Editing Systemic risk (section)

===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.