Inverse problems in finance (Q2849671)

The paper is a survey of the mathematical and computational methods for inverse problems in finance. The work starts with an introduction to financial option pricing and the direct or inverse problems in finance. Then the author gives an introduction into the regularization of ill-posed problems, sketching the main recent developments in both linear and nonlinear theory. In the next section, there are presented some numerical approaches (Newton's method and some gradient methods) for the identification of the implied (constant) volatility in the Black-Scholes model. The problem of identifiability of local volatility is also tackled. The reported results (the uniqueness theorem and computational methods for the local volatility) are based on the Dupire's equation. The last section of the paper considers two types of approaches to the calibration of the local volatility for vanilla call options: gradient methods and a method based on the Dupire's equation. A Kaczmarz-type method is also employed to calibrate the local volatilities.NEWLINENEWLINEFor the entire collection see [Zbl 1272.91008].