# Price of options when exercise date doesn't match nodes of BDT Tree

I think I'm missing something obvious here, but here I go. I'm studying pricing of bonds with embedded options using Black Derman Toy. I understand tree construction and its application for simple cases, by which I mean something like this:

Zero Coupon Bond maturing in 5 years with embedded call option expiring in 3 years, of strike price \$100

Here one would construct a binomial tree of step length, say, 0.25 years (3 months). Then from the end of the tree, apply backwards discounting using short rates calibrated on the tree. If the prices of bonds at option expiry is higher than 100 (strike) then 'fix' the price at that node to be 100;

This method works because the option expiry and bond maturity are both multiples of 0.25, so that the nodes coincide with respective dates.

But what if this wasn't the case? For example, what if bond matured in 5 years but option expires in some random number, say 3.12 years?

If we decied on the same strategy and generate a tree of step length 0.25 years, the option maturity will fall between the nodes placed at time 3 years and 3.25 years.

In this scenario, how do I determine if option is exercised or not? Of course, situation can be resolved by constructing a tree of step length 0.01 years, but is there simpler way?