Derivatives Models on Models

3

Back to Basics: A New Approach to the Discrete Dividend Problem^*

Together with Jørgen Haug^† and Alan Lewis^‡

1 Introduction

Stocks frequently pay dividends, which has implications for the value of options on these stocks. For options on a large portfolio of stocks, one can approximate discrete dividend payouts with a dividend yield and use the generalized Black-Scholes-Merton (BSM) model. For options on one stock, this is not a viable approximation, and the discreteness of the dividend has to be modeled explicitly.¹ We discuss how to properly make the necessary adjustments.

It might come as a surprise to many readers that we write an entire chapter about a supposedly mundane issue – which is treated thoroughly in any decent derivatives text books (including, but not limited to Cox and Rubinstein, 1985; Chriss, 1997; Haug, 1997; Hull, 2000; McDonald, 2003; Stoll and Whaley, 1993; Wilmott, 2000). It turns out, however, that some of the adjustments suggested in the extant literature admit arbitrage – which is fine if all your competitors use these models, but you know how to do the arbitrage-free adjustment.

1.1 Existing Methods

Escrowed Dividend Model: The simplest escrowed dividend approach makes a simple adjustment to the BSM formula. The adjustment consists of replacing the stock price S₀ by the stock price minus the present value of the dividend S₀ − e^−rt_D D, where D is the size of the cash dividend to be paid at time t_D. Because the stock price is lowered, the approach will ...