Sergey Shorokhov

The relationship between neural ordinary differential equations and inverse problems of dynamics is identified. This relationship opens up opportunities fo

We develop loss functionals for training physics–informed neural networks using variational principles for nonpotential operators. Generally, a quasi

We study neural network approximation of the solution to boundary value problem for Black-ScholesMerton partial differential equation for a European call option price, when model volatility is afunction of underlying asset price and time (local volatility model). Strike-price and expiry day of theoption are assumed to be fixed. An approximation to option price in local volatility model is obtainedvia deep learning with deep Galerkin method (DGM), making use of the neural network of specialarchitecture and stochastic gradient descent on a sequence of random time and underlying price points.Architecture of the neural network and the algorithm of its training for option pricing in local volatilitymodels are described in detail. Computational experiment with DGM neural network is performed toevaluate the quality of neural network approximation for hyperbolic sine local volatility model withknown exact closed form option price. The quality of the neural network approximation is estimatedwith mean absolute error, mean squared error and coefficient of determination. The computationalexperiment demonstrates that DGM neural network approximation converges to a European calloption price of the local volatility model with acceptable accuracy.

We study the problem of measuring market risk of financial asset with stochastic pricing. Market risk metrics under study are Value-at- Risk and Expected Shortfall. The price of the financial asset is assumed to satisfy a given stochastic differential equation with diffusion coefficient being a function of asset price and time. We investigate various models of asset price dynamics, including well-known lognormal Black-Scholes model, shifted lognormal model, Bachelier and Cox-Ross normal models and a new stochastic model with hyperbolic sine function. For stochastic models under study we derive explicit analytic expressions for loss distribution function, Value-at- Risk and Expected Shortfall. Dependence of derived Value-at- Risk and Expected Shortfall functions on confidence level is shown on the plots. The possibility of using derived formulae for market risk estimation for equity traded on financial markets is demonstrated. We show that the highest estimate of market risk is given by hyperbolic sine and Cox-Ross models, the lowest estimate is given by Black-Scholes model and shifted lognormal model with negative model parameter.