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,

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

On Jun 26, 2020, 3:59 PM -0400, Klaus Spanderen <kl...@sp...>, wrote:
> Hi Shenze,
>
> thanks for the example, it nicely highlights the importance of dividend modelling for equity option.
>
> Within the spot dividend model the dividend of 10 is substracted from the equity process on the 31.12.2020. The option will be early exercised somewhere between today and the 30.12.2020 to earn the intrinsic value of 10, hence this model gives the expected result.
>
> The escrowed dividend model substracts the net dividend payment from the start value of 100 and starts a geometric brownian motion at S(t=0) = S - D = 90. As the volatility is almost zero this process ends at 90 on maturity and the american option expires worthless. Both engine, the old FDDividendAmericanEngine and the FdBlackScholesVanillaEngine produce this result. The numerics is ok but the model is not appicable for this problem and delivers results, which look "a little wired" to say the least;-).
>
> best
> Klaus
>
> On Freitag, 26. Juni 2020 20:50:54 CEST Shenze Wang wrote:
> > Hi Klaus,
> >
> > I understand that different dividend models have different assumptions about the underlying process of the stock price. So, the pricing results will be different.
> >
> > But could you please help me check the pricing results in the following screenshot? It is an American call option with extremely low volatility and a dividend right before the expiry date. S = 100 and K =90. I think the value of this option should be very close to 10, which matches the result of the Spot dividend model. However, the pricing result from Escrowed dividend model is 0, which looks a little wired.
> >
> > Thanks a lot for your patience.
> >
> >
> >
> > Best,
> > Shenze
> >
> >
> > On Jun 26, 2020, 1:32 PM -0400, Klaus Spanderen <kl...@sp...>, wrote:
> > > Hi
> > > Spot dividend and escrowed dividend model are different models and are supposed to give different results for the same parameter set. An implied volatility calculated with the escrowed dividend model will not give the same price back when used in a spot dividend model because the underlying stochastic differential equations are different.
> > > best regards
> > > Klaus
> > > On Freitag, 26. Juni 2020 17:34:57 CEST Shenze Wang wrote:
> > > Hi Klaus,
> > >
> > > Thanks a lot for you explanations.
> > >
> > > I made more calculations. It seems that the differences between Spot dividend model and Escrowed dividend model are not only for pathologic cases. And also a greater number of time steps and grid points do not help. The calculation results are attached. So I guess that there may be something goes wrong with Spot dividend model.
> > >
> > > The difference between the two dividend model also causes another bug in impliedVolatility. Because for American options, FdBlackScholesVanillaEngine with Spot dividend model is always being used, when calculating ivol. Here is the issue I opened, https://github.com/lballabio/QuantLib/issues/850.
> > >
> > > I think maybe it is necessary to take a closer look at the two dividend models.
> > >
> > >
> > > Best,
> > > Shenze
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