# Expected life (Fugit) of American Option

How can I use the binomial tree pricing method for American options to determine the expected time of exercise for the option (or "Fugit")? In particular, how would I modify the algirithm here to compute expected life?

• When you iterate backwards in the tree, suppose you track and store when the exercise occurs (i.e. the smallest time when you compare the continuation to the intrinsic value, and that the former is smaller). Now if you see each branch of the tree like a (risk-neutral) realisation of the future spot price ($N$ tree levels = $N$ time steps) of which you know the probability (Bernoulli law with parameter $(N,q)$), you can just average these exercise times to obtain the average duration? This would be like the sum of the exercise times for each path times the corresponding probability? – Quantuple Sep 11 '20 at 5:42
• Agree with this method; you need a parallel ('shadow') grid; terminal condition is full time and when you get to the floating boundary you ensure the value is current time from grid initiation. Just remember to remove the discounting term from the backward-pass, it's not a tradable asset. Discounting time would be slightly bizarre and definitely wrong. – James Spencer-Lavan Sep 11 '20 at 6:06
• Although actually I'd advocate moving away from binomial trees towards finite difference grids. Biggest issue with trees is they give you information at only one spot level on the final time level. So for T0 delta, you need to run another tree. FDM more elegant. – James Spencer-Lavan Sep 11 '20 at 6:31
• @Quantuple I am working on a similar problem. At each time slice one can calculate the probability of call which is the ration of nodes with exercise state to the total number of nodes but how would you average though? because if the probability of exercise at time t1 is 60%(6 nodes out of 10 are exercise) then it can't be 50%(half the nodes are exercise) at the next grid time, should it be 40%*50% as you have to make it conditional you did not exercise on the previous date? Those probabilities should sum up to 1 in order to be use in average time computation? – Medan Jul 14 at 13:17