Financial Contagion with Multiple Illiquid Assets Zach Feinstein Electrical and Systems Engineering, Washington University in St. Louis Western Conference in Mathematical Finance March 24-25, 2017
1. The Eisenberg & Noe Local Model The Eisenberg & Noe Local Model Z. Feinstein Financial Contagion with Multiple Illiquid Assets 2 / 24
1. The Eisenberg & Noe Local Model Network Model with Local Interactions Only : Eisenberg & Noe (2001) n financial firms Nominal liability matrix: ( L ij ) i,j =0 , 1 , 2 ,...,n p i = � n Total liabilities: ¯ j =0 L ij Relative liabilities: � L ij if ¯ p i > 0 , ¯ p i a ij = 0 if ¯ p i = 0 . Z. Feinstein Financial Contagion with Multiple Illiquid Assets 3 / 24
1. The Eisenberg & Noe Local Model 2 6 3 1 5 4 Z. Feinstein Financial Contagion with Multiple Illiquid Assets 4 / 24
1. The Eisenberg & Noe Local Model Network Model with Local Interactions Only : Liquid endowment: x ∈ R n + Obligations fulfilled via transfers of the liquid asset. Equilibrium computed as fixed point: p ∈ R n + : n � , p i = ¯ p i ∧ x i + a ji p j i = 1 , 2 , ..., n j =1 Existence: Tarski’s fixed point theorem: maximal and minimal fixed points p − ≤ p + . Z. Feinstein Financial Contagion with Multiple Illiquid Assets 5 / 24
1. The Eisenberg & Noe Local Model Network Model with Local Interactions Only : Uniqueness S ⊆ { 1 , 2 , ..., n } is a surplus set if L ij = 0 and � i ∈ S x i > 0 for all ( i, j ) ∈ S × S c o ( i ) = { j ∈ { 1 , 2 , ..., n } | ∃ directed path from i to j } If o ( i ) is a surplus set for every bank i then there exists a unique payment vector p := p + = p − (Banach fixed point theorem) Z. Feinstein Financial Contagion with Multiple Illiquid Assets 6 / 24
2. Multilayered Financial Networks Multilayered Financial Networks Z. Feinstein Financial Contagion with Multiple Illiquid Assets 7 / 24
2. Multilayered Financial Networks Multilayered Network Model : Montagna & Kok (2013), Poledna, Molina-Borboa, Martinez-Jaramillo, Leij & Thurner (2015), Battiston, Caldarelli & D’Errico (2016), Feinstein (2017) Endowment: x ∈ R n × m + Nominal liabilities: L ∈ R n × n × m + Obligations must be fulfilled via transfers of the physical assets. Assets may be transferred to cover obligations or maximize utility, but these are subject to price impact described by the inverse demand function. Inverse demand function: F : R m → R m + maps units of illiquid assets being sold (positive input) or bought (negative input) into corresponding prices in some (possibly fictitious) num´ eraire. Z. Feinstein Financial Contagion with Multiple Illiquid Assets 8 / 24
2. Multilayered Financial Networks Multilayered Network Model : Assume inverse demand function is continuous and q ] ⊆ R m nonincreasing with codomain [ q, ¯ ++ . Assume the network model in asset k follows the Eisenberg & Noe (2001) model: i := � n p k j =1 L k Total liabilities: ¯ ij � L k p k ij if ¯ i > 0 Relative liabilities: a k p k ij := ¯ i p k 0 if ¯ i = 0 Firm portfolio holdings: y ∈ R n × m + Initial portfolio wealth: for firm i in asset k is n � x k a k p k j ∧ y k i + ji [¯ j ] . j =1 Z. Feinstein Financial Contagion with Multiple Illiquid Assets 9 / 24
2. Multilayered Financial Networks Multilayered Network Model : Payments must be made so that positive equity only accumulates after all obligations are paid p i ∈ P i ( y ∗ , q ∗ ) m m n � � � q ∗ k p k q ∗ x k a k p k j ∧ y ∗ k ⊆ Eff p i | i ≤ i + ji [¯ j ] . k p i ∈ [0 , ¯ p i ] j =1 k =1 k =1 Holdings may involve futher transfers to maximize utility y i ∈ Y i ( y ∗ , q ∗ ) = arg max u i ( e i ; y ∗ − i , q ∗ ) | e i ∈ H i � � e i ∈ R m + � � P i ( y ∗ , q ∗ ) , p i ∧ e i ¯ ∈ � H i = e i � � � �� � m � m i + � n k =1 q ∗ k =1 q ∗ x k j =1 a k p k j ∧ y ∗ k = ¯ � k e i k ji j � Z. Feinstein Financial Contagion with Multiple Illiquid Assets 10 / 24
2. Multilayered Financial Networks Multilayered Network Model : Prices update based on asset transfers n n � � q ( y ∗ , q ∗ ) = F x k a k p k j ∧ y ∗ k j ] − y ∗ k i + ji [¯ i i =1 j =1 k =1 ,...,m � n � � p i ∧ y ∗ i ] − y ∗ = F ( x i + [¯ i ) i =1 Equilibrium computed as fixed point: ( y, q ) ∈ R n × m × [ q, ¯ q ] + � n � � ( y, q ) ∈ Y i ( y, q ) × { q ( y, q ) } i =1 Z. Feinstein Financial Contagion with Multiple Illiquid Assets 11 / 24
2. Multilayered Financial Networks Multilayered Network Model : Existence Let P i be given as the maximizer of a continuous regulatory function h i which is strictly increasing and strictly quasi-concave in the first component P i ( y ∗ , q ∗ ) = m m n � � � h i ( p i ; y ∗ , q ∗ ) | q ∗ k p k q ∗ x k a k p k j ∧ y ∗ k arg max i ≤ i + ji [¯ j ] k p i ∈ [0 , ¯ p i ] j =1 k =1 k =1 Let u i be jointly continuous and quasi-concave in the first component There exists an equilibrium solution (via Berge maximum theorem and Kakutani fixed point theorem) � n � � ( y, q ) ∈ Y i ( y, q ) × { q ( y, q ) } i =1 Z. Feinstein Financial Contagion with Multiple Illiquid Assets 12 / 24
2. Multilayered Financial Networks Multilayered Network Model : Existence Assume h i and u i satisfy a dynamic programming principle p ) = h i ( p − P i ( y ′ , q ); y, x − P i ( y ′ , q ) , ¯ p − P i ( y ′ , q )) h i ( p ; y, x, ¯ p − Y i ( y ′ , q )) + ) p ) = u i ( e − Y i ( y ′ , q ); y, x − Y i ( y ′ , q ) , (¯ u i ( e ; y, x, ¯ For every q : There exists a greatest and least equilibrium holdings y ↑ ( q ) ≥ y ↓ ( q ) Positive equity of all firms is equal for every fixed point p i ) + = ( y ↓ ( y ↑ p i ) + i ( q ) − ¯ i ( q ) − ¯ If every bank owes positive amount to sink node (0) in every asset, then y ↑ ( q ) = y ↓ ( q ) for every q . q ] → R n × m This unique equilibrium y : [ q, ¯ is continuous. + Z. Feinstein Financial Contagion with Multiple Illiquid Assets 13 / 24
2. Multilayered Financial Networks Multilayered Network Model : Case Study A m = 2 assets; F 1 ≡ 1 � f ( z 2 ) if z 2 ≥ 0 F 2 ( z ) = 1 if z 2 < 0 f ( α − 1 ( − z 2 )) f ( z ) = 3 tan − 1 ( − z ) + 2 π ; α ( z ) = zf ( z ) 2 π n = 20 firms and a society node 25% of connection of size 1 between firms in each asset independently All firms owe 1 in each asset to the society node Initial endowments uniformly chosen between 0 and 20 and split evenly between the two assets Z. Feinstein Financial Contagion with Multiple Illiquid Assets 14 / 24
2. Multilayered Financial Networks Case Study A Comparison of initial price q* to resultant price q(y(q*),q*) 4 3.5 3 2.5 surplus priority proportional 2 wealth maximizing 1.5 1 0.5 0.5 1 1.5 2 2.5 3 3.5 4 Figure: A comparison of different regulatory and utility schemes with two assets. Z. Feinstein Financial Contagion with Multiple Illiquid Assets 15 / 24
2. Multilayered Financial Networks Multilayered Network Model : Case Study B m = 3 assets; F 1 ≡ 1 F k ( z ) = tan − 1 ( − 1 . 5 z k ) + π tan − 1 ( − 1 . 5 z 1 ) + π n = 10 firms and a society node 50% of connection of size 1 between firms in each asset independently All firms owe 1 in each asset to the society node Initial endowments of 5 split between the three assets (uniform between 10 9 and 20 9 in first asset, remainder split evenly in 2nd and 3rd) Z. Feinstein Financial Contagion with Multiple Illiquid Assets 16 / 24
2. Multilayered Financial Networks Case Study B Minimum trading: initial price q* vs. resultant price q(y(q*),q*) 3 1 0.8 0.6 0.6 4 0.6 . 0 2.5 1 0.8 0.4 0 1 . 0.8 4 2 0.4 1.2 0.6 q ∗ 3 0.6 0.8 1.5 0.6 1 1.4 4 0.2 . 1.6 0.1 1 1.8 1 1.6 0.8 0.4 0.4 1 1.8 1 0.5 . 0.6 2 2 0.5 1 1.5 2 2.5 3 q ∗ 2 Figure: Proportional regulation and minimal trading utility with three assets. Z. Feinstein Financial Contagion with Multiple Illiquid Assets 17 / 24
2. Multilayered Financial Networks Case Study B Wealth maximizing: initial price q* vs. resultant price q(y(q*),q*) 3 3 3.5 3 2.5 2.5 2 . 2 5 3 2 2.5 q ∗ 3 1.5 1.5 2 2 1.5 2.5 1 1 1 . 5 2 2 5 2 . 0.5 3 5 2 . 0.5 1 1.5 2 2.5 3 q ∗ 2 Figure: Proportional regulation and wealth maximizing utility with three assets. Z. Feinstein Financial Contagion with Multiple Illiquid Assets 18 / 24
2. Multilayered Financial Networks Multilayered Network Model : Grexit Study m = 2 assets; F 1 ≡ 1 � f ( z 2 ) if z 2 ≥ 0 F 2 ( z ) = 1 if z 2 < 0 f ( α − 1 ( − z 2 )) f ( z ) = 4 tan − 1 ( − 10 − 4 z ) + 3 π ; α ( z ) = zf ( z ) 3 π n = 87 firms and a society node Calibrated to EBA data with Gandy & Veraart (2016) Under Eisenberg & Noe (2001) : No failures Z. Feinstein Financial Contagion with Multiple Illiquid Assets 19 / 24
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