Option Valuation through Deep Learning of Transition Probability Density

Transition probability densities are fundamental to option pricing. Advancing recent work in deep learning, we develop novel transition density function generators through solving backward Kolmogorov equations in parametric space for cumulative probability functions, using neural networks to obtain accurate approximations of transition probability densities, creating ultra-fast transition density function generators offline that can be trained for any underlying. These are 'single solve' , so they do not require recalculation when parameters are changed (e.g. recalibration of volatility) and are portable to other option pricing setups as well as to less powerful computers, where they can be accessed as quickly as closed-form solutions. We demonstrate the range of application for one-dimensional cases, exemplified by the Black-Scholes-Merton model, two-dimensional cases, exemplified by the Heston process, and finally for a modified Heston model with time-dependent parameters that has no closed-form solution.