Quantitative Finance

Chapter 6

Implied volatility and volatility smiles

Recall the Black and Scholes (1973) formula for a European call option in its simplest form,

$\begin{array}{l}\begin{array}{cc}C\left(t\right)=X\left(t\right)\mathcal{N}\left(h_1\right)-K{e}^{-r\left(T-t\right)}\mathcal{N}\left(h_2\right)& \left(6.1\right)\end{array}\hfill \\ h_{1,2}=\frac{\text{ln}\frac{X\left(t\right)}{K}-r\left(T-t\right)\pm \frac{1}{2}{\sigma }_X^2\left(T-t\right)}{{\sigma }_X\sqrt{T-t}}\hfill \end{array}$

on a non-dividend–paying asset X(t) with constant (scalar) volatility σX, strike K, time to maturity T − t and constant continuously compounded risk– free interest rate r. K and T − t are specified in the option contract, and X(t) and r can be assumed to be observable in the market.1 Only the volatility σX is not directly observable. In the absence of any other market information, a simple approach would be to estimate it as the annualised standard deviation of past logarithmic returns of the underlying asset. However, ...