Post on 03-Feb-2022
The modelOptimal policies in t1
Example
Interbank LendingModelling the Development of Interbank Markets during a Liquidity
Shock
Dominik Joos,Karlsruhe Institute of Technology
21st March 2011
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
Motivation: decoupling of interest rates
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
The model
We study a model with
three dates t = 0, t1, t2
a [0, 1] continuum of identical banks maximizing expected utility oftheir profit
households which have future liquidity needs and let banks managetheir funds
one risky illiquid asset with uncertain payoff at t = t2
a government bond with certain return at t = t2
costless storage
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
The model
We study a model with
three dates t = 0, t1, t2
a [0, 1] continuum of identical banks maximizing expected utility oftheir profit
households which have future liquidity needs and let banks managetheir funds
one risky illiquid asset with uncertain payoff at t = t2
a government bond with certain return at t = t2
costless storage
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
The model
We study a model with
three dates t = 0, t1, t2
a [0, 1] continuum of identical banks maximizing expected utility oftheir profit
households which have future liquidity needs and let banks managetheir funds
one risky illiquid asset with uncertain payoff at t = t2
a government bond with certain return at t = t2
costless storage
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
The model
We study a model with
three dates t = 0, t1, t2
a [0, 1] continuum of identical banks maximizing expected utility oftheir profit
households which have future liquidity needs and let banks managetheir funds
one risky illiquid asset with uncertain payoff at t = t2
a government bond with certain return at t = t2
costless storage
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
The model
We study a model with
three dates t = 0, t1, t2
a [0, 1] continuum of identical banks maximizing expected utility oftheir profit
households which have future liquidity needs and let banks managetheir funds
one risky illiquid asset with uncertain payoff at t = t2
a government bond with certain return at t = t2
costless storage
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
The model
We study a model with
three dates t = 0, t1, t2
a [0, 1] continuum of identical banks maximizing expected utility oftheir profit
households which have future liquidity needs and let banks managetheir funds
one risky illiquid asset with uncertain payoff at t = t2
a government bond with certain return at t = t2
costless storage
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
Management of funds
We assume that each bank has one unit of the households’ fundsunder management at t = 0 and offers claims that can be withdrawneither at t = t1 or t = t2 worth c1 and c2 respectively.
The demand for liquidity on individual bank level is determined by a[0, 1]-valued random variable Λ describing the fraction of capital,which is withdrawn in t = t1.
Thus banks have the following random cashflows:
t 0 t1 t2
deposit contracts 1 −Λc1 −(1− Λ)c2
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
Management of funds
We assume that each bank has one unit of the households’ fundsunder management at t = 0 and offers claims that can be withdrawneither at t = t1 or t = t2 worth c1 and c2 respectively.
The demand for liquidity on individual bank level is determined by a[0, 1]-valued random variable Λ describing the fraction of capital,which is withdrawn in t = t1.
Thus banks have the following random cashflows:
t 0 t1 t2
deposit contracts 1 −Λc1 −(1− Λ)c2
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
Management of funds
We assume that each bank has one unit of the households’ fundsunder management at t = 0 and offers claims that can be withdrawneither at t = t1 or t = t2 worth c1 and c2 respectively.
The demand for liquidity on individual bank level is determined by a[0, 1]-valued random variable Λ describing the fraction of capital,which is withdrawn in t = t1.
Thus banks have the following random cashflows:
t 0 t1 t2
deposit contracts 1 −Λc1 −(1− Λ)c2
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
External investment possibilities
The payoff of the illiquid asset is modelled by an R+-valued process(St)0≤t≤t2 and the price for the bond is denoted by P0 in t = 0 andP2 in t = t2.
Then we have the following assets and financial claims:
t 0 t1 t2
cash −δ0 δ0
−δ1 δ1
stock −α0 0 α0St2
S0
bond −β0 0 β0P2
P0
cash flow 1 −Λc1 −(1− Λ)c2
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
External investment possibilities
The payoff of the illiquid asset is modelled by an R+-valued process(St)0≤t≤t2 and the price for the bond is denoted by P0 in t = 0 andP2 in t = t2.
Then we have the following assets and financial claims:
t 0 t1 t2
cash −δ0 δ0
−δ1 δ1
stock −α0 0 α0St2
S0
bond −β0 0 β0P2
P0
cash flow 1 −Λc1 −(1− Λ)c2
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
Interbank markets
In order to smooth out the different liquidity levels in t = t1, we allow forinterbank markets to develop:
Secured lending is modelled by an outright sale of bonds at price P1
The unsecured market consits of borrowing and lending at aninterest rate r
We assume that both markets are anonymous and competitive
Banks are price takers, prices are set by a Walrasian auctioneer
Banks are completely diversified across unsecured interbank loans
We denote the probability that a borrower in the unsecured marketis solvent in t = t2 by p̂
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
Interbank markets
In order to smooth out the different liquidity levels in t = t1, we allow forinterbank markets to develop:
Secured lending is modelled by an outright sale of bonds at price P1
The unsecured market consits of borrowing and lending at aninterest rate r
We assume that both markets are anonymous and competitive
Banks are price takers, prices are set by a Walrasian auctioneer
Banks are completely diversified across unsecured interbank loans
We denote the probability that a borrower in the unsecured marketis solvent in t = t2 by p̂
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
Interbank markets
In order to smooth out the different liquidity levels in t = t1, we allow forinterbank markets to develop:
Secured lending is modelled by an outright sale of bonds at price P1
The unsecured market consits of borrowing and lending at aninterest rate r
We assume that both markets are anonymous and competitive
Banks are price takers, prices are set by a Walrasian auctioneer
Banks are completely diversified across unsecured interbank loans
We denote the probability that a borrower in the unsecured marketis solvent in t = t2 by p̂
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
Interbank markets
In order to smooth out the different liquidity levels in t = t1, we allow forinterbank markets to develop:
Secured lending is modelled by an outright sale of bonds at price P1
The unsecured market consits of borrowing and lending at aninterest rate r
We assume that both markets are anonymous and competitive
Banks are price takers, prices are set by a Walrasian auctioneer
Banks are completely diversified across unsecured interbank loans
We denote the probability that a borrower in the unsecured marketis solvent in t = t2 by p̂
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
Interbank markets
In order to smooth out the different liquidity levels in t = t1, we allow forinterbank markets to develop:
Secured lending is modelled by an outright sale of bonds at price P1
The unsecured market consits of borrowing and lending at aninterest rate r
We assume that both markets are anonymous and competitive
Banks are price takers, prices are set by a Walrasian auctioneer
Banks are completely diversified across unsecured interbank loans
We denote the probability that a borrower in the unsecured marketis solvent in t = t2 by p̂
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
Interbank markets
In order to smooth out the different liquidity levels in t = t1, we allow forinterbank markets to develop:
Secured lending is modelled by an outright sale of bonds at price P1
The unsecured market consits of borrowing and lending at aninterest rate r
We assume that both markets are anonymous and competitive
Banks are price takers, prices are set by a Walrasian auctioneer
Banks are completely diversified across unsecured interbank loans
We denote the probability that a borrower in the unsecured marketis solvent in t = t2 by p̂
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
Interbank markets
In order to smooth out the different liquidity levels in t = t1, we allow forinterbank markets to develop:
Secured lending is modelled by an outright sale of bonds at price P1
The unsecured market consits of borrowing and lending at aninterest rate r
We assume that both markets are anonymous and competitive
Banks are price takers, prices are set by a Walrasian auctioneer
Banks are completely diversified across unsecured interbank loans
We denote the probability that a borrower in the unsecured marketis solvent in t = t2 by p̂
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
This yields the following assets and financial claims:
t 0 t1 t2
cash −δ0 δ0
−δ1 δ1
stock −α 0 αSt2
S0
bond −β0 β0P1
P0
−β1 β1P2
P1
unsecured ib debt −γ+ γ+p̂(1 + r)
γ− −γ−(1 + r)
cash flow 1 −Λc1 −(1− Λ)c2
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
The optimization problem in t1
We consider the investment decision from t = 0, (α,β0,δ0) as given.Let λ, r and P1 be fixed, then the optimization problem in t1 is
maxβ1,γ+,γ−,δ1
E [u(V2(β1, γ+, γ−, δ1))]
Constraints are
β1 + γ+ + δ1 = δ0 + β0P1
P0+ γ− − λc1,
β1, γ+, γ−, δ1 ≥ 0,
where
V2(β1, γ+, γ−, δ1) = αSt2
S0+β1
P2
P1+γ+(1+r)p̂−γ−(1+r)+δ1−(1−λ)c2.
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
Optimal policies in t1
We have to optimize pointwise for every possible liquidity demand andevery price pair on the interbank market, thus we derive an optimalsupply and demand function, i.e. a function
(β1, γ+, γ−, δ1) : supp(Λ)× (−1,∞)× (0,∞)→ R4≥0
(λ, r ,P1) 7→ (βλ1 (r ,P1), γλ+(r ,P1), γλ−(r ,P1), δλ1 (r ,P1))
which (pointwise for every (λ, r ,P1)) solves our optimization problem.
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
Since expectation is monotone and the utility function is increasing theobjective function
E [u(αSt2
S0+ βλ1
P2
P1+ γλ+(1 + r)p̂ − γλ−(1 + r) + δλ1 − (1− λ)c2)]
becomes
βλ1P2
P1+ γλ+(1 + r)p̂ − γλ−(1 + r) + δλ1 ,
which is an affine function (as well as the constraint functions).Therefore Karush-Kuhn-Tucker conditions are necessary and sufficient foroptimality.
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
In equilibrium, aggregate demand equals aggregate supply.Aggregate net demand functions are
E [(βΛ1 (r ,P1)− β0
P1
P0)+], E [γΛ
+(r ,P1)],
net supply is
E [(β0P1
P0− βΛ
1 (r ,P1))+], E [γΛ−(r ,P1)].
Thus an equlibrium price pair (r∗,P∗1 ) satisfies
E [γΛ+(r∗,P∗1 )] = E [γΛ
−(r∗,P∗1 )],
E [βΛ1 (r∗,P∗1 )] = β0
P∗1P0.
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
In equilibrium, aggregate demand equals aggregate supply.Aggregate net demand functions are
E [(βΛ1 (r ,P1)− β0
P1
P0)+], E [γΛ
+(r ,P1)],
net supply is
E [(β0P1
P0− βΛ
1 (r ,P1))+], E [γΛ−(r ,P1)].
Thus an equlibrium price pair (r∗,P∗1 ) satisfies
E [γΛ+(r∗,P∗1 )] = E [γΛ
−(r∗,P∗1 )],
E [βΛ1 (r∗,P∗1 )] = β0
P∗1P0.
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
For δ0 < E (Λ)c1, there are no equilibrium prices for the interbankmarkets.
For δ0 > E (Λ)c1, there is secured interbank lending. An unsecuredinterbank market develops iff
β0P2
P0+ δ0 < c1ess sup(Λ).
The equilibrium prices are given by
r =1
p̂− 1, P1 = P2.
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
For δ0 < E (Λ)c1, there are no equilibrium prices for the interbankmarkets.
For δ0 > E (Λ)c1, there is secured interbank lending. An unsecuredinterbank market develops iff
β0P2
P0+ δ0 < c1ess sup(Λ).
The equilibrium prices are given by
r =1
p̂− 1, P1 = P2.
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
For δ0 = E (Λ)c1, there is secured interbank lending. An unsecuredinterbank market develops iff
β0P2
P0< c1(ess sup(Λ)− E (Λ)).
In this case the equilibrium prices satisfy
r ∈ [1
p̂− 1,∞), P1 =
P2
(1 + r)p̂.
If there is no unsecured lending, the bond price must satisfy
P1 ∈ [P0c1
β0(ess sup(Λ)− E (Λ)),P2].
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
In order to get a closed form for the optimal policies, we define
Ss := E[(Λc1−δ0)1{ 1c1δ0≤Λ≤ 1
c1(β0
P1P0
+δ0)}]+β0P1
P0P(Λ >
1
c1(β0
P1
P0+δ0)),
the aggregate supply of secured interbank loans
,
Su := E[(Λc1 − (β0P1
P0+ δ0))1{Λ> 1
c1(β0
P1P0
+δ0)}],
the aggregate supply of unsecured interbank loans and
D := E[(δ0 − Λc1)1{Λ< 1c1}]
the excess capital of banks with liquidity surplus, which equals theaggregate demand for interbank loans plus demand for liquid asset.
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
In order to get a closed form for the optimal policies, we define
Ss := E[(Λc1−δ0)1{ 1c1δ0≤Λ≤ 1
c1(β0
P1P0
+δ0)}]+β0P1
P0P(Λ >
1
c1(β0
P1
P0+δ0)),
the aggregate supply of secured interbank loans,
Su := E[(Λc1 − (β0P1
P0+ δ0))1{Λ> 1
c1(β0
P1P0
+δ0)}],
the aggregate supply of unsecured interbank loans
and
D := E[(δ0 − Λc1)1{Λ< 1c1}]
the excess capital of banks with liquidity surplus, which equals theaggregate demand for interbank loans plus demand for liquid asset.
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
In order to get a closed form for the optimal policies, we define
Ss := E[(Λc1−δ0)1{ 1c1δ0≤Λ≤ 1
c1(β0
P1P0
+δ0)}]+β0P1
P0P(Λ >
1
c1(β0
P1
P0+δ0)),
the aggregate supply of secured interbank loans,
Su := E[(Λc1 − (β0P1
P0+ δ0))1{Λ> 1
c1(β0
P1P0
+δ0)}],
the aggregate supply of unsecured interbank loans and
D := E[(δ0 − Λc1)1{Λ< 1c1}]
the excess capital of banks with liquidity surplus, which equals theaggregate demand for interbank loans plus demand for liquid asset.
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
βλ∗1 =
0 , λ > 1
c1(β0
P1
P0+ δ0)
β0P1
P0+ δ0 − λc1 , 1
c1δ0 ≤ λ ≤ 1
c1(β0
P1
P0+ δ0)
β0P1
P0+ Ss
D (δ0 − λc1) , λ < 1c1δ0
γλ∗+ =
0 , λ ≥ 1c1δ0
Su
D (δ0 − λc1) , λ < 1c1δ0
γλ∗− =
λc1 − (β0P1
P0+ δ0) , λ > 1
c1(β0
P1
P0+ δ0)
0 , λ ≤ 1c1
(β0P1
P0+ δ0)
δλ∗1 =
0 , λ ≥ 1c1δ0
(1− Su+Ss
D )(δ0 − λc1) , λ < 1c1δ0
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
For δ0 ≥ E (Λ)c1 the strategy (βλ∗1 , γλ∗+ , γλ∗− , δλ∗1 ) is optimal and profit in
t = t2 is given by
V2(α, β0, δ0) = αSt2
S0+ β0
P2
P0− (1− Λ)c2 + (δ0 − Λc1)
P2
P11{Λ≤ 1
c1δ0}+
+ (δ0 − Λc1)P2
P11{ 1
c1δ0<Λ≤ 1
c1(β0
P1P0
+δ0)} + (δ0 − Λc1)P2
P1p̂1{Λ> 1
c1(β0
P1P0
+δ0)}
where P2
P1= (1 + r)p̂.
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
Allowing for failure in t = t1
Now we allow for banks which are not able to fulfill the deposit claims(δ0 < λc1) to fail in t = t1. The cost of failure is the sum of depositcontracts not fulfilled in t1 and open contracts in t2.
Thus utility of a failed bank is
Fλ := u(δ0 − λc1 − (1− λ)c2).
Banks which fail do not take part in the interbank market.
As in the situation without failure we have to consider three caseslooking for possible equilibria.
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
Allowing for failure in t = t1
Now we allow for banks which are not able to fulfill the deposit claims(δ0 < λc1) to fail in t = t1. The cost of failure is the sum of depositcontracts not fulfilled in t1 and open contracts in t2.
Thus utility of a failed bank is
Fλ := u(δ0 − λc1 − (1− λ)c2).
Banks which fail do not take part in the interbank market.
As in the situation without failure we have to consider three caseslooking for possible equilibria.
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
Allowing for failure in t = t1
Now we allow for banks which are not able to fulfill the deposit claims(δ0 < λc1) to fail in t = t1. The cost of failure is the sum of depositcontracts not fulfilled in t1 and open contracts in t2.
Thus utility of a failed bank is
Fλ := u(δ0 − λc1 − (1− λ)c2).
Banks which fail do not take part in the interbank market.
As in the situation without failure we have to consider three caseslooking for possible equilibria.
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
Allowing for failure in t = t1
Now we allow for banks which are not able to fulfill the deposit claims(δ0 < λc1) to fail in t = t1. The cost of failure is the sum of depositcontracts not fulfilled in t1 and open contracts in t2.
Thus utility of a failed bank is
Fλ := u(δ0 − λc1 − (1− λ)c2).
Banks which fail do not take part in the interbank market.
As in the situation without failure we have to consider three caseslooking for possible equilibria.
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
For δ0 < E (Λ)c1 we get the following result:
If there is r̃ ∈ [ 1p̂ − 1,∞) s.t.
E(
Λ1{FΛ>E(u(V Λ2 )), Λc1>δ0}
)= E (Λ)− 1
c1δ0,
then (r̃ , P2
(1+r̃)p̂ ) is an equilibrium price pair and banks with
Fλ > E (u(V λ2 (α, β0, δ0))) decide to fail in t = t1.
The remaining banks choose the strategy (βλ∗1 , γλ∗+ , γλ∗− , δλ∗1 ).
Else all banks with λc1 > δ0 fail, there is no interbank trading and theremaining banks hold all their bonds and invest spare liquidity in theliquid asset.
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
For δ0 = E (Λ)c1 the interest rates can’t be arbitrarily large anymore:
If (α, β0, δ0) satisfy
E
(u(α
St2
S0+ β0
P2
P0− (1− λ)c2 + (δ0 − λc1)
1
p̂)
)≥ Fλ
for all λ > 1c1
(β0P2
P0+ δ0), then we get an upper bound for r
r̄ := inf{r ∈ [1
p̂− 1,∞) : P(FΛ > E (V2(α, β0, δ0)|Λ >
1
c1δ0)) > 0},
else all banks with λc1 > δ0 fail and there is no interbank market.
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
The optimization problem in t = 0
The objective function (δ0 ≥ E (Λ)c1) is∫ ∞0
∫ 1c1
(β0P2P1
+δ0)
0
u(αs+β0P2
P0+λ(c2−c1
P2
P1)−c2+δ0
P2
P1)PΛ(dλ)P
St2S0 (ds)
+
∫ ∞0
∫ 1
1c1
(β0P2P1
+δ0)
u(αs+β0P2
P0+λ(c2−c1
P2
P1p̂)−c2+δ0
P2
P1p̂)PΛ(dλ)P
St2S0 (ds).
We maximize w.r.t. (α, β0, δ0) under constraints
α, β0 ≥ 0, δ0 ≥ E (Λ)c1
α + β0 + δ0 = 1
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
CRR model for S , Binomial distribution for Λ
Let λl < λh, d < P2
P0< u and P(Λ = λl) = πl = 1− P(Λ = λh),
P(St2
S0= u) = p = 1− P(
St2
S0= d), with p, πl ∈ (0, 1).
Set λ̄ := πlλl + (1− πl)λh. Assume exponential utilityu(x) := − exp(−κx) with κ > 0.
In this case an unsecured interbank market develops and the optimalsolution is interior iff
1− p
p
P2
P0− d
u − P2
P0
∈ (exp(−(u−d)(1−λ̄)κ), exp(−(u−d)(1−λ̄−πl(λh−λl)P0
P2))κ)
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
CRR model for S , Binomial distribution for Λ
Let λl < λh, d < P2
P0< u and P(Λ = λl) = πl = 1− P(Λ = λh),
P(St2
S0= u) = p = 1− P(
St2
S0= d), with p, πl ∈ (0, 1).
Set λ̄ := πlλl + (1− πl)λh. Assume exponential utilityu(x) := − exp(−κx) with κ > 0.
In this case an unsecured interbank market develops and the optimalsolution is interior iff
1− p
p
P2
P0− d
u − P2
P0
∈ (exp(−(u−d)(1−λ̄)κ), exp(−(u−d)(1−λ̄−πl(λh−λl)P0
P2))κ)
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
CRR model for S , Binomial distribution for Λ
Let λl < λh, d < P2
P0< u and P(Λ = λl) = πl = 1− P(Λ = λh),
P(St2
S0= u) = p = 1− P(
St2
S0= d), with p, πl ∈ (0, 1).
Set λ̄ := πlλl + (1− πl)λh. Assume exponential utilityu(x) := − exp(−κx) with κ > 0.
In this case an unsecured interbank market develops and the optimalsolution is interior iff
1− p
p
P2
P0− d
u − P2
P0
∈ (exp(−(u−d)(1−λ̄)κ), exp(−(u−d)(1−λ̄−πl(λh−λl)P0
P2))κ)
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
CRR / Binomial
Then the optimal solution is
(α∗, β∗0 , δ∗0 ) =
− log
(1−pp
P2P0−d
u− P2P0
)κ(u − d)
, 1− λ̄c1 +
log
(1−pp
P2P0−d
u− P2P0
)κ(u − d)
, λ̄c1
For κ = 1, λl = 14 , λh = 3
4 , u = 1.2, d = 1, P2
P0= 1.1 and p = 20
39 we get
(α∗, β∗0 , δ∗0 ) = (− log(0.95),
1
2+ log(0.95),
1
2)
≈ (0.0222763947, 0.477723605, 0.5)
An equilibirum price pair is (1.145, 1) and the solvency probability forlenders in the unsecured interbank market is 20
39 .
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
CRR / Binomial
Then the optimal solution is
(α∗, β∗0 , δ∗0 ) =
− log
(1−pp
P2P0−d
u− P2P0
)κ(u − d)
, 1− λ̄c1 +
log
(1−pp
P2P0−d
u− P2P0
)κ(u − d)
, λ̄c1
For κ = 1, λl = 1
4 , λh = 34 , u = 1.2, d = 1, P2
P0= 1.1 and p = 20
39 we get
(α∗, β∗0 , δ∗0 ) = (− log(0.95),
1
2+ log(0.95),
1
2)
≈ (0.0222763947, 0.477723605, 0.5)
An equilibirum price pair is (1.145, 1) and the solvency probability forlenders in the unsecured interbank market is 20
39 .
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
CRR / Binomial
Then the optimal solution is
(α∗, β∗0 , δ∗0 ) =
− log
(1−pp
P2P0−d
u− P2P0
)κ(u − d)
, 1− λ̄c1 +
log
(1−pp
P2P0−d
u− P2P0
)κ(u − d)
, λ̄c1
For κ = 1, λl = 1
4 , λh = 34 , u = 1.2, d = 1, P2
P0= 1.1 and p = 20
39 we get
(α∗, β∗0 , δ∗0 ) = (− log(0.95),
1
2+ log(0.95),
1
2)
≈ (0.0222763947, 0.477723605, 0.5)
An equilibirum price pair is (1.145, 1) and the solvency probability forlenders in the unsecured interbank market is 20
39 .
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending
The modelOptimal policies in t1
Example
References
Heider, F. and Hoerova, A. (2009). Interbank Lending, CreditRisk Premia and Collateral,ECB Working Paper
Freixas, X. and Holthausen, C. (2005). Interbank MarketIntegration under Asymmetric Information,Review of Financial Studies 18, 459-490
Hanson, M. (1981). On Sufficiency of the Kuhn-TuckerConditions,Journal of Mathematical Analysis and Applications 80, 545-550
Dominik Joos, Karlsruhe Institute of Technology Interbank Lending