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A Response function of Merton model and Kinetic Ising model

Masato Hisakado and Takuya Kaneko

posted on 13 December 2019

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We study contagious defaults of banks by
 applying  a voting model.
The network of the banks  are created by the relation, lending and borrowing among banks.
We introduce the response function from Merton model.
Using this response function we calculate the  probability of default (PD) which
includes not only   changes of asset values  but also  the effects of  connected  banks' defaults using
the mean field approximation.
If we approximate the normal distribution which Merton model uses by $\tanh$ function, we can obtain the kinetic Ising model which represents  phase transition.
The asset volatility plays the role of  temperature.
In the low temperature limit, the model becomes the threshold model.
We calculate   PD which shows   an effect of the situations around the bank as the additional PD using the self consistent equation.