In this research, we investigate the pricing issue for the Loan CDS and its index portfolio LCDX tranche swap under intensity-base model and extended one factor Gaussian copula respectively. Compared to a standard CDS, the most difficult task for pricing loan credit derivatives is to incorporate the cancellation feature into the model.
For modeling the Loan CDS, we consider the stochastic intensities rather than deterministic intensities, since the model with deterministic intensities neglect the cancellation affect. On the other hand, for the LCDX tranche swap, we propose an extended model of the classical single factor Gaussian copula framework which has the advantage of adhering to the current market convention of pricing CDX indices.
Due to the implement of the Bullet version of the LCDS contract in 2010, the current newly issued LCDS contract has a bullet maturity which is not subject to early termination. In this paper, we are interested in comparing the spread for the cancellable Legacy LCDS with the non-cancellable Bullet LCDS priced under our models. A comparison for the index product LCDX tranche swap is also shown in our results. From our results in Chapter 5, we find out that the spread for Legacy LCDS is generally lower than the Bullet LCDS. Furthermore, from the sensitivity analysis, it shows that as the correlation between default and cancellation approaches to -1, the spread for Legacy LCDS becomes higher until it is identical to the Bullet LCDS.
However, there is one exception. For the super senior tranche of the LCDX tranche swap, the Legacy LCDX spread happen to exceed the Bullet LCDX spread when the correlation between default and cancellation is negatively high.
In this research, we also test the sensitivity of the spread when different ρ and υ is involved (recall that the correlation parameter ρ stands for the correlation between the underlying asset value and the macro factor M, while the correlation parameter υ stands for the correlation between default and cancellation). This allows us to take a look on the spread for the Legacy LCDS and LCDX tranche swap under different market conditions. From our results, we find that there is a negative relationship between correlation parameter ρ and the spread for the equity tranche. However, the relationship turn out to be positive for the more senior tranches, including the junior senior tranche and the super senior tranche. But for the senior mezzanine and junior mezzanine tranche, the relationship is not consistent, which the spread first increase as the correlation parameter ρ increases from zero, then decreases as the correlation parameter ρ approaches to one. As for the correlation parameter υ, our findings for different tranches are consistent, all of them have a negative relationship with the spread, that is, tranche spread become higher as υ is negatively higher.
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Besides, we also examined the risk characteristics of the loan credit derivatives through computation of the credit risk measures, which can be categorized into expected risk and unexpected risk measures. From the results, we find that the equity tranche bears more risk during the initial period than the later period of the contract due to a decrease in leverage ratios throughout the term of the contract, while more senior tranches are just the opposite, which bears more risk during the later period of the contract due to a increase in leverage ratios.
Other than pricing, we make an effort on the hedging issue of the LCDS as well.
From our results, we conclude that although selling the LCDX index swap at the amount of the tranche Delta can achieve complete hedging, it also results in great hedging costs, causing negative net tranche return for the mezzanine and senior tranches. Only the equity tranche can still receive positive return after tranche Delta hedging. Therefore, we suggest tranche investor to sell single-name LCDSs to rather than the LCDX index swap to hedge the tranche loss because the tranche Delta computed using the single-name LCDSs is smaller. Generally, the sensitivity of the MTM of a tranche to the change of spread of a single-name LCDS is very small, usually smaller than 1; as a result, using single-name LCDSs to hedge tranche loss could be more efficient.
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