CDO TRANCHE PRICING BASED ON THE STABLE LAW VOLUME II: R ELAXING THE LHP. Abstract

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1 CDO TRANCHE PRICING BASED ON THE STABLE LAW VOLUME II: R ELAXING THE ASSUMPTION German Bernhart XAIA Investment GmbH Sonnenstraße 9, 833 München, Germany german.bernhart@xaia.com First Version: July 26, 23. This Version: January, 24. Abstract For many applications, e.g. the aily tracking of market prices in front office systems, it is sufficient to consier a simplifie CDO pricing moel. In Volume I of this series such a simplifie moel was presente base on the assumption of a very large, homogeneous portfolio. However, e.g. for more etaile scenario analyses an the computation of hege ratios, one has to relax these simplifying assumptions. In the current paper, several extensions of the CDO pricing moel base on a Gumbel copula are iscusse an possible implementations are presente. These extensions can be use to price CDO contracts on non-homogeneous pools. Introuction In Volume I of this series, we presente a moel for the pricing of CDO tranches base on a latent market factor with heavy-taile α-stable istribution. The pricing routines evelope in that paper were built on several simplifying assumptions renering the numerical implementation efficient. However, for several further applications an investigations an extension of the moel to nonhomogeneous portfolios is neee. Possible applications inclue, e.g., etaile scenario analyses or hege ratio computations. The purpose of this article is to escribe possible ways to relax some of the simplifying assumptions an to investigate the resulting changes for moel prices. Therefore, the paper is more concerne with numerical algorithms an a bit technical. The large homogeneous portfolio () approximation relies on the following battery of simplifying assumptions. All notations are the same as in Volume I, which we presuppose as known by the reaer. (i) The portfolio is very large an all portfolio weights are the same, i.e. 2 an ω =... = ω = /. (ii) All recovery rates are ientical an eterministic, i.e. R =... = R =: R [, ]. (iii) All efault times Xk of the portfolio constituents have the same marginal istribution function, which we enote by p(t), i.e. F (t) =... = F (t) =: p(t) for all t. (iv) The components of (X,..., X ) are conitionally inepenent given a latent market factor. We will present several ways to relax some of the above assumptions:

2 (a) In Section 2, we will relax the assumption of a very large portfolio an (iii). (b) In Section 3, we will sketch how to aitionally relax the assumptions of equal portfolio weights an ientical an eterministic recovery rates. Furthermore, we will briefly explain how one coul inclue a non-homogeneous epenence structure, whose implementation, however, then woul have to rely on time-consuming Monte Carlo techniques. We start with a short introuction of the moel consiere in the following. The current paper is base on the Gumbel copula moel, which is equivalent to the moel consiere in Vol. I, in the sense as escribe in Remark 3. of Vol. I, but consierably easier to exten to the non-homogeneous case. Depenence is introuce by a common factor M S(, α). The vector of efault times (X,..., X ) is formally efine by Xk := inf{t > : M hk (t) > k }, k =,...,, where,..., are ii unit exponential ranom variables an hk (t) := ( log( Fk (t)))/α. Consequently, the components of (X,..., X ) are conitionally inepenent given M an have the Gumbel copula as survival copula. 2 marginals We now consier the following setting: The portfolio size is fixe, e.g., = 25, an the marginal istribution functions, respectively efault probabilities, are not necessary ientical, given by pi (t), i. All other simplifying assumptions still hol. This extension is quite natural whenever CDS contracts on the portfolio constituents are available, which then allow to raw conclusions about the corresponing marginal efault probabilities. Our scenario is the following: We want to price several (here five for simplicity) tranches of a CDO at once with attachment points (l, u ),..., (l5, u5 ) an relevant payment ates T,..., Tk. As explaine in Vol. I of this series, require are the values in the following matrix: T E := T El,u (T )... T El5,u5 (T ) T El,u (Tk )... T El5,u5 (Tk ) One coul also compute the portfolio loss istribution for every Ti, i k, an compute the above matrix from those istributions. Here, however, we want to compute the ifferent relevant tranche expectations irectly. As usual, this is one by consiering everything conitione on the common factor an integrate out. The common factor consiere here is actually only one αstable ranom variable M. We have T Eli,ui (Tj ) {M = m} () = E min ui li, max, LTj li = X n= M = m R min ui li, max, n li P(L Tj = n M = m), 2

3 where L Tj := k= {Xk Tj }, i.e. the number of companies efaulte until Tj. Conitione on M, the {Xk Tj } are inepenent with efault probabilities P pk (Tj M = m) = exp m ( log( Fk (Tj )))/α. Thus, it is theoretically quite easy to compute the istribution of L Tj conitione on M = m, an from that the value of the tranche expectations for Tj conitione on M = m. In practice, to compute the istribution of L Tj conitione on M = m efficiently, we will use the first approach presente in Hull, White (24). For a etaile explanation of the algorithm use to compute the istribution of a sum of inepenent Bernoulli ranom variables, see Hull, White (24). The main iea is to use the representation P(L Tj = n M = m) = P(L Tj = M = m) U (n), where U (n) := X Y I {,...,}: r I wr, I =n with wr = pr (Tj M = m). pr (Tj M = m) This representation can be easily erive starting from P(L Tj = n M = m)! = X Y I {,...,}: r I pr (Tj M = m) Y ( pr (Tj M = m)), r I c I =n factoring out the require quantities. For an efficient computation of U, Hull, White (24) use an algorithm base on the so-calle Newton-Girar formulas. Having a closer look at their algorithm, it becomes clear that it runs into numerical instabilities as soon as wr > for at least one r. A solution to this problem is foun by iviing the portfolio into two subportfolios, one for which all wr s are smaller than one, an the rest. For the first portfolio, we can irectly apply the algorithm of Hull, White (24). For the secon, we apply the analogous algorithm to erive the istribution of the number of surviving companies, instea of the number of efaulte companies. This switches the wr to /wr, making the algorithm stable again. In a last step, one has to merge the istributions of the two inepenent subportfolios. Base on the previous proceure, we are able to compute T E {M = m}. In a last step, we have to compute the vector-value integral Z TE = T E {M = m} fm (m) m by numerical quarature. To compute the require ensity fm of the α-stable ranom variable M, we implement the formula evelope in Bernhart et al. (23). Of course, one has to truncate the integral at some point. However, it is nice that one can 3

4 estimate the resulting error for each entry in the matrix when approximating the integral by UB Z T Eli,ui (Tj ) {M = m} fm (m) m T E li,ui (Tj ) := + P(M > U B) T Eli,ui (Tj ) {M = U B}, i.e. assuming T Eli,ui (Tj ) {M = m} to be constant in m above the upper boun U B. This is possible as we know that this function is increasing in m with upper boun ui li an thus T E li,ui (Tj ) T Eli,ui (Tj ) P(M > U B) ui li T Eli,ui (Tj ) {M = U B}. As a consequence, we are able to choose U B accoring to a specific error boun if require. 2. Comparison of base correlations In the present paragraph, the base correlations of the inhomogeneous moel are compare with the base correlations using the assumptions, which were compute in Vol. I. For a etaile escription of the market ata use, the intereste reaer is referre to that ocument. Now, the marginal efault probabilities are chosen such that the market quotes for ten year single name CDS contracts are matche. The results can be foun in Figure. One can observe that the general level of base α s is 3 Mar 2 29 Oct 2 Base Alphas () Base Alphas () 8 base (in %) base (in %) Base Alphas () Base Alphas () [%,3%][%,6%] [%,9%] [%,2%] [%,22%] [%,3%][%,6%] 9 Sep 2 Base Alphas () Base Alphas () 8 base (in %) base (in %) [%,22%] [%,2%] 6 Nov 22 8 [%,9%] Base Alphas () Base Alphas () [%,3%][%,6%] [%,9%] [%,2%] [%,22%] [%,3%][%,6%] [%,9%] [%,2%] [%,22%] Fig. : Selecte ates of the time series of base α s for the inhomogeneous moel an uner the assumption. consierably higher in the inhomogeneous moel. There is an 4

5 intuitive explanation for that. Compare to the homogeneous case, the value of all base tranches is increase when consiering an inhomogeneous moel with the same average efault probability. This is ue to the fact that base tranches essentially equal a basket of k th-to-efault swaps, incluing k = until k = n for some n. The value of those swaps is consierably increase in the inhomogeneous case as, e.g., a first-to-efault swap on a portfolio of one very risky an nine almost riskless companies is worth more than a first-to-efault swap on a homogeneous pool of ten companies with smaller efault risk, as it is basically a protection against the efault of the most risky company. Since the value of all base tranches is higher in the inhomogeneous case, one nees a higher correlation to match the market price, as the value of those tranches is ecreasing in the correlation parameter. 2.2 Tranche sensitivity with respect to the correlation In the present paragraph, the sensitivity of the value of the ifferent tranches with respect to changes of the correlation parameter α is investigate. The investigation is carrie out with the ata of one ay only, here November 6, 22 is investigate. Values of α between an.9 are consiere an the result can be foun in Figure 2. It is very interesting that the behaviour of the with changing correlation can iffer between the case an the inhomogeneous case as can be seen here for tranches 3 an 4. Furthermore, consiering the α with the best overall fit (error measure in ters of the sum of square tranche values), one can observe that in the inhomogenous moel the error is reuce by 5%. As expecte, taking into account more information about the unerlying structure allows for a better fit. 2.3 Tranche sensitivity with respect to single name CDS In a last paragraph, the sensitivity of a with respect to changes in the corresponing single name CDS is investigate. Such investigations are the main reason for consiering inhomogeneous moels, as one might be intereste in analysing the impact of ifferent possible scenarios. Furthermore, the results are interesting for the purpose of heging. Here, we exemplarily investigate the equity tranche. The correlation parameter α is set to the base correlation for the first tranche. For simplicity, we consier Vl,u, si i.e. the erivative of the value of the equity tranche Vl,u with respect to the running sprea si of a single name CDS, which we will approximate with the absolute change in value when increasing the running sprea by bp. The result can be foun in Figure 3, where it is aitionally compare to the eltas for a ifferent value of α. One can observe that the elta is increasing in the sprea as expecte, an that the steepness of the curve epens on the value of the correlation parameter α. Repeating the investigation for other tranches than base tranches reveals that the monotonicity is estroye in those cases. 5

6 Tranche [%,3%] Tranche 2 [3%,6%] Tranche 3 [6%,9%].8 Tranche 4 [9%,2%] Tranche 5 [2%,22%] Fig. 2: The value of the five ifferent tranches with changing α is shown for both the case an the inhomogeneous case. 3 Further extensions There is obviously still a lot of room for relaxations of the previously mae assumptions. In this section, we shortly want to introuce the necessary ieas an comment on possible implementations without a etaile analysis. In a first step, the assumption of equal weights an equal an eterministic recovery rates is relaxe. One might allow the recovery rates to epen on the common factor M, or at least to be inhomogeneous. This can be realize almost analogously to our previous proceure. Again, one will consier everything conitione on the common 6

7 6.2 x =.432 =.3 elta equity tranche single name CDS sprea in bp 2 Fig. 3: The eltas of the equity tranche with respect to the running sprea of one single name CDS, plotte versus the corresponing running sprea. Since the consiere portfolio consists of 25 single names, we en up with 25 ifferent eltas, which are all visualize. The investigation is conucte for two ifferent values of α, the actual base correlation an an arbitrarily chosen value. factor, i.e. () Lt {M = m} := X ωk ( Rk (m)) {Xk t} {M = m}. k= The istribution of this object can be approximate using the secon algorithm presente in Hull, White (24). It is base on the iea of bucketing the loss istribution, i.e. one has to efine K buckets B := [, b ), B := [b, b ),..., BK := [bk, ) (actually, as we consiere a normalize portfolio, we coul have chosen instea of ). We will then compute the K probabilities () () () P(Lt Bi M = m) an also Ai := E[Lt Lt Bi, M = m], i.e. the expecte loss for each bucket. These quantities will be compute by one-by-one aing the ifferent constituents to the portfolio (starting from an empty portfolio) an upating the require quanitities accoringly. This will serve as an approxima7

8 tion of the true loss istribution an using this approach, again integrating over the ensity of M, allows to price CDOs in this context. If one aitionally wants to make the epenence structure inhomogeneous, a possible iea woul be to replace the Gumbel copula by a hierarchical Gumbel copula. By that, one can increase the epenence in between groups (e.g. the inustry sectors) while keeping the epenence between companies of ifferent groups small. A more etaile introuction into the usage of those epenence structures for CDO pricing can be foun in Hofert, Scherer (2). However, this approach requires the usage of Monte Carlo techniques, making it slower an more ifficult to calibrate. It can be spe up significantly by putting some effort into the implementation using importance sampling or other techniques, but a escprition of those techniques lies outsie the scope of this short note. 4 Conclusion In the present note, we extene the CDO pricing moel presente in Vol. I to non-homogeneous portfolios. The extension to finite portfolios with inhomogeneous marginals was investigate in more etail, explaining the impact on pricing an sensitivities. Furthermore, other ways of relaxing the simplifying assumptions were skeche. References G. Bernhart, J.-F. Mai, S. Schenk, M. Scherer, The ensity for istributions of the Bonesson class, Journal of Computational Finance (to appear) (23). J. Hull, A. White, Valuation of a CDO an an n-th to efault CDS without Monte Carlo simulation, Journal of Derivatives 2 (2) (24). M. Hofert, M. Scherer, CDO pricing with neste Archimeean copulas, Quantitative Finance (5) (2) pp