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On Esscher Transforms in Discrete Finance Models

  • Hans Bühlmann (a1), Freddy Delbaen (a1), Paul Embrechts (a1) and Albert N. Shiryaev (a2)
Extract

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).

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Copyright
Corresponding author
Department of Mathematics, ETH Zürich, CH – 8092 Zürich, Switzerland
Steklov Mathematical Institute, Ulitza Vavilova 42, Moscow 117966, GSP-1, Russia
References
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[1]Borch, K. (1962). Equilibrium in a reinsurance market. Econometrica 30, 424444.
[2]Bühlmann, H. (1984). The general economic premium principle. ASTIN Bulletin 14, 1321.
[3]Duffie, D. (1996). Dynamic Asset Pricing Theory, 2nd edition, Princeton University Press.
[4]Gerber, H.U. and Shiu, Elias S.W. (1994). Option pricing by Esscher transforms. Transactions of the Society of Actuaries, vol. XLVI, pp. 99140.
[5]Rogers, L.C.G. (1994). Equivalent martingale measures and no-arbitrage. Stochastics and Stochastics Reports 51, 4149.
[6]Rogers, L.C.G. (1997). The potential approach to the term structure of interest rates and their foreign exchange rates. Mathematical Finance 7, 157176.
[7]Schachermayer, W. (1992). A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time. Insurance: Mathematics and Economics 11, 249257.
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ASTIN Bulletin: The Journal of the IAA
  • ISSN: 0515-0361
  • EISSN: 1783-1350
  • URL: /core/journals/astin-bulletin-journal-of-the-iaa
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