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eredhuin
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Joined: July 14th, 2002, 3:00 am

Binomial & dividends

August 7th, 2002, 4:10 pm

QuoteOriginally posted by: alkurInteresting approach, but doesn't O(n^(2+j)) seem scary ?Can we apply it for j>2 on PC really ?It's worse than that, actually. I had to make certain assumptions about when dividends occur to come up with O(n^{2+j}). As j approaches n, the complexity approaches O(2^n), the non-recombinant case complexity. So the answer is "not in general". I did a quick experiment using n=500 and a maturity of 1.1ynumber of divs, elapsed (s)0, 01, 12, 73, 3364, much longerMoral: It's way faster to use (1) or (2) when j is bigger than 2.
 
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qquant
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Joined: March 4th, 2002, 1:31 pm

Binomial & dividends

August 8th, 2002, 11:43 pm

Beware. This procedure "redefines" volatility in the tree, when you hit a dividend. No big deal for small dividends, but for big dividends, you can have a problem in interpreting the value of the option.
 
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qquant
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Joined: March 4th, 2002, 1:31 pm

Binomial & dividends

August 8th, 2002, 11:50 pm

Sorry, I should say my comment about volatility redefinition refers to model (1).
 
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alkur
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Joined: January 21st, 2002, 1:33 pm

Binomial & dividends

August 9th, 2002, 11:35 am

QuoteOriginally posted by: qquantSorry, I should say my comment about volatility redefinition refers to model (1).quant,sorry, haven't quite got. Model (1) assumes that clean spot undergoes some stochastic process, while the dividends are just deterministic addition. So why one should redefine volatility in frames of this model ?Alexei.
 
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qquant
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Joined: March 4th, 2002, 1:31 pm

Binomial & dividends

August 12th, 2002, 6:35 pm

Hull discusses this very briefly, at least in the 3rd addition of his textbook. Using the method in sec 15.3 of that book, he says that we assume that the clean process S* has a constant volatility, but it will generally be different than the vol of the process S (full stock price process including dividends). I read an article (by Eric Berger and David Klein in July 1998 Bloomberg Magazine, pp 116-120 describing how this can create errors in valuation/intrepretation. They pick an example with a single dividend of $10 on a $100 stock - and of course it's a big effect. In practice, this is perhaps only something to be aware of in unusual cases of large dividends.
 
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alkur
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Joined: January 21st, 2002, 1:33 pm

Binomial & dividends

August 13th, 2002, 2:04 pm

qquant,yes, sure I agree that this model is not strictly correct (some will say altogether incorrect) and can give crazy results for large dividends.But this seems to be rather rare case anyway ...
 
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svunt

Binomial & dividends

September 3rd, 2002, 8:22 am

Hull also presents an extension of the Roll, Geske, Whaley formula for a call option on a stock that pays a single cash dividend during the life of the option. The RGW formula goes like this:C = [ – D1exp(-rt1)] * N(b1) + [S – D1exp(-rt)]*M[a1, -b1; - (t1/T)^1/2] – (K – D)exp(-rt1)N(b2)D1 is the single cash dividend and t1 the time of the payment. N(x) is the cumulative normal distribution, and M(a,b;p) the cumulative bivariate normal distribution.If there are several dividends, one can deduct the PV of the cash dividends excluding the last cash dividend payment and replace D1 with the last dividend payment and t1 becomes the time of the last payment. I have tried it and compared it to the binomial tree approach presented in Hull, p. 399 - 400 (I think reza mentioned this approach already). The values are quite different for in-the-money calls. Since I do not have a true value to compare these two methods, I wonder which one is generally more accurate. Is the extended RGW really an alternative to binomial trees when pricing calls on stocks that pay more than one dividend during the life of the option?Svunt