A Brownian sheet martingale with the same marginals as the arithmetic average of geometric Brownian motion

David Baker (Université Pierre et Marie Curie)
Marc Yor (Université Pierre et Marie Curie)

Abstract


We construct a martingale which has the same marginals as the arithmetic average of geometric Brownian motion.This provides a short proof of the recent result due to P. Carr et al that the arithmetic average of geometric Brownian motion is increasing in the convex order. The Brownian sheet plays an essential role in the construction. Our method may also be applied when the Brownian motion is replaced by a stable subordinator.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1532-1540

Publication Date: May 19, 2009

DOI: 10.1214/EJP.v14-674

References

  1. D.B. Madan, M. Yor. Making Markov martingales meet marginals: with explicit constructions. Bernoulli 8 (2002), 509-536. Math. Review 2003h:60062
  2. S.Pal, P. Protter. Strict local martingales, bubbles, and no early exercise. Preprint (2007).
  3. F. Delbaen, W. Schachermayer. Arbitrage possibilities in Bessel processes and their relation to local martingales. Prob. Theory Relat. Fields 102 (1995), 357-366. Math. Review 97b:90018
  4. M. Rothschild, J. Stiglitz. Increasing Risk: I. A definition. Journal of Economic Theory Vol 2, No 3 (1970). Math. Review 0503565
  5. M. Shaked; J.G. Shanthikumar. Stochastic orders with applications Academic press, (1994).
  6. M. Shaked; J.G. Shanthikumar. Stochastic orders Springer Statistics (2006).
  7. G. Samorodnitsky; M.S. Taqqu. Stable non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman-Hall (1994)
  8. P. Carr, C.-O. Ewald, Y. Xiao. On the qualitative effect of volatility and duration on prices of Asian options. D.B. Madan, M. Yor Making Markov martingales meet marginals: with explicit constructions. Finance research Letters Vol. 5, Issue 3 (Sep 2008), 162-171.
  9. D. Baker, M. Yor. A proof of the increase in maturity of the expectation of a convex function of the arithmetic average of geometric brownian motion. Preprint (Sep. 2002)
  10. F. Hirsch, M. Yor. A construction of processes with one dimensional martingale marginals, based upon path-space Ornstein-Uhlenbeck and the Brownian sheet. to appear in Journal Math. Kyoto University (2009),
  11. M. Jeanblanc, J. Pitman, M. Yor. Self-similar processes with independent increments associated with Levy and Bessel processes. Stoch. Proc. and their Appl. Vol. 100 (2002), 223-232. Math. Review 2003e:60081
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.