Assume Black-Scholes accurately models stock price movements. Assume you know the future volatility "sigma" of the stock's price action. Assume the stock price today is "P". Assume the price-to-be-touched is "S" (the "strike price").
The probability "X" that the stock will touch or exceed the strike price S, within T days, can be found thus:
Z = ln(S/P) / (sigma * sqrt(T/365))
X = CNDF(Z)
ln() = natural logarithm = log to the base e
Z = Zscore = size of price move from P to S, in standard deviations
CNDF() = Cumulative Normal Distribution Function
Now you just construct a summation.
The first term is the probability that the stock will touch or exceed the strike price within 1 day (T=1).
The second term is the probability that the stock DOES NOT touch or exceed the strike price withing 1 day, times the probability that the stock touches or exceeds the strike price within 2 days.
The third term is the probability that the stock DOES NOT touch or exceed within 2 days, times the probability that the stock does touch or exceed within 3 days.
Create 90 terms (for 3 months), add them up, done.
