I'm trying to do some convertible bond pricing. Typically, there's a soft call provision which is triggered if the value of the underlying equity is above x% of par for m of the last n days. (Usually 130% of par and 20 of the last 30 days). Bloomberg's convertible model calculates an effective trigger and then uses that to do pricing via the finite difference method. Their effective trigger is defined to be the value such that the probability that the underlying stock is about that for one day is equal to the probability that the underlying stock is above x% of par for m of the last n days. Can the effective trigger be approached analytically within the Black-Scholes framework? I reached out to Bloomberg for a white paper on the calculation, but there doesn't seem to be one available.
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