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Walter Schachermayer

Asymptotic Arbitrage in Non-Complete Large Financial Markets.

I. Klein, W. Schachermayer
Theory of Probability and its Applications, Vol. 41 (1996), No. 4, pp. 927-934.

Abstract:

Kabanov and Kramkov introduced the notion of "large financial markets". Instead of considering--as usual in mathematical finance--a stochastic stock price process $S$ based on a filtered probability space $(\Omega,\Cal F,(\Cal F_t)_{t\in I},\Bbb P)$ one considers a sequence $(S^n)_{n\ge1}$ of such processes based on a sequence $(\Omega^n,\Cal F^n,(\Cal F^n_t)_{t\in I^n},\Bbb P^n)_{n\ge1}$ of filtered probability spaces. The interpretation is that an investor can invest not only in one stock exchange but in several (in the model countably many) stock exchanges.

The usual notion of arbitrage then may be interpreted by "asymptotic" arbitrage concepts, where it is essential to distinguish between two different kinds introduced by Kabanov and Kramkov. If for each $n\in\Bbb N$ the market is complete i.e., there is exactly one local martingale measure $Q^n$ for the process $S^n$ on $\Cal F^n$ which is equivalent to $\Bbb P^n$, then Kabanov and Kramkov showed that contiguity of $(\Bbb P^n)_{n\ge1}$ with respect to $(Q^n)_{n\ge1}$ (respectively vice versa) is equivalent to the absence of asymptotic arbitrage of first (respectively second) kind.

In the present paper we extend this result to the non-complete case i.e., where for each $n\in\Bbb N$ the set of equivalent local martingale measures for the process $S^n$ is non-empty but not necessarily a singleton. The question arises whether we can extend the theorem of Kabanov and Kramkov to this situation by selecting a proper sequence $(Q^n)_{n\ge1}$ of equivalent local martingale measures.

It turns out that the theorem characterising asymptotic arbitrage of first kind may be directly extended to this setting while for the theorem characterising asymptotic arbitrage of second kind some modifications are needed. We also provide an example showing that these modifications cannot be avoided.

Keywords:

arbitrage, asymptotic arbitrage, contiguity of measures, equivalent martingale measure, free lunch, free lunch with vanishing risk, large financial market

Preprints:

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