Mar
18

## Binomial Tree Option Pricing with Discrete Dividends

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

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

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. Specifically, as pointed by Hull, when i = k+m, there are m(k+2) rather than k+m+1 nodes. The issue becomes more challenging when we increase the number of dividends. Fortunately, there is a simpler way to get around of this difficulty by dividing the stock price into two components: an uncertain part and a part that is the present value of all future dividends during the life of the option. Please check the book for detail Options, Futures, and Other Derivatives, 7th Economy Edition with CD.

(source from

Should you are interested into a sample implementation in Matlab of Binomial Tree Option Pricing with Discrete Dividends, take a look at the file http://www.ualberta.ca/dept/aict/bluejay/usr/local/matlab-6.5/toolbox/finance/finance/binprice.m.

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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. Specifically, as pointed by Hull, when i = k+m, there are m(k+2) rather than k+m+1 nodes. The issue becomes more challenging when we increase the number of dividends. Fortunately, there is a simpler way to get around of this difficulty by dividing the stock price into two components: an uncertain part and a part that is the present value of all future dividends during the life of the option. Please check the book for detail Options, Futures, and Other Derivatives, 7th Economy Edition with CD.

(source from

*Options, futures and other derivatives*)Should you are interested into a sample implementation in Matlab of Binomial Tree Option Pricing with Discrete Dividends, take a look at the file http://www.ualberta.ca/dept/aict/bluejay/usr/local/matlab-6.5/toolbox/finance/finance/binprice.m.

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