Hongjoong Kim

Adaptive lattice methods for multi-asset models

Adaptive lattice methods are developed to compute the price of multivariate contingent claims. A simple coordinate representation is used to extend one dimensional lattice methods to multivariate asset models. Two algorithms are proposed, one performing several levels of refinement for a time interval [T-@Dt,T] and the other performing one level of refinement for @l% of a given time domain [0,T], where T is the time to maturity, @Dt is the time step size and @l>0 is a constant. Numerical experiments are carried out for the European and American barrier-type options with one, two, or three underlying assets. In our numerical experiments, both adaptive algorithms improve efficiency over lattice methods with a uniform time step for the same level of accuracy.

A multi-dimensional local average lattice method for multi-asset models

We develop a multi-dimensional local average lattice method in order to compute efficiently and accurately the price of multivariate contingent claims. The proposed method improves the accuracy of the standard lattice method by considering the local averages of option prices around each node at the final time, rather than the prices at the nodes. The average value smooths the oscillatory behavior of the lattice method, which leads to fast convergence of the option values. Numerical computations show that the proposed local average lattice method is more efficient than other lattice methods for a given level of accuracy.

An adaptive averaging binomial method for option valuation

Abstract We introduce efficient accurate binomial methods for option pricing. The standard binomial approximation converges to continuous Black–Scholes values with the saw-tooth pattern in the error as the number of time steps increases. When we introduce local averages of payoffs at expiry, the saw-tooth pattern in the error has been reduced and the approximation becomes reliable. Furthermore, we employ adaptive meshes around non-smooth regions for efficiency. Numerical experiments illustrate that the proposed method gives more accurate values with less computational work compared to other methods.

AN IMPROVED BINOMIAL METHOD FOR PRICING ASIAN OPTIONS

We present an improved binomial method for pricing European- and American-type Asian options based on the arithmetic average of the prices of the underlying asset. At each node of the tree we propose a simple algorithm to choose the representative averages among all the effective averages. Then the backward valuation process and the interpolation are performed to compute the price of the option. The simulation results for European and American Asian options show that the proposed method gives much more accurate price than other recent lattice methods with less computational effort.

Stability of Symmetric Solitary Wave Solutions of a Forced Korteweg-de Vries Equation and the Polynomial Chaos

In this paper, we consider the numerical stability of gravity-capillary waves generated by a localized pressure in water of finite depth based on the forced Korteweg-de Vries (FKdV) framework and the polynomial chaos. The stability studies are focused on the symmetric solitary wave for the subcritical flow with the Bond number greater than one third. When its steady symmetric solitarywave-like solutions are randomly perturbed, the evolutions of some waves show stability in time regardless of the randomness while other waves produce unstable fluctuations. By representing the perturbation with a random variable, the governing FKdV equation is interpreted as a stochastic equation. The polynomial chaos expansion of the random solution has been used for the study of stability in two ways. First it allows us to identify the stable solution of the stochastic governing equation. Secondly it is used to construct upper and lower bounding surfaces for unstable solutions, which encompass the fluctuations of waves. AMS subject classifications: 65C20, 65C30