e are interested in evaluation of
when
where the
is the optimal stopping strategy. The sup is taken over all functional forms
of
.
In the section on backward induction (
Backward
induction
) and Bellman equation
(
Bellman equation section
) we saw
that the
is a function of the state variable and the final condition,
.
Here
represent the time and the process state variables. The
is the final time and
is the payoff. In particular, the optimal stopping rule does not depend on the
initial condition. Since we recover the
when evaluating the
itself we simply use the
that we already have.
For valuation of Vega (or similar sensitivity) we need a different argument
but arrive to the same result. Assuming that the sup is attained on some
,
we
have


(Optimal stopping)

for any variation
.
Hence, when evaluating
Vega,
where we abuse the notation slightly: the
is the derivative with respect to any direction in
space
or a derivative with respect to any parameterization of
.
In any case, by the (
Optimal stopping
),
.
Hence, again we may assume that the optimal stopping rule does not change.
The rest of the calculation may follow the section
(
Pathwise differentiation
).
