UBC Math Biology Seminar: Elias Mwakilama

Abstract: In recent years, nonlinear Black–Scholes equations have gained significant attention for their improved accuracy in option pricing, as they account for more realistic market conditions. These include factors such as transaction costs, unhedged portfolio risks, market illiquidity, the impact of large investors on stock prices, and dependencies on volatility, drift, and option values. In this talk, I discuss methods to solve nonlinear forms of the Black–Scholes partial differential equation numerically, using spectral methods, but focusing on the European call options. The multi-domain spectral collocation method is employed, integrating stochastic volatility and transaction cost effects. The quasilinearization method is utilized to linearize the nonlinear Black-Scholes model. Initially, Barles and Soner's volatility model is examined, and the results are compared with those from the linear Black–Scholes model that ignores transaction costs. The problem is reformulated as a convection-diffusion equation, and numerical simulations are carried out using MATLAB to visualize solutions through graphs and tables. Further, I demonstrate how the proposed scheme is applied to solve well known theoretical and computational biology problems.