Efficient Pricing Routines of Credit Default Swaps in a Structural Default Model with Jumps Matthias Scherer Department of Financial Mathematics University of Ulm Working Paper Preprint: This draft: December 2, 2005 First draft: October 2, 2005

Abstract In this paper, we present two efficient algorithms for pricing credit default swaps based on a structural default model. In our model, the value of the firm is assumed to be the exponential of a jump-diffusion process. Our first algorithm to price a credit default swap within this framework is an efficient and unbiased Monte Carlo simulation. An excellent performance is obtained by first simulating the jump component and then analytically evaluating the pricing formula conditioned on the simulated jumps. This technique is flexible enough to allow all possible jump distributions. The second approach uses the Laplace transform of first-passage times. This transform is analytically tractable for a jump-diffusion process with two-sided exponentially distributed jumps. Combined with the Gavard-Stehfest algorithm for Laplace inversion, this provides a very fast method for finding default probabilities based on which credit default swap prices can be computed. Within the framework of a jump diffusion with two-sided exponentially distributed jumps, we present a sensitive analysis of the parameters and compute the exact limit of resulting par spreads at the short end of the term structure. In contrast to pure diffusion models, this term structure of par spreads has a non-zero limit that matches empirical observations.

1

2

1

Introduction to credit default swaps and structural default models

Credit default swaps (CDS) can heuristically be described as insurance contracts against credit risk. The protection buyer makes periodic payments over a predetermined number of years, while the protection seller commits to make a payment in the event of credit default of a reference entity. This contractual structure implies that default probabilities are the crucial factor in pricing CDS. We derive those default probabilities from a structural firm-value model which is introduced below. CDS are the most important credit derivatives, in both market activity and notational amount. Additionally to the original idea of buying or selling default risk, CDS are building blocks for different portfolio strategies and complex credit derivatives. A comprehensive introduction to the common use and to contractual variants of CDS is given in Bomfim 2005.

105

110

In a structural firm-value model default occurs when a company cannot meet its financial obligations, or in other words, when the firm value falls below a certain threshold. Based on this model, default probabilities needed to compute CDS prices are derived. In a traditional pure diffusion model the firm-value process is assumed to follow a geometric Brownian motion. In this scenario, the distribution of first-passage times is well known, which allows closed form expressions of CDS 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, as pure diffusion models systematically underestimate CDS spreads at the short end of the term structure. The reason behind this discrepance is that the firm value of a company may be subject to sudden major changes, due to external shocks or other unpredicted events. Such incidents can not be captured in a pure diffusion model as all trajectories of a geometric Brownian motion are continuous. To overcome this shortfall, we 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 solvent companies to default within any interval of time at a realistic rate. 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 tractable which complicates the derivation of bond and derivative prices. Our model takes up the approach of Zhou 2001; we add a poisson process to an existing pure diffusion model. However, 95

100

Asset Values

90

Default Threshold

85

Default

0.0

0.2

0.4

0.6

Time

0.8

1.0

3 while Zhou works with Gaussian jumps, we primarily focus on two-sided exponentially distributed jumps. This jump distribution is not symmetric, has semi-heavy tails and, most important, admits an analytical approximation of the distribution of first-passage times.

2

The firm-value model with jumps

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)

94

96

98

V(t)

100

102

104

106

Jump sizes, Nt and Wt are assumed to be 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 as a model for stock prices. The L´evy density of the jump-diffusion process X is given by Figure 1: A realization of V ν(dx) = λf (x)dx . The n -th absolute moment of Xt exists for some t > 0 or, equivalently for all t ≥ 0 , if and only if R |x|n ν(dx) < ∞ . This is guaranteed for all n ∈ N by the exponential tails of |x|≥1 the jump distribution. We let µc = γ + λ (p/λ⊕ − (1 − p)/λ⊖ ) be the center of X and obtain    p 1−p 2 IE[Xt ] = tµc and Var(Xt ) = t σ + λ + 2 . (2) λ2⊕ λ⊖ 0.0

0.2

0.4

0.6

Time

0.8

1.0

2.1 First-passage times

4

  The moment-generating function satisfies IE eθXt = eG(θ)t , where 1 G(x) = xγ + x2 σ 2 + λ 2



 pλ⊕ (1 − p)λ⊖ + −1 . λ⊕ − x λ⊖ + x

(3)

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

2.1



 (1 − p)λ⊖ pλ⊕ + −1 . λ⊕ − iz λ⊖ + iz

(4)

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

0≤s≤t



= IP





inf Xs > log (d/v0 ) .

0≤s≤t

(5)

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. 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}. 2.1.1

The first-passage time in a pure diffusion model

The firm-value process simplifies to Vt = v0 exp(γt + σWt ) in a pure diffusion model, which corresponds to λ = 0 in a jump-diffusion model. Not only is the pure diffusion model a special case of our model, the primary reason for studying its properties is the idea to reduce the jump diffusion to a pure diffusion model by conditioning on the number, the location and the size of possible jumps. Let us therefore collect some results.

5 Lemma 2.1 (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 and Φ(x) is the cumulative density function of the standard normal distribution. Hence, with x = − log (d/v0 )     x + γt −x + γt −2γxσ−2 √ √ IP(τ > t) = Φ −e Φ . (6) σ t σ t A proof of this lemma is given in Musiela and Rutkowski 2004 on page 581. Later, we additionally 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 running maximum of a Brownian bridge, the corresponding version for the running minimum is given below. Lemma 2.2 (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) = 1{x>b,y>b} 1 − exp − , (7) tσ 2 where IPx denotes the measure under which W0 = x .

3 3.1

Pricing credit default swaps The pricing formula

Having the structure of an insurance policy against default risk of a reference entity, pricing CDS is done using the actuarial principle of equivalence. Therefore, we consider the payment streams of both insurance buyer and insurance seller.

3.1 The pricing formula

6

To simplify calculations we assume the insurance buyer to continuously pay c as long as the reference entity is solvent, whereas the insurance seller indemnifies the insurance buyer by paying the difference of face value minus recovery R in the event of default. We discount all payments using a flat risk-free yield of r > 0 and obtain for a contract with face value one and maturity T :   Z T −rτ −rt CDS(0, T ) = IE e (1 − R)1{τ ≤T } − ce 1{τ >t} dt 0

= (1 − R)

Z

T

−rt

e

0

dIP(τ ≤ t) − c

Z

T

e−rt IP(τ > t)dt.

(8)

0

This formula reflects the view of the insurance buyer, the insurance seller uses the same formula with opposite signs. Using integration by parts, we establish Z T Z T −rt −rT e dIP(τ ≤ t) = 1 − e IP(τ > T ) − r IP(τ > t)e−rt dt, 0

0

which allows us to rewrite the CDS pricing formula as follows: Z T
. . . = ((R − 1)r − c) e−rt IP(τ > t)dt + (1 − R) 1 − e−rT IP(τ > T ) (9) 0

Z 
c c  T −rt e dIP(τ ≤ t) − 1 − e−rT IP(τ > T ) . = 1−R+ r 0 r

(10)

In practice, the insurance premium is usually chosen such that the contract can be entered at par. Therefore, we solve for c and consider this premium that allows both parties to enter the contract at zero cost as a function of time to maturity. So, the par spread of a contract with maturity T is defined as RT (1 − R) 0 e−rt dIP(τ ≤ t) , (11) cT = RT e−rt IP(τ > t)dt 0

We observe that in order to evaluate either one of the pricing Formulae (8)-(11), knowing the distribution of τ is indispensable. Evaluating these formulae in a pure diffusion model is relatively simple, due to Lemma 2.1. In what follows, we present two methods to efficiently evaluate the pricing formulae in a jump diffusion model.

3.1.1

The value process of a CDS

So far, we only considered the problem of pricing a CDS when its issued. As long as the reference entity is solvent, the market price of the CDS at 0 < t0 < T is given by Z T Z T −rt e−rt IP(τ > t)dt. e dIP(τ ≤ t) − c CDS(t0 , T ) = (1 − R) t0

t0

3.2 CDS pricing in a pure diffusion model

7

We can omit this formula by considering the replacement costs of this contract, i.e. we compute what it costs to replace the old contract by a new contract with time to maturity T − t0 . For the protection seller, the market value of the old contract agrees with the negative of its replacement costs. As the new contract is entered at zero cost with a par spread of cnew , we find T CDS(t0 , T ) =

3.2

(cnew T

− c)

Z

T

e−rt IP(τ > t)dt.

t0

CDS pricing in a pure diffusion model

In a pure diffusion model the distribution of τ is known to be inverse Gaussian, which makes it possible to evaluate Equation (10) explicitly. Neverteless, the calculation is long but simplifies significantly if we use Lemma 3.2.1 of Bielecki and Rutkowski 2002. We obtain 
c c CDS P D (0, T ) = 1 − R + A− 1 − e−rT B , (12) r r p where b = log(d/v0 ) , γ˜ = γ 2 + 2rσ 2 and −bσ−2 (˜ γ −γ)

A = e



b − γ˜ T √ Φ σ T



−bσ−2 (˜ γ −γ)

+e


−2 = e−bσ (˜γ −γ) 1 − ΦBM (0, T ) , b,γ7→γ ˜     −b + γT b + γT 2γbσ−2 √ √ B = Φ −e Φ σ T σ T



b + γ˜ T √ Φ σ T



,

= ΦBM (0, T ). b

The notation γ 7→ γ˜ in ΦBM b,γ7→γ ˜ denotes a changed drift in the computation of the respective survival probability in Lemma 2.1. The premium that allows both parties to enter the contract at par is given by cT =

3.3

r(1 − R)A . 1 − e−rT B − A

(13)

CDS pricing via Monte Carlo simulation

The idea of our Monte Carlo simulation to efficiently estimate CDS prices in a jump-diffusion model is to reduce the pricing problem to an analytically tractable problem in a pure diffusion scenario. This can be achieved using an algorithm that simulates the times of the jumps 0 < τ1 < . . . < τNT < T , the jump sizes Xτi −Xτi − and the increments of the diffusion in between two jumps Xτi − −Xτi−1 .

3.3 CDS pricing via Monte Carlo simulation

8

If we now let F ∗ = σ {0 < τ1 < . . . < τNT < T ; X0 , . . . , Xτi − , Xτi , . . . , XT } it is possible to evaluate the pricing formula conditioned on F ∗ . This technique is explained for instance in Metwally and Atiya 2002 in the context of pricing barrier options. Moreover, they derive the density of first-passage times conditioned on the endpoints of a Brownian bridge connecting two jumps of the process X . 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 − ) =

2yπσ 2 (t

Xτi−1 − b · − 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 where

  (Xτi−1 − Xτi − − γ∆τi )2 1 exp − . y=√ 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.

We can now evaluate Equation (10) conditioned on F ∗ and obtain 

−rτ

IE e

(1 − R)1{τ ≤T } −

Z

T −rt

ce

0

   c ˆ c  ˆ , 1{τ >t} dt F ∗ = 1 − R + A− 1 − e−rT B r r

BB where we use the abbreviations ΦBB b (j) = Φb (Xτj−1 , Xτj − , ∆τj ) and

Aˆ =

U i−1 X Y i=1

ΦBB b (j)

j=1

ˆ = 1{I=0} B

NY T +1

!Z

τi

τi−1

−rt

e

−rτI

gi (t)dt + 1{I6=0} e

I Y

ΦBB b (j),

(14)

j=1

ΦBB b (j).

j=1

Algorithm 3.1 (Monte Carlo pricing of CDS) Choose the number of simulation runs K and estimate CDS(0, T ) via   c ˆ c  ˆ , A− 1 − e−rT B CDS(0, T ) ≈ 1 − R + r r

(15)

3.4 CDS pricing via the Laplace transform

9

P PK ˆ 1 ˆ ˆ ˆ where Aˆ = K1 K n=1 An and B = K n=1 Bn . In each step of the simulation, An ˆn are calculated as described in Equation (14) and (15) from a new set of and B simulated jumps with default threshold b = log (d/v0 ) . The par spread cT is then estimated via r(1 − R)Aˆ cT ≈ . ˆ − Aˆ 1 − e−rT B R Let us remark that evaluating e−rt gi (t)dt 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.

3.4

CDS pricing via the Laplace transform

In a jump-diffusion model with two-sided exponentially distributed jumps it is possible to calculate the Laplace transform of survival probabilities. Those probabilities can numerically be recovered and then be used to calculate CDS prices. More precisely, we approximate the Riemann-Stieltjes integral of Equation (9) as a Riemann-Stieltjes sum. We obtain the following algorithm: Algorithm 3.2 (Pricing of CDS via the Laplace transform) Partition the interval [0, T ] equidistant with mesh T /n . Then
CDS(0, T ) ≈ ((R − 1)r − c) Dn + (1 − R) 1 − e−rT IP(τ > T ) ,
(1 − R) 1 − e−rT IP(τ > T ) − rDn , cT ≈ Dn RT where we approximate the integral 0 IP(τ > t)e−rt dt via

(16) (17)

n TX Dn = IP(τ > (j − 0.5)T /n)e−r(j−0.5)T /n . n j=1

The respective survival probabilities are obtained from the inverse Laplace transform as explained in what follows.

3.4.1

The Laplace transform of IP(τ ≤ t)

Due to the memoryless property of the exponential distribution it is possible to calculate the Laplace transform of IP(τb ≤ t) explicitly. Using integration by parts,

3.4 CDS pricing via the Laplace transform

10

we find ϕ(α) =

Z

∞ −αt

e

0

1 IP(τb ≤ t)dt = α

Z

0

∞

e−αt dIP(τb ≤ t) =

1 IE[e−ατb ]. α

(18)

To derive an analytical expression of this transform, we begin with a result involving the function G(x) of Equation (3). 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 < ∞ . Kou and Wang 2003 derive the Laplace transform of the running supremum of X . We alter their proof and find that for α > 0 and b < 0 , we have IE[e−ατb ] = A2 ebβ3,α + B2 ebβ4,α , (19) where A2 =

λ⊖ − β3,α β4,α λ⊖ β4,α − β3,α

and B2 =

β4,α − λ⊖ β3,α . λ⊖ β4,α − β3,α

The Laplace transform of IP(τb ≤ t) is now easily derived from Equation (18). 3.4.2

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 Gaver-Stehfest 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 3.1 (Gaver 1966) For a bounded and real-valued function f , continuous at t , we have     n (n + k) log 2 log 2 (2n)! X n k ϕ (−1) , f (t) = lim n→∞ k t n!(n − 1)! k=0 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 3.2 (Stehfest 1970) A better sequence of weights was found by Stehfest. He showed that with f˜n , defined as before in Lemma 3.2, 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)!

11 He also presents the asymptotic result fn∗ (t) − f (t) = o(n−k ) for all k. Algorithm 3.3 (Laplace transform of first-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 in 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.

4

Pure diffusion versus jump-diffusion models

0.02 0.01 0.00

Par Spread

0.03

0.04

Let us begin with an illustration that provides us with some intuition about the resulting term structure of parspread curves in the respective models. We calculate par spreads in a pure diffusion model using Equation (12) and with Algorithm 3.2 in the jumpdiffusion model; time to maturity varies within zero and five years. The parameters of X are chosen to be γ = .02 , Figure 2: Term structure of par spreads σ = .05 , p = .5 , λ = 2 , λ⊕ = λ⊖ = 30 and d/v0 = 90% . The parameters of the pure diffusion XtD = µD t + σ D Wt are chosen such that IE[Xt ] = IE[XtD ] and Var(Xt ) = Var(XtD ) . The resulting parspread curves for R = 50% and r = .02 are given in Figure 2. We observe that in the pure diffusion model par spreads vanish at the short end of the term structure, while in the presence of jumps, par spreads have a positive limit. This difference is crucial, as the jump-diffusion model matches empirical results while a zero limit of par spreads is not observed in reality. In what follows, we explain and discuss this difference in detail. 0

1

2

3

Time to Maturity

4

5

4.1 The local default rate of τ

4.1

12

The local default rate of τ

A common feature of all structural default models is that the conditional probability for default 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 discontinuous models from pure diffusion models is the rate of convergence given full information F . We apply Equation (6) and l’Hospital’s rule to observe that a solvent company with xt = − log (d/Vt ) > 0 has a local default rate of zero in a pure diffusion model. More precisely, we have      −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 (20) This fact forces CDS par spreads to tend to zero as maturity decreases to zero; as we shall see in Equation (21). In our jump-diffusion model we obtain a positive local default rate which 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 ) . Given 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 A proof of this theorem is presented in Scherer 2005.

4.2 4.2.1

The limit of par spreads for small maturities The pure diffusion case

In a pure diffusion model it is virtually impossible for a solvent company to default within a small interval of time. Moreover, just as for bond spreads we can show that the limit of par spreads is zero as time to maturity decreases to zero. To prove this claim, consider the abbreviations A and B from Equation (13) as functions of T . One can show that as long as the company is solvent, or in other words, as long as the distance to default x = − log (d/v0 ) is positive, we have lim A(T ) = lim A′ (T ) = lim B ′ (T ) = 0 and

T ց0

T ց0

T ց0

lim B(T ) = 1.

T ց0

L’Hospital’s rule finally establishes r(1 − R)A′ (T ) = 0. T ց0 re−rT B(T ) − e−rT B ′ (T ) − A′ (T )

lim cT = lim

T ց0

(21)

13 4.2.2

The jump-diffusion model

The limit of CDS par spreads at the short end of the term structure can be found using Theorem 4.1 in our jump-diffusion model. This theorem justifies the approximation IP(τ ∈ (0, dt]) = λ(1 − p) exp(−λ⊖ x)dt , where x = − log (d/v0 ) . We can now let T decrease to zero and obtain RT (1 − R) T1 0 e−rt dIP(τ ≤ t) lim cT = lim = (1 − R)λ(1 − p)e−λ⊖ x . RT 1 −rt T ց0 T ց0 e IP(τ > t)dt T 0

We observe that this limit only depends on the parameters for negative jumps, the recovery rate and the distance to default at time zero, but not on the diffusion component. This fact is also reflected in several figures of the following sensitivity analysis.

5

Sensitivity analysis of the parameters

In this section we present the results of several numerical experiments on how sensitive CDS par spreads are with respect to changes in model parameters.

5.1

Dominance of jumps or diffusion component

Equation (2) allows us to determine what percentage of the overall volatility is explained by jumps. We fix the parameters γ = 0 , p = .5 , λ = .5 and d/v0 = 85% . We assume that λ⊕ = λ⊖ and Var(Xt ) = (.05)2 t . If we now assume this volatility to be explained to a certain percentage by jumps, we can solve for σ and λ⊕ = λ⊖ . The resulting term structures of par spreads corresponding to 0%, 25%, 50% and 75% are given in Figure 3. To distinguish the curves, let us remark that the par spread at zero increases with the influence of jumps. [Insert Figure 3 here.] We observe that especially for small maturities, an increasing probability to default by jumps results in a significantly larger par spread. This effect remains valid for longer maturities, but the gap between the curves dwindles at the long end of the term structure.

5.2 The leverage ratio of the reference entity

5.2

14

The leverage ratio of the reference entity

We assume a variance of Var(Xt ) = (.05)2 t , explained to equal parts by jumps and diffusion. We choose γ = 0 , r = .02 , R = 50% and vary d/v0 in {70%, 80%, 90%, 92.5%, 95%}. It is obvious that par spreads are increasing in d/v0 , surprising is how sensitive par spreads are for highly leveraged companies; compare Figure 4. We also notice that par spreads of highly leveraged companies have the typical hump size structure while the term structure of par spreads of companies with more equity capital is upward sloping. Both is supported by empirical evidence. [Insert Figure 4 here.]

5.3

The drift of the diffusion

Our last experiment involves the drift of the diffusion component, the other parameters are chosen as for Figure 4 with a leverage ratio of d/v0 = 85% . The drift varies within {-0.04, -.02, 0, .02, 0.04}, the resulting par spreads are given in Figure 5. Especially for companies with a negative drift, par spreads increase rapidly. This is not surprising, as such a negative drift (if not compensated by jumps) forces the company to default with probability one. [Insert Figure 5 here.]

15

0.010 0.000

0.005

Par Spread

0.015

0.020

5.3 The drift of the diffusion

0

2

4

6

8

10

Time to Maturity

0.08 0.06 0.00

0.02

0.04

Par Spread

0.10

0.12

Figure 3: The contribution of jumps to the overall volatility

0

2

4

6

8

10

Time to Maturity

Figure 4: The term structure of par spreads for different leverage ratios

16

0.06 0.04 0.00

0.02

Par Spread

0.08

0.10

References

0

2

4

6

8

10

Time to Maturity

Figure 5: The term structure of par spreads for different drifts

References [1] Abate, J., Whitt, W., 1992. The Fourier-series method for inverting transforms of probability distributions, Queueing Systems 10, pp. 5-88. [2] Bomfim, A., 2005. Understanding credit derivatives and related instruments, Elsevier Academic Press. [3] Bielecki, T. and Rutkowski, M., 2002. Credit risk: Modeling, valuation and hedging, Springer. [4] Borodin, A. and Salminen, P., 1996. Handbook of Brownian motion: Facts and formulae, Birkhauser. [5] Gaver Jr., D. P., 1966. Observing stochastic processes and approximate transform inversion, Operations Res. 14 pp. 444-459. [6] Kou, S., 2002. A jump-diffusion model for option pricing, Management Science, Vol. 48, pp. 1086-1101. [7] Kou, S. and Wang, H., 2003. First passage times of a jump diffusion process, Adv. Appl. Prob. 35, pp. 504-531.

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[8] 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. [9] Musiela, M. and Rutkowski, M., 2004. Martingale methods in financial modelling, Springer, 2nd ed. [10] Scherer, M., 2005. A structural credit-risk model based on a jump diffusion, Universit¨at Ulm, Working paper. [11] Stehfest, H., 1970. Algorithm 368. Numerical inversion of Laplace transforms, Comm. ACM 13 pp. 479-49 (erratum 13, 624). [12] 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]