Volume 25, Issue 4

Practitioners sometimes suggest to use a combination of Sobol sequences and orthonormal polynomials when applying an LSMC algorithm for evaluation of option prices or in the context of risk capital calculation under the Solvency II regime. In this paper, we give a theoretical justification why good implementations of an LSMC algorithm should indeed combine these two features in order to assure numerical stability. Moreover, an explicit bound for the number of outer scenarios necessary to guarantee a prescribed degree of numerical stability is derived. We embed our observations into a coherent presentation of the theoretical background of LSMC in the insurance setting.

We develop a numerical algorithm for predicting prices and Greeks of equity-linked securities (ELS) with a knock-in barrier at any time over the total time period from issue date to maturity by using Monte Carlo simulation (MCS). The ELS is one of the most important financial derivatives in Korea. In the proposed algorithm, first we calculate the probability (0≤p≤1{0\leq p\leq 1}) that underlying asset price never hits the knock-in barrier up to the intermediate evaluation date. Second, we compute two option prices Vn⁢k{V_{nk}} and Vk{V_{k}}, where Vn⁢k{V_{nk}} is the option value which knock-in event does not occur and Vk{V_{k}} is the option value which knock-in event occurs. Finally, we predict the option value with a weighted average. We apply the proposed algorithm to two- and three-asset ELS. We provide the pseudo-numerical algorithm and computational results to demonstrate the usefulness of the proposed method.

Multi-stage models have been used to describe progression of individuals which develop through a sequence of discrete stages. We focus on the multi-stage model in which the number of individuals in each stage is assessed through destructive samples for a sequence of sampling time. Moreover, the stage duration distributions of the model are effected by a time-dependent hazard rate. The multi-stage models become complex with a stage having time-dependent hazard rate. The main aim of this paper is to derive analytically the approximation of the likelihood of the model. We apply the approximation to the Metropolis–Hastings (MH) algorithm to estimate parameters for the model. The method is demonstrated by applying to simulated data which combine non-hazard rate, stage-wise constant hazard rate and time-dependent hazard rates in stage duration distributions.

The standard coupling from the past (CFTP) algorithm is an interesting tool to sample from exact Markov chain steady-state probability. The CFTP detects, with probability one, the end of the transient phase (called burn-in period ) of the chain and consequently the beginning of its stationary phase. For large and/or stiff Markov chains, the burn-in period is expensive in time consumption. In this work, we propose a kind of dual form for CFTP called D-CFTP that, in many situations, reduces the Monte Carlo simulation time and does not need to store the history of the used random numbers from one iteration to another. A performance comparison of CFTP and D-CFTP will be discussed, and some numerical Monte Carlo simulations are carried out to show the smooth running of the proposed D-CFTP.

One method of computing the electrostatic energy of a biomolecule in a solution uses a continuum representation of the solution via the Poisson–Boltzmann equation. This can be solved in many ways, and we consider a Monte Carlo method of our design that combines the Walk-on-Spheres and Walk-on-Subdomains algorithms. In the course of examining the Monte Carlo implementation of this method, an issue was discovered in the Walk-on-Subdomains portion of the algorithm which caused the algorithm to sometimes take an abnormally long time to complete. As the problem occurs when a walker repeatedly oscillates between two subdomains, it is something that could cause a large increase in runtime for any method that used a similar algorithm. This issue is described in detail and a potential solution is examined.

The paper proposes a new second-order discretization method for forward-backward stochastic differential equations. The method is given by an algorithm with polynomials of Brownian motions where the local approximations using Malliavin calculus play a role. For the implementation, we introduce a new least squares Monte Carlo method for the scheme. A numerical example is illustrated to check the effectiveness.