Pricing Vanilla Options with Cash Dividends

Abstract

The pricing of vanilla options on underliers with cash dividends is a surprisingly contentious and active research subject, for both European or American exercise style. Neither on the listed options side (calls and puts) nor on the flow/structured side of longer-term vanillas or light exotics are market participants in agreement on what model to use, nor on what an efficient practical implementation of the chosen model would be. The modeling problem boils down to the question of what a proper generalization of the Black-Scholes model to the case of cash dividends is, i.e. what should replace simple geometric Brownian motion (GBM).

We discuss this question with the aim of taking a first step towards a rationalization and normalization of the equity volatility market. We compare the two main classes of models in use, namely the "spot model" (piecewise GBM) and several "hybrid models" (shifted GBM). We are interested in consistency, simplicity, speed, and generality (covering all traded vanilla options, dividend and borrow rate assumptions, as well as easy modeling of business time, events, term-structure, credit, light exotics, etc). We also discuss the calibration problems that market participants face in some detail.

We show that: (i) all hybrid models are closely related on a mathematical level -- despite qualitatively different financial properties -- with simple and accurate relationships between calibrated parameters (borrow costs and volatilities) for both European and American options with cash dividends; (ii) all hybrid models allow accurate and very fast pricing of vanilla options using fine-tuned tree methods; (iii) some hybrid models have essentially all the desired properties outlined above; in particular, we describe a hybrid model closely related to the spot model, motivated by the spot-strike adjustment idea of Bos and Vandermark.

Keywords: Options pricing, American options, cash dividends, implied volatility, borrow cost, calibration, options market making

JEL Classification: G12, G13, C51, C60, C63