Volume 20, Issue 4

In the present paper, we introduce a numerical scheme for the price of a barrier option when the price of the underlying follows a diffusion process. The numerical scheme is based on an extension of a static hedging formula of barrier options. To get the static hedging formula, the underlying process needs to have a symmetry. We introduce a way to “symmetrize” a given diffusion process. Then the pricing of a barrier option is reduced to that of plain options under the symmetrized process. To show how our symmetrization scheme works, we will present some numerical results of path-independent Euler–Maruyama approximation applied to our scheme, comparing them with the path-dependent Euler–Maruyama scheme when the model is of the type Black–Scholes, CEV, Heston, and (λ)-SABR, respectively. The results show the effectiveness of our scheme.

This paper develops some non-parametric simultaneous confidence bands for survival function when data are randomly censored on the right. To construct the confidence bands, a computer-assisted method is utilized and this approach requires no distributional assumptions, so the confidence bands can be easily estimated. The procedures are based on the integrated martingale whose distribution is approximated by a Gaussian process. The supremum distribution of the Gaussian process generated by simulation techniques leads to the construction of the confidence bands. To improve the estimation procedures for the finite sample sizes, the log-minus-log transformation is employed. The proposed confidence bands are assessed using numerical simulations and applied to a real-world data set regarding leukemia.

In this paper we analyze and compare the use of Monte Carlo, quasi-Monte Carlo and hybrid Monte Carlo methods in the credit risk management system “Credit Metrics” by J. P. Morgan. We show that hybrid sequences, used suitably for simulations, perform better, in many relevant situations, than pure Monte Carlo and pure quasi-Monte Carlo methods, and they only rarely perform worse than these methods.

The uncertainty of world population growth represents a serious global problem. Existing methods for quantifying this uncertainty face a variety of questions. An essential problem of these methods is the lack of direct evidence for their validity, for example by means of comparisons with independent observations like measurements. A way to support the validity of such forecast methods is to validate these models with reference models, which play the role of independent observations. Desired properties of such a reference model are formulated here. A new reference world population model is formulated by a probabilistic extension of recent deterministic UN projections. This model is validated in terms of theory and observations: it is shown that the model has all desired properties of a reference model, and its predictions are very well supported by the known world population development from 1980 till 2010. Applications of this model as a reference model demonstrate the advantages of the stochastic world population model presented here.

A generalization of a polynomial chaos-based algorithm for solving PDEs with random input data is suggested. The input random field is assumed to be defined by its mean and correlation function. The method uses the Karhunen–Loève expansion, in its analytical form, for the input random field. Potentially, however, if desired, the Karhunen–Loève expansion can be also constructed by a randomized singular value decomposition of the correlation function recently suggested in our paper [Math. Comput. Simulation 82 (2011), 295–317]. The polynomial chaos expansion is then constructed by resolving a probabilistic collocation-based system of linear equations. The method is compared against a direct Monte Carlo method which solves repeatedly many times the PDE for a set of samples of the input random field. Along with the commonly used statistical characteristics like the mean and variance of the solution, we were able to calculate more sophisticated functionals like the instant velocity samples and the mean for Eulerian and Lagrangian velocity fields.