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### UnRisk Support: Sometimes a Matter of Solving a Riddle

Recently one of our customers had the following support question (Support Case #12341)
„We want to price a Knock-In / Knock-Out FX Call Option having the following properties:
• Property 1: Strike FX Rate = Knock-Out FX Rate = x
• Property 2: Knock-In FX Rate = y (y > x)
• Property 3: As soon as Knock-Out Level x is reached, the option is knocked out, no matter if it had been knocked in before or not
How can we set up this Instrument in UnRisk?“
By the use of UnRisk barrier Options being either of type single / double barrier knock-in or knock-out my be priced, but there is no function for pricing a mixed barrier option.
After some thinking (it was some kind of solving a small riddle) we had the following conclusion:

The described instrument can be split into the following two instruments:

Instrument 1: Long Position in a Single Down & Out FX Call Option, with
• Strike FX Rate = Knock-Out FX Rate = x
Instrument 2: Short Position in a Double Knock-Out FX Call Option, with
• Strike FX Rate = Lower Knock-Out FX Rate = x
• Upper Knock-Out FX Rate = y
At the end of our „Thinking Process“ we checked if this replication behaves in the same way as the original instrument – so we had to compare the following cases:
Case 1: The FX Rate never reaches Knock-In FX Rate y
• The value of the given instrument is 0
• Instrument 1 and Instrument 2 of the replication have the same Pay-Off (either 0 or (final FX Rate - x)), thus the value of the replication is also 0
Case 2: The FX Rate reaches Knock-Out FX Rate x and Knock-In FX Rate y
• The value of the given instrument is 0 due to Property 3
• Both, Instrument 1 and Instrument 2, have a value of 0, thus the value of the replication is also 0
Case 3: The FX Rate reaches Knock-In FX Rate y but does not reach Knock-Out FX Rate x
• The Pay-Off of the considered instrument is given by (final FX Rate – x)
• The Pay-Off of Instrument 1 of the replication is given by (final FX Rate – x), while the Pay-Off of Instrument 2 is 0 (Knock-Out FX Rate y has been reached), thus the Pay-Off of the replication is (final FX Rate – x)
So in all possible cases the replication delivers the same result as the original Instrument - Support Case #12341 closed.