A remark on the set of arbitrage‐free prices in a multi‐period model. (21st March 2013)

• Record Type: Journal Article
• Authors: Cornet, Bernard; Ranjan, Abhishek
• Is Part Of: International journal of economic theory. Volume 9:Number 1(2013:Mar.)
• Page Start: 35
• Page End: 43
• Publication Date: 2013-03-21
• Subjects: Economics -- Periodicals; 330.01
• DOI: 10.1111/j.1742-7363.2013.12004.x
• Languages: English
• ISSNs: 1742-7355

Abstract

<abstract abstract-type="main" xml:lang="en"> <title>Abstract</title> <sec id="ijet12004-sec-0001" sec-type="section"> <p>We study the convexity property of the set <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg1x5p1dwn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="urn:x-wiley:17427355:media:ijet12004:ijet12004-math-0001" display="inline" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi>ℱ</mml:mi></mml:msub></mml:mrow></mml:math> of arbitrage‐free prices of a multi‐period financial structure <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg1x5p1dv3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="urn:x-wiley:17427355:media:ijet12004:ijet12004-math-0002" display="inline" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ℱ</mml:mi></mml:math>. The set of arbitrage‐free prices is shown to be a convex cone under conditions on the financial structure <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg1x5p1dtj" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="urn:x-wiley:17427355:media:ijet12004:ijet12004-math-0003" display="inline" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ℱ</mml:mi></mml:math> that hold in particular for short‐lived assets. Furthermore, we provide examples of equivalent financial structures <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg1x5p1ds0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="urn:x-wiley:17427355:media:ijet12004:ijet12004-math-0004" display="inline" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ℱ</mml:mi></mml:math> and <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg1x5p1drf" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="urn:x-wiley:17427355:media:ijet12004:ijet12004-math-0005" display="inline" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>ℱ</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math> such that <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg1x5p1dqw" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="urn:x-wiley:17427355:media:ijet12004:ijet12004-math-0006" display="inline" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi>ℱ</mml:mi></mml:msub></mml:mrow></mml:math> is a convex cone, but <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg1x5p1dpb" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="urn:x-wiley:17427355:media:ijet12004:ijet12004-math-0007" display="inline" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:msup><mml:mi>ℱ</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:msub></mml:mrow></mml:math> is neither convex nor a cone.</p> </sec> </abstract>