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Finite Differences

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Financial Modeling

Part of the book series: Springer Finance ((SFTEXT))

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Abstract

This book emphasizes the use of backward stochastic differential equations (BSDEs) for financial modeling. The BSDE perspective is useful for various reasons. This said, in Markov setups, BSDEs are nothing than the stochastic counterparts of the more familiar, deterministic, partial differential equations (PDEs, or PIDEs for partial integro-differential equations if there are jumps), for representing prices and Greeks of financial derivatives (have you already heard of the Black–Scholes equation?). This means that prices and Greeks can also be computed by good old deterministic numerical schemes for the corresponding P(I)DEs, such as finite differences (for pricing problems which are typically posed on rectangular domains, the additional complexity of using potentially more powerful finite element methods is often not justified). Like tree methods and as opposed to simulation methods, finite difference methods can easily cope with early exercise features, and in low dimension they can give an accurate and robust computation of an option price and Greeks (delta, gamma, and theta at time 0). However, they are not practical for dimension greater than three or four, for then too many grid points are required for achieving satisfactory accuracy.

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Notes

  1. 1.

    See [124] for a PDE version of the Girsanov transformation.

  2. 2.

    Recalling that the pricing equations are posed in backward time, with a terminal condition at time T.

  3. 3.

    See Sect. 8.2.1.2 for the notions of stability and consistency.

  4. 4.

    See the equation (5.71).

  5. 5.

    However, the sparseness of the corresponding matrix may be exploited in an iterative solution.

  6. 6.

    Adequately adjusting the pricing formula (4.62) for the dividends of S; see the related comment in Sect. 4.3.1.

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  1. Département de mathématiques, Laboratoire Analyse & Probabilités, Université d’Evry Val d’Essone, Evry, France

    Prof. Stéphane Crépey

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Crépey, S. (2013). Finite Differences. In: Financial Modeling. Springer Finance(). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37113-4_8

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