Trying to price a Partial Time End Barrier Option. Can someone please tell me why it's using a bivarite distribution function in the model when it's only for a single asset? I don't have 2 random variables, only 1. Thanks.

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.

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.

Ohhhh ok. This barrier expires at T expiration so it's a little different but nonetheless this explains the 2 random integrals. I was confused that I was looking at a 2 asset barrier which uses the multivariate. This helps.

Yeah, see my edit.

Attempting the Latex again:

[$]V(t_0,S_0) = e^{-r t_1} E_0[V(t_1,S_1)] = e^{-r t_1} \int V(t_1,S_1) \, q(S_1 | S_0) \, dS_1 [$]

and

[$] V(t_1,S_1) = e^{-r (T-t_1)} \int w(S_T) \, p(S_T | S_1) \, dS_T[$]

Attempting the Latex again:

[$]V(t_0,S_0) = e^{-r t_1} E_0[V(t_1,S_1)] = e^{-r t_1} \int V(t_1,S_1) \, q(S_1 | S_0) \, dS_1 [$]

and

[$] V(t_1,S_1) = e^{-r (T-t_1)} \int w(S_T) \, p(S_T | S_1) \, dS_T[$]

Last edited by Alan on May 24th, 2019, 3:25 pm, edited 6 times in total.

Thanks Alan. This helps and I jumped the gun since I can see now the arguments being passed into the bivariate that you described above. Old dog > new tricks

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