Credit Default Swap Case Study

2.2 Credit Default Swap pricing formula.

The pricing formula of Credit Default Swap incorporates two components: the fixed leg, which refers to the periodic payment the protection buyer has to honor, and the default leg which refers to the payment of the protection seller in case of a default event. We define N as the notional amount of the CDS contract, D(t) the risk free discount factor at time t, R the recovery rate and Delta t the time interval between the payments.

The present value of the fixed leg and is defined as:

PV_{fixed}=Ssum_{i=1}^{n}ND(t_{i})P_{survival}(t_{i})Delta t_{i}

The default leg is defined as:

PV_{default}=(1-R)sum_{i=1}^{n}ND(t_{i})(P_{surv}(t_{i-1})-P_{surv}(t_{i}))

The price of the CDS contract at the time t=0

Numerous extensions of the two models have been published. One of them , Credit Grades model is a generalization of the previous two and is widely accepted by practitioners. The model 's main difference compared to its predecessors is that it allows for instantaneous default by incorporating the barrier L as stochastic, resulting into higher short term spreads compared to implementing a deterministic barrier.

Nelson on 1993 and Longstaff and Schwartz on 1995, attempted to avoid some of the shortcomings of the Merton 's model. They approach the reference equity 's structure in a different way and default can happen at any point in time. In case of a default event, the entity 's debt is paid partly or in fixed amount. Their models, implement interest rates as stochastic, accounting for interest rate risk as well. As we have seen, the equity value of a financial entity depends on the stock prices, while the interest rates depend on the level of the interest on government

The underlying distribution of stock returns may exhibit fat tails, be skewed and may have positive excess kurtosis. In their paper, Cariboni and Schoutens (2004) based on the intuition of the Merton 's model as well, attempt to overcome the flaws of Black - Scholes normality assumptions and suggest a model in which, the stock price process is described by en exponential of a Lévy process driven entirely by jumps. The underlying distributions of the Lévy processes are more flexible and capable of better reflecting the variations of the underlying. This specification allows for instantaneous default without the need for a stochastic barrier