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Billy7
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Joined: March 30th, 2016, 2:12 pm

Numerical challenge: Discrete barriers (Black-Scholes / Heston)

January 17th, 2017, 9:30 pm

As was done in another thread here for American vanillas, I thought of trying to get 6-7 decimals of accuracy for a discretely monitored double knock-out barrier. This is a difficult problem to solve with general tree or PDE methods because the periodically imposed discontinuities tend to cause spurious oscillations, and even if your scheme safeguards against those, the accuracy and order of convergence will generally suffer. An efficient Monte Carlo implementation with variance reduction may get you 4 (if that) confirmed decimals. With the recent emergence of characteristic function-based methods though, this previously difficult problem is now apparently solvable to very high accuracy and almost instantaneously. For example see the work by Fang & Oosterlee here and here.

So here's a sample problem: 
Euro put option that knocks out if either down barrier of 80, or up barrier of 120 is reached, S = 90, K = 100, T=0.5, r = 5.07%, div yield = 4.69%, with 125 equi-spaced monitoring dates until and including maturity (daily monitoring).

Heston: spot variance = 0.0625, long-run variance = 0.06, reversion rate = 2.5, vol of vol = 0.5, correlation = -0.1
Value: 2.725XXXX

B-S: vol = 25%
Value = 2.209XXXXXX

I'll tell you what I've got but I'd be curious to see what other people get with any method. Also wondering if someone here has implemented the Fang & Oosterlee method. If so then please impress us with the results! I'm wondering if I should throw my FDM solver out of the window...
 
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Collector
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Re: Numerical challenge: Discrete barriers (Black-Scholes / Heston)

January 18th, 2017, 5:02 pm

Possiby useful http://www.columbia.edu/~sk75/sinica.pdf   can it be extended to also hold for Americano Barriers? 
 
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Billy7
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Re: Numerical challenge: Discrete barriers (Black-Scholes / Heston)

January 18th, 2017, 9:14 pm

Possiby useful http://www.columbia.edu/~sk75/sinica.pdf   can it be extended to also hold for Americano Barriers? 
Hi Collector:)
AFAIK, barrier shifting a la BGK can often give a pretty good approximation but the error becomes significant when the underlying is close to the barrier. In the sample problem I posted though it isn't close and it's daily monitored, so it may get the first couple of decimals right, maybe.