Systemic Risk and Stochastic Games with Delay Jean-Pierre Fouque - PowerPoint PPT Presentation

Systemic Risk and Stochastic Games with Delay Jean-Pierre Fouque (with Ren´ e Carmona, Mostafa Mousavi and Li-Hsien Sun) 8th WCMF, University of Washington Seattle, March 24-25, 2017 JP Fouque (UC Santa Barbara) Stochastic Games with Delay UW Seattle, March 25, 2017 1 / 49

Coupled Diffusions: Liquidity Model X ( i ) t , i = 1 , . . . , N denote log-monetary reserves of N banks dX ( i ) = b ( i ) t dt + σ ( i ) t dW ( i ) i = 1 , . . . , N , t t which are non-tradable quantities . Assume independent Brownian motions W ( i ) t and identical constant volatilities σ ( i ) = σ . t Model borrowing and lending through the drifts: N = a dX ( i ) ( X ( j ) − X ( i ) t ) dt + σ dW ( i ) � , i = 1 , . . . , N . t t t N j =1 The overall rate of borrowing and lending a / N has been normalized by the number of banks. JP Fouque (UC Santa Barbara) Stochastic Games with Delay UW Seattle, March 25, 2017 2 / 49

Coupled Diffusions: Liquidity Model X ( i ) t , i = 1 , . . . , N denote log-monetary reserves of N banks dX ( i ) = b ( i ) t dt + σ ( i ) t dW ( i ) i = 1 , . . . , N , t t which are non-tradable quantities . Assume independent Brownian motions W ( i ) t and identical constant volatilities σ ( i ) = σ . t Model borrowing and lending through the drifts: N = a dX ( i ) ( X ( j ) − X ( i ) t ) dt + σ dW ( i ) � , i = 1 , . . . , N . t t t N j =1 The overall rate of borrowing and lending a / N has been normalized by the number of banks. JP Fouque (UC Santa Barbara) Stochastic Games with Delay UW Seattle, March 25, 2017 3 / 49

Fully Connected Symmetric Network N = a dX ( i ) ( X ( j ) − X ( i ) t ) dt + σ dW ( i ) � , i = 1 , . . . , N . t t t N j =1 N=10 Denote the default level by D < 0 and simulate the system for various values of a : 0 , 1 , 10 , 100 with fixed σ = 1 JP Fouque (UC Santa Barbara) Stochastic Games with Delay UW Seattle, March 25, 2017 4 / 49

Weak Coupling: a = 1 One realization of the trajectories of the coupled diffusions X ( i ) with a = 1 (left t plot) and trajectories of the independent Brownian motions ( a = 0) (right plot) using the same Gaussian increments. Solid horizontal line: default level D = − 0 . 7 JP Fouque (UC Santa Barbara) Stochastic Games with Delay UW Seattle, March 25, 2017 5 / 49

Moderate Coupling: a = 10 One realization of the trajectories of the coupled diffusions X ( i ) with a = 10 (left t plot) and trajectories of the independent Brownian motions ( a = 0) (right plot) using the same Gaussian increments. Solid horizontal line: default level D = − 0 . 7 JP Fouque (UC Santa Barbara) Stochastic Games with Delay UW Seattle, March 25, 2017 6 / 49

Strong Coupling: a = 100 One realization of the trajectories of the coupled diffusions X ( i ) with a = 100 (left t plot) and trajectories of the independent Brownian motions ( a = 0) (right plot) using the same Gaussian increments. Solid horizontal line: default level D = − 0 . 7 JP Fouque (UC Santa Barbara) Stochastic Games with Delay UW Seattle, March 25, 2017 7 / 49

Loss Distributions These simulations “show” that STABILITY is created by increasing the rate of borrowing and lending. Next, we compare the loss distributions for the coupled and independent cases. We compute these loss distributions by Monte Carlo method using 10 4 simulations, and with the same parameters as previously. In the independent case, the loss distribution is Binomial( N , p ) with parameter p given by � � 0 ≤ t ≤ T ( σ W t ) ≤ D p = I P min � D � √ = 2Φ , σ T where Φ denotes the N (0 , 1) cdf. With our choice of parameters, we have p ≈ 0 . 5 JP Fouque (UC Santa Barbara) Stochastic Games with Delay UW Seattle, March 25, 2017 8 / 49

Loss Distributions These simulations “show” that STABILITY is created by increasing the rate of borrowing and lending. Next, we compare the loss distributions for the coupled and independent cases. We compute these loss distributions by Monte Carlo method using 10 4 simulations, and with the same parameters as previously. In the independent case, the loss distribution is Binomial( N , p ) with parameter p given by � � 0 ≤ t ≤ T ( σ W t ) ≤ D p = I P min � D � √ = 2Φ , σ T where Φ denotes the N (0 , 1) cdf. With our choice of parameters, we have p ≈ 0 . 5 JP Fouque (UC Santa Barbara) Stochastic Games with Delay UW Seattle, March 25, 2017 9 / 49

Loss Distribution: weak coupling 0.25 0.2 0.2 0.15 prob of # of default prob of # of default 0.15 0.1 0.1 0.05 0.05 0 0 0 5 10 6 8 10 # of default # of default On the left, we show plots of the loss distribution for the coupled diffusions with a = 1 (solid line) and for the independent Brownian motions (dashed line). The plots on the right show the corresponding tail probabilities. JP Fouque (UC Santa Barbara) Stochastic Games with Delay UW Seattle, March 25, 2017 10 / 49

Loss Distribution: moderate coupling 0.5 0.2 0.4 0.15 prob of # of default prob of # of default 0.3 0.1 0.2 0.05 0.1 0 0 0 5 10 6 8 10 # of default # of default On the left, we show plots of the loss distribution for the coupled diffusions with a = 10 (solid line) and for the independent Brownian motions (dashed line). The plots on the right show the corresponding tail probabilities. JP Fouque (UC Santa Barbara) Stochastic Games with Delay UW Seattle, March 25, 2017 11 / 49

Loss Distribution: strong coupling 1 0.2 0.8 0.15 prob of # of default prob of # of default 0.6 0.1 0.4 0.05 0.2 0 0 0 5 10 6 8 10 # of default # of default On the left, we show plots of the loss distribution for the coupled diffusions with a = 100 (solid line) and for the independent Brownian motions (dashed line). The plots on the right show the corresponding tail probabilities. JP Fouque (UC Santa Barbara) Stochastic Games with Delay UW Seattle, March 25, 2017 12 / 49

Mean Field Limit Rewrite the dynamics as: N a dX ( i ) ( X ( j ) − X ( i ) t ) dt + σ dW ( i ) � = t t t N j =1     N  1 X ( j )  − X ( i )  dt + σ dW ( i ) � = a t .  t t N j =1 The processes X ( i ) ’s are “OUs” mean-reverting to the ensemble average which satisfies � N � � N � 1 σ X ( i ) W ( i ) � � d = d . t t N N i =1 i =1 JP Fouque (UC Santa Barbara) Stochastic Games with Delay UW Seattle, March 25, 2017 13 / 49

Mean Field Limit Assuming for instance that x ( i ) = 0 , i = 1 , . . . , N , we obtain 0 N N 1 = σ X ( i ) W ( i ) � � t , and consequently t N N i =1 i =1     N  σ dX ( i ) W ( j )  − X ( i )  dt + σ dW ( i ) � = a . t  t t t N j =1 Note that the ensemble average is distributed as a Brownian motion with √ diffusion coefficient σ/ N . In the limit N → ∞ , the strong law of large numbers gives N 1 W ( j ) � → 0 a . s . , t N j =1 and therefore, the processes X ( i ) ’s converge to independent OU processes with long-run mean zero. JP Fouque (UC Santa Barbara) Stochastic Games with Delay UW Seattle, March 25, 2017 14 / 49

Mean Field Limit In fact, X ( i ) is given explicitly by t X ( i ) j =1 W ( j ) = σ � N + t t N σ e − at � t e − at � t � � 0 e as dW ( i ) 0 e as dW ( j ) � N − σ , s s j =1 N converges to σ e − at � t and therefore, X ( i ) 0 e as dW ( i ) which are independent s t OU processes. This is a simple example of a mean-field limit and propagation of chaos studied in general by Sznitman (1991). JP Fouque (UC Santa Barbara) Stochastic Games with Delay UW Seattle, March 25, 2017 15 / 49

Systemic Risk Using classical equivalent for the Gaussian cumulative distribution function, we obtain the large deviation estimate � � N � � D 2 N →∞ − 1 σ � W ( i ) lim N log I P min ≤ D = 2 σ 2 T . t N 0 ≤ t ≤ T i =1 For a large number of banks, the probability that the ensemble average � � − D 2 N reaches the default barrier is of order exp 2 σ 2 T N N 1 = σ X ( i ) W ( i ) � � Since t , we identify t N N i =1 i =1 � � N � � 1 � X ( i ) as a systemic event min ≤ D t N 0 ≤ t ≤ T i = 1 This event does not depend on a . In fact, once in this event, increasing a creates more defaults by “flocking to default” . JP Fouque (UC Santa Barbara) Stochastic Games with Delay UW Seattle, March 25, 2017 16 / 49

Systemic Risk Using classical equivalent for the Gaussian cumulative distribution function, we obtain the large deviation estimate � � N � � D 2 N →∞ − 1 σ � W ( i ) lim N log I P min ≤ D = 2 σ 2 T . t N 0 ≤ t ≤ T i =1 For a large number of banks, the probability that the ensemble average � � − D 2 N reaches the default barrier is of order exp 2 σ 2 T N N 1 = σ X ( i ) W ( i ) � � Since t , we identify t N N i =1 i =1 � � N � � 1 � X ( i ) as a systemic event min ≤ D t N 0 ≤ t ≤ T i = 1 This event does not depend on a . In fact, once in this event, increasing a creates more defaults by “flocking to default” . JP Fouque (UC Santa Barbara) Stochastic Games with Delay UW Seattle, March 25, 2017 17 / 49