Simultaneous identification of time-dependent volatility and interest rate for European options

We investigate a specific ill-posed nonlinear inverse problem that arise on the financial markets. Precisely, as a benchmark problem, we consider the simultaneous recovery of implied volatility and interest rate functions over a finite interval corresponding to call and put prices for idealized continuous European vanilla options over the same maturity. In order to find these time-dependent functions, we minimize the cost functional which is the sum of the squared error between the market values and the computed values, obtained by implicit difference scheme for the Black–Scholes model. Furthermore, results of numerical experiments are discussed.