Monte Carlo Implementation in Pricing 5th-to-Default Basket of CDS Using Gaussian and Student T’s Copula
Introduction
A credit derivative is a contract that transfers credit risk from one party (the buyer) to the other (the seller). Such credit risks may be linked to events that pose risk on the drastic devaluation of the underlying asset or worse, the non-repayment of the underlying asset. These products allow trading and hedging of credit risk. The most popular credit derivatives contracts currently traded in the market are credit default swaps (CDS), collateralized debt obligations and credit-linked notes. In this paper, we will focus on the pricing dynamics of a basket CDS.
A Credit Default Swap (CDS) is a financial product that provides insurance to the buyer against certain pre-determined credit event that will happen to the issuer of the underlying product until the contract matures. The underlying product may be a bond issued by company X, in which case the CDS protects the buyer on any drastic event that affects company X’s ability to repay the bond at maturity. As in any insurance, the buyer of the CDS must pay the seller a premium in a regular basis until the CDS matures or until the credit event happens. This “insurance payment” is called the CDS spread premium and is computed at the beginning of the contract. If a credit event happens, e.g. the issuer of the underlying defaults before maturity, the seller will pay the buyer the differential between the face value of the underlyinf asset and the recoverable value of the underlying asset. Thus, in valuing this CDS spread premium, we have two legs to consider: the periodic spread payment (premium leg) and default payment (default leg).
In this paper, we will focus on the generalized multi-asset version of the CDS contract called kth-to-default basket CDS (from hereon will be referred to as KTD). This contract provides protection over the credit risk of a basket of >k obligors. Let T denote the maturity of a KTD contract. As in the basic CDS, the buyer pays the seller the predetermined premium periodically until T. However, the seller will only pay the buyer after the kth default occurs (the seller will not pay the 1st, 2nd, …, (k-1)th defaults), in that case the seller will pay the face value of the asset that defaulted on the kth default, not the face value of the entire basket of underlying assets.
Factors that affect the price of KTD
The value of the KTD depends on the following risk factors:
Basket size. All other terms in the contract equal, if one KTD contains one additional underlying asset compared to another KTD (the underlying assets are the same between the two KTDs, except for the inclusion of an additional underlying in the first), then the kth default will happen earlier to the first KTD. Thus, the first KTD will have a greater spread premium than the second.
Value of k. All other terms in the contract equal, if one KTD has higher value of k than another KTD, then the first KTD has higher spread premium than the second.
Maturity. The effect of the maturity of a KTD is not straightforward since some obligors have a higher chance to default in the nearest years (a start-up’s critical years, for example) compared to the succeeding years while others are becoming riskier as the time passes.
Credit rating. The higher credit rating of an underlying, the less likely is the default to occur; thus the lower the premium and vice versa.
Recovery rates. If the underlying’s recovery rate is higher than another underlying, the spread will be lower.
Correlation. Note that correlated assets will tend to default at the same time. Suppose a KTD contract contains k underlying assets with pairwise correlation of 1. This can be considered as k basic

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