Numerical Problems And Models In Actuarial And Risk Management

In this paper, we mainly consider the computational problems in the fnanceindustry. These problems include the numerical calculation of Panjer recursion,the pricing problem of BCVA for CDS contract and the valuation process of CDSproduct.Chapter 1 discusses a new numerical method of calculating Panjer recursion.In actuarial science, Panjer recursion (1981) is used to compute the loss distributionof the compound risk models. When the severity loss is continuous with densityfunction, numerical calculation for the compound distribution by applying Panjerrecursion will involve an approximation of the integration. In order to simplifythe numerical algorithms, we apply Bernstein approximation for the continuousseverity distribution function and obtain approximated recursive equations, whichare used for computing the approximated values of the compound distribution. Thetheoretical error bound for the approximation is also obtained. Numerical resultsshow that our algorithm provides reliable results.Chapter 2 develops and tests a fast and accurate semi-analytical formula forthe computation of bilateral counterpart risk credit value adjustment (BCVA). We illustrate the efect of correlation between investor, reference asset and counter-party by C^A,Bcopula in the context of a Cox-Ingersoll-Ross jump-difusion (JCIR)stochastic-intensity model of default. This model can be calibrated to the BCVAdata by solving an optimization problem. Credit default swap (CDS) is a specifcclass of fnancial instruments, the value of which is derived from an underlying mar-ket value that is driven by the credit risk of private or government entities. Theemergence of credit default swap has been proven to be as important as the adventof credit derivatives themselves and their use in capital markets. In this chapter,we introduce a new model by assuming the correlation of default events with C^A,Bcopula which is projected by Yang, Qi and Wang (2008) based on two realistic as-sumptions. A semi-analytical formula is faster and convenient is derived. Numericalresults show that the formula is reliable and efcient.Chapter 3 introduces an accurate, efcient and robust pricing model of col-lateralized debt obligations (CDO). Due to the underlying of CDO is a portfoliowith a lot of bonds, the valuation is a complex but interesting topic. How to modelthe correlation between risks is a problem in the central of CDO valuation. In thispaper, we will focus on the pricing of CDO under C^A,Bcopula.