Why Vanilla Implied Volatility Does Not Directly Price Barrier Options

Summary

The document explains why a vanilla option’s implied volatility is not, by itself, a reliable way to price a knock-out barrier option with a rebate. Market implied volatility is defined through the Black–Scholes formula for a vanilla contract. A volatility smile across strikes signals that the constant-volatility Black–Scholes dynamics do not match observed vanilla prices, so applying those dynamics to a path-dependent barrier payoff does not guarantee a market-consistent price.

It also cautions against treating barrier implied volatility as a straightforward risk measure. Under Black–Scholes, a barrier price need not move monotonically with volatility: rising volatility can increase the terminal payoff while also increasing the chance of hitting the barrier. Consequently, a market price may have no matching Black–Scholes volatility, or may correspond to more than one value. The answer gives conceptual reasoning but no pricing alternative or numerical example; barrier valuation requires a model that accounts for path dependence and suitable market calibration.

Key ideas

• Vanilla implied volatility is a Black–Scholes quoting convention for vanilla option prices.
• A volatility smile indicates that constant-volatility Black–Scholes dynamics do not reproduce vanilla market prices across strikes.
• A vanilla volatility surface alone does not ensure market-consistent prices for path-dependent barrier options.
• Barrier prices may be non-monotonic in volatility because payoff and barrier-hit effects compete.
• A Black–Scholes implied volatility for a barrier may have zero, one, or two solutions.

Tags

# Barrier option with Rebate


# 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)

## Answer by q.t.f. (score 1)

https://quant.stackexchange.com/a/36921

No.

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.