Re: [Quantlib-dev] 答复: Grids shift trick to improve the convergence speed PDE method

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This looks great. Some mundane comments:

- requireCPoint and shiftLocation are somewhat contracting
requirements, so we should forbid both being true by a QL_REQUIRE
- the loop in L92 could easily be replaced by a simple
lower_bound(...) - call to get (i+1), which does not help much in
terms of performance, but is nicer code
- cPoint is already ensured to be in [start, end], so i can not be =
locations_size(), the check is not required I think. cPoint may be =
start however, in which case i+1 = 0, in this case we should increase
i by 1, the remaining code then works as desired.

Also I think that this kind of adjustment - another example would
possibly be the smoothed payoff function described just in the section
just before the one you refer to (does that improve the vanilla case
btw ?) - should maybe be triggered from the Payoff, i.e. via
optionally implemented functions in classes derived from Payoff like

void adaptGrid( ... )        // given a fd grid, return a modification
(with same start, end, #points) adapted to the payoff
Real fdValue( ... , Real)  // return a payoff value smoothed w.r.t. a
given fd grid

Just a spontaneous idea, I don't know.

Peter

On 14 November 2014 14:17, cheng li <scr...@gm...> wrote:
> Hi Spanderen,
>
> The numerical result says there is no much difference between these 2
> methods for vanilla option:
>
> The parameters are same as my previous example
>
> Grids   Shift Grids + Damping   Damping
> 10      4.049015807        4.023573966
> 20      3.888731876     3.883259801
> 30      3.859347301     3.860949207
> 40      3.853164351     3.853494439
> 50      3.850117743     3.850127399
> 60      3.848403455     3.848320323
> 70      3.847346072     3.847233176
> 80      3.846453732     3.846532679
> 90      3.846028584     3.84605574
> 100     3.845716741     3.845717576
> 110     3.845481428     3.845467956
> 120     3.845299598     3.845277305
> 130     3.845111453     3.845129343
> 140     3.845005422     3.845012225
> 150     3.844918509     3.84491863
> 160     3.844846402     3.844841907
> 170     3.844785935     3.844778105
> 180     3.844717978     3.844724605
> 190     3.844676773     3.844679439
> 200     3.844641179     3.844641171
> 210     3.844307792     3.844307792
>
> Regards,
> Cheng
>
> -----邮件原件-----
> 发件人: Cheng Li [mailto:scr...@gm...]
> 发送时间: 2014年11月14日 20:09
> 收件人: 'Klaus Spanderen'; qua...@li...
> 主题: 答复: [Quantlib-dev] Grids shift trick to improve the convergence
> speed PDE method
>
> Hi Spanderen,
>
> Let me do one for us. I'll come back to you when it is finished. IMO I think
> the convergence speed won't improve as much as the case for digital since
> there is no discontinuity in plain vanilla payoff.
>
> BTW, I agree with you that this will change the default behavior if using my
> implementation. That is the reason I sent this all the developers to see
> other's opinion before raise a pull request in github.
>
> Regards,
> Cheng
>
> -----邮件原件-----
> 发件人: Klaus Spanderen [mailto:kl...@sp...]
> 发送时间: 2014年11月14日 14:42
> 收件人: qua...@li...
> 主题: Re: [Quantlib-dev] Grids shift trick to improve the convergence speed
> PDE method
>
> Hi
>
> cool!
>
> The results are looking promising. Do you get a similar improvement if you
> apply the shift trick to plain vanilla option.
>
> The only thing we might consider is that we change behavior if the shift
> becomes the default.
>
> regards
> Klaus
>
> On Wednesday, November 12, 2014 12:01:52 PM cheng li wrote:
>> Hi Team,
>>
>>
>>
>> Current I am in an effort to improve the convergence speed of
>> FdBlackScholesVanillaEngine to digital option. The background is as
> follows:
>>
>>
>>
>> For digital option there is some discontinuity in the initial condition.
>> Such discontinuity will cause the straight Crank-Nicolson method to
>> show severe odd-even effect. In current implementation, the pricing
>> engine will use several damping steps to suppress some effect. However
>> as claimed in Interest Rate Modeling. Section 2.5 p60 in V1 by
>> Anderson and Piterbarg, Damping steps is not adequate for such task.
>> Referencing to Tavella and Randall's book, they suggest to shift the
>> grids to let the discontinuity lies exactly half way between 2 grids.
>> They claimed that this will greatly improve the convergence.
>>
>>
>>
>> I have completed such experimental implementation in QuantLib. Besides
>> I do a comparison between "Grids Shift + Damping" and "Damping"
>> methods. The result is promising. The convergence speed is greatly
>> improved and is consistent with what Anderson and Piterbarg claimed.
>> The result can be referred from the attached Shift_experiment.xlsx file.
>>
>>
>>
>> To replicate my result, you can replacing the original files in
>> ql\methods\finitedifferences\meshers with the files I attached
>> (concentrating1dmesher.hpp, concentrating1dmesher.cpp). And run the
>> test program testPDEScheme.cpp
>>
>>
>>
>> The results is based on "Grids shift + 2 damping steps". Also you can
>> run the test program without replacing the mesher file to see what is
>> result of
>> "2 damping steps". Besides I also run through the QuantLIb test suite.
>> This change won't break any existing test case.
>>
>>
>>
>> Is anyone have some interest to have a look at this? I'd like to
>> create a pull request for this change if no one is against it.
>>
>>
>>
>> Regards,
>>
>> Cheng
>
>
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