Defaults risk-neutral probability

The default swap market is not unlike the lottery ticket. What if the shipping and handling fee for the winning ticket was unknown or turned out to be zero In that case, if an investor observed these lottery tickets trading at a price of 4, it may appear that the probability of winning was simply 4.00%. In the case of a default swap this is what is referred to as the risk-neutral probability of default. The risk-neutral probability of our default swap is approximately equal to the premium of 4.00%. By applying the lottery ticket example to our default swap, it is easy to see how the hazard rate is dependent on both the risk-neutral probability as well as the recovery value assumption, and thus can be approximated hy X = P/(l - R). 

Default can take place randomly over time, and the default probability can be determined using the risk-neutral transition matrix. 

The JLT model allows the pricing of default swaps, as the risk neutral transition matrix can be used to determine the probability of... 

Various credit derivatives may be priced using this model for example, credit default swaps, total return swaps, and credit spread options. The pricing of these products requires the generation of the appropriate credit dependent cash flows at each node on a lattice of possible outcomes. The fair value may be determined by discounting the probability-weighted cash flows. The probability of the outcomes would be determined by reference to the risk neutral transition matrix. 

If the time of default for a reference credit is l, then let us denote the cumulative (risk neutral) default probabilities for the reference credit as... 

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