Abstract
In this article we study the completion by options of a two-period security market in which the space of marketed securities is a subspace X of \(\mathbb{R}^m\). Although there are important results about the completion (by options) Z of X, the problem of the determination of Z in its general form is still open. In this paper we solve this problem by determining a positive basis of Z. This method of positive bases simplifies the theory of security markets and also answers other open problems of this theory. In the classical papers of this subject, call and put options are taken with respect to the riskless bond 1 of \(\mathbb{R}^m\). In this article we generalize this theory by taking call and put options with respect to different risky vectors u from a fixed vector subspace U of \(\mathbb{R}^m\). This generalization was inspired by certain types of exotic option in finance.
Mathematics Subject Classification (2000): 46B40, 46A35, 91B28, 91B30
Journal of Economic Literature Classification: G190, D520
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References
1. Aliprantis, C.D., Burkinshaw, O. (1985): Positive operators. Academic Press, Orlando, FL
2. Aliprantis, C.D., Tourky, R. (2002): Markets that don't replicate any option. Economics Letters 76, 443–447
3. Arditti, F.D., John, K. (1980): Spanning the state space with options. Journal of Financial and Quantitative Analysis 15, 1–9
4. Brown, D.J., Ross, S.A. (1991): Spanning, valuation and options. Economic Theory 1, 3–12
5. Green, R.C., Jarrow, R.A. (1987): Spanning and completeness in markets with contingent claims. Journal of Economic Theory 41, 202–210
6. John, K. (1981): Efficient funds in a financial market with options: a new irrelevance proposition. The Journal of Finance 36, 685–695
7. LeRoy, S.F., Werner, J. (2001): Principles of financial economics. Cambridge University Press, Cambridge, MA
8. Magill, M., Quinzii, M. (1996): Theory of incomplete markets. I. MIT Press, Cambridge, MA
9. Musiela, M., Rutkowski, M. (1997): Martingale methods in financial modelling. Springer, Berlin
10. Nachman, D.C. (1988): Spanning and completeness with options. The Review of Financial Studies 1, 311–328
11. Polyrakis, I.A. (1996): Finite-dimensional lattice-subspaces of C(Ω) and curves of R n. Transactions of the American Mathematical Society 348, 2793–2810
12. Polyrakis, I.A. (1999): Minimal lattice-subspaces. Transactions of the American Mathematical Society 351, 4183–4203
13. Ross, S.A. (1976): Options and efficiency. The Quarterly Journal of Economics 90, 75–89
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Kountzakis, C., Polyrakis, I.A. The completion of security markets. Decisions Econ Finan 29, 1–21 (2006). https://doi.org/10.1007/s10203-006-0059-z
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DOI: https://doi.org/10.1007/s10203-006-0059-z