University of Sussex
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High-order compact finite difference schemes for parabolic partial differential equations with mixed derivative terms and applications in computational finance

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posted on 2023-06-08, 18:13 authored by Christof Heuer
This thesis is concerned with the derivation, numerical analysis and implementation of high-order compact finite difference schemes for parabolic partial differential equations in multiple spatial dimensions. All those partial differential equations contain mixed derivative terms. The resulting schemes have been applied to equations appearing in computational finance. First, we develop and study essentially high-order compact finite difference schemes in a general setting with option pricing in stochastic volatility models on non-uniform grids as application. The schemes are fourth-order accurate in space and second-order accurate in time for vanishing correlation. In the numerical study we obtain high-order numerical convergence also for non-zero correlation and non-smooth payoffs which are typical in option pricing. In all numerical experiment a comparative standard second-order discretisation is significantly outperform. We conduct a numerical stability study which indicates unconditional stability of the scheme. Second, we derive and analyse high-order compact schemes with n-dimensional spatial domain in a general setting. We obtain fourth-order accuracy in space and second-order accuracy in time. A thorough von Newmann stability analysis is performed for spatial domain with dimensions two and three. We prove that a necessary stability condition holds unconditionally without additional restrictions on the choice of the discretisation parameters for vanishing mixed derivative terms. We also give partial results for non-vanishing mixed derivative terms. As first example Black-Scholes Basket options are considered. In all numerical experiments, where the initial conditions were smoothened using the smoothing operators developed by Kreiss, Thomée and Widlund, a comparative standard second-order discretisation is significantly outperformed. As second example the multi-dimentional Heston basket option is considered for n independent Heston processes, where for each Heston process there is a non-vanishing correlation between the stock and its volatility.


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University of Sussex

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