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SWilson
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Joined: February 13th, 2018, 5:27 pm

Partial Time Barrier Option

May 24th, 2019, 2:26 pm

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.  
 
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Alan
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Re: Partial Time Barrier Option

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.
 
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SWilson
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Joined: February 13th, 2018, 5:27 pm

Re: Partial Time Barrier Option

May 24th, 2019, 3:05 pm

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.  
 
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Alan
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Re: Partial Time Barrier Option

May 24th, 2019, 3:08 pm

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

Re: Partial Time Barrier Option

May 24th, 2019, 3:19 pm

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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Alan
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Re: Partial Time Barrier Option

May 24th, 2019, 3:23 pm

Good. Finally got the eqns to display.
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