No-Arbitrage Principle in Conic Finance

Mehdi Vazifedan and Qiji Jim Zhu
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Mehdi Vazifedan: Department of Mathematics, Western Michigan University, 1903 West Michigan Avenue, Kalamazoo, MI 49008, USA
Qiji Jim Zhu: Department of Mathematics, Western Michigan University, 1903 West Michigan Avenue, Kalamazoo, MI 49008, USA

Risks, 2020, vol. 8, issue 2, 1-34

Abstract: In a one price economy, the Fundamental Theorem of Asset Pricing (FTAP) establishes that no-arbitrage is equivalent to the existence of an equivalent martingale measure. Such an equivalent measure can be derived as the normal unit vector of the hyperplane that separates the attainable gain subspace and the convex cone representing arbitrage opportunities. However, in two-price financial models (where there is a bid–ask price spread), the set of attainable gains is not a subspace anymore. We use convex optimization, and the conic property of this region to characterize the “no-arbitrage” principle in financial models with the bid–ask price spread present. This characterization will lead us to the generation of a set of price factor random variables. Under such a set, we can find the lower and upper bounds (supper-hedging and sub-hedging bounds) for the price of any future cash flow. We will show that for any given cash flow, for which the price is outside the above range, we can build a trading strategy that provides one with an arbitrage opportunity. We will generalize this structure to any two-price finite-period financial model.

Keywords: conic finance; convex optimization; arbitrage pricing (search for similar items in EconPapers)
JEL-codes: C G0 G1 G2 G3 K2 M2 M4 (search for similar items in EconPapers)
Date: 2020
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