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Martingale Modeling

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Financial Modeling

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

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Abstract

In this chapter, we show how the task of pricing and hedging financial derivatives can be reduced to that of solving related backward stochastic differential equations, called stochastic pricing equations in this book, equivalent to the deterministic pricing equations that arise in Markovian setups. The deterministic pricing equations, starting with the celebrated Black–Scholes equation, are better known to practitioners. However, these deterministic partial-differential equations, also including integral terms in models with jumps, are more “model dependent” than the stochastic pricing equations. Moreover, the deterministic pricing equations are less general since they are only available in Markovian setups. In addition, the mathematics of pricing and hedging financial derivatives is simpler in terms of the stochastic pricing equations. Indeed, rigorous demonstrations based on the deterministic pricing equations involve technical notions of viscosity or Sobolev solutions (at least as soon as the problem is nonlinear, e.g. when early exercise clauses and related obstacles in the deterministic equations come into the picture).

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Notes

  1. 1.

    As soon as early exercise clauses and related obstacles in the deterministic equations come into the picture.

  2. 2.

    This classical assumption is, of course, seriously challenged and, in fact, unrealistic since the crisis; see Remark 4.1.1.

  3. 3.

    Typically nonnegative, but not necessarily so, as seen with the Swiss currency during the 2011 Euro sovereign debt crisis.

  4. 4.

    Default-free and without call protection and so, in particular, ϑ≡0.

  5. 5.

    With priority of a put over a call here, though this is immaterial in terms of pricing and hedging.

  6. 6.

    Equation (4.11) being here assumed to yield a well-defined semimartingale.

  7. 7.

    For an appropriately chosen call time θ ∗; see Proposition 4.1.15 below for a precise statement.

  8. 8.

    Starting from 0 by definition; see Sect. 3.1.5.

  9. 9.

    Not necessarily continuous.

  10. 10.

    Special semimartingale with additional integrability properties; see Sect. 12.1.2.2.

  11. 11.

    See Definition 12.2.4 for formal statements.

  12. 12.

    At least in principle, leaving alone the computational issues; these will be dealt with in Chap. 9.

  13. 13.

    Or , upon exercise at a stopping time ν, for American or game claims.

  14. 14.

    See Example 12.4.6 for a precise statement.

  15. 15.

    Otherwise a more general but less constructive representation for M t can be given in terms of Malliavin calculus.

  16. 16.

    See Sect. 2.2.

  17. 17.

    See Sect. 6.9.3 for an example of the dividend-adjustment which is otherwise required in the equations.

  18. 18.

    In which β now refers to the discount factor associated with an arbitrary numéraire B.

  19. 19.

    Otherwise a more general but less constructive representation for m t can be given in terms of Malliavin calculus.

  20. 20.

    Or upon exercise at a stopping time ν, in the cases of American or game claims.

  21. 21.

    See Sect. 14.1 for the case of discrete coupons.

  22. 22.

    Since the 2011 Euro sovereign debt crisis, defaultability is also a feature of many Euro government bonds.

  23. 23.

    Otherwise a more general but less constructive representation for \(\widetilde{M}_{t}\) can be given in terms of Malliavin calculus.

  24. 24.

    Boolean-valued processes.

  25. 25.

    Model calibration will be the topic of Chap. 9.

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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). Martingale Modeling. In: Financial Modeling. Springer Finance(). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37113-4_4

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