Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Crises and Liquidity in Over-the-Counter Markets Ricardo Lagos NYU
Guillaume Rocheteau
Pierre-Olivier Weill
UC Irvine
UCLA
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Over-the-counter markets Trade in asset markets is often distinctively non-Walrasian Over-the-counter markets 1
decentralized, no formal organization
2
trade is bilateral, prices and quantities negotiated
Many assets are traded in over-the-counter markets Some stocks Currencies Federal funds Bonds ABS, MBS...
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Over-the-counter markets Trade in asset markets is often distinctively non-Walrasian Over-the-counter markets 1
decentralized, no formal organization
2
trade is bilateral, prices and quantities negotiated
Many assets are traded in over-the-counter markets Some stocks Currencies Federal funds Bonds ABS, MBS...
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Over-the-counter markets Trade in asset markets is often distinctively non-Walrasian Over-the-counter markets 1
decentralized, no formal organization
2
trade is bilateral, prices and quantities negotiated
Many assets are traded in over-the-counter markets Some stocks Currencies Federal funds Bonds ABS, MBS...
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Over-the-counter markets Trade in asset markets is often distinctively non-Walrasian Over-the-counter markets 1
decentralized, no formal organization
2
trade is bilateral, prices and quantities negotiated
Many assets are traded in over-the-counter markets Some stocks Currencies Federal funds Bonds ABS, MBS...
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Over-the-counter markets Trade in asset markets is often distinctively non-Walrasian Over-the-counter markets 1
decentralized, no formal organization
2
trade is bilateral, prices and quantities negotiated
Many assets are traded in over-the-counter markets Some stocks Currencies Federal funds Bonds ABS, MBS...
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Over-the-counter markets Trade in asset markets is often distinctively non-Walrasian Over-the-counter markets 1
decentralized, no formal organization
2
trade is bilateral, prices and quantities negotiated
Many assets are traded in over-the-counter markets Some stocks Currencies Federal funds Bonds ABS, MBS...
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Over-the-counter markets Trade in asset markets is often distinctively non-Walrasian Over-the-counter markets 1
decentralized, no formal organization
2
trade is bilateral, prices and quantities negotiated
Many assets are traded in over-the-counter markets Some stocks Currencies Federal funds Bonds ABS, MBS...
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Over-the-counter markets Trade in asset markets is often distinctively non-Walrasian Over-the-counter markets 1
decentralized, no formal organization
2
trade is bilateral, prices and quantities negotiated
Many assets are traded in over-the-counter markets Some stocks Currencies Federal funds Bonds ABS, MBS...
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Over-the-counter markets Trade in asset markets is often distinctively non-Walrasian Over-the-counter markets 1
decentralized, no formal organization
2
trade is bilateral, prices and quantities negotiated
Many assets are traded in over-the-counter markets Some stocks Currencies Federal funds Bonds ABS, MBS...
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Over-the-counter markets Trade in asset markets is often distinctively non-Walrasian Over-the-counter markets 1
decentralized, no formal organization
2
trade is bilateral, prices and quantities negotiated
Many assets are traded in over-the-counter markets Some stocks Currencies Federal funds Bonds ABS, MBS...
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Liquidity provision in over-the-counter markets
Financial liquidity: The ability to trade cheaply (e.g., fast and at low cost) Broker-dealers—provide liquidity (“immediacy”): Finding counterparts for trade Becoming counterparts for trade
Liquidity provision is relevant in: OTC-style markets—all the time Particularly during times of large trade “imbalances”
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Liquidity provision in over-the-counter markets
Financial liquidity: The ability to trade cheaply (e.g., fast and at low cost) Broker-dealers—provide liquidity (“immediacy”): Finding counterparts for trade Becoming counterparts for trade
Liquidity provision is relevant in: OTC-style markets—all the time Particularly during times of large trade “imbalances”
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Liquidity provision in over-the-counter markets
Financial liquidity: The ability to trade cheaply (e.g., fast and at low cost) Broker-dealers—provide liquidity (“immediacy”): Finding counterparts for trade Becoming counterparts for trade
Liquidity provision is relevant in: OTC-style markets—all the time Particularly during times of large trade “imbalances”
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Liquidity provision in over-the-counter markets
Financial liquidity: The ability to trade cheaply (e.g., fast and at low cost) Broker-dealers—provide liquidity (“immediacy”): Finding counterparts for trade Becoming counterparts for trade
Liquidity provision is relevant in: OTC-style markets—all the time Particularly during times of large trade “imbalances”
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Liquidity provision in over-the-counter markets
Financial liquidity: The ability to trade cheaply (e.g., fast and at low cost) Broker-dealers—provide liquidity (“immediacy”): Finding counterparts for trade Becoming counterparts for trade
Liquidity provision is relevant in: OTC-style markets—all the time Particularly during times of large trade “imbalances”
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Liquidity provision in over-the-counter markets
Financial liquidity: The ability to trade cheaply (e.g., fast and at low cost) Broker-dealers—provide liquidity (“immediacy”): Finding counterparts for trade Becoming counterparts for trade
Liquidity provision is relevant in: OTC-style markets—all the time Particularly during times of large trade “imbalances”
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Liquidity provision in over-the-counter markets
Financial liquidity: The ability to trade cheaply (e.g., fast and at low cost) Broker-dealers—provide liquidity (“immediacy”): Finding counterparts for trade Becoming counterparts for trade
Liquidity provision is relevant in: OTC-style markets—all the time Particularly during times of large trade “imbalances”
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Motivating example I: Crash of 1987 NASDAQ: market makers did not provide liquidity: “During the week of October 19, some market makers formally withdrew from making markets. In addition, some market makers ceased performing their function, merely by not answering their telephones during this period. (...) Other market makers who were willing to trade were unreachable when they were overwhelmed by the volume of telephone orders, many of which normally would have been executed by the automated systems.” Report of the Presidential Task Force on Market Mechanisms (1988)
Why didn’t dealers “lean against the wind”?
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Motivating example I: Crash of 1987 NASDAQ: market makers did not provide liquidity: “During the week of October 19, some market makers formally withdrew from making markets. In addition, some market makers ceased performing their function, merely by not answering their telephones during this period. (...) Other market makers who were willing to trade were unreachable when they were overwhelmed by the volume of telephone orders, many of which normally would have been executed by the automated systems.” Report of the Presidential Task Force on Market Mechanisms (1988)
Why didn’t dealers “lean against the wind”?
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Motivating example II: Financial crisis 2007–2009
Liquidity dried up in many OTC markets (e.g., MBS) Fed responded agressively: In 2008: liquidity facilities to channel funds directly to dealers (some to entice them to purchase particular asset classes) In November 2008: direct asset purchases
Why didn’t dealers buy these assets themselves?
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Motivating example II: Financial crisis 2007–2009
Liquidity dried up in many OTC markets (e.g., MBS) Fed responded agressively: In 2008: liquidity facilities to channel funds directly to dealers (some to entice them to purchase particular asset classes) In November 2008: direct asset purchases
Why didn’t dealers buy these assets themselves?
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Motivating example II: Financial crisis 2007–2009
Liquidity dried up in many OTC markets (e.g., MBS) Fed responded agressively: In 2008: liquidity facilities to channel funds directly to dealers (some to entice them to purchase particular asset classes) In November 2008: direct asset purchases
Why didn’t dealers buy these assets themselves?
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Motivating example II: Financial crisis 2007–2009
Liquidity dried up in many OTC markets (e.g., MBS) Fed responded agressively: In 2008: liquidity facilities to channel funds directly to dealers (some to entice them to purchase particular asset classes) In November 2008: direct asset purchases
Why didn’t dealers buy these assets themselves?
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Motivating example II: Financial crisis 2007–2009
Liquidity dried up in many OTC markets (e.g., MBS) Fed responded agressively: In 2008: liquidity facilities to channel funds directly to dealers (some to entice them to purchase particular asset classes) In November 2008: direct asset purchases
Why didn’t dealers buy these assets themselves?
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Liquidity facilities: Primary Dealer Credit Facility collateralized loans to primary dealers Term Securities Lending Facility collateralized loans of treasury securities to primary dealers ABCP MMMF Liquidity Facility direct loans to depository institutions and bank-holding companies to finance purchases of ABCP held by MMMFs Money Market Investor Funding Facility direct loans to MM investors to finance purchases of certain assets from eligible investors Direct asset purchases plans: Up to $100bn in GSE debt (up to $200bn in March 2009) Up to $500bn in MBS (up to $1.25 trillion in March 2009)
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
What we do
We study liquidity provision by dealers, following a “crisis” in a theoretical trading model that incorporates the key frictions characteristic of OTC markets (search and bargaining)
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
The questions
1
Should dealers provide liquidity in times of large imbalances?
2
What determines their incentives to do so?
3
Could there be an efficiency rationale for the government to “act as a dealer”?
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
The questions
1
Should dealers provide liquidity in times of large imbalances?
2
What determines their incentives to do so?
3
Could there be an efficiency rationale for the government to “act as a dealer”?
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
The questions
1
Should dealers provide liquidity in times of large imbalances?
2
What determines their incentives to do so?
3
Could there be an efficiency rationale for the government to “act as a dealer”?
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Related work Duffie, Gˆ arleanu and Pedersen (2005) Restrictions on asset holdings, a ∈ {0, 1} Steady state Dealers cannot hold inventories
Weill (2007) Restrictions on asset holdings, a ∈ {0, 1} Dynamic path to the steady state Dealers hold inventories
Lagos and Rocheteau (2009) Asset holdings a ∈ R + Steady state and dynamics Dealers cannot hold inventories
Efficiency
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Relative to the existing search-based OTC literature... Richer dynamics: stochastic transitions Dealers can hold inventories Asset holdings a ∈ R + Government interventions Unrestricted asset holdings ⇒ 1
Investors can supply liquidity to other investors
2
Demand for liquidity is endogenous Investors mitigate trading frictions by choosing asset holdings that prevent them from having to unwind large positions
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Environment
Continuous time, infinite horizon Two types of infinitely-lived agents (unit measure of each) dealers investors
An asset (e.g., a tree) perfectly divisible, in fixed supply A ∈ R + yields a nontradable dividend flow to its owner (e.g., fruit)
A numeraire good consumed and produced by all agents
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Preferences
Investors’ instantaneous utility: θ (t ) ui (a) + c a ∈ R + dividend flow generated by a units of asset c ∈ R flow net consumption of numeraire good i ∈ X = {1, ..., I } index for idiosyncratic preference shock θ (t ) aggregate preference shock (to create a crisis)
Idiosyncratic preference shocks at Poisson rate δ Probability of preference type i is π i Dealers’ instantaneous utility: c All agents discount at rate r
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Crisis scenario
At t = 0, θ (t ) jumps down from θ (t ) = 1 to θ (t ) = θ < 1 At a random Poisson time (intensity ρ), θ (t ) jumps back to 1 θ (t ) θ (t ) = 1 θ (t ) < 1 time random recovery time
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Trading arrangement
Dealers have continuous access to a competitive interdealer market may hold any nonnegative asset position
Investors contact dealers at random with Poisson rate α may hold any nonnegative asset position
When a dealer and an investor make contact, they trade bilaterally with terms of trade determined by Nash bargaining
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Trading arrangement
Dealers have continuous access to a competitive interdealer market may hold any nonnegative asset position
Investors contact dealers at random with Poisson rate α may hold any nonnegative asset position
When a dealer and an investor make contact, they trade bilaterally with terms of trade determined by Nash bargaining
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Trading arrangement
Dealers have continuous access to a competitive interdealer market may hold any nonnegative asset position
Investors contact dealers at random with Poisson rate α may hold any nonnegative asset position
When a dealer and an investor make contact, they trade bilaterally with terms of trade determined by Nash bargaining
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Trading arrangement
Dealers have continuous access to a competitive interdealer market may hold any nonnegative asset position
Investors contact dealers at random with Poisson rate α may hold any nonnegative asset position
When a dealer and an investor make contact, they trade bilaterally with terms of trade determined by Nash bargaining
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Trading arrangement
Dealers have continuous access to a competitive interdealer market may hold any nonnegative asset position
Investors contact dealers at random with Poisson rate α may hold any nonnegative asset position
When a dealer and an investor make contact, they trade bilaterally with terms of trade determined by Nash bargaining
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Trading arrangement
Dealers have continuous access to a competitive interdealer market may hold any nonnegative asset position
Investors contact dealers at random with Poisson rate α may hold any nonnegative asset position
When a dealer and an investor make contact, they trade bilaterally with terms of trade determined by Nash bargaining
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Trading arrangement
Dealers have continuous access to a competitive interdealer market may hold any nonnegative asset position
Investors contact dealers at random with Poisson rate α may hold any nonnegative asset position
When a dealer and an investor make contact, they trade bilaterally with terms of trade determined by Nash bargaining
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Trading arrangement
Dealers have continuous access to a competitive interdealer market may hold any nonnegative asset position
Investors contact dealers at random with Poisson rate α may hold any nonnegative asset position
When a dealer and an investor make contact, they trade bilaterally with terms of trade determined by Nash bargaining
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Trading arrangement
Dealers have continuous access to a competitive interdealer market may hold any nonnegative asset position
Investors contact dealers at random with Poisson rate α may hold any nonnegative asset position
When a dealer and an investor make contact, they trade bilaterally with terms of trade determined by Nash bargaining
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Market
α
Investors
Inter-dealer
Dealers
α
Dealers
Investors
Trading arrangement
Efficiency
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
We construct the equilibrium in two steps: Step 1: Characterize equilibrium following the realization of Tρ Step 2: Characterize equilibrium before the realization of Tρ (taking as given the continuation recovery path of Step 1, and the fact that the recovery time, Tρ , is random)
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
We construct the equilibrium in two steps: Step 1: Characterize equilibrium following the realization of Tρ Step 2: Characterize equilibrium before the realization of Tρ (taking as given the continuation recovery path of Step 1, and the fact that the recovery time, Tρ , is random)
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Investor’s problem following recovery For t ≥ Tρ : Z
T
e −r (s −t ) uk (s ) (a)ds t n −r (T −t ) +e V k ( T ) [ ak ( T ) ( T ) , T ]
Vi (a, t ) = Ei
−p (T )[ak (T ) (T ) − a] − φk (T ) (a,T )
o
p (t ) : competitive price of the asset at time t ai (t ) : asset holdings chosen by an investor of type i at time t φi (a,t ) : intermediation fee paid to the dealer by an investor with preference type i and asset holdings a at time t
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Investor’s problem following recovery For t ≥ Tρ : Z
T
e −r (s −t ) uk (s ) (a)ds t n −r (T −t ) +e V k ( T ) [ ak ( T ) ( T ) , T ]
Vi (a, t ) = Ei
−p (T )[ak (T ) (T ) − a] − φk (T ) (a,T )
o
p (t ) : competitive price of the asset at time t ai (t ) : asset holdings chosen by an investor of type i at time t φi (a,t ) : intermediation fee paid to the dealer by an investor with preference type i and asset holdings a at time t
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Investor’s problem following recovery For t ≥ Tρ : Z
T
e −r (s −t ) uk (s ) (a)ds t n −r (T −t ) +e V k ( T ) [ ak ( T ) ( T ) , T ]
Vi (a, t ) = Ei
−p (T )[ak (T ) (T ) − a] − φk (T ) (a,T )
o
p (t ) : competitive price of the asset at time t ai (t ) : asset holdings chosen by an investor of type i at time t φi (a,t ) : intermediation fee paid to the dealer by an investor with preference type i and asset holdings a at time t
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Investor’s problem following recovery For t ≥ Tρ : Z
T
e −r (s −t ) uk (s ) (a)ds t n −r (T −t ) +e V k ( T ) [ ak ( T ) ( T ) , T ]
Vi (a, t ) = Ei
−p (T )[ak (T ) (T ) − a] − φk (T ) (a,T )
o
p (t ) : competitive price of the asset at time t ai (t ) : asset holdings chosen by an investor of type i at time t φi (a,t ) : intermediation fee paid to the dealer by an investor with preference type i and asset holdings a at time t
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Dealer’s problem following recovery Z W (at , t ) = max − q (s )
s.t.
∞
e −r (s −t ) p (s )q (s )ds
t
a˙ d (s ) = q (s ) ,
ad (s ) ≥ 0,
+ Φ (t )
ad ( t ) = at
ad (s ) : dealer’s asset inventory at time s q (s ) : change in the dealer’s asset inventory at time s Φ (t ) = E {e −r (T −t ) [φ¯ (T ) + Φ (T )]} R φ¯ (t ) = φj (a, t )dHt
Ht (A, I) : time-t measure of investors with asset holding a in the set A ⊆R + and preference type i in the set I ⊆X
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Dealer’s problem following recovery Z W (at , t ) = max − q (s )
s.t.
∞
e −r (s −t ) p (s )q (s )ds
t
a˙ d (s ) = q (s ) ,
ad (s ) ≥ 0,
+ Φ (t )
ad ( t ) = at
ad (s ) : dealer’s asset inventory at time s q (s ) : change in the dealer’s asset inventory at time s Φ (t ) = E {e −r (T −t ) [φ¯ (T ) + Φ (T )]} R φ¯ (t ) = φj (a, t )dHt
Ht (A, I) : time-t measure of investors with asset holding a in the set A ⊆R + and preference type i in the set I ⊆X
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Dealer’s problem following recovery Z W (at , t ) = max − q (s )
s.t.
∞
e −r (s −t ) p (s )q (s )ds
t
a˙ d (s ) = q (s ) ,
ad (s ) ≥ 0,
+ Φ (t )
ad ( t ) = at
ad (s ) : dealer’s asset inventory at time s q (s ) : change in the dealer’s asset inventory at time s Φ (t ) = E {e −r (T −t ) [φ¯ (T ) + Φ (T )]} R φ¯ (t ) = φj (a, t )dHt
Ht (A, I) : time-t measure of investors with asset holding a in the set A ⊆R + and preference type i in the set I ⊆X
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Dealer’s problem following recovery Z W (at , t ) = max − q (s )
s.t.
∞
e −r (s −t ) p (s )q (s )ds
t
a˙ d (s ) = q (s ) ,
ad (s ) ≥ 0,
+ Φ (t )
ad ( t ) = at
ad (s ) : dealer’s asset inventory at time s q (s ) : change in the dealer’s asset inventory at time s Φ (t ) = E {e −r (T −t ) [φ¯ (T ) + Φ (T )]} R φ¯ (t ) = φj (a, t )dHt
Ht (A, I) : time-t measure of investors with asset holding a in the set A ⊆R + and preference type i in the set I ⊆X
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Dealer’s problem following recovery Z W (at , t ) = max − q (s )
s.t.
∞
e −r (s −t ) p (s )q (s )ds
t
a˙ d (s ) = q (s ) ,
ad (s ) ≥ 0,
+ Φ (t )
ad ( t ) = at
ad (s ) : dealer’s asset inventory at time s q (s ) : change in the dealer’s asset inventory at time s Φ (t ) = E {e −r (T −t ) [φ¯ (T ) + Φ (T )]} R φ¯ (t ) = φj (a, t )dHt
Ht (A, I) : time-t measure of investors with asset holding a in the set A ⊆R + and preference type i in the set I ⊆X
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Dealer’s problem following recovery Z W (at , t ) = max − q (s )
s.t.
∞
e −r (s −t ) p (s )q (s )ds
t
a˙ d (s ) = q (s ) ,
ad (s ) ≥ 0,
+ Φ (t )
ad ( t ) = at
ad (s ) : dealer’s asset inventory at time s q (s ) : change in the dealer’s asset inventory at time s Φ (t ) = E {e −r (T −t ) [φ¯ (T ) + Φ (T )]} R φ¯ (t ) = φj (a, t )dHt
Ht (A, I) : time-t measure of investors with asset holding a in the set A ⊆R + and preference type i in the set I ⊆X
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Brokered trades Dealer’s bargaining power: η ∈ [0, 1]
[ai (t ), φi (a, t )] = arg max [Vi (a′ , t ) − Vi (a, t ) − p (t ) (a′ − a) − φ]1−η φη (a ′ ,φ)
Lemma ai (t ) = arg max Vi ( a ′ , t ) − p ( t ) a ′ ′ a ≥0
φi (a, t ) = η {Vi [ai (t ) , t ] − Vi (a, t ) − p (t ) [ai (t ) − a]} ai (t ) maximizes the total gains from trade φi (a, t ) splits the gains from trade according to η
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Brokered trades Dealer’s bargaining power: η ∈ [0, 1]
[ai (t ), φi (a, t )] = arg max [Vi (a′ , t ) − Vi (a, t ) − p (t ) (a′ − a) − φ]1−η φη (a ′ ,φ)
Lemma ai (t ) = arg max Vi ( a ′ , t ) − p ( t ) a ′ ′ a ≥0
φi (a, t ) = η {Vi [ai (t ) , t ] − Vi (a, t ) − p (t ) [ai (t ) − a]} ai (t ) maximizes the total gains from trade φi (a, t ) splits the gains from trade according to η
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Brokered trades Dealer’s bargaining power: η ∈ [0, 1]
[ai (t ), φi (a, t )] = arg max [Vi (a′ , t ) − Vi (a, t ) − p (t ) (a′ − a) − φ]1−η φη (a ′ ,φ)
Lemma ai (t ) = arg max Vi ( a ′ , t ) − p ( t ) a ′ ′ a ≥0
φi (a, t ) = η {Vi [ai (t ) , t ] − Vi (a, t ) − p (t ) [ai (t ) − a]} ai (t ) maximizes the total gains from trade φi (a, t ) splits the gains from trade according to η
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Brokered trades Dealer’s bargaining power: η ∈ [0, 1]
[ai (t ), φi (a, t )] = arg max [Vi (a′ , t ) − Vi (a, t ) − p (t ) (a′ − a) − φ]1−η φη (a ′ ,φ)
Lemma ai (t ) = arg max Vi ( a ′ , t ) − p ( t ) a ′ ′ a ≥0
φi (a, t ) = η {Vi [ai (t ) , t ] − Vi (a, t ) − p (t ) [ai (t ) − a]} ai (t ) maximizes the total gains from trade φi (a, t ) splits the gains from trade according to η
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Solution to the investor’s problem
max [u¯ i (a) − ξ (t )a] a ≥0
where: u¯ i (a) ≡ (r + κ ) Ei
Z T˜ t
e −r (s −t ) uk (s ) (a)ds =
(r +κ )ui (a )+ δ ∑Ij =1 πj uj (a ) r +κ + δ
h i ˜ ξ (t ) ≡ (r + κ ) E p (t ) − e −r (T −t ) p (T˜ ) where T˜ − t ∼ exponential with parameter κ ≡ α(1 − η )
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Solution to the dealer and the investor problems Dealer holdings 0 ≤ rp (t ) − p˙ (t )
(necessary for a solution)
ad ( t ) = 0
if
0 < rp (t ) − p˙ (t )
ad ( t ) ≥ 0
if
0 = rp (t ) − p˙ (t )
Investor holdings u¯ i′ [ai (t )] = ξ (t ) ξ˙ (t ) rp (t ) − p˙ (t ) = ξ (t ) − | {z } r +κ dealer’s cost
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Solution to the dealer and the investor problems Dealer holdings 0 ≤ rp (t ) − p˙ (t )
(necessary for a solution)
ad ( t ) = 0
if
0 < rp (t ) − p˙ (t )
ad ( t ) ≥ 0
if
0 = rp (t ) − p˙ (t )
Investor holdings u¯ i′ [ai (t )] = ξ (t ) ξ˙ (t ) rp (t ) − p˙ (t ) = ξ (t ) − | {z } r +κ dealer’s cost
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Solution to the dealer and the investor problems Dealer holdings 0 ≤ rp (t ) − p˙ (t )
(necessary for a solution)
ad ( t ) = 0
if
0 < rp (t ) − p˙ (t )
ad ( t ) ≥ 0
if
0 = rp (t ) − p˙ (t )
Investor holdings u¯ i′ [ai (t )] = ξ (t ) ξ˙ (t ) rp (t ) − p˙ (t ) = ξ (t ) − | {z } r +κ dealer’s cost
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Solution to the dealer and the investor problems Dealer holdings 0 ≤ rp (t ) − p˙ (t )
(necessary for a solution)
ad ( t ) = 0
if
0 < rp (t ) − p˙ (t )
ad ( t ) ≥ 0
if
0 = rp (t ) − p˙ (t )
Investor holdings u¯ i′ [ai (t )] = ξ (t ) ξ˙ (t ) rp (t ) − p˙ (t ) = ξ (t ) − | {z } r +κ dealer’s cost
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Clearing of the interdealer market
assets held by investors
A˙ (t ) = | d{z }
dealer’s purchases
α |
z }| { I [A − Ad (t )] − α ∑i =1 π i u¯ i′−1 [ξ (t )] {z } | {z }
assets supplied by investors
assets demanded by investors
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Equilibrium conditions in the recovery phase
u¯ i′ [ai (t )] = ξ (t )
ξ˙ (t ) ξ (t ) − Ad ( t ) = 0 r +κ h i I A˙ d (t ) = α A − Ad (t ) − ∑i =1 π i ai (t )
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Equilibrium following the recovery Given the realization of Tρ and given Ad Tρ ≥ 0...
...there are two possible paths (depending on the initial condition): 1
If Ad Tρ = 0, then ξ (t ) = ξ¯ and Ad (t ) = 0 for all t ≥ Tρ (ξ¯ solves ∑Ii =1 π i u¯ ′−1 ξ¯ = A) i
2
If Ad Tρ > 0, then there exists a unique T > Tρ such that: Ad (t ) > 0 for t ∈ [Tρ , T ) and Ad (t ) = 0 for t ≥ T A˙ d (t ) < 0 for t ∈ [Tρ , T ) ξ Tρ = ψ Ad Tρ , where ψ ′ < 0
ξ˙ (t ) /ξ (t ) = r + κ for t ∈ [Tρ , T ) and ξ (t ) = ξ¯ for t ≥ T
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Equilibrium following the recovery Given the realization of Tρ and given Ad Tρ ≥ 0...
...there are two possible paths (depending on the initial condition): 1
If Ad Tρ = 0, then ξ (t ) = ξ¯ and Ad (t ) = 0 for all t ≥ Tρ (ξ¯ solves ∑Ii =1 π i u¯ ′−1 ξ¯ = A) i
2
If Ad Tρ > 0, then there exists a unique T > Tρ such that: Ad (t ) > 0 for t ∈ [Tρ , T ) and Ad (t ) = 0 for t ≥ T A˙ d (t ) < 0 for t ∈ [Tρ , T ) ξ Tρ = ψ Ad Tρ , where ψ ′ < 0
ξ˙ (t ) /ξ (t ) = r + κ for t ∈ [Tρ , T ) and ξ (t ) = ξ¯ for t ≥ T
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Equilibrium following the recovery Given the realization of Tρ and given Ad Tρ ≥ 0...
...there are two possible paths (depending on the initial condition): 1
If Ad Tρ = 0, then ξ (t ) = ξ¯ and Ad (t ) = 0 for all t ≥ Tρ (ξ¯ solves ∑Ii =1 π i u¯ ′−1 ξ¯ = A) i
2
If Ad Tρ > 0, then there exists a unique T > Tρ such that: Ad (t ) > 0 for t ∈ [Tρ , T ) and Ad (t ) = 0 for t ≥ T A˙ d (t ) < 0 for t ∈ [Tρ , T ) ξ Tρ = ψ Ad Tρ , where ψ ′ < 0
ξ˙ (t ) /ξ (t ) = r + κ for t ∈ [Tρ , T ) and ξ (t ) = ξ¯ for t ≥ T
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Equilibrium following the recovery Given the realization of Tρ and given Ad Tρ ≥ 0...
...there are two possible paths (depending on the initial condition): 1
If Ad Tρ = 0, then ξ (t ) = ξ¯ and Ad (t ) = 0 for all t ≥ Tρ (ξ¯ solves ∑Ii =1 π i u¯ ′−1 ξ¯ = A) i
2
If Ad Tρ > 0, then there exists a unique T > Tρ such that: Ad (t ) > 0 for t ∈ [Tρ , T ) and Ad (t ) = 0 for t ≥ T A˙ d (t ) < 0 for t ∈ [Tρ , T ) ξ Tρ = ψ Ad Tρ , where ψ ′ < 0
ξ˙ (t ) /ξ (t ) = r + κ for t ∈ [Tρ , T ) and ξ (t ) = ξ¯ for t ≥ T
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Equilibrium following the recovery Given the realization of Tρ and given Ad Tρ ≥ 0...
...there are two possible paths (depending on the initial condition): 1
If Ad Tρ = 0, then ξ (t ) = ξ¯ and Ad (t ) = 0 for all t ≥ Tρ (ξ¯ solves ∑Ii =1 π i u¯ ′−1 ξ¯ = A) i
2
If Ad Tρ > 0, then there exists a unique T > Tρ such that: Ad (t ) > 0 for t ∈ [Tρ , T ) and Ad (t ) = 0 for t ≥ T A˙ d (t ) < 0 for t ∈ [Tρ , T ) ξ Tρ = ψ Ad Tρ , where ψ ′ < 0
ξ˙ (t ) /ξ (t ) = r + κ for t ∈ [Tρ , T ) and ξ (t ) = ξ¯ for t ≥ T
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Equilibrium following the recovery Given the realization of Tρ and given Ad Tρ ≥ 0...
...there are two possible paths (depending on the initial condition): 1
If Ad Tρ = 0, then ξ (t ) = ξ¯ and Ad (t ) = 0 for all t ≥ Tρ (ξ¯ solves ∑Ii =1 π i u¯ ′−1 ξ¯ = A) i
2
If Ad Tρ > 0, then there exists a unique T > Tρ such that: Ad (t ) > 0 for t ∈ [Tρ , T ) and Ad (t ) = 0 for t ≥ T A˙ d (t ) < 0 for t ∈ [Tρ , T ) ξ Tρ = ψ Ad Tρ , where ψ ′ < 0
ξ˙ (t ) /ξ (t ) = r + κ for t ∈ [Tρ , T ) and ξ (t ) = ξ¯ for t ≥ T
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Equilibrium following the recovery Given the realization of Tρ and given Ad Tρ ≥ 0...
...there are two possible paths (depending on the initial condition): 1
If Ad Tρ = 0, then ξ (t ) = ξ¯ and Ad (t ) = 0 for all t ≥ Tρ (ξ¯ solves ∑Ii =1 π i u¯ ′−1 ξ¯ = A) i
2
If Ad Tρ > 0, then there exists a unique T > Tρ such that: Ad (t ) > 0 for t ∈ [Tρ , T ) and Ad (t ) = 0 for t ≥ T A˙ d (t ) < 0 for t ∈ [Tρ , T ) ξ Tρ = ψ Ad Tρ , where ψ ′ < 0
ξ˙ (t ) /ξ (t ) = r + κ for t ∈ [Tρ , T ) and ξ (t ) = ξ¯ for t ≥ T
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Recovery dynamics if dealers do not hold assets
p (t )
time Ad ( t )
0
random recovery time
time
Efficiency
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Recovery dynamics if dealers hold assets
p (t )
time Ad ( t )
0
random recovery time
time
Efficiency
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Investor’s problem during the crisis The investor who contacts a dealer at t ∈ [0, Tρ ) solves i h max u¯ iC (a) − ξ C (t ) a a ≥0
where u¯ iC (a) ≡ (r + κ ) Ei
Z T˜ t
θ + (1 − θ )I {s ≥Tρ } uk (s ) (a)e −r (s −t ) ds
h i ˜ ξ C (t ) ≡ (r + κ ) Ei p C (t ) − e −r (T −t ) p (T˜ , Tρ ) p (T˜ , Tρ ) ≡ I {T˜
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Investor’s problem during the crisis The investor who contacts a dealer at t ∈ [0, Tρ ) solves i h max u¯ iC (a) − ξ C (t ) a a ≥0
where u¯ iC (a) ≡ (r + κ ) Ei
Z T˜ t
θ + (1 − θ )I {s ≥Tρ } uk (s ) (a)e −r (s −t ) ds
h i ˜ ξ C (t ) ≡ (r + κ ) Ei p C (t ) − e −r (T −t ) p (T˜ , Tρ ) p (T˜ , Tρ ) ≡ I {T˜
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Investor’s problem during the crisis The investor who contacts a dealer at t ∈ [0, Tρ ) solves i h max u¯ iC (a) − ξ C (t ) a a ≥0
where u¯ iC (a) ≡ (r + κ ) Ei
Z T˜ t
θ + (1 − θ )I {s ≥Tρ } uk (s ) (a)e −r (s −t ) ds
h i ˜ ξ C (t ) ≡ (r + κ ) Ei p C (t ) − e −r (T −t ) p (T˜ , Tρ ) p (T˜ , Tρ ) ≡ I {T˜
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Dealer’s problem during the crisis
max E
q C (s )
s.t.
Z
Tρ t
i h −e −r (s −t ) p C (s )q C (s )ds + e −r (Tρ −t ) W adC Tρ , Tρ | Tρ
a˙ dC (s ) = q C (s ) ,
adC (s ) ≥ 0,
adC (t ) given
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Dealer’s problem during the crisis
max E
q C (s )
s.t.
Z
Tρ t
i h −e −r (s −t ) p C (s )q C (s )ds + e −r (Tρ −t ) W adC Tρ , Tρ | Tρ
a˙ dC (s ) = q C (s ) ,
adC (s ) ≥ 0,
adC (t ) given
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Solution to the dealer and the investor problems Dealer holdings h i 0 ≤ rp C (t ) − p˙ C (t ) − ρ p (t |t ) − p C (t ) adC (t ) = 0
if
adC (t ) ≥ 0
if
(necessary)
p˙ C (t ) + ρ p (t |t ) − p C (t )
Investor holdings: u¯ iC ′ [ai (t )] = ξ C (t ) i h C ξ˙ (t )+ ρ[ξ (t | t )− ξ C (t ) ] rp C (t ) − p˙ C (t ) − ρ p (t | t ) − p C (t ) = ξ C (t ) − r +κ | {z } dealer’s opportunity cost
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Solution to the dealer and the investor problems Dealer holdings h i 0 ≤ rp C (t ) − p˙ C (t ) − ρ p (t |t ) − p C (t ) adC (t ) = 0
if
adC (t ) ≥ 0
if
(necessary)
p˙ C (t ) + ρ p (t |t ) − p C (t )
Investor holdings: u¯ iC ′ [ai (t )] = ξ C (t ) i h C ξ˙ (t )+ ρ[ξ (t | t )− ξ C (t ) ] rp C (t ) − p˙ C (t ) − ρ p (t | t ) − p C (t ) = ξ C (t ) − r +κ | {z } dealer’s opportunity cost
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Solution to the dealer and the investor problems Dealer holdings h i 0 ≤ rp C (t ) − p˙ C (t ) − ρ p (t |t ) − p C (t ) adC (t ) = 0
if
adC (t ) ≥ 0
if
(necessary)
p˙ C (t ) + ρ p (t |t ) − p C (t )
Investor holdings: u¯ iC ′ [ai (t )] = ξ C (t ) i h C ξ˙ (t )+ ρ[ξ (t | t )− ξ C (t ) ] rp C (t ) − p˙ C (t ) − ρ p (t | t ) − p C (t ) = ξ C (t ) − r +κ | {z } dealer’s opportunity cost
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Solution to the dealer and the investor problems Dealer holdings h i 0 ≤ rp C (t ) − p˙ C (t ) − ρ p (t |t ) − p C (t ) adC (t ) = 0
if
adC (t ) ≥ 0
if
(necessary)
p˙ C (t ) + ρ p (t |t ) − p C (t )
Investor holdings: u¯ iC ′ [ai (t )] = ξ C (t ) i h C ξ˙ (t )+ ρ[ξ (t | t )− ξ C (t ) ] rp C (t ) − p˙ C (t ) − ρ p (t | t ) − p C (t ) = ξ C (t ) − r +κ | {z } dealer’s opportunity cost
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Equilibrium conditions in the crisis phase
h i u¯ i′ aiC (t ) = ξ C (t ) h i C ξ˙ (t ) + ρ ξ (t | t ) − ξ C (t ) ACd (t ) = 0 ξ C (t ) − r +κ
h i I α A − ACd (t ) − ∑i =1 π i aiC (t ) = A˙ Cd (t )
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Equilibrium path
C ¯C The previous system of ODEs has a unique steady state ξ¯ , A d ...there are two possibilities (depending on the parametrization)
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Equilibrium path when dealers do not provide liquidity
¯ C = 0, then: If A d Ad (t ) = 0 for all t C ξ (t ) jumps down to ξ¯ when the crisis starts
ξ (t ) jumps back up to ξ¯ when the crisis ends
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Equilibrium path when dealers do not provide liquidity
¯ C = 0, then: If A d Ad (t ) = 0 for all t C ξ (t ) jumps down to ξ¯ when the crisis starts
ξ (t ) jumps back up to ξ¯ when the crisis ends
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Equilibrium path when dealers do not provide liquidity
¯ C = 0, then: If A d Ad (t ) = 0 for all t C ξ (t ) jumps down to ξ¯ when the crisis starts
ξ (t ) jumps back up to ξ¯ when the crisis ends
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Equilibrium path when dealers do not provide liquidity
p (t )
time Ad ( t )
0
random recovery time
time
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Equilibrium path when dealers provide liquidity ¯ C > 0, then there exists a unique T > Tρ such that: If A d > 0 for t < Tρ A˙ d (t ) < 0 for Tρ ≤ t < T = 0 for T ≤ t Ad ( t )
> 0 for t < T = 0 for T ≤ t
p (t ) jumps down <0 p˙ (t ) p (t ) jumps up >0 =0
at t = 0 for t < Tρ at t = Tρ for Tρ ≤ t < T for T < t
Efficiency
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Equilibrium path when dealers provide liquidity ¯ C > 0, then there exists a unique T > Tρ such that: If A d > 0 for t < Tρ A˙ d (t ) < 0 for Tρ ≤ t < T = 0 for T ≤ t Ad ( t )
> 0 for t < T = 0 for T ≤ t
p (t ) jumps down <0 p˙ (t ) p (t ) jumps up >0 =0
at t = 0 for t < Tρ at t = Tρ for Tρ ≤ t < T for T < t
Efficiency
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Equilibrium path when dealers provide liquidity ¯ C > 0, then there exists a unique T > Tρ such that: If A d > 0 for t < Tρ A˙ d (t ) < 0 for Tρ ≤ t < T = 0 for T ≤ t Ad ( t )
> 0 for t < T = 0 for T ≤ t
p (t ) jumps down <0 p˙ (t ) p (t ) jumps up >0 =0
at t = 0 for t < Tρ at t = Tρ for Tρ ≤ t < T for T < t
Efficiency
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Equilibrium path when dealers provide liquidity
p (t )
time Ad ( t )
0
random recovery time
time
Efficiency
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Equilibrium path ξC
A˙ d (t ) = 0 A˙ Cd (t ) = 0
ξ¯ ξ C ( 0) C ξ¯ C ξ˙ (t ) = 0
ACd ¯C A d Figure: Phase diagram for the crisis path.
Efficiency
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
When do dealers provide liquidity? From dealer’s FOC: C ξ˙ (t ) C
ξ (t )
+ρ
ψ [Ad (t )] − ξ C (t ) C
ξ (t )
C < r + κ ⇒ ACd (t ) = 0 ⇒ ξ˙ (t ) = 0
where ψ [Ad (t )] = ξ (t |t ), with ψ′ < 0 and ψ (0) = ξ¯
⇒ Necessary condition for dealers not to provide liquidity: C ξ¯ − ξ¯ C ξ¯ | {z }
capital gain at recovery with no liquidity provision
It is also a sufficient condition.
<
r +κ ρ
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
When do dealers provide liquidity? From dealer’s FOC: C ξ˙ (t ) C
ξ (t )
+ρ
ψ [Ad (t )] − ξ C (t ) C
ξ (t )
C < r + κ ⇒ ACd (t ) = 0 ⇒ ξ˙ (t ) = 0
where ψ [Ad (t )] = ξ (t |t ), with ψ′ < 0 and ψ (0) = ξ¯
⇒ Necessary condition for dealers not to provide liquidity: C ξ¯ − ξ¯ C ξ¯ | {z }
capital gain at recovery with no liquidity provision
It is also a sufficient condition.
<
r +κ ρ
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
When do dealers provide liquidity? From dealer’s FOC: C ξ˙ (t ) C
ξ (t )
+ρ
ψ [Ad (t )] − ξ C (t ) C
ξ (t )
C < r + κ ⇒ ACd (t ) = 0 ⇒ ξ˙ (t ) = 0
where ψ [Ad (t )] = ξ (t |t ), with ψ′ < 0 and ψ (0) = ξ¯
⇒ Necessary condition for dealers not to provide liquidity: C ξ¯ − ξ¯ C ξ¯ | {z }
capital gain at recovery with no liquidity provision
It is also a sufficient condition.
<
r +κ ρ
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
When do dealers provide liquidity? From dealer’s FOC: C ξ˙ (t ) C
ξ (t )
+ρ
ψ [Ad (t )] − ξ C (t ) C
ξ (t )
C < r + κ ⇒ ACd (t ) = 0 ⇒ ξ˙ (t ) = 0
where ψ [Ad (t )] = ξ (t |t ), with ψ′ < 0 and ψ (0) = ξ¯
⇒ Necessary condition for dealers not to provide liquidity: C ξ¯ − ξ¯ C ξ¯ | {z }
capital gain at recovery with no liquidity provision
It is also a sufficient condition.
<
r +κ ρ
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Example Assume ui (a) = θ (t ) εi a1−σ / (1 − σ ). Dealers do not provide liquidity if and only if h
where:
iσ 1/σ ∑Ii =1 π i (ε¯ i ) r +κ h iσ − 1 < 1/σ ρ ∑Ii =1 π i ε¯Ci
ε¯i
=
I r +κ δ εi + π j εj ∑ r +κ+δ r + κ + δ j =1
ε¯ Ci
=
(r +κ )[(r +κ + δ)θ + ρ] ε (r +κ + δ)(r +κ + ρ+ δ) i
+
δ{(r +κ + ρ+ δ)ρ+(r +κ)[(r +κ + δ) θ + ρ]} (r +κ + δ)(r +κ + ρ)(r +κ + ρ+ δ)
I
∑ π j εj j =1
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Example Assume ui (a) = θ (t ) εi a1−σ / (1 − σ ). Dealers do not provide liquidity if and only if h
where:
iσ 1/σ ∑Ii =1 π i (ε¯ i ) r +κ h iσ − 1 < 1/σ ρ ∑Ii =1 π i ε¯Ci
ε¯i
=
I r +κ δ εi + π j εj ∑ r +κ+δ r + κ + δ j =1
ε¯ Ci
=
(r +κ )[(r +κ + δ)θ + ρ] ε (r +κ + δ)(r +κ + ρ+ δ) i
+
δ{(r +κ + ρ+ δ)ρ+(r +κ)[(r +κ + δ) θ + ρ]} (r +κ + δ)(r +κ + ρ)(r +κ + ρ+ δ)
I
∑ π j εj j =1
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
When do dealers provide liquidity?
0.03
1 0.8
0.02
ρ
θ
0.6 0.4
0.01
0.2 0
0.5
1
α
1.5
0
2
0.5
1
1.5
2
1.6
1.8
2
α
1 0.8
η
0.6 0.4 0.2 0
0.2
0.4
0.6
0.8
1
α
1.2
1.4
Efficiency
Conclusion
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Planner’s problem
Planner has: State: Ad∗ (t ) ←→ equilibrium dealer holdings Ad (t ) Co-state: λ (t ) ←→ analogous to the equilibrium “price” ξ (t )
Planner’s solution coincides with equilibrium iff η = 0
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Planner’s problem
Planner has: State: Ad∗ (t ) ←→ equilibrium dealer holdings Ad (t ) Co-state: λ (t ) ←→ analogous to the equilibrium “price” ξ (t )
Planner’s solution coincides with equilibrium iff η = 0
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Planner’s problem
Planner has: State: Ad∗ (t ) ←→ equilibrium dealer holdings Ad (t ) Co-state: λ (t ) ←→ analogous to the equilibrium “price” ξ (t )
Planner’s solution coincides with equilibrium iff η = 0
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Planner’s problem
Planner has: State: Ad∗ (t ) ←→ equilibrium dealer holdings Ad (t ) Co-state: λ (t ) ←→ analogous to the equilibrium “price” ξ (t )
Planner’s solution coincides with equilibrium iff η = 0
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Equilibrium vs. Efficient Liquidity Provision
1 0.9
too little liquidity −→
0.8 0.7 0.6
η 0.5 0.4 0.3 0.2 0.1 0
0.2
0.4
0.6
0.8
1
α
1.2
1.4
1.6
1.8
2
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Why do dealers provide liquidity? C C Liquidity provision ⇔ ξ¯ − ξ¯ /ξ¯ large Equilibrium logic: Dealers buy inventories to make capital gains in recovery phase (advantage over investors due to “∞ contact rate”) Dealers bear opportunity cost of buying the asset but get no holding utility
Planner’s logic: MU is currently low, will be high in recovery phase, so planner wants to be able to transfer more assets to each “cohort” of those investors faster Along the stochastic transition, planner trades off “MU smoothing motive” against the cost of having dealers hold the asset rather than investors
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Why do dealers provide liquidity? C C Liquidity provision ⇔ ξ¯ − ξ¯ /ξ¯ large Equilibrium logic: Dealers buy inventories to make capital gains in recovery phase (advantage over investors due to “∞ contact rate”) Dealers bear opportunity cost of buying the asset but get no holding utility
Planner’s logic: MU is currently low, will be high in recovery phase, so planner wants to be able to transfer more assets to each “cohort” of those investors faster Along the stochastic transition, planner trades off “MU smoothing motive” against the cost of having dealers hold the asset rather than investors
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Why do dealers provide liquidity? C C Liquidity provision ⇔ ξ¯ − ξ¯ /ξ¯ large Equilibrium logic: Dealers buy inventories to make capital gains in recovery phase (advantage over investors due to “∞ contact rate”) Dealers bear opportunity cost of buying the asset but get no holding utility
Planner’s logic: MU is currently low, will be high in recovery phase, so planner wants to be able to transfer more assets to each “cohort” of those investors faster Along the stochastic transition, planner trades off “MU smoothing motive” against the cost of having dealers hold the asset rather than investors
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Why do dealers provide liquidity? C C Liquidity provision ⇔ ξ¯ − ξ¯ /ξ¯ large Equilibrium logic: Dealers buy inventories to make capital gains in recovery phase (advantage over investors due to “∞ contact rate”) Dealers bear opportunity cost of buying the asset but get no holding utility
Planner’s logic: MU is currently low, will be high in recovery phase, so planner wants to be able to transfer more assets to each “cohort” of those investors faster Along the stochastic transition, planner trades off “MU smoothing motive” against the cost of having dealers hold the asset rather than investors
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Why do dealers provide liquidity? C C Liquidity provision ⇔ ξ¯ − ξ¯ /ξ¯ large Equilibrium logic: Dealers buy inventories to make capital gains in recovery phase (advantage over investors due to “∞ contact rate”) Dealers bear opportunity cost of buying the asset but get no holding utility
Planner’s logic: MU is currently low, will be high in recovery phase, so planner wants to be able to transfer more assets to each “cohort” of those investors faster Along the stochastic transition, planner trades off “MU smoothing motive” against the cost of having dealers hold the asset rather than investors
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Why do dealers provide liquidity? C C Liquidity provision ⇔ ξ¯ − ξ¯ /ξ¯ large Equilibrium logic: Dealers buy inventories to make capital gains in recovery phase (advantage over investors due to “∞ contact rate”) Dealers bear opportunity cost of buying the asset but get no holding utility
Planner’s logic: MU is currently low, will be high in recovery phase, so planner wants to be able to transfer more assets to each “cohort” of those investors faster Along the stochastic transition, planner trades off “MU smoothing motive” against the cost of having dealers hold the asset rather than investors
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Why do dealers provide liquidity? C C Liquidity provision ⇔ ξ¯ − ξ¯ /ξ¯ large Equilibrium logic: Dealers buy inventories to make capital gains in recovery phase (advantage over investors due to “∞ contact rate”) Dealers bear opportunity cost of buying the asset but get no holding utility
Planner’s logic: MU is currently low, will be high in recovery phase, so planner wants to be able to transfer more assets to each “cohort” of those investors faster Along the stochastic transition, planner trades off “MU smoothing motive” against the cost of having dealers hold the asset rather than investors
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Open market operations Consider parametrization with η > 0, and such that dealers do not provide liquidity when the planner would. Can a government improve welfare by acting as a dealer? Suppose: Buys assets in the interdealer market (big player, price impact) Investors still bargain with dealers (η friction is still there) Financed by lump-sum taxes of numeraire good
Answer: Yes (At this point we do not solve for the optimal policy, but can construct marginal interventions that increase efficiency.)
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Open market operations Consider parametrization with η > 0, and such that dealers do not provide liquidity when the planner would. Can a government improve welfare by acting as a dealer? Suppose: Buys assets in the interdealer market (big player, price impact) Investors still bargain with dealers (η friction is still there) Financed by lump-sum taxes of numeraire good
Answer: Yes (At this point we do not solve for the optimal policy, but can construct marginal interventions that increase efficiency.)
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Open market operations Consider parametrization with η > 0, and such that dealers do not provide liquidity when the planner would. Can a government improve welfare by acting as a dealer? Suppose: Buys assets in the interdealer market (big player, price impact) Investors still bargain with dealers (η friction is still there) Financed by lump-sum taxes of numeraire good
Answer: Yes (At this point we do not solve for the optimal policy, but can construct marginal interventions that increase efficiency.)
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Open market operations Consider parametrization with η > 0, and such that dealers do not provide liquidity when the planner would. Can a government improve welfare by acting as a dealer? Suppose: Buys assets in the interdealer market (big player, price impact) Investors still bargain with dealers (η friction is still there) Financed by lump-sum taxes of numeraire good
Answer: Yes (At this point we do not solve for the optimal policy, but can construct marginal interventions that increase efficiency.)
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Open market operations Consider parametrization with η > 0, and such that dealers do not provide liquidity when the planner would. Can a government improve welfare by acting as a dealer? Suppose: Buys assets in the interdealer market (big player, price impact) Investors still bargain with dealers (η friction is still there) Financed by lump-sum taxes of numeraire good
Answer: Yes (At this point we do not solve for the optimal policy, but can construct marginal interventions that increase efficiency.)
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Conclusion
Dealers will/should provide liquidity during “crises” when: crash is severe crash is expected to be short-lived trading frictions (κ) are moderate
There appears to be an efficiency rationale for the government to “act as a dealer”
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Conclusion
Dealers will/should provide liquidity during “crises” when: crash is severe crash is expected to be short-lived trading frictions (κ) are moderate
There appears to be an efficiency rationale for the government to “act as a dealer”
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Conclusion
Dealers will/should provide liquidity during “crises” when: crash is severe crash is expected to be short-lived trading frictions (κ) are moderate
There appears to be an efficiency rationale for the government to “act as a dealer”
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Conclusion
Dealers will/should provide liquidity during “crises” when: crash is severe crash is expected to be short-lived trading frictions (κ) are moderate
There appears to be an efficiency rationale for the government to “act as a dealer”
Introduction
Model
Outline
Recovery phase
Crisis phase
Equilibrium
Liquidity provision
Efficiency
Conclusion
Conclusion
Dealers will/should provide liquidity during “crises” when: crash is severe crash is expected to be short-lived trading frictions (κ) are moderate
There appears to be an efficiency rationale for the government to “act as a dealer”
