The loss density function is useful for pricing credit derivatives and risk management of credit portfolio. The loss density function concludes estimating the joint default probability and survival probability. Under the structural-form model, computation of the loss density function relies on the estimates of some mixture of joint default probability and joint survival probability. We provide several efficient importance sampling methods to estimate the probability of defaultable and/or survival event. We also provide an empirical study of a correlated hedging for CDS spread.

1 Introduction and Problem Formulation
1.1 Loss Density Function Estimation
2 Structural-Form Models
2.1 Compound Poisson Process}
2.2 Jump-Diffusion Model
2.3 Probability of Default
2.3.1 Default at Maturity
2.3.2 Default before Maturity: First-Passage Time Problem
3 Variance Reduction Method
3.1 Monte Carlo Simulations
3.2 Importance Sampling
4 Importance Sampling and Its Efficiency
4.1 Efficient Importance Sampling on Diffusion model
4.1.1 Default Probability
4.1.2 Survival Probability
4.2 Efficient Importance Sampling on Jump-Diffusion Model
5 Numerical Results
5.1 Single-Name Case
5.1.1 Diffusion Model: Default Probability
5.1.2 Diffusion Model: Survival Probability
5.1.3 Jump Diffusion Model: First-Passage Time Problem
5.2 Multi-Dimension
5.2.1 Diffusion Model: Default Probability
5.2.2 Diffusion Model: Survival Probability
5.2.3 Jump Diffusion Model: First-Passage Time Problem
6 Loss Density Function
6.1 Diffusion model
6.2 Jump Diffusion Model
7 Hedging Credit Derivative

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