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Cuchulainn
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Benchmark tests: KO European and American double barrier options

January 10th, 2021, 1:32 pm

// Call up-and-out-down-and-out
value_type K = 100.0;
value_type T = 0.25;
value_type r = 0.1;
value_type d = 0.0;
value_type sig = 0.25;

// Barrier parameters
value_type L = 50.0;
value_type H = 150.0;

// Rebates
value_type RebateL = 0.0;
value_type RebateH = 0.0;

S = 100
Euro = 6.1644XY
Amer = 6.2538WZ

What are  XY , WZ ?
And if time allows, what's delta and gamma at S = 100?
 
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bearish
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Re: Benchmark tests: KO European and American double barrier options

January 10th, 2021, 3:36 pm

XY=53
WZ=94

European delta and gamma: .5792 .025
American delta and gamma: .6033 .031
 
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bearish
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Re: Benchmark tests: KO European and American double barrier options

January 10th, 2021, 9:03 pm

Does the non-existence of an optimal exercise strategy for the American barrier option bother you? Should it?
 
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Cuchulainn
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Re: Benchmark tests: KO European and American double barrier options

January 11th, 2021, 1:37 pm

XY=53
WZ=94

European delta and gamma: .5792  .025
American delta and gamma: .6033  .031
European
6.164452, 0.57919, 0.025073

American
6.2543881, 0.603344, 0,03079

I am not using Crank Nicolson and etc. but Method of Lines  and then an adaptive ODE solver in time. For early exercise, I used both the Brennan Schwartz trick (it is first-order) and the barrier term in the  ODE. So I don't explicitly track the moving boundary. A number of "moving parts" as it were.
 For the greeks I use cubic spline Interpolation and not 'raw' divided differences.

BTW are you using Ikeda/Kunimoto?

Thanks!

edit: in short, the PDE is on a bounded domain with zero Dirichlet boundary conditions which in a sense is the easiest of all scenarios. i.e. solve directly without any assumptions/preprocessing.
Last edited by Cuchulainn on January 11th, 2021, 2:01 pm, edited 3 times in total.
 
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bearish
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Re: Benchmark tests: KO European and American double barrier options

January 11th, 2021, 1:53 pm

OK - I’ll confess to cheating. The lower barrier is so far away as to be practically irrelevant, so I treated them as single barrier problems. For the American option, I assumed exercise at the barrier, which is the same as plugging in a rebate of $50 to the standard barrier option formula. I’m pretty sure I can prove that it is suboptimal to exercise the option for any stock price strictly below the barrier, but that leads to a certain awkwardness when combined with the fact that the option is worthless once the barrier is reached.
 
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Alan
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Re: Benchmark tests: KO European and American double barrier options

January 11th, 2021, 3:36 pm

Although common, it's an abuse to use "American/European" for barriers, as the words usually have nothing to do with early exercise. Instead: just define the contract terms. For the double barrier under GBM, which is extinguished at the moment of the first hit, there is an exact sum (pretty sure), so easily checkable numerics to a zillion places. Simple BS-type formulas for the other case, too. 

So, better exercise for students: what are the results to 30 digits? Now there's a benchmark.  :D
 
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bearish
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Re: Benchmark tests: KO European and American double barrier options

January 11th, 2021, 4:01 pm

I agree that people abuse the terminology here, but in this case we seem to have on our hands two call options, both with a continuously monitored up-and-out barrier, with one being exercisable only at maturity and the other being exercisable at any time up to and including maturity as long as it has not been knocked out.
 
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Cuchulainn
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Re: Benchmark tests: KO European and American double barrier options

January 11th, 2021, 5:44 pm

So, better exercise for students: what are the results to 30 digits? Now there's a benchmark.  Image
No. Better to expose them to as many numerical methods as possible to many kinds of problems.
// 30 digits implies going down the multiprecision road. It is outside the scope.

edit: Alan, if you give 30 digits I can test it. You can probably do it in one line of MM. I don't have MM.
Last edited by Cuchulainn on January 11th, 2021, 6:49 pm, edited 1 time in total.
 
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Paul
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Re: Benchmark tests: KO European and American double barrier options

January 11th, 2021, 6:40 pm

No, best exercise for students is what is vol to nearest 5%? And how much are you willing to bet?
 
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Cuchulainn
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Re: Benchmark tests: KO European and American double barrier options

January 11th, 2021, 6:48 pm

No, best exercise for students is what is vol to nearest 5%? And how much are you willing to bet?
UVM case? Worst and best? 
If yes, then yess.
 
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Alan
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Re: Benchmark tests: KO European and American double barrier options

January 11th, 2021, 8:08 pm

So, better exercise for students: what are the results to 30 digits? Now there's a benchmark.  Image
No. Better to expose them to as many numerical methods as possible to many kinds of problems.
// 30 digits implies going down the multiprecision road. It is outside the scope.

edit: Alan, if you give 30 digits I can test it. You can probably do it in one line of MM. I don't have MM.
Sorry, but I have other stuff going on. The hard part would be nailing down the putative exact sum. 
 
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Cuchulainn
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Re: Benchmark tests: KO European and American double barrier options

January 11th, 2021, 8:15 pm

Actually, I would be expecting 100 digits accuracy. Something like
262537412640768743.999999999999250072597198185688879353856337336990862707537410378210647910118607313
 
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Alan
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Re: Benchmark tests: KO European and American double barrier options

January 11th, 2021, 8:24 pm

Or, in Paul's spirit, the BS value with no barrier, is 6.25, so the barriers don't much matter and let's all go home.   :D
 
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Cuchulainn
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Re: Benchmark tests: KO European and American double barrier options

January 11th, 2021, 8:33 pm

Or, in Paul's spirit, the BS value with no barrier, is 6.25, so the barriers don't much matter and let's all go home.   :D
That's a good answer to a question that was not asked. I asked to 6 digits. My final American price is 6,25389 Vote agree/don't agree and then I can go home  :roll:

Remember the old days? viewtopic.php?f=34&t=90957&hilit=heston

Some poetic license

www.youtube.com/watch?v=XMxSOiOWdvA
 
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bearish
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Re: Benchmark tests: KO European and American double barrier options

January 11th, 2021, 10:13 pm

Or, in Paul's spirit, the BS value with no barrier, is 6.25, so the barriers don't much matter and let's all go home.   :D
That's a good answer to a question that was not asked. I asked to 6 digits. My final American price is 6,25389 Vote agree/don't agree and then I can go home  :roll:

Remember the old days? viewtopic.php?f=34&t=90957&hilit=heston

Some poetic license

www.youtube.com/watch?v=XMxSOiOWdvA
Agree. The closed form solution for the up and out with a rebate of 50 comes to 6.25389 using a calculator found on a random website (coggit.com).