Implied Volatility For European options we take the valuation formula of Black and Scholes of the type V = v(S,τ,K,r,σ), where τ denotes the time to maturity, τ := T −t.

For the definition of the function v see Appendix A4, equation (A4.10). If actual market data V mar of the price are known, then one of the parameters considered known so far can be viewed as unknown and fixed via the implicit equation V mar − v(S,τ,K,r,σ)=0 . (∗)

In this calibration approach the unknown parameter is calculated iteratively as solution of equation (∗). Consider σ to be in the role of the unknown parameter. The volatility σ determined in this way is called implied volatility and is zero of f(σ) := V mar − v(S,τ,K,r,σ).