A set of standard European options of different strikes and maturities on the same underlying has two related volatility surfaces: the implied volatility surface and the local volatility surface. When we use the market prices of options to derive these surfaces we call them the *market implied volatility* surface and the *market local volatility* surface. When we use option prices based on a stochastic volatility model we call them the *model implied volatility* surface and the *model local volatility* surface.^{1}

Implied volatility is a transformation of a standard European option price. It is the volatility that, when input into the Black–Scholes–Merton (BSM) formula, yields the price of the option. In other words, it is the constant volatility of the underlying process that is implicit in the price of the option. For this reason some authors refer to implied volatility as *implicit volatility*.

The BSM *model* implied volatilities are constant. And if the assumptions of the BSM model were valid then all options on the same underlying would have the same *market* implied volatility. However, traders do not believe in these assumptions, hence the market prices of options yield a surface of market implied volatilities, by strike (or moneyness) and maturity of the option, that is *not* flat. In particular, the market implied volatility of all options of the same maturity but different strikes has a skewed smile shape when plotted as a function of the strike (or ...

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