Towards Fast Computation of Exit Probabilities for Maxima of Randomized Brownian Bridges

The probability that the maximum price that a stock could take between now and the end of the trading period exceeds a given value K is of interest to those trading (American) options for this stock. Suppose a trader knew that the probability that the maximum future stock price (given the current price) exceeds Ps, the strike price, plus Po, the current option price, were greater than .5. Then it is in the trader’s interest to buy options (exactly how many is left to the trader’s discretion, although the larger the aforementioned probability, the more options one would consider buying), as it is more likely than not that the maximum future stock price will be some number Pmax > Ps + Po, in which case the trader could exercise the purchased options, sell the corresponding stocks, and obtain a profit 1 of N (