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)

  • $\begingroup$ 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. $\endgroup$ Nov 14, 2017 at 15:56
  • $\begingroup$ @ 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. $\endgroup$ Nov 15, 2017 at 8:26
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    $\begingroup$ 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. $\endgroup$ Nov 15, 2017 at 9:53
  • $\begingroup$ @ 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? $\endgroup$ Nov 15, 2017 at 16:07
  • $\begingroup$ 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. $\endgroup$ Nov 16, 2017 at 7:43

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Implied vol as used in the market is purely a convention to express prices of vanilla options. The definition of implied vol is the number to plug into the Black-Scholes option pricing formula to get the right price for a vanilla option.

The fact that options at different strikes have different implied vols proves that the Black-Scholes dynamics (i.e. the assumption that spot follows a geometric Brownian motion) are incompatible with market pricing. If the Black-Scholes dynamics were correct, then the implied vol smiles would be constant.

Given that Black-Scholes market dynamics are incorrect, there is no reason they should give a correct price for any kind of path dependent option. Indeed market prices for barrier options do not match those that come from the Black-Scholes model.

One cannot even define an "implied vol" analogue for barrier options. This is because, unlike vanilla options, the Black-Scholes price of a barrier option is not necessarily monotonic in volatility. For low levels of volatility, the option price is near intrinsic value, but for high volatility there are counteracting effects of increased final payoff value but also increased probability of a barrier hit. The market price of the option may be higher than the highest possible Black-Scholes price. So there may be zero, one, or two levels of vol to correctly price the option.

  • $\begingroup$ Thanks for the answer, I understand from your paper about the GAP RISK that you have modelled the Turbo type warrants in the BS framework, with a constant volatility. But I have read from other sources that atleast one should use local volatility model to price the same. How much of an error will I make if I am using the constant vol instead of a local vol model to value the Leveraged certificates? I am using it only for risk calculation and not for market making . $\endgroup$ Nov 21, 2017 at 9:44

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