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Discrete Time Term Structure Theory and Consistent Recalibration Models

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  • Anja Richter
  • Josef Teichmann

Abstract

We develop theory and applications of forward characteristic processes in discrete time following a seminal paper of Jan Kallsen and Paul Kr\"uhner. Particular emphasis is placed on the dynamics of volatility surfaces which can be easily formulated and implemented from the chosen discrete point of view. In mathematical terms we provide an algorithmic answer to the following question: describe a rich, still tractable class of discrete time stochastic processes, whose marginal distributions are given at initial time and which are free of arbitrage. In terms of mathematical finance we can construct models with pre-described (implied) volatility surface and quite general volatility surface dynamics. In terms of the works of Rene Carmona and Sergey Nadtochiy, we analyze the dynamics of tangent affine models. We believe that the discrete approach due to its technical simplicity will be important in term structure modelling.

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  • Anja Richter & Josef Teichmann, 2014. "Discrete Time Term Structure Theory and Consistent Recalibration Models," Papers 1409.1830, arXiv.org.
    • Handle: RePEc:arx:papers:1409.1830
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    References listed on IDEAS

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    1. Heath, David & Jarrow, Robert & Morton, Andrew, 1990. "Bond Pricing and the Term Structure of Interest Rates: A Discrete Time Approximation," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 25(4), pages 419-440, December.
    2. Christa Cuchiero & Josef Teichmann, 2011. "Path properties and regularity of affine processes on general state spaces," Papers 1107.1607, arXiv.org, revised Jan 2013.
    3. Robert A. Jarrow & Stuart M. Turnbull, 2008. "Pricing Derivatives on Financial Securities Subject to Credit Risk," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 17, pages 377-409, World Scientific Publishing Co. Pte. Ltd..
    4. René Carmona & Sergey Nadtochiy, 2009. "Local volatility dynamic models," Finance and Stochastics, Springer, vol. 13(1), pages 1-48, January.
    5. David Heath & Robert Jarrow & Andrew Morton, 2008. "Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 13, pages 277-305, World Scientific Publishing Co. Pte. Ltd..
    6. Martin Schweizer & Johannes Wissel, 2008. "Term Structures Of Implied Volatilities: Absence Of Arbitrage And Existence Results," Mathematical Finance, Wiley Blackwell, vol. 18(1), pages 77-114, January.
    7. Martin Schweizer & Johannes Wissel, 2008. "Arbitrage-free market models for option prices: the multi-strike case," Finance and Stochastics, Springer, vol. 12(4), pages 469-505, October.
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    Cited by:

    1. Laurini, Márcio Poletti & Ohashi, Alberto, 2015. "A noisy principal component analysis for forward rate curves," European Journal of Operational Research, Elsevier, vol. 246(1), pages 140-153.
    2. Rene Carmona & Yi Ma & Sergey Nadtochiy, 2015. "Simulation of Implied Volatility Surfaces via Tangent Levy Models," Papers 1504.00334, arXiv.org.

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