Structural jump-diffusion model for pricing
collateralized debt obligations tranches
YANG Rui-cheng
Abstract. This paper considers the pricing problem of collateralized debt obligations tranches under a structural jump-diffusion model, where the asset value of each reference entity is gener-ated by a geometric Brownian motion and jump with an asymmetric double exponential distri-bution. Conditioned on the common factor of individual entity, this paper gets the conditional distribution, and further obtains the loss distribution of the whole reference portfolio. Based on the semi-analytic approach, the fair spreads of collateralized debt obligations tranches, i.e., the prices of collateralized debt obligations tranches, are derived.
§
1
Introduction
Collateralized debt obligations triggered a worldwide financial crisis in 2008. The reason was mainly due to the irrational prices of the collateralized debt obligations that resulted from the portfolio of reference entities jumped down heavily. This paper introduces the structural jump-diffusion model to describe the jump feature of the individual asset value and further discuss the pricing problem of the collateralized debt obligations tranches.
More specifically, a collateralized debt obligation, or CDO, consists of a portfolio of reference entities (e.g. bonds, loans, residential and commercial mortgages) whose credit risk is sold to investors who, in return for an agreed payment (usually a periodic fee), will bear the losses of the portfolio derived from the default of the reference entities. Through a securitization technique, CDOs repackage a portfolio credit risk into tranches with varying seniority. During the life of the transaction the resulting losses affect first the so called equity piece and then, after the equity tranche has been exhausted, the mezzanine tranches. Further losses, due to credit events on a large number of reference entities, are supported by senior and super senior tranches. The credit risk of the portfolio underlying the CDO is sold in these tranches. Generally, a tranche
Received: 2009-02-16.
MR Subject Classification: 91B70, 91B28.
Keywords: Structural jump-diffusion model, Brownian motion, asymmetric double exponential distribution, collateralized debt obligations, loss distribution.
Digital Object Identifier(DOI): 10.1007/s11766-010-2196-y.
Supported by the National Natural Science Foundation of China (70771018), the Natural Science Foundation of Shandong Province (2009ZRB019AV), Mathematical Subject Construction Funds and the Key Laboratory of Financial Information Engineering of Ludong University (2008).
is defined by a lower and an upper attachment points. The buyers of the tranche with lower attachment pointKLand upper attachment pointKU will bear all losses in the portfolio value in excess of KL, and up to KU, percent of the initial value of the portfolio. CDO tranching allows the holders of each tranche to limit their loss exposure toKU−KL percent of the initial portfolio value.
The central problem for pricing CDO tranches is the calculation of loss distributions of the reference portfolio over different time horizons, and the key issue is how to model the individual asset value, especially in structural models. The earlier works for modelling the individual asset value can be found in [1] where the evolution of asset value was described by a diffusion process driven by a geometric Brownian motion. But the empirical application of a diffusion approach had yielded very disappointing results for fitting the market data, so some better models were proposed. Examples of these models can be found in [2-9]. Albrecher et al. [2] addressed a L´evy model for CDO pricing. Zhou [3] used a jump-diffusion approach with a Poisson process to analyze credit spreads, and Willemann [4] used the jump-diffusion model to analyze a CDO pricing. [5] to [8] considered the CDO pricing under the copula model. Wang et al. reviewed the CDO theory in recent years. They derived some relatively good results for matching the market data of asset value. However, in these models, there still exists a leptokurtic feature that the return distribution of assets may have a higher peak and two asymmetric heavier tails. For analyzing an option pricing, Kou [10] presented a jump-diffusion asset model with asymmetric double exponential distribution and derived some ideal results. Motivated by this, we introduce the idea of Kou into CDO pricing. Using the asset process in [10] and [11], we analyze in detail the loss distribution of the reference portfolio and derive the fair spreads of CDO tranches.
The paper is organized as follows. Section 2 summarizes a semi-analytic approach for pricing CDO tranche. Section 3 presents the structural jump-diffusion model with asymmetric double exponential distribution, and derives the loss distribution of the whole reference portfolio.
§
2
Semi-analytic approach
For pricing a CDO tranche [KL, KU], that is, finding the fair spread of the tranche, we must analyze the values of two legs: default leg (DL) and premium leg (PL). The value of DL presents the value of tranche losses triggered by credit events during the CDO lifetime, and the value of PL is the premium payments weighted by the outstanding asset (original tranche amount minus accumulated losses). Under the risk-neutral measure, the expected value of both legs should be equal, i.e.,
EDL[KL,KU]= EP L[KL,KU], (1) from which we can derive the fair spread of tranche [KL, KU].
Now let us describe in detail the method to calculate the values of DL and PL for CDO tranche [KL, KU]. For convenience, we assume that the reference portfolio consists ofNentities. Let
2. Mi= the notional of thei-th reference entity, Mi>0;
3. ai= the lower default barrier of the thei-th reference entity,ai>0;
4. τi= the default time for thei-th reference entity;
5. T = the maturity time;
6. r= the risk-free discount rate,r >0.
The cumulative loss at time tis given by
L(t) =
N
i=1
Mi(1−Ri)1{τi≤t}, where the indicator function1{·} is given by
1{·}=
1, x∈ {·}, 0, otherswise, and the cumulative tranche loss of [KL, KU] is
L[Kl,KU](t) = min{L(t), KU} −min{L(t), KL}.
For the default leg, we assume that
0≤t0< t1<· · ·< tM−1< tM =T
is the spread payment dates. Then, under the risk neutral probability measure, the expected value of the default leg for the tranche [KL, KU] is
EDL[KL,KU] = tM t0 e−rtudEL [KL,KU](tu) = E M m=1 e−rtmL [KL,KU](tm)−L[KL,KU](tm−1) . (2)
On the other hand, the expected value of the premium leg of tranche [KL, KU] is the present value of all expected spread payments, and it is given by
EP L[KL,KU]= E M
m=1
SCDOΔtme−rtmminmax[K
U−L(tm),0], KU−KL, (3) where Δtm=tm−tm−1.
Substituting (2) and (3) into (1) yields
SCDO= E M m=1e −rtmL [KL,KU](tm)−L[KL,KU](tm−1) E M m=1Δtme −rtmmin{max[KU−L[K L,KU](tm),0], KU −KL} . (4) Eq.(4) shows that the key issue for pricing the CDO tranche [KL, KU] is to compute the loss distributionL[KL,KU](t) of the reference portfolio. The next section will present a structural jump-diffusion form to model the individual asset value and analyze the loss distribution of the underlying portfolio.
§
3
Loss distribution of reference portfolio of CDO
3.1
Structural jump-diffusion model
Referring to the model in [10] or [11], we assume that the i-th asset value Zi(t) of the reference portfolio is given by
dZi(t) Zi(t−)=μidt+σidW(t) +d N˜(t) j=1 (Vj−1) , i= 1, . . . , N,
where W(t) is a standard Brownian motion, ˜N(t) is a Poisson process with intensity λ, and
{Vj, j = 1, . . . , N(t)} is a sequence of independent identically distributed (i.i.d.) nonnegative
random variables such that Υ = ln(Vj) has an asymmetric double exponential distribution with the density
fΥ(y) =pη1e−η1y1
{y≥0}+ (1−p)η2eη2y1
{y<0}, η1>0, η2>0,
wherep≥0 and 1−p≥0 represent the probabilities of upward and downward jumps respec-tively. In other words,
ln(Vj) = Υ =
ξ+, with probabilityp,
ξ−, with probability 1−p,
where ξ+ and ξ− are exponential random variables with means 1/η1 and 1/η2, respectively. In the model, all sources of randomness, ˜N(t), W(t), and Υ, are assumed to be independent. Then
E[Vj] = E[eΥ] =p η1
η1−1 + (1−p)
η2
η2+ 1, η1>1, η2>0. The requirementη1>1 is needed to ensure thate(Vj)<+∞ande(Zi(t))<∞.
Letai be the lower barrier of thei-th entity’s asset. Then default is defined to occur when
Zi(t)≤ai,that is, the default time is
τi= inf{t≥0 :Zi(t)≤ai}, i= 1, . . . , N.
Applying the generalized Itˆo formula (see [11]) to Zi(t) yields
Zi(t) =Zi(0)exp μi−σ 2 i 2 t+σiW(t) N˜(t) j=1 Vj, i.e., lnZi(t) = lnZi(0) + μi−σ 2 i 2 t+σiW(t) + ˜ N(t) j=1 lnVj. The risk-neutral dynamics for the asset valueZi(t) are given by
ln Z i(t) Zi(0) = r−σ 2 i 2 t+σiW(t) + ˜ N(t) j=1 lnVj. LetXi(t) = Zia(t) i . Then lnXi(t) = lnXi(0) + r−σ 2 i 2 t+σiW(t) + ˜ N(t) j=1 lnVj
withXi(0) = Zi(0)
ai , and the default timeτi of thei-th entity can be expressed as
τi= inf{t≥0 :Xi(t)≤1}= inf{t≥0 : lnXi(t) ≤0}. (5)
Now we decompose the driving standard Brownian motionW(t) of thei-th reference asset as
W(t) =ρiWC(t) +
1−ρ2iWi(t),
where ρi ∈ [0,1], and Wi(t) and WC(t) are independent standard Brownian motions. Here
ρiρ is thecorrelationbetween the asset value processes of two entities kand,dWC(t) is the macroeconomic factor (in whichWC is the common risk factor) of the asset value which is the same for all entities, anddWi(t) is the continuous change of idiosyncratic factor (in whichWi is the idiosyncratic uncertain factor) of thei-th entity asset value.
So the dynamics of ln(Xi(t)) is lnXi(t) = lnXi(0) + r−σ 2 i 2 t+σiρiWC(t) +σi 1−ρ2iWi(t) + ˜ N(t) j=1 lnVj. (6)
3.2
Conditional default probability of individual asset
Conditioning the common factorWC, we will derive the conditional probability distribution of default time for an individual asset in this section.
Following the ideas of Kou [11], we give the computation method for conditional default probability distribution givenWC as follows.
Let M(x) =1 2σ 2 i(1−ρ2i)x2+ r−σi2+ 1 2ρ 2 i x+λ (1−p)η 2 x+η2 − pη1 x−η1−1 .
For any twice continuously differentiable functionu(x), the operatorG is defined by
Gu(x) = 1 2σ 2 i(1−ρ2i)u(x) + r−σi2+1 2ρ 2 i u(x) +λ ∞ −∞ u(x+y)−u(x)fΥ(y)dy. Lemma 3.1. For any s∈ (0,+∞), there exist two positive roots θ1,s and θ2,s of M(x) = s withθ2,s<−η2< θ1,s<0. Let v(x) = B1eθ1,sx+ (1−B1)eθ2,sx, x >0, 1, x≤0 with B1=θ2,s(η2+θ1,s) η2(θ2,s−θ1,s). Then v(x)is continuous on (0,∞)and for x >0,
Gv(x)−sv(x) = 0.
Proof. The first statement is obvious because M(x) < s when x ↑ 0 and M(x) > s when
x↓ −η2, andM(x)> swhenx↓ −∞andM(x)< swhen x↑ −η2. Now we show that Gv(x)−sv(x) = 0 forx >0. In fact,
Gv(x)−sv(x) =λ ∞ −∞v (x+y)fΥ(y)dy+B1eθ1,sx1 2σ 2 i(1−ρ2i)θ21,s+ r−σ2i + 1 2ρ 2 i θ1,s−s + (1−B1)eθ2,sx1 2σ 2 i(1−ρ2i)θ22,s+ r−σi2+ 1 2ρ 2 i θ2,s−s , (7)
and ∞ −∞v (x+y)fΥ(y)dy = −x −∞ (1−p)η2eη2ydy+ 0 −x (1−p)η2B1eθ1,s(x+y)+ (1−B 1)eθ2,s(x+y) eη2ydy + ∞ 0 pη1 B1eθ1,s(x+y)+ (1−B1)eθ2,s(x+y) e−η1ydy = 1− B1η2 θ1,s+η2− (1−B1)η2 θ2,s+η2 (1−p)e−η2x+B 1 (1−p)η 2 θ1,s+η2 − pη1 θ1,s−η1 eθ1,sx + (1−B1) (1−p)η2 θ2,s+η2 − pη1 θ2,s−η1 eθ2,sx. (8) Since B1η2 θ1,s+η2 + (1−B1)η2
θ2,s+η2 −1 = 0, substituting (8) into (7) yields
Gv(x)−sv(x) =B1eθ1,sx1 2σ 2 i(1−ρ2i)θ21,s+ r−σ2i + 1 2ρ 2 i θ1,s−s+λ (1−p)η 2 θ1,s+η2 − pη1 θ1,s−η1 −1 + (1−B1)eθ2,sx1 2σ 2 i(1−ρ2i)θ22,s+ r−σ2i+ 1 2ρ 2 i θ2,s−s+λ (1−p)η 2 θ2,s+η2− pη1 θ2,s−η1−1 =B1eθ1,sxM(θ 1,s)−s + (1−B1)eθ2,sxM(θ2,s)−s = 0. Let τi= inf t≥0 : z+ r−σ 2 i 2 t+σi 1−ρ2iWi(t) + ˜ N(t) j=1 lnVj≤0 (9)
withz= lnXi(0) +σiρiw. From Lemma 3.1, we have Theorem 3.1. For anys >0, we have
E[e−sτi] =B
1eθ1,sz+ (1−B
1)eθ2,sz, z >0, (10)
and the Laplace transform ofP{τi≤t}is given by ∞ 0 e −stP{τ i≤t}dt=1 sE[e −sτi], i= 1, . . . , N.
Furthermore, withF(s) =1sE[e−sτi], then
P{τi≤t}=L−1(F(s)) (11)
whereL−1 is the Laplace inversion.
Proof. From Lemma 3.1, we haveGv(z)−sv(z) = 0 for anyz >0.Applying the generalized Itˆo formula to the processe−stv(ln(Xi(t))) in which ln(Xi(t)) is given by (6) and taking expectation yield Ee−s(t∧τi)vln(X i(t∧τi)) =v(z) + E t∧τi 0 e −suGvln(X i(u)) −svln(Xi(u)) du.
Since ln(Xi(u))>0 for u∈(0, t∧τi), we have Ee−s(t∧τi)v(ln(X
i(t∧τi))=v(z). Lettingt→ ∞yields
Ee−sτi=v(z) =B
Applying the Laplace transform to P{τi≤t}yields ∞ 0 e −stP{τ i≤t}dt=1 s ∞ 0 e −stdP{τ i≤t}=1 sE[e −sτi].
The equality (11) involving the Laplace inversion is obvious (see [12]). Remarks 3.1. Theorem 3.1 shows how to get the default probabilityP{τi ≤t} easily. We only have to find E[e−sτi] by (10) and then apply (11).
SinceWCis independent ofWiand Υ, from (5), (6) and (9) we have
Theorem 3.2. Let Pti|WC be the conditional default distribution given on WC for the i-th reference entity. Then
Pti|WC =P{τi≤t|WC(t) =w}=P{τi≤t}, i= 1, . . . , N.
3.3
Loss distribution of the whole portfolio
In this section we are going to consider the algorithm for computing the loss distribution of the whole reference portfolio.
Conditioning the common factor WC, the individual default time τi for i = 1, . . . , N are independent. Then the conditional characteristic function ofL(t) givenWC can be expressed by EeIuL(t)|WC=w = E e N
i=1IuMi(1−Ri)1{τi≤t} |WC=w
= N i=1 EeIuMi(1−Ri)1{τi≤t} |WC=w, whereI is the imaginary unit withI2=−1. Thus
EeIuL(t)|WC=w=
N
i=1
EeIuMi(1−Ri)1{τi≤t} |WC =w.
Note that the individual conditional characteristic function for thei-th reference entity is EeIuMi(1−Ri)1{τi≤t} |WC=w = eIuMi(1−Ri)pi|WC
t + 1−pit|WC = 1 +eIuMi(1−Ri)−1 pi|WC t . Hence EeIuL(t)|WC =w= N i=1 1 + (eIuMi(1−Ri)−1)pi|WC t .
By taking out the common factorWC we get the unconditional characteristic function
EeIuL(t)= ∞ −∞ N i=1 1 + (eIuMi(1−Ri)−1)pi|WC t fWC(w)dw,
wherefWC(w) is the density of the standard Brownian motionWC(t) with fWC(w) = √1
2πte
−w2 2t.
prob-ability distributionFL(t)(x) of the lossL(t) is FL(t)(x) =P(L(t)≤x) = limU→∞ 1 2π U −UH (u)e −Ixu Iu du.
This computation of the loss distribution of the whole reference portfolio is not only suitable for inhomogeneous portfolio, but also for homogeneous one, i.e., whenρi =ρ∈[0,1],ai=a >0,
Mi =M >0 andRi =R∈[0,1] for alli= 1,2, . . . , N. However, there is a simpler computation
for homogeneous portfolio.
Theorem 3.3. Assume that the reference portfolio is homogeneous. Let Lk = Nk(1−R)M. Then the loss distribution P{L(t) =Lk} of the the whole reference portfolio is given by
P{L(t) =Lk}= N k ∞ −ln(Xi(0)) Pti|WC k 1−Pti|WC N−kfWC(w)dw,
and the distribution function FL(t)of the cumulative loss is
FL(t)(Lk) =P{L(t)≤Lk}= k
l=0
P{L(t) =Ll}.
Proof. Since Lk = Nk(1−R)M is equal to the probability that exactly k out of N entities default, and the defaults are independent when conditioned onWC(t), then
P{L(t) =Lk|WC(t) =w}= N k Pti|WC k 1−Pti|WC N−k and P{L(t) =Lk}= ∞ −∞P{L (t) =Lk|WC(t) =w}fWC(w)dw.
From the fact thatPti|WC = 1 forw∈(−∞,−ln(Xi(0))], we have
P{L(t) =Lk} = −ln(Xi(0)) −∞ P{L (t) =Lk|WC(t) =w}fWC(w)dw + ∞ −ln(Xi(0)) P{L(t) =Lk|WC(t) =w}fWC(w)dw = N k ∞ −ln(Xi(0)) Pti|WC k1−Pti|WC N−kfWC(w)dw.
The cumulative loss follows readily from the definition of cumulative distribution function. So far we have been computing the loss distribution of the whole reference portfolio. Once we have found the loss distribution ofL(t), we can easily get the fair spreadsSCDO of CDO tranche [KL, KU] by (4).
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School of Mathematics and Information, Ludong University, Yantai 264025, China Email: [email protected]