Pricing with a smile

The Black-Scholes model1 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-money options on the Nikkei

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