How do I calculate the theoretical "fair value" of an oanda.com box option?
More specifically, how do I calculate the probability that a given FOREX parity will enter a given range in a given period of time.
Example: probability that USDCAD will trade between 1.0200 and 1.0250 sometime between noon GMT this Monday afternoon and 7pm GMT this Monday afternoon?
Calculating the probability that USDCAD will be between 1.0200 and 1.0250 at noon (hitting the left edge of the box) is a fairly easy Black-Scholes computation, assuming you have a good source for time/price based volatility.
It's also easy (or at least possible) to calculate the probability that USDCAD will be outside that range at noon, but in that range at 7pm (missing the left edge of the box, but hitting the right edge). This requires integrating for all prices outside 1.0200-1.0250 and is ugly, but not uncomputable.
The problem: if USDCAD is at 1.0190 at noon, rises to 1.0210 at 3pm, and sinks back to 1.0190 at 7pm, that still counts as a hit, but neither case above will catch it.
I've played around with binary/ternary methods (which also require knowing how frequently the market trades), but am wondering if there's a simpler approach I'm overlooking.
If anyone's deeply interested, oanda.com offers free trial accounts, and provides instantaneous pricing on most box options (which means there's definitely a server-side formula for them-- it's not hand-computed each time). I've requested their formula, but am not sure how they'll respond.