A N I NTRODUCTION TO S TRUCTURAL C REDIT-E QUITY M ODELS

Transcription

1 A N I NTRODUCTION TO S TRUCTURAL C REDIT-E QUITY M ODELS Amelie Hüttner XAIA Investment GmbH Sonnenstraße München Germany ameliehuettner@xaiacom October 8 14 Abstract This article aims to give an introduction to the concept of structural credit-equity models where both debt and equity products are treated as derivatives on the underlying firm value These models allow for the valuation of credit and equity instruments in a unified framework Moreover when taxes and bankruptcy costs are incorporated they can be used to identify a firm s optimal capital structure Starting with the first structural model introduced by Merton (1974) several diffusion-based models that extend 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 and also consider the possibility of default When the perfect market assumptions are relaxed they even allow for an answer to the question of optimal capital structure The central idea is to model the firm s assets with a stochastic process and then treat all the firm s securities debt and equity instruments as derivatives on the total firm value Default happens when the firm s asset value falls to a certain level Thus in contrast to the intensity-based approach to credit-equity modeling presented in a previous article see Mai (1) structural models offer an economic explanation for the default time On the other hand they are usually more complicated when it comes to the calibration to market data This article attempts to give an overview of several structural models based on Brownian motion as well as the development of 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) where default happens when the firm s asset value falls below a certain barrier - to the relaxation of perfect market assumptions and the question of optimal capital structure discussed in a series of papers by Leland (Leland (1994a) Leland (1994b) Leland Toft (1996)) - Finally we briefly discuss the CreditGrades model by Finger et al () which extends the concept to a random default barrier All models presented in this chapter assume that under the riskneutral measure the firm s asset value (Vt )t follows a geo1

2 metric Brownian motion of the following form (unless otherwise stated): Vt := V ext t Xt := θt + Wt θ := r δ (1) Here (Wt )t is a standard Brownian motion is the instantaneous return volatility r is the risk-free interest rate and δ represents the proportional payout rate to investors comprised of dividend and interest payments Claims on the firm s asset value will be priced using the risk-neutral valuation approach Before discussing the models stated above we start with a short review of several important properties of Brownian motion which are essential to the derivation of survival probabilities and pricing formulas Mathematical background Let (Wt )t denote a standard Brownian motion Z a standard normal random variable and T > Considering the Brownian motion with drift (Xt )t in (1) we know x θt x + θt P (XT > x) = P Z > =Φ x R T T where Φ() denotes the standard normal distribution function Its running minimum is distributed according to P θx x + θt x + θt min Xt > x = Φ e Φ () t T T T for x < see Musiela Rutkowski (5 Lemma A18) The joint distribution of (Xt )t and its running minimum is given by P XT > x min Xt > y t T θy x + θt y x + θt =Φ e Φ T T (3) for y y x and x R; see Musiela Rutkowski (5 Proposition A183) Now define τb := inf{t : Xt = b} for some b R Shreve (4 Theorem 83) gives the Laplace transform of τb as b E[e uτb ] = e ( θ+ θ +u ) b > (4) It will also be required to compute the expectation E[e uτb 1{τb T } ] for b < For this we need the following result on the evaluation of Riemann Stieltjes integrals with respect to the standard normal distribution function see Bielecki Jeanblanc Rutkowski (8): Z T a x e dφ b c x x d + c b (c d) b dt Φ = e d T d c b (c +d) b + dt + e Φ d T

3 with d = c a We find Z T uτ b e us dp (τb s) E e 1{τb T } = Z T us = e d 1 P min Xt > b t s Z T Z T bθ b θs b + θs () e us dφ = + e e us dφ (5) s s! b b θ + u T = e (θ θ +u ) Φ T! + u T b θ b + + e (θ+ θ +u ) Φ T In the context of the structural models presented in the following it will be required to work with the distribution of the minimum and the first passage time of the geometric Brownian motion (Vt )t introduced in (1) The above results on (Xt )t can be transferred to the corresponding results on (Vt )t as Xt = Vt and ln() is a strictly increasing function so {Vt b} nv o = Xt ln Vb ln 3 Merton s model 31 Model specification Assumptions: The model presented in Merton (1974) is considered to be the first structural credit model It makes use of some results already presented in Black Scholes (1973) in the context of option pricing and is based on similar assumptions: 1 There are no taxes transaction costs or bankruptcy costs Individual investors investment decisions do not influence the market price 3 There is a unique interest rate r for borrowing and lending It is 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 assumptions from now on Due to the absence of taxes and bankruptcy costs stated in Assumption 1 the total firm value is always equal to the value 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 firm is not able to repay its debt therefore it is necessary to assume a certain capital structure to be able to express mathematically the default probability and the terminal payoffs of the firm s securities The firm s assets are composed of equity (Et )t and debt instruments with total value (Dt )t where Dt is identified with a single zero-coupon bond issue with maturity T and total 3

4 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 in Equation (1) represents the dividend rate Default happens if the firm 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 of the firm Note that default is only observable at the maturity date of debt since only then it is checked whether the firm can repay its debt or not 3 Applications Due to the fact that the default event depends on the (admittedly very restrictive) capital structure assumption and that default is only possible at the debt s maturity date pricing of credit instruments other than the zero-coupon bond under consideration does not make sense in Merton s model What can be done is the valuation of debt and equity in total and the derivation of default probabilities for a fixed time point T Valuation of debt and equity: At T the bondholders receive the amount B and the shareholders regain control over the firm s assets If payment is not possible the firm defaults and is turned over to the bondholders Looking at the terminal payoffs of creditors and equityholders DT = B1{VT >B} + VT 1{VT B} = min{b VT } ET = (VT B)1{VT >B} = max{vt B } one observes that the value of the firm s equity corresponds to the value of a European call on the firm s assets when the firm pays a continuous dividend δ One can therefore use an extension of the well-known valuation formula presented in Black Scholes (1973): E = E[e rt ET ] = V e δt Φ(d1 ) Be rt Φ(d ) ln + (θ + )T d1 = d = d1 T T (6) Looking at the terminal value of debt one notes that it can be formulated as DT = B max{b VT } Recalling the formula for the value of a European put in the Black-Scholes framework with continuous dividends we determine the value of debt as: D = E[e rt DT ] = Be rt Put Put = Be rt Φ( d ) V e δt Φ( d1 ) D = Be rt Φ(d ) + V e δt (7) Φ( d1 ) Note that the balance sheet equality Vt = Et +Dt has to be modified for the firm s payouts: Et + Dt = Vt e δ(t t) for all t T Default probability: The risk-neutral probability of default is the probability that the firm cannot repay its debt at T : P (τ = T ) = P (VT B) = Φ( d ) (8) 4

5 where τ denotes the time of default For T the default probability approaches zero Since a potential default is only considered at maturity T the probabilities P (τ t) t T are not defined at all Optimal capital structure: The Modigliani Miller Theorem in presence of bankruptcy: The Modigliani Miller Theorem was derived in Modigliani Miller (1958) and states that a firm s value is invariant to its capital structure under certain conditions The assumptions taken in their paper are similar to those in Merton s framework with the additional assumption that all investors view the income generated by bonds as certain regardless of issuer In other words they rule out the possibility of default The statement is then proven by looking at two firms with identical investment decisions one with debt and the other purely equity-financed and arguing via arbitrage arguments that those two firms have to have the same value Merton (1977) proves the Modigliani Miller Theorem in the presented framework that incorporates possibility of default: Essentially considering two firms with identical investment decisions one purely equity-financed (unlevered) the other with both equity and debt (levered) he shows that one can replicate the debt and equity of the levered firm with portfolios consisting of equity of the unlevered firm and riskless debt Combining the two portfolios yields the value of equity of the unlevered firm which is due to arbitrage arguments equal to the sum of debt and equity of the levered firm Thus the two firms have the same value This proof is rather technical but in our framework one sees clearly in the balance sheet equality that as Dt +Et = Vt e δ(t t) at any time and Vt is random capital structure optimization in order to achieve a maximal firm value does not make sense Modigliani Miller (1958) already identify taxes as a possible factor that invalidates their argumentation and indeed we will see in Section 5 that the firm value is no longer independent of the capital structure when taxes and bankruptcy costs are introduced 33 Possible extensions In Merton s framework the event of default {VT B} has a very simple form If we choose a more complicated model for the firm s debt than a single zero-coupon bond it is no longer possible to express the default probability as simple as in (8) which significantly hinders the pricing of securities Then we also need to model how the remaining assets in case of default will be distributed among different debt instruments However some of the assumptions on the debt structure can be relaxed for example the single homogeneous bond issue can be replaced by a bond issue containing different seniorities as shown in Black Cox (1976 p 358f) Merton (1974) also considers debt in form of a single homogeneous coupon-paying bond issue but is only able to state an analytical valuation formula for the case T Nevertheless efficient numerical routines can be applied to the case T < Jones Mason Rosenfeld (1984) extend the analysis to a debt 5

6 structure consisting of several callable coupon-paying bonds with sinking funds 34 Shortfalls From the previous paragraphs one sees immediately the shortfalls of Merton s model: 1 Default is only considered at maturity 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 certainly not the same for borrowers and lenders1 assets are not infinitely divisible and in some of the smaller derivative markets there exist individual investors whose decisions influence the market price 4 Short-term credit spreads and default probabilities are significantly underestimated since the probability that the firm value process reaches the level of the debt s face value in short time periods by diffusion only is small The models presented in the following sections gradually overcome the above shorfalls 4 Black Cox s model One of the biggest shortfalls of Merton s model is that default is only observable at a single maturity Black Cox (1976) extend Merton s model via a deterministic exogenously given default barrier (t) Default happens at the first time when the firm s asset value crosses this barrier therefore the Black Cox model is a so-called first passage time model 41 Model specification Like Merton Black Cox (1976) assume a perfect market and the asset value is assumed to follow the geometric Brownian motion given in (1) The important difference to Merton s model is that default is no longer observable only at maturity but also during the lifetime of debt: Default occurs at the first point in time t where the asset value crosses a (possibly time-dependent) barrier (t) ie when Vt (t) (here Vt = (t) as our asset value process and the barrier process are both continuous): τ = inf{t : Vt (t)} = inf{t : Vt = (t)} where τ denotes again the time of default The values of debt and equity at τ are given by Dτ = and Eτ = ie in case of default the remaining firm value is handed over to the debtholders An economic explanation for such a barrier is given by safety covenants that give the debtholders the right to force default or restructuring of the firm if it performs badly according to some pre-specified criteria For simplicity we assume a constant default barrier (t) = 1 t T One can introduce deterministic or even stochastic interest rates in several of the presented models but for the sake of simplicity we stick to the assumption of a constant predictable interest rate r 6

7 and that the firm is not in default when debt is issued ie V > Note that we do not assume (T ) = = B See Figure 1 for an illustration of the differences of the default conditions in the Black Cox model and Merton s model 18 (Vt)t B 16 no default in Merton s model! firm value 14 1 default time in Black Cox s model time Fig 1: Introducing a default barrier allows us to observe default also during the lifetime of debt We gain in accuracy in the valuation of debt as there may be scenarios where there is a default in the framework of Black and Cox that would not be observed in Merton s model As for the assumptions concerning the firm s debt Black and Cox stick with Merton s assumption of debt being represented by a single zero-coupon bond of maturity T Consequently Merton 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 Shortfalls -4 still apply to the Black-Cox model Further note that as Black Cox (1976) assume perfect market conditions the Modigliani Miller Theorem as proved by Merton (1977) holds and the question of optimal capital structure does not arise in their model 4 Applications We define p η := θ + r z(a B C) := ln A B + CT T Survival probability: Starting at time t = the risk-neutral survival probability is the probability that the firm s asset value process does not hit the barrier up to time T and that the asset value at T is bigger than the face value of debt: P (τ > T VT > B) = P min Vt > VT > B t T B = P min Xt > ln XT > ln t T V V θ (3) = Φ(z(V B θ)) Φ(z( BV θ)) V (9) 7

8 Comparing this with the survival probability at T in Merton s model which is given by P (VT > B) = Φ(d ) = Φ(z(V B θ)) confirms that survival is less likely in Black Cox s model Another useful result is the probability that the asset value process does not reach the barrier before T (leaving the terminal default condition aside): P (τ > T ) = P =P min Vt > t T () = Φ(z(V θ)) (1) V min Xt > ln t T V θ Φ(z( V θ)) (1) Like in Merton s model we note that the default probability approaches zero for T Also the probability that (Vt )t crosses the barrier before T goes to zero for T See Figure for a comparison of the default probabilities default probabilities over time 7 6 default probability 5 4 = r = 1 δ = 5 V = 1 = 5 B = 8 3 default probability Merton default probability Black Cox probability that first barrier passage after T T Fig : Comparison of default probabilities in Merton (1974) (blue) and Black Cox (1976) (green) The probability that the first passage to the default barrier occurs later than T as given in (1) is also included (green dotted) Valuation of debt and equity: Using the survival probabilities (9) and (1) and the conditional Laplace transformation of the first passage time given in (5) the valuation formulas for the firm s equity and debt can be stated as E = E e rt max{vt B }1{τ >T } = V e δt Φ(z(V B θ + )) ) (θ+ δt V e Φ(z( BV θ + )) V! θ V B e rt B Φ(z(V B θ)) Φ(z( BV θ)) V 8

9 and D = E e rτ 1{τ T } + e rt 1{τ >T } min{b VT }! θ V B = Be rt Φ(z(V B θ)) Φ(z( BV θ)) V! θ η θ+η + Φ(z( V η)) + Φ(z( V η)) V V + V e δt Φ(z(V θ + )) Φ(z(V B θ + )) ) (θ+ Φ(z( V θ + )) Φ(z( BV θ + )) V Compared to Merton s model the value of debt is larger in the Black Cox model and it is larger the higher the default barrier A higher makes pre-maturity default more likely but since we assumed the bondholders receive at default they profit from being close to the face value of the debt B CDS curve: In Black Cox s model the CDS spread v is given as follows : EDDL := (1 R)E e rτ 1{τ T } Z T ve rt 1{τ >t} dt EDP L := E v = 1 e rt P (τ > T ) E e rτ 1{τ T } r r(1 R)E e rτ 1{τ T } v= (11) 1 e rt P (τ > T ) E e rτ 1{τ T } Here R denotes the constant recovery rate paid in case of default Assuming that one recovery rate applies to all classes of debt and matching the recovery on bond nominal with the remaining firm value in case of default yields RB = Explicit formulas for the survival probability and the expectation can be stated with the help of the formulas in Section Also in this method the spread vanishes for T This can be seen in Figure 3 which also illustrates how the CDS curve reacts to changes in the parameters and 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 and assume that the asset value process (Vt )t is given by (1) The time of default is τ = inf{t : Vt = } and the corresponding survival probability at t is given by (1) The value of a CDS at some time point t is given by the expected value of payments received in case of default (EDDL) minus the expected value of premium payments (EDPL) Its fair spread (par spread) is determined such that the initial value of the contract is zero For a comprehensive introduction to the valuation of CDS see eg Mai (14) 9

10 CDS spread term structures for different =1 = =3 =4 =5 r = 1 δ = 5 V = 1 = R = 65 bps T CDS spread term structures for different 14 =4 r = 1 δ = 5 = V = 1 R = 65 1 =5 =6 =7 1 =8 bps T Fig 3: CDS spreads in the Black Cox model for different values of and The difference to the default event in the Black Cox model is that now the firms can endogenously choose the optimal value of the default barrier CDS spread calculation is similar to the Black Cox model and will therefore not be presented Instead we focus on the calculation of the optimal barrier One of the shortfalls that is gradually overcome in this section is the restrictive assumption that debt is represented by a zerocoupon bond of a certain maturity The first models considering endogenous default Black Cox (1976) and Leland (1994a) consider debt in form of a single coupon-paying bond issue of infinite maturity Later on this is replaced by the more sophisticated assumption of debt being constantly rolled over see eg Leland (1994b) and Leland Toft (1996) Both assumptions on the composition of the firm s debt help us to get rid of the terminal default condition VT B and result in debt and equity valuation formulas that depend only on the asset value and not explicitly on time The models by Leland also partially relax Shortfall 3 the perfect market assumptions and introduce taxes and bankruptcy costs which gives rise to the question of optimal capital structure 51 Pioneering endogenous default Black Cox (1976) pioneer endogenous default models in a later paragraph of their paper where they treat the influence of financ1

11 ing restrictions They assume that the firm s debt is represented by a single coupon-paying perpetual bond issue The coupon is paid continuously with rate C Further the perfect market assumptions hold For now assume is an exogenously given constant Today s value of debt depending on V is: Z C τ rt D(V ) = E re dt + E e rτ r C = 1 E e rτ + E e rτ r θ+η V C (4) C = + r r (1) The first part of the expectation corresponds to the discounted value of coupon payments (up to default) and the second part is the discounted value of the payment received in case of default (ie 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 want to maximize the value of equity As Black Cox (1976) assumed perfect market conditions the Modigliani Miller Theorem holds and the firm value is invariant to capital structure and to the choice of Consequently the maximization of equity value is equivalent to the minimization of debt value with respect to This gives as optimal value for the default barrier: = 5 Extension to taxes and bankruptcy costs C(θ + η) r(θ + η + ) Leland (1994a) extends the above model to include taxes and bankruptcy costs As already discussed in Section 3 this raises again the question of optimal capital structure Debt is again assumed to consist of a single coupon-paying bond of infinite lifetime The fraction α of the firm value at bankruptcy α 1 is lost due to bankruptcy costs Further corporate taxes with tax rate ν are introduced As we are no longer in a perfect market the Modigliani Miller Theorem does not hold and the firm value changes for different choices of leverage (ie depending on the amount of debt issued) This is due to the tax benefits on debt which vary with the amount of debt and due to the bankruptcy costs The value of debt is calculated similarly to (1): θ+η C C V D(V ) = + (1 α) r r The value of equity is determined as the difference between total firm value and debt value Due to the effects of taxes and bankruptcy costs the total firm value is now F (V ) = V + T B(V ) BC(V ) where T B(V ) represents the benefits generated by the tax shield3 and BC(V ) are the bankruptcy costs These two components 3 Interest payments on corporate debt are in most cases tax deductable which results in a benefit for the firm 11

12 can also be determined via risk-neutral valuation: Z νc T B(V ) = E νce 1{τ >t} dt = 1 r θ+η V rτ BC(V ) = E α e = α rt V! θ+η Therefore total firm value and equity value are: θ+η θ+η! V V 1 α θ+η V C C E(V ) = V (1 ν) + (1 ν) r r νc F (V ) = V + r (13) The optimal is again chosen such that equity value is maximized Default occurs when the firm cannot issue additional equity to finance its payouts ie when the value of equity falls to zero Consequently E(V ) = for V and E(V ) > for V > So the condition that equity value must always be positive prior to default limits the range of admissible The corresponding optimization problem is max E(V ) E(V ) V V B A sufficient condition for the existence of an optimal solution is the so-called smooth-pasting condition4 which ensures that the equity value as a function of V is not only continuous but also continuously differentiable at V = : de(v ) = dv V = (14) de(v ) An equivalent condition in this framework is d = which uses the fact that here equity is a concave function of all other parameters being equal The endogenous bankruptcy level is computed as = (1 ν)c(θ + η) r(θ + η + ) Note that is proportional to C independent of V and α and decreasing in ν r and Optimal capital structure: In the endogenous framework debt is a function of the asset value process its parameter and the known quantities r δ ν and α In order to determine the value of debt that maximizes the total firm value the only variable we can optimize is the coupon C Looking again at the total firm value in the endogenous default framework we note that it is concave in C The first order 4 According to Chen Kou (9) the smooth-pasting condition requires local convexity in For diffusion-based models this was verified numerically in Leland Toft (1996) 1

13 df condition dc = thus gives us the optimal coupon C that maximizes firm value: r(θ + η + ) C = V (1 ν)(θ + η) ν (θ + η)(α + ν αν) + ν θ+η The maximal firm value and corresponding debt value are obtained by inserting and C in the above formulas Figure 4 D (V ) shows the firm value as function of leverage F (V ) : The lower the volatility parameter the higher is optimal leverage and also maximal firm value total firm value vs leverage for varying firm value = =15 = leverage ratio Fig 4: Maximal firm value as a function of leverage shown for different asset volatilities The lower the volatility the higher is the optimal leverage and also the maximal possible firm value 53 A more realistic capital structure assumption Leland (1994b) extends his model presented above by introducing a more realistic assumption on the composition of the firm s debt: It is now assumed to be continuously rolled over with the amount outstanding B being constant over time The coupon payment rate on B is denoted by c Basic assumptions and notation 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 issues bonds of total face value mb whose maturities are random and exponentially distributed with parameter m The expected maturity of one of the firm s bonds is thus 1/m Random maturities seem to be a rather technical construct but in fact this corresponds to a sinking fund where the firm retires debt at fractional rate m: The firm continuously replaces the constant fraction mb of outstanding debt principal by newly issued debt with exactly the same face value and other conditions Newly issued debt is technically of infinite lifetime but at each instant t the fraction me mt B of outstanding face value is retired via the sinking fund To calculate the value of total debt we first state the value d of newly issued debt at time : Let p[t] denote the nominal of debt 13

14 issued at time which is still outstanding at time t We have p[] = mb As debt is retired with rate m we have p[t] = mp[t] t p[t] = e mt p[] = e mt mb see Leland (1994b p 9) Thus at time t the amount of notional issued at time that is still outstanding equals e mt mb Regarding the redemption at time t of debt issued at our assumption states that at each instant t the fraction m of the nominal still outstanding at t is retired m e mt mb dt Coupon payments in [t t + dt) on the still outstanding nominal issued at are c e mt mb dt Thus in case of no bankruptcy debt issued at time receives at each instant [t t + dt) a cash flow of me mt (c + m)b dt from coupon payments and debt retirement In case of bankruptcy at t the remaining firm value is distributed to the different issues according to the fractions of outstanding nominal ie still outstanding bonds issued at time correspond to the fraction me mt of total nominal thus they receive me mt (1 α) as recovery value Together this yields Z τ d = E me (r+m)t (c + m)bdt + me (r+m)τ (1 α) i i h h (c + m)mb 1 E e (r+m)τ + m(1 α) E e (r+m)τ r+m θ+ηm! θ+ηm (4) (c + m)mb = 1 + m(1 α) r+m V V = p with θ as defined in (1) and ηm := θ + (m + r) Debt that was issued at time s in the past and is still outstanding has value ems d as all fragments of debt have the same characteristics regardless of time of issuance and thus sell at the same price The total debt value is then calculated by integrating over the values of outstanding principal: Z ems d ds = D(V ) = (c + m)b = r+m 1 V d m θ+ηm! + (1 α) V θ+ηm The total firm value is the same as in Leland see Equation (13) as the values of tax benefits and bankruptcy costs do not depend on the expected debt maturity but only on the coupon payments cb = C the tax rate ν the bankruptcy costs α and the default barrier and the distribution of the first passage time to the barrier The value of equity is again given by the difference between firm value and debt value Just like above we can calculate the endogenous bankruptcy level with the help of the smooth-pasting condition (14) see Leland (1994b) p 16: = (c+m)b θ+ηm θ+η νcb r+m r m 1 + α θ+η + (1 α) θ+η 14

15 The case m = corresponds to debt of infinite lifetime and therefore the debt structure in Leland (1994a) Leland Toft (1996) incorporate a similar roll-over model for the firm s debt but in their case the maturity of newly issued debt is not random but some T the firm can optimally choose (ie no sinking fund) They analyze the influence of the chosen debt maturity on optimal leverage by choosing the coupon in a way such that the price of newly issued bonds equals their face value and note that in this case optimal leverage increases with debt maturity Leland (1994b p 3f) similarly analyzes the influence of the parameter m on the firm value at optimal leverage and notes that it is larger for larger values of m 6 The CreditGradesTM model None of the previously discussed models was able to overcome the issue of too low short-term spreads The CreditGradesTM model adresses this shortfall by introducing randomness in the default barrier the economic interpretation being that the exact leverage of the firm may not be known to investors due to loans that are off the balance sheet Introduced in Finger et al () this model aims at providing an analytically tractable framework of single-name credit risk A positive default probability at t = is achieved by manipulation of the process that enters the pricing equations Said manipulation originates from the assumption that the default barrier is not a constant but a lognormal random variable with known mean This results in a time shift in the asset value process assigning a nonzero default probability to a time interval before zero The underlying framework is basically the same as in the previously presented models see Finger et al ( p6ff): The underlying process is described by Vt = V e +Wt where (Wt )t is a standard Brownian motion and default happens when Vt crosses the default barrier for the first time The difference to the previous models is that now the default barrier is driven by a lognormal random variable L LN (ln(l ) γ γ ): = DL = DL eγz γ where D is interpreted as the firm s debt-per-share ratio and E[L] = L represents the average recovery rate on total debt We assume γ and Z is a standard normal random variable independent of the Brownian motion (Wt )t The default time is then given by γz γ Wt DL e τ = inf{t : Vt } = inf t : V e t + γ DL = inf t : Wt γz =: X t ln γ V t + γ We approximate the Obviously X t N t+γ process (X t )t with a Brownian motion X t := Wt t started 15

16 at time s = γ This approximation is justified by the fact that X t and X t have the same first and second moments for all t The formula for the default probability is then given as follows: L D P (τ T ) = 1 P (τ > T ) = 1 P min X t > ln γ t T V L D 1 P min X t > ln γ t T +s V ln L Dγ + 1 ( t + γ ) V e p = Φ t + γ L D 1 t + γ) ln ( γ V e V eγ p Φ + L D t + γ We observe that the default probability at T = is positive This may help us to achieve a positive CDS spread for short maturities Note that the default probability for short maturities depends strongly on the level of the parameter γ For a comparison of default probabilities in the CreditGradesTM model with the model by Black Cox (1976) see Figure 5 A formula for the CDS spread is also presented: RT r(1 R) P (τ ) + e rt fτ (t)dt v(t ) = RT P (τ > ) e rt P (τ > T ) e rt fτ (t)dt d where fτ (t) = dt P (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 s debt that is protected by the CDS This is analogous to the CDS spread in the Black Cox model (11) Finger et al () take the following formula for the above integral from Reiner Rubinstein (1991): Z T rγ e rt fτ (t)dt = e γ γ G T + G with t + 8r V e G(t) = Φ L D t! +8r L D t ln + + 8r γ V e V eγ + Φ L D t γ! + +8r ln L D V eγ Looking at the limit of the CDS spread for T reveals one shortfall of the CreditGradesTM model: As T RT T e rt fτ (t)dt T and e rt P (τ > T ) P (τ > ) we have v(t ) Thus the model-generated CDS spread for maturities very close to zero is unrealistic see Figure 5 Also the estimation of the random default threshold is not intuitive when one wants to calibrate the model to market data 16

17 comparison of default probabilities in BlackCox and CreditGrades V = 1 r = 1 δ= = 5 L D = 5 λ = 5 R = default probability in BlackCox default probability in CreditGrades T comparison of CDS spreads in BlackCox and CreditGrades 6 5 bps 4 V = 1 r = 1 δ= = 5 L D = 5 λ = 5 R = CDS spreads in BlackCox CDS spreads in CreditGrades T Fig 5: Comparison of default probabilities and CDS spreads in the CreditGradesTM model (green) and the Black Cox model (blue) Due to the random default barrier the CreditGradesTM model achieves higher short-term default probabilities and CDS spreads but at the very short end the credit curve is unrealistic 7 Outlook: Models with jumps Introducing jumps in the asset value process is a way of achieving higher short-term default probabilities and CDS spreads that stands theoretically on a more rigorous ground compared to the CreditGradesTM model Unlike in diffusion models there is a significant probability that the asset value descends to a much lower level in short time namely by a jump and there is no need for an artifice like in the CreditGradesTM model Positive CDS spreads for short maturities can be generated with those models and model-generated credit curves do not exhibit the unrealistic behavior at the very short end that is observed in the CreditGradesTM model The first model of this type was presented in Zhou (1) He uses the jump diffusion with normally distributed jump sizes featured in Merton (1976) to model the evolution of the firm s assets Other references include Hilberink Rogers () who incorporate processes with only downward jumps and Chen Kou (9) who employ a double-exponential jump diffusion The 17

18 general form of the asset value process is Vt := V exp µt + Wt + Nt X! Yk k=1 where (Wt )t denotes again a standard Brownian motion (Nt )t is a Poisson process and Yk k = 1 Nt are the independent and identically distributed jump sizes These sources of randomness are assumed to be independent Again the default event is defined as the first passage of (Vt )t below some barrier value The mathematics involved with jump models are more complicated and often closed-form pricing formulas are not available An exception is the model with double-exponentially distributed jumps which is able to provide a quasi-closed form pricing formula for the CDS spread In Figure 6 we show model-generated CDS spreads with varying diffusion volatility and default barrier Note that the CDS spread for very short maturities is significantly bigger than zero so Shortfall 4 is finally overcome Jump diffusion structural models can indeed produce a wide variety of different shapes for the CDS curve CDS curve for varing 9 κ = 3 λ = 3 ξ = spread in bps = 1 = = 3 = 4 = CDS maturity CDS curve for varing κ 1 = λ = 3 ξ = 3 1 spread in bps κ = 3 κ = 4 κ = 5 κ = 6 κ = CDS maturity Fig 6: CDS spreads in a model with double-exponentially distributed jumps with varying and κ := V 8 Conclusions A comprehensive introduction to diffusion-based structural models was given The development of this approach starting from the first structural model by Merton (1974) to the CreditGradesTM 18

19 model by Finger et al () was illustrated and basic pricing formulas for all models were derived Gradually all shortfalls of the first structural model were eliminated except for the failure to provide realistic CDS spreads for short maturities A brief outlook to structural models with jumps which finally are able to solve this problem was included References Black F Cox J: Valuing corporate securities: Some effects of bond indenture provisions Journal of Finance Vol 31 No 1976 pp Black F Scholes M: The pricing of options and corporate liabilities Journal of Political Economy Vol 81 No pp Bielecki T Jeanblanc M Rutkowski M: Credit Risk Modeling Lecture notes of M Jeanblanc Université d Evry 8 available at perso/jeanblanc/cours/credit_risk_modeling_notes pdf Chen N Kou SG: Credit spreads optimal capital structure and implied volatility with endogenous default and jump risk Mathematical Finance Vol 19 No 3 9 pp Finger C Finkelstein V Lardy JP Pan G Ta T Tierney J: CreditGrades Techincal Document Hilberink B Rogers LCG: Optimal capital structure and endogenous default Finance and Stochastics Vol 6 No pp Jones EP Mason SP Rosenfeld E: Contingent claims analysis of corporate capital structure: An empirical investigation Journal of Finance Vol 39 No pp Leland H: Corporate debt value bond covenants and optimal capital structure Journal of Finance Vol 49 No pp Leland H: Bond prices yield spreads and optimal capital structure with default risk Working Paper Haas School of Business UC Berkeley 1994 Leland H Toft KB: Optimal capital structure endogenous bankruptcy and the term structure of credit spreads Journal of Finance Vol 51 No pp Mai J-F: Pricing single-name CDS options: A review of standard approaches XAIA homepage article 14 available at: XAIACDSOptionspdf Mai J-F: The joint modeling of debt and equity: An introduction XAIA homepage article 1 available at: xaiacom/uploads/media/creditequityengl_1pdf 19

20 Merton RC: On the pricing of corporate debt: The risk structure of interest rates Journal of Finance Vol 9 No 1974 pp Merton RC: Option pricing when underlying stock returns are discontinuous Journal of Financial Economics Vol 3 No pp Merton RC: On the pricing of contingent claims and the Modigliani-Miller theorem Journal of Financial Economics Vol 5 No 1977 pp Modigliani F Miller MH: The cost of capital corporation finance and the theory of investment American Economic Review Vol 48 No pp Musiela M Rutkowski M: Martingale methods in financial modelling nd ed Springer Series in Stochastic Modelling and Applied Probability 5 Reiner E Rubinstein M: Breaking down the barriers Risk Magazine Vol 4 No pp 8 35 Shreve S: Stochastic calculus for finance II - Continuous time models Springer 4 Zhou C: The term structure of credit spreads with jump risk Journal of Banking and Finance Vol 5 No 11 1 pp 15 4