Prom a pricing perspective, a basket default swap written as notional loss is essentially an option on the aggregate loss for a given portfolio. Loss amount is a function of default timing and recovery rate. Default timing in turn is driven by hazard rate level and default correlation. Therefore the main risk factors that impact the price of a basket default swap are the hazard rates for each name, the default correlation and the recovery rate distribution.
Analytic solutions for the price of the basket default swap may run up against a combinatorial explosion in the number of possible orderings of defaulting assets. In this paper we present a practical Monte Carlo approach to valuation, where the default times of multiple assets are simulated. We first describe the simulation of default times for a single asset.
3.1. Simulation of default time for a single asset
A "compensator" method may achieve simulation of the time of default by means of path-wise simulation of the hazard rate. We suggest in this paper that the computationally intensive method of simulating hazard rates may be replaced with a faster algorithm termed the (deterministic proxy algorithm). The proposed deterministic proxy is equivalent to a lookup function for the convexity adjusted integrated hazard rate, as compared with the actual integrated hazard rate which is a random variable. We note that
£ > - l o g E exp - hu
Jo
du • E exp • / K Jo du where £ is an exp(l) random variable. Therefore, if we define
T * ^ i n f U : £ < - l o g E exp / hu
Jo du
(r > t),
(10)
then for any t > 0,
F(r*>t) = P(Z>-logE exp / K
Jo du E exp / hu
Jo du (r>t)
This demonstrates that by using a convexity adjusted lookup in place of simulation of the hazard rate, one may achieve simulation of the default time T* with the same marginal distribution as the default time r. For affine processes, we have closed form expression for the right hand side of equation (10), i.e.
E exp - hu Jo
du _ e-a(t)-b(t)h0
with deterministic a(t) and b(t). Therefore simulating the default time T using formula (10) requires only the simulation of exp(l) variable £ and simulating the process ht can be avoided. Figure 1 illustrates the algorithm for finding the default arrival time r.
Figure 1. Fast Algorithm: A sample path for finding T
3.2. Modeling default correlation and and simulating joint default times
The deterministic proxy method described above is now adapted for mul- tivariate default time simulation in a general setting.
To simulate joint default arrival times, we need a joint distribution P(Ti < *i,T2 < t2,--- ,Td < td) of default times, where n is the default time for the ith reference entity. In our case, the marginal distributions P(TJ < t) are given but the joint distribution is not. Intuitively speaking, if we are only concerned with the correlation among the n's, we can use the compensator algorithm to simulate each T* but with correlated exp(l) random variables & in formula (10). We can formulate this idea as follows.
Observe that if Z is a standard Gaussian random variable with distri- bution function $ ( i ) , i.e. Z ~ N(0,1), then - l o g ( l - $(Z)) is an exp(l) random variable. Set & = - log(l-$(Zj)) and {Zi,Z2, ...,Zd)~ JVd(0, E), where E = (pij)dxd and pa = 1, then the exp(l) random variables
£1,62, • • • ,fd are correlated. The joint distribution of default arrival times is given by
P(TI <h,...,Td<td) = QdiQ-^ih)),...,S-^te))) (11) where Fi(t) = P(TJ <t),i = l,...,d. This is equivalent to use the Gaussian
copula function (see Appendix § Appendix B for more detail):
C(xu...,xd) = $d($-l(l-e-xi),...,$-1(l-e-x*)). (12) The default indicator correlation is given by
CorriHU),^)) = ^•••Mti),...,Pj(tj),...,l)-Pi(ti)Pj(tj) VPt(*i)Pj(*i)(l -pi(u)){l -Pj(tj))
(13) where pi(t) = P(TJ < t) and pj{t) — P(TJ < t), and the conditional default probabilities are
(14) 3.3. The pricing algorithm
In combination with the compensator proxy algorithm described above, we can price a basket default swap as follows:
O Generate (ui,...,un) ~ $n( *_ 1( u i ) , . . . , $_ n( un) , E ) by
® Generating (j/i,..., yn) ~ $ „ ( j / i , . . . , y„, E) from the correlated normal distribution.
© Setting Ui = *_1(?/i)-
Set & = - log(l - Ui). Then the default time of the ith name is given n - i n ft > 0{ t : & < o<(t) + 6ằ(t)/ii,o}, where a*, 6,, hj)0 are the parameters for the generating function of the affine process described in the previous section.
@ Feed the default time into the cashflow generator (varies accord- ing to the deal microstructure and waterfall specifications)
© Discount all cash flows using the risk free discount curve to ob- tain the present value for the sample path.
© Repeat for a pre-defined number of sample paths. The basket price is the average over all paths.
Figure 2 is a numerical example showing how correlation affects the de- fault distribution for a 30 name basket. In this example, the average hazard rate is about 300 basis point. Figure 3 shows the impact of correlation on the premium for the first 10 to default and last 20 to default basket based on the same portfolio.
Histogram of Defaults (Corr=0%) Histogram of Defaults (Corr=10%)
0.5
10 20 Number of Names Histogram of Defaults (Corr=20%)
L^ 10 20
Number of Names
0.4
t:
0.1
l l l l l l l — —
0.5 0.4
I:;
0.1 0
10 20 Number of Names Histogram of Defaults (Corr=40%)
10 20 Number of Names
Figure 2. Impact of correlation on default distribution, 30 reference entities, average spread ss 170bp, maturity = 3.5 years
A Graph of First-to-Default Par Premium vs Correlation
0.1 0.2 0.3 0.4 Default Indicator Correlation
0.5
A Graph of Second-to-Default Par Premium vs Correlation 0.6
0.1 0.2 0.3 0.4 Default Indicator Correlation
0.5 0.6
Figure 3. First (10/30) and second (20/30) to default swap - average spread w 170bp, maturity = 3.5 years