# Abstracts

**Efficient importance sampling for estimating lower tail probabilities under Gaussian and Student t distributions**

*Chuan-Hsiang Han (National Tsing-Hua University, Taiwan)*

Thursday June 5, 16:30-17:00 | session 9.1 | Computational Finance | room AB

In credit risk modeling, Gaussian and Student t variates arise primarily from the copula method for retaining certain correlation structures among defaultable assets. We propose efficient importance sampling algorithms to estimate lower tail probabilities of these two variates in any finite dimension. Variances of importance sampling estimators are shown to be zero asymptotically by means of the large deviation theory and a truncation argument. Moreover these algorithms are suitable for parallel computing so that their standard errors can be dramatically reduced by small standard deviation and large sample size. Numerical comparisons with commercial codes, such as mvncdf.m and mvtcdf.m in Matlab, demonstrate robustness and efficiency of our proposed algorithms. Furthermore, generalizations of these algorithms can be seen from the following two applications: (1) probability estimation for the nth-to-default, i.e., the nth order statistic, given a credit portfolio of homogeneous assets, and (2) the default probability estimation of an inhomogeneous portfolio. The latter is related to a relative entropy minimization problem.

**A comparative study on time-efficient methods to price compound options in the heston model**

*Carl Chiarella (The University of Technology, Sydney, Australia)*

Thursday June 5, 17:00-17:30 | session 9.1 | Computational Finance | room AB

The primary purpose of this paper is to provide an in-depth analysis of a number of structurally different methods to numerically evaluate European compound option prices under Heston’s stochastic volatility dynamics. Therefore, we first outline several approaches that can be used to price these type of options in the Heston model: a modified sparse grid method, a fractional fast Fourier transform technique, a (semi-)analytical valuation formula using the Green’s function of logarithmic spot and volatility and a Monte Carlo simulation. Then we compare the methods on a theoretical basis and report on their numerical properties with respect to computational times and accuracy. One key element of our analysis is that the analyzed methods are extended to incorporate piecewise time-dependent model parameters, which allows for a more realistic compound option pricing. The results in the numerical analysis section are important for practitioners in the financial industry to identify under which model prerequisites (for instance, Heston model where Feller condition is fulfilled or not, Heston model with piecewise time-dependent parameters or with stochastic interest rates) it is preferable to use and which of the available numerical methods.

**A Least-Squares Monte Carlo Approach to the Calculation of Capital Requirements**

*Hongjun Ha (Georgia State University, USA)*

Thursday June 5, 17:30-18:00 | session 9.1 | Computational Finance | room AB

The calculation of capital requirements for financial institutions usually entails a reevaluation of the company’s assets and liabilities at some future point in time for a (large) number of stochastic forecasts of economic and firm-specific variables. The complexity of this nested valuation problem leads many companies to struggle with the implementation. Relying on a well-known method for pricing non-European derivatives, the current paper proposes and analyzes a novel approach to this computational problem based on least-squares regression and Monte Carlo simulations. We show convergence of the algorithm, we analyze the resulting estimate for practically important risk measures, and we derive optimal basis functions based on spectral methods. Our numerical examples demonstrate that the algorithm can produce accurate results at relatively low computational costs, particularly when relying on the optimal basis functions.