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All basic rules to update DX prices for FX options are already described in these two posts : FX options pricing part I and part II.
In order to price exotic options, you just need to add an extra DX.PRICE.SOURCE record, and linked it to the DX.CONTRACT.MASTER keys defining all the exotic options you have to deal. In the definition of this extra DX.PRICE.SOURCE, you will specify a pricing routine specific to barrier options in the fields 10 (Build.Pgm) and 3 (Program).
Vanilla options are priced using Garman & Kohlhagen model. These two articles (FX options pricing part I and part II) are detailing how to include a model in T24 prices updates.
For barrier options, a specific model must be used instead. Read the full article on barrier options here: Merton, Reiner and Rubinstein.
You will also need the Normal cumulative distribution described in this post. It could be insteresting to program it as a function to make formulas easier to read, as explained here.
The closed forms to be used:
Barrier option price
The revaluation price is determined by combining these 4 forms, depending on the type of option (Call / Put), the type of barrier (knock-in / knock-out), and the level of the barrier vs the strike.
Knock-out Prices
For Knock-out CALL options:
Down and Out: barrier is out of the money (S > H)
Knock-out CALL with strike < barrier (X < H): B – D
Knock-out CALL with strike > barrier (X > H): A – C
Up and Out: barrier is in the money (S < H)
Reverse Knock-out CALL with strike < barrier (X < H): A – B + C – D
Reverse Knock-out CALL with strike > barrier (X > H): 0
For Knock-out PUT options:
Up and out: barrier is out of the money (S < H)
Knock-out PUT with strike < barrier (X < H): A – C
Knock-out PUT with strike > barrier (X > H): B – D
Down and out: barrier is in the money (S > H)
Reverse Knock-out PUT with strike < barrier (X < H): 0
Reverse Knock-out PUT with strike > barrier (X > H): A – B + C – D
Knock-in Prices
For Knock-in CALL options
Down and in: barrier is out of the money (S > H)
Knock- in CALL with strike < barrier (X < H): A- B + D
Knock- in CALL with strike > barrier (X > H): C
Up and in: barrier is in the money (S < H)
Reverse Knock-in CALL with strike < barrier (X < H): B – C + D
Reverse Knock- in CALL with strike > barrier (X > H): A
For Knock-in PUT options:
Up and in: barrier is out of the money (S < H)
Knock- in PUT with strike < barrier (X < H): C
Knock- in PUT with strike > barrier (X > H): A – B + D
Down and in: barrier is in the money (S > H)
Reverse Knock- in PUT with strike < barrier (X < H): A
Reverse Knock- in PUT with strike > barrier (X > H): B – C + D
Delta
For Delta sensitivity, same combinations must be used, with A, B, C and D below forms:
