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Pricing FX Window Barrier Options and Their Numerical Alternatives

Article Quant Q&A · Author: Candidate
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Summary

The document discusses pricing single- and double-window barrier options, whose barriers are monitored only during a specified interval within the option’s life. It cautions that closed-form formulas are generally limited to Black–Scholes settings with constant volatility and simplifying assumptions such as deterministic interest rates. Even there, the formulas can involve a three-dimensional normal cumulative distribution function, making them much more complicated than standard full-time barrier formulas.

Key ideas

  • Window barriers are active only during a defined portion of an option’s life.
  • Closed-form pricing is generally associated with constant-volatility Black–Scholes assumptions and deterministic rates.
  • The analytic formulas can require higher-dimensional normal distributions and become complex.
  • Monte Carlo or PDE methods are suggested for pricing under more realistic models.
  • As the monitoring window passes, a contract may become knocked out or reduce to a partial-time barrier option.

Tags

ForexOptionsVolatilityDerivatives pricing
Full text
# Pricing (Single and Double) Window Barrier FX Options


# Pricing (Single and Double) Window Barrier FX Options












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!

## Answer by Kurt G. (score 4, accepted)

https://quant.stackexchange.com/a/68882

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.