A novel option pricing framework using Pell-Locas collocation method under the stochastic local volatility model

Abstract

In this paper, we obtain the European option price using the Pell-Lucas collocation numerical method. To do this, we present the long-memory version of the hybrid stochastic local volatility model to forecast asset prices based on futures market data. Next, we apply financial market concepts such as the self-financing portfolio and the no-arbitrage theorem to derive a partial differential equation (PDE) for evaluating the option price. The structure of the PDE is complex, so to solve it, we employ a spectral collocation method based on Pell-Lucas polynomials as basis functions. Since the coefficients of the governing PDE are variable, the collocation method offers several advantages over the Galerkin and Tau methods. To implement this approach, we compute the operational matrix of Pell-Lucas polynomials and approximate the first, second, and mixed partial derivatives of the model. By collocating the equation, we obtain a system of algebraic equations that can be solved using traditional numerical methods.