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2015 | OriginalPaper | Chapter

The Joint Law of the Extrema, Final Value and Signature of a Stopped Random Walk

Authors : Moritz Duembgen, L. C. G. Rogers

Published in: In Memoriam Marc Yor - Séminaire de Probabilités XLVII

Publisher: Springer International Publishing

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Abstract

A complete characterization of the possible joint distributions of the maximum and terminal value of uniformly integrable martingale has been known for some time, and the aim of this paper is to establish a similar characterization for continuous martingales of the joint law of the minimum, final value, and maximum, along with the direction of the final excursion. We solve this problem completely for the discrete analogue, that of a simple symmetric random walk stopped at some almost-surely finite stopping time. This characterization leads to robust hedging strategies for derivatives whose value depends on the maximum, minimum and final values of the underlying asset.

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If either of p _± is zero, then the inequalities (13), (14) have to be understood in cross-multiplied form, when they state vacuously that 0 ≤ 0.

There is no reason why v need be a multiple of h, but this does not matter; if s = +, say, we shall use the Brownian motion living in the original probability space, starting at b and run until it first hits either the upper barrier b + h or the lower barrier, which will be randomized, taking value v ₊ with suitably-chosen probability \(\theta\), otherwise taking value − a − h.

We provide details of what happens if \(S_{\tau _{n}} =\xi _{\tau _{n}}\); the treatment of the case \(I_{\tau _{n}} =\xi _{\tau _{n}}\) is analogous.

We shall establish in the inductive proof that ψ _± are non-negative.

The functions ψ _± are defined in terms of m by (4), (5), (8), (9).

The papers ([7][1][2]) give examples of this kind. The recent paper of Galichon, Henry-Labordère and Touzi [6] strengthens [1] to multiple time points.

Let us suppose that the expiry is 1.

As before, when the time subscript of a process is omitted, we understand it to be 1.

This linear programming approach to the problem is also used in [4].

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Title: The Joint Law of the Extrema, Final Value and Signature of a Stopped Random Walk
Authors: Moritz Duembgen
L. C. G. Rogers
Publisher: Springer International Publishing
Book: In Memoriam Marc Yor - Séminaire de Probabilités XLVII
Print ISBN: 978-3-319-18584-2

Electronic ISBN: 978-3-319-18585-9

Copyright Year: 2015
DOI: https://doi.org/10.1007/978-3-319-18585-9_15