Loan credit default swaps (LCDSs) is a agreement between two parties to exchange credit risk at the occurrence of a credit event, which acts as a insurance to the protection buyer. LCDSs are very similar to standard credit default swaps (CDSs), except that other than default risk, the LCDS contract is also exposed to cancellation risk due to the possibility of prepayment of loans before maturity. Hence, the LCDS contract is cancellable and have higher recovery rates.
Even though LCDS and CDS operate on two separate markets, since they have the same underlying reference entity, Morgan and Zheng (2007) supposed that the both LCDS and CDS must also share the same probabilities of default (or default intensities). Consequently, the relationship for the two instruments can be expressed as
𝑠𝑝𝑟𝑒𝑎𝑑𝐿𝐶𝐷𝑆
1 − 𝑅𝑒𝑐𝑜𝑣𝑒𝑟𝑦𝐿𝐶𝐷𝑆 = 𝑠𝑝𝑟𝑒𝑎𝑑𝐶𝐷𝑆
1 − 𝑅𝑒𝑐𝑜𝑣𝑒𝑟𝑦𝐶𝐷𝑆 (2.1) Equation 2.1 gives a rough approximation between the two different instruments. However, from market spreads, it is shown that the equation above clearly does not hold, which the LCDS spreads are sometimes much higher than the recovery-adjusted CDS spreads. In order to solve this puzzle, Peter Dobranszky (2008) tried to find stochastic risk factors that may provide the link between CDS and LCDS markets and explain the current market figures and their dynamics by incorporating the possibility of negative correlation between default and cancellation intensities and the possibility of stochastic recovery rate correlated with the default intensity.
The inequality of the above equation possibly stream from different conventions in the contracts between CDSs and LCDSs, resulting in more difficult modeling issues for LCDSs, simply not only higher recovery rates. From an LCDS modeling perspective, here we summarize several key factors that need to be considered:
(1) default risk (2) cancellation risk (3) higher recovery rate
(4) correlation between cancellation and probability of default
The correlation aspect is the most difficult part, while, intuitively, a negative correlation between cancellation and the probability of default can be observed from
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empirical results. For the pricing of LCDSs, prevailing approaches follow the industry standard approach with deterministic intensity that JP Morgan (2001) originally introduced in the valuation of standard CDSs. In 2007, Scott et al (2007b), from JP Morgan, considered a simple mark-to-market approach using a historical cancellation probability matrix. Also, Elizalde et al (2007) from Merrill Lynch introduced a generic LCDS modeling framework that incorporated a refinancing rate into the modeling of the cancellation feature. The same year, Morgan and Zheng (2007) of Lehman Brothers designed a generalized model based on the hazard rate model in pricing standard CDSs while assuming independence between cancellation and default. Later, Wei (2007) pointed out the limitations of the prior approaches and extended this framework further by incorporating stochastic intensity, which he borrowed the short rate model for the pricing of bonds to derive a closed-form partial differential equation solution under CIR specification. Under the stochastic intensity framework, Wang and Liang (2012) consider the pricing of both a single-name and a two-reference basket LCDS and Wang, Liang and Yang (2012) extended this framework into N baskets as well. From a different aspect, Wu and Liang (2012) extended the model into a multifactor case by incorporating stochastic recovery. Other non-standard approaches are developed by Bandreddi et al (2007) from Merrill Lynch who used a double-barrier model with Gaussian distribution instead of Poisson distribution so that the cancellation barrier and the default barrier behave as two competing factors. A survey of the LCDS pricing models can be found in Ong, Li and Lu (2012).
Furthermore, for the pricing of the index product of the credit derivatives, the industry has accepted the copula method to model joint default behavior of names in the basket, which was introduced by Li (2000). One of the key assumptions of the one factor Gaussian copula model is that the asset values of the names in the underlying reference portfolio are correlated to a common macroeconomic factor with the same correlation for all single names. Using the one factor Gaussian copula framework, Shek et al (2007) extend the model with a new variable to incorporate the cancellation feature for the index loan derivatives, including the LCDX index swap and LCDX tranche swap. While, Dobranszky and Schoutens (2008) extended the approach to the generic one-factor Levy copula. However, in the double barrier one-factor copula approach, the default barrier needs to be lower than the prepayment barrier. This requirement could be violated when the default probability and the cancellation probability are high. Michael Liang (2009), from industrial bank in China provide a simple two-factor Gaussian copula model to price the LCDX tranche swap with cancellation risk.
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In this article we do not intend to develop a new model for LCDSs, rather we assume that a model for LCDSs is already in use. For the valuation of LCDSs, we consider a industry standard intensity model incorporated with stochastic intensity which was first introduced by Wei (2007). Similarly, to model a tranched portfolio of LCDSs, the LCDX tranche swap, here we refer to Shek et al (2007), who introduced a extended double barrier single factor Gaussian copula model to deal with the cancellation issue.
For the risk measurement of the more complicated LCDX tranche swap, we follow Gibson (2004) who introduced the expected and unexpected risk measures to examine tranche risk of the synthetic CDOs. Moreover, Chiang, M., Yueh, M., and Lin, A. (2009) proposed a Delta hedge analysis to determine hedging costs for synthetic CDO tranches under a hypothesis portfolio. In this thesis, we conduct a hedging analysis based on the methods referring to their works.
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