Journal of Applied Probability

On the uniqueness of martingales with certain prescribed marginals

Michael R. Tehranchi

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This note contains two main results. (i) (Discrete time) Suppose that S is a martingale whose marginal laws agree with a geometric simple random walk. (In financial terms, let S be a risk-neutral asset price and suppose that the initial option prices agree with the Cox-Ross-Rubinstein binomial tree model.) Then S is a geometric simple random walk. (ii) (Continuous time) Suppose that S=S0eσ X2X〉/2 is a continuous martingale whose marginal laws agree with a geometric Brownian motion. (In financial terms, let S be a risk-neutral asset price and suppose that the initial option prices agree with the Black-Scholes model with volatility σ>0.) Then there exists a Brownian motion W such that Xt=Wt+o(t1/4+ ε) as t↑∞ for any ε> 0.

Article information

J. Appl. Probab., Volume 50, Number 2 (2013), 557-575.

First available in Project Euclid: 19 June 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G42: Martingales with discrete parameter 60G44: Martingales with continuous parameter 91B25: Asset pricing models

Fake Brownian motion binomial tree model geometric Brownian motion weak convergence to Brownian motion


Tehranchi, Michael R. On the uniqueness of martingales with certain prescribed marginals. J. Appl. Probab. 50 (2013), no. 2, 557--575. doi:10.1239/jap/1371648961.

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  • Albin, J. M. P. (2008). A continuous non-Brownian motion martingale with Brownian motion marginal distributions. Statist. Prob. Lett. 78, 682–686.
  • Billingsley, P. (1999). Convergence of Probability Measures, 2nd end. John Wiley, New York.
  • Carmona, R. and Nadtochiy, S. (2009). Local volatility dynamic models. Finance Stoch. 13, 1–48.
  • Carmona, R. and Nadtochiy, S. (2012). Tangent Lévy market models. Finance Stoch. 16, 63–104.
  • Cox, J. C., Ross, S. A. and Rubinstein, M. (1979). Option pricing: a simplified approach. J. Financial Econom. 7, 229–263.
  • Derman, E. and Kani, I. (1994). The volatility smile and its implied tree. Goldman Sachs Quantitative Strategies Research Notes.
  • Dupire, B. (1994). Pricing with a smile. Risk 7, 32–39.
  • Durrleman, V. (2008). Convergence of at-the-money implied volatilities to the spot volatility. J. Appl. Prob. 45, 542–550.
  • Filipović, D. (2001). Consistency Problems for Heath–Jarrow–Morton Interest Rate Models (Lecture Notes Math. 1760). Springer, Berlin.
  • Hamza, K. and Klebaner, F. C. (2007). A family of non-Gaussian martingales with Gaussian marginals. J. Appl. Math. Stoch. Anal. 2007, article ID 92723, 19 pp.
  • Heath, D., Jarrow, R. and Morton, A. (1992). Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation. Econometrica 60, 77–105.
  • Hull, J. C. and White, A. (1990). Pricing interest-rate-derivative securities. Rev. Financial Studies 3, 573–592.
  • Kallsen, J. and Krühner, P. (2010). On a Heath–Jarrow–Morton approach for stock options. Preprint.
  • Kellerer, H. G. (1972). Markov-Komposition und eine Anwendung auf Martingale. Math. Ann. 198, 99–122.
  • Kunita, H. (1997). Stochastic Flows and Stochastic Differential Equations. Cambridge University Press.
  • Madan, D. B. and Yor, M. (2002). Making Markov martingales meet marginals: with explicit constructions. Bernoulli 8, 509–536.
  • Musiela, M. (1993). Stochastic PDEs and term structure models. In Journées Internationales de Finance, IGR-AFFI, La Baule.
  • Oleszkiewicz, K. (2008). On fake Brownian motions. Statist. Prob. Lett. 78, 1251–1254.
  • Rogers, L. C. G. (2009). A martingale with binomial marginals is a simple random walk. Personal communication.
  • Schweizer, M. and Wissel, J. (2006). Term structures of implied volatilities: absence of arbitrage and existence results. Math. Finance 18, 77–114.
  • Schweizer, M. and Wissel, J. (2008). Arbitrage-free market models for option prices: the multi-strike case. Finance Stoch. 12, 469–505.
  • Tehranchi, M. R. (2009). Symmetric martingales and symmetric smiles. Stoch. Process. Appl. 119, 3785–3797.
  • Xu, X. (2011). Fake geometric Brownian motion and its option pricing. MSc Thesis, Oxford University.