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Functional convergence of Snell envelopes: Applications to American options approximations

Author

Listed:
  • Maurizio Pratelli

    (Dipartimento di Matematica, UniversitÁ di Pisa, Via Buonarroti 2, I-56100 Pisa, Italy Manuscript)

  • Sabrina Mulinacci

    (Istituto di Econometria e Matematica per le Decisioni Economiche, UniversitÁ Cattolica del S. Cuore, Via Necchi 9, I-20131 Milano, Italy)

Abstract

The main result of the paper is a stability theorem for the Snell envelope under convergence in distribution of the underlying processes: more precisely, we prove that if a sequence $(X^n)$ of stochastic processes converges in distribution for the Skorokhod topology to a process $X$ and satisfies some additional hypotheses, the sequence of Snell envelopes converges in distribution for the Meyer-Zheng topology to the Snell envelope of $X$ (a brief account of this rather neglected topology is given in the appendix). When the Snell envelope of the limit process is continuous, the convergence is in fact in the Skorokhod sense. This result is illustrated by several examples of approximations of the American options prices; we give moreover a kind of robustness of the optimal hedging portfolio for the American put in the Black and Scholes model.

Suggested Citation

  • Maurizio Pratelli & Sabrina Mulinacci, 1998. "Functional convergence of Snell envelopes: Applications to American options approximations," Finance and Stochastics, Springer, vol. 2(3), pages 311-327.
  • Handle: RePEc:spr:finsto:v:2:y:1998:i:3:p:311-327
    Note: received: January 1996; final version received: July 1997
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    Citations

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    Cited by:

    1. Dietmar P.J. Leisen, 1997. "The Random-Time Binomial Model," Finance 9711005, EconWPA, revised 29 Nov 1998.
    2. Ross A. Maller & David H. Solomon & Alex Szimayer, 2006. "A Multinomial Approximation For American Option Prices In Lévy Process Models," Mathematical Finance, Wiley Blackwell, vol. 16(4), pages 613-633.
    3. repec:dau:papers:123456789/5374 is not listed on IDEAS
    4. Blanka Horvath & Antoine Jacquier & Aitor Muguruza, 2017. "Functional central limit theorems for rough volatility," Papers 1711.03078, arXiv.org, revised Nov 2017.
    5. Leisen, Dietmar P. J., 1999. "The random-time binomial model," Journal of Economic Dynamics and Control, Elsevier, vol. 23(9-10), pages 1355-1386, September.
    6. RØdiger Frey, 2000. "Superreplication in stochastic volatility models and optimal stopping," Finance and Stochastics, Springer, vol. 4(2), pages 161-187.
    7. Yan Dolinsky, 2009. "Applications of weak convergence for hedging of game options," Papers 0908.3661, arXiv.org, revised Nov 2010.
    8. Szimayer, Alex & Maller, Ross A., 2007. "Finite approximation schemes for Lévy processes, and their application to optimal stopping problems," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1422-1447, October.

    More about this item

    Keywords

    American options; Snell envelopes; convergence in distribution; optimal stopping times;

    JEL classification:

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

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