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(Elle)
#1
American Exercise
The valuation of options with American exercise — or Bermudan exercise, to this end
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— is much more involved than with European exercise. Therefore, we have to introduce a
bit more valuation theory first before proceeding to the valuation class.
Least-Squares Monte Carlo
Although Cox, Ross, and Rubinstein (1979) presented with their binomial model a simple
numerical method to value European and American options in the same framework, only
with the Longstaff-Schwartz (2001) model was the valuation of American options by
Monte Carlo simulation (MCS) satisfactorily solved. The major problem is that MCS per
se is a forward-moving algorithm, while the valuation of American options is generally
accomplished by backward induction, estimating the continuation value of the American
option starting at maturity and working back to the present.
The major insight of the Longstaff-Schwartz (2001) model is to use an ordinary least-
squares regression
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to estimate the continuation value based on the cross section of all
available simulated values — taking into account, per path:
The simulated value of the underlying(s)
The inner value of the option
The actual continuation value given the specific path
In discrete time, the value of a Bermudan option (and in the limit of an American option)
is given by the optimal stopping problem, as presented in Equation 17-3 for a finite set of
points in time 0 < t 1 < t 2 < ... < T.
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Equation 17-3. Optimal stopping problem in discrete time for Bermudan option
Equation 17-4 presents the continuation value of the American option at date 0 ≤ tm < T. It
is just the risk-neutral expectation at date tm+1 under the martingale measure of the value
of the American option at the subsequent date.
Equation 17-4. Continuation value for the American option
The value of the American option at date can be shown to equal the formula in
Equation 17-5 — i.e., the maximum of the payoff of immediate exercise (inner value) and
