Calibration, local and stochastic volatility

Review of the delta hedging of options, hedging with Black-Scholes formulas, and hedging with vanillas.

Local Volatility (LV) models: option replications, evaluation partial differential equations (EDP). Forward EDP for the prices of calls/puts in an LV model and applications to model calibration. Dupire formula and its proof. Extension to surfaces of regular and arbitrage-free calls/puts prices. Markovian projection of a stochastic volatility model, connection with the Gyongy theorem.

Implied variance and volatility. Static no-arbitrage conditions on the implied volatility surface, some asymptotic properties. SVI and SSVI parameterizations, an example of calibration to equity data.

Forward variance of the log-contract, the VIX index. Stochastic models of forward variance (especially Bergomi models). Other instruments in volatility markets: variance swaps. Forward variance of the var swap, connection with the log-contract.

A precise bibliography will be provided during the course.

• J. Guyon’s course, “Advanced Calibration Methods and Derivatives on the VIX,” is highly complementary to the MAP655B course, “Calibration, LV, and SV,” and serves as an excellent continuation. It addresses topics not covered in MAP655B (calibration using particle methods), introduces new tools (the theory of martingale optimal transport as an instrument for model construction and hedging), and delves deeper into the analysis of the VIX market and its joint calibration with the SPX. The two courses can be taken sequentially (and I actually recommend taking both if these topics interest you), with almost no redundancy.

• The same comment applies to E. Abi Jaber’s course, “Fractional Processes and Volterra Processes in Finance”: it covers complementary topics to the MAP655B calibration course, including classical affine models (including the Heston model), which is an important subject, and affine Volterra processes, representing a recent evolution.