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Mar
18
How to value a stock option with discrete dividend was briefly introduced at http://www.mathfinance.cn/valuation-of-stock-option-with-discrete-dividend/, where the main goal is to compare the performance of different methods, namely, Escrowed dividend model, Chriss volatility adjustment model, Haug & Haug volatility adjustment model, Bos volatility adjustment model, and Haug, Haug and Lewis method. I didn't include lattice method for comparison because non-recombining binomial tree is computer intensive, especially when the number of dividends is large.
In the book Options, futures and other derivatives by John Hull, how to deal with discrete dividend with a binomial tree is explained in detail, see page 402, fifth version, where future discrete dividend is divided into two types:
1, known dividend yield. For instance, there will be a 3% dividend 3 months later (3% of the stock price), it is straightforward to handle it as the binomial tree is recombined when the nodes are multiplied by a percentage, so basically what we need to do is to construct a tree like usual before ex-dividend date, and then shift all the left tree nodes down by (1-dividend yield), that's it, the number of nodes are the same as for non-dividend binomial tree;
(source from Options, futures and other derivatives)
2, known dollar dividend. For instance, there will be a 2.5 dollar dividend 3 months later, so before ex-dividend date the binomial tree is constructed as usual but exactly at the date after ex-dividend, the whole nodes are shifted down by 2.5 dollar, and then a new binomial tree is constructed, because the nodes are shifted by an absolute amount number, the new binomial tree is not recombined any more, which means much more nodes than the non-dividend case.
In the book Options, futures and other derivatives by John Hull, how to deal with discrete dividend with a binomial tree is explained in detail, see page 402, fifth version, where future discrete dividend is divided into two types:
1, known dividend yield. For instance, there will be a 3% dividend 3 months later (3% of the stock price), it is straightforward to handle it as the binomial tree is recombined when the nodes are multiplied by a percentage, so basically what we need to do is to construct a tree like usual before ex-dividend date, and then shift all the left tree nodes down by (1-dividend yield), that's it, the number of nodes are the same as for non-dividend binomial tree;
(source from Options, futures and other derivatives)
2, known dollar dividend. For instance, there will be a 2.5 dollar dividend 3 months later, so before ex-dividend date the binomial tree is constructed as usual but exactly at the date after ex-dividend, the whole nodes are shifted down by 2.5 dollar, and then a new binomial tree is constructed, because the nodes are shifted by an absolute amount number, the new binomial tree is not recombined any more, which means much more nodes than the non-dividend case.
Feb
3
When asked how to value a stock option without dividend or with continuous dividend, many people would refer to Black Scholes formula, but how to price an option with discrete dividend then? certainly Black Scholes model can't be used directly since one of its assumptions is continuous payout. Paper Back to Basics: a new approach to the discrete dividend problem by Haug, Haug and Lewis summarizes the following ways:
1, Escrowed dividend model, which is the simplist and the least accurate way as a result. The basic idea of Escrowed dividend model is to adjust the current stock price by deducting the present value of future dividends, and plug in the replaced stock price to Black Scholes formula;
2, Chriss volatility adjustment model, besides replacing current stock price, this model adjusts volatility as well because the Escrowed dividend model alone decreases the absolute price standard deviation, hence underestimates an option's value. However, Chriss model yields too high volatility if the dividend is paid out early in the option’s lifetime, which generally overprices call options;
3, Haug & Haug volatility adjustment model; which is more sophisticated than Chriss model and takes into account the timing of the dividend, unfortunately, the authors show this method performs particularly poorly for multiple dividends stock option;
4, Bos volatility adjustment model, a even more sophiscated model than Haug & Haug, but still, it performs poorly for large dividends or long term options;
5, Lattice method, for example, non-recombining binomial tree introduced in the bible book Options, Futures, and Other Derivatives with Derivagem CD (7th Edition)
, we all know it is time-consuming;
6, Haug, Haug and Lewis method introduced in the above-mentioned paper, the basic idea is to calculate first the ex-dividend option price by Black Scholes model, then discount back the ex-dividend value under equivalent martingale measure. The authors demonstrate the high accuracy of their model with several examples afterwards.
Below is sample Matlab codes I wrote for comparision, a single dividend is used for simplicity, results similar to the table listed in the paper
1, Escrowed dividend model, which is the simplist and the least accurate way as a result. The basic idea of Escrowed dividend model is to adjust the current stock price by deducting the present value of future dividends, and plug in the replaced stock price to Black Scholes formula;
2, Chriss volatility adjustment model, besides replacing current stock price, this model adjusts volatility as well because the Escrowed dividend model alone decreases the absolute price standard deviation, hence underestimates an option's value. However, Chriss model yields too high volatility if the dividend is paid out early in the option’s lifetime, which generally overprices call options;
3, Haug & Haug volatility adjustment model; which is more sophisticated than Chriss model and takes into account the timing of the dividend, unfortunately, the authors show this method performs particularly poorly for multiple dividends stock option;
4, Bos volatility adjustment model, a even more sophiscated model than Haug & Haug, but still, it performs poorly for large dividends or long term options;
5, Lattice method, for example, non-recombining binomial tree introduced in the bible book Options, Futures, and Other Derivatives with Derivagem CD (7th Edition)
, we all know it is time-consuming;6, Haug, Haug and Lewis method introduced in the above-mentioned paper, the basic idea is to calculate first the ex-dividend option price by Black Scholes model, then discount back the ex-dividend value under equivalent martingale measure. The authors demonstrate the high accuracy of their model with several examples afterwards.
Below is sample Matlab codes I wrote for comparision, a single dividend is used for simplicity, results similar to the table listed in the paper
