Filtration shrinkage, the structure of deflators, and failure of market completeness
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Filtration shrinkage, the structure of deflators, and failure of market completeness Constantinos Kardaras1 · Johannes Ruf2
Received: 21 December 2019 / Accepted: 2 June 2020 © The Author(s) 2020
Abstract We analyse the structure of local martingale deflators projected on smaller filtrations. In a general continuous-path setting, we show that the local martingale parts in the multiplicative Doob–Meyer decomposition of projected local martingale deflators are themselves local martingale deflators in the smaller information market. Via use of a Bayesian filtering approach, we demonstrate the exact mechanism of how updates on the possible class of models under less information result in the strict supermartingale property of projections of such deflators. Finally, we demonstrate that these projections are unable to span all possible local martingale deflators in the smaller information market, by investigating a situation where market completeness is not retained under filtration shrinkage. Keywords Bayes rule · Brownian motion · Deflator · Lévy transform · Local martingale · Market completeness · Predictable representation property Mathematics Subject Classification (2010) 60G44 · 60H10 · 91G20 JEL Classification C11 · G13 · G14
B C. Kardaras
[email protected] J. Ruf [email protected]
1
Department of Statistics, London School of Economics and Political Science, 10 Houghton St., London WC2A 2AE, UK
2
Department of Mathematics, London School of Economics and Political Science, 10 Houghton St., London WC2A 2AE, UK
C. Kardaras, J. Ruf
1 Introduction Optional projections of martingales onto smaller filtrations retain the martingale property; for the class of local martingales, this preservation may fail. For instance, the projection of a nonnegative local martingale can only be guaranteed to be a supermartingale in the smaller filtration, but might fail to be a local martingale; see Stricker [41] and Föllmer and Protter [14]. Positive local martingales appear naturally as deflators in arbitrage theory. (See Sect. 2 for definitions and a review of classical concepts in the theory of noarbitrage.) Consider two nested, right-continuous filtrations F ⊆ G and a continuous and F-adapted process S, having the interpretation of the discounted price of a financial asset. Then the existence of a strictly positive G-local martingale Y such that Y S is also a G-local martingale is equivalent to the so-called absence of arbitrage of the first kind. If no such arbitrage opportunities are possible under G, then the same is true under the smaller filtration F; we refer to Sect. 2 for a rigorous argument for this assertion. Hence, there must exist an F-local martingale L such that LS is an F-local martingale. Let now Y G and Y F denote the set of all G-adapted and F-adapted such local martingale deflators, respectively. The above no-arbitrage considerations yield the implication Y G = ∅
=⇒
Y F = ∅.
It is natural to ask at this point if there is a direct way to construct an element of Y F from a given Y ∈ Y G . The optiona
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