# Barrier option with Rebate

Can I use the Implied vol surface from the plain vanilla options to price the Knock out Barrier options with Rebate?. In addition, for risk management purpose, can I just imply the volatility from the Barrier option prices like in plain vanilla options (Black and Scholes Framework)

• If you want to conform to the full implied vol surface you need at a minimum to build a local vol model or to use replication. – Antoine Conze Nov 14 '17 at 15:56
• @ Antoine Conze Thanks for the comment. The model I am using is based on Black and scholes framework. Just to be more clear, if I have implied smile for a particular maturity is there a way that I can use this smile to price Barrier options on same expiry without building local vol or stochastic vol surface. – Thiyagu Dhandapani Nov 15 '17 at 8:26
• Unfortunately no. First you need the smile for all maturities (because the barrier applies at all times prior to expiry), then even if you try replication you need the local volatility to compute the weights (see Andersen et al Static replication of barrier options some general results). The easiest route is probably to apply Dupire's formula to get the local volatility and then to solve the PDE for the barrier options. – Antoine Conze Nov 15 '17 at 9:53
• @ Antoine Conze, thanks again. One final question, if I imply the volatility of barrier option from market price and use Black and scholes closed form equation just like the plain vanilla, what does that volatility denote. Can it be used in risk management just like the impl vols of plain vanilla? – Thiyagu Dhandapani Nov 15 '17 at 16:07
• you can call it the implied volatility of a barrier option - not to be confused with the implied volatility of a vanilla option - but it might not even be unique as barrier options prices are not necessarily monotonous with respect to volatility. As for risk management computing the greeks assuming BS with this flat "implied vol" might be ok for simplified reporting but probably not for hedging. – Antoine Conze Nov 16 '17 at 7:43