# Nonexpansive maps and option pricing theory

Kybernetika (1998)

• Volume: 34, Issue: 6, page [713]-724
• ISSN: 0023-5954

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## Abstract

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The famous Black–Sholes (BS) and Cox–Ross–Rubinstein (CRR) formulas are basic results in the modern theory of option pricing in financial mathematics. They are usually deduced by means of stochastic analysis; various generalisations of these formulas were proposed using more sophisticated stochastic models for common stocks pricing evolution. In this paper we develop systematically a deterministic approach to the option pricing that leads to a different type of generalisations of BS and CRR formulas characterised by more rough assumptions on common stocks evolution (which are therefore easier to verify). On the other hand, this approach is more elementary, because it uses neither martingales nor stochastic equations.

## How to cite

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Kolokoltsov, Vassili N.. "Nonexpansive maps and option pricing theory." Kybernetika 34.6 (1998): [713]-724. <http://eudml.org/doc/33400>.

@article{Kolokoltsov1998,
abstract = {The famous Black–Sholes (BS) and Cox–Ross–Rubinstein (CRR) formulas are basic results in the modern theory of option pricing in financial mathematics. They are usually deduced by means of stochastic analysis; various generalisations of these formulas were proposed using more sophisticated stochastic models for common stocks pricing evolution. In this paper we develop systematically a deterministic approach to the option pricing that leads to a different type of generalisations of BS and CRR formulas characterised by more rough assumptions on common stocks evolution (which are therefore easier to verify). On the other hand, this approach is more elementary, because it uses neither martingales nor stochastic equations.},
author = {Kolokoltsov, Vassili N.},
journal = {Kybernetika},
keywords = {option pricing; stocks pricing evolution; Black-Scholes formula; option pricing; stocks pricing evolution; Black-Scholes formula},
language = {eng},
number = {6},
pages = {[713]-724},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Nonexpansive maps and option pricing theory},
url = {http://eudml.org/doc/33400},
volume = {34},
year = {1998},
}

TY - JOUR
AU - Kolokoltsov, Vassili N.
TI - Nonexpansive maps and option pricing theory
JO - Kybernetika
PY - 1998
PB - Institute of Information Theory and Automation AS CR
VL - 34
IS - 6
SP - [713]
EP - 724
AB - The famous Black–Sholes (BS) and Cox–Ross–Rubinstein (CRR) formulas are basic results in the modern theory of option pricing in financial mathematics. They are usually deduced by means of stochastic analysis; various generalisations of these formulas were proposed using more sophisticated stochastic models for common stocks pricing evolution. In this paper we develop systematically a deterministic approach to the option pricing that leads to a different type of generalisations of BS and CRR formulas characterised by more rough assumptions on common stocks evolution (which are therefore easier to verify). On the other hand, this approach is more elementary, because it uses neither martingales nor stochastic equations.
LA - eng
KW - option pricing; stocks pricing evolution; Black-Scholes formula; option pricing; stocks pricing evolution; Black-Scholes formula
UR - http://eudml.org/doc/33400
ER -

## References

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1. Baccelli F., Cohen G., Olsder G., Quadrat J.-P., Synchronisation and Linearity: An Algebra for Discrete Event Systems, Wiley, New York 1992 MR1204266
2. Cox J. C., Ross S. A., Rubinstein M., 10.1016/0304-405X(79)90015-1, J. Financial Economics 7 (1979), 229–263 (1979) Zbl1131.91333DOI10.1016/0304-405X(79)90015-1
3. (Ed.) J. Gunawardena, Proceedings of the International Workshop “Idempotency”, Bristol 1994, Cambridge Univ. Press, Cambridge 1998 MR1608365
4. Kolokoltsov V. N., A Formula for Option Prices on a Market with Unknown Volatility, Research Report No. 9/96, Dep. Math. Stat. and O. R., Nottingham Trent University 1996
5. Kolokoltsov V. N., Maslov V. P., Idempotent Analysis and its Applications, Kluwer Academic Publishers, Dordrecht 1997 Zbl0941.93001MR1447629
6. Lions T., 10.1080/13504869500000007, Appl. Math. Finance 2 (1995), 117–133 (1995) DOI10.1080/13504869500000007
7. McEneaney W. M., 10.1287/moor.22.1.202, Math. Oper. Research 22 (1997), 202–221 (1997) Zbl0871.90010MR1436580DOI10.1287/moor.22.1.202

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