Equivalent Black volatilities

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Abstract
We consider European calls and puts on an asset whose forward price F(t) obeys dF(t)=α(t)A(F)dW(t,) under the forward measure. By using singular perturbation techniques, we obtain explicit algebraic formulas for the implied volatility σB in terms of today's forward price F0 ≡ F(0), the strike K of the option, and the time to expiry tex. The price of any call or put can then be calculated simply by substituting this implied volatility into Black's formula. For example, for a power law (constant elasticity of variance) model dF(t)=aFβdW(t) we obtain σB = a/faυ1-β {1 + (1-β)(2+β)/24 (F0 - K/faυ)2 + (1 - β)2/24 a2tex/faυ2-2β +…} where faυ = ½(F0 + K). Our formula for the implied volatility is not exact. However, we show that the error is insignificant, rarely approaching 1/1000 of the time value of the option. We also present more accurate (albeit more complicated) formulas which can be used for the implied volatility.Skews, Smiles, Implied Volatility, Black-scholes, Options,

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