Equivalent Black volatilities

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 σ B in terms of today's forward price F 0 ≡ 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)=aFβ dW(t) we obtain σ B = a/f aυ 1− β {1 + (1−β)(2+β)/24 (F 0 − K/f aυ)2 + (1 − β)2/24 a 2 tex /f aυ 2−2β +…} where f aυ = ½(F 0 + 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.