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LSV Models with Stochastic Interest Rates and Correlated Jumps

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Pricing Derivatives Under Lévy Models

Part of the book series: Pseudo-Differential Operators ((PDO,volume 12))

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Abstract

Pricing and hedging exotic options using local stochastic volatility models has drawn serious attention within the past decade, and nowadays it has become almost a standard approach to this problem. In this chapter, we show how this framework can be extended by adding to the model stochastic interest rates and correlated jumps in all three components. We also propose a new fully implicit modification of the popular Hundsdorfer and Verwer and modified Craig–Sneyd finite difference schemes that provides a second-order approximation in space and time, is unconditionally stable, and preserves nonnegativity of the solution, while still having linear complexity in the number of grid nodes.

To answer the world’s material needs, technology has to be not only beautiful but also cheap.

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Notes

  1. 1.

    Here we don’t discuss this conclusion. However, for the sake of reference, note that this can be dictated by some inflexibility of the Heston model in which vol-of-vol (volatility of volatility) is proportional to v 0. 5. More flexible models that consider the vol-of-vol power to be a parameter of calibration [13, 21] might not need jumps in v. See also [37] and the discussion therein.

  2. 2.

    If, however, one wants to determine these parameters by calibration, she has to be careful, because having both vol-of-vol and a power constant in the same diffusion term brings an ambiguity into the calibration procedure. Nevertheless, this ambiguity can be resolved if for calibration, some additional financial instruments are used, e.g., exotic option prices are combined with the variance swaps prices; see [21].

  3. 3.

    This can always be achieved by choosing a relatively small Δ τ.

  4. 4.

    Since the flop counts rarely predict accurately an elapsed time, this statement should be further verified.

  5. 5.

    Note, that, e.g., for the HV scheme we need two sweeps per step in time.

  6. 6.

    In this chapter we don’t analyze the convergence and order of approximation of the FD scheme, since the convergence in time is the same as in the original HV scheme, and approximation was proved by the theorem. For the jump FD schemes, the convergence and approximation are considered in [23].

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Authors and Affiliations

  1. Department of Finance and Risk Engineering, NYU Tandon School of Engineering, Brooklyn, NY, USA

    Andrey Itkin

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Itkin, A. (2017). LSV Models with Stochastic Interest Rates and Correlated Jumps. In: Pricing Derivatives Under Lévy Models . Pseudo-Differential Operators, vol 12. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-6792-6_9

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