A Model of Systemic Risk Andrey Sarantsev University of California, - PowerPoint PPT Presentation

A Model of Systemic Risk Andrey Sarantsev University of California, Santa Barbara March 24, 2017 Andrey Sarantsev 8th Western Conference in Mathematical Finance 1 / 21

The Financial System We have N private banks and the central bank. Central bank chooses the discount rate: interest rate of lending to private banks There can be cash flows between each pair of private banks Private banks borrow from central bank and pay back interest Private banks invest in risky assets Andrey Sarantsev 8th Western Conference in Mathematical Finance 2 / 21

Private Banks The i th private bank has capital X i ( t ), with Y i ( t ) = log X i ( t ) It borrows the amount Z i ( t ) ≥ 0 from the central bank, under discount interest rate r ( t ) Therefore, it pays interest r ( t ) Z i ( t ) d t during [ t , t + d t ] The case Z i ( t ) ≤ 0 is when the bank does not borrow anything but sets aside money − Z i ( t ) in cash, earning zero interest. In both cases, the bank invests X i ( t ) + Z i ( t ) in a risky asset S i ( t ), and pays interest r ( t )( Z i ( t )) + d t Andrey Sarantsev 8th Western Conference in Mathematical Finance 3 / 21

Risky Assets The i th risky asset is given by S i ( t ) = exp ( M i ( t ) − � M i � t ) where M = ( M 1 , . . . , M N ) is a Brownian motion with drift vector µ = ( µ 1 , . . . , µ N ) and covariance matrix A = ( a ij ). In particular, the component M i is a Brownian motion with drift µ i and diffusion a ii = σ 2 i : d S i ( t ) S i ( t ) = d M i ( t ) = µ i d t + σ i d W i ( t ) Andrey Sarantsev 8th Western Conference in Mathematical Finance 4 / 21

Equation Without Interbank Flows Combining investment and borrowing for the i th bank: d X i ( t ) = ( X i ( t ) + Z i ( t )) d S i ( t ) S i ( t ) − r ( t )( Z i ( t )) + d t By Itˆ o’s formula, for Y i ( t ) = log X i ( t ) and α i ( t ) := Z i ( t ) / X i ( t ) � σ 2 d Y i ( t ) = (1 + α i ( t )) d S i ( t ) � 2 (1 + α i ( t )) 2 + r ( t )( α i ( t )) + i S i ( t ) − d t Andrey Sarantsev 8th Western Conference in Mathematical Finance 5 / 21

Equation with Added Interbank Flows N d Y i ( t ) = 1 � c ij ( t )( Y j ( t ) − Y i ( t )) d t N j =1 � σ 2 + (1 + α i ( t )) d S i ( t ) � 2 (1 + α i ( t )) 2 + r ( t )( α i ( t )) + i S i ( t ) − d t Here, c ij ( t ) = c ji ( t ) ≥ 0 are controlled by the central bank and are used to keep banks close to one another, to minimize the possibility of bankruptcy This model is taken from (Carmona, Fouque, Sun, 2013), where they had c ij ( t ) ≡ c > 0 Andrey Sarantsev 8th Western Conference in Mathematical Finance 6 / 21

Objective of Each Private Bank The i th bank chooses the amount Z i of borrowing (or, equivalently, α i = Z i / X i ) to maximize its expected terminal logarithmic utility: E log X i ( T ) = E Y i ( T ) The i th bank takes as given X j ( t ) , Z j ( t ) for j � = i (of other banks) and r ( t ) , c ij ( t ) (instruments of the central bank) Andrey Sarantsev 8th Western Conference in Mathematical Finance 7 / 21

Objective of Central Bank Central bank chooses the discount interest rate r to control (as we see below) the overall size of the system: N Y ( t ) = 1 � Y i ( t ) , N i =1 and the rates c ij to make Y i closer to this average Y by directing flow of cash to this bank from other banks (or vice versa). Objective: to prevent Y i ( t ) from becoming too small (which corresponds to bankruptcy) Andrey Sarantsev 8th Western Conference in Mathematical Finance 8 / 21

Opitmal Control for Private Bank Φ i ( t , y ) := sup E ( Y i ( T ) | Y ( t ) = y ) α i satisfies Φ i ( T , y ) = y i , and HJB equation: � N 0 = ∂ Φ i h j ( α j ) ∂ Φ i � ∂ t + sup ∂ y j α i j =1 N N ∂ 2 Φ i � + 1 � � a jk (1 + α j )(1 + α k ) 2 ∂ y j ∂ y k j =1 k =1 σ 2 2 (1 + α j ) 2 − r ( α j ) + j h j ( α j ) := (1 + α j ) µ j − Andrey Sarantsev 8th Western Conference in Mathematical Finance 9 / 21

Optimal Control for Private Bank N � Try anzats Φ i ( t , y ) = g i 0 ( t ) + g ij ( t ) y j j =1 Because the objective and dynamics are linear in Y j , we can solve this problem explicitly. Just find α i which maximizes h i ( α i ) := (1 + α i ) µ i − σ 2 2 (1 + α i ) 2 − r ( α i ) + i  � � µ i − r ( t ) µ i ≥ σ 2 − 1 + , i ;  σ 2 This is α ∗ i := i µ i − σ 2 µ i ≤ σ 2 2 , i  i Andrey Sarantsev 8th Western Conference in Mathematical Finance 10 / 21

Liquidity Trap If µ i ≤ σ 2 i for all i , then banks do not borrow from the central bank; rather, they set aside some cash Even setting r = 0 cannot induce banks to borrow Below, we assume that µ i ≥ σ 2 i for all i Andrey Sarantsev 8th Western Conference in Mathematical Finance 11 / 21

Dynamics Under Optimal Control Under optimal choice α i = α ∗ i , i = 1 , . . . , N , we have: N i ( t ) + 1 � d Y i ( t ) = d M ∗ c ij ( t )( Y j ( t ) − Y i ( t )) d t N j =1 d M ∗ i ( t ) = h i ( α ∗ i ( t )) d t + σ i (1 + α ∗ i ( t )) d W i ( t ) i are too, and M ∗ = ( M ∗ If r is constant, then α ∗ 1 , . . . , M ∗ N ) is an N -dimensional Brownian motion Andrey Sarantsev 8th Western Conference in Mathematical Finance 12 / 21

Dynamics of Total Size of System Y ( t ) = g ( r ( t )) d t + ρ ( r ( t )) d W ( t ) ,  ( µ i − r ) 2 N r ≤ µ i − σ 2 + r , i ; g ( r ) = 1  2 σ 2 � g i ( r ) , g i ( r ) := i µ i − σ 2 N r ≥ µ i − σ 2 2 , i i .  i =1 � µ i − r N N r ≤ µ i − σ 2 i , i ; ρ 2 ( r ) := 1 σ 2 � � a ij ρ i ( r ) ρ j ( r ) , ρ i ( r ) := N 2 r ≥ µ i − σ 2 1 , i . i =1 j =1 Andrey Sarantsev 8th Western Conference in Mathematical Finance 13 / 21

Goals of Central Bank Maximize E U λ ( Y ( T )) for U λ ( y ) := − e − λ y , λ > 0. Central bank is even more risk-averse than private banks. Parameter of risk aversion: λ The HJB equation becomes g ( r ) − λ 2 ρ 2 ( r ) → sup r ≥ 0 Andrey Sarantsev 8th Western Conference in Mathematical Finance 14 / 21

The Case of the Same Asset S 1 = . . . = S N Then µ 1 = . . . = µ N = µ , σ 1 = . . . = σ N = σ � µ � − 1 . Then the maximum is attained at Let λ ∗ := 1 − 2 σ 2 + 1 � 0 , λ < λ ∗ ; r = µ − σ 2 , λ > λ ∗ λ < λ ∗ : a less risk-averse central bank, slashes the rate to zero λ > λ ∗ : a more risk-averse central bank, increases the rate Andrey Sarantsev 8th Western Conference in Mathematical Finance 15 / 21

The Case of Independent Assets Assume µ 1 = . . . = µ N = µ and σ 1 = . . . = σ N = σ Then same holds true for N λ ∗ instead of λ ∗ Even a more risk-averse central bank (than in case of same asset) can slash rate r to zero Independence of assets creates diversification and reduces risk Andrey Sarantsev 8th Western Conference in Mathematical Finance 16 / 21

The Case of Correlated Assets W 0 ( t ) + ρ ′ ˜ W i ( t ) = ρ ˜ W i ( t ) , ρ 2 + ρ ′ 2 = 1, ˜ W i i.i.d. Brownian motions If µ 1 = . . . = µ N = µ and σ 1 = . . . = σ N = σ , then same holds true for N λ ∗ / (( N − 1) ρ + 1) instead of λ ∗ Here, the room for risk is less than in case of independent assets, but more than in case of the same asset Andrey Sarantsev 8th Western Conference in Mathematical Finance 17 / 21

Stability of the System Let ˜ Y i ( t ) = Y i ( t ) − Y ( t ). Then ˜ Y = ( ˜ Y 1 , . . . , ˜ Y N ) is the solution of an SDE on the hyperplane Π = { y 1 + . . . + y N = 0 } . Theorem Assume c ij ( t ) = c ij do not depend on t, and the graph G on { 1 , . . . , N } defined as i ↔ j iff c ij > 0 is connected. Then ˜ Y has a unique stationary distribution π on Π . For any bounded measurable f : Π → R , we have: � T 1 � f ( ˜ lim Y ( t )) d t = f ( y ) π ( d y ) . T T →∞ 0 Π Andrey Sarantsev 8th Western Conference in Mathematical Finance 18 / 21

The Case of Similar Flows Assume all c ij ( t ) = c > 0. Then ˜ Y is an Orntsein-Uhlenbeck process on Π. And π is a multivariate normal distribution on Π with i th marginal � ˜ σ 2 c , ˜ � g i i π i ∼ N , 2 c where ˜ g i , ˜ σ i can be explicitly found. Andrey Sarantsev 8th Western Conference in Mathematical Finance 19 / 21

Control Problem for Flow Rate This allows to formulate control problem, assuming the process ˜ Y is in the stationary distribution π : � � y � 2 π ( d y ) + k ( c ) = M 1 c − 1 + M 2 c − 2 + k ( c ) → min c > 0 Π N N M 1 := 1 � σ 2 � g 2 ˜ i , M 2 := ˜ i . 2 i =1 i =1 k ( c ): cost of maintaining flow rate c Andrey Sarantsev 8th Western Conference in Mathematical Finance 20 / 21

Thanks! Andrey Sarantsev 8th Western Conference in Mathematical Finance 21 / 21