The Annals of Applied Probability

Simple arbitrage

Christian Bender

Full-text: Open access

Abstract

We characterize absence of arbitrage with simple trading strategies in a discounted market with a constant bond and several risky assets. We show that if there is a simple arbitrage, then there is a 0-admissible one or an obvious one, that is, a simple arbitrage which promises a minimal riskless gain of $\varepsilon$, if the investor trades at all. For continuous stock models, we provide an equivalent condition for absence of 0-admissible simple arbitrage in terms of a property of the fine structure of the paths, which we call “two-way crossing.” This property can be verified for many models by the law of the iterated logarithm. As an application we show that the mixed fractional Black–Scholes model, with Hurst parameter bigger than a half, is free of simple arbitrage on a compact time horizon. More generally, we discuss the absence of simple arbitrage for stochastic volatility models and local volatility models which are perturbed by an independent 1$/$2-Hölder continuous process.

Article information

Source
Ann. Appl. Probab., Volume 22, Number 5 (2012), 2067-2085.

Dates
First available in Project Euclid: 12 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1350067994

Digital Object Identifier
doi:10.1214/11-AAP830

Mathematical Reviews number (MathSciNet)
MR3025689

Zentralblatt MATH identifier
1266.91092

Subjects
Primary: 91G10: Portfolio theory
Secondary: 60G44: Martingales with continuous parameter 60G22: Fractional processes, including fractional Brownian motion

Keywords
Arbitrage simple strategies fractional Brownian motion law of the iterated logarithm conditional full support

Citation

Bender, Christian. Simple arbitrage. Ann. Appl. Probab. 22 (2012), no. 5, 2067--2085. doi:10.1214/11-AAP830. https://projecteuclid.org/euclid.aoap/1350067994


Export citation

References

  • [1] Bayraktar, E. and Sayit, H. (2010). No arbitrage conditions for simple strategies. Ann. Finance 6 147–156.
  • [2] Bender, C., Sottinen, T. and Valkeila, E. (2008). Pricing by hedging and no-arbitrage beyond semimartingales. Finance Stoch. 12 441–468.
  • [3] Bender, C., Sottinen, T. and Valkeila, E. (2011). Fractional processes as models in stochastic finance. In Advanced Mathematical Methods for Finance (G. Di Nunno and B. Øksendal, eds.) 75–103. Springer, Heidelberg.
  • [4] Cheridito, P. (2001). Mixed fractional Brownian motion. Bernoulli 7 913–934.
  • [5] Cheridito, P. (2003). Arbitrage in fractional Brownian motion models. Finance Stoch. 7 533–553.
  • [6] Comte, F. and Renault, E. (1998). Long memory in continuous-time stochastic volatility models. Math. Finance 8 291–323.
  • [7] Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300 463–520.
  • [8] Delbaen, F. and Schachermayer, W. (1994). Arbitrage and free lunch with bounded risk for unbounded continuous processes. Math. Finance 4 343–348.
  • [9] Delbaen, F. and Schachermayer, W. (1995). The existence of absolutely continuous local martingale measures. Ann. Appl. Probab. 5 926–945.
  • [10] Dudley, R. M. and Norvaiša, R. (1998). An Introduction to $p$-Variation and Young Integrals. MaPhySto Lecture Note. Available at http://www.maphysto.dk/cgi-bin/gp.cgi?publ=60.
  • [11] Dupire, B. (1994). Pricing with a smile. Risk January 18–20.
  • [12] Guasoni, P. (2006). No arbitrage under transaction costs, with fractional Brownian motion and beyond. Math. Finance 16 569–582.
  • [13] Guasoni, P., Rásonyi, M. and Schachermayer, W. (2008). Consistent price systems and face-lifting pricing under transaction costs. Ann. Appl. Probab. 18 491–520.
  • [14] Guasoni, P., Rasonyi, M. and Schachermayer, W. (2010). The fundamental theorem of asset pricing for continuous processes under small transaction costs. Ann. Finance 6 157–191.
  • [15] Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6 327–343.
  • [16] Jarrow, R. A., Protter, P. and Sayit, H. (2009). No arbitrage without semimartingales. Ann. Appl. Probab. 19 596–616.
  • [17] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [18] Musiela, M. and Rutkowski, M. (2005). Martingale Methods in Financial Modelling, 2nd ed. Stochastic Modelling and Applied Probability 36. Springer, Berlin.
  • [19] Pakkanen, M. S. (2010). Stochastic integrals and conditional full support. J. Appl. Probab. 47 650–667.
  • [20] Rogers, L. C. G. and Williams, D. (1994). Diffusions, Markov Processes, and Martingales. Vol. 1, 2nd ed. Wiley, Chichester.