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Si us plau utilitzeu sempre aquest identificador per citar o enllaçar aquest document: https://hdl.handle.net/2445/230613
Option Pricing under Black–Scholes and Jump-Diffusion Models: A Partial Differential Equation Approach
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This thesis studies the pricing of European options using partial differential equation methods. The starting point is the Black–Scholes model, where the underlying asset is assumed to follow a geometric Brownian motion. Under this assumption, the option price satisfies a parabolic partial differential equation, and for European call and put options this equation can be solved explicitly to obtain the classical Black–Scholes formula.
The first part of the thesis develops this model from the basic financial and probabilistic ideas. We introduce options, the no-arbitrage principle, Brownian motion, geometric Brownian motion and Itô’s lemma, before deriving the Black–Scholes equation by a delta-hedging argument. The analytical solution is then obtained by transforming the Black–Scholes PDE into the heat equation. Since closed-form solutions are not available in many more general situations, we also study finite difference methods for parabolic PDEs, including the explicit, implicit, Crank–Nicolson and $(\theta)$-schemes, together with the ideas of consistency, stability and convergence.
The second part considers models where the asset price is allowed to jump. This is motivated by the fact that real financial markets can experience sudden movements which are not well described by continuous Brownian paths. We introduce Poisson and compound Poisson processes, and then study the Merton and Kou jump-diffusion models. In this setting, the option price satisfies a partial integro-differential equation, where the integral term represents the effect of jumps. We derive this equation and discuss its numerical solution using an explicit-implicit scheme, treating the differential part implicitly and the integral part explicitly.
The thesis finishes with numerical studies implemented in MATLAB. These are used to compare finite difference prices with the Black–Scholes formula, to test the IMEX method for the Merton model against the corresponding series representation, and to compare the implied volatility smiles generated by the Black–Scholes, Merton and Kou models. The overall aim is to show how the PDE approach to option pricing extends from the classical continuous-path model to jump-diffusion models, and how numerical methods are used when explicit formulae are no longer available.
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Treballs finals del Màster en Matemàtica Avançada, Facultat de Matemàtiques, Universitat de Barcelona: Any: 2026. Director: Josep Vives Santa Eulàlia
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HONEY, Jenna. Option Pricing under Black–Scholes and Jump-Diffusion Models: A Partial Differential Equation Approach. [consulted: 9 of September of 2026]. Available at: https://hdl.handle.net/2445/230613