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

[Shenze W. <she...@da...>]

Hi Klaus & Luigi,

Thanks a lot for your explanations and referring to the papers. They are very helpful and I learnt a lot.

After studying the paper Back to basic, I believe the dividend model in that paper (Spot model in QuantLib) is good model with the correct assumption about the process of the underlying, when there are dividends. We’ll probably go with this one.

I also think it’s a good idea disable the Escrowed model for path dependent options.

Here is one observations about Escrowed model.
Even for European options, FdBalckScholesVanilaEngine with Escrowed model and AnalyticDividendEuropeanEngine will give wrong or poor results when the dividend is huge, as shown in the following screen shot. This call option has a big dividend right before expiry and a strike price of 0. You may argue that this is a wired option. But actually you can consider this option as the equity part in the Merton’s model of credit risk. The reason is that in Escrowed model, the dividend part of the underlying price has 0 volatility.
Here is one thought about the impliedVolatility.
I agree that FdBalckScholesVanilaEngine is a good default engine when calculating ivol for American options. But it would be really nice if I can specify the parameters of this engine when using impliedVolatility(). Because for options with strike far from spot, the results of FdBalckScholesVanilaEngine with default grid_points are not accurate enough. I can use a larger grid_points when calculating the price, but not the impliedVolatility for now. A $30 strike put on a $300 underlying stock is not rare in the market.

Here is one more questions.
When the stock price is lower than the dividend on the dividend date, what is current behavior of the FdBalckScholesVanilaEngine. Will it pay a dividend equal to the current stock price, or will it pay nothing at all?

Thanks again.


Best,
Shenze

On Jun 26, 2020, 8:30 PM -0400, Klaus Spanderen <kl...@sp...>, wrote:
> 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