AN INTRODUCTION TO

STRUCTURAL

CREDIT-EQUITY MODELS

Amelie HüttnerXAIA Investment GmbHSonnenstraße 19, 80331 München, Germanyamelie.huettner@xaia.com

October 28, 2014

Abstract This article aims to give an introduction to the concept of struc-tural credit-equity models, where both debt and equity productsare treated as derivatives on the underlying firm value. Thesemodels allow for the valuation of credit and equity instrumentsin a unified framework. Moreover, when taxes and bankruptcycosts are incorporated, they can be used to identify a firm’s op-timal capital structure. Starting with the first structural model in-troduced by Merton (1974), several diffusion-based models thatextend the concept to a more realistic framework are surveyed.

1 Introduction Structural models are a powerful tool in Quantitative Finance.They allow for a unified pricing of credit and equity products, andalso consider the possibility of default. When the perfect marketassumptions are relaxed, they even allow for an answer to thequestion of optimal capital structure. The central idea is to modelthe firm’s assets with a stochastic process and then treat all thefirm’s securities, debt and equity instruments, as derivatives onthe total firm value. Default happens when the firm’s asset valuefalls to a certain level. Thus, in contrast to the intensity-based ap-proach to credit-equity modeling presented in a previous article,see Mai (2012), structural models offer an economic explanationfor the default time. On the other hand, they are usually morecomplicated when it comes to the calibration to market data.

This article attempts to give an overview of several structuralmodels based on Brownian motion as well as the developmentof the central ideas in this field:

- From the first structural model proposed by Merton (1974),which employs the framework introduced by Black, Scholes(1973) for option pricing on firm value level, ...

- ... continuing with its extension by Black, Cox (1976), wheredefault happens when the firm’s asset value falls below a cer-tain barrier, ...

- ... to the relaxation of perfect market assumptions and thequestion of optimal capital structure, discussed in a series ofpapers by Leland (Leland (1994a), Leland (1994b), Leland,Toft (1996)).

- Finally we briefly discuss the CreditGrades model by Fingeret al. (2002), which extends the concept to a random defaultbarrier.

All models presented in this chapter assume that, under the risk-neutral measure, the firm’s asset value (Vt)t≥0 follows a geo-

111

metric Brownian motion of the following form (unless otherwisestated):

Vt := V0eXt , Xt := θt+ σWt, t ≥ 0, (1)

θ := r − δ − σ2

2.

Here, (Wt)t≥0 is a standard Brownian motion, σ is the instanta-neous return volatility, r is the risk-free interest rate and δ rep-resents the proportional payout rate to investors, comprised ofdividend and interest payments. Claims on the firm’s asset valuewill be priced using the risk-neutral valuation approach.

Before discussing the models stated above, we start with a shortreview of several important properties of Brownian motion, whichare essential to the derivation of survival probabilities and pricingformulas.

2 Mathematical background Let (Wt)t≥0 denote a standard Brownian motion, Z a standardnormal random variable, and T > 0. Considering the Brownianmotion with drift (Xt)t≥0 in (1) we know

P (XT > x) = P

(Z >

x− θTσ√T

)= Φ

(−x+ θT

σ√T

), x ∈ R,

where Φ(.) denotes the standard normal distribution function.Its running minimum is distributed according to

P

(min

0≤t≤TXt > x

)= Φ

(−x+ θT

σ√T

)− e

2θxσ2 Φ

(x+ θT

σ√T

), (2)

for x < 0, see Musiela, Rutkowski (2005, Lemma A.18.2). Thejoint distribution of (Xt)t≥0 and its running minimum is given by

P

(XT > x, min

0≤t≤TXt > y

)= Φ

(−x+ θT

σ√T

)− e

2θy

σ2 Φ

(2y − x+ θT

σ√T

),

(3)

for y ≤ 0, y ≤ x, and x ∈ R; see Musiela, Rutkowski (2005,Proposition A.18.3).

Now define τb := inf{t ≥ 0 : Xt = b} for some b ∈ R. Shreve(2004, Theorem 8.3.2) gives the Laplace transform of τb as

E[e−uτb ] = e−bσ2

(−θ+√θ2+2uσ2), b > 0. (4)

It will also be required to compute the expectation E[e−uτb1{τb≤T}]for b < 0. For this, we need the following result on the evalua-tion of Riemann–Stieltjes integrals with respect to the standardnormal distribution function, see Bielecki, Jeanblanc, Rutkowski(2008):∫ T

0eaxdΦ

(b− cx√

x

)=d+ c

2deb(c−d)Φ

(b− dT√

T

)+d− c

2deb(c+d)Φ

(b+ dT√

T

),

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with d =√c2 − 2a. We find

E[e−uτb1{τb≤T}

]=

∫ T

0e−usdP (τb ≤ s)

=

∫ T

0e−usd

(1− P

(mint≤s

Xt > b

))(2)=

∫ T

0e−usdΦ

(b− θsσ√s

)+ e

2bθσ2

∫ T

0e−usdΦ

(b+ θs

σ√s

)= e

bσ2

(θ−√θ2+2uσ2)Φ

(b−√θ2 + 2uσ2T

σ√T

)

+ ebσ2

(θ+√θ2+2uσ2)Φ

(b+√θ2 + 2uσ2T

σ√T

).

(5)

In the context of the structural models presented in the follow-ing, it will be required to work with the distribution of the mini-mum and the first passage time of the geometric Brownian mo-tion (Vt)t≥0 introduced in (1). The above results on (Xt)t≥0 canbe transferred to the corresponding results on (Vt)t≥0 as Xt =

ln(VtV0

)and ln(.) is a strictly increasing function, so {Vt ≥ b}

={Xt ≥ ln

(bV0

)}.

3 Merton’s model

3.1 Model specification Assumptions:The model presented in Merton (1974) is considered to be thefirst structural credit model. It makes use of some results alreadypresented in Black, Scholes (1973) in the context of option pric-ing and is based on similar assumptions:

1. There are no taxes, transaction costs or bankruptcy costs.

2. Individual investors’ investment decisions do not influence themarket price.

3. There is a unique interest rate r for borrowing and lending. Itis static and known in advance.

4. Short-selling is allowed.

5. Assets are infinitely divisible and traded continuously.

They will be referred to as the perfect market assumptionsfrom now on. Due to the absence of taxes and bankruptcy costsstated in Assumption 1, the total firm value is always equal to thevalue of the firm’s assets (adjusted for payouts).

Default event and capital structure assumptions:The condition for default in Merton’s framework is that the firmis not able to repay its debt, therefore it is necessary to assumea certain capital structure to be able to express mathematicallythe default probability and the terminal payoffs of the firm’s se-curities. The firm’s assets are composed of equity (Et)t≥0 anddebt instruments with total value (Dt)t≥0, where Dt is identifiedwith a single zero-coupon bond issue with maturity T and total

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face value B. All bonds are assumed to be of equal seniority.As there is no payout to the bondholders in Merton’s model, δ inEquation (1) represents the dividend rate. Default happens if thefirm is not able to repay the face value of debt at time T :

{VT ≤ B},

and in this case the bondholders receive the remaining assets ofthe firm. Note that default is only observable at the maturity dateof debt, since only then it is checked whether the firm can repayits debt or not.

3.2 Applications Due to the fact that the default event depends on the (admit-tedly very restrictive) capital structure assumption, and that de-fault is only possible at the debt’s maturity date, pricing of creditinstruments other than the zero-coupon bond under considera-tion does not make sense in Merton’s model. What can be doneis the valuation of debt and equity in total, and the derivation ofdefault probabilities for a fixed time point T .

Valuation of debt and equity:At T , the bondholders receive the amount B and the sharehold-ers regain control over the firm’s assets. If payment is not pos-sible, the firm defaults and is turned over to the bondholders.Looking at the terminal payoffs of creditors and equityholders,

DT = B1{VT>B} + VT1{VT≤B} = min{B, VT },ET = (VT −B)1{VT>B} = max{VT −B, 0},

one observes that the value of the firm’s equity corresponds tothe value of a European call on the firm’s assets when the firmpays a continuous dividend δ. One can therefore use an ex-tension of the well-known valuation formula presented in Black,Scholes (1973):

E0 = E[e−rTET ] = V0e−δTΦ(d1)−Be−rTΦ(d2),

d1 =ln(V0B

)+ (θ + σ2)T

σ√T

, d2 = d1 − σ√T .

(6)

Looking at the terminal value of debt, one notes that it can beformulated as DT = B−max{B−VT , 0}. Recalling the formulafor the value of a European put in the Black-Scholes frameworkwith continuous dividends, we determine the value of debt as:

D0 = E[e−rTDT ] = Be−rT − Put0,

Put0 = Be−rTΦ(−d2)− V0e−δTΦ(−d1),

⇒ D0 = Be−rTΦ(d2) + V0e−δTΦ(−d1).

(7)

Note that the balance sheet equality Vt = Et+Dt has to be mod-ified for the firm’s payouts: Et +Dt = Vte

−δ(T−t) for all t ≤ T .

Default probability:The risk-neutral probability of default is the probability that thefirm cannot repay its debt at T :

P (τ = T ) = P (VT ≤ B) = Φ(−d2), (8)444

where τ denotes the time of default. For T → 0, the default prob-ability approaches zero. Since a potential default is only consid-ered at maturity T , the probabilities P (τ ≤ t), t ≤ T , are notdefined at all.

Optimal capital structure: The Modigliani–Miller Theorem in pres-ence of bankruptcy:The Modigliani–Miller Theorem was derived in Modigliani, Miller(1958) and states that a firm’s value is invariant to its capitalstructure under certain conditions. The assumptions taken intheir paper are similar to those in Merton’s framework, with theadditional assumption that all investors view the income gener-ated by bonds as certain, regardless of issuer. In other words,they rule out the possibility of default. The statement is thenproven by looking at two firms with identical investment deci-sions, one with debt and the other purely equity-financed, and ar-guing via arbitrage arguments that those two firms have to havethe same value.Merton (1977) proves the Modigliani–Miller Theorem in the pre-sented framework that incorporates possibility of default: Essen-tially, considering two firms with identical investment decisions,one purely equity-financed (unlevered), the other with both eq-uity and debt (levered), he shows that one can replicate the debtand equity of the levered firm with portfolios consisting of equityof the unlevered firm and riskless debt. Combining the two port-folios yields the value of equity of the unlevered firm, which isdue to arbitrage arguments equal to the sum of debt and equityof the levered firm. Thus, the two firms have the same value.This proof is rather technical, but in our framework one seesclearly in the balance sheet equality that, asDt+Et = Vte

−δ(T−t)

at any time, and Vt is random, capital structure optimization in or-der to achieve a maximal firm value does not make sense.

Modigliani, Miller (1958) already identify taxes as a possible fac-tor that invalidates their argumentation, and indeed we will see inSection 5 that the firm value is no longer independent of the cap-ital structure when taxes and bankruptcy costs are introduced.

3.3 Possible extensions In Merton’s framework, the event of default {VT ≤ B} has avery simple form. If we choose a more complicated model forthe firm’s debt than a single zero-coupon bond, it is no longerpossible to express the default probability as simple as in (8),which significantly hinders the pricing of securities. Then we alsoneed to model how the remaining assets in case of default willbe distributed among different debt instruments.However, some of the assumptions on the debt structure can berelaxed, for example the single homogeneous bond issue canbe replaced by a bond issue containing different seniorities, asshown in Black, Cox (1976, p. 358f).Merton (1974) also considers debt in form of a single homoge-neous coupon-paying bond issue, but is only able to state ananalytical valuation formula for the case T → ∞. Neverthelessefficient numerical routines can be applied to the case T <∞.Jones, Mason, Rosenfeld (1984) extend the analysis to a debt

555

structure consisting of several callable coupon-paying bonds withsinking funds.

3.4 Shortfalls From the previous paragraphs, one sees immediately the short-falls of Merton’s model:

1. Default is only considered at maturity.

2. The model for the firm’s capital structure is overly simplistic.

3. The perfect market assumptions are unrealistic: In practice,one has to consider taxes, transaction and bankruptcy costs,the interest rate is not fixed and known in advance and cer-tainly not the same for borrowers and lenders1, assets are notinfinitely divisible and in some of the smaller derivative mar-kets there exist individual investors whose decisions influencethe market price.

4. Short-term credit spreads and default probabilities are sig-nificantly underestimated, since the probability that the firmvalue process reaches the level of the debt’s face value inshort time periods by diffusion only is small.

The models presented in the following sections gradually over-come the above shorfalls.

4 Black, Cox’s model One of the biggest shortfalls of Merton’s model is that default isonly observable at a single maturity. Black, Cox (1976) extendMerton’s model via a deterministic, exogenously given defaultbarrier VB(t). Default happens at the first time when the firm’sasset value crosses this barrier, therefore the Black–Cox modelis a so-called first passage time model.

4.1 Model specification Like Merton, Black, Cox (1976) assume a perfect market, and theasset value is assumed to follow the geometric Brownian motiongiven in (1). The important difference to Merton’s model is thatdefault is no longer observable only at maturity, but also dur-ing the lifetime of debt: Default occurs at the first point in time twhere the asset value crosses a (possibly time-dependent) bar-rier VB(t), i.e when Vt ≤ VB(t) (here Vt = VB(t), as our assetvalue process and the barrier process are both continuous):

τ = inf{t ≥ 0 : Vt ≤ VB(t)} = inf{t ≥ 0 : Vt = VB(t)},

where τ denotes again the time of default. The values of debtand equity at τ are given by Dτ = VB and Eτ = 0, i.e. incase of default the remaining firm value is handed over to thedebtholders. An economic explanation for such a barrier is givenby safety covenants that give the debtholders the right to forcedefault or restructuring of the firm if it performs badly according tosome pre-specified criteria. For simplicity, we assume a constantdefault barrier,

VB(t) = VB, ∀ 0 ≤ t ≤ T,1One can introduce deterministic or even stochastic interest rates in several

of the presented models, but for the sake of simplicity we stick to the as-sumption of a constant predictable interest rate r.

666

and that the firm is not in default when debt is issued, i.e. V0 >VB . Note that we do not assume VB(T ) = VB = B. See Figure1 for an illustration of the differences of the default conditions inthe Black–Cox model and Merton’s model.

0 10 20 30 40 50 60 70 80 90 10060

80

100

120

140

160

180

time

firm

val

ue

(Vt)tVB

B

no defaultinMerton’smodel!

default timein Black,Cox’smodel

Fig. 1: Introducing a default barrier allows us to observe defaultalso during the lifetime of debt. We gain in accuracy in thevaluation of debt, as there may be scenarios where thereis a default in the framework of Black and Cox that wouldnot be observed in Merton’s model.

As for the assumptions concerning the firm’s debt, Black andCox stick with Merton’s assumption of debt being represented bya single zero-coupon bond of maturity T . Consequently, Mer-ton’s terminal default condition, VT ≤ B, applies here as well,additional to the barrier condition.Concerning the shortfalls of Merton’s model, we find that Short-falls 2.-4. still apply to the Black-Cox model. Further note that, asBlack, Cox (1976) assume perfect market conditions, the Modigliani–Miller Theorem as proved by Merton (1977) holds and the ques-tion of optimal capital structure does not arise in their model.

4.2 Applications We define

η :=√θ2 + 2rσ2, z(A,B,C) :=

ln(AB

)+ CT

σ√T

.

Survival probability:Starting at time t = 0, the risk-neutral survival probability is theprobability that the firm’s asset value process does not hit thebarrier VB up to time T , and that the asset value at T is biggerthan the face value of debt:

P (τ > T, VT > B) = P

(min

0≤t≤TVt > VB, VT > B

)= P

(min

0≤t≤TXt > ln

(VBV0

), XT > ln

(B

V0

))(3)= Φ(z(V0, B, θ))−

(VBV0

) 2θσ2

Φ(z(V 2B, BV0, θ)).

(9)

777

Comparing this with the survival probability at T in Merton’s model,which is given by P (VT > B) = Φ(d2) = Φ(z(V0, B, θ)), con-firms that survival is less likely in Black, Cox’s model.Another useful result is the probability that the asset value pro-cess does not reach the barrier before T (leaving the terminaldefault condition aside):

P (τ > T ) = P

(min

0≤t≤TVt > VB

)(1)= P

(min

0≤t≤TXt > ln

(VBV0

))(2)= Φ(z(V0, VB, θ))−

(VBV0

) 2θσ2

Φ(z(VB, V0, θ)).

(10)

Like in Merton’s model, we note that the default probability ap-proaches zero for T → 0. Also the probability that (Vt)t≥0 crossesthe barrier VB before T goes to zero for T → 0.See Figure 2 for a comparison of the default probabilities.

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7default probabilities over time

T

defa

ult p

roba

bilit

y

default probability Mertondefault probability Black−Coxprobability that first barrier passage after T

σ = 0.2r = 0.01δ = 0.005V0 = 100VB = 50

B = 80

Fig. 2: Comparison of default probabilities in Merton (1974)(blue) and Black, Cox (1976) (green). The probability thatthe first passage to the default barrier occurs later than T ,as given in (10), is also included (green, dotted).

Valuation of debt and equity:Using the survival probabilities (9) and (10), and the conditionalLaplace transformation of the first passage time given in (5), thevaluation formulas for the firm’s equity and debt can be stated as

E0 = E[e−rT max{VT −B, 0}1{τ>T}

]= V0e

−δTΦ(z(V0, B, θ + σ2))

− V0e−δT

(VBV0

) 2(θ+σ2)

σ2

Φ(z(V 2B, BV0, θ + σ2))

− e−rTB

(Φ(z(V0, B, θ))−

(VBV0

) 2θσ2

Φ(z(V 2B, BV0, θ))

),

888

and

D0 = E[e−rτ1{τ≤T}VB + e−rT1{τ>T}min{B, VT }

]= Be−rT

(Φ(z(V0, B, θ))−

(VBV0

) 2θσ2

Φ(z(V 2B, BV0, θ))

)

+ VB

((VBV0

) θ−ησ2

Φ(z(VB, V0,−η)) +

(VBV0

) θ+η

σ2

Φ(z(VB, V0, η))

)+ V0e

−δT (Φ(z(V0, VB, θ + σ2))− Φ(z(V0, B, θ + σ2))

−(VBV0

) 2(θ+σ2)

σ2 (Φ(z(VB, V0, θ + σ2))− Φ(z(V 2

B, BV0, θ + σ2))) .

Compared to Merton’s model, the value of debt is larger in theBlack–Cox model, and it is larger the higher the default barrierVB . A higher VB makes pre-maturity default more likely, butsince we assumed the bondholders receive VB at default, theyprofit from VB being close to the face value of the debt B.

CDS curve:In Black, Cox’s model, the CDS spread v is given as follows2:

EDDL0 := (1−R)E[e−rτ1{τ≤T}

],

EDPL0 := E[∫ T

0ve−rt1{τ>t}dt

]=v

r

(1− e−rTP (τ > T )− E

[e−rτ1{τ≤T}

]),

⇒ v =r(1−R)E

[e−rτ1{τ≤T}

]1− e−rTP (τ > T )− E

[e−rτ1{τ≤T}

] . (11)

Here, R denotes the constant recovery rate paid in case of de-fault. Assuming that one recovery rate applies to all classes ofdebt and matching the recovery on bond nominal with the re-maining firm value in case of default yields RB = VB . Explicitformulas for the survival probability and the expectation can bestated with the help of the formulas in Section 2. Also in thismethod, the spread vanishes for T → 0. This can be seenin Figure 3, which also illustrates how the CDS curve reacts tochanges in the parameters σ and VB .

5 Endogenous default and optimal

capital structure

In this section, we discuss several endogenous default models.All are first passage time models with constant default barrier VBand assume that the asset value process (Vt)t≥0 is given by (1).The time of default is

τ = inf{t ≥ 0 : Vt = VB},

and the corresponding survival probability at t is given by (10).

2The value of a CDS at some time point t is given by the expected valueof payments received in case of default (EDDL) minus the expected valueof premium payments (EDPL). Its fair spread (par spread) is determinedsuch that the initial value of the contract is zero. For a comprehensiveintroduction to the valuation of CDS, see, e.g., Mai (2014).

999

0 1 2 3 4 5 6 7 8 9 100

200

400

600

800

1000

1200

CDS spread term structures for different σ

T

bps

σ=0.1σ=0.2σ=0.3σ=0.4σ=0.5

r = 0.01δ = 0.005V0 = 100VB = 50

R = 0.625

0 1 2 3 4 5 6 7 8 9 100

200

400

600

800

1000

1200

1400

CDS spread term structures for different VB

T

bps

VB=40

VB=50

VB=60

VB=70

VB=80

r = 0.01δ = 0.005σ = 0.2V0 = 100

R = 0.625

Fig. 3: CDS spreads in the Black–Cox model for different valuesof σ and VB .

The difference to the default event in the Black–Cox model isthat now the firms can endogenously choose the optimal valueof the default barrier VB . CDS spread calculation is similar to theBlack–Cox model and will therefore not be presented. Insteadwe focus on the calculation of the optimal barrier VB .

One of the shortfalls that is gradually overcome in this sectionis the restrictive assumption that debt is represented by a zero-coupon bond of a certain maturity. The first models consideringendogenous default, Black, Cox (1976) and Leland (1994a), con-sider debt in form of a single coupon-paying bond issue of infinitematurity. Later on, this is replaced by the more sophisticated as-sumption of debt being constantly rolled over, see, e.g. Leland(1994b) and Leland, Toft (1996). Both assumptions on the com-position of the firm’s debt help us to get rid of the terminal defaultcondition VT ≤ B, and result in debt and equity valuation formu-las that depend only on the asset value and not explicitly on time.The models by Leland also partially relax Shortfall 3, the perfectmarket assumptions, and introduce taxes and bankruptcy costs,which gives rise to the question of optimal capital structure.

5.1 Pioneering endogenous default Black, Cox (1976) pioneer endogenous default models in a laterparagraph of their paper, where they treat the influence of financ-

101010

ing restrictions. They assume that the firm’s debt is representedby a single, coupon-paying perpetual bond issue. The couponis paid continuously with rate C. Further, the perfect market as-sumptions hold.For now, assume VB is an exogenously given constant. Today’svalue of debt, depending on V0, is:

D(V0) = E[C

r

∫ τ

0re−rtdt

]+ VBE

[e−rτ

]=C

r

(1− E

[e−rτ

])+ VBE

[e−rτ

](4)=C

r+

(VB −

C

r

)(V0

VB

)− θ+ησ2

.

(12)

The first part of the expectation corresponds to the discountedvalue of coupon payments (up to default), and the second part isthe discounted value of the payment received in case of default(i.e the remaining firm value).The optimal default barrier is chosen by the firm’s management.As they uphold the interests of the firm’s equityholders, they wantto maximize the value of equity. As Black, Cox (1976) assumedperfect market conditions, the Modigliani–Miller Theorem holdsand the firm value is invariant to capital structure and to thechoice of VB . Consequently the maximization of equity value isequivalent to the minimization of debt value with respect to VB .This gives as optimal value for the default barrier:

V ∗B =C(θ + η)

r(θ + η + σ2).

5.2 Extension to taxes and bankruptcycosts

Leland (1994a) extends the above model to include taxes andbankruptcy costs. As already discussed in Section 3, this raisesagain the question of optimal capital structure. Debt is againassumed to consist of a single, coupon-paying bond of infinitelifetime. The fraction αVB of the firm value at bankruptcy, 0 ≤α ≤ 1, is lost due to bankruptcy costs. Further, corporate taxeswith tax rate ν are introduced. As we are no longer in a per-fect market, the Modigliani–Miller Theorem does not hold, andthe firm value changes for different choices of leverage (i.e. de-pending on the amount of debt issued). This is due to the taxbenefits on debt, which vary with the amount of debt, and due tothe bankruptcy costs.The value of debt is calculated similarly to (12):

D(V0) =C

r+

((1− α)VB −

C

r

)(V0

VB

)− θ+ησ2

.

The value of equity is determined as the difference between to-tal firm value and debt value. Due to the effects of taxes andbankruptcy costs, the total firm value is now

F (V0) = V0 + TB(V0)−BC(V0),

where TB(V0) represents the benefits generated by the tax shield3,and BC(V0) are the bankruptcy costs. These two components

3Interest payments on corporate debt are in most cases tax deductable, whichresults in a benefit for the firm.

111111

can also be determined via risk-neutral valuation:

TB(V0) = E[∫ ∞

0νCe−rt1{τ>t}dt

]=νC

r

(1−

(V0

VB

)− θ+ησ2

),

BC(V0) = E[αVBe

−rτ ] = αVB

(V0

VB

)− θ+ησ2

.

Therefore total firm value and equity value are:

F (V0) = V0 +νC

r

(1−

(V0

VB

)− θ+ησ2

)− αVB

(V0

VB

)− θ+ησ2

,

E(V0) = V0 − (1− ν)C

r+

((1− ν)

C

r− VB

)(V0

VB

)− θ+ησ2

.

(13)

The optimal VB is again chosen such that equity value is maxi-mized. Default occurs when the firm cannot issue additional eq-uity to finance its payouts, i.e. when the value of equity falls tozero. Consequently, E(V ) = 0 for V ≤ VB and E(V ) > 0for V > VB . So the condition that equity value must always bepositive prior to default limits the range of admissible VB . Thecorresponding optimization problem is max

VBE(V )

E(V ) ≥ 0 ∀ V ≥ VB.

A sufficient condition for the existence of an optimal solution isthe so-called smooth-pasting condition4, which ensures thatthe equity value as a function of V is not only continuous, butalso continuously differentiable at V = VB :

dE(V )

dV

∣∣∣∣V=VB

= 0. (14)

An equivalent condition in this framework is dE(V )dVB

= 0, whichuses the fact that here, equity is a concave function of VB , allother parameters being equal. The endogenous bankruptcy levelis computed as

V ∗B =(1− ν)C(θ + η)

r(θ + η + σ2).

Note that VB is proportional to C, independent of V0 and α, anddecreasing in ν, r and σ2.

Optimal capital structure:In the endogenous framework debt is a function of the assetvalue process, its parameter σ and the known quantities r, δ, νand α. In order to determine the value of debt that maximizes thetotal firm value, the only variable we can optimize is the couponC. Looking again at the total firm value in the endogenous de-fault framework, we note that it is concave in C. The first order

4According to Chen, Kou (2009), the smooth-pasting condition requires localconvexity in VB . For diffusion-based models, this was verified numericallyin Leland, Toft (1996).

121212

condition dFdC = 0 thus gives us the optimal coupon C∗ that max-

imizes firm value:

C∗ = V0r(θ + η + σ2)

(1− ν)(θ + η)

(νσ2

(θ + η)(α+ ν − αν) + νσ2

) σ2

θ+η

.

The maximal firm value and corresponding debt value are ob-tained by inserting V ∗B and C∗ in the above formulas. Figure 4shows the firm value as function of leverage D∗(V )

F ∗(V ) : The lowerthe volatility parameter σ, the higher is optimal leverage and alsomaximal firm value.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 140

50

60

70

80

90

100

110

120

130

140total firm value vs. leverage for varying σ

leverage ratio

firm

val

ue

σ=0.2

σ=0.15

σ=0.25

Fig. 4: Maximal firm value as a function of leverage, shown fordifferent asset volatilities. The lower the volatility, thehigher is the optimal leverage, and also the maximal pos-sible firm value.

5.3 A more realistic capital structureassumption

Leland (1994b) extends his model presented above by introduc-ing a more realistic assumption on the composition of the firm’sdebt: It is now assumed to be continuously rolled over with theamount outstanding B being constant over time. The couponpayment rate on B is denoted by c. Basic assumptions and no-tation is similar to the previous model unless stated otherwise.For simplicity, all debt is assumed to be of equal seniority.

Technically, this means that at each instant of time the firm issuesbonds of total face value mB whose maturities are random andexponentially distributed with parameter m. The expected matu-rity of one of the firm’s bonds is thus 1/m. Random maturitiesseem to be a rather technical construct, but in fact this corre-sponds to a sinking fund where the firm retires debt at fractionalrate m: The firm continuously replaces the constant fraction mBof outstanding debt principal by newly issued debt with exactlythe same face value and other conditions. Newly issued debtis technically of infinite lifetime, but at each instant t the fractionme−mtB of outstanding face value is retired via the sinking fund.

To calculate the value of total debt, we first state the value d0 ofnewly issued debt at time 0: Let p[0,t] denote the nominal of debt

131313

issued at time 0 which is still outstanding at time t ≥ 0. We havep[0,0] = mB. As debt is retired with rate m, we have

∂p[0,t]

∂t= −mp[0,t] ⇔ p[0,t] = e−mtp[0,0] = e−mtmB,

see Leland (1994b, p. 9). Thus at time t, the amount of no-tional issued at time 0 that is still outstanding equals e−mtmB.Regarding the redemption at time t of debt issued at 0, our as-sumption states that at each instant t, the fraction m of the nom-inal still outstanding at t is retired, m · e−mtmB dt. Coupon pay-ments in [t, t + dt) on the still outstanding nominal issued at 0are c · e−mtmB dt. Thus, in case of no bankruptcy, debt is-sued at time 0 receives at each instant [t, t + dt) a cash flowof me−mt(c + m)B dt from coupon payments and debt retire-ment. In case of bankruptcy at t, the remaining firm value isdistributed to the different ’issues’ according to the fractions ofoutstanding nominal, i.e. still outstanding bonds issued at time 0correspond to the fraction me−mt of total nominal, thus they re-ceive me−mt · (1−α)VB as recovery value. Together, this yields

d0 = E[∫ τ

0me−(r+m)t(c+m)Bdt+me−(r+m)τ (1− α)VB

]=

(c+m)mB

r +m

(1− E

[e−(r+m)τ

])+m(1− α)VBE

[e−(r+m)τ

](4)=

(c+m)mB

r +m

(1−

(VBV0

) θ+ηmσ2

)+m(1− α)VB

(VBV0

) θ+ηmσ2

,

with θ as defined in (1) and ηm :=√θ2 + 2(m+ r)σ2.

Debt that was issued at time s in the past and is still outstandinghas value emsd0, as all fragments of debt have the same char-acteristics, regardless of time of issuance, and thus sell at thesame price. The total debt value is then calculated by integratingover the values of outstanding principal:

D(V0) =

∫ 0

−∞emsd0 ds =

d0

m

=(c+m)B

r +m

(1−

(VBV0

) θ+ηmσ2

)+ (1− α)VB

(VBV0

) θ+ηmσ2

.

The total firm value is the same as in Leland, see Equation (13),as the values of tax benefits and bankruptcy costs do not dependon the expected debt maturity, but only on the coupon paymentscB = C, the tax rate ν, the bankruptcy costs α and the defaultbarrier VB , and the distribution of the first passage time to thebarrier VB . The value of equity is again given by the differencebetween firm value and debt value.

Just like above, we can calculate the endogenous bankruptcylevel with the help of the smooth-pasting condition (14), see Le-land (1994b), p. 16:

V ∗B =

(c+m)Br+m

θ+ηmσ2 − νcB

rθ+ησ2

1 + α θ+ησ2 + (1− α) θ+ηm

σ2

.

141414

The casem = 0 corresponds to debt of infinite lifetime and there-fore the debt structure in Leland (1994a). Leland, Toft (1996) in-corporate a similar roll-over model for the firm’s debt, but in theircase the maturity of newly issued debt is not random, but someT the firm can optimally choose (i.e. no sinking fund). They an-alyze the influence of the chosen debt maturity on optimal lever-age by choosing the coupon in a way such that the price of newlyissued bonds equals their face value, and note that in this caseoptimal leverage increases with debt maturity. Leland (1994b, p.30f) similarly analyzes the influence of the parameter m on thefirm value at optimal leverage and notes that it is larger for largervalues of m.

6 The CreditGradesTM model None of the previously discussed models was able to overcomethe issue of too low short-term spreads. The CreditGradesTM

model adresses this shortfall by introducing randomness in thedefault barrier, the economic interpretation being that the exactleverage of the firm may not be known to investors due to loansthat are off the balance sheet.Introduced in Finger et al. (2002), this model aims at providingan analytically tractable framework of single-name credit risk. Apositive default probability at t = 0 is achieved by manipulationof the process that enters the pricing equations. Said manipula-tion originates from the assumption that the default barrier is nota constant, but a lognormal random variable with known mean.This results in a time shift in the asset value process, assigninga nonzero default probability to a time interval before zero.

The underlying framework is basically the same as in the pre-viously presented models, see Finger et al. (2002, p.6ff): Theunderlying process is described by

Vt = V0e−σ

2

2+σWt ,

where (Wt)t≥0 is a standard Brownian motion, and default hap-pens when Vt crosses the default barrier VB for the first time.The difference to the previous models is that now, the default bar-rier is driven by a lognormal random variable L ∼ LN (ln(L) −γ2

2 , γ2):

VB = DL = DLeγZ−γ2

2 ,

where D is interpreted as the firm’s debt-per-share ratio, andE[L] = L represents the average recovery rate on total debt.We assume γ ≥ 0, and Z is a standard normal random variableindependent of the Brownian motion (Wt)t≥0. The default timeis then given by

τ = inf{t ≥ 0 : Vt ≤ VB} = inf

{t ≥ 0 : V0e

σWt−σ2

2 ≤ DLeγZ−γ2

2

}= inf

{t ≥ 0 : σWt − γZ −

σ2t+ γ2

2=: Xt ≤ ln

(DLV0

)− γ2

}.

Obviously, Xt ∼ N(−σ2t+γ2

2 , σ2t+ γ2)

. We approximate the

process (Xt)t≥0 with a Brownian motion Xt := σWt− σ2

2 t started

151515

at time −s = − γ2

σ2 . This approximation is justified by the fact thatXt and Xt have the same first and second moments for all t ≥ 0.The formula for the default probability is then given as follows:

P (τ ≤ T ) = 1− P (τ > T ) = 1− P(

mint≤T

Xt > ln

(LDV0

)− γ2

)≈ 1− P

(mint≤T+s

Xt > ln

(LDV0

)− γ2

)

= Φ

ln(

LDV0eγ

2

)+ 1

2(σ2t+ γ2)√σ2t+ γ2

+V0e

γ2

LDΦ

ln(

LDV0eγ

2

)− 1

2(σ2t+ γ2)√σ2t+ γ2

.

We observe that the default probability at T = 0 is positive. Thismay help us to achieve a positive CDS spread for short maturi-ties. Note that the default probability for short maturities dependsstrongly on the level of the parameter γ. For a comparison of de-fault probabilities in the CreditGradesTM model with the model byBlack, Cox (1976) see Figure 5. A formula for the CDS spread isalso presented:

v(T ) =r(1−R)

(P (τ ≤ 0) +

∫ T0 e−rtfτ (t)dt

)P (τ > 0)− e−rTP (τ > T )−

∫ T0 e−rtfτ (t)dt

,

where fτ (t) = ddtP (τ ≤ t) is the density of the first passage time

to the barrier, and R is the recovery rate on the class of the firm’sdebt that is protected by the CDS. This is analogous to the CDSspread in the Black–Cox model (11). Finger et al. (2002) take thefollowing formula for the above integral from Reiner, Rubinstein(1991):∫ T

0e−rtfτ (t)dt = e

rγ2

σ2

(G(T +

γ2

σ2

)− G

(γ2

σ2

)),

with

G(t) =

(V0e

γ2

LD

)σ+√σ2+8r2σ

Φ

ln(

LDV0eγ

2

)− σt

2

√σ2 + 8r

σ√t

+

(V0e

γ2

LD

)σ−√σ2+8r2σ

Φ

ln(

LDV0eγ

2

)+ σt

2

√σ2 + 8r

σ√t

.

Looking at the limit of the CDS spread for T → 0 reveals oneshortfall of the CreditGradesTM model: As

∫ T0 e−rtfτ (t)dt

T→0−→ 0

and e−rTP (τ > T )T→0−→ P (τ > 0), we have v(T )

T→0−→ ∞.Thus the model-generated CDS spread for maturities very closeto zero is unrealistic, see Figure 5. Also the estimation of the ran-dom default threshold is not intuitive when one wants to calibratethe model to market data.

161616

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35comparison of default probabilities in Black,Cox and CreditGrades

T

default probability in Black,Coxdefault probability in CreditGrades

V0 = 100

r = 0.01δ = 0σ = 0.25L D = 50λ = 0.25R = 0.3

0 1 2 3 4 5 6 7 8 9 100

100

200

300

400

500

600comparison of CDS spreads in Black,Cox and CreditGrades

T

bps

CDS spreads in Black,CoxCDS spreads in CreditGrades

V0 = 100

r = 0.01δ = 0σ = 0.25L D = 50λ = 0.25R = 0.3

Fig. 5: Comparison of default probabilities and CDS spreads inthe CreditGradesTM model (green) and the Black–Coxmodel (blue). Due to the random default barrier, theCreditGradesTM model achieves higher short-term defaultprobabilities and CDS spreads, but at the very short endthe credit curve is unrealistic.

7 Outlook: Models with jumps Introducing jumps in the asset value process is a way of achiev-ing higher short-term default probabilities and CDS spreads thatstands theoretically on a more rigorous ground compared to theCreditGradesTM model. Unlike in diffusion models, there is asignificant probability that the asset value descends to a muchlower level in short time, namely by a jump, and there is no needfor an artifice like in the CreditGradesTM model. Positive CDSspreads for short maturities can be generated with those mod-els, and model-generated credit curves do not exhibit the un-realistic behavior at the very short end that is observed in theCreditGradesTM model.

The first model of this type was presented in Zhou (2001). Heuses the jump diffusion with normally distributed jump sizes fea-tured in Merton (1976) to model the evolution of the firm’s as-sets. Other references include Hilberink, Rogers (2002), who in-corporate processes with only downward jumps, and Chen, Kou(2009), who employ a double-exponential jump diffusion. The

171717

general form of the asset value process is

Vt := V0 exp

(µt+ σWt +

Nt∑k=1

Yk

),

where (Wt)t≥0 denotes again a standard Brownian motion, (Nt)t≥0

is a Poisson process, and Yk, k = 1, . . . , Nt, are the independentand identically distributed jump sizes. These sources of random-ness are assumed to be independent. Again, the default event isdefined as the first passage of (Vt)t≥0 below some barrier valueVB .

The mathematics involved with jump models are more compli-cated, and often closed-form pricing formulas are not available.An exception is the model with double-exponentially distributedjumps, which is able to provide a quasi-closed form pricing for-mula for the CDS spread. In Figure 6, we show model-generatedCDS spreads with varying diffusion volatility σ and default barrierVB . Note that the CDS spread for very short maturities is signif-icantly bigger than zero, so Shortfall 4 is finally overcome. Jumpdiffusion structural models can indeed produce a wide variety ofdifferent shapes for the CDS curve.

0 1 2 3 4 5 6 7 8 9 100

100

200

300

400

500

600

700

800

900CDS curve for varing σ

CDS maturity

spre

ad in

bps

σ = 0.1σ = 0.2σ = 0.3σ = 0.4σ = 0.5

κ = 0.3λ = 0.3ξ2 = 3

0 1 2 3 4 5 6 7 8 9 100

200

400

600

800

1000

1200CDS curve for varing κ

CDS maturity

spre

ad in

bps

κ = 0.3κ = 0.4κ = 0.5κ = 0.6κ = 0.7

σ = 0.2λ = 0.3ξ2 = 3

Fig. 6: CDS spreads in a model with double-exponentially dis-tributed jumps, with varying σ and κ := VB

V0.

8 Conclusions A comprehensive introduction to diffusion-based structural mod-els was given. The development of this approach, starting fromthe first structural model by Merton (1974) to the CreditGradesTM

181818

model by Finger et al. (2002), was illustrated and basic pricingformulas for all models were derived. Gradually, all shortfalls ofthe first structural model were eliminated, except for the failureto provide realistic CDS spreads for short maturities. A brief out-look to structural models with jumps, which finally are able tosolve this problem, was included.

References

Black, F., Cox, J.: Valuing corporate securities: Some effects ofbond indenture provisions, Journal of Finance, Vol. 31, No. 2,1976, pp. 351–367

Black, F., Scholes, M.: The pricing of options and corporateliabilities, Journal of Political Economy, Vol. 81, No. 3, 1973,pp. 637–654

Bielecki, T., Jeanblanc, M., Rutkowski, M.: Credit Risk Mod-eling, Lecture notes of M. Jeanblanc, Université d’Evry, 2008,available at http://www.maths.univ-evry.fr/pages_

perso/jeanblanc/cours/Credit_Risk_Modeling_Notes.

pdf

Chen, N., Kou, S.G.: Credit spreads, optimal capital structureand implied volatility with endogenous default and jump risk,Mathematical Finance, Vol. 19, No. 3, 2009, pp. 343–378

Finger, C., Finkelstein, V., Lardy, J.P., Pan, G., Ta, T., Tierney,J.: CreditGrades Techincal Document, 2002

Hilberink, B., Rogers, L.C.G.: Optimal capital structure andendogenous default, Finance and Stochastics, Vol. 6, No. 2,2002, pp. 237–263

Jones, E.P., Mason, S.P., Rosenfeld, E.: Contingent claimsanalysis of corporate capital structure: An empirical investi-gation, Journal of Finance, Vol. 39, No. 3, 1984, pp. 611–625

Leland, H.: Corporate debt value, bond covenants and optimalcapital structure, Journal of Finance, Vol. 49, No. 4, 1994, pp.1213–1252

Leland, H.: Bond prices, yield spreads and optimal capital struc-ture with default risk, Working Paper, Haas School of Busi-ness, U.C. Berkeley, 1994

Leland, H., Toft, K.B.: Optimal capital structure, endogenousbankruptcy and the term structure of credit spreads, Journalof Finance, Vol. 51, No. 3, 1996, pp. 987–1019

Mai, J.-F.: Pricing single-name CDS options: A re-view of standard approaches, XAIA homepage article,2014, available at: http://www.xaia.com/uploads/media/

XAIACDSOptions.pdf

Mai, J.-F.: The joint modeling of debt and equity: An introduc-tion, XAIA homepage article, 2012, available at: http://www.xaia.com/uploads/media/CreditEquityengl_01.pdf

191919

Merton, R.C.: On the pricing of corporate debt: The risk struc-ture of interest rates, Journal of Finance, Vol. 29, No. 2, 1974,pp. 449–470

Merton, R.C.: Option pricing when underlying stock returns arediscontinuous Journal of Financial Economics, Vol. 3, No. 1-2,1976, pp. 125–144

Merton, R.C.: On the pricing of contingent claims and theModigliani-Miller theorem, Journal of Financial Economics,Vol. 5, No. 2, 1977, pp. 241–249

Modigliani, F., Miller, M.H.: The cost of capital, corporation fi-nance and the theory of investment, American Economic Re-view, Vol. 48, No. 3, 1958, pp. 261–297

Musiela, M., Rutkowski, M.: Martingale methods in financialmodelling, 2nd ed., Springer Series in Stochastic Modellingand Applied Probability, 2005

Reiner, E., Rubinstein, M.: Breaking down the barriers, RiskMagazine, Vol. 4, No. 8, 1991, pp. 28–35

Shreve, S.: Stochastic calculus for finance II - Continuous timemodels, Springer, 2004

Zhou, C: The term structure of credit spreads with jump riskJournal of Banking and Finance, Vol. 25, No. 11, 2001, pp.2015–2040

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