No arbitrage
Freedom from arbitrage is understood to mean the lack of any possibility of (economic) arbitrage . This term was coined especially for the financial markets . Through international, electronic trading in these markets and the rapid, worldwide dissemination of new information, market participants adjust the prices of their products so quickly that arbitrage opportunities usually only exist for very short periods of time. As a result, the transaction costs are usually higher than the profits that can be achieved through arbitrage.
meaning
The freedom from arbitrage is one of the basic assumptions of modern financial mathematics : In equilibrium models, prices are determined as endogenous variables; H. the prices are adjusted depending on the supply and demand quantities until the market is in equilibrium. This adjustment process has no effect on the prices of other goods. In 1980 years, the shortcomings of these models were clear when the based on them yield curve for fixed-income derivatives did not comply with the actual curves and were unusable for trade, as they are not the law of the unit price ( Law Of One Price , in short: LOOP ) corresponded.
Arbitrage -free (also English no-arbitrage or arbitrage-free ) models, however, determine the prices exogenously : The market prices flow into the model directly, and the yield curves developed from them correspond to reality. The first rate structure-compliant valuations were made possible by the work of Thomas Ho and Sang-Bin Lee and later David Heath , Robert Jarrow and Andrew Morton . All models used in practice today to value derivatives are arbitrage-free.
Formal representation
Formally, freedom from arbitrage can be described as a condition:
- There is no portfolio with a value of zero at the point in time that has a certainly non-negative value and a positive value with a positive probability.
Definitions
The following table shows the different definitions of freedom from arbitrage, which, when read from left to right, become stronger in their requirements. No free lunch means that there cannot be a position in which there is safe consumption today without any consideration today or tomorrow. No free lottery, that there cannot be a position in which you have a chance of winning without investing money.
| Definitions of No Arbitrage | ||
|---|---|---|
| Type 1: No free lottery ("no free lottery ticket with a positive chance of winning") | Type 2: No free lunch ("no free lunch") | Law of the unit price |
| A portfolio with a positive profit probability without net capital employed at time t = 0 is not possible. | A portfolio with a certain net inflow in t = 0 without future payment obligations is not possible. | Two instruments with identical future cash flows must have the same price today. |
literature
- Hansjörg Albrecher, Andreas Binder, Philipp Mayer: Introduction to financial mathematics. Birkhäuser, Basel / Boston / Berlin 2009, ISBN 978-3-7643-8783-9 , Chapter III.
- Freddy Delbaen, Walter Schachermayer: The Mathematics of Arbitrage. Springer, Berlin / Heidelberg 2010, ISBN 978-3-642-06030-4 .
- Hans Föllmer , Alexander Schied: Stochastic Finance. An Introduction in Discrete Time. 2. revised and extended edition. de Gruyter, Berlin et al. 2004, ISBN 3-11-018346-3 ( De Gruyter Studies in Mathematics ).
Individual evidence
- ↑ Thomas SY Ho, Sang-Bin Lee: Term structure movements and pricing interest rate contingent claims. In: Journal of Finance. 41, 5, 1986, ISSN 0022-1082 , pp. 1011-1029, online (PDF; 533 kB) ( Memento of the original from March 9, 2013 in the Internet Archive ) Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. .
- ^ David Heath, Robert Jarrow, Andrew Morton: Bond pricing and the term structure of interest rates. In: Econometrica. 60, 1, 1992, ISSN 0012-9682 , pp. 77-105 online (PDF; 442 kB) .
