The algorithm we describe in the following is called Least-Squares Monte Carlo (LSM)
and is from the paper by Longstaff and Schwartz (2001). It can be shown that the value of
an American (Bermudan) option at any given date t is given as Vt(s) = max(ht(s),Ct(s)),
where is the so-called continuation value of the option
given an index level of St = s.
Consider now that we have simulated I paths of the index level over M time intervals of
equal size . Define Yt,i ≡ e
–rt
Vt+ t,i to be the simulated continuation value for path i at
time t. We cannot use this number directly because it would imply perfect foresight.
However, we can use the cross section of all such simulated continuation values to
estimate the (expected) continuation value by least-squares regression.
Given a set of basis functions bd, d = 1,...,D, the continuation value is then given by the
regression estimate , where the optimal regression parameters
*
are the
solution of the least-squares problem stated in Equation 10-13.
Equation 10-13. Least-squares regression for American option valuation
The function gbm_mcs_amer implements the LSM algorithm for both American call and
put options:
[ 39 ]
In [ 64 ]: def gbm_mcs_amer(K, option=‘call’):
”’ Valuation of American option in Black-Scholes-Merton
by Monte Carlo simulation by LSM algorithm
Parameters
==========
K : float
(positive) strike price of the option
option : string
type of the option to be valued (‘call’, ‘put’)
Returns
=======
C0 : float
estimated present value of European call option
”’
