6.3 Algorithms under Student-T Copula
6.3.2 Conditional Importance Sampling
Basic MC method is inaccurate when default is very rare, so we resort to importance sampling. We do not know how to perform change of measure efficiently with multivariate Student-T variables., but we can first condition on χ2ν, which then makes S Gaussian. Then we can apply our results from the Gaussian case as before. This means we perform conditional importance sampling first by conditioning on χ2ν. Then
SLLLχ2ν = X
Let A& = ' ν
χ2νA. Let Σ& = χν2
νΣ. Note that now, SLLLχ2ν = [S1&, ..., Sn&] ∼ N (0, Σ&) and A& is Cholesky decomposition of Σ& and A&−1 =
'χ2
ν A−1. Let W = [W1, ..., Wn]T where W1, ..., Wnare i.i.d. N(0, 1) Then A&W and SLLLχ2ν are equal in distribution. Let D = [d1, ..., dn]T = [t−1ν (F1(T )), ..., t−1ν (Fn(T ))]T. Let D& = [d&1, ..., d&n]T = A&−1[t−1ν (F1(T )), ..., t−1ν (Fn(T ))]T Now, we are ready to for-mulate our double expectation and conditional importance sampling. We can write DL as,
Condition on χ2ν = χ, τ is dependent on W in that it is the kthorder statis-tics of {τ1, ..., τn} and τi = Fi−1Φ(A&W [i]), where A&W [i] is the i-th element of A&W for i = 1, ..., n
DL = E.
(1− R)B(0, τ)I(τ≤T )
/
= E= E=
(1− R)B(0, τ(W ))I(τ (W )≤T )
LL
Lχ2ν = χ>>
= E=
E(d#1,...,d#n)=
(1− R)B(0, τ(W&))I(τ (W#)≤T )Q(W&)LLLχ2ν = χ>>
(6.10) Here, the likelihood ratio for change of measure is,
Q(W&) = 1n i=1
exp$
−12W&2i% exp$
−12(W&i− d&i)2%
Now we are ready to present our conditional importance sampling algorithm, Algorithm 6.6 (conditional Student-T).
1. Perform Cholesky decomposition on Σ to find A such that Σ = AAT. 2. Find A−1
3. For l = 1 to m2
(1) Generate χ2(l)ν
(2) Let A&(l) =' ν
χ2(l)ν A. Let A&(l)−1 = '
χ2(l)ν
ν A−1.
(3) Let D&(l) = A&(l)−1D Let d&(l)i be the i-th element of D&(l) for i =
(4) For j = 1 to m1
We compare Basic MC and conditional importance sampling under Student-T copula. First we compare SE of DL estimates under different correlation strengths, ρ, then under different default intensity λ−1. We calculate SE re-duction ratio in the same way as before.
We can observe that conditional importance sampling is able to consis-tently reduce SE in Table 6.3 and Table 6.4. We also discovered that SE reduction ratio increases as degrees of freedom increased in Table 6.5. This implies that with very low degrees of freedom, it is better to use Basic MC while with higher degrees of freedom, it is better to use conditional impor-tance sampling. However, in most cases, it is still better to use conditional importance sampling as SE reduction ratio is greater than 1 for degrees of freedom greater than 3.
Table 6.3: Evaluating DL with Different Correlation Strengths
parameters df n k T λ−1 r R m m1 m2
10 5 3 2 100 0.05 0.4 100000 10 10000 Basic MC Conditional Importance Sampling
ρ DL SE DL SE Ratio
0.00 3.73E-04 4.59E-05 3.73E-04 3.01E-05 1.53 0.05 3.45E-04 4.41E-05 3.77E-04 3.65E-05 1.21 0.10 4.79E-04 5.20E-05 4.64E-04 3.51E-05 1.48 0.15 4.73E-04 5.16E-05 4.28E-04 3.64E-05 1.42 0.20 4.85E-04 5.23E-05 5.03E-04 3.67E-05 1.43 0.25 7.01E-04 6.29E-05 5.85E-04 3.39E-05 1.85 0.30 6.48E-04 6.05E-05 7.79E-04 5.00E-05 1.21 0.35 8.94E-04 7.11E-05 8.66E-04 4.05E-05 1.75 0.40 1.06E-03 7.75E-05 9.94E-04 3.96E-05 1.96 0.45 1.39E-03 8.86E-05 1.30E-03 4.76E-05 1.86 0.50 1.74E-03 9.91E-05 1.75E-03 5.90E-05 1.68 0.55 2.14E-03 1.10E-04 2.09E-03 5.84E-05 1.88 0.60 2.62E-03 1.21E-04 2.61E-03 6.82E-05 1.78 0.65 3.17E-03 1.34E-04 3.26E-03 7.78E-05 1.72 0.70 4.07E-03 1.52E-04 3.88E-03 8.45E-05 1.79 0.75 4.91E-03 1.66E-04 4.96E-03 1.02E-04 1.63 0.80 5.91E-03 1.83E-04 5.96E-03 1.10E-04 1.67 0.85 7.01E-03 1.99E-04 6.95E-03 1.18E-04 1.68 0.90 7.89E-03 2.11E-04 8.39E-03 1.37E-04 1.54 0.95 9.83E-03 2.35E-04 9.89E-03 1.56E-04 1.51
Table 6.4: Evaluating DL with Different Default Intensity
parameters df n k T ρ r R m m1 m2
10 5 3 2 0.5 0.05 0.4 100000 10 10000 Basic MC Conditional Importance Sampling
λ−1 DL SE DL SE Ratio
20.00 1.86E-02 3.19E-04 1.90E-02 2.31E-04 1.38 40.00 6.67E-03 1.93E-04 6.66E-03 1.27E-04 1.52 60.00 3.55E-03 1.41E-04 3.59E-03 9.20E-05 1.54 80.00 2.45E-03 1.18E-04 2.40E-03 7.12E-05 1.65 100.00 1.76E-03 9.98E-05 1.71E-03 5.65E-05 1.77 120.00 1.33E-03 8.69E-05 1.39E-03 5.30E-05 1.64 140.00 1.24E-03 8.40E-05 1.08E-03 4.17E-05 2.01 160.00 9.58E-04 7.37E-05 8.61E-04 3.44E-05 2.14 180.00 8.61E-04 6.98E-05 8.43E-04 4.06E-05 1.72 200.00 6.45E-04 6.04E-05 6.25E-04 2.99E-05 2.02 220.00 6.67E-04 6.14E-05 5.94E-04 2.97E-05 2.07 240.00 4.77E-04 5.21E-05 4.99E-04 2.73E-05 1.91 260.00 4.14E-04 4.85E-05 4.92E-04 2.86E-05 1.70 280.00 4.06E-04 4.78E-05 3.82E-04 2.16E-05 2.21 300.00 3.40E-04 4.39E-05 3.73E-04 2.60E-05 1.69 320.00 4.06E-04 4.79E-05 3.39E-04 2.28E-05 2.10 340.00 3.30E-04 4.33E-05 3.36E-04 2.38E-05 1.82 360.00 3.33E-04 4.34E-05 2.75E-04 1.90E-05 2.29 380.00 2.85E-04 4.04E-05 2.96E-04 2.04E-05 1.98 400.00 2.90E-04 4.06E-05 2.52E-04 1.70E-05 2.39 420.00 2.67E-04 3.90E-05 2.40E-04 1.84E-05 2.12 440.00 2.32E-04 3.63E-05 2.41E-04 1.92E-05 1.89 460.00 2.26E-04 3.58E-05 2.28E-04 1.56E-05 2.30 480.00 1.92E-04 3.30E-05 1.95E-04 1.51E-05 2.19 500.00 2.27E-04 3.59E-05 1.76E-04 1.20E-05 2.99 520.00 1.43E-04 2.87E-05 2.32E-04 2.36E-05 1.22 540.00 1.59E-04 3.00E-05 1.78E-04 1.55E-05 1.93 560.00 1.31E-04 2.74E-05 1.68E-04 1.48E-05 1.84 580.00 1.18E-04 2.58E-05 1.39E-04 1.23E-05 2.10 600.00 1.52E-04 2.93E-05 1.56E-04 1.32E-05 2.23 620.00 1.20E-04 2.63E-05 1.42E-04 1.67E-05 1.57 640.00 1.37E-04 2.80E-05 1.41E-04 1.22E-05 2.30 660.00 1.02E-04 2.41E-05 1.22E-04 9.85E-06 2.45 680.00 1.20E-04 2.62E-05 1.11E-04 1.17E-05 2.25 700.00 1.41E-04 2.83E-05 1.17E-04 1.17E-05 2.43 720.00 1.13E-04 2.52E-05 1.14E-04 9.82E-06 2.57 740.00 9.09E-05 2.27E-05 1.27E-04 1.32E-05 1.72 760.00 1.26E-04 2.69E-05 1.38E-04 1.23E-05 2.19 780.00 1.03E-04 2.43E-05 9.93E-05 9.34E-06 2.61 800.00 6.24E-05 1.88E-05 1.13E-04 1.18E-05 1.59
Table 6.5: Evaluating DL with Different Degrees of Freedom
parameters n k T ρ λ−1 r R m m1 m2
5 3 2 0.5 400 0.05 0.4 100000 10 10000 Basic MC Conditional Importance Sampling
df DL SE DL SE Ratio
1 2.02E-03 1.07E-04 2.33E-03 2.26E-04 0.47 2 1.34E-03 8.76E-05 1.45E-03 1.26E-04 0.69 3 9.53E-04 7.37E-05 9.81E-04 8.84E-05 0.83 4 7.79E-04 6.65E-05 6.87E-04 6.38E-05 1.04 5 5.98E-04 5.84E-05 5.19E-04 4.18E-05 1.40 6 4.62E-04 5.14E-05 4.54E-04 3.67E-05 1.40 7 3.42E-04 4.41E-05 3.76E-04 2.88E-05 1.53 8 3.34E-04 4.35E-05 3.70E-04 2.90E-05 1.50 9 3.06E-04 4.16E-05 2.90E-04 2.22E-05 1.87 10 2.26E-04 3.57E-05 2.23E-04 1.44E-05 2.48 11 2.28E-04 3.61E-05 2.01E-04 1.14E-05 3.18 12 1.86E-04 3.25E-05 2.33E-04 1.69E-05 1.92 13 2.28E-04 3.60E-05 2.06E-04 1.33E-05 2.70 14 1.83E-04 3.23E-05 1.86E-04 1.25E-05 2.58 15 1.19E-04 2.60E-05 1.74E-04 1.35E-05 1.93 16 1.98E-04 3.34E-05 1.68E-04 9.67E-06 3.45 17 2.25E-04 3.55E-05 1.58E-04 8.57E-06 4.14 18 9.57E-05 2.32E-05 1.58E-04 1.03E-05 2.25 19 1.64E-04 3.05E-05 1.38E-04 7.61E-06 4.01 20 1.63E-04 3.03E-05 1.36E-04 7.79E-06 3.88
Chapter 7 Conclusion
In our study, we have confirmed the effectiveness of direct importance sam-pling under Gaussian copula and conditional importance samsam-pling under Student-T copula for evaluating DL in BDS. We use the idea of efficient importance sampling under Large Deviation Theory in estimating joint default probabil-ity and pricing basket default swaps. We first extend CYH’s conditional on marginal factors approach to the conditional on common factor approach.
Then we formulate our own direct importance sampling scheme. Under Gaus-sian copula, we test these different algorithms with different correlation strengths and default intensity. We discover that direct change of measure is a stable approach. It is more accurate except in extreme cases when correlation is very high or low. Also, we discover that consistent with large deviation theory, as reciprocal of default intensity λ−1 increases, SE reduction ratio increases as well. This allows us to effectively use the direct importance sampling algo-rithm when default is extremely rare, causing default threshold to be very small.
Also, we prefer importance sampling because of its versatility. The impor-tance sampling algorithm we used for estimating joint default probability is easily extendable to more complicated problems such as working with order statistics and evaluating BDS. Though Quasi MC is more effective in evalu-ating multivariate normal and Student-T CDF, it is not easily extendable to evaluating BDS. We compared importance sampling with Quasi MC method and discover that though importance sampling is less accurate, it nonetheless performs reasonably well. Therefore, we adopt importance sampling as the main method in this study.
Another advantage of the direct importance sampling scheme is that it only involves one expectation instead of double expectation. Monte Carlo methods become less stable and more inaccurate when working with multiple layers of expectations. Direct importance sampling removes one layer of expectation.
We take advantage of this feature and apply it to Student-T copula. Under Student-T copula, our importance sampling scheme under Gaussian Copula cannot be applied directly, so we first condition on the χ2 variable. Then we are back to working with multivariate normal variable. Now we can apply our importance sampling scheme. We do not apply the condition on common or marginal factors scheme because that would require three layers of ex-pectations. We discovered that our conditional importance sampling scheme consistently reduced SE under different correlation strengths and default in-tensity. We also discovered that SE reduction ratio increases as degrees of freedom increased. This implies that with very low degrees of freedom, it is better to use Basic MC while with higher degrees of freedom, it is better to use conditional importance sampling. However, in most cases, it is still better to use conditional importance sampling.
Here, we conclude this study by recommending direct importance sam-pling under Gaussian copula and conditional importance samsam-pling under Student-T copula for pricing BDS.
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