May 24th, 2019, 2:48 pm
If the barrier disappears at some time [$]t_1 < T[$], where [$]T[$] is expiration, the natural way to price the option is to take the
time-0 expectation of [$]V(t_1,S_1)[$], so
[$]V(t_0,S_0) = e^{-r t_1} E_0[V(t_1,S_1)] = e^{-r t_1} \int V(t_1,S_1) q(S_1|S_0) dS_1[$],
where [$]q[$] is the probability density to reach [$]t_1[$] with price [$]S_1[$] without ever touching the barrier. That's one integral.
As
[$]V(t_1,S_1) = e^{-r (T-t_1)} \int w(S_T) p(S_T|S_1) dS_T[$],
where [$]w(S)[$] is the payoff function, and [$]p[$] is the log-normal density, that's the second integral.
A similar argument would hold if the barrier starts at [$]t_1 < T[$].
For some reason, the LaTex is not compiling correctly; I hope it's readable anyway.
Last edited by
Alan on May 24th, 2019, 3:07 pm, edited 1 time in total.