Quantifying contagion among European banks from the Great Recession to the Pandemic crisis using Network Analysis

Contagion risk can be defined as the probability that the instability of the given institution will spread to other parts of the financial system with a negative effect Smaga [2014].
Contagion risk is based on two assumptions:
• Contagion cannot occur without the initial shock.
• The scale of its transmission between institutions is higher during financial turmoil given the tendency of correlation to become positive during financial turmoil.

The simplest theoretical framework of contagion risk is with the interconnections of credit risk between different financial institutions. It manifests itself according to the scheme of the domino effect. The domino effect is based on the negative impact of the failure of a single financial institution on the entire financial system, causing a chain re- action of further defaults Smaga [2014].
Taking as an example the interconnections in interbank market payments, the spread of the suffering of an insolvent bank to other banks in the interbank market is explained in Figure1.3. The mechanism is quite simple. Given the idiosyncratic failure of a bank, it is no longer able to repay its interbank obligations by causing losses to the creditor banks. If one of the creditor banks becomes insolvent due to the losses incurred by the bankruptcy of the first bank, the mechanism repeats itself until further bank failures occur.
The procedure of the domino effect is in Figure1.3 Smaga [2014].
The idea that the bank's bankruptcy is the requirement to have a domino effect has been criticized by some researchers. Losses could materialize even in absence of default, due to the downgrading of the bank's credit rating Caccioli et al. [2018].
For example, given two banks x and y with x exposed to bank z which suffers a large loss. The loss will reduce the cash flow of x from z. This has increased the probability that x defaults and therefore the expected cash flow associated with the exposures between x and y is reduced Caccioli et al. [2018].
In a contest of market-to-market valuation of inter-bank assets were to be market to mar- ket, this would mean that the value of the inter-bank asset of y that is associated with its exposure to x, is reduced Caccioli et al. [2018].

In the case of the equity financial market, the concept of contagion risk is slightly different. The assumption individuated before are always valid.
The difference lies in the type of risk, namely market risk. Contagion risk implies that losses in the stock values suffered by one or a group of listed companies may be passed on to other listed companies because of links in the stock market.
Concerning market risk, a second definition of the risk of contagion is proposed. The latter, in this case, can be explained as the increase in share price co-movements: this means that they move more synchronously. [...]