Seminar “Nonpotential dynamical systems and neural network technologies”

The report examines the inverse problem of finance, which consists of constructing (calibrating) a volatility function using available financial data. An analytical approach to solving the problem is the Dupire formula, which allows to construct a volatility function for given option prices. Examples of application of the Dupire formula for various models of local volatility are presented. As an alternative to Dupire’s approach, a neural network approach using neural networks of CaNN (Calibration Neural Network) architecture is discussed. In CaNN neural networks, calibration of the volatility function of a given structure occurs in two stages — first, the neural network is trained to approximate option prices with a loss function based on the residual for the Black-Scholes-Merton partial differential equation, then a special calibrating layer of the neural network is trained, in which the weights are unknown coefficients of the volatility function. The results of calibration of volatility functions using CaNN neural networks for specific local volatility models are presented.