The least squares method for option pricing revisited

It is shown that the the popular least squares method of option pricing converges even under very general assumptions. This substantially increases the freedom of creating different implementations of the method, with varying levels of computational complexity and flexible approach to regression. It is also argued that in many practical applications even modest non-linear extensions of standard regression may produce satisfactory results. This claim is illustrated with examples

Pricing of American call options using regression and numerical integration

Consider the American basket call option in the case where there are N underlying assets, the number of possible exercise times prior to maturity is finite, and the vector of N asset prices is modeled using a Levy process. A numerical method based on regression and numerical integration is proposed to estimate the price of the American option. In the proposed method, we first express the asset prices as nonlinear functions of N uncorrelated standard normal random variables. For a given set of time-t asset prices, we next determine the time-t continuation value by performing a numerical integration along the radial direction in the N-dimensional polar coordinate system for the N uncorrelated standard normal random variables, expressing the integrated value via a regression procedure as a function of the polar angles, and performing a numerical integration over the polar angles. The larger value of the continuation value and the time-t immediate exercise value will then be the option value. The time-t option values over the N-dimensional space may be represented by a quadratic function of the radial distance, with the coefficients of the quadratic function given by second degree polynomials in N-1 polar angles. Partitioning the maturity time T into k* intervals of length Δt, we obtain the time-(k-1)Δt option value from the time-kΔt option values for k= k*, k*-1,…, 1. The time-0 option value is then the price of the American option. It is found that the numerical results for the American option prices based on regression and numerical integration agree well with the simulation results, and exhibit a variation of the prices as we vary the non-normality of the underlying distributions of the assets. To assess the accuracy of the computed price we may use estimated standard error of the computed American option price. The standard error will help us gauge whether the number of selected points along the radial direction and the number of selected polar angles are large enough to achieve the required level of accuracy for the computed American option price

Semi-tractability of optimal stopping problems via a weighted stochastic mesh algorithm

In this article we propose a Weighted Stochastic Mesh (WSM) algorithm for approximating the value of discrete and continuous time optimal stopping problems. It is shown that in the discrete time case the WSM algorithm leads to semi-tractability of the corresponding optimal stopping problem in the sense that its complexity is bounded in order by $\varepsilon^{-4}\log^{d+2}(1/\varepsilon)$ with $d$ being the dimension of the underlying Markov chain. Furthermore we study the WSM approach in the context of continuous time optimal stopping problems and derive the corresponding complexity bounds. Although we can not prove semi-tractability in this case, our bounds turn out to be the tightest ones among the complexity bounds known in the literature. We illustrate our theoretical findings by a numerical example