elle
(Elle)
#1
the expected payoff of not exercising (continuation value).
Equation 17-5. Value of American option at any given date
In Equation 17-5, the inner value is of course easily calculated. The continuation value is
what makes it a bit trickier. The Longstaff-Schwartz (2001) model approximates this value
by a regression, as presented in Equation 17-6. There, i stands for the current simulated
path, D is the number of basis functions for the regression used,
*
are the optimal
regression parameters, and bd is the regression function numbered d.
Equation 17-6. Regression-based approximation of continuation value
The optimal regression parameters are the result of the solution of the least-squares
regression problem presented in Equation 17-7. Here, is the actual
continuation value at date tm for path i (and not a regressed/estimated one).
Equation 17-7. Ordinary least-squares regression
This completes the basic (mathematical) tool set to value an American option by MCS.
The Valuation Class
Example 17-4 presents the class for the valuation of options and derivatives with
American exercise. There is one noteworthy step in the implementation of the LSM
algorithm in the present_value method (which is also commented on inline): the optimal
decision step. Here, it is important that, based on the decision that is made, the LSM
algorithm takes either the inner value or the actual continuation value — and not the
