Patrick Weke

Construction of moment-matching multinomial lattices using Vandermonde matrices and Gröbner bases

In order to describe and analyze the quantitative behavior of stochastic processes, such as the process followed by a financial asset, various discretization methods are used. One such set of methods are lattice models where a time interval is divided into equal time steps and the rate of change for the process is restricted to a particular set of values in each time step. The well-known binomial- and trinomial models are the most commonly used in applications, although several kinds of higher order models have also been examined. Here we will examine various ways of designing higher order lattice schemes with different node placements in order to guarantee moment-matching with the process.In order to describe and analyze the quantitative behavior of stochastic processes, such as the process followed by a financial asset, various discretization methods are used. One such set of methods are lattice models where a time interval is divided into equal time steps and the rate of change for the process is restricted to a particular set of values in each time step. The well-known binomial- and trinomial models are the most commonly used in applications, although several kinds of higher order models have also been examined. Here we will examine various ways of designing higher order lattice schemes with different node placements in order to guarantee moment-matching with the process.

Asian Options, Jump-Diffusion Processes on a Lattice and Vandermonde Matrices

Asian options are options whose value depends on the average asset price during its lifetime. They are useful because they are less subject to price manipulations. We consider Asian option pricing on a lattice where the underlying asset follows the Merton–Bates jump-diffusion model. We describe the construction of the lattice using the moment matching technique which results in an equation system described by a Vandermonde matrix. Using some properties of Vandermonde matrices we calculate the jump probabilities of the resulting system. Some conditions on the possible jump sizes in the lattice are also given.

Pricing Asian options using moment matching on a multinomial lattice

Pricing Asian options is often done using bi- or trinomial lattice methods. Here some results for generalizing these methods to lattices with more nodes are presented. We consider Asian option pricing on a lattice where the underlying asset follows Merton–Bates jump-diffusion model and describe the construction of a lattice using the moment matching technique which results in an equation system described by a rectangular Vandermonde matrix. The system is solved using the explicit expression for the inverse of the Vandermonde matrix and some restrictions on the jump sizes of the lattice and the distribution of moments are identified. The consequences of these restrictions for the suitability of the multinomial lattice methods are also discussed.

Common Nearly Best Linear Estimates of Location and Scale Parameters: Normal and Logistic Distributions

Common nearly best linear estimates of location and scale parameters of normal and logistic distributions, which are based on complete samples, are considered. Here, the population from which the samples are drawn is either normal or logistic population or a fusion of both distributions and the estimates are computed when it is not yet known which of the two populations (between the normal and logistic) is true. The problem discussed in this paper involves two possible population types in a given sample. Samples of sizes n=5,6,8,10 and are used to validate these estimates and a comparison of their variances is made with those of the best linear unbiased estimators (BLUEs) for normal and logistic distributions. Keywords : Estimation; order statistics; unbiased and nearly unbiased estimators Discovery and Innovation Vol. 19 (2) 2007: pp. 133-139