Mean Field Games and. Systemic Risk

Transcription

1 Mean Field Games and Sysemic Risk Jean-Pierre Fouque Universiy of California Sana Barbara Financial Mahemaics Tuorials IPAM-UCLA March 10-11, 2015

2 2010: Dodd-Frank Bill includes he creaion of he Office of Financial Research (OFR) Ediors: J.-P. Fouque and J. Langsam Cambridge Universiy Press (2013)

3 Coupled Diffusions: Liquidiy Risk X (i), i = 1,...,N denoe log-moneary reserves of N banks dx (i) = b (i) d + σ (i) dw (i) i = 1,...,N, which are non-radable quaniies. Assume independen Brownian moions W (i), i = 1,...,N and idenical consan volailiies σ (i) = σ Model borrowing and lending hrough he drifs: dx (i) = α N j=1 (X (j) X (i) ) d + σdw (i), i = 1,...,N. The overall rae of borrowing and lending α/n has been normalized by he number of banks and we assume α > 0

4 Coupled Diffusions: Liquidiy Risk X (i), i = 1,...,N denoe log-moneary reserves of N banks dx (i) = b (i) d + σ (i) dw (i) i = 1,...,N, which are non-radable quaniies. Assume independen Brownian moions W (i), i = 1,...,N and idenical consan volailiies σ (i) = σ Model borrowing and lending hrough he drifs: dx (i) = α N j=1 (X (j) X (i) ) d + σdw (i), i = 1,...,N. The overall rae of borrowing and lending α/n has been normalized by he number of banks and we assume α > 0

5 Coupled Diffusions: Liquidiy Risk X (i), i = 1,...,N denoe log-moneary reserves of N banks dx (i) = b (i) d + σ (i) dw (i) i = 1,...,N, which are non-radable quaniies. Assume independen Brownian moions W (i), i = 1,...,N and idenical consan volailiies σ (i) = σ Model borrowing and lending hrough he drifs: dx (i) = a N j=1 (X (j) X (i) ) d + σdw (i), i = 1,...,N. The overall rae of borrowing and lending a/n has been normalized by he number of banks.

6 Fully Conneced Symmeric Nework dx (i) = a N j=1 (X (j) X (i) ) d + σdw (i), i = 1,...,N. N=10 Denoe he defaul level by D < 0 and simulae he sysem for various values of a: 0, 1, 10, 100 wih fixed σ = 1

7 One realizaion of he rajecories of he coupled diffusions X (i) wih a = 1 (lef plo) and rajecories of he independen Brownian moions (a = 0) (righ plo) using he same Gaussian incremens. Solid horizonal line: defaul level D = 0.7

8 One realizaion of he rajecories of he coupled diffusions X (i) wih a = 10 (lef plo) and rajecories of he independen Brownian moions (a = 0) (righ plo) using he same Gaussian incremens. Solid horizonal line: defaul level D = 0.7

9 One realizaion of he rajecories of he coupled diffusions X (i) wih a = 100 (lef plo) and rajecories of he independen Brownian moions (a = 0) (righ plo) using he same Gaussian incremens. Solid horizonal line: defaul level D = 0.7

10 These simulaions show ha STABILITY is creaed by increasing he rae of borrowing and lending. Nex, we compare he loss disribuions for he coupled and independen cases. We compue hese loss disribuions by Mone Carlo mehod using 10 4 simulaions, and wih he same parameers as previously. In he independen case, he loss disribuion is Binomial(N, p) wih parameer p given by ( ) p = IP min (σw ) η 0 T ( ) η = 2Φ σ, T where Φ denoes he N(0, 1)-cdf, and we used he disribuion of he minimum of a Brownian moion (see Karazas-Shreve 2000 for insance). Wih our choice of parameers, we have p 0.5

11 These simulaions show ha STABILITY is creaed by increasing he rae of borrowing and lending. Nex, we compare he loss disribuions for he coupled and independen cases. We compue hese loss disribuions by Mone Carlo mehod using 10 4 simulaions, and wih he same parameers as previously. In he independen case, he loss disribuion is Binomial(N, p) wih parameer p given by ( ) p = IP min (σw ) D 0 T ( ) D = 2Φ σ, T where Φ denoes he N(0, 1)-cdf, and we used he disribuion of he minimum of a Brownian moion (see Karazas-Shreve 2000 for insance). Wih our choice of parameers, we have p 0.5

12 These simulaions show ha STABILITY is creaed by increasing he rae of borrowing and lending. Nex, we compare he loss disribuions for he coupled and independen cases. We compue hese loss disribuions by Mone Carlo mehod using 10 4 simulaions, and wih he same parameers as previously. In he independen case, he loss disribuion is Binomial(N, p) wih parameer p given by ( ) p = IP min (σw ) D 0 T ( ) D = 2Φ σ, T where Φ denoes he N(0, 1) cdf. Wih our choice of parameers, we have p 0.5

13 prob of # of defaul prob of # of defaul # of defaul # of defaul On he lef, we show plos of he loss disribuion for he coupled diffusions wih a = 1 (solid line) and for he independen Brownian moions (dashed line). The plos on he righ show he corresponding ail probabiliies.

14 prob of # of defaul prob of # of defaul # of defaul # of defaul On he lef, we show plos of he loss disribuion for he coupled diffusions wih a = 10 (solid line) and for he independen Brownian moions (dashed line). The plos on he righ show he corresponding ail probabiliies.

15 1 0.2 prob of # of defaul prob of # of defaul # of defaul # of defaul On he lef, we show plos of he loss disribuion for he coupled diffusions wih a = 100 (solid line) and for he independen Brownian moions (dashed line). The plos on he righ show he corresponding ail probabiliies.

16 Rewrie he dynamics as: dx (i) = a N j=1 = a 1 N Mean-field Limi (X (j) j=1 X (i) ) d + σdw (i) X (j) X (i) d + σdw (i). The processes X (i) s are OUs mean-revering o he ensemble average which saisfies ( ) ( ) 1 d X (i) σ = d W (i). N N i=1 i=1

17 Rewrie he dynamics as: dx (i) = a N j=1 = a 1 N Mean-field Limi (X (j) j=1 X (i) ) d + σdw (i) X (j) X (i) d + σdw (i). The processes X (i) s are OUs mean-revering o he ensemble average which saisfies ( ) ( ) 1 d X (i) σ = d W (i). N N i=1 i=1

18 Assuming for insance ha x (i) 0 = 0, i = 1,...,N, we obain 1 N dx (i) i=1 X (i) = σ N = a σ N j=1 i=1 W (i), and consequenly W (j) X (i) d + σdw (i). Noe ha he ensemble average is disribued as a Brownian moion wih diffusion coefficien σ/ N. In he limi N, he srong law of large numbers gives 1 N j=1 W (j) 0 a.s., and herefore, he processes X (i) s converge o independen OU processes wih long-run mean zero.

19 Assuming for insance ha x (i) 0 = 0, i = 1,...,N, we obain 1 N dx (i) i=1 X (i) = σ N = a σ N j=1 i=1 W (i), and consequenly W (j) X (i) d + σdw (i). Noe ha he ensemble average is disribued as a Brownian moion wih diffusion coefficien σ/ N. In he limi N, he srong law of large numbers gives 1 N j=1 W (j) 0 a.s., and herefore, he processes X (i) s converge o independen OU processes wih long-run mean zero.

20 Assuming for insance ha x (i) 0 = 0, i = 1,...,N, we obain 1 N dx (i) i=1 X (i) = σ N = a σ N j=1 i=1 W (i), and consequenly W (j) X (i) d + σdw (i). Noe ha he ensemble average is disribued as a Brownian moion wih diffusion coefficien σ/ N. In he limi N, he srong law of large numbers gives 1 N j=1 W (j) 0 a.s., and herefore, he processes X (i) s converge o independen OU processes wih long-run mean zero.

21 In fac, X (i) X (i) = σ N j=1 is given explicily by W (j) +σe a 0 e as dw s (i) σ N (e a j=1 0 ) e as dw s (j), and herefore, X (i) converges o σe a 0 eas dw s (i) independen OU processes. which are This is a simple example of a mean-field limi and propagaion of chaos sudied in general by Szniman (1991).

22 Sysemic Risk Using classical equivalen for he Gaussian cumulaive disribuion funcion, we obain he large deviaion esimae ( ( ) ) lim 1 σ log IP min W (i) D = D2 N N 0 T N 2σ 2 T. For a large number of banks, he probabiliy ha ( he ensemble ) average reaches he defaul barrier is of order exp η2 N 2σ 2 T { Since min 0 T ( σ N 1 N i=1 i=1 X (i) X (i) ) = σ N η } i=1 i=1 W (i), we idenify as a sysemic even Observe ha his even does no depend on α > 0

23 Sysemic Risk Using classical equivalen for he Gaussian cumulaive disribuion funcion, we obain he large deviaion esimae ( ( ) ) lim 1 σ log IP min W (i) D = D2 N N 0 T N 2σ 2 T. For a large number of banks, he probabiliy ha ( he ensemble ) average reaches he defaul barrier is of order exp D2 N 2σ 2 T { Since min 0 T ( σ N 1 N i=1 i=1 X (i) X (i) ) = σ N η } i=1 i=1 W (i), we idenify as a sysemic even Observe ha his even does no depend on α > 0

24 Sysemic Risk Using classical equivalen for he Gaussian cumulaive disribuion funcion, we obain he large deviaion esimae ( ( ) ) lim 1 σ log IP min W (i) D = D2 N N 0 T N 2σ 2 T. For a large number of banks, he probabiliy ha ( he ensemble ) average reaches he defaul barrier is of order exp D2 N 2σ 2 T { Since min 0 T ( 1 N 1 N i=1 i=1 X (i) X (i) ) = σ N D } i=1 i=1 W (i), we idenify as a sysemic even Observe ha his even does no depend on a > 0

25 The probabiliy exp ( D2 N 2σ 2 T of a sysemic even does no depend on a > 0, in oher words: Increasing sabiliy by increasing he rae of borrowing and lending does no preven a sysemic even where a large number of banks defaul In fac, once in his even, increasing a creaes even more defauls by flocking o defaul. This is illusraed in he simulaion wih a = 100 where he probabiliy of sysemic risk is roughly 3%. )

26 dx i = a 1 N Sysemic Risk and Common Noise X j X i d+σ j=1 ( ρdw 0 + ) 1 ρ 2 dw i, i = 1,,N, The ensemble average: ( 1 X i = σ 1 ρ W i = σ ρw N N N i=1 i=1 i=1 = (in D) σ ρ 2 + (1 ρ2 ) N B, The probabiliy of he sysemic even becomes ( ) ( ) 1 ( ) IP min Xs i D < D = 2Φ 0 s T N σ N D 2Φ T Nρ 2 + (1 ρ 2 ) σ ρ T i=1 W i )

27 dx i = a 1 N Sysemic Risk and Common Noise X j X i d+σ j=1 ( ρdw 0 + ) 1 ρ 2 dw i, i = 1,,N, The ensemble average: ( 1 X i = σ 1 ρ W i = σ ρw N N N i=1 i=1 i=1 = (in D) σ ρ 2 + (1 ρ2 ) N B, The probabiliy of he sysemic even becomes ( ) ( ) 1 ( ) IP min Xs i D < D = 2Φ 0 s T N σ N D 2Φ T Nρ 2 + (1 ρ 2 ) σ ρ T i=1 W i )

28 dx i = a 1 N Sysemic Risk and Common Noise X j X i d+σ j=1 ( ρdw 0 + ) 1 ρ 2 dw i, i = 1,,N, The ensemble average: ( 1 X i = σ 1 ρ W i = σ ρw N N N i=1 i=1 i=1 = (in D) σ ρ 2 + (1 ρ2 ) N B, The probabiliy of he sysemic even becomes ( ) ( ) 1 ( ) IP min Xs i D < D = 2Φ 0 s T N σ N D 2Φ T Nρ 2 + (1 ρ 2 ) σ ρ T i=1 W i )

29 ρ = 0 ρ = 0.5

30 So far we have seen ha: Lending and borrowing improves sabiliy bu also conribues o sysemic risk bf Bu how abou if he banks compee? bf (minimizing coss, maximizing profis,...) Can we find an equilibrium in which he previous analysis can sill be performed? Can we find and characerize a Nash equilibrium? Wha follows is a work in progress wih René Carmona

31 So far we have seen ha: Lending and borrowing improves sabiliy bu also conribues o sysemic risk Bu how abou if he banks compee? (minimizing coss, maximizing profis,...) Can we find an equilibrium in which he previous analysis can sill be performed? Can we find and characerize a Nash equilibrium? Wha follows is from Mean Field Games and Sysemic Risk R. Carmona, J. P. Fouque and L.-H. Sun

32 So far we have seen ha: Lending and borrowing improves sabiliy bu also conribues o sysemic risk Bu how abou if he banks compee? (minimizing coss, maximizing profis,...) Can we find an equilibrium in which he previous analysis can sill be performed? Can we find and characerize a Nash equilibrium? Wha follows is a work in progress wih René Carmona

33 So far we have seen ha: Lending and borrowing improves sabiliy bu also conribues o sysemic risk Bu how abou if he banks compee? (minimizing coss, maximizing profis,...) Can we find an equilibrium in which he previous analysis can sill be performed? Can we find and characerize a Nash equilibrium? Wha follows is from Mean Field Games and Sysemic Risk by R. Carmona, J.-P. Fouque and L.-H. Sun (2013)

34 Mean Field Game Banks are borrowing from and lending o a cenral bank: dx i = α i d + σdw i, i = 1,,N where α i is he conrol of bank i which wans o minimize { } T J i (α 1,,α N ) = IE f i (X, α)d i + g i (X T ), 0 wih running cos [ 1 f i (x, α i ) = 2 (αi ) 2 qα i (x x i ) + ǫ ] 2 (x xi ) 2, q 2 ǫ, and erminal cos g i (x) = c 2 ( x x i ) 2. This is an example of Mean Field Game (MFG) sudied exensively by P.L. Lions and collaboraors (see also he recen work of R. Carmona and F. Delarue).

35 Nash Equilibria (FBSDE Approach) The Hamilonian: H i (x, y i,1,,y i,n, α 1 (, x),,α, i,α N (, x)) = α k (, x)y i,k + α i y i,i k i (αi ) 2 qα i (x x i ) + ǫ 2 (x xi ) 2, Minimizing H i over α i gives he choices: ˆα i = y i,i + q(x x i ), i = 1,,N, Ansaz: Y i,j = η ( 1 N δ i,j ) (X X i ), where η is a deerminisic funcion saisfying he erminal condiion η T = c.

36 Nash Equilibria (FBSDE Approach) The Hamilonian: H i (x, y i,1,,y i,n, α 1 (, x),,α, i,α N (, x)) = α k (, x)y i,k + α i y i,i k i (αi ) 2 qα i (x x i ) + ǫ 2 (x xi ) 2, Minimizing H i over α i gives he choices: ˆα i = y i,i + q(x x i ), i = 1,,N, Ansaz: Y i,j = η ( 1 N δ i,j ) (X X i ), where η is a deerminisic funcion saisfying he erminal condiion η T = c.

37 Nash Equilibria (FBSDE Approach) The Hamilonian: H i (x, y i,1,,y i,n, α 1 (, x),,α, i,α N (, x)) = α k (, x)y i,k + α i y i,i k i (αi ) 2 qα i (x x i ) + ǫ 2 (x xi ) 2, Minimizing H i over α i gives he choices: ˆα i = y i,i + q(x x i ), i = 1,,N, Ansaz: Y i,j = η ( 1 N δ i,j ) (X X i ), where η is a deerminisic funcion saisfying he erminal condiion η T = c.

38 Forward Equaion: Forward-Backward Equaions dx i = y i,ih i d + σdw i [ = q + (1 1 ] N )η (X X)d i + σdw i, wih iniial condiions X i 0 = x i 0. Backward Equaion: dy i,j = x jh i d + k=1 Z i,j,k dw k = ( 1 [ N δ i,j)(x X) i (a + q)η 1 N ( 1 ] N 1)η2 + q 2 ǫ d + k=1 Z i,j,k dw k, Y i,j T = c( 1 N δ i,j)(x T X i T).

39 Forward Equaion: Forward-Backward Equaions dx i = y i,ih i d + σdw i [ = q + (1 1 ] N )η (X X)d i + σdw i, wih iniial condiions X i 0 = x i 0. Backward Equaion: dy i,j = x jh i d + k=1 Z i,j,k dw k = ( 1 [ N δ i,j)(x X) i qη 1 N ( 1 ] N 1)η2 + q 2 ǫ d + k=1 Z i,j,k dw k, Y i,j T = c( 1 N δ i,j)(x T X i T).

40 Soluion o he Forward-Backward Equaions By summaion of he forward equaions: dx = σ N N k=1 dw k. ( Differeniaing he ansaz Y i,j = η dy i,j = 1 N δ i,j ) (X X i ), we ge: ( ) [ 1 N δ i,j (X X) i η η (a + q + (1 1 ) ] N )η d +η ( 1 N δ i,j)σ ( 1 N δ i,k)dw k. k=1 Idenifying wih he backward equaions: Z i,j,k = η σ 1 ρ 2 ( 1 N δ i,j)( 1 N δ i,k) for k = 1,,N, and η mus saisfy he Riccai equaion η = 2(a + q)η + (1 1 N 2 )η2 (ǫ q 2 ), wih he erminal condiion η T = c.

41 Soluion o he Forward-Backward Equaions By summaion of he forward equaions: dx = σ N N k=1 dw k. ( Differeniaing he ansaz Y i,j = η dy i,j = 1 N δ i,j ) (X X i ), we ge: ( ) [ 1 N δ i,j (X X) i η η (q + (1 1 ) ] N )η d +η ( 1 N δ i,j)σ ( 1 N δ i,k)dw k. k=1 Idenifying wih he backward equaions: Z i,j,k = η σ( 1 N δ i,j)( 1 N δ i,k) for k = 1,,N, and η mus saisfy he Riccai equaion η = 2(a + q)η + (1 1 N 2 )η2 (ǫ q 2 ), wih he erminal condiion η T = c.

42 Soluion o he Forward-Backward Equaions By summaion of he forward equaions: dx = σ N N k=1 dw k. ( Differeniaing he ansaz Y i,j = η dy i,j = 1 N δ i,j ) (X X i ), we ge: ( ) [ 1 N δ i,j (X X) i η η (q + (1 1 ) ] N )η d +η ( 1 N δ i,j)σ ( 1 N δ i,k)dw k. k=1 Idenifying wih he backward equaions: Z i,j,k = η σ( 1 N δ i,j)( 1 N δ i,k) for k = 1,,N, and η mus saisfy he Riccai equaion η = 2(a + q)η + (1 1 N 2 )η2 (ǫ q 2 ), wih he erminal condiion η T = c.

43 Soluion o he Forward-Backward Equaions By summaion of he forward equaions: dx = σ N N k=1 dw k. ( Differeniaing he ansaz Y i,j = η dy i,j = 1 N δ i,j ) (X X i ), we ge: ( ) [ 1 N δ i,j (X X) i η η (q + (1 1 ) ] N )η d +η ( 1 N δ i,j)σ ( 1 N δ i,k)dw k. k=1 Idenifying wih he backward equaions: Z i,j,k = η σ( 1 N δ i,j)( 1 N δ i,k) for k = 1,,N, and η mus saisfy he Riccai equaion η = 2qη + (1 1 N 2 )η2 (ǫ q 2 ), wih he erminal condiion η T = c.

44 η = Soluion o he Riccai Equaion ( ) ) (ǫ q 2 ) e (δ+ δ )(T ) 1 c (δ + e (δ+ δ )(T ) δ ( δ e (δ+ δ )(T ) δ +) c(1 1 N ) ( e 2 (δ+ δ )(T ) 1 ), wih he noaions δ ± = q ± R, ( R = q ) N 2 (ǫ q 2 ) > 0. Observe ha η is well defined for any T since he denominaor can be wrien as ( ) ( ( e (δ+ δ )(T ) + 1 R a + q + c 1 1 )) ( N 2 e (δ+ δ )(T ) 1), which says negaive because δ + δ = 2 R > 0. In fac, using q 2 ǫ, we see ha η is posiive wih η T = c.

45 η = Soluion o he Riccai Equaion ( ) ) (ǫ q 2 ) e (δ+ δ )(T ) 1 c (δ + e (δ+ δ )(T ) δ ( δ e (δ+ δ )(T ) δ +) c(1 1 N ) ( e 2 (δ+ δ )(T ) 1 ), wih he noaions δ ± = q ± R, ( R = q) ) N 2 (ǫ q 2 ) > 0. Observe ha η is well defined for any T since he denominaor can be wrien as ( ) ( ( e (δ+ δ )(T ) + 1 R q + c 1 1 )) ( N 2 e (δ+ δ )(T ) 1), which says negaive because δ + δ = 2 R > 0, and in fac, using q 2 ǫ, we see ha η is posiive wih η T = c.

46 Financial Implicaions 1. Once he funcion η has been obained, bank i implemens is sraegy by using is conrol [ ˆα i = Y i,i + q(x X) i = q + (1 1 ] N )η (X X), i I requires is own log-reserve X i bu also he average reserve X which may or may no be known o he individual bank i. Observe ha he average X is given by dx = σ N N k=1 dw k, and is idenical o he average found in he unconrolled case. Therefore, sysemic risk occurs in he same manner as in he case of unconrolled dynamics. However, he conrol affecs he rae of borrowing and lending by adding he ime-varying componen q + (1 1 N )η o he unconrolled rae a.

47 Financial Implicaions 1. Once he funcion η has been obained, bank i implemens is sraegy by using is conrol [ ˆα i = Y i,i + q(x X) i = q + (1 1 ] N )η (X X), i I requires is own log-reserve X i bu also he average reserve X which may or may no be known o he individual bank i. Observe ha he average X is given by dx = σ N N k=1 dw k, and is idenical o he average found in he unconrolled case. Therefore, sysemic risk occurs in he same manner as in he case of unconrolled dynamics. However, he conrol affecs he rae of borrowing and lending by adding he ime-varying componen q + (1 1 N )η o he unconrolled rae a.

48 Financial Implicaions 2. In fac, he conrolled dynamics can be rewrien dx i = ( q + (1 1 ) 1 N )η N (X j X)d i + σdw i. The effec of he banks using heir opimal sraegies corresponds o iner-bank borrowing and lending a he effecive rae j=1 A := q + (1 1 N )η. Under his equilibrium, he cenral bank is simply a clearing house, and he sysem is operaing as if banks were borrowing from and lending o each oher a he rae A, and he ne effec is creaing liquidiy quanified by he rae of lending/borrowing.

49 Financial Implicaions 2. In fac, he conrolled dynamics can be rewrien dx i = ( q + (1 1 ) 1 N )η N (X j X)d i + σdw i. The effec of he banks using heir opimal sraegies corresponds o iner-bank borrowing and lending a he effecive rae j=1 A := q + (1 1 N )η. Under his equilibrium, he cenral bank is simply a clearing house, and he sysem is operaing as if banks were borrowing from and lending o each oher a he rae A, and he ne effec is creaing liquidiy quanified by he rae of lending/borrowing.

50 Financial Implicaions 3. For T large (mos of he ime T large), η is mainly consan. For insance, wih c = 0, lim T η = ǫ q2 δ := η Plos of η wih c = 0, q = 1, ǫ = 2 and T = 1 on he lef, T = 100 on he righ wih η 0.24 (here we used 1/N 0). A := a + q + (1 1 N )η.

51 Financial Implicaions 3. For T large (mos of he ime T large), η is mainly consan. For insance, wih c = 0, lim T η = ǫ q2 δ := η Plos of η wih c = 0, q = 1, ǫ = 2 and T = 1 on he lef, T = 100 on he righ wih η 0.24 (here we used 1/N 0). Therefore, in his infinie-horizon equilibrium, banks are borrowing and lending o each oher a he consan rae A := q + (1 1 N )η = q + η in he Mean Field Limi.

52 Equilibrium: Fully Conneced Symmeric Nework dx (i) = A N j=1 (X (j) ( X (i) ) d+σ ρdw ρ 2 dw (i) ), i = 1,...,N. N=10 Bu, how o generae a cenrally conneced nework?

53 Game wih a Cenral Bank Banks (i = 2,,N) are borrowing from and lending o a cenral bank (i = 1): ( dx i = αd i + σ ρdw 0 + ) 1 ρ 2 dw i, i = 2,,N ( N ( ) dx 1 = α )d i + σ 1 ρ 1 dw ρ 2 1 dw 1 i=2 where α i is he conrol of bank i which wans o minimize { } T J i (α 2,,α N ) = IE f i (X, α)d i + g i (X T ), 0 wih running cos [ 1 f i (x, α i ) = 2 (αi ) 2 qα i (x 1 x i ) + ǫ ] 2 (x1 x i ) 2, q 2 ǫ, and erminal cos g i (x) = c 2 ( x 1 x i) 2.

54 Equilibrium (skipping deails of he derivaion) dx i = A(X 1 X i )d + σ dx 1 = A ( X i X 1 i=2 ( ρdw 0 + ) 1 ρ 2 dw i, i = 2,,N ) ) d + σ1 ( ρ 1 dw ρ 2 1 dw 1 N=10

55 Comparison using Sysemic Risk Measures Join work in progress wih F. Biagini, M. Frielli and T. Meyer-Brandis Le X = (X 1,, X N ) be a risky posiion. Aggregae firs versus Allocae firs Aggregae firs: ρ(x) = inf { } m IR Λ 1 ( X i + m) A γ. i=1 For insance: Λ 1 (Y ) = (Y d) and A γ = {Z L 0 IE(Z) γ}. Allocae firs: { N } ρ(x) = inf m i Λ(X + m), m = (m 1,,m N ) IR N. i=1

56 Comparison using Sysemic Risk Measures Join work in progress wih F. Biagini, M. Frielli and T. Meyer-Brandis Le X = (X 1,, X N ) be a risky posiion. Aggregae firs versus Allocae firs Aggregae firs: ρ(x) = inf { m IR Λ 1 ( For insance: Λ 1 (Y ) = (Y d) } X i + m) A γ. i=1 and A γ = {Z L 0 IE(Z) γ}. Allocae firs: { N } ρ(x) = inf m i Λ(X + m), m = (m 1,,m N ) IR N. i=1 For insance: Λ(Y ) = (Y i d i ) and A γ = {Z L 0 IE(Z) γ}. i=1

57 Comparison using Sysemic Risk Measures Join work in progress wih F. Biagini, M. Frielli and T. Meyer-Brandis Le X = (X 1,, X N ) be a risky posiion. Aggregae firs versus Allocae firs Aggregae firs: ρ(x) = inf { m IR Λ 1 ( For insance: Λ 1 (Y ) = (Y d) } X i + m) A γ. i=1 and A γ = {Z L 0 IE(Z) γ}. Allocae firs: { N } ρ(x) = inf m i Λ(X + m) A γ, m = (m 1,,m N ) IR N. i=1 For insance: Λ(Y ) = (Y i d i ) and A γ = {Z L 0 IE(Z) γ}. i=1

58 The Gaussian Case Le X = (X 1,, X N ) be a Gaussian vecor wih marginal disribuions N(µ i, s 2 i), hen by allocaing firs we ge: ρ(x) = inf { N m i i=1 i=1 } IE[(X i + m i d i ) ] γ. = i=1 Ψ(m) := IE[(X i + m i d i ) ] i=1 [ s i exp ( (d i µ i m i ) 2 ) ( )] di µ i m i + (d i µ i m i )Φ. 2π 2s 2 i s i

59 Noing ha Opimal Allocaion Ψ(m) m i ( ) di µ i m i = Φ, a direc compuaion via Lagrange muliplier gives: where R solves s i m i = d i µ i s i R P(R) := 1 exp( R 2 /2) RΦ(R) = γ 2π N i=1 s i Noe ha his allocae firs approach gives a way o rank banks in erms of heir conribuions o sysemic risk. A direc analysis of m i µ i and m i σ i and increases wih s i. shows ha m i decreases wih µ i

60 Fully Conneced Symmeric Nework Here we consider he fully homogeneous case where x i 0 = x 0, p i,j = a/n, σ i = σ, ρ i = ρ, d i = d, so ha he model becomes dx i = a ( (X j X i N ) d+σ ρdw 0 + ) 1 ρ 2 dw i, i = 1,,N, j=1 N=10

61 Fully Conneced Symmeric Nework (coninued) X = (X 1,,X N ) is Gaussian wih marginal disribuions characerized by he means µ i = IE(X i ) = x 0, and he variances s 2 i = σ 2 (1 ρ 2 )(1 1 ( ) 1 e 2a N ) + σ (ρ ) ρ2. 2a N

62 Cenral Clearing Symmeric Nework dx i = a(x 1 X i )d + σ dx 1 = a ( X i X 1 i=2 ( ρdw 0 + ) 1 ρ 2 dw i, i = 2,,N ) ) d + σ1 ( ρ 1 dw ρ 2 1 dw 1 N=10

63 Cenral Clearing Symmeric Nework (coninued) X = (X 1,,X N ) is again Gaussian. Assuming X0 i = x 0, i = 1,,N, a edious compuaion gives IE(X ) i = x 0 and he marginal variances for i 2: s 2 i = σ 2 (1 ρ 2 )(1 1 ( ) 1 e 2a N ) + σ (ρ ) ρ2 2a N + 2σρ N [σ 1ρ 1 σρ] + O( 1 N 2 ), which shows ha he sign of [σ 1 ρ 1 σρ] decides which nework is more risky (a leas a he order 1/N).

64 Mean Field Game (N ) In mos cases he finie player game will no be solved explicily. For insance, wih a sligh generalizaion such as: dx i = a(x ) α i d + σdw i, i = 1,,N This is where MFG is needed.

65 Mean Field Game (N ) wih Common Noise 1. Fix (m ) 0 (he limi of X as N which depends on W 0 ) 1. Solve he conrol problem { T [ 1 2 (α ) 2 qα (m X ) + ǫ ] } 2 (m X ) 2 d + c 2 (m T X T ) 2 inf IE (α ) 0 subjec o: dx = [a(m X ) + α ]d + σ (ρdw 0 + ) 1 ρ 2 dw 2. Find m so ha m = IE[X (W 0 s ) s ] for all. Hamilonian: H(, x, y, α) = [a(m x) + α] y α2 qα(m x) H α ˆα = q(m x) y

66 Mean Field Game (N ) Fix (m ) 0 (he limi of X as N which depends on W 0 ) Solve he conrol problem { T [ 1 2 (α ) 2 qα (m X ) + ǫ ] } 2 (m X ) 2 d + c 2 (m T X T ) 2 inf IE (α ) 0 subjec o: dx = α d + σ (ρdw 0 + ) 1 ρ 2 dw Find m so ha m = IE[X (W 0 s ) s ] for all. Hamilonian: H(, x, y, α) = [a(m x) + α] y α2 qα(m x) H α ˆα = q(m x) y

67 Mean Field Game (N ) Fix (m ) 0 (he limi of X as N which depends on W 0 ) Solve he conrol problem { T [ 1 2 (α ) 2 qα (m X ) + ǫ ] } 2 (m X ) 2 d + c 2 (m T X T ) 2 inf IE (α ) 0 subjec o: dx = α d + σ (ρdw 0 + ) 1 ρ 2 dw Find m so ha m = IE[X (W 0 s ) s ] for all. Hamilonian: H(, x, y, α) = [a(m x) + α] y α2 qα(m x) H α ˆα = q(m x) y

68 Mean Field Game (N ) Fix (m ) 0 (he limi of X as N which depends on W 0 ) Solve he conrol problem { T [ 1 2 (α ) 2 qα (m X ) + ǫ ] } 2 (m X ) 2 d + c 2 (m T X T ) 2 inf IE (α ) 0 subjec o: dx = α d + σ (ρdw 0 + ) 1 ρ 2 dw Find m so ha m = IE[X (W 0 s ) s ] for all. Hamilonian: H(, x, y, α) = αy α2 qα(m x) + ǫ 2 (m x) 2 H α ˆα = q(m x) y See also he recen paper by Carmona-Delarue-Lacker.

69 Adjoin Equaions dx = [q(m X ) Y ]d + σ (ρdw 0 + ) 1 ρ 2 dw, X 0 = ξ dy = H x d + Z0 dw 0 + Z dw, Y T = c(x T m T ) = [ qy + (ǫ q 2 )(m X ) ] d + Z 0 dw 0 + Z dw. Use he noaion m X = IE[X (W 0 s ) s ], m Y = IE[Y (W 0 s ) s ]. Taking condiional expecaion given (Ws 0 ) s in he second equaion and using m = m X for all T and consequenly m Y T = c(mx T m T) = 0, gives: m Y = T e (a+q)(s ) Z 0 sdw 0 s. Then, aking condiional expecaions in he firs equaion gives: dm X = m Y d + ρσdw 0.

70 Adjoin Equaions dx = [q(m X ) Y ]d + σ (ρdw 0 + ) 1 ρ 2 dw, X 0 = ξ dy = H x d + Z0 dw 0 + Z dw, Y T = c(x T m T ) = [ qy + (ǫ q 2 )(m X ) ] d + Z 0 dw 0 + Z dw. Use he noaion m X = IE[X (W 0 s ) s ], m Y = IE[Y (W 0 s ) s ]. Taking condiional expecaion given (Ws 0 ) s in he second equaion and using m = m X for all T and consequenly m Y T = c(mx T m T) = 0, gives: m Y = T e (a+q)(s ) Z 0 sdw 0 s. Then, aking condiional expecaions in he firs equaion gives: dm X = m Y d + ρσdw 0.

71 Adjoin Equaions dx = [q(m X ) Y ]d + σ (ρdw 0 + ) 1 ρ 2 dw, X 0 = ξ dy = H x d + Z0 dw 0 + Z dw, Y T = c(x T m T ) = [ qy + (ǫ q 2 )(m X ) ] d + Z 0 dw 0 + Z dw. Use he noaion m X = IE[X (W 0 s ) s ], m Y = IE[Y (W 0 s ) s ]. Taking condiional expecaion given (Ws 0 ) s in he second equaion and using m = m X for all T and consequenly m Y T = c(mx T m T) = 0, gives: m Y = T e q(s ) Z 0 sdw 0 s. Then, aking condiional expecaions in he firs equaion gives: dm X = m Y d + ρσdw 0.

72 Adjoin Equaions dx = [q(m X ) Y ]d + σ (ρdw 0 + ) 1 ρ 2 dw, X 0 = ξ dy = H x d + Z0 dw 0 + Z dw, Y T = c(x T m T ) = [ qy + (ǫ q 2 )(m X ) ] d + Z 0 dw 0 + Z dw. Use he noaion m X = IE[X (W 0 s ) s ], m Y = IE[Y (W 0 s ) s ]. Taking condiional expecaion given (Ws 0 ) s in he second equaion and using m = m X for all T and consequenly m Y T = c(mx T m T) = 0, gives: m Y = T e q(s ) Z 0 sdw 0 s. Then, aking condiional expecaions in he firs equaion gives: dm X = m Y d + ρσdw 0.

73 Adjoin Equaions (coninued) Ansaz: Y = η (m X ) wih η deerminisic. Differeniaing his ansaz and using he forward equaion leads o dy = η (m X )d η d(m X ) = [ ( η + η (a + q + η )) (m X ) + η m Y ] d + η σ 1 ρ 2 dw. Plugging he ansaz in he backward equaion gives dy = [ (a + q)η + (ǫ q 2 ) ] (m X )d + Z 0 dw 0 + Z dw. Idenifying he wo Iô decomposiions, we deduce from he maringale erms ha Z 0 0 and Z = η σ 1 ρ 2. From m Y = T e (a+q)(s ) Z 0 sdw 0 s, we obain m Y = 0.

74 Adjoin Equaions (coninued) Ansaz: Y = η (m X ) wih η deerminisic. Differeniaing his ansaz and using he forward equaion leads o dy = η (m X )d η d(m X ) = [ ( η + η (q + η )) (m X ) + η m Y ] d + η σ 1 ρ 2 dw. Plugging he ansaz in he backward equaion gives dy = [ (a + q)η + (ǫ q 2 ) ] (m X )d + Z 0 dw 0 + Z dw. Idenifying he wo Iô decomposiions, we deduce from he maringale erms ha Z 0 0 and Z = η σ 1 ρ 2. From m Y = T e (a+q)(s ) Z 0 sdw 0 s, we obain m Y = 0.

75 Adjoin Equaions (coninued) Ansaz: Y = η (m X ) wih η deerminisic. Differeniaing his ansaz and using he forward equaion leads o dy = η (m X )d η d(m X ) = [ ( η + η (q + η )) (m X ) + η m Y ] d + η σ 1 ρ 2 dw. Plugging he ansaz in he backward equaion gives dy = [ qη + (ǫ q 2 ) ] (m X )d + Z 0 dw 0 + Z dw. Idenifying he wo Iô decomposiions, we deduce from he maringale erms ha Z 0 0 and Z = η σ 1 ρ 2. From m Y = T e (a+q)(s ) Z 0 sdw 0 s, we obain m Y = 0.

76 Adjoin Equaions (coninued) Ansaz: Y = η (m X ) wih η deerminisic. Differeniaing his ansaz and using he forward equaion leads o dy = η (m X )d η d(m X ) = [ ( η + η (q + η )) (m X ) + η m Y ] d + η σ 1 ρ 2 dw. Plugging he ansaz in he backward equaion gives dy = [ qη + (ǫ q 2 ) ] (m X )d + Z 0 dw 0 + Z dw. Idenifying he wo Iô decomposiions, we deduce from he maringale erms ha Z 0 0 and Z = η σ 1 ρ 2. From m Y = T e (a+q)(s ) Z 0 sdw 0 s, we obain m Y = 0.

77 Adjoin Equaions (coninued) Ansaz: Y = η (m X ) wih η deerminisic. Differeniaing his ansaz and using he forward equaion leads o dy = η (m X )d η d(m X ) = [ ( η + η (q + η )) (m X ) + η m Y ] d + η σ 1 ρ 2 dw. Plugging he ansaz in he backward equaion gives dy = [ qη + (ǫ q 2 ) ] (m X )d + Z 0 dw 0 + Z dw. Idenifying he wo Iô decomposiions, we deduce from he maringale erms ha Z 0 0 and Z = η σ 1 ρ 2. From m Y = T e q(s ) Z 0 sdw 0 s, we obain m Y = 0.

78 Mean Field Game Soluion Equaing he drifs in he wo Iô decomposiions, we ge η = η 2 + 2qη (ǫ q 2 ), η T = c, which is he same Riccai equaion as before bu wih N =. From m Y = 0, we deduce ha m X = IE(ξ) + ρσw 0 which will ener in he opimal conrol (q + η )(m X X ). Once a soluion o he MFG is found, on can use i o consruc approximae Nash equilibria for he finiely many players games. Here, if one assumes ha each player is given he informaion X, and if player i uses he sraegy α i = (q + η )(X X i ), which is he limi as N of he sraegy used in he finie players game, one sees how solving he limiing MFG problem can provide approximae Nash equilibria for which he financial implicaions are idenical as he ones given for he exac Nash equilibria.

79 Mean Field Game Soluion Equaing he drifs in he wo Iô decomposiions, we ge η = η 2 + 2qη (ǫ q 2 ), η T = c, which is he same Riccai equaion as before bu wih N =. From m Y = 0, we deduce ha m X = IE(ξ) + ρσw 0 which will ener in he opimal conrol (q + η )(m X X ). Once a soluion o he MFG is found, on can use i o consruc approximae Nash equilibria for he finiely many players games. Here, if one assumes ha each player is given he informaion X, and if player i uses he sraegy α i = (q + η )(X X), i which is he limi as N of he sraegy used in he finie players game, one sees how solving he limiing MFG problem can provide approximae Nash equilibria for which he financial implicaions are idenical as he ones given for he exac Nash equilibria.

80 Mean Field Game Soluion Equaing he drifs in he wo Iô decomposiions, we ge η = η 2 + 2qη (ǫ q 2 ), η T = c, which is he same Riccai equaion as before bu wih N =. From m Y = 0, we deduce ha m X = IE(ξ) + ρσw 0 which will ener in he opimal conrol (q + η )(m X X ). Once a soluion o he MFG is found, on can use i o consruc approximae Nash equilibria for he finiely many players games. Here, if one assumes ha each player is given he informaion X, and if player i uses he sraegy α i = (q + η )(X X i ), which is he limi as N of he sraegy used in he finie players game, one sees how solving he limiing MFG problem can provide approximae Nash equilibria for which he financial implicaions are idenical as he ones given for he exac Nash equilibria.

81 Mean Field Equilibrium As N, banks become independen, ˆα = q(0 x) y = (q + η)x, and hey follow he dynamics dx = (q + η )X d + σdw Bu observe ha he large deviaion probabiliy of sysemic risk has been los.

82 MFG wih Common Noise / HJB Approach For Markovian sraegies α(, x), he dynamics are given by dx = α(, X )d + σ (ρdw 0 + ) 1 ρ 2 dw. Forward equaion for he condiional densiy of X given W 0 : dp = { x [(a(m x) + α(, x)) p ] + 12 } σ2 (1 ρ 2 ) xx p d ρσ( x p )dw 0 wih α(, x) given and m = xp (x)dx. The condiional mean m of X given W 0 is Markovian wih infiniesimal generaor denoed by L m d + ρσ( m )dw 0.

83 MFG wih Common Noise / HJB Approach For Markovian sraegies α(, x), he dynamics are given by dx = α(, X )d + σ (ρdw 0 + ) 1 ρ 2 dw. Forward equaion for he condiional densiy of X given W 0 : dp = { x [α(, x)p ] + 12 } σ2 (1 ρ 2 ) xx p d ρσ( x p )dw 0 wih α(, x) given and m = xp (x)dx. The condiional mean m of X given W 0 is Markovian wih infiniesimal generaor denoed by L m d + ρσ( m )dw 0.

84 Sochasic HJB Equaion The HJB equaion for he value funcion V (, x, m) is [ ] 1 dv + 2 σ2 (1 ρ 2 ) xx V + L m d m, X V + ( xm V ) d d + inf {α x V + α2 α 2 qα(m x) + ǫ } (m x)2 d 2 + ρσ( m V )dw 0 + ρσ( x V )dw 0 = 0. Minimize in α o ge ˆα = q(m x) x V Make he ansaz V (, x, m) = η 2 (m x) 2 + µ Plug in he forward equaion for p, muliply by x, and inegrae wih respec o x gives: dm = ρσdw 0

85 Mean Field Game Soluion Therefore, condiionally in W 0, L m V = 0 and d m, X = 0. Then, verifying ha he ansaz saisfies he HJB equaion, by canceling erms in (m x) 2 we obain ha η mus saisfy he same Riccai equaion as before (no affeced by ρ): η = η 2 + 2qη (ǫ q 2 ), η T = 0. Canceling sae-independen erms leads o µ = 1 2 σ2 (1 ρ 2 )η and herefore µ = 1 2 σ2 (1 ρ 2 ) T η s ds.

86 THANKS FOR YOUR ATTENTION