Recursive valuation of Basket Default Swaps - CiteSeerX

Recursive valuation of Basket Default Swaps Ian Iscoe Algorithmics Inc., 185 Spadina Avenue, Toronto, Ontario, Canada M5T 2C6

Alex Kreinin Algorithmics Inc., 185 Spadina Avenue, Toronto, Ontario, Canada M5T 2C6

In this paper we consider an analytical valuation of Basket Default Swaps. Our solution is based on a continuous-time model in a conditional independence framework. We use the order statistics of the default times of the names in the basket to find a recursive algorithm for computation of the risk-neutral distribution of the default process of the basket. We derive an analytical expression for the value of the first-to-default swap, which leads to a solution for an mth-to-default swap, using the recursive algorithm. The accuracy and performance of the analytical method are compared with that obtained using Monte Carlo simulation.

1 Introduction In this paper we consider a simple continuous-time model for analytical valuation of Basket Default Swaps (BDS). A BDS is a credit derivative security whose underlying reference assets are usually corporate bonds. The contingent payment is triggered by a combination of default events of the reference entities. The simplest example of a BDS is the first-to-default contract under which the protection seller is obliged to pay compensation after the first default event (if it occurred prior to the maturity of the BDS). Among many different models and methods proposed for analysis of basket credit derivatives, the Gaussian multi-factor models with constant factor loadings have become very popular (see Andersen et al. 2003; Bielecki and Rutkowski 2002; Finger 2002; Huge 2002; Hull and White 2004; Kijima 2000; Kijima and Muromachi 2000; Lando 2004; Laurent 2003; Laurent and Gregory 2003; Madan et al. 2004; Schönbucher 2003). This model, with a piecewise constant hazard rate process, allows one to value the BDS and to compute a constant BDS premium. We find a closed-form solution for the first-to-default and for the second-to-default We are very grateful to our colleagues Ben De Prisco, Asif Lakhany, Helmut Mausser and Jennifer Lai for many comments and suggestions that helped to improve the paper. We would like to thank David Lando and Brian Huge for providing us with Dr Huge’s thesis and for taking an interest in our research. We would also like to thank N. Balakrishnan for useful discussions on order statistics and in particular for the reference to his paper, cited in the bibliography. We discussed computational aspects of the recursion, proposed in this paper, with Luis Seco and David Saunders. Finally, we would like to thank Mark Broadie and a referee for their critical comments and suggestions.

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contracts without assuming any homogeneity of the basket. We find a recursion for probabilities of terminal default events1 in the basket in terms of first-todefault contracts with reduced sets of names. These default probabilities play the key role in pricing different types of basket credit derivatives. We illustrate our approach by pricing mth-to-default swaps using a one-factor Gaussian model for the conditional default probabilities. The recursion for the probabilities of the terminal default events implies a similar relation for the value of the BDS. Similar results for the value of a BDS were obtained by Huge (2002) for a cumulative first-m-defaults contract using arbitrage arguments (see also Lando (2004)). The paper is organized as follows. In Section 2 we describe the pricing2 of a BDS and derive the main pricing equations for contracts slightly more general than a BDS (this generalization includes, in particular, call and put options on mth-to-default losses). In Section 3 we review the model for correlated default events, including the specification of the default-time process and default intensities, to set the notation for our results. This specification allows one to calibrate the risk-neutral default probability curves. This model is identical to that described in Andersen et al. 2003; Finger 2002; Huge 2002; Hull and White 2004; Kijima 2000; Kijima and Muromachi 2000; Lando 2004; Laurent 2003; Laurent and Gregory 2003; Madan et al. 2004; Schönbucher 2003. Valuation of the BDS is considered in Section 4. We solve the valuation problem explicitly for the first-to-default contract. Then we present an algorithm for valuation of the mth-to-default contract for m > 1. The algorithm is a recursive formula for the computation of the risk-neutral terminal default probabilities. It is important to mention that this recursion does not depend on the model of the dependence structure of default events; it does not even rely on the conditional independence framework. Calculation of the credit spread sensitivities is one of the most important and the most challenging problems in risk management of portfolios of credit derivatives (Andersen et al. 2003). In Section 4.4 we briefly discuss the benefits of our analytical solution for BDS, for computation of the BDS sensitivities to the Credit Default Swaps (CDS) spreads. In Section 5 we discuss performance of the recursive valuation algorithm and the accuracy of the analytical solution described in Section 4. The paper concludes with remarks related to the extension of the proposed approach to valuation of more complex credit derivatives and to future research directions.

2 Pricing equation We consider a continuous-time model for the pricing of a BDS. Let T = {t1 , t2 , . . . , ti , . . . , tn = T } denote the set of premium times, with T denoting the maturity of the swap; let t0 denote today. The basket of securities contains 1 Those which trigger compensation payment. 2 More precisely, we price the credit-risk premium.

Journal of Computational Finance

Recursive valuation of Basket Default Swaps

K instruments (whose individual maturities are greater than T ). Denote by N (k) the recovery-adjusted notional value of the kth instrument initially in the basket, k = 1, 2, . . . , K. We assume that the recovery rates are deterministic. The buyer of protection against the mth default within the basket pays regular premiums at the premium times ti ∈ T, (i = 1, 2, . . . , n), prior to the time of the mth default, expressed as an annualized rate s against a notional value N. The premium, s, is determined to balance the expected payout in the default event. We make the following standing, financial assumptions. (A1) There is no replacement of the underlying instruments in the basket. (A2) The premium s of the basket does not depend on time. We consider two variants of the protection payment. (C1) An amount of compensation is paid out at the terminal default time provided it occurs by the swap maturity. (C2) An amount of compensation is paid out at the nearest premium date following (or equal to) the terminal default time, provided it occurs by the swap maturity. With frequent premium payments (eg, quarterly), it is most likely that defaults will only be detected at a premium date; so (C2) is the appropriate variant in this case. We also consider two variants of the premium payments. (P1) Accrued interest is paid out at the terminal default time. (P2) No accrued interest is paid out. The inclusion or exclusion of accrued interest is a feature of the particular contract. In order to mathematically express the value of the default and premium legs, for each of the variants, we introduce the following notation. Let τ (k) denote the default time of the kth instrument (with a value +∞ if default never occurs). The terminal default time, τ , triggering the compensation payment, is a function of the random variables τ (1) , τ (2) , . . . , τ (K) . For example, in the first-to-default contract, τ = min1≤k≤K τ (k) . For the contract variants (C2) and (P2), we introduce the notation i(τ ) = max{i : ti < τ } and also for the variant (C2) we introduce the notation ti(τ )+1 if τ < T τ¯ = T if τ ≥ T Denote by L the loss of the BDS at the terminal default time: g(N (k) ) if τ = τ (k) ≤ T L= 0 otherwise Volume 9/Number 3, Spring 2006

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where g(·) is a payoff function. In particular, if g(x) = x and τ = min1≤k≤K τ (k) , we obtain the payoff function of the first-to-default contract. If g(x) = max(x − g∗ , 0) and τ is the time of the mth default, we obtain the payoff function of a call option on the mth-to-default basket loss with the strike g∗ . If g(x) = g0 , and τ is the time of the mth default, we obtain the payoff of the digital mth-to-default contract. We assume that the interest rate process is deterministic. It is not a significant loss of generality. If one assumes the hypothesis that the interest rate process is independent of the default process of the basket, the effect of stochasticity of interest rates disappears. Let D(t) = e−r(t)·(t−t0 ) be the discount factor corresponding to time t; r(t) is the risk-free interest rate corresponding to maturity t, linearly interpolated between the premium dates. Thus, r(t) = r(ti−1 ) +

ri (t − ti−1 ), ti

ti−1 ≤ t ≤ ti

where ri = ri − ri−1 , ri = r(ti ), ti = ti − ti−1 , and r(t0 ) is approximated by the overnight rate. We denote the slopes of r(t) by ci : ci =

ri , ti

i = 1, 2, . . . , n

(2.1)

We also introduce the discount factors, D(τ ) and D(τ¯ ), corresponding to the default time of the contract. Throughout the paper, E denotes the risk-neutral expectation with respect to a risk-neutral probability, P. The value of the default leg in the case (C1) is E[L · D(τ )]. The value of the default leg in the case τ ∧T(C2) is E[L · D(τ¯ )]. The value of the premium leg in the case (P1) is E[sN 0 D(t) dt]; in the case i(τ ) (P2) the value is E[ i=1 s · ti · N · D(ti )]. As in Lando (2004) and Laurent and Gregory (2003), for the purpose of numerical computation of the risk-neutral expectations, we introduce the probabilities (k) (k) i = P(τ = τ , τ ∈ (ti−1 , ti ]),

i = 1, 2, . . . , n, k = 1, 2, . . . , K (2.2) (k)

and corresponding probability density functions, pτ (t), satisfying the relation t+h P(τ = τ (k) , τ ∈ (t, t + h]) = pτ(k)(u) du t

The following proposition is similar to the result obtained in Lando (2004) and in Laurent and Gregory (2003) for the mth-to-default contracts. P ROPOSITION 1 Under the condition (C1), the value of the default leg is

E[L · D(τ )] =

K k=1

g(N (k) ) 0

T

pτ(k)(t)D(t) dt

(2.3)



Under the condition (C2), it is

E[L · D(τ¯ )] =

K

g(N (k) )

k=1

n

(k)

D(ti )i =

i=1

n

D(ti )

i=1

K

(k)

g(N (k) )i

The value of the premium leg under the condition (P1) is T τ ∧T E sN D(t) dt = sN F¯τ (t)D(t) dt 0

(2.4)

k=1

(2.5)

0

where F¯τ (t) = P(τ > t). Under the condition (P2), the value of the premium leg is i(τ ) n ¯i E sN ti · D(ti ) = sN ti · D(ti ) i=1

(2.6)

i=1

¯ i = P(τ > ti ) satisfy the relations where the probabilities ¯0=1 ¯i = ¯ i−1 − i , i =

K

(k)

i ,

i = 1, 2, . . . , n i = 1, 2, . . . , n

k=1

The proof of Proposition 1 is straightforward and, therefore, is omitted. R EMARK . The default time, τ , can be somewhat general in Proposition 1. We need only the following technical assumption: the probability sample space decomposes as

K

{τ = τ (k) ≤ T } ∪ {τ > T }

k=1

(the event, {τ > T }, is the possibility that no compensation is ever paid).

3 Conditional independence framework In this section we review the conditional independence framework based on a multi-factor Gaussian model with constant factor loadings.

3.1 Conditional default probabilities Denote the risk-neutral, cumulative default probabilities of the kth name by πˆ (k) (t): P(τ (k) ≤ t) = πˆ (k) (t), k = 1, 2, . . . , K Let X denote the vector of jointly normally distributed credit drivers X := (X1 , . . . , XC ) with each Xc standardized and denote by R the correlation matrix of X. Volume 9/Number 3, Spring 2006

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Let x = (x1 , . . . , xC ) be a particular value of X. The conditional risk-neutral default probabilities are given by

C (k) −1 (k) (k) (k) πˆ (t, x) = (πˆ (t)) − σ (3.1) βc xc c=1

where denotes the standard normal cumulative distribution function, and the (k) coefficients βc and σ (k) satisfy the relation (σ (k))2 +

C C c=1 c =1

βc(k) Rcc βc(k) =1

for each k = 1, 2, . . . , K. In this framework, the default events of the names are conditionally independent. This assumption allows us to reduce the problem of computation of the (k) densities pτ (·) in Proposition 1 to the case of independent names. The latter satisfy the relation pτ(k)(t, x) dϕ(x), k = 1, 2, . . . , K (3.2) pτ(k)(t) = RC

(k)

where pτ (t, x) denotes the conditional density conditioned on X = x (ie, with default probabilities given by (3.1)) and ϕ is the joint distribution of the credit drivers. We denote by Px the risk-neutral probability measure, conditional on X = x.

3.2 Default intensities The conditional probability distribution function of τ (k) can be represented in the form F (k) (t, x) ≡ Px (τ (k) ≤ t) = 1 − exp(− (k) (t, x)) t where (k) (t, x) = t0 λ(k)(u, x) du and the function λ(k) (·) is the conditional default intensity of the kth name. The unconditional distribution of the default time of the kth name can be determined from the CDS credit spread quotes having maturities among t1 , t2 , . . . , tn . Let πˆ i(k) be the cumulative unconditional risk-neutral probability of default of (k) the kth name by time ti : πˆ i = P(τ (k) ≤ ti ), i = 1, 2, . . . , n. Using (3.1) we (k) compute the conditional risk-neutral default probabilities, πˆ i (x) at t = ti , and (k) set F (ti , x) to be the latter, for i = 1, 2, . . . , n, in a given scenario, x, so that Px (τ (k) ≤ ti ) = πˆ i(k) (x). The function (k) (t, x) satisfies the equation (k)

(k) (ti , x) = −ln π¯ i (x), (k)

k = 1, 2, . . . , K, i = 1, 2, . . . , n

(k)

where π¯ i (x) = 1 − πˆ i (x). Since (k) (t, x) is determined only at times ti , we have to interpolate the values of (k) (t, x) to compute it at an arbitrary value of t. Journal of Computational Finance


If we choose linear interpolation, then λ(k)(t, x) is a piecewise constant function: (k)

λ(k)(t, x) = λi (x)

for t ∈ (ti−1 , ti ]

Then we obtain that, for i = 1, 2, . . . , n, (k) π¯ (x) 1 (k) λi (x) = log i−1 (k) ti π¯ i (x)

where ti = ti − ti−1

(3.3)

Linear interpolation of (k) (t, x) is not the only way to construct the function λ(k) (t, x). An alternative method of continuation can be based on linear interpolation of the distribution function F (k) (t, x), which has the following remarkable property: it maximizes the entropy of the random variable τ (k) . Any choice of interpolation introduces a numerical (as opposed to financial) approximation because conditional and unconditional default probabilities cannot be set arbitrarily and remain probabilistically consistent. In this paper we adopt linear interpolation.

4 Valuation of BDS In this section we continue the evaluations of the right-hand sides of (2.2)–(2.6), in the setting of Section 3. Let τ1 < τ2 < · · · < τK be the order statistics of the default times3 τ1 = min τ (k) , 1≤k≤K

τ2 = min {τ (k) : τ (k) > τ1 }, . . . 1≤k≤K

. . . , τm = min {τ (k) : τ (k) > τm−1 }, . . . , τK = max τ (k) 1≤k≤K

1≤k≤K

If the BDS is structured as an mth-to-default contract, the default time τ ≡ τm is the mth of the order statistics of τ (k) (k = 1, 2, . . . , K).

4.1 First-to-default contract In this case τ ≡ τ1 = min1≤k≤K τ (k) . To formulate the result for the first-to-default contract we introduce the special functions Daw± , usually called Dawson’s integrals:4 u 2 2 Daw+ (u) = e−u ev dv Daw− (u) = eu

2

0 u

e−v dv 2

0

3 Note that the default times τ (k) are distinct almost surely. For this reason, the definition of the

order statistics requires strict inequality. 4 The properties of the Dawson’s integrals can be found in Abramovitz and Stegun (1972) and Spanier and Oldham (1987). Volume 9/Number 3, Spring 2006

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P ROPOSITION 2 (k) (1) The risk-neutral density pτ (t) of the first-to-default contract is (k) λ(k) (t, x) e− (t,x) dϕ(x) pτ (t) = RC

(4.1)

(k) (t, x). where (t, x) = K k=1 T (k) (2) The integrals 0 pτ (t)D(t) dt, k = 1, 2, . . . , K, are given by

T

0

pτ(k)(t)D(t) dt =

n i=1

¯ λ(k) i (x)i−1 (x)D(ti−1 )J (i, x) dϕ(x)

RC

(4.2)

where J (i, x)  −1/2 (exp(−t (µ (x) + r )) Daw (b + ) − Daw (b ∗ ))  i i i − i − i |ci | = |ci |−1/2 (exp(−ti (µi (x) + ri )) Daw+ (bi− ) + Daw+ (bi∗ ))   (µi (x))−1(1 − exp(−µi (x)ti ))

if ri > 0 if ri < 0 if ri = 0

√ µi (x) = λi (x) + ri−1 + ti−1 ri /ti , bi∗ = (1/2)|ci |−1/2 µi (x), b± = |ci |ti ± bi∗ , and ci is given by (2.1). (k) (3) The risk-neutral probabilities (k) i = P(τ1 = τ , τ1 ∈ (ti−1 , ti ]) are given by

(k)

i =

(k)

RC

λi (x) ¯ i (x)) dϕ(x), ¯ i−1 (x) − ( λi (x) i = 1, . . . , n; k = 1, . . . , K

(4.3)

(k) (k) ¯ i (x) = K λi (x) and ¯ i (x). where λi (x) = K k=1 k=1 π T (4) The integral 0 F¯τ (t)D(t) dt is given by 0

T

F¯τ (t)D(t) dt =

n i=1

RC

¯ i−1 (x)D(ti−1 )J (i, x) dϕ(x)

(4.4)

Proposition 2 is proved in Appendix A.

4.2 Second-to-default contract Let τ = τ2 be the second-to-default time. (k)

P ROPOSITION 3 The risk-neutral densities pτ (t), k = 1, 2, . . . , K, of the second-to-default contract are given by F (j) (t, x) (k) pτ (t) = λ(k) (t, x) e− (t,x) dϕ(x) ¯ (j) (t, x) RC j=k F Journal of Computational Finance

Recursive valuation of Basket Default Swaps (k)

The risk-neutral probabilities i (k) i

=

RC 1≤j≤K j=k

are given by

Qkj (i, x) dϕ(x),

k = 1, 2, . . . , K, i = 1, 2, . . . , n (4.5)

where Qkj (i, x) =

λ(k) i (x)

¯ ˆ (x) − ¯ ˆ (x)) ( i i−1 (j) λi (x) − λi (x) (k) λi (x) ¯ i (x)) ¯ i−1 (x) − − ( λi (x)

(4.6)

and ¯ ˆ (x) = i

(k) π¯ i (x),

λi (x) =

1≤k≤K k=j

K

(k)

λi (x)

k=1

Using Equations (2.4), (2.6), (4.5) and (4.6), we can find simple expressions for the risk-neutral credit-risk premium or the value of the second-to-default contract of variant (C2) and (P2). For other variants, one can use Proposition 2 and Theorem 1 of the next section, but the resulting expressions are complicated and therefore not detailed here. Proposition 3 is a corollary of Proposition 2 and Theorem 1 in the next section. The structure of the expression for the probabilities Qkj (i, x) in Proposition 3 (k) reveals that the computation of the probabilities, i , for the second-to-default (k) contract is reduced to computation of the probabilities i for the first-to-default  , (j = 1, 2, . . . , K, j = k) containing all contract on two types of baskets: (i) B but the j th name from the original basket; and (ii) the original basket. We make direct use of this idea in handling the general case of an mth-to-default contract in the next section.

4.3 Valuation of mth-to-default contract We first derive a recurrence relation (in m) for the risk-neutral probability (k) densities, pτ(k) m , and the risk-neutral probabilities, i , associated with the mthto-default BDS, for m ≥ 2.  , containing all but the j th Let us introduce again the modified basket, B  the mth-to-default time in name from the original basket. Denote by τm B . the basket B (k) (k) We use the notation pm (t; B ) for the risk-neutral density pτm (t; B ) in the orig(k)  ) for the risk-neutral density p (k) (t; B  ) inal basket, and the notation pm (t; B τm  in the basket B . Volume 9/Number 3, Spring 2006

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I. Iscoe and A. Kreinin  ) satisfy the T HEOREM 1 The risk-neutral densities pm (t; B ) and pm−1 (t; B relation (k) (k) (k)  (m − 1)pm (t; B ) = pm−1 (t; B ) − (K − m + 1)pm−1 (t; B ) (4.7) (k)

(k)

j=k

Theorem 1 is proved in Appendix B. Let us also introduce the probabilities Pm (B ) = P(τm = τ (k) , τm ∈ (ti−1 , ti ])  ) = P(τ (B  ) = τ (k) , τ ∈ in the original basket, and the probabilities Pm (B m m (k)  (ti−1 , ti ]) in the basket B . Note that Pm is the same as i for the mth-to-default contract, but when k and i are fixed, we use the former notation to focus on the . variable parameter, m, and also the choice of the basket, B or B ) = ¯ Denote Fτm (t; B ) = P(τm > t) in the original basket, and F¯τm (t; B  P(τm > t) in the basket B . The following corollary is immediate from Theorem 1.  ) satisfy the relation C OROLLARY 1 The probabilities Pm (B ) and Pm (B

(m − 1)Pm (B ) =  Pm−1 (B ) − (K − m + 1)Pm−1 (B ),

m = 2, 3, . . . , K

(4.8)

j=k (k)

The integrals Im (B ) = recursion

T

(m − 1)Im(k) (B ) = The integrals Iˆm (B ) = Iˆ substituted for I (k) .

T 0

0

(k)

pτm (t; B )D(t) dt, k = 1, 2, . . . , K, satisfy the

j=k

(k) (k)  Im−1 (B ) − (K − m + 1)Im−1 (B )

(4.9)

F¯τm (t; B )D(t) dt satisfy the same relation (4.9) with

Theorem 1 and Corollary 1 allow us to compute recursively the risk-neutral default probabilities and expectations for the mth-to-default BDS, from which we can then find the risk-neutral credit-risk premium or the value of the contract, using the results (2.3)–(2.6). When the recursion arrives at m = 1, we can use Proposition 2 to finish computation. R EMARK . The recursion in Theorem 1 does not depend explicitly on the choice of interpolation used to construct the default distributions, F (k) , nor does it depend on the conditional independence assumption. The latter follows from the proof in Appendix B. This is useful for implementations, as Theorem 1 is valid for m = 2, so that only the simplest case of the first-to-default BDS need be worked out in detail (although performance must also be considered). For the first-todefault BDS, the choice of interpolation must come into play; and it is for the first-to-default BDS result that we have used the conditional independence. Journal of Computational Finance


It is at least of theoretical interest to find an explicit expression that reduces the general case down to the case of first-to-default contracts. To do that, we must generalize our notation for reduced baskets. With the names in the original basket enumerated by K = {1, 2, . . . , K}, for each subset J ⊂ K, we let |J | denote the number of names in J. Denote by B J , the reduced basket obtained by omitting from B , the names in J. When |J | = 1,  . When |J | = 0, ie, when J is empty, with say J = {j }, then B J is simply B B J ≡ B . The complete reduction for the probabilities Pm (B ) can now be expressed as

Pm (B ) =

m−1

(−1)

m−ν−1

ν=0

K −ν −1 m−ν −1

P1 (B J )

(4.10)

J ⊂K:|J |=ν

This result follows from Theorem 1, by induction on m. Naive implementation of recursion (4.8) leads to recalculation of the same probabilities many times for m > 2. Therefore, it is recommended to cache the results or, alternatively, to implement (4.10) for which there is no redundancy.

4.4 Computation of sensitivities In this section we address the problem of computation of sensitivities (greeks) for a BDS, ie, the partial derivatives of the value function, V , with respect to the basic parameters, on which it depends. In order to make the discussion reasonably general but also explicit, we shall restrict our attention to those parameters that (k) influence the conditional default probabilities, πˆ i (x). These parameters are the (k) factor loadings (β (k) ), the volatilities (σ (k) ), and the par CDS credit spreads (si ) which determine the unconditional default probabilities from the bootstrapped default intensity. (Thus we are excluding the recovery rates, the risk-neutral interest rates, and the time to maturity, but only for the sake of concreteness. There is no essential difficulty in handling them.) Sensitivities to the par CDS credit spread quotes are perhaps the most complex and perhaps also of the greatest interest. We denote by θ any one of the scalar parameters that we are considering. Having analytic solutions for the valuation of a BDS reduces the computation of sensitivities to nothing more than a lengthy exercise in differential calculus. This approach is described in general terms in Andersen et al. (2003), which we make more explicit for our results. There are two key aspects of the computation for an mth-to-default contract: (i) a constant-coefficient, linear recursion for m > 1; and (ii) an explicit result for m = 1. These two aspects apply not only for valuation; they can be used in exactly the same way for the computation of sensitivities. Indeed, the linear recursion(s) for the case of an mth-to-default contract (see Corollary 1), can be differentiated with respect to the parameter, θ, yielding recursions required Volume 9/Number 3, Spring 2006

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for the desired sensitivity – recursions of the same algebraic form as the undifferentiated ones. In particular, the problem is reduced to that for a firstto-default contract, for which we have an explicit result for differentiation – see Propositions 1 and 2. Together with (3.1), those results show that, in taking the derivative, ∂/∂θ, we can apply the chain rule, going through the (k) intermediate variables, πˆ i (x). In other words, the formula for the value of a first-to-default contract is of the form, RC G(x) dϕ(x), where G is an alge(k) braic function (including an exponential composite) of the variables, πˆ i (x). Thus we obtain the desired sensitivity as

RC k,i

(k) ∂G(x) ∂ πˆ i (x) dϕ(x) (k) ∂[πˆ (x)] ∂θ i

where the partials, ∂G(x)/∂[πˆ i(k)(x)], will be explicit. The quantities ∂ πˆ i(k) (x)/∂θ are calculated with the chain rule applied to (3.1). For θ being one of the factor loadings or volatilities, the latter derivatives are explicit and therefore immediate. In the case that θ is one of the par CDS credit spread quotes, θ appears only implicitly in πˆ i(k) (x), through −1 (πˆ i(k)): for each name k, πˆ i(k) = t 1 − exp(− t0i λ(k) (t) dt); and λ(k) (t) is piecewise constant and bootstrapped from the par CDS credit spread quotes in the usual way. Suppose we have B spread B quotes, (sb )B b=1 , corresponding to maturities (Tb )b=1 . (With k fixed, we suppress it from the notation.) Denoting the theoretical value of the bth CDS by Vb , we have the following system of equations for (λb )B b=1 , λb being the constant value of λ(t) for t in (Tb−1 , Tb ]: 0 = V1 (λ1 ; s1 )

⇒ λ1 ≡ λ1 (s1 )

0 = V2 (λ1 , λ2 ; s2 )

⇒ λ2 ≡ λ2 (s1 , s2 )

.. . 0 = VB (λ1 , . . . , λB ; sB )

⇒ λB ≡ λB (s1 , . . . , sB )

from which it follows by implicit differentiation that, for 1 ≤ b ≤ B and 1 ≤ j < b, ∂λb ∂Vb /∂sb =− ∂sb ∂Vb /∂λb b−1 (∂Vb /∂λ )(∂λ /∂sj ) ∂λb =− ∂sj ∂Vb /∂λb =j (k)

The quantities ∂G(x)/∂[πˆ i (x)] are common to large parts of these computations, and can be precalculated for use with all θs of interest. For credit-spread sensitivities, the partials, ∂Vb /∂λ , 1 ≤ ≤ b, are common to all computations, ∂λb /∂sj . Journal of Computational Finance


TABLE 1 Parameters of the BDS. Name

Notional

Credit rating

Correlation

1 2 3 4 5 6 7 8 9 10

190 80 70 360 100 200 150 123 95 107

C7 C2 C4 C6 C5 C6 C6 C5 C6 C3

0.5 0.6 0.9 0.6 0.5 0.4 0.7 0.64 0.55 0.22

5 Accuracy and performance In this section we compare the accuracy and performance of Monte Carlo simulation of BDS with that of the analytical solution. All examples follow the contract variant with the assumptions (C2) and (P2). The method of valuation of BDS contracts invokes numerical integration in Equations (4.2)–(4.5). Clearly the computation time is proportional to the number of points in the numerical quadrature. In the context of risk management, where each BDS will be valued under thousands of scenarios, the choice of numerical quadrature becomes important. An optimal choice of the numerical quadrature depends on the type of the model describing the joint distribution of default times. Both the Gauss–Legendre quadrature and the Hermite quadratures (Press et al. 2002) can be used for pricing BDS in the Gaussian one-factor and multi-factor models. In our numerical experiments, we used the Gaussian one-factor model for the joint distribution of default times. We also used a special quadrature originally designed for pricing of interest rate derivatives (Curran 2001). It performs well in the computation of Gaussian integrals and often outperforms the Gauss–Legendre and Hermite quadratures.

5.1 Accuracy The first basket used in our numerical experiments contains 10 names with the parameters shown in Table 1. This BDS has an inhomogeneous structure: all the instruments in the basket have different notionals as well as different correlations β (k) with the credit driver and different risk-neutral default probability curves. The maturity of the BDS is 5 years and ti = i, (i = 1, 2, 3, 4, 5). The recovery rates in the basket are 15%. The interest rate curve (rates are continuously compounded) is r(t1 ) = 0.046,

r(t2 ) = 0.05,

Volume 9/Number 3, Spring 2006

r(t3 ) = 0.056,

r(t4 ) = 0.058,

r(t5 ) = 0.06

107

108


TABLE 2 Risk-neutral cumulative default probabilities. Time

Credit rating

1st year

2nd year

3rd year

4th year

5th year

C1 C2 C3 C4 C5 C6 C7 C8

0.0041 0.0071 0.0072 0.0258 0.0305 0.0420 0.0501 0.0571

0.0052 0.0185 0.0225 0.0575 0.0616 0.0713 0.0802 0.0872

0.0069 0.0328 0.0439 0.0930 0.0936 0.0953 0.1062 0.1132

0.0217 0.0495 0.0692 0.1304 0.1464 0.1661 0.2171 0.2241

0.0288 0.0682 0.0967 0.1683 0.1702 0.195 0.303 0.310

TABLE 3 Confidence interval on BDS risk premium (bps): positive correlations. S

m=1

m=2

m=3

m=4

100,000 1,000,000 2,000,000 ∞ (analytic)

[235.3, 246.5] [239.4, 243.7] [239.8, 242.1] 240.2

[98.3, 103.3] [100.0, 101.7] [100.0, 101.5] 101.0

[52.2, 53.4] [52.5, 53.2] [52.6, 53.1] 52.8

[29.6, 30.5] [29.8, 30.4] [30.0, 30.3] 30.2

and the discount factors are d1 = 0.955, d2 = 0.905, d3 = 0.845, d4 = 0.792, d5 = 0.741. The risk-neutral cumulative default probability curves are described in Table 2. Table 3 contains results for the risk-neutral credit-risk premium for the mth-todefault contracts (m = 1, 2, 3, 4). We show the 95% confidence interval for the BDS credit-risk premium. The confidence intervals were computed as follows: we repeated each of the 12 Monte Carlo experiments (m = 1, 2, 3, 4; S = number of scenarios: 105 , 106 , 2 × 106 ) 500 times. After that, for each experiment, we computed the non-parametric confidence interval of the risk-neutral credit-risk premium, from the empirical distribution based on the sample of 500 values. The risk premium for the mth-to-default contract was also computed using the analytical technique in Section 4. The values of the credit-risk premium in basis points (bps), after rounding to the nearest integer, are 240, 101, 53 and 30 bps, respectively. They are included in the table as the line S = ∞. We show the results of Monte Carlo estimation of the risk premium to one decimal place, in order to indicate the rate of convergence as the number of scenarios increases. The results in this table allow one to conclude that if the number of scenarios, S, is less than 1,000,000 then the relative error of the estimation of the credit-risk premium of the first-to-default and of the secondto-default BDS may become greater than 2%. Next, the risk-neutral credit-risk premium for a small homogeneous basket was calculated. In this basket we have only six names, each of which has the notional 100. The risk-neutral default probability curve of the names is C4 in Table 2. Journal of Computational Finance


TABLE 4 Confidence interval on BDS risk premium (bps): signed correlations. S

m=1

m=2

m=3

m=4

100,000 1,000,000 4,000,000 ∞ (analytic)

[222.9, 232.1] [226.2, 229.6] [227.0, 228.6] 227.6

[98.1, 104.5] [100.0, 102.4] [100.6, 102.1] 101.3

[43.9, 45.2] [44.2, 44.9] [44.3, 44.8] 44.6

[0, 0.1] [0, 0] [0, 0] 0

The names in the basket have the following correlations with the credit driver: β (1) = −0.94, β (4) = −0.87,

β (2) = 0.92, β (5) = 0.85,

β (3) = 0.91 β (6) = −0.79

Thus, the names in the basket are split into two subgroups (according to the sign of β), both having strong correlations with the credit driver. The maturity of each of the contracts is 5 years. The discount curve and premium dates in this experiment are the same as in the previous one. Table 4 contains the results of Monte Carlo simulation of the risk-neutral creditrisk premium for four mth-to-default contracts (m = 1, 2, 3, 4). Again, the results would be rounded to the nearest integer in practice. In this case, the analytical values of the credit-risk premium are: 228 bps for the first-to-default contract; 101 bps for the second-to-default contract; 45 bps for the third-to-default contract. Since the probability of four defaults during the lifetime of the contract is very small, the risk-neutral credit-risk premium for the fourthto-default contract is zero after rounding. (Four defaults entails the same behavior between two highly negatively correlated groups.) Again, the relative error in the Monte Carlo estimation of the credit-risk premium may become greater than 2% if the number of scenarios S < 1,000,000. Figure 1 represents the credit-risk premiums of the mth-to-default contract as a function of m. The first (solid) curve is the credit-risk premiums of the first BDS, described in this section. The second curve represents the credit-risk premiums of the contracts having the same notionals and the same correlation coefficients, β (k) but with the risk-neutral default probabilities of the names increased as follows: if the default probability curve of a name is Cj in the original basket then in the modified basket this name will be attached to the curve Cj+1 . The results in Figure 1 demonstrate that in this case the credit-risk premium sensitivity (relative difference) of the mth-to-default contract is a monotonically increasing function of the parameter m.

5.2 Performance of recursive algorithm The algorithm described in Theorem 1 has a combinatorial nature. Indeed, to compute the risk-neutral default probabilities for the second-to-default contract,  (j = 1, 2, . . . , K), K valuations of first-to-default contracts on K baskets, B Volume 9/Number 3, Spring 2006

109


FIGURE 1 Credit-risk premium of the mth-to-default contract. 350

Original Modified 300

250

Spread (bps)

110

200

150

100

50

0

1

2

4

3

5

6

m

are required. It is not difficult to see that, for the mth-to-default contract, the number of valuations of first-to-default contracts will be O(K m−1 ). In Figure 2 the natural logarithm of the performance ratio, Perf = TMC /Tan , is displayed, where TMC is the time for Monte Carlo pricing of the first basket, described in Section 5.1, and Tan is the computation time of the analytical solution for this basket. The number of samples, S95% = 400,000 and S99% = 3,200,000 used in Monte Carlo pricing, allows us to obtain the Monte Carlo spread estimate within 95% and 99% confidence intervals. Although the valuation of a BDS using the Monte Carlo method takes much longer for m = 1 and m = 2, its complexity is O(K · n · ε−2 ), where ε is an acceptable error level (ε−2 indicating the number of scenarios for the simulation). This bound practically does not depend on m. Performance of the recursive algorithm is significantly better than that of a Monte Carlo simulation for sufficiently small values of the parameter m (see Figure 2). However, if m = 7 the computation time of the risk-neutral terminal default probabilities analytically becomes comparable with that of a Monte Carlo simulation, for a BDS of a Journal of Computational Finance


FIGURE 2 Performance comparison. 9

95% CI 99% CI 8

7

log(Perf)

6

5

4

3

2

1

1

2

3

4 m

5

6

7

medium size. For m > 7, the valuation approach based on the Monte Carlo method may become preferable.

6 Conclusion In this paper we proposed a recursive valuation of basket default swaps in the conditional independence framework for a continuous-time model. Our solution is based on explicit formulae for the conditional probabilities of terminal default events and associated expectations, in the basket for the first-to-default and for the second-to-default contracts. Performance of the recursive algorithm depends on the value of the parameter m (of the mth-to-default contract) and the number of names in the basket. There are several research areas left untouched in this paper. Among them are: (1) calibration of correlations in the BDS pricing model; some results in this direction were obtained in Andersen et al. (2003) although there are many open questions in this area; Volume 9/Number 3, Spring 2006

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(2) analysis and valuation of BDS options; (3) extension of the approach to stochastic recoveries and a random interest rate process. Our approach to pricing of BDS contracts can be significantly generalized to more complex contracts including options on basket losses and Asian-type options. These are directions for future research.

Appendix A Proof of Proposition 2 Equation (4.1) for the density can be found in Lando (2004). Consider the conditional probability density function, q (k)(t, x), of the random variable τ (k) . We have q (k) (t, x) =

∂ (k) (t, x) exp(− (k) (t, x)), ∂t

k = 1, 2, . . . , K

(A.1)

The conditional probability that the index of the first-to-default name is k and the default time τ1 ∈ (ti−1 , ti ] is (k)

i (x) ≡ Px (τ1 = τ (k) ∈ (ti−1 , ti ]) ti (k) (k ) = q (t, x)Px min τ > t dt k =k

ti−1

ti

=

q (k) (t, x)

exp(− (k ) (t, x)) dt

(A.2)

k =k

ti−1

From (A.1) and (A.2) we obtain (k) i (x) =

where (t, x) =

K k=1

ti

ti−1

∂ (k) (t, x) exp(− (t, x)) dt ∂t

(A.3)

(k) (t, x). The function (t, x) satisfies the equation

(t, x) = (ti−1 , x) + λi (x) · (t − ti−1 ),

t ∈ (ti−1 , ti ]

(A.4)

(k) where λi (x) = K k=1 λi (x). ¯ i (x) = Px (τ1 > ti ). Then we have Denote ¯ i (x) =

K

(k)

π¯ i (x) = exp(− (ti , x))

k=1

The latter relation, (A.3) and (A.4) yield ti (k) (k) ¯ exp(−λi (x)(t − ti−1 )) dt i (x) = λi (x)i−1 (x)

(A.5)

ti−1



and, finally, (k)

λi (x) ¯ i (x)] ¯ i−1 (x) − [ λi (x)

(k) i (x) =

(A.6)

Equation (4.3) follows from (A.6) by integrating. The proof of (4.2) is similar to that of (4.3): starting from Equation (3.2), one reduces the calculation to 0

T

pτ(k) (t, x)D(t) dt =

n

¯ i−1 (x)D(ti−1 ) λi (x) (k)

i=1

×

ti

exp(λi (x)(t − ti−1 )) exp(−ci · t · (t − ti−1 )) dt

ti−1

which follows in the same way as was done for (A.5). The rest is calculus. Proof of (4.4) goes along the same lines as those for (4.2), starting from the representation

T

F¯τ (t)D(t) dt =

0

RC

dϕ(x)

n i=1

ti

e− (t,x) D(t) dt

2

ti−1

Appendix B Proof of Theorem 1 Theorem 1 follows immediately from a very general result on order statistics, which we first establish. Let (, F , P) be a probability space and let A1 , . . . , An be n events. For 1 ≤ r ≤ n, denote by r and Pr , the probabilities r = P(exactly r of the Aj occur)

Pr = P(at least r of the Aj occur) so that

Pr =

n

R ,

1≤r ≤n

R=r

r = Pr − Pr+1 ,

1≤r ≤n−1

(B.1)

[i] We similarly denote by [i] r and Pr the probabilities corresponding to the reduced list of n − 1 events, obtained by omitting Ai from the original list, A1 , . . . , An . The indicator technique used by Balasubramanian and Balakrishnan to prove their theorem in Balasubramanian and Balakrishnan (1992) can also be used to establish the following more general result. (Indeed, the proof essentially follows verbatim their proof, with only a small change in notation: replace their F with our P . See also the Remark at the end of this appendix, for an alternative approach.)

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T HEOREM 2 For 1 ≤ r ≤ n − 1, (r + 1)r+1 + (n − r)r =

n

[i] r

(B.2)

Pr[i]

(B.3)

i=1

r Pr+1 + (n − r)Pr =

n i=1

The result in Balasubramanian and Balakrishnan (1992) is a special case of this theorem, corresponding to the choice Aj = {Xj ≤ x}, where X1 , . . . , Xn are random variables on and x is any fixed real number. Continuing with this notation, we let Xr:n denote the rth-order statistic from this family of random [i] variables, and Xr:n the rth-order statistic from the same family but with Xi omitted. The following corollary generalizes the result in Balasubramanian and Balakrishnan (1992), in the case where the joint distribution of the random variables is continuous, so that we may speak of the rth-order statistic coinciding with one of the original random variables. C OROLLARY 2 Assuming that the joint distribution of the random variables X1 , . . . , Xn is continuous, the following identity holds for 1 ≤ r ≤ n − 1, 1 ≤ k ≤ n, and x ∈ R: r P(Xr+1:n = Xk ≤ x) + (n − r)P(Xr:n = Xk ≤ x) [i] P(Xr:n = Xk ≤ x) = 1≤i≤n,i=k

P ROOF. If P(Xk ≤ x) = 0, then the corollary is trivial; so we assume otherwise and apply the identity (B.2) of Theorem 2 to the probability measure, P(· | Xk ≤ x), and the events Aj = {Xj ≤ Xk }. The corollary then follows from the following three observations: (1) the two events, {exactly r Xj ’s ≤ Xk } and {Xr:n = Xk }, coincide; [i] (2) for i = k, {exactly r Xj ’s (j = i) ≤ Xk } = {Xr:n = Xk }; (3) {exactly r Xj ’s (j = k) ≤ Xk } = {Xr+1:n = Xk }; so [k] r = r+1 .

2

P ROOF OF T HEOREM 1. We need only make the following identifications in Corollary 2: n = K,

i = k,

r = m − 1,

Xj = τ (j) ,

x=t

and then differentiate with respect to t. (For (4.8) in Corollary 1, one can directly take a difference, in Corollary 2, at the two times ti−1 , ti in (2.2).) 2 R EMARK . The identity (B.3) in Theorem 2 can be completely reduced to the following form: r−1 r−ν−1 n − ν − 1 Pr = (−1) P1[J ] (B.4) r − ν − 1 |J |=ν ν=0 Journal of Computational Finance


where J ranges over subsets of {1, 2, . . . , n}, |J | denotes the number of elements in J , and P1[J ] denotes the probability of occurrence of at least one of the events A1 , A2 , . . . , An , other than (Aj )j∈J . Consequently, by (B.1), r =

r n − ν [J ] (−1)r−ν−1 P r − ν |J |=ν 1 ν=0

(B.5)

Identities of the type (B.2), (B.3), (B.4) and (B.5) are a subject unto themselves. The interested reader may consult Rényi (1970, Section 2.6). In particular, the identity (2.6.23) therein, due to Jordan, is a different (unreduced) version of (B.5). An alternative proof of Theorem 2 can be based on Rényi (1970, Theorem 2.6.1). REFERENCES Abramovitz, M., and Stegun, I. (1972). Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9th edn. Dover, New York. Andersen, L., Sidenius, J., and Basu, S. (2003). All your hedges in one basket. RISK 16(11), 67–72. Balasubramanian, K., and Balakrishnan, N. (1992). Indicator method for a recurrence relation for order statistics. Statistics and Probability Letters 14, 67–69. Bielecki, T. R., and Rutkowski, M. (2002). Credit Risk: Modeling, Valuation and Hedging. Springer, Berlin. Curran, M. (2001). Willow power: optimizing derivatives pricing trees. Algo Research Quarterly 4(4), 15–23. Finger, C. (2002). Closed-form Pricing of Synthetic CDO’s and Applications. ICBI Risk Management, Geneva, December 2002. Huge, B. (2002). On defaultable claims and credit derivatives. PhD Thesis, Department of Statistics and Operations Research, University of Copenhagen. Hull, J., and White, A. (2004). Valuation of a CDO and an nth to default CDS without Monte Carlo simulation. Journal of Derivatives 12(2), 8–23. Kijima, M. (2000). Valuation of credit swap of the basket type. Review of Derivatives Research 4, 81–97. Kijima, M., and Muromachi, Y. (2000). Credit events and the valuation of credit derivatives of basket type. Review of Derivatives Research 4, 55–79. Lando, D. (2004). Credit Risk Modeling: Theory and Applications. Princeton University Press, Princeton. Laurent, J.-P. (2003). Accurately Valuing Basket Default Swaps and CDO’s using Factor Models (Quant’03). London, September (http://www.defaultrisk.com). Laurent, J.-P., and Gregory, J. (2003). Basket default swaps, CDO’s and factor copulas. Working Paper, 21 pp. (http://www.defaultrisk.com). Madan, D., Konikov, M., and Marinescu, M. (2004). Credit and basket default swaps, Robert H. Smith School of Business and Bloomberg L. P. Working Paper, 23 pp. (http://www.defaultrisk.com). Volume 9/Number 3, Spring 2006

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Press, W., Teukolsky, S, Vetterling, W., and Flannery B. (2002). Numerical Recipes in C++: The Art of Scientific Computing. Cambridge University Press, Cambridge. Rényi, A. (1970). Foundations of Probability. Holden-Day, San Francisco, CA. Schönbucher, P. (2003). Credit Derivatives Pricing Models: Models, Pricing and Implementation. Wiley, Chichester. Spanier, J., and Oldham, K. (1987). An Atlas of Functions. Hemisphere, Washington, DC.
