A note on uniqueness of clearing prices in financial systems
Summary (2 min read)
1 Introduction
- In the aftermath of the financial crisis in 2008 the delicate ways the players in the financial industries are intertwined is seen as the main source of the world wide spread of the shock caused by the subprime mortgage crises.
- The major lesson for the architecture of the financial network is that it cannot only be seen as a means by which institutions and firms may diversify their risk exposures, but that instead it may also be the main cause for the amplification of risk.
- More specifically, given such payment scheme, there are two types of nodes – the ones with a positive net value who will be able to pay all their liabilities and those with a negative net value that cannot.
- The analysis for other monotonic bankruptcy rules is similar as the induced games also show strategic complementarities.
- The question of uniqueness of clearing prices is also addressed by Csósak and Herings (2018), who present a discrete model that allows for decentralized clearing of the financial system.
2.1 Mathematical prerequisities
- Let Rn denote the n-dimensional Euclidean vector space.
- Special vector is the zero vector 0 with all zero coordinates.
2.2 Bankruptcy rules
- 1Here I chose to use the term bankruptcy problem, but in fact the rationing problems as in Moulin (2002) or taxation problems in Young (1988) are of the same mathematical structure.
- Well-known in the literature on taxation problems (see Young (1988), Lambert and Naughton (2009)) is the class of strictly monotonic rules which are referred to as equal sacrifice rules.
- Basically, strong monotonicity rules out a kind of exotic rules which are strictly monotonic and do allow for zero derivatives.
2.3 The economic model
- Where the connections or relations of agents within the network are shaped through the nominal liabilities an agent has to other agents in the system.the authors.
- Let τ ∈ Rn+ be the vector that summarizes the total nominal obligations of the agents in the system, i.e., for i ∈ N let τi := n∑ j=1 Lij. (4) This total obligation vector τ summarizes agent-wise the payment levels required to satisfy all the contractual liabilities in the network.
- The authors will assume that all liabilities have the same maturity date at which they become due and should be paid for.
- On the other hand, each agent i has some justified claim Lji on pj.
- This means that from payment pj by agent j, agent i obtains ri(Lj, pj).
2.4 Clearing payment vectors for a financial system
- Below the authors will focus on the question whether payment vectors exist, that see to a clearing of the financial system such that two minimal requirements are satisfied.
- First the authors will require from a payment vector that it expresses the idea of limited liability: no agent should pay more than the total of his cash inflow.
- This holds for the partially ordered set [0, τ ].
- For part (b) let p′ be any clearing vector.
3 Characterizing the clearing prices
- Eisenberg and Noe (2001) characterize vectors of clearing prices using the notion of a surplus set, i.e., a set of agents S with no external obligations and a positive aggregate operation cash flow: Definition 2 Then Lemma 1 shows O(i) should contain an agent with positive equity.
- If the set of defaulting agents is larger under pk+1 than under pk, then it must be that some agents pay their obligations in full under pk and default under pk+1, and the other agents either default or not both under pk+1 and pk.
- And this concludes their proof by induction as also the authors have shown that {pj} is a weakly decreasing sequence.
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"A note on uniqueness of clearing pr..." refers methods in this paper
...Consider for example the model including defaulting costs by Rogers and Veraart (2013), or the model where financial institutes reinsure themselves through credit default swaps as in Schuldenzucker et al. (2016) (see also Elliott et al. (2014))....
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"A note on uniqueness of clearing pr..." refers background in this paper
...Other measures of financial instability and assessment of systemic risk are found in Elsinger et al. (2006), Acemoglu et al. (2015), Battiston et al. (2012)....
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369 citations
"A note on uniqueness of clearing pr..." refers methods in this paper
...Consider for example the model including defaulting costs by Rogers and Veraart (2013), or the model where financial institutes reinsure themselves through credit default swaps as in Schuldenzucker et al. (2016) (see also Elliott et al. (2014))....
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320 citations
"A note on uniqueness of clearing pr..." refers background in this paper
...For overviews, see Thomson (2015) and Moulin (2002)....
[...]
...(1)Here I chose to use the term bankruptcy problem, but in fact the rationing problems as in Moulin (2002) or taxation problems in Young (1988) are of the same mathematical structure....
[...]
...(1)Here I chose to use the term bankruptcy problem, but in fact the rationing problems as in Moulin (2002) or taxation problems in Young (1988) are of the same mathematical structure. Solution concepts within these fields of the literature on distributive justice can usually easily be transfered and interpreted. For overviews, see Thomson (2015) and Moulin (2002)....
[...]
...1Here I chose to use the term bankruptcy problem, but in fact the rationing problems as in Moulin (2002) or taxation problems in Young (1988) are of the same mathematical structure....
[...]