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Equilibrium State Prices In A Stochastic Volatility Model1

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  • Huyěn Pham
  • Nizar Touzi

Abstract

In a stochastic volatility model, the no‐free‐lunch assumption does not induce a unique arbitrage price because of market incompleteness. In this paper, we consider a contingent claim on the primitive asset, traded in zero net supply. Given a system of Arrow‐Debreu state prices, we provide necessary and sufficient conditions for consistency with an intertemporal additive equilibrium model that we fully characterize. We show that the risk premia corresponding to the minimal martingale of Föllmer and Schweizer (1991) are consistent with logarithmic preferences, while the Hull and White model (1987) (volatility risk premium independent of the asset price) is consistent with a class of utility functions including constant relative risk aversion (CRRA) ones.

Suggested Citation

  • Huyěn Pham & Nizar Touzi, 1996. "Equilibrium State Prices In A Stochastic Volatility Model1," Mathematical Finance, Wiley Blackwell, vol. 6(2), pages 215-236, April.
    • Handle: RePEc:bla:mathfi:v:6:y:1996:i:2:p:215-236
    DOI: 10.1111/j.1467-9965.1996.tb00078.x
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    Cited by:

    1. Paola Zerilli, 2005. "Option pricing and spikes in volatility: theoretical and empirical analysis," Money Macro and Finance (MMF) Research Group Conference 2005 76, Money Macro and Finance Research Group.
    2. Carpinteyro, Martha & Venegas-Martínez, Francisco & Martínez-García, Miguel Ángel, 2018. "Modeling Returns of Stock Indexes through Fractional Brownian Motion Combined with Jump Processes and Modulated by Markov Chains," MPRA Paper 90549, University Library of Munich, Germany.
    3. Bertram Düring, 2009. "Asset pricing under information with stochastic volatility," Review of Derivatives Research, Springer, vol. 12(2), pages 141-167, July.
    4. Broadie, Mark & Detemple, Jerome & Ghysels, Eric & Torres, Olivier, 2000. "American options with stochastic dividends and volatility: A nonparametric investigation," Journal of Econometrics, Elsevier, vol. 94(1-2), pages 53-92.
    5. Carpinteyro, Martha & Venegas Martínez, Francisco & Martínez García, Miguel Ángel, 2019. "Modelado de rendimientos de índices bursátiles mediante movimiento fraccional browniano combinado con procesos de saltos y modulado por cadenas de Markov / Modeling Returns of Stock Indexes through Fr," Estocástica: finanzas y riesgo, Departamento de Administración de la Universidad Autónoma Metropolitana Unidad Azcapotzalco, vol. 9(2), pages 163-180, julio-dic.
    6. Ang, Andrew & Liu, Jun, 2007. "Risk, return, and dividends," Journal of Financial Economics, Elsevier, vol. 85(1), pages 1-38, July.
    7. Srdjan Stojanovic, 2006. "Pricing and Hedging of Multi Type Contracts under Multidimensional Risks in Incomplete Markets Modeled by General Itô SDE Systems," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 13(4), pages 345-372, December.
    8. Elyès Jouini & Clotilde Napp, 1998. "Contiuous Time Equilibrium Pricing of Nonredundant Assets," Working Papers 98-30, Center for Research in Economics and Statistics.
    9. Pan, Jun, 2002. "The jump-risk premia implicit in options: evidence from an integrated time-series study," Journal of Financial Economics, Elsevier, vol. 63(1), pages 3-50, January.
    10. Thai Huu Nguyen & Serguei Pergamenshchikov, 2015. "Approximate hedging problem with transaction costs in stochastic volatility markets," Papers 1505.02546, arXiv.org.
    11. Elyès Jouini & Clotilde Napp, 2002. "Arbitrage Pricing And Equilibrium Pricing: Compatibility Conditions," World Scientific Book Chapters, in: Marco Avellaneda (ed.), Quantitative Analysis In Financial Markets Collected Papers of the New York University Mathematical Finance Seminar(Volume III), chapter 6, pages 131-158, World Scientific Publishing Co. Pte. Ltd..
    12. Lüders, Erik, 2002. "Asset Prices and Alternative Characterizations of the Pricing Kernel," ZEW Discussion Papers 02-10, ZEW - Leibniz Centre for European Economic Research.
    13. Rüdiger Frey & Carlos A. Sin, 1999. "Bounds on European Option Prices under Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 9(2), pages 97-116, April.
    14. Lüders, Erik & Peisl, Bernhard, 2001. "How do investors' expectations drive asset prices?," ZEW Discussion Papers 01-15, ZEW - Leibniz Centre for European Economic Research.
    15. Chourdakis, Kyriakos & Dotsis, George, 2011. "Maximum likelihood estimation of non-affine volatility processes," Journal of Empirical Finance, Elsevier, vol. 18(3), pages 533-545, June.
    16. Schweizer, Martin, 1999. "A guided tour through quadratic hedging approaches," SFB 373 Discussion Papers 1999,96, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.

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