Valuation of Barrier Options

There are many publications on how to price barrier options. 


In general, the valuation of barrier options needs to take into account stock-level dependency of volatility dynamics, that is, local volatility surface. 

This requires using numerical methods like PDE or MC simulation. In certain situations, the barrier options can be priced by using the static replication method. 


In-Out Parity


Being long a KO option and a KI option with the same features is equivalent to owning a comparable vanilla option independently from the behaviour of the spot with respect to the barrier level.

Knock-in (K,T,H) + Knock-out (K,T,H) = Vanilla (K,T)



Constant volatility 

In the limit of constant volatility, Merton provided the first analytical formula for DO call option. 

Later on, Reiner and Rubinstein extended the formula for all eight combinations of barriers. 




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