Dividends

We extend the previously derived theory to include the case when the underlying assets are paying dividends. After a short discussion of discrete dividends we mainly study the case of continuous dividends. The theory is derived by reducing the dividend-paying model to an equivalent standard model with no dividends. For the case of a constant dividend yield we derive explicit option pricing formulas.

Pricing American Options under High-Dimensional Models with Recursive Adaptive Sparse Expectations*

We introduce a novel numerical framework for pricing American options in high dimensions. Our scheme manages to alleviate the problem of dimension scaling through the use of adaptive sparse grids. We approximate the value function with a low number of points and recursively apply fast approximations of the expectation operator from an exercise period to the previous period. Given that available option databases gather several thousands of prices, there is a clear need for fast approaches in empirical work. Our method processes an entire cross section of options in a single execution and offers an immediate solution to the estimation of hedging coefficients through finite differences. It thereby brings valuable advantages over Monte Carlo simulations, which are usually considered to be the tool of choice in high dimensions, and satisfies the need for fast computation in empirical work with current databases containing thousands of prices. We benchmark our algorithm under the canonical model of Black and Scholes and the stochastic volatility model of Heston, the latter in the presence of discrete dividends. We illustrate the massive improvement of complexity scaling over dense grids with a basket option study including up to eight underlying assets. We show how the high degree of parallelism of our scheme makes it suitable for deployment on massively parallel computing units to scale to higher dimensions or further speed up the solution process.

Analytical Approximations of American Call Options with Discrete Dividends

In this study, we assume that stock prices follow piecewise geometric Brownian motion, a variant of geometric Brownian motion except the ex-dividend date, and find pricing formulas of American call options. While piecewise geometric Brownian motion can effectively incorporate discrete dividends into stock prices without losing consistency, the process results in the lack of closed-form solutions for option prices. We aim to resolve this by providing analytical approximation formulas for American call option prices under this process. Our work differs from other studies using the same assumption in at least three respects. First, we investigate the analytical approximations of American call options and examine European call options as a special case, while most analytical approximations in the literature cover only European options. Second, we provide both the upper and the lower bounds of option prices. Third, our solutions are equal to the exact price when the size of the dividend is proportional to the stock price, while binomial tree results never match the exact option price in any circumstance. The numerical analysis therefore demonstrates the efficiency of our method. Especially, the lower bound formula is accurate, and it can be further improved by considering second order approximations although it requires more computing time.

Pricing barrier options with discrete dividends

The presence of discrete dividends complicates the derivation and form of pricing formulas even for vanilla options. Existing analytic, numerical, and theoretical approximations provide results of varying quality and performance. Here, we compare the analytic approach, developed and effective for European puts and calls, of Buryak and Guo with the formulas, designed in the context of barrier option pricing, of Dai and Chiu.

HIGH ORDER SPLITTING METHODS FOR FORWARD PDEs AND PIDEs

This paper is dedicated to the construction of high order (in both space and time) finite-difference schemes for both forward and backward PDEs and PIDEs, such that option prices obtained by solving both the forward and backward equations are consistent. This approach is partly inspired by Andreassen & Huge (2011) who reported a pair of consistent finite-difference schemes of first-order approximation in time for an uncorrelated local stochastic volatility (LSV) model. We extend their approach by constructing schemes that are second-order in both space and time and that apply to models with jumps and discrete dividends. Taking correlation into account in our approach is also not an issue.