Abstract: A financial option is a contractual agreement that grants the holder the right to buy or sell an underlying asset at a predetermined price within/at a specified time frame. Although there are analytical formulas available under certain restrictive assumptions and semi-analytical transform methods, such as the Fourier transform, that can be used when the characteristic function of the underlying asset process is known, many real-world problems are non-linear and cannot be solved analytically. Therefore, numerical methods are necessary to solve complex models and option types. The most common method used is Monte Carlo simulation, but it suffers from the curse of dimensionality when dealing with problems in multiple dimensions.
In this project, we aimed to numerically solve non-linear option pricing problems by utilizing Backward Stochastic Differential Equations (BSDEs). We employed Markovian BSDEs to formulate nonlinear pricing and hedging problems, which is crucial in pricing financial instruments as it enables the consideration of market imperfections and computations in high dimensions. The solutions to the processes involve conditional expectations. Thus, we employed the least squares Monte Carlo and deep neural network methods to approximate the solutions numerically.
Keywords: Backward stochastic differential equations, non-linear pricing problems, pricing in high dimensions, deep BSDE approach
