recently I have been trying to understand how to price FX options with single and double window barriers. Could someone please recommend a source (e.g., book, article, etc.), where I can find the pricing formulas. Thank you in advance!
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1 Answer
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A window barrier option is one where the barrier is monitored only during an interval starting after "today" and ending before the option matures. Closed form pricing formulas will
(typically) only exist for the Black-Scholes model with constant volatility and further simplistic assumptions such as deterministic interest rates;
(if they ever got published) be horrendously complex -even under the above simplistic assumptions.
For example, instead of the CDF of the one-dimensional normal distribution we are familiar with in full time barrier options, the true window barrier option will require the three dimensional normal CDF.
It is far better to apply a more realistic model and solve numerically for the price for example by Monte Carlo or a PDE method. See for example this paper.
In addition: a window barrier option will (as time passes) sooner or later become knocked out, or turn into an early ending partial time barrier option.
Black-Scholes formulas for those were published by Heynen and Kat and can be found in the book by
E.G. Haug, The Complete Guide to Option Pricing Formulas.
The paper by T. Guillaume that you found is quite good but seems to have a few typos:
$\widetilde{\mu}$ and $\overline{\mu}$ should be flipped. In formula (1) the $\mu$ with the $+$ sign in front of $\sigma^2$ definitely belongs to the $S$ term which we know from the vanilla Black Scholes formula.
The second exponential term in formula (2) should be $$ e^{\frac{2\mu}{\sigma^2}(h_1-nh)} $$ without a 2 infront of $nh\,.$
I wrote some Python code and matched the results in his table quite well.
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$\begingroup$ Thank you for the reply @Kurt G. I will look deep into the two sources you referred to. Thank you again! $\endgroup$ Commented Nov 27, 2021 at 20:24
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$\begingroup$ I came across this paper. Have you seen it? In general, I have a number of questions, but shortly – I am trying to compute the analytical values of SWDKOP from Table 1 on page 27 (of course using the Proposition 1 on pages 5-6). I would appreciate any help. $\endgroup$ Commented Dec 11, 2021 at 19:45
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1$\begingroup$ Thanks for that paper. I was not aware of it. Best to compare Guillaume's magnificent formulas to, say, a monte carlo simulation. Not because I have any doubts but I know from experience that many typos can occur. $\endgroup$– Kurt G.Commented Dec 13, 2021 at 9:05
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$\begingroup$ I have tried a few different ways to implement the formula, but unfortunately I am not able to come to the results provided by the author. I get nonsense, i.e. negative values of the option price. Either I do not understand the formula correctly or there is a typo in the paper. I am stuck, but I will keep digging. Anyway, thank you for the time! :) $\endgroup$ Commented Dec 13, 2021 at 22:02
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