A Structural Credit-Risk Model based on a Jump Diffusion Matthias Scherer Department of Financial Mathematics University of Ulm Working Paper Preprint: This draft: December 2, 2005 First draft: July 1, 2005 Abstract In this paper, we generalize the pure diffusion approach for structural credit risk modeling by including jumps in the firm-value process. In pure diffusion models, the probability for a solvent company to default within a small interval of time is negligible, whereas a real company may face sudden financial distress. Our generalization allows those unpredicted extremal events, raising the probability for a solvent company to default within a small interval of time to a realistic level. Compared to a pure diffusion model, including jump risk increases credit spreads especially for small maturities. The resulting term structure of credit spreads is extremely flexible, hence, our model provides a powerful tool to fit a real spread curve. Evaluating bond prices in a jump-diffusion model is complicated, as the distribution of first-passage times is not available in closed form. We present two approaches to overcoming this problem. First, we derive a semi-analytical Monte Carlo simulation which is unbiased and efficient and allows all possible jump distributions. The algorithm only requires us to simulate the firm value at the times of jumps and not on a fine grid. Then, we analytically calculate bond prices conditioned on the simulated jumps. The second approach to first-passage times and bond pricing uses specific properties of jumps with two-sided exponential distribution. In this scenario it is possible to calculate the Laplace transform of survival probabilities. Those survival probabilities are then recovered numerically and used to price corporate bonds. The last section of this paper presents a method of obtaining parameter estimates based on observed bond prices. Our first results indicate that the overall volatility of the firm-value process is explained to a large extent by jumps, supporting the need for unpredicted jump risk in a realistic firm-value model.

1

2

1 1.1

Introduction Structural default models

105

110

In a structural credit-risk model default occurs when a company cannot meet its financial obligations, or in other words, when the firm value falls below a certain threshold. The company’s total liabilities are often used as default threshold; other interpretations are weighted averages of short- and long-term liabilities (KMV) or a minimum firm value which is required to operate the company (Black and Cox 1976). Based on this model, default probabilities needed to compute bond prices are derived. 95

100

Asset Values

90

Default Threshold

85

Default

0.0

0.2

0.4

0.6

0.8

1.0

Time

1.2

Pure diffusion models

0.00

0.05

Credit Spread

0.10

0.15

In a pure diffusion model the firm-value process is assumed to follow a geometric Brownian motion. Another issue is the amount of information available to an investor who is not involved with the company. From a mathematical point of view, it is convenient to assume the value process of the company to be perfectly observable. In this scenario, the distribution of first-passage times is well known, which allows closed-form expressions of zero-coupon bond prices depending essentially on the parameters of the diffusion. This simplicity turns out to be at the same time an advantage and a major weakness of pure diffusion models. If we compare the upper figure, showing a typical term structure of credit spreads in a pure diffusion model, to an observed term structure of real credit spreads in the lower figure, we notice that the pure diffusion model systematically underestimates credit spreads for 0

2

4

6

0.008 0.006 0.004 0.002

Industrial BBB Industrial BBB + 14.09.2005

0.000

Credit Spread

0.010

0.012

Time to Maturity

0

2

4

6

Time to Maturity

8

10

1.3 Improving a pure diffusion model

3

bonds with small maturities. The reason behind this discrepancy is that even for small maturities, investors demand compensation for investing in corporate bonds and not in risk-free alternatives, whereas in a pure diffusion model the limit of credit spreads as maturity tends to zero is zero. Both the hump size structure and the vanishing limit of credit spreads at zero maturity are inherent features of pure diffusion models. The empirical credit spreads are as observed for BBB+ and BBB- rated companies in Europe. Data is obtained from Bloomberg at September 14, 2005.

1.3

Improving a pure diffusion model

Different approaches have been taken to overcome the shortfalls of pure diffusion modes. We introduce and briefly summarize them in the following categories.

1.3.1

Including jumps

The firm value of a company may be subject to sudden major changes, due to external shocks or other unpredicted events. Such an incident cannot be captured by a pure diffusion model, as all trajectories of a geometric Brownian motion are continuous. Therefore, it is reasonable to leave the class of continuous processes and use a discontinuous L´evy process as model for the value of the firm. Allowing jumps makes it possible for companies to default within any interval of time, and the resulting default time is no longer predictable. L´evy processes can be obtained by subordinating a pure diffusion model or by directly modeling the firm value as a L´evy process. In both approaches, the distribution of first-passage times is no longer analytically obtainable, which complicates the derivation of bond prices. A reference for a jump-diffusion model is Zhou 2001; Cariboni and Schoutens 2004 present a variance-gamma approach.

1.3.2

Working with a reduced or blurred filtration

Another approach is to consider the question of what investors actually observe and how this can be modeled. Companies have the obligation to publish their balance sheet every quarter. For outsiders, it is difficult to obtain reliable information about the firm value in between those dates. Hence, a reasonable model assumption resulting in a reduction of information is to allow observations of the value process only at preassigned dates. A second information-based approach is to blur the value process by adding noise which can be interpreted as accounting inaccuracy. The pioneering publication in this field is Duffie and Lando 1997, who allow investors

4 to observe a disturbed version of the value process at different times. The resulting default intensity is positive and so is the limit of credit spreads for small maturities. In Giesecke 2001, the default threshold is assumed to be an unobservable random variable. Here, the default intensity is positive as long as the value process is around its running minima. C ¸ etin, Jarrow, Protter and Yildirim 2004 model default based on the company’s cash-flow process and allow investors to observe the signum of this process.

2

Model description

We model the value of a company as a stochastic process V = {Vt }t≥0 on the filtered probability space (Ω, F , F, IP) , where Vt = v0 exp(Xt ),

v0 > 0.

We denote by F = {Ft }t≥0 the natural filtration of the process V , augmented so to satisfy the usual conditions of completeness and right continuity, i.e. Ft = σ(Vs : 0 ≤ s ≤ t) = σ(Xs : 0 ≤ s ≤ t). The process X = {Xt }t≥0 is a jump-diffusion process given by Nt X Yi . Xt = γt + σWt + i=1

The sequence of jump sizes {Yi}i≥1 is i.i.d. with two-sided exponential density f (x) = pλ⊕ e−λ⊕ x 1{x>0} + (1 − p)λ⊖ eλ⊖ x 1{x<0} .

(1)

V(t)

94

96

98

100

102

104

106

Jump sizes, Nt and Wt are mutually independent. Figure 1 presents a realization of the process V with parameters γ = .02 , σ = .05 , p = .5 , λ = 3 , λ⊕ = λ⊖ = 30 and v0 = 100 . This jump-diffusion process was introduced in the financial literature by Kou 2002, who used it as a model for stock prices. We assume the company to default when its value process falls below Figure 1: A realization of V its debt level d . In this case, the holder of a bond receives some recovery payment. As recovery scheme, we use fractional recovery of face value at the time of default. Hence, our model consists of the following parameters: 0.0

0.2

0.4

0.6

Time

γ = The linear trend of the diffusion component.

0.8

1.0

2.1 Basic properties of X and V

5

σ = The volatility of the diffusion component. λ = The jump intensity. p = The probability for a jump to be positive. λ⊕ , λ⊖ = The parameters for positive resp. negative jumps. v0 = The initial value of the company. d = The debt level. R = The recovery rate. r = The risk-free interest rate.

2.1

Basic properties of X and V

The following section provides a summary of all properties of X and V which are later used in our credit-risk model. Most results are obtained from elementary computations, so proofs are omitted or shortened.

2.1.1

The L´ evy triple of X

The L´evy density of the jump-diffusion process X is given by ν(dx) = λf (x)dx . The n -th absolute moment of Xt exists for some t > 0 or, equivalently for all R t ≥ 0 , if and only if |x|≥1 |x|n ν(dx) < ∞ . This is guaranteed for all n ∈ N by the exponential tails of the jump distribution. We let   p 1−p µc = γ + λ − (2) λ⊕ λ⊖ be the center of the process X , and obtain   p 2 IE[Xt ] = tµc , and Var(Xt ) = t σ + λ + λ2⊕   The moment-generating function satisfies IE eθXt = eG(θ)t ,  1 2 2 (1 − p)λ⊖ pλ⊕ G(x) = xγ + x σ + λ + 2 λ⊕ − x λ⊖ + x

1−p λ2⊖



.

where  −1 .

(3)

(4)

The L´evy-Khinchin representation of the characteristic function is given by   IE eizXt = etΨ(z) z ∈ R, with characteristic exponent

1 Ψ(z) = − z 2 σ 2 + izγ + λ 2



 (1 − p)λ⊖ pλ⊕ + −1 . λ⊕ − iz λ⊖ + iz

(5)

6 R In order to complete the L´evy triple of X , we first notice that |x|≥1 |x|ν(dx) < ∞ . This is convenient as we do not have to truncate large jumps, see Cont and Tankov 2004 p. 83. We therefore consider the center of the process X instead of a L´evy drift that depends on the choice of truncation function. This center µc is linked to the L´evy drift µ that corresponds to the often-used truncation function 1{|x|≤1} via      Z 1 1 −λ⊖ −λ⊕ 1+ − (1 − p)e . 1+ µc = µ + xν(dx) = µ + λ pe λ⊕ λ⊖ |x|≥1 Hence, we obtain      1 e−λ⊕ 1 e−λ⊖ −λ⊕ −λ⊖ µ=γ+λ p −e − + (p − 1) −e − λ⊕ λ⊕ λ⊖ λ⊖ Let us summarize the observations made above in the following lemma. Lemma 2.1 (L´ evy triple of X ) The L´evy triple of X is given by (µc , σ 2 , ν(dx)) , where the centered drift of Equation (2) is calculated without truncation function and the L´evy density is given by λf (x)dx .

2.1.2

Moments of V

The n -th moment of V exists if and only if the corresponding exponential moment of X is finite. IE[enXt ] is finite for some t or, equivalently for all t ≥ 0 , if and R only if |x|>1 enx ν(dx) < ∞ . In our scenario, this is fulfilled as long as λ⊕ > n , degenerated cases as p = 0 or λ = 0 excluded. The moment-generating function of Equation (4) and the characteristic exponent of Equation (5) provide two convenient methods for calculating these moments. We obtain    1 2 pλ⊕ (1 − p)λ⊖ IE[Vt ] = v0 exp γt + σ t + λt + −1 , 2 λ⊕ − 1 λ⊖ + 1
Var(Vt ) = v02 etG(2) − e2tG(1) .

3

First-passage times

In structural credit-risk models the problem of calculating the probability for the firm-value process not to fall below the default threshold arises naturally. We define the first-passage time τ = inf{t ≥ 0 : Vt ≤ d} and observe that     IP(τ > t) = IP inf Vs > d = IP inf Xs > log (d/v0 ) . (6) 0≤s≤t

0≤s≤t

3.1 The pure diffusion model

7

We denote the term x = − log (d/v0 ) as distance to default for X . Let us remark that this expression is not well-defined in the financial literature: sometimes a normalized distance is also referred to as distance to default. Due to Equation (6) we can work with the process X instead of the process V . Therefore, we define the running infimum and supremum of X as X∗t = inf Xs 0≤s≤t

and X ∗ t = sup Xs . 0≤s≤t

We also define the stopping times τb = inf{t ≥ 0 : Xt ≤ b, b < 0} and τ b = inf{t ≥ 0 : Xt ≥ b, b > 0}.

3.1

The pure diffusion model

The process V simplifies to Vt = v0 exp(γt + σWt ) in a pure diffusion model, which corresponds to λ = 0 in a jump-diffusion model. The pure diffusion model is not only contained in our model, the main reason for studying its properties is the idea to reduce the jump diffusion to a pure diffusion model by conditioning on the number and size of possible jumps. Let us therefore collect some results. Lemma 3.1 (The minimum of a Brownian motion) The running minimum of a Brownian motion with drift is inverse Gaussian distributed, and so are first-passage times in a pure diffusion model. More precisely, the probability for a Brownian motion with drift starting at x to remain above the threshold b over an interval of length t is given by   BM Φb (x, t) = IPx min σWs + γs > b 0≤s≤t      x − b + γt −(x − b) + γt −2γ(x−b)σ−2 √ √ = 1{x>b} Φ −e Φ , σ t σ t where IPx denotes the measure under which W0 = x . Hence, with x = − log (d/v0 ) , IP(τ > t) = Φ



x + γt √ σ t



−2γxσ−2

−e

Φ



−x + γt √ σ t



.

(7)

A proof of this lemma is given in Musiela and Rutkowski 2004 on page 581. Later, we will also need the probability for a Brownian bridge not to fall below a certain threshold. This result can be found in Borodin and Salminen 1996 on page 63 for the maximum of a Brownian bridge or in Karatzas and Shreve 1997 on page 265. Let us state it for the reader’s convenience.

3.2 Monte Carlo simulation of first-passage times

8

Lemma 3.2 (The minimum of a Brownian bridge) The probability for a Brownian bridge pinned at x and y spanning over an interval of length t not to fall below the threshold b is given by   BB Φb (x, y, t) = IPx min σWs + γs > b σWt + γt = y 0≤s≤t    2(y − b)(x − b) , (8) = 1{x>b,y>b} 1 − exp − tσ 2 where IPx denotes the measure under which W0 = x .

3.2

Monte Carlo simulation of first-passage times

Finding the distribution of τ in a jump-diffusion setting is a delicate issue. In general, closed-form solutions are not known for commonly used jump distributions. A probabilistic approach to estimating survival probabilities is to perform a Monte Carlo simulation. The algorithm of our choice is a variant of an algorithm for pricing barrier options; a description in this context can be found in Cont and Tankov 2004 on page 177 or in Metwally and Atiya 2002. The idea of our algorithm is as follows. To efficiently estimate passage probabilities of X it is sufficient to simulate the times of the jumps 0 < τ1 < . . . < τNT < T and the process X , respectively its left limit, at those times. In a second step, we calculate the probabilities for Brownian bridges that connect those jumps not to fall below the passage level. More precise, we rewrite the survival probability of the process X conditioned on the number of jumps. This gives ∞   X IP(τb > T ) = IE 1{X∗T >b} = IP(X∗T > b|NT = k)IP(NT = k). k=0

Knowing the number of jumps allows us to rewrite IP(X∗T > b|NT = k) by conditioning on the location of the jumps, the size of the jumps and the increments of the pure diffusion in between two jumps. Conditioned on the number of jumps the jump times are distributed as order statistics. Jump sizes are assumed to be i.i.d. with density f , and the increments of the pure diffusion are Gaussian distributed with mean γ∆tj and variance σ 2 ∆tj . This yields for k ≥ 1 Z

Z

Z

(t1 ,...,tk ) (x1 ,...,xk ) (y1 ,...,yk ) ∈[0,T ]k ∈(−∞,∞)k ∈(−∞,∞)k

k k! Y BB Φ (Xtj−1 , Xtj − , ∆tj )· 1{0
ΦBM (Xtk , T − tk )ϕγ∆tj ,σ2 ∆tj (yj )f (xj )d(y1 , . . . , yk )d(x1 , . . . , xk )d(t1 , . . . , tk ), b

3.2 Monte Carlo simulation of first-passage times

9

P where t0 = 0 and we use the abbreviations ∆tj = tj − tj−1 , Xtj = ji=1 (xi + yi ) Pj−1 (xi + yi) + yj . The density of a normal distribution is denoted by and Xtj − = i=1 ϕµ,σ2 . This artificial reformulation of the survival probability not only motivates the following simulation, it also shows that the algorithm is unbiased. Algorithm 3.1 (Monte Carlo simulation of first-passage times) Repeat the following steps K times and calculate the average over the resulting conditioned survival probabilities {SPn }n=1,...,K . We then obtain the estimate IP(τb > T ) ≈

K 1 X SPn . K n=1

1. Simulate the pure jump part (a)-(c) and the diffusion component (d) of X : (a) The number of jumps NT ∼ Poi(λT ) . (b) The jump times 0 < τ1 < . . . < τNT < T , uniformly distributed over the interval [0, T ] . (c) The jump sizes at τi , distributed with density f from Equation (1) or, for a different model with the respective distribution. (d) Let τ0 = 0 and τNT +1 = T . Simulate the increments of the diffusion part Xτi − − Xτi−1 ∼ N (γ∆τi , σ 2 ∆τi ) . 2. Calculate each conditioned survival probability SPn : (a) Let F ∗ be given by F ∗ = σ {0 < τ1 < . . . < τNT < T ; X0 , . . . , Xτi − , Xτi , . . . , XT } . (b) Check whether Xτi − ≤ b or Xτi ≤ b for some i ∈ {0, 1, . . . , NT + 1} . If so, let SPn = 0 and skip (c). (c) If no passage is observed at any τi , calculate the probability of each Brownian bridge connecting Xτi−1 with Xτi − not to fall below the level b . This yields SPn = IP(X∗T > b|F ∗ )   NY T +1  2(Xτi−1 − b)(Xτi− − b) = 1 − exp − σ 2 ∆τi i=1 =

NY T +1 i=1

ΦBB b (Xτi−1 , Xτi − , ∆τi ).

3.3 The Laplace transform of first-passage times

10

If we compare Algorithm 3.1 to a Monte Carlo simulation that is based on simulations of complete trajectories of X on a discrete grid, we observe two advantages. First of all, our algorithm is much faster. In each iteration run, X has to be simulated only at a few points and not on a fine grid. Moreover, even on the smallest grid one would overestimate the survival probability IP(τb > T ) , as there is always a small probability for X to fall below b between two points of the grid where X lies above b . This results in a systematic bias, whereas our algorithm is unbiased. In the next chapter, we introduce a second non-probabilistic approach to challenge the problem of finding the distribution of τb . Those approaches are compared in Table 6.1 for different choices of parameters.

3.3

The Laplace transform of first-passage times

Due to the memoryless property of the exponential distribution it is possible to calculate the Laplace transform of IP(τb ≤ t) explicitly. Later, we numerically recover IP(τb ≤ t) from this transform. Using integration by parts, we obtain ϕ(α) =

Z

0

∞ −αt

e

1 IP(τb ≤ t)dt = α

Z

0

∞

e−αt dIP(τb ≤ t) =

1 IE[e−ατb ]. α

(9)

Theorem 3.1 presents an analytical expression of the expectation in Equation (9). To derive this result, we begin with the following lemma, which involves the function G(x) of Equation (4). Lemma 3.3 (Roots of G(x) − α ) Kou and Wang 2003 establish that for α > 0 , the function G(x) − α has exactly four roots. We denote them by β1,α , β2,α , −β3,α and −β4,α . Moreover, all roots are real and satisfy 0 < β1,α < λ⊕ < β2,α < ∞ and 0 < β3,α < λ⊖ < β4,α < ∞. Later, we need an efficient method of calculating those roots with high precision and speed. We therefore propose to solve the equivalent polynomial equation p(x) = ax4 + bx3 + cx2 + dx + e = 0, with coefficients a = − 21 σ 2 , b = 21 σ 2 (λ⊕ − λ⊖ ) − γ , c = γ(λ⊕ − λ⊖ ) + 21 σ 2 λ⊕ λ⊖ + λ + α , d = λp(λ⊕ + λ⊖ ) + (γλ⊕ + α)λ⊖ − λ⊕ (λ + α) and e = −αλ⊕ λ⊖ instead. Kou and Wang 2003 obtained the expectation i h b IE e−ατ = A1 e−bβ1,α + B1 e−bβ2,α ,

3.3 The Laplace transform of first-passage times where A1 =

λ⊕ − β1,α β2,α , λ⊕ β2,α − β1,α

B1 =

11

β2,α − λ⊕ β1,α . λ⊕ β2,α − β1,α

We alter their proof to obtain the Laplace transforms of IP(τb ≤ t) . This yields Theorem 3.1 (The Laplace transform of IP(τb ≤ t) ) Fix α > 0 and b < 0 . Then IE[e−ατb ] = A2 ebβ3,α + B2 ebβ4,α ,

(10)

where A2 =

β4,α λ⊖ − β3,α λ⊖ β4,α − β3,α

and B2 =

β4,α − λ⊖ β3,α . λ⊖ β4,α − β3,α

Also the Laplace transform of IP(τb ≤ t) is easily obtained from Equation (9). Sketch of the proof: The proof works similarly to the one presented by Kou and Wang 2003 for the running maximum. We write for short βi = βi,α and define the function  1 x ≤ b, u(x) = β3 (b−x) β4 (b−x) A2 e + B2 e x > b. After some lengthy algebraic manipulations, we find −αu(x) + L(u)(x) = 0 ∀x > b, where the infinitesimal generator L of the jump diffusion X is given by Z ∞ 1 2 ′′ ′ L(u)(x) = σ u (x) + γu (x) + λ (u(x + y) − u(x)) f (y)dy 2 −∞ with jump density f from Equation (1). We approximate u using a sequence {un }n∈N of C 2 functions with properties un = u on x ≥ b , un = 1 on x ≤ b−1/n and un ≤ 2 . This gives for all x > b L(un )(x) = αu(x) + λ

Z

b−x

b−x−1/n

un (x + y)f (y)dy − λ

Z

b−x

u(x + y)f (y)dy,

b−x−1/n

which we use to establish | − αun (x) + Lun (x)| ≤

λλ⊖ n

∀x > b.

An application of Itˆo’s formula for jump processes gives Z t∧τb −α(t∧τb ) e−αs (−αun (Xs ) + Lun (Xs )) ds, e un (Xt∧τb ) = un (X0 ) + 0

3.3 The Laplace transform of first-passage times

12

from which it follows that Mtn

−α(t∧τb )

=e

un (Xt∧τb ) −

Z

0

t∧τb

e−αs (−αun (Xs ) + Lun (Xs )) ds

is a local martingale starting at un (0) = u(0) . By dominated convergence, M n is even a martingale. Hence,   Z t∧τb n −α(t∧τb ) −αs IE[Mt ] = IE e un (Xt∧τb ) − e (−αun (Xs ) + Lun (Xs )) ds = u(0). 0

By uniform convergence, we observe that the second summand vanishes as n tends to infinity. This gives   u(0) = IE e−α(t∧τb ) u(Xt∧τb )     = IE e−α(t∧τb ) u(Xt∧τb )1{τb <∞} + IE e−α(t∧τb ) u(Xt∧τb )1{τb =∞} .

Finally, we let t tend to infinity and use that u is bounded and u(Xτb ) = 1 on the set {τb < ∞} . We conclude     u(0) = IE e−ατb u(Xτb ) = IE e−ατb . 3.3.1

Gaver-Stehfest algorithm for Laplace inversion

Now that we found an explicit expression of the Laplace transform of IP(τb ≤ t) , we need an algorithm that recovers this probability from the transform. The GaverStehfest Algorithm has the advantage over most Laplace inversion algorithms that it purely works on the real line, which is convenient when implementing it. Advantages and disadvantages of this algorithm, and the following lemmata on which this method is based, are described in Abate and Whitt 1991. Lemma 3.4 (Gaver 1966) For a bounded and real-valued function f , continuous at t , we have     n log 2 (2n)! X (n + k) log 2 n k (−1) ϕ , f (t) = lim n→∞ t n!(n − 1)! k=0 k t where ϕ denotes the Laplace transform of f . In what follows, we denote the sequence of functions inside the limit by f˜n . Lemma 3.5 (Stehfest 1970) A better sequence of weights was found by Stehfest. He showed that with f˜n , defined as before in Lemma 3.5, we can approximate f using fn∗ (t)

=

n X k=1

w(k, n)f˜k (t),

(−1)n−k k n . where w(k, n) = k!(n − k)!

13 He also presents the asymptotic result fn∗ (t) − f (t) = o(n−k ) for all k. Algorithm 3.2 (Laplace transformation of passage times) We approximate IP(τb ≤ t) by IP(τb ≤ t) ≈

fn∗ (t)

=

n X

w(k, n)f˜k+B (t),

B = 2,

k=1

where B ≥ 0 is the burning-out number as discussed by Kou and Wang 2003. This approximation converges very quickly: we found that n = 9 is accurate enough for our problem. Nevertheless, the algorithm is sensitive to the precision of which the roots of G(x) − α are calculated. Therefore, we propose to apply Lemma 3.3 for more numerical stability, as finding the roots of a polynomial can be done efficiently and with high precision. We obtained a good performance using the Pegasus Algorithm, which is described in Engeln-M¨ ullges and Reuter 1991 on page 34. Table 6.1 at the end of this text compares Algorithm 3.2 with the Monte Carlo simulation of Algorithm 3.1 for a selection of different parameters.

4 4.1

Bond pricing Coupon bonds

Pricing a corporate bond with face value F and promised coupon payments qj at 0 < t1 < . . . < tn = T reduces to the problem of pricing several zero-coupon bonds, as we can replicate each coupon bond using a linear combination of zero-coupon bonds. More precisely, we have B(t1 < . . . < tn , q) =

n−1 X

qj φ(0, tj ) + (F + qn )φ(0, tn ),

(11)

j=1

where φ(0, tj ) denotes a zero-coupon bond with face value one and maturity tj . Therefore, we can restrict our focus on the analysis of zero-coupon bonds, even if the vast majority of corporate bonds promise coupon payments periodically.

4.2

Zero-coupon bonds

We now consider the problem of pricing zero-coupon bonds within our framework. We assume that the holder of a zero-coupon bond receives one Euro if the company

4.2 Zero-coupon bonds

14

survives up to maturity T . Otherwise, the investor receives as recovery payment a fraction R ∈ [0, 1) of his outstanding face value at the time of default. This yields for the price of a zero coupon with maturity T   φ(0, T ) = IE 1{τ >T } e−rT + R1{τ ≤T } e−rτ , (12)

where r is the risk-free interest rate and τ the time of default. This recovery scheme is often called fractional recovery of face value at default.

4.2.1

Pricing in a pure diffusion model

In a pure diffusion model, it is possible to explicitly evaluate Equation (12). To begin with, we rewrite the pricing formula as follows: Z T −rT φ(0, T ) = e IP(τ > T ) + R e−rt dIP(τ ≤ t). 0

The distribution of τ was derived in Equation (7), and is now used to evaluate the expression above. We recall that with x = − log (d/v0 )     −x + γT x + γT −2γxσ−2 √ √ −e Φ , IP(τ > T ) = Φ σ T σ T     −x − γt −x + γt −2γxσ−2 √ √ IP(τ ≤ t) = Φ +e Φ . σ t σ t To evaluate the integral, we make use of a result which is presented in Bielecki and Rutkowski 2002 on page 74. Lemma 4.1 For real numbers a , b and c , satisfying b < 0 and c2 > a , we have   Z y d+c b − cx d−c ax √ = e dΦ g(y) + h(y), x 2d 2d 0 where we use the abbreviations   √ b − dy b(c−d) Φ d = c2 − 2a, g(y) = e √ y

b(c+d)

and h(y) = e



b + dy Φ √ y

Finally, a lengthy but straightforward calculation shows      −b + γT b + γT −rT 2γbσ−2 √ √ Φ φ(0, T ) = e −e Φ + σ T σ T      b + γ˜ T b − γ˜ T −bσ−2 (˜ γ −γ) −bσ−2 (γ−˜ γ) √ √ +e Φ R e Φ σ T σ T
−b(˜ γ −γ)σ−2 BM = e−rT ΦBM (0, T ) + Re 1 − Φ (0, T ) . b b,γ7→γ ˜



.

(13)

4.2 Zero-coupon bonds

15

p where γ˜ = γ 2 + 2rσ 2 and b = log(d/v0 ) . The notation γ 7→ γ˜ in ΦBM b,γ7→γ ˜ denotes a changed drift in the computation of the respective survival probability in Lemma 3.2.

4.2.2

Zero-coupon bond pricing via Monte Carlo simulation

The idea of our Monte Carlo simulation to efficiently estimate φ(0, T ) in a jumpdiffusion model is described in Metwally and Atiya 2002 in the context of pricing barrier options. We use most of their notations and adapt their algorithm to our problem of pricing a zero-coupon bond. The outline of the algorithm is as follows. We only simulate the times of the jumps 0 < τ1 < . . . < τNT < T and the jump diffusion X , respectively its left limits, at those times, i.e. {X0 , . . . , Xτi − , Xτi , . . . , XT } . Here, we use that Xτi − − Xτi−1 ∼ N (γ∆τi , σ 2 ∆τi ) and Xτi − Xτi − is distributed with density f from Equation (1). We again let F ∗ be given by F ∗ = σ {0 < τ1 < . . . < τNT < T ; X0 , . . . , Xτi − , Xτi , . . . , XT } . The probability for a Brownian bridge anchored at Xτi−1 and Xτi − not to fall BB below the default level b is denoted by ΦBB b (i) = Φb (Xτi−1 , Xτi − , ∆τi ) ; an explicit formula is presented in Equation (8). Metwally and Atiya 2002 calculate the density of the first-passage time conditioned on the endpoints of a Brownian bridge. They show that with Ct , defined to be the event that the process passes the barrier b for the first time in the interval [t, t + dt] , we have gi(t) = IP(Ct ∈ dt|Xτi−1 , Xτi − ) =

where

Xτi−1 − b · 2yπσ 2 (t − τi−1 )3/2 (τi − t)1/2   (Xτi − − b − γ(τi − t))2 (Xτi−1 − b + γ(t − τi−1 ))2 , − exp − 2(τi − t)σ 2 2(t − τi−1 )σ 2   (Xτi−1 − Xτi − − γ∆τi )2 1 y=√ . exp − 2σ 2 ∆τi 2πσ 2 ∆τi

We let I be the index of the first jump such that XτI crosses the barrier, i.e.  I = min i ∈ N : Xτj− > b, j = 1, . . . , i; Xτj > b, j = 1, . . . , i − 1; Xτi ≤ b ,

and let I = 0 if no passage is observed at any jump time. We also introduce U=



I if I 6= 0, NT + 1 if I = 0.

4.2 Zero-coupon bonds

16

We can now evaluate Equation (12) conditioned on the outcome of the simulated BB jumps. We obtain with ΦBB b (j) = Φb (Xτj−1 , Xτj − , ∆τj ) and b = log(d/v0 )   ˆ T ) = IE 1{τ >T } e−rT + R1{τ ≤T } e−rτ |F ∗ φ(0, ! Z τi U Y i−1 I X Y = R ΦBB e−rs gi (s)ds + 1{I6=0} e−rτI ΦBB b (j) b (j) τi−1

i=1 j=1

−rT

+1{I=0} e

NY T +1

j=1

ΦBB b (j).

(14)

j=1

Algorithm 4.1 (Monte Carlo pricing of zero-coupon bonds) Choose the number of simulation runs K and estimate φ(0, T ) as K 1 Xˆ φn (0, T ), φ(0, T ) ≈ K n=1

where each φˆn (0, T ) is calculated as described in Equation (14). R Let us remark that evaluating e−rs gi(s)ds is computationally expensive, due to the complicated structure of the function gi . Metwally and Atiya 2002 propose a Taylor approximation of the integral in r which can be integrated analytically after some algebraic manipulations. We compared a numerical integration with their Taylor approximation and found that, at least for reasonable and hence small interest rates, the second approach is only marginally biased downward but significantly faster.

4.2.3

Zero-coupon bond pricing via Laplace transform

In this approach, we approximate the Riemann-Stieltjes integral of Equation (12) as a simple Riemann-Stieltjes sum, where the integrator is evaluated at the points of the partition using the inverse Laplace method described in Algorithm 3.2. Algorithm 4.2 (Laplace pricing of zero-coupon bonds) Partition the interval [0, T ] equidistant with mesh T /n and use Algorithm 3.2 to calculate the respective probabilities of default within two points of the partition. The price of the bond is then approximated via Z T −rT φ(0, T ) = e IP(τ > T ) + R e−rt dIP(τ ≤ t) −rT

≈ e

IP(τ > T ) + R

0 n X j=1

T −r j−0.5 n

e



IP τ ∈



(j − 1)T jT , n n



. (15)

4.3 Credit spreads for small maturities

17

Table 6.2 provides a comparison of zero-coupon bond prices computed with Algorithm 4.1 and Algorithm 4.2 for different parameters.

4.2.4

The credit spread of a zero-coupon bond

The credit spread of a zero-coupon bond φ with maturity T is defined to be the real number ηT that solves the relation φ(0, T ) = e−(r+ηT )T .

(16)

0.06 0.04 0.00

0.02

Credit Spread

0.08

0.10

For both, the pure and the jumpdiffusion model we can calculate credit spreads from the respective pricing formulae. Typical term structures of credit spreads are given in Figure 2. As parameters for the jump diffusion, we choose γ = .02 , σ = .05 , p = .5 , λ = 2 , λ⊕ = λ⊖ = 20 , d/v0 = 90% , R = 42% and r = .02 . The parameters for the pure diffusion XtD = γD t+σD Wt Figure 2: Spreads with and w/o jumps are chosen such that γD = IE[X1 ] and 2 σD = Var(X1 ) to provide comparable results. We observe that unlike in the pure diffusion model, credit spreads for small maturities do not vanish in a jumpdiffusion model. Heuristically, this is due to the possibility of a default-triggering downward jump, even for small maturities. This feature of our model is discussed more detailed in the next section. 0

1

2

3

4

5

Time to Maturity

4.3

Credit spreads for small maturities

In the presence of negative jumps, the first-passage time is no longer a predictable stopping time. A detailed discussion of this issue can be found in Jarrow and Protter 2004. Therefore, we assume that credit spreads in our model do not vanish as time to maturity tends to zero. A less theoretical approach to the same presumption is simply to compute the term structure of credit spreads in our model and to observe the positive limit of credit spreads at zero, as we did in Figure 2. This feature of our model coincides with real credit spreads and fixes the often criticized small credit spreads for small maturities in a pure diffusion model. In what follows, we will even derive the exact limit of credit spreads in our model.

4.3 Credit spreads for small maturities 4.3.1

18

The local default rate of τ

In all structural credit-risk models, the conditional probability that default occurs within h units of time tends to zero as h does. This limit is even independent of the information on which the computation of the conditional probability is based. What distinguishes our model from pure diffusion models is the rate of convergence given full information F . In a pure diffusion model, we apply Equation (7) and l’Hospital’s rule and observe for a solvent company with xt = − log (d/Vt )      −xt + γh 1 −xt − γh 1 −2γxt σ−2 √ √ +e Φ = 0. lim IP(τ ≤ t + h|Ft ) = lim Φ hց0 h hց0 h σ h σ h (17) In pure diffusion models, this fact forces credit spreads of zero-coupon bonds to tend to zero as maturity decreases to zero; compare Duffie and Lando 2000. In our model we obtain a positive limit which only depends on the parameters of the jump component and the distance to default. Theorem 4.1 (The local default rate of τ ) At any time t ≥ 0 , the distance to default for X is given by xt = − log (d/Vt ) . Based on full information F , we obtain for τ > t 1 lim IP(τ ≤ t + h|Ft ) = λ(1 − p) (d/v0 )λ⊖ e−λ⊖ Xt = λ(1 − p)e−λ⊖ xt , IP-a.s. hց0 h Proof: X has independent and stationary increments, hence   IP(τ ≤ t + h|Ft ) = IP inf Xt+s − Xt ≤ −xt Ft = IP (X∗h ≤ −xt ) , IP-a.s. 0≤s≤h

This justifies that we may consider w.l.o.g. the limiting behavior of h1 IP(τ ≤ h) . We condition on the number of jumps that occurred up to time h and obtain ! Ns X 1 1 lim IP(τ ≤ h) = lim IP inf γs + σWs + Yj ≤ −x0 hց0 h hց0 h 0≤s≤h j=1 ∞   1X = lim IP(Nh = n)IP X∗h ≤ −x0 Nh = n . hց0 h n=0

The following conditional probabilities are examined separately. We condition on the events that up to time h , we observe either no jump, exactly one jump or more then one jump. For the case Nh = 1 , we even condition on whether the jump is positive or negative. This yields   e−λh IP inf γs + σWs ≤ −x0 . . . = lim 0≤s≤h hց0 h

4.3 Credit spreads for small maturities

19

 + lim λpe IP inf γs + σWs + 1{s≥τ1 } Y1 ≤ −x0 Nh = 1, Y1 > 0 hց0 0≤s≤h   −λh + lim λ(1 − p)e IP inf γs + σWs + 1{s≥τ1 } Y1 ≤ −x0 Nh = 1, Y1 < 0 hց0 0≤s≤h ! ∞ Ns X 1 X e−λh (λh)n + lim Yj ≤ −x0 Nh = n . IP inf γs + σWs + hց0 h 0≤s≤h n! n=2 j=1 −λh



We saw in Equation (17) that the first limit is zero, and obviously so is the second. Considering the last limit, a dominated convergence argument allows us to interchange limit and summation, establishing that this limit also equals zero. Let us now examine the third limit, the case of one negative jump. We approximate this probability from below by assuming that default can only occur at h . An upper bound is derived by adding the negative jump already at time zero. More precisely, we have  
⊖ ⊖ IP γh + σWh − Y ≤ −x0 ≤ IP inf γs + σWs − 1{s≥τ1 } Y ≤ −x0 Nh = 1 0≤s≤h   ⊖ ≤ IP inf γs + σWs − Y ≤ −x0 . 0≤s≤h

The sequence of events Ah = {ω ∈ Ω : inf 0≤s≤h γs + σWs − Y ⊖ ≤ −x0 } is decreasing in h . Therefore, by continuity of the probability measure we obtain for the limit of upper bounds lim IP(Ah ) = IP(A0 ) = IP(−Y ⊖ ≤ −x0 ) = e−λ⊖ x0 .

hց0

To show that the sequence of lower bounds has the same limit, we make use of Equation (B12)√of Kou 2002 and √ Lemma 4.1 of Kou and Wang 2003. We let −1 α = λ⊖ , β = (σ h) , δ = −σ hλ⊖ and c = −(xt + γh) and obtain 

 λ⊖ 1 σ2 hλ2⊖ lim IP(γh + σWh − Y ≤ −xt ) = lim 1 − √ e 2 I0 (c; α; β; δ) = hց0 hց0 2π !! √ α2 2π αδ λ⊖ 1 σ2 hλ2⊖ eαc α + lim 1 − √ e 2 e β 2β 2 Φ(−βc + δ + ) = − Hh0 (βc − δ) + hց0 α α β 2π !! √ √ α2 2πeαc 2π αδ λ⊖ 1 σ2 hλ2⊖ α + Φ(βc − δ) + e β 2β 2 Φ(−βc + δ + ) = − lim 1 − √ e 2 hց0 α α β 2π ⊖

. . . replace α, β, δ, c and let h tend to 0 . . .

=

e−λ⊖ xt .

20

0.3 0.1

95

Asset Values

0.2

100

Local Default Rate

105

110

0.4

4.4 The limit of credit spreads at the short end of the term structure

90

Default Threshold

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

Time

0.6

0.8

1.0

Time

Figure 3: A sample path of V and the corresponding local default rate of τ . As parameters, we choose γ = .02 , σ = .05 , p = .5 , λ = 2 and λ⊕ = λ⊖ = 30 .

4.4

The limit of credit spreads at the short end of the term structure

In the next theorem, we calculate the limit of credit spreads in our jump-diffusion model as time to maturity decreases to zero. This limit is found using the local default rate of τ , which allows us to write IP(τ ∈ (0, ds]) = λ(1−p) exp(−λ⊖ x0 )ds . Theorem 4.2 (Credit spreads at zero) ˆ = λ(1 − p)e−λ⊖ x0 and obtain We let λ ˆ − R). lim ηh = λ(1

hց0

Proof: We obtain from Equation (16) and l’Hospital’s rule lim ηh

hց0

  Z h 1 −rh −rs = lim − log e p(0, h) + R e p(0, ds) − r hց0 h 0   Z h 1 −rh −rs ˆ ˆ e λds − r = lim − log e (1 − λh) + R hց0 h 0 ˆ + λe ˆ −rh − Re−rh λ ˆ re−rh (1 − λh) = lim −r Rh hց0 ˆ + R e−rs λds ˆ e−rh (1 − λh) 0 ˆ − R). = λ(1

21

5

Calibration of the model

We now discuss an approach on how parameter estimates can be obtained from market data for a given company. We denote the issued bonds of this company by B1 , . . . , Bn . For each bond Bi , we obtain model prices BiM from Equation (11) that depend on the set of parameters (γ, σ, λ, λ⊕ , λ⊖ , p) . The idea of our calibration is to choose the parameters such that model prices agree with market prices. More precisely, we seek to minimize the sum of squared distances of model to market prices. This estimate can be interpreted as the market’s agreement of the parameters. We obtain the minimization problem n   X
2 ˆ ˆ ˆ γˆ , σ ˆ , λ, λ⊕ , λ⊖ , pˆ = argmin Bi − BiM . i=1

All other parameters that occur in the pricing formula are obtained either from current market data, from publications of the company, or are estimated a priori.

5.1

Numerical details

Our optimization problem is to minimize a function of several variables with unknown partial derivatives. The parameters show functional dependence and different local minima make this computation difficult. Time needed to evaluate the sum of squared differences for a set of parameters depends heavily on the accuracy of the Laplace inversion algorithm. We obtained a good performance by gradually increasing this accuracy. The numerical routine of our choice is nag opt bounds no deriv of the NAG software packet.

5.2

Setup and results

We slightly modify the pricing formulae (11) and (15), replacing the constant risk-free interest r by a deterministic risk-free yield curve rt . This yield curve is obtained from market prices of ’Bundesanleihen’, as published by Stuttgart’s stock exchange1 for maturities t ∈ {0, . . . , 10} . We interpolate linearly for noninteger maturities. Companies publish their debt-to-value ratio every quarter; we obtained current numbers from Bloomberg. As recovery rates, one can either use estimates which are published by different rating agencies or historical values. In our analysis, we rely on historical numbers as published by Altman and Kishore 1996. We run an estimation for DaimlerChrysler and General Motors; the results are presented in the next section. 1

www.boerse-stuttgart.de

5.2 Setup and results 5.2.1

22

Setup and results for DaimlerChrysler (DCX)

Bond quotes are obtained from Stuttgart’s stock exchange; all prices include brokenperiod interest. Quotation date is 25.08.2005, 1pm. As input parameters we use a debt-to-value ratio of d/v0 = 82.3913% . This ratio is obtained from DCX’s balance sheet of the second quarter 2005. Altman and Kishore 1996 report an average recovery rate of 41% , averaged over all industrial sectors. Considering the business structure of DCX and reported recoveries for different industries, we felt we should choose R for DCX a notch higher, so we let R = 42% . The risk-free yield curve rt is obtained from ’Bundesanleihen’. Using this setup, we obtain Table 1: Estimated parameters for DCX γˆ

σ ˆ

pˆ

ˆ λ

λˆ⊕

λˆ⊖

.004449 .020260 .476894 .852775 35.7906 28.5123

P14

M i=1 (Bi

− Bi )2

.390455

Table 2: Model and market prices: DCX WKN

Coupon

Maturity

Model Price

Market Price Difference

369293 4.625 10.03.2006 103.306861 103.19 -.001132 611867 6.125 21.03.2006 104.712257 104.67 -.000404 689080 5.625 06.07.2006 103.498493 103.42 -.000759 A0DB7Z 2.000 05.09.2006 101.547979 101.54 -.000079 907882 3.750 02.10.2006 104.799059 104.73 -.000659 829942 5.625 16.01.2007 107.656132 107.46 -.001825 A0DHP3 2.475 16.03.2007 101.017892 101.22 .001997 851890 6.125 27.03.2007 108.038199 107.80 -.002210 A0BD90 2.608 02.07.2007 100.416606 100.78 .003606 A0DZP6 3.125 10.03.2008 102.317418 102.39 .000709 765013 3.750 04.06.2008 103.311009 103.19 -.001173 A0ACD4 4.125 23.01.2009 106.146911 106.19 .000406 611868 7.000 21.03.2011 121.048224 120.84 -.001723 A0DDFR 4.250 04.10.2011 107.844546 108.02 .001624 The difference is calculated as (Market Price-Model Price)/Market Price .

5.2 Setup and results 5.2.2

23

Setup and results for General Motors (GM)

The setup and data sources for GM are as those of DCX. GM reports in 2/2005 a debt-to-value ratio of d/v0 = 94.47131% . We again choose R = 42% . The list of bonds used to estimate the parameters is given below, omitting bonds in currencies other than Euro and longer times to maturity. We obtain Table 3: Estimated parameters for GM γˆ

σ ˆ

pˆ

ˆ λ

λˆ⊕

λˆ⊖

-.003913 .012313 .505527 .373096 96.5989 38.3502

P20

M i=1 (Bi

− Bi )2

2.497470

Table 4: Model and market prices: GM WKN

Coupon

Maturity

Model Price

Market Price

Difference

183098 7.000 15.11.2005 106.142388 106.21 .000637 291815 4.000 09.02.2006 102.234274 102.48 .002398 610260 5.750 14.02.2006 103.897810 104.15 .002421 776306 4.174 03.03.2006 102.130616 102.22 .000874 A0BC23 2.625 14.06.2006 99.430427 99.43 -.000004 908510 4.375 26.09.2006 104.190749 104.50 .002959 748413 6.000 16.10.2006 107.077676 107.15 .000675 A0DACL 3.040 15.02.2007 99.598528 98.60 -.010127 850892 6.125 15.03.2007 105.091735 105.23 .001314 A0E8A5 4.125 02.06.2007 99.988892 100.45 .004590 A0E7D3 5.625 13.07.2007 102.130365 102.16 .000290 819413 4.750 16.07.2007 100.433439 100.22 -.002130 A0DCTX 2.923 14.09.2007 98.939518 99.27 .003329 A0DG6B 3.674 03.12.2007 99.648395 99.45 -.001995 894454 6.000 03.07.2008 101.925706 101.32 -.005978 905302 3.920 12.09.2008 98.760729 98.76 -.000007 A0BEAR 3.429 30.06.2009 90.925046 91.51 .006392 A0DCTY 4.750 14.09.2009 98.718413 98.60 -.001201 908511 5.750 27.09.2010 100.522887 100.24 -.002822 A0AWBL 5.375 06.06.2011 93.247923 93.43 .001949 The difference is calculated as (Market Price-Model Price)/Market Price .

5.2 Setup and results 5.2.3

24

A note on the results of the estimation

We first observe the excellent fitting capability of our model, both for small and long maturities. For all bonds, the difference of model to market prices is below bid-ask spreads. The estimated parameters for DCX (resp. GM) correspond to IE[X1 ] = .000166 and Var(X1 ) = .001277 (resp. IE[X1 ] = −.006771 and Var(X1 ) = .000297 ). It is interesting to notice that the total variance of Xt is explained to 32.15% (resp. 51.00% ) by the diffusion, the jump component accounting for the remainder. This indicates that including jumps in a traditional diffusion model is a realistic and useful generalization. The diffusion component explains the longterm behavior of credit spreads, while the jump component corrects unrealistically small credit spreads for small maturities.

25

6 6.1

Numerical examples Passage probabilities of the jump diffusion X

The following table compares the Monte Carlo simulation of Algorithm 3.1 with the inverse Laplace transform of Algorithm 3.2. The Monte Carlo simulation is based on 100,000 simulation runs, the inverse Laplace transform is evaluated up to summand n = 9 . We fix p = .5 and γ = .05 and vary the remaining parameters; the threshold is chosen to be b = −.2 . Let us remark that in the upper part of the table, where λ⊕ = λ⊖ = 2 , the jumps dominate the overall variance, compare Equation (3), while in the lower part with λ⊕ = λ⊖ = 30 the diffusion component is dominant. Table 5: Calculated and simulated passage probabilities IP(τb > t) Eq. (7)

Alg. 3.1

Alg. 3.2 Alg. 3.1 Alg. 3.2

λ⊖ = 2 t=1 t=5 t=1 t=5

λ=0 .999999 .999746 .756894 .512585

λ = .05 .983693 .934113 .747532 .491378

λ = .05 .983353 .933801 .746222 .491614

λ=3 .488645 .223129 .421392 .193907

λ=3 .489167 .225705 .421267 .192604

λ⊕ = 30 λ⊖ = 30 σ = .05 t=1 t=5 σ = .2 t=1 t=5

λ=0 .999999 .999746 .756894 .512585

λ = .05 .999884 .999294 .756055 .511517

λ = .05 .999895 .999355 .756314 .511945

λ=3 .983211 .922313 .723714 .477441

λ=3 .983234 .922913 .723888 .476614

λ⊕ = 2 σ = .05 σ = .2

6.2 Estimated and approximated bond prices

6.2

26

Estimated and approximated bond prices

In this table, we compare the Monte Carlo method 4.1 with the inverse Laplace approximation 4.2 for pricing a zero-coupon bond with face value one. Additionally, we calculate treasury and pure diffusion prices from Equation (13). As parameters for the value process of the underlying company, we choose γ = .02 , σ = .05 and R = 50% . The risk-free interest rate is assumed to be r = .02 . Each Monte R Carlo simulation is based on 100,000 simulation runs; the integrals gi (s)e−rs ds are numerically evaluated using the NAG routine d01ajc. The Laplace inversion is performed up to summand n = 9 ; the integral in the pricing formula is approximated with a mesh of 1/100 . Table 6: Estimated and approximated zero-coupon bond prices Treasury

Eq. (13) Alg. 4.1

Alg. 4.2 Alg. 4.1 Alg. 4.2

λ⊖ = 2 t=1 t=5 t=1 t=5

−rt

e .980199 .904837 .980199 .904837

λ=0 .980198 .902314 .973264 .852492

λ = .05 .972334 .870844 .963313 .822096

λ = .05 .972181 .870890 .963445 .821632

λ=3 .733749 .579972 .701662 .567019

λ=3 .734532 .579859 .702112 .566023

λ⊕ = 30 λ⊖ = 30 d = .8 t=1 v0 t=5 d = .9 t=1 v0 t=5

e−rt .980199 .904837 .980199 .904837

λ=0 .980198 .902314 .973264 .852492

λ = .05 .980142 .901643 .971927 .849456

λ = .05 .980149 .901612 .972038 .849276

λ=3 .972087 .840852 .904750 .730477

λ=3 .972141 .840733 .905834 .730540

λ⊕ = 2 d = .8 v0 d v0

= .9

References

27

References [1] Abate, J., Whitt, W., 1992. The Fourier-series method for inverting transforms of probability distributions, Queueing Systems 10, pp. 5-88. [2] Altman, E., Kishore, M., 1996. Almost everything you wanted to know about recoveries on defaulted bonds, Financial Analysts Journal, Vol. 52, No. 6, pp. 57-64. [3] Bielecki, T. and Rutkowski, M., 2002. Credit risk: Modeling, valuation and hedging, Springer. [4] Black, F. and Cox, C., 1976. Valuing corporate securities: Some effects of bond indenture provisions, Journal of Finance 31, pp. 351-367. [5] Borodin, A. and Salminen, P., 1996. Handbook of Brownian motion: Facts and formulae, Birkhauser. [6] Cariboni, J. and Schoutens, W., 2004. Pricing credit default swaps under L´evy models, UCS Report 2004-07. [7] C ¸ etin, U., Jarrow, R., Protter, P. and Yildrim, Y., 2004. Modeling credit risk with partial information, The Annals of Applied Probability, Vol. 14, No. 3, pp. 1167-1178. [8] Cont, R. and Tankov, P., 2004. Financial modelling with jump processes, Chapman & Hall/CRC. [9] Duffie, D. and Lando, D., 2000. Term structure of credit spreads with incomplete accounting information, Econometrica, Econometric Society, Vol. 69 (3), pp. 633-664. [10] Engeln-M¨ ullges, G. and Reutter, F., 1993. Numerik-Algorithmen mit ANSI C-Programmen, BI Wissenschaftsverlag. [11] Gaver Jr., D. P., 1966. Observing stochastic processes and approximate transform inversion, Operations Res. 14 pp. 444-459. [12] Giesecke, K., 2001. Default and information, Working Paper, Stanford University. www.stanford.edu/dept/MSandE/people/faculty/giesecke/publications.html

References

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[13] Jarrow, R. and Protter, P., 2004. Structural versus reduced form models: A new information based perspective, JOIM Vol. 2, No. 2. pp. 1-10. [14] Karatzas, I. and Shreve, S., 1997. Brownian motion and stochastic calculus, Springer, 2nd ed. [15] Kou, S., 2002. A jump-diffusion model for option pricing, Management Science, Vol. 48, pp. 1086-1101. [16] Kou, S. and Wang, H., 2003. First passage times of a jump diffusion process, Adv. Appl. Prob. 35, pp. 504-531. [17] Metwally, S. and Atiya, A., 2002. Using brownian bridge for fast simulation of jump-diffusion processes and barrier options, The Journal of Derivatives. Vol. 10, pp. 4354. [18] Musiela, M. and Rutkowski, M., 2004. Martingale methods in financial modelling, Springer, 2nd ed. [19] Protter, P., 2003. Stochastic integration and differential equations, Springer, 2nd ed. [20] Stehfest, H., 1970. Algorithm 368. Numerical inversion of Laplace transforms, Comm. ACM 13 pp. 479-49 (erratum 13, 624). [21] Zhou, C., 2001. The term structure of credit spreads with jump risk, Journal of Banking and Finance, Vol. 25, pp. 2015-2040.

Address for correspondence: Matthias Scherer University of Ulm - Department of Financial Mathematics Faculty of Mathematics and Economics Helmholtzstr. 18 89069 Ulm, Germany Phone: +49-731-5023517, Fax: +49-731-5031096 [email protected]