Value of vanilla put as value of 2 barrier puts

[inik]

@David Harper CFA FRM CIPM

Quick question / reassurance --- Everywhere I read, the mention is that

• Value(call_{vanilla}) = Value(call_{up-n-in}) + Value(call_{up-n-out}).

where call_{vanilla} is a European call w/o rebate or any other weirdness.

As a sanity check, similar concept is applicable to puts as well, no?

• Value(put_{vanilla}) = Value(put_{down-n-in}) + Value(put_{down-n-out})

 

[David Harper CFA FRM]

Hi @inik

Yes, it's true, see 413.1 @ https://forum.bionicturtle.com/threads/p2-t5-413-exotic-options-barrier-binary-and-lookback.7612/

It generalizes to calls/puts and up/down; e.g., Peter James http://www.amazon.com/Option-Theory-Peter-James/dp/0471492892 :

The reader should now be in a position to derive a formula for any knock-in option. If he really enjoys integration, he can work out the integral results for all the puts and calls with barriers in different positions. Without showing all the detailed workings, we give the results in the next subsection. First, however, we take note of a simple but powerful relationship:
Knock-in Option + Knock-out Option = European Option
This result is obvious if we consider a portfolio consisting of two options which are the same except that one knocks in and the other knocks out. Whether or not the barrier is crossed, the payoff is that of a European option. This relationship allows us to calculate all the knock-out formulas from the knock-in results.

 

[inik]

Awesome; thank you, David!

(your reference to the 413.1 further underscores that I need to move up freshly-posted PQs on my todo list. t-y!)