Discrete dividends and the FTSE-100 index options valuation

Abstract

This paper studies the effect of discrete dividends on the FTSE-100 index options valuation, following closely Harvey and Whaley's [J. Fut. Mkts, 1992, 12(2), 123–137] study on the S&P-100 index. To the best of our knowledge, no such study has ever been performed on FTSE-100 options, where the dividends have a discreteness pattern different from the S&P-100. Unlike the Harvey and Whaley study, both American and European options are considered, a more accurate benchmark is proposed, and a comprehensive comparison of the accuracy of a larger set of valuation methods is performed. It is shown that there are significant differences in accuracy and speed among different methods, and that, for both American and European options, a great deal of accuracy can be gained by using an approximation that takes into account the discrete nature of the FTSE-100 index option dividends.

Keywords:

• American options
• Computational finance
• Derivative pricing models
• Monte Carlo methods
• Numerical methods for option pricing
• Options pricing

JEL Classification:

• G1
• G13

Acknowledgements

Support from the Portuguese Foundation for Science and Technology (project PTDC/GES/78033/2006) is acknowledged. We thank Ana Carvalho for helpful comments. Participants at the Portuguese Finance Network 2010 Conference and FMA 2010 Annual Meeting provided useful comments. The report of an anonymous referee substantially improved the article.

Notes

†Under the geometric Brownian motion assumption, there is a known closed-form solution for European options (Black and Scholes 1973, Merton 1973), and for the perpetual American-style options there is also the closed-form solution proposed by Samuelson (1965).

‡These authors used the Black and Scholes (1973) European option valuation method to value American options.

†Note that we are interested in valuing options on a stock index, therefore the option valuation problem we are concerned with is for options on assets that pay discrete known cash dividends. The problem of valuing options on assets with known discrete proportional dividends, or with a continuous dividend yield, can be solved quite easily. Chance et al. (2002) propose an approach to deal with options on assets with stochastic discrete dividends.

‡Semi-numerical methods are methods that propose a formula as a solution to the option valuation problem, but that also require the use of some numerical methods to estimate the option value.

†The stock price is assumed to go down by the amount of the dividend d _i when it goes ex-dividend. Due to tax considerations, this may not always be the case. If the stock price is expected to go down by the amount of αd _i , the same applies with d _i replaced by α d _i .

‡For most financial applications, this is a problem with little effect since the probability of the asset price becoming lower than a discrete dividend is very small, especially because dividends are usually not very large.

§This will be the approach followed when implementing the method suggested by Wilmott et al. (1998) binomial trees with discrete payments of dividends. All the other methods are implemented with the liquidator approach.

†This model is usually implemented by adding the value of dividends at maturity to the exercise price of the option.

‡See also Geske (1981).

§See Perrakis and Lefoll (2000) and references therein, for the valuation of American call options with transaction costs and discrete known dividends.

†Haug et al.'s (2003) comparison is done with a small sample of options with parameters defined by themselves.

†Refer to Veiga and Wystup (2009) for further details.

‡Fleming (1998) and Harvey and Whaley (1991) use this methodology.

§Convergence of the algorithm has been analysed by Tsitsiklis and Van Roy (2001), Clément et al. (2002), Glasserman and Yu (2004), and Stentoft (2004). Several improvements to the LSMC are proposed by Areal et al. (2008).

†The number of dividends had to be limited to three, since it is not possible to compute the option value using a non-recombining tree for more dividends in a timely manner. From our results it is possible to infer that simulation accuracy is not affected by increasing the number of dividends.

‡The year is assumed to have 360 days.

§Valkanov et al. (2005) also use the LSMC method to value American FTSE-100 options with discrete dividends.

†Generally, the exercise prices are set in increments of 100 or 50 index points, but the exchange reserves the right to set a tighter increment of 25 index points.

‡The return consists of capital gains alone.

†Harvey and Whaley (1992) assume the same.

‡The company Markit provides forecasts on individual dividend amounts and ex-dates for up to 4 years.

†Since the instantaneous risk-free interest rate is used, it is necessary to convert the annual nominal Euro–Sterling currency interest rate before using it as an option valuation parameter, using the following expression:

where i is the instantaneous interest rate, i _n is the annual nominal interest rate, and c _f is the compounding frequency over the year.

‡All computations were performed with a computer with a 2 × Quad Core Intel Xeon~5450 3.0 GHz processor, 24 GB of RAM, and a Linux kernel. All programs were written in C and compiled with the GNU GCC compiler.

§Typically, FTSE-100 options have a minimum price fluctuation of 0.5 index points.

†Detailed results are available upon request.

†Detailed results are available upon request.