Question 1

We have a first to default derivative written on two obligors, A and B. The survival probabilities are described by QA(t) = exp( - λA t) and QB(t) = exp( - λB t), where λA denotes the hazard function (instantaneous probability of default) for obligor A, with a similar definition for λB. We want to use a Gaussian copula to examine how default dependence affects the distribution of the default times. The correlation coefficient for the Gaussian copula is ρ.

Part A

If λA = 0.03 and λB = 0.03 and ρ = 0.20, plot a graph of the default time for A on the horizontal axis and the default time for B on the vertical axis.

(Hint: A fast method to generate two variables from a zero-mean, unit variance bi-variate normal distribution is to use:
X1 = ρZ + (1 - ρ2)1/2 e1 and X2 = ρZ + (1 - ρ2)1/2 e2

where Z, e1 and e2 are drawn from a univariate, zero mean, unit variance normal distribution.) Use 500 simulations.

It is essential that you provide, in English, a description of the algorithm.

Part B

If λA = 0.03 and λB = 0.03 and ρ = 0.90, plot a graph of the default time for A on the horizontal axis and the default time for B on the vertical axis.

Comment of the difference of your results with Part A.

Question 2 Pricing a First to Default Derivative

We have a first to default derivative, maturity 5 years. The two underlying assets are as described in Question 1. The recovery rate for asset A is 40% and for B 35%. The notional for each asset is $10 million. The protection buyer makes an up-front payment; there are no quarterly payments. The protection seller will pay the protection buyer 10 * ( 1 - R), where R is the recovery rate of the asset that defaults first. The price of a zero coupon bond Z(0, T) = exp (- r * T), where r is the reference risk free rate and is equal to 1%. Compute the value for the first to default derivative for Part A and Part B, as described in Question 1.

If you used 1000 simulations, what is the average time of default for A and B? What percentage of time did A default before B?

Examine the sensitivity of your answers as you vary the number of simulations from 1000, 10,000, 100,000 and 250,000.

Question 3 Pricing a Second to Default Derivative

Given the information in Question 2, determine the value of a second to default derivative. Compute the value for the second to default derivative for Part A and Part B, as described in Question 1.

Examine the sensitivity of your answers as you vary the number of simulations from 1000, 10,000, 100,000 and 250,000.