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SWilson
Topic Author
Posts: 33
Joined: February 13th, 2018, 5:27 pm

### Partial Time Barrier Option

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.

Alan
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Joined: December 19th, 2001, 4:01 am
Location: California
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### Re: Partial Time Barrier Option

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.

SWilson
Topic Author
Posts: 33
Joined: February 13th, 2018, 5:27 pm

### Re: Partial Time Barrier Option

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.

Alan
Posts: 9779
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: Partial Time Barrier Option

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

$V(t_1,S_1) = e^{-r (T-t_1)} \int w(S_T) \, p(S_T | S_1) \, dS_T$
Last edited by Alan on May 24th, 2019, 3:25 pm, edited 6 times in total.

SWilson
Topic Author
Posts: 33
Joined: February 13th, 2018, 5:27 pm

### Re: Partial Time Barrier Option

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

Alan
Posts: 9779
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: Partial Time Barrier Option

Good. Finally got the eqns to display.

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