# probability of exercise

I am pricing a bermudan call option using finite difference method. At each exercise time on the grid I have points I exercise and on some of those I don't. Thus, at any given call time there is a probability to exercise which is the ratio of number of points of call/all points. But how would I calculate how probable to exercise on a particular date? It should be in some way conditioned I did not exercise prior to that exercise. Should the probability to exercise at each date sum up to 1 and essentially be a decreasing function with the last value is the probability to hold till the end? Is there an approach for it? I have not seen this in the textbook so please point to the references if those exist.

• I think, if anything, optionality will be worth less and less as time goes by and maturity approaches, so probability of exercise will tend to increase with time. How much so will depend on how the underlying fluctuates in price, and hence I don't think you can associate qualities such as "the probability to exercise at each date sum up to 1". Jul 13 at 19:16
• yes, if not one but do I have something like 70%,60%,50%? I don't think it is descriptive. My exercise decision boundary is a distribution so there should be some type of pdf for the time points. I just can't find it in any books! Jul 13 at 19:43
• The exercise boundary should be unknown and found by solving the free boundary PDE problem, with finite differences. If you already know it, all the better, you know in advance whether each node in the grid is exercised/not exercised. Why would you want to know the analytical form of the exercise boundary? Jul 13 at 19:58
• @Giogre yes, I do know a state for each node if it is call or hold. The reason is I need to calculate the Expected call time. Each call date has a probability and it is the ration of nodes where I exercise to the total number of nodes. The question becomes then what is the expected call date then? Thus, I am thinking it can be represented as a pdf across call dates that I can use to compute the average. Jul 14 at 13:08
• You know the distribution at discrete points, perhaps you can pick some distribution that qualitatively looks like your exercise boundary and try moment matching its parameters? Jul 14 at 14:42