Abstract
We show that several general classes of stochastic processes satisfy a functional co-monotony principle, including processes with independent increments, Brownian bridge, Brownian diffusions, Liouville processes, fractional Brownian motion. As a first application, we recover and extend some recent results about peacock processes obtained by Hirsch et al. in (Peacocks and Associated Martingales, with Explicit Constructions, Bocconi & Springer, 2011, 430p) [see also (Peacocks sous l’hypothèse de monotonie conditionnelle et caractérisation des 2-martingales en termes de peacoks, thèse de l’Université de Lorraine, 2012, 169p)] which were themselves motivated by a former work of Carr et al. in (Finance Res. Lett. 5:162–171, 2008) about the sensitivities of Asian options with respect to their volatility and residual maturity (seniority). We also derive semi-universal bounds for various barrier options.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Stands for the French acronym PCOC (Processus Croissant pour l’Ordre Convexe).
- 2.
For every α, β, γ ∈ E and every \(\lambda \in \mathbb{R}_{+}\), α ≤ β ⇒ α + γ ≤ β + γ and λ ≥ 0 ⇒ λ α ≤ λ β.
- 3.
The fact that \(A(L_{_{T}}^{2})\) is the Cameron–Martin space i. e. the reproducing space of the covariance operator, which is obvious here, is a general fact for any such decomposition (see [16]).
References
P. Bauman, M. Émery, Peut-on “voir” dans l’espace à n dimensions, L’ouvert 116, 1–8 (2008)
P. Billingsley, Convergence of Probability Measures, 2nd edn. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication (Wiley, New York, 1999), 277p
A.M. Bogso, Peacocks sous l’hypothèse de monotonie conditionnelle et caractérisation des 2-martingales en termes de peacoks, thèse de l’Université de Lorraine, 2012, 169p
P. Carr, C.-O. Ewald, Y. Xiao, On the qualitative effect of volatility and duration on prices of Asian options. Finance Res. Lett. 5, 162–171 (2008)
P.H. Diananda, On nonnegative forms in real variables some or all of which are nonnegative. Proc. Cambridge Philos. Soc. 58, 17–25 (1962)
H.J. Engelbert, On the theorem of T. Yamada and S. Watanabe. Stochast. Stochast. Rep. 36 (3–4), 205–216 (1991)
M. Hall, Discrete problems, in A Survey of Numerical Analysis, ed. by John Todd (McGraw-Hill, New York, 1962)
F. Hirsch, B. Roynette, A new proof of Kellerer’s Theorem. ESAIM: P&S 16, 48–60 (2012)
F. Hirsch, C. Profeta, B. Roynette, M. Yor, Peacocks and Associated Martingales, with Explicit Constructions Bocconi and Springer Series, 3 (Springer, Milan; Bocconi University Press, Milan, 2011), 430p
J. Jacod, A.N. Shiryaev, Limit Theorems for Stochastic Processes, 2nd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288 (Springer, Berlin, 2003), 661p
K. Joag-Dev, M.D. Perlman, L.D. Pitt, Association of normal random variables and Slepian’s inequality. Ann. Probab. 11(2), 451–455 (1983)
I. Karatzas, S.E. Shreve, Brownian Motion and Stochastic Calculus, 2nd edn. Graduate Texts in Mathematics, vol. 113 (Springer, New York, 1991), 470p
H.G. Kellerer, Markov-Komposition und eine Anwendung auf Martingale. (German) Math. Ann. 198, 99–122 (1972)
S. Laruelle, C.A. Lehalle, Optimal posting price of limit orders: learning by trading. arXiv preprint arXiv:1112.2397 (2011)
M. Ledoux, M. Talagrand, Probability in Banach Spaces. Isoperimetry and Processes, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 23 (Springer, Berlin, 1991), 480p
H. Luschgy, G. Pagès, Expansions for Gaussian processes and Parseval frames. Electron. J. Probab. 14(42), 1198–1221 (2009)
L.D. Pitt, Positively correlated normal variables are associated. Ann. Probab. 10(2), 496–499 (1982)
D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293 (Springer, Berlin, 1999), 560p
K.I. Sato, Lévy Distributions and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics (Cambridge University Press, London, 1999), 486p (first japanese edition in 1990)
G. Zbaganu, M. Radulescu, Trading prices when the initial wealth is random. Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci. 10(1), 1–8 (2009)
Acknowledgements
The author thanks the anonymous referee for constructive suggestions and F. Panloup for his help.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Pagès, G. (2013). Functional Co-monotony of Processes with Applications to Peacocks and Barrier Options. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLV. Lecture Notes in Mathematics(), vol 2078. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00321-4_15
Download citation
DOI: https://doi.org/10.1007/978-3-319-00321-4_15
Published:
Publisher Name: Springer, Heidelberg
Print ISBN: 978-3-319-00320-7
Online ISBN: 978-3-319-00321-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)