In this paper we consider the following inverse problem for the first hitting time distribution: given a Wiener process with a random initial state, probability distribution, F (t), and a linear boundary, b(t) = µt, find a distribution of the initial state such that the distribution of the first hitting time is F (t). This problem has important applications in credit risk modeling where the process represents, so-called, distance to default of an obligor, the first hitting time represents a default event and the boundary separates the healthy states of the obligor from the default state. We show that randomization of the initial state of the process makes the problem analytically tractable.Primary Subjects: 60G40 Secondary Subjects: 91B70
A forward starting CDO is a single tranche CDO with a specified premium starting at a specified future time. Pricing and hedging forward starting CDOs has become an active research topic. We present a method for pricing a forward starting CDO by converting it to an equivalent synthetic CDO. The value of the forward starting CDO can then be computed by the well developed methods for pricing the equivalent synthetic one.We illustrate our method using the industry-standard Gaussian-factor-copula model.Numerical results demonstrate the accuracy and efficiency of our method.
The Gaussian factor copula model is the market standard model for multi-name credit derivatives. Its main drawback is that factor copula models exhibit correlation smiles when calibrating against market tranche quotes. To overcome the calibration deficiency, we introduce a multi-period factor copula model by chaining one-period factor copula models. The correlation coefficients in our model are allowed to be timedependent, and hence they are allowed to follow certain stochastic processes. Therefore, we can calibrate against market quotes more consistently. Usually, multi-period factor copula models require multi-dimensional integration, typically computed by Monte Carlo simulation, which makes calibration extremely time consuming. In our model, the portfolio loss of a completely homogeneous pool possesses the Markov property, thus we can compute the portfolio loss distribution analytically without multi-dimensional integration. Numerical results demonstrate the efficiency and flexibility of our model to match market quotes.
A basket default swap (BDS) is a credit derivative with contingent payments that are triggered by a combination of default events of the reference entities. A forwardstarting basket default swap (FBDS) is a BDS starting at a specified future time.Existing analytic or semi-analytic methods for pricing FBDS are time consuming due to the large number of possible default combinations before the BDS starts. This paper develops a fast approximation method for FBDS based on the conditional independence framework. The method converts the pricing of a FBDS to an equivalent BDS pricing problem and combines Monte Carlo simulation with an analytic approach to achieve an effective method. This hybrid method is a novel technique which can be viewed either as a means to accelerate the convergence of Monte Carlo simulation or as a way to estimate parameters in an analytic method that are difficult to compute directly. Numerical results demonstrate the accuracy and efficiency of the proposed hybrid method.
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