Editing Systemic risk (section)

==Valuation of assets and derivatives under systemic risk==

===Inadequacy of classic valuation models===
One problem when it comes to the valuation of derivatives, debt, or equity under systemic risk is that financial interconnectedness has to be modelled. One particular problem is posed by closed valuations chains, as exemplified here for four firms A, B, C, and D:

:B might hold shares of A, C holds some debt of B, D owns a derivative issued by C, and A owns some debt of D.<ref name="Fischer (2014b)">{{cite journal|last=Fischer|first=Tom|title=Valuation in the structural model of systemic interconnectedness|journal=Presentation at the Frankfurt MathFinance Colloquium, November 27, 2014|year=2014|url=http://www.statistik-mathematik.uni-wuerzburg.de/fileadmin/10040800/user_upload/Fischer/Frankfurt_Fischer_handout.pdf}}</ref>

For instance, the share price of A could influence all other asset values, including itself.

====The Merton (1974) model====
Situations as the one explained earlier, which are present in mature financial markets, cannot be modelled within the single-firm [[Merton model]],<ref name="Merton (1974)">{{cite journal|last=Merton|first=R.C.|title=On the pricing of corporate debt: the risk structure of interest rates|journal=Journal of Finance|year=1974|volume=29|issue=2|pages=449–470|doi=10.1111/j.1540-6261.1974.tb03058.x|doi-access=free|hdl=1721.1/1874|hdl-access=free}}
</ref> but also not by its straightforward extensions to multiple firms with potentially correlated assets.<ref name="Fischer (2014b)"/> To demonstrate this, consider two financial firms, <math>i = 1, 2</math>, with limited liability, which both own system-exogenous assets of a value <math>a_i \geq 0</math> at a maturity <math>T \geq 0</math>, and which both owe a single amount of zero coupon debt <math>d_i \geq 0</math>, due at time <math>T</math>. "System-exogenous" here refers to the assumption, that the business asset <math>a_i</math> is not influenced by the firms in the considered financial system.
In the classic single firm Merton model,<ref name="Merton (1974)"/> it now holds at maturity for the equity <math>s_i \geq 0</math> and for the recovery value <math>r_i \geq 0</math> of the debt, that

:<math>r_i = \min\{d_i, a_i\}</math>

and

:<math>s_i = (a_i - d_i)^+.</math>

Equity and debt recovery value, <math>s_i</math> and <math>r_i</math>, are thus uniquely and immediately determined by the value <math>a_i</math> of the exogenous business assets. Assuming that the <math>a_i</math> are, for instance, defined by a Black-Scholes dynamic (with or without correlations), risk-neutral no-arbitrage pricing of debt and equity is straightforward.

====Non-trivial asset value equations====
Consider now again two such firms, but assume that firm 1 owns 5% of firm two's equity and 20% of its debt. Similarly, assume that firm 2 owns 3% of firm one's equity and 10% of its debt. The equilibrium price equations, or liquidation value equations,<ref name="Fischer (2014a)">{{cite journal|last=Fischer|first=Tom|title=No-arbitrage pricing under systemic risk: Accounting for cross-ownership|journal=Mathematical Finance|year=2014|volume=24|issue=1|pages=97–124|doi=10.1111/j.1467-9965.2012.00526.x|arxiv=1005.0768|s2cid=153225655}}</ref> at maturity are now given by

:<math>r_1 = \min\{d_1, a_1 + 0.05s_2 + 0.2r_2\}</math>
:<math>r_2 = \min\{d_2, a_2 + 0.03s_1 + 0.1r_1\}</math>
:<math>s_1 = (a_1  + 0.05s_2 + 0.2r_2 - d_1)^+</math> 
:<math>s_2 = (a_2 + 0.03s_1 + 0.1r_1 - d_2)^+.</math>

This example demonstrates, that systemic risk in the form of financial interconnectedness can already lead to a non-trivial, non-linear equation system for the asset values if only two firms are involved.

====Over- and underestimation of default probabilities====
It is known that modelling credit risk while ignoring cross-holdings of debt or equity can lead to an under-, but also an over-estimation of default probabilities.<ref>{{cite journal |last=Karl|first=S.|author2=Fischer, T. |year=2014|title=Cross-ownership as a structural explanation for over- and underestimation of default probability|journal=Quantitative Finance|volume=14 |issue=6 |pages=1031–1046 (Published online: 18 Nov 2013)|doi=10.1080/14697688.2013.834377|arxiv=1301.6069|citeseerx=10.1.1.768.3101|s2cid=155177007}}</ref> The need for proper structural models of financial interconnectedness in quantitative risk management – be it in research or practice – is therefore obvious.

===Structural models under financial interconnectedness===
The first authors to consider structural models for financial systems where each firm could own the debt of other firms were Eisenberg and Noe in 2001.<ref>{{cite journal |last=Eisenberg|first=L.|author2=Noe, T.H. |year=2001|title=Systemic Risk in Financial Systems|journal=Management Science|volume=47 |issue=2 |pages=236–249|doi=10.1287/mnsc.47.2.236.9835}}</ref> Suzuki (2002) extended the analysis of interconnectedness by modeling the cross ownership of both debt and equity claims.<ref name="Suzuki (2002)2">{{cite journal|last=Suzuki|first=T.|year=2002|title=Valuing Corporate Debt: The Effect of Cross-Holdings of Stock and Debt|url=http://www.orsj.or.jp/~archive/pdf/e_mag/Vol.45_2_123.pdf|journal=Journal of the Operations Research Society of Japan|volume=45|issue=2|pages=123–144|doi=10.15807/jorsj.45.123|doi-access=free}}</ref> Building on Eisenberg and Noe (2001), Cifuentes, Ferrucci, and Shin (2005) considered the effect of costs of default on network stability.<ref name="Cifuentes, Ferrucci, and Shin (2005)2">{{cite journal|last=Cifuentes|first=R.|author2=Ferrucci, G.|author3=Shin, H. S.|year=2005|title=Liquidity Risk and Contagion|journal=Journal of the European Economic Association|volume=3|issue=2/3|pages=556–566|doi=10.1162/jeea.2005.3.2-3.556}}</ref> Elsinger's further developed the Eisenberg and Noe (2001) model by incorporating financial claims of differing priority.<ref name="Elsinger (2009)3">{{cite journal|last=Elsinger|first=H.|year=2009|title=Financial Networks, Cross Holdings, and Limited Liability|url=http://www.oenb.at/Publikationen/Volkswirtschaft/Working-Papers/2009/Working-Paper-156.html|journal=Working Paper 156, Oesterreichische Nationalbank, Wien}}</ref>

Acemoglu, Ozdaglar, and Tahbaz-Salehi, (2015) developed a structural systemic risk model incorporating both distress costs and debt claim with varying priorities and used this model to examine the effects of network interconnectedness on financial stability. They showed that, up to a certain point, interconnectedness enhances financial stability. However, once a critical threshold density of connectedness is exceeded, further increases in the density of the financial network propagate risk.<ref name="Acemoglu, Ozdaglar, and Tahbaz-Salehi, (2001)2">{{cite journal|last=Acemoglu|first=D.|author2=Ozdaglar, A.|author3=Tahbaz-Salehi, A.|year=2015|title=Systemic Risk and Stability in Financial Networks|journal=American Economic Review|volume=105|issue=2|pages=564–608|doi=10.1257/aer.20130456|hdl=1721.1/100979|s2cid=7447939|hdl-access=free}}</ref>

Glasserman and Young (2015) applied the  Eisenberg and Noe (2001) to modelling the effect of shocks to banking networks. They develop general bounds for the effects of network connectivity on default probabilities. In contrast to most of the structural systemic risk literature, their results are quite general and do not require assuming a specific network architecture or specific shock distributions.<ref name="Glasserman and Young(2015)3">{{cite journal|last=Glasserman|first=P.|author2=Young, H. P.|year=2015|title=How Likely Is Contagion in Financial Networks?|journal=Journal of Banking & Finance|volume=50|pages=383–399|doi=10.1016/j.jbankfin.2014.02.006|url=http://www.economics.ox.ac.uk/materials/papers/12619/paper642.pdf}}</ref>

===Risk-neutral valuation: price indeterminacy and open problems===
Generally speaking, risk-neutral pricing in structural models of financial interconnectedness requires unique equilibrium prices at maturity in dependence of the exogenous asset price vector, which can be random. While financially interconnected systems with debt and equity cross-ownership without derivatives are fairly well understood in the sense that relatively weak conditions on the ownership structures in the form of ownership matrices are required to warrant uniquely determined price equilibria,<ref name="Fischer (2014b)"/><ref name="Suzuki (2002)2"/><ref name="Elsinger (2009)3"/> the Fischer (2014) model needs very strong conditions on derivatives – which are defined in dependence on any other liability of the considered financial system – to be able to guarantee uniquely determined prices of all system-endogenous liabilities. Furthermore, it is known that there exist examples with no solutions at all, finitely many solutions (more than one), and infinitely many solutions.<ref name="Fischer (2014b)"/><ref name="Fischer (2014a)"/> At present, it is unclear how weak conditions on derivatives can be chosen to still be able to apply risk-neutral pricing in financial networks with systemic risk. It is noteworthy, that the price indeterminacy that evolves from multiple price equilibria is fundamentally different from price indeterminacy that stems from market incompleteness.<ref name="Fischer (2014a)"/>