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Erschienen in: Mathematics and Financial Economics 3/2019

01.01.2019

Bubbles in assets with finite life

verfasst von: Henri Berestycki, Cameron Bruggeman, Regis Monneau, José A. Scheinkman

Erschienen in: Mathematics and Financial Economics | Ausgabe 3/2019

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Abstract

We study the speculative value of a finitely lived asset when investors disagree and short sales are limited. In this case, investors are willing to pay a speculative value for the resale option they obtain when they acquire the asset. Using martingale arguments, we characterize the equilibrium speculative value as a solution to a fixed point problem for a monotone operator \(\mathbb F\). A Dynamic Programming Principle applies and is used to show that the minimal solution to the fixed-point problem is a viscosity solution of a naturally associated (non-local) obstacle problem. Combining the monotonicity of the operator \({\mathbb {F}}\) and a comparison principle for viscosity solutions to the obstacle problem we obtain several comparison of solution results. We also use a characterization of the exercise boundary of the obstacle problem to study the effect of an increase in the costs of transactions on the value of the bubble and on the volume of trade, and in particular to quantify the effect of a small transaction (Tobin) tax.

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Fußnoten
1
See Carlos et al. [4] for the increase in trading volume during the South Sea Bubble, Hong and Stein [10] for the Roaring Twenties, Cochrane [6], Lamont and Thaler [11] and Ofek and Richardson [12] for the internet bubble, and Xiong and Yu [19] for evidence concerning a recent Chinese warrants bubble.
 
2
An exception is Weil [18] that explains a bubble in an asset that pays dividends for a bounded amount of time, by assuming that agents would value the asset-stub after all the dividends have been paid at a constant positive price \({{\hat{p}}}.\) This hypothesis is even less reasonable nowadays, when most traded securities are held as electronic entries.
 
3
These two basic assumptions—heterogeneous beliefs and higher costs of going short—are far from standard in the asset-pricing literature. The existence of differences in beliefs is obvious for the vast majority of market practitioners, but economists have produced a myriad of results showing that “rational” investors cannot agree to disagree. Similarly, there are good economic reasons why investors should have more difficulty going short than going long, but most economic models assume no asymmetry.
 
4
An alternative mathematical theory of bubbles postulates only no-arbitrage. In the simplest case of an asset that pays only a final dividend and complete markets, the price process of the asset, \(S_t,\) is a local martingale under the unique risk-neutral measure Q. The fundamental value of the asset \(S^*_t\) is the expected value (under Q) of the payoff, and necessarily a martingale. The difference between S and \(S^*\) is non-negative and is defined as the bubble. For a summary and further references see Protter [14]. Because it is a no-arbitrage theory, there are no implications concerning trading volume.
 
5
See also Chen and Kohn [5] and Dumas et al. [7]. As it is common in economics, the model of asset price bubbles in this paper deals with a single asset while in reality bubbles typically involve whole classes of assets.
 
6
In Scheinkman and Xiong [16] the focus is also on a minimum solution, because it minimizes the size of the bubble.
 
7
Zariphopoulou [20] and Theorem VIIII.5.1 in Fleming and Soner [8] are early examples of the use of the dynamic programming principle to establish the necessity of viscosity solutions.
 
8
The condition \(\lambda +r>0\) is necessary for the existence of an equilibrium of the associated stationary problem (see Scheinkman and Xiong [16]).
 
9
The assumption that the value of the asset at the terminal date is zero is not essential, what is needed is that at maturity the asset’s value is common knowledge.
 
10
In principle, there could exist more complicate equilibria including non-markovian ones.
 
11
A detailed argument for this and the previous assertion would follow a similar result for discrete-time in Shiryaev [17] (Theorem 21 on page 91), by discretizing the possible stopping times.
 
12
See e.g. Lemma 1.3, Chapter 1 in Peskir and Shiryaev [13].
 
13
See e.g. Theorem VIIII.5.1 in Fleming and Soner [8], Zariphopoulou [20] or Bouchard [3].
 
14
Berestycki et al. [2] shows that the comparison theorem is false for \(c=0\).
 
15
Notice that monotonicity with respect to t and the monotonicity properties of the exercise boundary are not immediate consequences of properties of the operator \( {\mathbb {F}}\).
 
16
In the results in this section increasing should be read as non-decreasing etc.
 
17
See Scheinkman [15] Section 2.2 for examples and references.
 
18
Recall that \({{\mathcal {F}}}_\tau \) is the collection of all \(A \in {{\mathcal {F}}}\) such that \(A \bigcap \{\omega \in \Omega : \tau \le t\} \in {{\mathcal {F}}}_t\), for all \(t \in [0,T]\).
 
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Metadaten
Titel
Bubbles in assets with finite life
verfasst von
Henri Berestycki
Cameron Bruggeman
Regis Monneau
José A. Scheinkman
Publikationsdatum
01.01.2019
Verlag
Springer Berlin Heidelberg
Erschienen in
Mathematics and Financial Economics / Ausgabe 3/2019
Print ISSN: 1862-9679
Elektronische ISSN: 1862-9660
DOI
https://doi.org/10.1007/s11579-018-0233-4

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