8. (Efficiency Improvement)

The code in Section 12.4.2 that computes the price of a discrete dividend-paying option is inefficient in the following sense: first, the (default) forward induction algorithm updates the lattice from the current time to expiration and second, the lattice is again updated to reflect the presence of a discrete dividend at the ex-dividend date. Modify the code in the framework to optimise the code by updating the lattice only once, for example:

 ‘Normal’ forward induction up to but not including the ex-dividend date.

 Implementing formula (12.8) starting from the ex-dividend date to expiration.