- Original article
- Open Access
On the one-sided maximum of Brownian and random walk fragments and its applications to new exotic options called “meander option”
© Fujita et al.; licensee Springer 2014
Received: 28 March 2014
Accepted: 9 April 2014
Published: 20 September 2014
We consider some distributions of one sided maxima of excursions and related variables for standard random walk and Brownian motion. We propose some new exotic options called meander options related to one of the fragments: the meander. We discuss the prices of meander options in a Black-Scholes market.
AMS Subject classification 60J55; 60J65; 91B28
1.1 Two-sided maxima for BW and RW
For , and is independent of BM, whereas in the RW case for , and θ∼ Geom (1−q) i.e: P(θ=k)=(1−q)q k for k) is independent of RW.
In this paper, instead of two-sided maxima, we shall consider one sided maxima for these fragments and investigate their distributions.
1.2 New exotic options called “Meander Option”
Payoff of “meander lookback call option” where and S t is a stock value process, .
The financial meaning of meander lookback option is the following: If we consider a usual lookback option (with payoff: the price of this option is sometimes extremely high. So partial lookback option (with payoff: where is considered and sometimes traded. Meander lookback option is one example of this partial lookback option with closed price formula.
1.3 Self-explanatory tables for computations
We now present our results in the form of two self-explanatory Tables.
1.4 Organisation of our paper
A list of interesting maxima
A list of joint distributions
Computations of distributions for the six maxima
Notation: For clarity, we write: P(Γ||Λ) for and P(Γ||X=x) for the conditional law of Γ, given X.
Computations of joint distributions
where we put , and X∼Y∼E x p(λ), X and Y are independent.
In the following section, we state applications of these exact computations to price some exotic options which we call “Meander Options”.
Price of some meander options
4.1 Option price at independent exponential time
If Y is of the form ϕ(F T ), instead of fixed time T, it may be more convenient to work at time θ, an independent exponential time, because using such θ often makes expressions simpler than at fixed time T.
There are 2 ways to access such results.
to obtain the law of F t ;
in fact, very often for this, it is simpler to consider F θ , θ∼Exp(λ), and to invert the Laplace transform to get the law of F t . Then, compute E(ϕ(F t )) for the particular ϕ of interest.
- b)second attitude: Start directly with(33)
In fact, there is the commutative diagram:
which indicates that we may use either route from NW to SE.
Generally we get that where we used that for .
We get the usual Black-Scholes formula by inverting the above with respect to λ.
4.2 Price of meander lookback option
In the following, we calculate the above in two cases: a) and b)
For call option i.e. f(x)=(x−K)+, we obtain that by some elementary calculation,
Especially, if σ2=2r, the price equals .
b) when , the price equals: .
Prof. Marc Yor passed away suddenly on January 9 2014. He brought so many gifts to our mathematics. We will never forget him.
- Bertoin J., Fujita T., Roynette B., Yor M.: On a particular class of self-decomposable random variables: the durations of a Bessel excursion straddling an independent exponential time. Prob. Math. Stat. 2006, 26 (2): 315-366.MathSciNetMATHGoogle Scholar
- Biane P., Pitman J., Yor M.: Probability laws related to Riemann zeta and theta functions. Bull. AMS. 2001, 38: 435-469. 10.1090/S0273-0979-01-00912-0.View ArticleMathSciNetMATHGoogle Scholar
- Fujita T., Yor M.: On the remarkable distributions of maxima of some fragments of the standard reflecting random walk and Brownian Motion. Prob. Math. Stat. 2007, 27 (1): 89-104.MathSciNetMATHGoogle Scholar
- Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, Third edition. Springer Velrag (2005)Google Scholar
- Yor M., Chesney M., Geman H., Jeanblanc-Picqué M.: Some combinations of Asian, Parisian and barrier options. Mathematics of derivative securities. Edited by: Dempster MH., Pliska SR. 1997, Cambridge University Press, UKGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.