l=1
1 m1
m1
5
j=1
0 n 1
i=1
I(AW#[i](j)≤d(l)i )exp
:d&(l)2i
2 − Wi(j)d&(l)i ; LLLχ2(l)ν
2
where for each i, W&(j)i are i.i.d. samples of Wi& for j = 1, ..., m1, and d&(l)i for l = 1, ..., m2 are samples of d&i which is dependent on samples of χ2ν in the outer expectation. Let Standard error will be,
SE =&
var(ˆp)
Recall in (3.3), calculating joint default probability is equivalent to evaluating the multivariate student-T CDF.
p = TΣ,ν(t−1ν (F1(T )), ..., t−1ν (Fn(T )))
There is no close form solution but Genz and Bretz [8] (1999) proposed Quasi MC to evaluate this CDF when n is large. We will compare our importance sampling scheme to Quasi MC.
5.3 Numerical Comparison
We compare the performance of Basic MC, Importance Sampling and Quasi MC under different scenarios for Gaussian copula and Stuent-T copula. First we take into consideration the rarity of the default event by testing different threshold D. Recall that D = [d1, ..., dn]T = [Φ−1(F1(T )), ..., Φ−1(Fn(T ))]T. But for simplicity, we let d1, ..., dn be equal to one constant and call it D. For our purpose, we just need to test performance as D decreases, causing default to be more rare so we do not need to construct D from Φ(.) and Fi(.) at the moment. Then we compare performance of different methods with different number of firms, n. For constructing covariance matrix Σ we use the factor copula model and assume ρ1, ..., ρn are all equal and call it ρ.
In the following Tables,
• D = default threshold for each firm.
• ρ = correlation coefficient used to construct Σ under factor copula model
• df = degree of freedom for chi-square variable under Student-T copula
• m = number of iterations in Basic Monte Carlo and Importance Sam-pling method for Gaussian copula.
• m1 = number of iterations in evaluating inner expectation under Condi-tional Importance Sampling for Student-T copula.
• m2 = number of iterations in evaluating outer expectation under Condi-tional Importance Sampling for Student-T copula.
Under both Gaussian and Student-T copula, we can see importance sam-pling and Quasi MC both performed significantly better than Basic MC method.
As D gets smaller, Basic MC method is no longer capable of sampling from such rare events. Importance sampling and Quasi MC are comparable in terms of performance when D gets very small and when n increases. Quasi MC seems more accurate overall, but importance sampling is a simpler, eas-ily modifiable and versatile approach. Given that the importance sampling method performs reasonably well under both Gaussian and Student-T copula, we take advantage of it’s simplicity and later apply it to more complex prob-lems such as calculating tail probability for some order statistics of default time, where Quasi MC method is not easily applicable.
Table 5.1: Estimating Joint Default Probability with Different Default Thresh-old under Gaussian Copula
parameters n ρ m
5 0.5 25000
Basic MC Importance Sampling Quasi MC
D p SE p SE p Error
0.0 9.00E-02 1.81E-03 8.90E-02 1.80E-03 9.07E-02 6.91E-05 -0.5 2.30E-02 9.47E-04 2.13E-02 4.61E-04 2.10E-02 5.61E-05 -1.0 2.84E-03 3.37E-04 3.07E-03 8.25E-05 3.05E-03 1.36E-05 -1.5 2.00E-04 8.94E-05 2.60E-04 9.01E-06 2.68E-04 1.76E-06 -2.0 0.00E+00 0.00E+00 1.34E-05 5.53E-07 1.40E-05 1.05E-07 -2.5 0.00E+00 0.00E+00 4.61E-07 2.65E-08 4.24E-07 4.51E-09 -3.0 0.00E+00 0.00E+00 7.73E-09 5.39E-10 7.39E-09 2.37E-10 -3.5 0.00E+00 0.00E+00 7.27E-11 6.26E-12 7.17E-11 9.68E-13 -4.0 0.00E+00 0.00E+00 3.75E-13 4.05E-14 3.96E-13 5.64E-15 -4.5 0.00E+00 0.00E+00 1.33E-15 1.46E-16 1.21E-15 1.41E-17 -5.0 0.00E+00 0.00E+00 2.26E-18 3.86E-19 2.11E-18 4.19E-20 -5.5 0.00E+00 0.00E+00 3.20E-21 6.69E-22 1.97E-21 5.82E-23 -6.0 0.00E+00 0.00E+00 1.03E-24 1.85E-25 1.04E-24 4.29E-26 -6.5 0.00E+00 0.00E+00 2.66E-28 4.20E-29 2.89E-28 1.09E-29 -7.0 0.00E+00 0.00E+00 5.44E-32 1.28E-32 4.58E-32 1.48E-33 -7.5 0.00E+00 0.00E+00 6.50E-36 2.29E-36 3.83E-36 1.52E-37 -8.0 0.00E+00 0.00E+00 1.64E-40 5.34E-41 1.42E-40 1.35E-42 -8.5 0.00E+00 0.00E+00 4.06E-45 9.79E-46 6.29E-46 8.76E-48 -9.0 0.00E+00 0.00E+00 1.24E-49 7.58E-50 7.49E-52 9.33E-54 -9.5 0.00E+00 0.00E+00 3.29E-55 9.99E-56 3.56E-58 1.89E-60 -10.0 0.00E+00 0.00E+00 6.45E-61 2.46E-61 4.60E-65 2.34E-67
Table 5.2: Estimating Joint Default Probability with Different Number of Firms under Gaussian Copula
parameters D ρ m
-2 0.5 25000
Basic MC Importance Sampling Quasi MC
n p SE p SE p Error
5 4.00E-05 4.00E-05 1.35E-05 5.46E-07 1.40E-05 1.31E-07 6 0.00E+00 0.00E+00 4.74E-06 2.69E-07 4.77E-06 1.14E-07 7 0.00E+00 0.00E+00 1.78E-06 1.03E-07 1.85E-06 3.93E-08 8 0.00E+00 0.00E+00 7.50E-07 5.57E-08 8.12E-07 2.39E-08 9 0.00E+00 0.00E+00 3.96E-07 2.94E-08 3.81E-07 3.49E-08 10 0.00E+00 0.00E+00 2.13E-07 1.71E-08 2.01E-07 1.62E-08 11 0.00E+00 0.00E+00 1.02E-07 1.10E-08 1.07E-07 8.07E-09 12 0.00E+00 0.00E+00 8.55E-08 9.43E-09 6.30E-08 4.77E-09 13 0.00E+00 0.00E+00 3.43E-08 4.36E-09 3.65E-08 2.83E-09 14 0.00E+00 0.00E+00 2.09E-08 2.32E-09 2.24E-08 1.71E-09 15 0.00E+00 0.00E+00 1.67E-08 2.46E-09 1.52E-08 2.30E-09 16 0.00E+00 0.00E+00 8.83E-09 1.73E-09 9.77E-09 1.73E-09 17 0.00E+00 0.00E+00 6.17E-09 8.85E-10 7.61E-09 2.59E-09 18 0.00E+00 0.00E+00 4.41E-09 5.78E-10 4.61E-09 1.08E-09 19 0.00E+00 0.00E+00 2.94E-09 5.34E-10 3.80E-09 2.15E-09 20 0.00E+00 0.00E+00 2.54E-09 4.02E-10 2.56E-09 7.42E-10 21 0.00E+00 0.00E+00 1.47E-09 3.62E-10 1.64E-09 3.34E-10 22 0.00E+00 0.00E+00 1.45E-09 2.89E-10 1.35E-09 3.99E-10 23 0.00E+00 0.00E+00 1.28E-09 2.18E-10 1.07E-09 4.47E-10 24 0.00E+00 0.00E+00 1.05E-09 1.89E-10 6.23E-10 1.67E-10 25 0.00E+00 0.00E+00 4.42E-10 1.10E-10 7.26E-10 3.44E-10
Table 5.3: Estimating Joint Default Probability with Different Default Thresh-old under Student-T Copula
parameters n ρ df m m1 m2
5 0.5 10 25000 10 2500
Basic MC Importance Sampling Quasi MC
D p SE p SE p Error
0.0 8.99E-02 1.81E-03 9.09E-02 1.80E-03 9.07E-02 6.43E-05 -0.5 2.39E-02 9.66E-04 2.27E-02 5.38E-04 2.31E-02 9.38E-05 -1.0 4.68E-03 4.32E-04 4.62E-03 1.70E-04 4.82E-03 8.65E-05 -1.5 9.60E-04 1.96E-04 8.27E-04 4.67E-05 9.52E-04 4.39E-05 -2.0 2.80E-04 1.06E-04 2.11E-04 2.27E-05 1.92E-04 1.19E-05 -2.5 4.00E-05 4.00E-05 3.79E-05 7.12E-06 4.39E-05 6.97E-06 -3.0 4.00E-05 4.00E-05 1.36E-05 3.25E-06 1.22E-05 3.72E-06 -3.5 0.00E+00 0.00E+00 3.85E-06 1.23E-06 3.72E-06 2.51E-06 -4.0 0.00E+00 0.00E+00 1.94E-06 1.13E-06 1.46E-06 1.06E-06 -4.5 0.00E+00 0.00E+00 2.03E-07 6.64E-08 5.81E-07 5.18E-07 -5.0 0.00E+00 0.00E+00 2.24E-08 1.79E-08 1.24E-07 1.76E-07 -5.5 0.00E+00 0.00E+00 8.40E-08 8.02E-08 9.17E-09 9.45E-09 -6.0 0.00E+00 0.00E+00 6.82E-09 6.36E-09 8.38E-08 2.49E-07 -6.5 0.00E+00 0.00E+00 5.64E-09 5.04E-09 5.08E-09 8.56E-09 -7.0 0.00E+00 0.00E+00 3.33E-11 3.05E-11 6.34E-10 8.39E-10 -7.5 0.00E+00 0.00E+00 1.39E-11 1.33E-11 7.08E-09 2.45E-08 -8.0 0.00E+00 0.00E+00 7.75E-09 7.75E-09 1.80E-11 2.68E-11 -8.5 0.00E+00 0.00E+00 7.47E-14 7.45E-14 1.17E-10 3.65E-10 -9.0 0.00E+00 0.00E+00 7.10E-15 5.42E-15 1.45E-11 2.58E-11 -9.5 0.00E+00 0.00E+00 9.25E-13 9.25E-13 3.44E-12 8.76E-12 -10.0 0.00E+00 0.00E+00 3.82E-13 3.80E-13 3.44E-10 1.18E-09
Table 5.4: Estimating Joint Default Probability with Different Number of Firms under Student-T Copula
parameters D ρ df m m1 m2
-2 0.5 10 25000 10 2500
Basic MC Importance Sampling Quasi MC
n p SE p SE p Error
5 3.20E-04 1.13E-04 2.01E-04 1.78E-05 2.00E-04 1.54E-05 6 0.00E+00 0.00E+00 9.36E-05 1.19E-05 1.00E-04 1.06E-05 7 0.00E+00 0.00E+00 5.30E-05 1.09E-05 5.64E-05 1.26E-05 8 0.00E+00 0.00E+00 3.46E-05 5.09E-06 2.91E-05 5.83E-06 9 0.00E+00 0.00E+00 1.72E-05 3.77E-06 2.11E-05 7.28E-06 10 0.00E+00 0.00E+00 1.74E-05 3.19E-06 1.17E-05 2.38E-06 11 0.00E+00 0.00E+00 1.08E-05 3.39E-06 8.91E-06 2.22E-06 12 0.00E+00 0.00E+00 8.29E-06 2.17E-06 6.01E-06 2.12E-06 13 0.00E+00 0.00E+00 3.79E-06 1.14E-06 4.56E-06 1.69E-06 14 0.00E+00 0.00E+00 1.96E-06 4.34E-07 3.37E-06 1.06E-06 15 0.00E+00 0.00E+00 2.42E-06 1.06E-06 2.21E-06 8.64E-07 16 0.00E+00 0.00E+00 9.02E-07 3.35E-07 3.41E-06 3.05E-06 17 0.00E+00 0.00E+00 1.14E-06 3.25E-07 1.58E-06 1.25E-06 18 0.00E+00 0.00E+00 1.82E-06 9.91E-07 1.09E-06 8.00E-07 19 0.00E+00 0.00E+00 9.04E-07 2.96E-07 6.47E-07 2.79E-07 20 0.00E+00 0.00E+00 6.33E-07 2.46E-07 6.01E-07 3.48E-07 21 0.00E+00 0.00E+00 4.14E-07 2.92E-07 5.63E-07 3.37E-07 22 0.00E+00 0.00E+00 3.53E-07 1.88E-07 5.44E-07 5.20E-07 23 0.00E+00 0.00E+00 5.44E-07 3.36E-07 3.54E-07 1.96E-07 24 0.00E+00 0.00E+00 2.51E-07 1.68E-07 2.62E-07 1.76E-07 25 0.00E+00 0.00E+00 4.75E-08 3.07E-08 3.58E-07 2.50E-07
Chapter 6
Application II: Basket Default Swap
6.1 Introduction to Basket Default Swaps
We now wish to apply our importance sampling scheme in evaluating joint de-fault probability to evaluating multi-name credit derivatives. This is motivated by CYH’s study [5] on BDS. CYH mainly employed conditional importance sampling under the Gaussian copula factor model. We will first correct his ap-proach which seems to only consider the outer expectation. Then we suggest a different conditional importance sampling scheme which improves accuracy under specified conditions. In this case, we perform change of measure as suggest by results in Section 4.3. Finally, we introduce direct importance sam-pling based on the method used in evaluating joint default probability. Lastly, we will compare performance of these methods under Gaussian copula model.
As an extension, we apply similar method to evaluating BDS under Student-T copula and compare it with Basic MC.
The mechanism of a credit default swap (CDS) is similar to that of an insur-ance. The protection buyer makes periodical premium payments (protection leg or PL) until some credit events happen. Then swap issuer compensates for the non-recovered part of the reference entities’ notional amounts (de-fault leg or DL). CDS provides credit protection only for a single underlying.
Multi-name credit derivatives, such as BDS and CDO have gained increasing popularity in recent years because they extend credit protection to a pool of underlying. We focus on BDS in this study which provides protection to a
pool of underlying until default of one underlying. A kth-to-default BDS offers protection only against the event of the kthdefault on a pool of n underlying.
Pricing BDS is equivalent to determining the fair premium a BDS buyer needs to pay. Under risk-neutral measure, the fair premium is determined by equating protection leg or PL to default leg or DL. Here we introduce some notations and assumptions of our pricing model.
• n: Number of names in one basket, usually 5 or 6.
• T : Terminal time of a BDS contract.
• R: Recovery rate.
• M: Notional amount.
• *j−1, j, j = 1, 2,· · · , N: The time increment tj− tj−1.
• τi: Default time of the ith company.
• τ: kth default time.
• B(0, τ): exp$
−Nτ
o r(u)du%
, the discount factor, (or price of zero coupon bond) where r(·) denotes risk free interest rate.
• prem: Fair premium for protection buyer.
The default leg when kth default takes place is DL =E.
(1− R) · M · B(0, τ) · I(τ≤T )
/ (6.1)
where E is the expectation under risk neutral measure. When kthdefault does not take place, then protection leg is
P L = E 0 N
5
j=1
*j−1,j· M · prem · B(0, tj)· I(τ≥tj)
2
(6.2)
We equate DL (6.1) and PL (6.2) to derive premium:
prem = E {(1 − R) · B(0, τ) · I(τ ≤ T )}
E=<N
j=1*j−1,j· B(0, tj)· I(τ > tj)>. (6.3)
To be more precise, we take accrued interests into considerations, modify-ing (6.2) by:
P Lacc =E 0 N
5
j=1
3τ − tj−1
tj − tj−1*j−1, j
4
· M · prem · B(0, τ) · I(tj−1 ≤ τ ≤ tj) 2
, (6.4) when defaults happen between two payment dates.
To derive fair premium (or spread), we need to evaluate both PL and DL.
As suggested by CYH, we will mainly focus on evaluation of DL, which is what mainly affects accurate evaluation of fair premium because default events are extremely rare.