicolas.benielli@crisil.com

1 Introduction

Some of the challenges involved in pricing convertible bonds (CB) are mainly due to the factthe they are hybrid instruments depending on at least three factors and their interactions thatrequires the modelling of three correlated stochastic processes: the underlying equity price, therisk-free interest rate and the credit risk. Of all the risk factors affecting the CB price themost important ones are the equity price – the basic underling variable– and the credit risk,and there are indications that stochasticity of interest rates is secondary (see [Brennan andSchwartz(1980)], and [Grimwood and Hodges(2002)]).

We propose here a two factor model where the CB price is a function of the stock price andthe short credit spread, and we use a multinomial pricing scheme based on results of [He(1990)].The contributions of this work are as follows:

• A model for pricing CB with negatively correlated equity and credit spread risks.

• The dynamics of the spread is assumed to satisfy a log-normal diffusion with boundedvolatility.

• Default is triggered the first time that the spread hits an upper barrier.

• The numerical pricing scheme is a multinomial tree that allows us to remain in a non-arbitrage context and it is flexible enough to handle any real-world restrictions including,besides conversion, call and put options, discrete cash flows (coupons or dividends) andrecovery rates.

The remainder of this paper is structured as follows. Section 2 briefly review the literatureon CB models. In Section 3 we introduce our model and the numerical pricing schema. Resultsof numerical applications are shown in Section 4, and the last section contains conclusions andcomments on future research.

2 Literature review

The first rigorous theoretical bases to value convertible bonds were developed independentlyby [Ingersoll(1977)] and [Brennan and Schwartz(1977)] who apply a firm–value approach,when [Brennan and Schwartz(1980)] extend their model by incorporating a stochastic risk–freerate as a second factor, but conclude that for practical purposes the stochasticity of the ratecan be disregarded.

Modern valuation models introduce credit risk using a risky discount rate instead of arisk–free rate. [McConnell and Schwartz(1986)] assume a constant credit spread; Bardham et. al.(1994) choose a discount rate given by the weighted average of the risk-free rate and the discountrate of a corporate bond of the same credit quality, and [Tsiveriotis and Fernandes(1998)]

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also use a constant spread, but split the convertible into two components with different creditqualities: a share–only part discounted at risk–free rate, and a debt–only part subjected to creditrisk and discounted at risk–free rate plus a constant spread. As mentioned in [Ygitbasioglu andAlexander(2006)], none of these papers deals explicitly with default, and assume implicitly thatthe stock price does not fall at bankruptcy.

Following the introduction of reduced–form models in [Jarrow and Turnbull(1995)], [Jarrowet al.(1997)Jarrow, Lando, and Turnbull], [Lando(1998)], [Madan and Unal(1998)] and [Duffie andSingleton(1999)], more recent papers on CB models incorporate default as a jump process with astochastic intensity. The driving factor in these models is the equity price, and in most of themthe occurrence of a jump to default is linked with a downward jump of the equity price. [Davisand Lischka(2002)], [Takahashi et al.(2001)Takahashi, Kobayashi, and Nakagawa], [Hung andWang(2002)], [Andersen and Buffum(2004)] and [Chambers and Lu(2007)] include the risk–freerate as an additional factor. [Ayache et al.(2003)Ayache, Forsyth, and Vetzal] provide a one factormodel that extends [Tsiveriotis and Fernandes(1998)] by considering the implications of a defaultevent, modelling this as an exogenous Poisson process.. [Ygitbasioglu and Alexander(2006)]employ a multifactor reduced model with uncertain long term equity volatility, and [Das andSundaram(2007)] present a three factor model (equity, risk–free rate and intensity process fordefault) that enjoy properties of both structural and reduced–form models, and accounts forcorrelations between the three sources of risk.

Our model is motivated both by [Cathcart and El-Jahel(1998)], who propose that defaultoccurs when some signaling process hits a constant barrier, and also by the observation in [Duffieand Singleton(1996)] that credit spread reflects also non default factors. Further studies alongthis line address the problem of how market prices credit risk, and some of the authors produceempirical evidence of non–default determinants of credit spread, both of corporate bonds( [Delianedis and Geske(2001)], [Driessen(2005)], [Chebbi(2009)]) and of corporate credit swaps( [Longstaff et al.(2005)Longstaff, Mithal, and Neis], [Bernt et al.(2005)Bernt, Douglas, Duffie,Ferguson, and Schranz] and [Liu et al.(2006)Liu, Longstaff, and Mandell]). A summary onestimates of credit spread components is given in [Chebbi(2009)]. One implication of theseresults is that reduced–form CB models disregard market premia that compensate investors forbearing non default components in credit spread. This realization togheher with the implicitassumption of reduced–form CB approaches that default occurs exclusively as a jump, led usto choose a structural two factor model with stock price and stochastic credit spread as statevariables that allows

i. To consider non default components of risk reflected in the spread.

ii. To incorporate the correlation between market risk and credit risk.

iii. To address not unusual situations in which default does not occur unexpectedly (jump),but involve a progressive increase of spread associated with a gradual decrease of the stockprice.

As an example, Figure 1 shows the equity price and the spread of an AMR corporate bond,its estimated correlation being −0.81, by far more significant that the equity–risk-free rate

3

correlation (see, e.g [Chambers and Lu(2007)] that report an equity–risk-free rate correlation of0.1).

Figure 1: AMR Stock Price and Spread

01/01/2006 01/01/2008 01/01/20100

10

20

30

40 S h

AM

R S

tock

Pric

e ($

)

500

1000

1500

2000

2500

3000

It is worth pointing out that we have not found so far a thorough empirical research on therelationship between the credit quality of corporate bonds and the correlation equity–spreadnor any CB model that contemplates this correlation.

References in the literature on credit risk related to our paper are, among others, [Longstaffand Schwartz(1995)], [Cathcart and El-Jahel(1998)], [Tauren(1999)], [Cathcart and El-Jahel(2002)],[Hui and Lo(2002)], [Peterson and Stapleton(2003)], [Finnerty(2007)], [Cane de Estrada et al.(2005)Cane deEstrada, Cortina, Ferro-Fontan, and di Fiori] and [Chebbi(2009)].

3 The model

In this section we derive a two factor model for the valuation of convertible bonds: the priceof the underlying equity – the main driving factor– and the short credit spread implied bynon-convertible bonds of the same issuer for similar maturities. We work in a continuous timeframework where S(t) denotes the price of the equity at time t, rd(t) is the defultable short rateif a default has not occurred until time t, r(t) is the risk-free rate, and the credit spread h(t) isdefined as

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h(t) = rd(t)− r(t).

Our assumptions are:

Assumption 1 At any time t it is possible to borrow and lend cash at a known constantrisk-free interest rate, and to buy and sell any amount of stock (including short selling).

Assumption 2 At any time t risk-free discount bonds and corporate bonds of all maturitiesare are available, and can be traded (short and long) without any of the normal costs (bid-askspread or transaction consts).

Assumption 3 h(t) is positive and bounded by a finite upper barrier 0 < h(t) ≤ H <∞∀t, and a default occurs the first time that h(t) hits the barrier.

Assumption 4 The dynamics of S(t) and h(t) are governed by diffusion equations

{dS(t) = µS(t)Sdt+ σS(t)SdZ1,

dh(t) = µh(t)hdt+ σh(h, t)hdZ2,(1)

with

σh(h, t) = min (H, h(t))σh(t), (2)

where Z1 y Z2 are negative correlated standard Wiener processes

dZ1dZ1 = dZ2dZ2 = dt

dZ1dZ2 = ρdt, −1 < ρ < 0,

ρ and H are constants, and µS(t), µh(t), σS(t) and σh(t) are deterministic functions of time.Note that imposing an upper bound to a log–normal diffusion is equivalent to define a boundedvolatility process.

Assumption 5 If a default event happens at time 0 < t ≤ T , the equity value reduces to afraction of its level S− immediately before default

S+ = εS−, 0 ≤ ε < 1,

and the recovery value of the convertible is a fraction Q(t), 0 ≤ Q < 1, of par value plus thelast coupon F + CT .

Assumption 6 r(t) is a known function of time.

5

To simplify the calculations we make the additional hypothesis that µS, µh, σS and σh areconstants. Along with these assumptions on the parameters we make the additional assumptionof flat yield curves. Beside these assumptions, the typical CB final condition, contractualfeatures and default condition are held:

Final condition

V (S, h, T ) = F + CT (3)

if default has not occurred prior to maturity, introduces a discontinuity in V since theconversion right guarantees that at T−, immediately before maturity, the holder still keeps theright to convert

V (S, h, T−) = max (nS, F ) . (4)

Contractual features

In praxis coupons are usually paid at discrete equally spaced time points. We have thereforea jump condition on coupon dates tc

V (S, h, t−c , T ) = V (S, h, t+c , T ) + Ctc , (5)

Call provisions give the issuer the right to redeem a convertible before maturity at apredetermined specified amount Mc(t) that may be a function of time. Arbitrage argumentsimposes the restriction

V (S, h, T ) ≤ max (nS,MC) .

Likewise, the holder may have a put option to require the issuer to repurchase the bond at apredetermined amount MP (t) that may also depend on time

V (S, h, T ) ≥ max (nS,MP ) .

Default Condition

Admitting that even at default the holder keeps the right to convert, and may choose toreceive the recovery rate Q(r, t) or the fraction of n defaulted stocks, implies that

V (S,H, t, T ) = max (nεS,Q) . (6)

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3.1 Multinomial trees – discretization schema

To determine the no-arbitrage value of a convertible bond with embbeded call and put and discretecoupon payements, we adapt and apply the discretisation method introduced by [He(1990)].This model has been developed for a complete market of N stocks and a numeraire in whichthe economy may evolve to N + 1 states in each time step, and its weakly convergence tocountinuous diffusion precesses has been proved ( [Chen et al.(2002)Chen, Chung, and Yang]use a similar valuation model for multi–asset option). We outline here the relevant features ofHe’s method to contribute to the understanding of our calculations. The time interval [0, T ] isdivided in n subintervals of equal lengths, the distance between two periods being ∆t = T/n.The N stocks can take (N + 1) different states on each time step, and X i designs the randomvariable

X i =(xi1, x

i2, . . . , x

iN+1

),

that takes the value xij, (j = 1, 2, . . . , N) in the state j with probability 1N+1

, such that itsexpected value is zero its variance is equal 1

E[X i]

=1

N + 1

N+1∑j=1

xij = 0. (7)

V ar[X i]

=1

N + 1

N+1∑j=1

(xij − E

[X i])2

= 1. (8)

Additionally, it is required that the covariance matrix between two random variables X i yXk is zero

Cov[X i, Xk

]=

1

N + 1

N+1∑j=1

(xij − E

[X i]) (

xkj − E[Xk])

= 0 (9)

In our problem we must deal with the uncompletness of the market, since the spread is nota tradable asset, and risk–neutral valuation methods can not be applied for pricing derivatives.To obtain a unique price we need the risk adjusted processes instead the real ones given by(1), i.e. market prices of risk (MPRs) different from zero are required. Under the martingalemeasure defined by MPRs, the stock price and the spread satisfy the risk-adjusted dynamics

{dS = rSdt+ σSSdW1,

dh = (µh − σhρλ1 − σh√

1− ρ2λ2)hdt+ σhhρdW1 + σhh√

1− ρ2dW2.(10)

where dW1 y dW2 are uncorrelated Wiener processes,

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λ1 =µS − rσS

and λ2 is the market price of credit spread risk.

Now we can use He’s discretization method with N = 2, approximating increments of theWiener processess W1 and W2 by two random variables and three possible states for each one ofthem. It is easy to see that the following choice satisfies conditions (7), (8) and (9)

X1 = (x11, x12, x

13) =

(1√2,− 2√

2,

1√2

),

X2 =

(x21, x

22, x

23) = (

√3√2, 0,−

√3√2

)

The stochastic evolution of (10) is discretised and approximated by trinomial processes of nperiods

Sk+1 =

SkαS1

SkαS2

SkαS3 ,

(11)

hk+1 =

hkαh1

hkαh2

hkαh3 ,

(12)

where k + 1 is a discrete time step (0 ≤ k ≤ n− 1), and constants αS and αh that multiplySk y hk at each time step k are defined by

αS1 = 1 + r∆t+ σS√3√2

√∆t,

αS2 = 1 + r∆t,

αS3 = 1 + r∆t− σS√3√2

√∆t;

αh1 = 1 + bh∆t+√

1− ρ2σh 1√2

√∆t+ σhρ

√3√2

√∆t,

αh2 = 1 + bh∆t−√

1− ρ2σh 2√2

√∆t,

αh3 = 1 + bh∆t+√

1− ρ2σh 1√2

√∆t− σhρ

√3√2

√∆t,

(13)

with

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bh = µh − σhρλ1 − σh√

1− ρ2λ2.

In order to comply with the need that the tree recombines we assume that λ2 is constant.This is a critical feature to maintain the computational tractability. Figure 2 illustrates thetwo first time steps for the recombining equity tree (the recombining feature implies that thenumber of nodes at each time step k reduces to (k + 1)(k + 2)/2. Furthermore, algorithms tocompute S and h at these nodes are based on simpler expressions),

Sk = S0αk−jS1

αj−iS2αiS3,

hk = h0αk−jh1

αj−ih2αih3,

for j = 0, . . . , k and i = 0, . . . , j. This means that to build trees for S and h we may firstconstruct trees for the multiplying constants, and then multiply them by the initial values of Sand h, respectively.

Figure 2: Two steps tree for S

Numerical example of tree construction

Consider a convertible bond with maturity three years issued by a corporation that also issuesstocks and a 3-years corporate bond with parameters given in Table 1. For simplicity weillustrate the construction of stock and spread trees with only three time steps.

We initialise the algorithm by setting αS = 1 and αh = 1 at time zero, and we start bycomputing the constants values at the first time step, given by (13): αS1 = 1.42, αS2 = 1.05,αS3 = 0.68 and αh1 = 1.16, αh2 = 0.73, αh3 = 1.45. Values at the six nodes in the second stepcan be calculated by formula

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Table 1: Parameters of S and hS h

Initial value 100 0.08Drift 0.05 0.1173

Volatility 0.3 0.3ρ -0.4

αk−jS1

αj−iS2αiS3

and αk−jh1

αj−ih2αih3, (14)

for j = 0, . . . , 2, i = 0, . . . , j. We use again (14) with k = 3, j = 0, . . . , 2, and i = 0, . . . , j tocompute values at the ten nodes in the third time step. Table 2 shows the three steps treesfor the constants, and Table 3 contains the trees for S y h. This construction can be easilyextended to n steps.

Table 2: Trees for constants (a) aS and (b) ah

Step k 1 2 3 1 2 3

0.32 3.100.49 1.550.75 0.771.16 0.39

0.47 0.66 2.13 2.480.72 1.02 1.06 1.241.10 1.56 0.53 0.62

0.68 0.97 1.37 1.45 1.70 1.981.05 1.49 2.11 0.73 0.85 0.99

1 1.42 2.01 2.85 1 1.16 1.36 1.58

(a) (b)

3.2 The two factor trinomial tree and the backward induction algo-rithm

Once we have got trees for S and h, to calculate the CB price, V , we proceed according to thefollowing steps:

1. Given the terminal values of S and h, we start by computing the values of V at the final

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Table 3: Trees for (a) S and (b) h

Paso k 1 2 3 1 2 3

31.80 0.248348.92 0.124075.25 0.0619115.76 0.0309

46.59 66.04 0.1702 0.198271.67 101.59 0.0850 0.0990110.25 156.27 0.0424 0.0494

68.26 96.76 137.14 0.1167 0.1359 0.1583105.00 148.83 210.95 0.0583 0.0679 0.0790

100.00 141.74 200.90 284.77 0.0800 0.0932 0.1085 0.1264

(a) (b)

nodes, given by (3) if a default has not occurred until maturity, and by (6) when defaulthappens, i.e. at the nodes for which h > H,

2. Backward induction: We go back from k = (n− 1) to k = 1, and calculate the value Vi,kof the convertible bond at node i on time k by

Vi,k = max(nS, P,min

(C, Vi,k

))for hi,k < H,

Vi,k = max (nεSi,k, Q) for hi,k ≥ H,

for 1 ≤ i ≤ (k+1)(k+2)2

, where Vi,k is the average between the tree children nodes of thenode i at time step (k+ 1), discounted at risk-free rate. During the non-put period P = 0,and during the non-call period C −→∞. When k coincides with a coupon date the jumpcondition incorporates coupons in the tree.

4 Numerical results

We first analyse the convergence of the method for a convertible without optionalities and withparameters given in Table 4. Convergence is achieved at about 150 time steps, lower than thenumber of steps required for the convergence of traditional tree models. Figure 3 shows theconvergence results. The initial oscilation is explained by the presence of the spread barrier(a similar behaviour is found in binomial tree models for barrier options, see, e.g. [Dermanet al.(1995)Derman, Kani, Ergener, and Bardhan]).

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Table 4: Parameters for the first numerical exampleT 3 years

Convertion ratio 1.0Recovery rate Q 0.1

Face Value F 100Coupon rate 0.05 annualized

Coupon frequency 2 payments per yearr 0.05S0 100σS 0.3µS 0.08h0 0.08µh 0.03σh 0.3ρ -0.7η 0.1

Hdefault 0.15λ 0.01

Figure 3: CB price as a function of the number of time steps n

0 50 100 150 200 250 30080

90

100

110

120

In the second numerical example we incorporate coupon payments and call and put options.The input parameters shown in Table 5 constitutes the base case, and the price obtained is112.9071. Results of CB price sensitivities are displayed in Table 6.

Starting from the base case, we measure the CB price sensitivities to volatilities, risk–freerate and recovery rate. The price increases with σS because S may reach higher values turning

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Table 5: Parameters for the second numerical exampleT 3 years

Call 115 in years 1 and 2Put 110 in year 2r 0.05

Conversion ratio 1.0Recovery rate Q 0.1

Face value F 100Coupon rate 0.05 annualized

Coupon frequency 2 payments per yearη 0.1S0 100σS 0.3µS 0.15h0 0.08µh 0.1σh 0.3ρ -0.4

Hdefault 0.15λ 0.01

Table 6: Sensitivity analysisr CB Price σS CB Price σh CB Price Q CB Price

0.030 112.9145 0.22 109.0792 0.22 121.6614 0.1 112.90710.035 112.9125 0.24 110.0550 0.24 117.8221 0.2 113.01300.040 112.9105 0.26 110.9816 0.26 115.8134 0.3 113.15110.045 112.9085 0.28 111.9257 0.28 114.2507 0.4 113.30590.050 112.9071 0.30 112.9071 0.30 112.9071 0.5 113.4852

into higher conversion values. The reverse holds for σh: the price decreases because a higherσh implies higher probability that the spread hits the barrier. Rising rates are accompaniedby declining CB prices, as expected, but the absolute variations of prices are lower than thevariations caused by changes in σS and σh. This result is consistent with the findings of [Brennanand Schwartz(1977)] and [Grimwood and Hodges(2002)]. Also as expected, prices are higher forhigher values of Q.

5 Conclusions

In this article we have developed a two-factor model for valuing CBs. Besides the equity as statevariable, we have included the credit spread not only as signaling process but also to considerother risk factors which are not reflected in the commonly used default risk. In our model, both

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stochastic process for S and h are negatively correlated. Also, we assume a log-normal dynamicsfor the credit spread. The numerical schema is based on multinomial trees, following He (1990).We have shown the algorithm to construct the correlated trees for S and h, and the backwardinduction algorithm to value CBs. Both numerical results and sensitivity analysis are reasonableand behave as expected. Also, the convergence is achieved in about 150 time steps. The modelcan be extended to include more variables, for instance stochastic interest rate and volatility,but our discrete schema is restricted to recombinant trees. Thus, we believe that other methodscould be more appropiate in order to include more state variables.

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