Numerical Methods for HJB Equations in Option Pricing with Greek Sensitivities
| dc.contributor.author | Berikbayev K. | |
| dc.date.accessioned | 2026-10-15T05:41:09Z | |
| dc.date.available | 2026-10-15T05:41:09Z | |
| dc.date.issued | 2026 | |
| dc.description.abstract | Usually, optimal control problems in financial mathematics are solved by maximizing profit and using the Hamilton-Jacobi-Bellman equation. This leads to nonlinear PDE, the solution of which is based on numerical methods. In this paper, we use the method described in the article by Christara and Wu: second-order finite differences on a non-uniform grid. However, instead of the long position discussed there, we study the short position. Therefore, here too we use the same time step, calculated using the Crank-Nicolson-Rannacher method, and we employ an iterative solver with a penalty term. The nonuniform grid puts more nodes near the strike price. This is important for keeping second-order convergence when the straddle payoff is not smooth. Rannacher smoothing at the first time steps helps remove unwanted oscillations from the kink in the initial condition. The nonlinear penalty-like term comes from the optimal control in the stock borrowing fee model. We solve it at each time step with a Newton-type iterative algorithm that uses two stopping criteria. Our numerical tests show secondorder convergence for the short position. The results agree with the benchmarks of Forsyth and Labahn and also Christara and Wu. The main result of this work is the numerical calculation of the Greeks for this nonlinear HJB model: Delta, Gamma, and Theta. As far as we know, there is no other full numerical study of the Greeks for the short position using finite differences on HJB problems. This is the main applied result of our work. | |
| dc.identifier.citation | Berikbayev K / Numerical Methods for HJB Equations in Option Pricing with Greek Sensitivities / SDU University / Department of Mathematics and Natural Sciences | |
| dc.identifier.uri | https://repository.sdu.edu.kz/handle/123456789/2671 | |
| dc.language.iso | en | |
| dc.publisher | SDU University | |
| dc.subject | Derivative Matrices | |
| dc.subject | From Brownian motion to BSM | |
| dc.subject | Penalty Iteration Algorithm | |
| dc.title | Numerical Methods for HJB Equations in Option Pricing with Greek Sensitivities | |
| dc.type | Thesis |