Re: [Quantlib-users] The reason of differences between `FDDividendAmericanEngine` of QuantLib 1.12 and `FdBlackScholesVanillaEngine` of QuantLib 1.18

[Klaus S. <kl...@sp...>]

Hi Shenze,

I think, we have a naming clash here. The implementation of the escrowed dividend models follows chapter 9.1.1 "The Escrowed Dividend Model" in E. Haug's "Guide to Option Pricing Formulas", quote "...can be priced by the BSM formula, by simple replacing S with S minus the present value of the dividends". As we've said before and also stated in Haug's book, in this form the model is not really appicable for American/Bermudan options.

The escrowed model was added lately to FdBlackScholesVanillaEngine mainly, to have a dividend  model, which is compatible with the AnalyticDividendEuropeanEngine (for the spot adjustment, see line 50ff in analyticdividendeuropeanengine.cpp).

Sure, one can extend the original model definition from Haug's book by 
> The process should follow *S_t = 10 * x_t + GBM(90)*, in which 10 is the present value of dividends, and x_t is 1 before the dividend and 0 after the dividend.
and this will improve American/Bermudan pricing.

Looking at your explanations below, you should be fine with the default dividend model. Just to be precise, for your example the default model will start a GBM at 100 and adds a deterministic down jump at every dividend date. The default model also covers the case S < D.


hope that helps, regards
Klaus

On Freitag, 26. Juni 2020 22:48:17 CEST Shenze Wang wrote:


Hi, 


A correction for the last email.


About the Escrowed dividend model, I think:
The process should follow *S_t = 10 * x_t + GBM(90)*, in which 10 is the present value of dividends, and x_t is 1 before the dividend and 0 after the dividend.






Best,
Shenze
On Jun 26, 2020, 4:43 PM -0400, Shenze Wang <she...@da...>, wrote:


Hi Klaus,  


I think I have a different understanding about Escrowed dividend model from you. 


Let GBM(S_0) represents a Geometric Brownian motion with initial price S_0. In the previous example, with Escrowed dividend model, 
     *  You think, the process of the stock *S = GBM(90)*;


     *  However, I think, the process should follow *S = 10 + GBM(90)*, in which 10 is the present value of dividends;
The paper /Back to Basics/ proposes that *S = GBM(100)*, but with proper adjustment or assumption when* *S < D at dividend time.




If the Spot dividend model is following that paper, then the difference between the Spot and Escrowed should be that Spot handles the situation when S<D, while Escrowed does not. I do not think throwing away the dividend totally from the process of the price is the right way. If the Escrowed dividend is programmed by my understanding, I believe it also can provide fairly accurate results as long as the dividend is not huge compared to the stock price.


Some other things:
     1. The option mentioned in my previous email will be always exercised on 30.12.2020, right before the dividend, because it’s an American call.
     2. I think the old FDDividendAmericanEngine does not provide the same results as the Escrowed model in the new engine. Codes and results are in attached files.




Best,
Shenze