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On Esscher Transforms in Discrete Finance Models

Published online by Cambridge University Press:  29 August 2014

Hans Bühlmann ,
Freddy Delbaen ,
Paul Embrechts  and
Albert N. Shiryaev
Hans Bühlmann*
Affiliation:
Department of Mathematics, ETH Zürich
Freddy Delbaen*
Affiliation:
Department of Mathematics, ETH Zürich
Paul Embrechts*
Affiliation:
Department of Mathematics, ETH Zürich
Albert N. Shiryaev*
Affiliation:
Steklov Mathematical Institute, Moscow
*
Department of Mathematics, ETH Zürich, CH – 8092 Zürich, Switzerland
Department of Mathematics, ETH Zürich, CH – 8092 Zürich, Switzerland
Department of Mathematics, ETH Zürich, CH – 8092 Zürich, Switzerland
Steklov Mathematical Institute, Ulitza Vavilova 42, Moscow 117966, GSP-1, Russia
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The object of our study is

where each Sn is a m-dimensional stochastic (real valued) vector, i.e.

denned on a probability space (Ω, , P) and adapted to a filtration (n)0≤n≤N with 0 being the σ-algebra consisting of all null sets and their complements. In this paper we interpret as the value of some financial asset k at time n.

Remark: If the asset generates dividends or coupon payments, think of as to include these payments (cum dividend process). Think of dividends as being reinvested immediately at the ex-dividend price.

Definition 1

(a) A sequence of random vectors

where

is called a trading strategy. Since our time horizon ends at time N we must always have ϑN ≡ 0.

The interpretation is obvious: stands for the number of shares of asset k you hold in the time interval [n,n + 1). You must choose ϑn at time n.

(b) The sequence of random variables

where Sn stands for the payment stream generated by ϑ (set ϑ−1 ≡ 0).

Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 28 , Issue 2 , November 1998 , pp. 171 - 186
Copyright
Copyright © International Actuarial Association 1998

References

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