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Robustness of the Black-Scholes approach in the case of options on several assets

Author

Listed:
  • Tiziano Vargiolu

    (Dipartimento di Matematica Pura ed Applicata, Universitá di Padova, Via Belzoni 7, 35131 Padova, Italy Manuscript)

  • Silvia Romagnoli

    (Istituto di Matematica Generale e Finanziaria, Universitá di Bologna, Piazza Scaravilli 2, 40139 Bologna, Italy)

Abstract

In this paper we analyse a stochastic volatility model that is an extension of the traditional Black-Scholes one. We price European options on several assets by using a superstrategy approach. We characterize the Markov superstrategies, and show that they are linked to a nonlinear PDE, called the Black-Scholes-Barenblatt (BSB) equation. This equation is the Hamilton-Jacobi-Bellman equation of an optimal control problem, which has a nice financial interpretation. Then we analyse the optimization problem included in the BSB equation and give some sufficient conditions for reduction of the BSB equation to a linear Black-Scholes equation. Some examples are given.

Suggested Citation

  • Tiziano Vargiolu & Silvia Romagnoli, 2000. "Robustness of the Black-Scholes approach in the case of options on several assets," Finance and Stochastics, Springer, vol. 4(3), pages 325-341.
    • Handle: RePEc:spr:finsto:v:4:y:2000:i:3:p:325-341
    Note: received: April 1998; final revision received: May 1999
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    Citations

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    Cited by:

    1. Joel Vanden, 2006. "Exact Superreplication Strategies for a Class of Derivative Assets," Applied Mathematical Finance, Taylor & Francis Journals, vol. 13(1), pages 61-87.
    2. Daniel Fernholz & Ioannis Karatzas, 2012. "Optimal arbitrage under model uncertainty," Papers 1202.2999, arXiv.org.
    3. Peter Bank & Yan Dolinsky & Ari-Pekka Perkkiö, 2017. "The scaling limit of superreplication prices with small transaction costs in the multivariate case," Finance and Stochastics, Springer, vol. 21(2), pages 487-508, April.
    4. Rasmussen, Nicki Søndergaard, 2002. "Hedging with a Misspecified Model," Finance Working Papers 02-15, University of Aarhus, Aarhus School of Business, Department of Business Studies.
    5. Takeru Matsuda & Akimichi Takemura, 2018. "Game-theoretic derivation of upper hedging prices of multivariate contingent claims and submodularity," Papers 1806.07626, arXiv.org.
    6. Giorgio Fabbri & Fausto Gozzi & Andrzej Swiech, 2017. "Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations," Post-Print hal-01505767, HAL.

    More about this item

    Keywords

    stochastic volatility; superreplication; stochastic optimal control; Hamilton-Jacobi-Bellman equations;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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