The objective of this dissertation is to develop a probabilistic solution of a semi-linear Partial Differential Equation (PDE) to be applied to the non-linear pricing of financial contingent claims by constructing a Forward Backward (FBSDEs) frame-work. Our method relies on the Markov operator that approximates the solution of the forward SDE as a truncated stochastic series. We put forward new insights into how a stochastic Stratonovich–Taylor expansion for composite functions with Magnus series can build an SDE weak solution. We apply this framework to both the
forward and backward steps and obtain gradient bounds, as well as the convergence order for a multi-step, non-linear FBSDE scheme. We use this setup to incorporate asymmetric lending and borrowing rates into the Black–Scholes framework and derive no-arbitrage pricing formulas in Lending and Borrowing measures for the upper and lower bounds of the option price. In conclusion, we implement pricing routines for the Premium and the Delta of Vanilla, Asian, American, and American Asian options in the presence of funding rates to test the robustness and scalability of the framework.