The various concepts of volatility (realized, local, stochastic, implied), well defined or depending on a given model and/or statistical estimates, will be discussed. Backward and forward equations for call-option payoffs (Black-Scholes and Dupire equations) will be revisited. We will show that, besides the Black-Scholes model with constant volatility, fast mean reverting stochastic volatility models can reconcile local and implied volatilities. If time permits we will also look at the relation between volatility and correlation in the multidimensional case.
The talk is addressed to a general audience in Probability without any particular deep background in financial mathematics.