Pricing with a smile

In the January 1994 issue of Risk, Bruno Dupire showed how the Black-Scholes model can be extended to make it compatible with observed market volatility smiles, allowing consistent pricing and hedging of exotic options

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The Black-Scholes model gives options prices as a function of volatility. If an option price is given by the market we can invert this relationship to get the implied volatility.

If the model were perfect, this implied value would be the same for all option market prices, but reality shows this is not the case. Implied Black-Scholes volatilities strongly depend on the maturity and the strike of the European option under scrutiny. If the implied volatilities of at-the

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