Ebook Valuation of Credit Default Swap and Parameter Estimation for Vasicek-type Hazard Rate Model
The starting point of our researdh is to consider how credit default swap. a kind of credit derivatives, is modeled for estimating the value. Roughly speaking, credit default swap is, like a sort of Insurance, defined as the contract, made with the holder of a defaultable bond, which obliges the counter party of the swap to compensate t.he loss that the holder suffers at the default in return for a regular premium income. Since the demand for such contract, for the purpose of hedging credit risk, is more and more increasing and appear some new versions, for example, one for more than two defaultable bonds and one with counter party risk,
There are several problems for modeling the credit derivatives, however. It is in general the most significant the way the event of default should be characterized. Duffie and Singleton(l994) propose the model in which the default is specified by a hazard rate process deeply connected with the distribution of default time. In their model it is possible to price the defaultable claims almost the same way as non-default claims. Davis and Mavroidis(l997) utilize the above framework to study the valuation of credit default swap by supposing that the hazard rate is a Gaussian model with time dependent deterministic drift. In this article we also consider the default model which takes the hazard rate as principal factor following these previous model,
This paper consists of two parts --- theoretical one and empirical one. The first half is the theoretical one which explains the scheme of credit default swap that we take up (Section 2) and gives the general valuation of default swap which contains the case of so-called basket type and counter party risk type (Section 3). The evaluation formula obtained there seems to be too complicated for numerical computation, but under the hypothesis that the hazard rate is modeled in the form of affine type or quadratic Gaussian type term structure model such as Vasicek model, CIR model and so on, the expectation can be numerically calculated by solving some Riccaci type ordinary differential equations. It is often effective in thinking the case where there is a correlation among the issuers of defaultable bonds. (See [3],[4] and [9].) Anyway it is concluded that the choice of hazard rate model is important in order to evaluate the value of credit default swap. Mathematical discussion is omitted to the utmost extent,
In the latter half, we devote ourselves to estimate parameters of hazard rate model, It is suitable for the setting that the hazard rate is specified in terms of a stochastic differential equation such as term structure models, say, Vasicek model, CIR model and so on. The first is the question if such SDE models are really suitable to hazard rate process. We believe that this approach is rather efficient since it is possible to see the hazard rate indirectly through the spread between riskless interest rate and default adjusted interest rate. Besides the hazard rate is expected to have such properties as risk-free interest rate, for instance, mean-reverting and positive. Vasicek model is often used as risk-free interest rate model since it is mean-reverting and Gaussian, although it has a serious defect of taking negative value with a positive probability.
It is also noted that the mean-reverting property lessens the possibility of negative hazard rate in comparison with the non mean-reverting Gaussian case. That is why Vasicek type is allowed to use as hazard rate model in simulation and is our main object. On the other hand, hazard rate is guaranteed to be positive, for example, when choosing CIR model or Quadratic-Gaussian model. CIR model is also widely used because it is mean reverting and affine as well as positive. Quadratic-Gaussian model is tractable when considering the correlation among riskless interest rate and each issuer’s hazard rate. The way of parameter estimation for these models is analogue to that for Vasicek type.
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