Numerical Analysis: Calibration in mathematical finance
The calibration problem is posed as an optimization problem under pde-constraints, and solved using e.g. techinques from optimal control.
The so called local volatility model (Dupire 1994, Derman and Kani 1994) is one of the most widely used models in option pricing among practitioners in the financial markets. The model gives the instantaneous standard deviation of the log-spot process of an asset as a deterministic function of the spot price and of time. Given some rather mild regularity assumptions on quoted put- and call options in the market, a local volatility function that will make the model reproduce all observed option prices can – in theory – be chosen. In practice, this is a ill posed inverse problem that is difficult to solve. We pose this calibration problem as an optimization problem under pde-constraints. One of our approaches to solve this optimzation exploits techinques from optimal control and the Hamilton-Jacobi-Bellman equation.