investment-consumption problem in illiquid ... - uni-jena.de · u(c)−c ∂v i ∂x = x j6= i q...
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![Page 1: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/1.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Investment-consumption problem in illiquidmarkets with regime switching
Paul Gassiat
LPMA - Universite Paris VII
Spring School ”Stochastic Models in Finance and Insurance”,22 March 2011
Joint work with Huyen Pham and Fausto Gozzi
![Page 2: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/2.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Introduction
An important aspect of market liquidity : restriction ontrading times.
We assume the investor can trade the risky asset only atrandom times, corresponding to the arrival of buy/sell orders.
Empirical evidence for liquidity fluctuation : market withregime-switching liquidity/price dynamics.
In this context, we study the problem of optimalinvestment/consumption over an infinite horizon.
![Page 3: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/3.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Previous works
Rogers, Zane (02); Matsumoto (06); Pham , Tankov (08) :single-regime case. In all of these papers, the trading times arethe jump times of a Poisson process with constant intensity λ.
Diesinger, Kraft, Seifried (10); Ludkovski, Min (10) : Tworegimes, fully liquid (continuous trading) and fully illiquid (notrading).
![Page 4: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/4.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Outline
1 The illiquid market model
2 HJB equations and characterization of the solution
3 Power utility functions and numerical results
![Page 5: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/5.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Outline
1 The illiquid market model
2 HJB equations and characterization of the solution
3 Power utility functions and numerical results
![Page 6: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/6.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Filtered probability space (Ω,F , (Ft)t > 0,P).
Liquidity regime cycles : Markov chain (It)t > 0 with statespace Id = 1, . . . , d, and intensity matrix Q = (qij). N
ij
associated Poisson processes.
Risky asset of price process (St)t > 0.
The investor is restricted to trading S only at random timesτn, n > 1, jump times of a Cox process (Nt)t > 0 withintensity (λIt ).
![Page 7: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/7.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Hybrid regime-switching jump diffusion model
In the liquidity regime It = i ,
dSt = St(bidt + σidWt),
where W is a (Ft)-Brownian Motion, and bi ∈ R, σi > 0.At the times of transition from It− = i to It = j , ∆St = −St−γij ,where γij < 1.Overall :
dSt = St−(bI
t−dt + σI
t−dWt − γ
It−
,ItdN
It−
,Itt
).
![Page 8: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/8.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Trading strategies
We consider an agent investing and consuming in this market.Trading strategy : pair (c , ζ), where
(ct) nonnegative adapted process is the consumption process,
(ζt) predictable is the investment strategy : at t = τn, theagent buys an amount ζt in the risky asset.
(X c,ζt ,Y
c,ζt ) amounts invested in cash and in the risky asset.
dXc,ζt = −ctdt − ζtdNt .
dYc,ζt = Yt−
(bI
t−dt + σI
t−dWt − γ
It−
,ItdN
It−
,Itt
)+ ζtdNt ,
![Page 9: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/9.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Admissible strategies
Total wealth Rt := Xt + Yt .Under initial conditions (i , x , y),(c , ζ) ∈ Ai (x , y) (set of admissible strategies) if Rτn > 0 a.s.,n > 1.This is equivalent to a no-short sale constraint :
Xt > 0 ; Yt > 0,
or in terms of the controls :
−Yt− 6 ζt 6 Xt− ,∫ τn+1
t
csds 6 Xt , τn 6 t < τn+1, n > 1.
![Page 10: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/10.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Optimal investment/consumption
U increasing, concave function on [0,∞) with U(0) = 0.ρ > 0 discount factor.Value functions :
vi (x , y) = sup(ζ,c)∈Ai (x ,y)
E
[∫ ∞
0e−ρtU(ct)dt
], i ∈ Id , (x , y) ∈ R
2+,
vi (r) = supx+y=r
vi (x , y) i ∈ Id , r ∈ R+.
![Page 11: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/11.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Outline
1 The illiquid market model
2 HJB equations and characterization of the solution
3 Power utility functions and numerical results
![Page 12: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/12.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Assumptions :
There exists p ∈ (0, 1), K1 > 0 s.t.
U(x) 6 K1xp, x > 0.
ρ > k(p), where
k(p) := supi∈Id ,z∈[0,1]
pbiz −σ2i
2p(1− p)z2 +
∑
j 6=i
qij((1− zγij)p − 1).
![Page 13: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/13.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Proposition
For some positive constant C,
vi (x , y) 6 C (x + y)p, (i , x , y) ∈ Id × R+ × R+. (1)
vi is concave in (x , y), increasing in both variables, and continuous
on R2+.
vi is increasing, concave and continuous on R+.
![Page 14: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/14.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
HJB System
The Hamilton-Jacobi-Bellman equation for this problem is :
ρvi − biy∂vi
∂y−
1
2σ2i y
2∂2vi
∂y2− sup
c > 0
[U(c)− c
∂vi
∂x
]
−∑
j 6=i
qij
[vj
(x , y(1− γij)
)− vi (x , y)
](2)
− λi
[sup
−y 6 ζ 6 x
vi (x − ζ, y + ζ)− vi (x , y)]
= 0.
on Id × (0,∞)× R+, together with the boundary conditions :
vi (0, 0) = 0 (3)
vi (0, y) = Ei
sup0 6 ζ 6 y
Sτ1S0
vIτ1
(ζ, y
Sτ1S0
− ζ
) (4)
![Page 15: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/15.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Viscosity characterization of the value function
Theorem
The value function v is a viscosity solution to the HJB system (2)and the boundary conditions (3) and (4).It is the unique solution satisfying the growth condition
vi (x , y) 6 C (x + y)p.
![Page 16: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/16.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Outline
1 The illiquid market model
2 HJB equations and characterization of the solution
3 Power utility functions and numerical results
![Page 17: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/17.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Power utility functions
We consider the case U(c) = cp
p, 0 < p < 1.
Scaling property for the value function : vi (kx , ky) = kpvi (x , y).Change of variables
r = x + y
z =y
x + y.
The value function for (r , z) can be written as
ui (r , z) =rp
pϕi (z)
![Page 18: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/18.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
The HJB equation for the ϕi is the system of IODEs :
(ρ− qii + λi − pbiz +1
2p(1− p)σ2
i z2)ϕi − z(1− z)(bi − z(1− p)σ2
i )ϕ′i
−
1
2z2(1− z)2σ2
i ϕ′′i − (1− p)
(
ϕi −z
pϕ
′i
)−p
1−p (5)
−
∑
j 6=i
qij
[
(1− zγij)pϕj
(
z(1− γij)
1− zγij
)
]
− λi supπ∈[0,1]
ϕi (π) = 0,
together with the boundary conditions
(ρ− qii + λi )ϕi (0)− (1− p)ϕi (0)−
p1−p =
∑
j 6=i
qijϕj (0) + λi supπ∈[0,1]
ϕi (π), (6)
(ρ− qii + λi − pbi +1
2p(1− p)σ2
i )ϕi (1) =∑
j 6=i
qij (1− γij )pϕj (1) + λi sup
π∈[0,1]ϕi (π). (7)
![Page 19: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/19.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Regularity of the value function
Proposition
(ϕi )i=1,...,d is concave and continuous on [0, 1], C2 on (0, 1) and is
a classical solution of (5) on (0, 1), with boundary conditions
(6)-(7).
![Page 20: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/20.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Optimal Control
Define
c∗(i , z) =
(ϕi (z)−
zpϕ′i (z)
) −11−p
when 0 < z < 1
(ϕi (0))−11−p when z = 0
0 when z = 1
,
π∗(i) = arg supπ∈[0,1]
ϕi (π).
Proposition
Given initial conditions i , r , z, there exists an admissible control
(c , ζ) such that :
ct = Rt−c∗(It−,Zt−),
ζt = Rt−(π∗(It−)− Zt−),
and this control is optimal.
![Page 21: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/21.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Numerical analysis : iterative method
Recall the HJB system :
(ρ− qii + λi )vi − biy∂vi
∂y−
1
2σ2i y
2 ∂2vi
∂y2− sup
c > 0
[U(c)− c
∂vi
∂x
]
=∑
j 6=i
qijvj
(x , y(1− γij)
)+ λi sup
−y 6 ζ 6 x
vi (x − ζ, y + ζ)
![Page 22: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/22.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Numerical analysis : iterative method
We solve it by iteration :
v 0 = 0,
Given vn,
(ρ− qii + λi )vn+1i − biy
∂vn+1i
∂y−
1
2σ2i y
2 ∂2vn+1
i
∂y 2− sup
c > 0
[
U(c)− c∂vn+1
i
∂x
]
=∑
j 6=i
qijvnj
(
x , y(1− γij))
+ λi sup−y 6 ζ 6 x
vni (x − ζ, y + ζ) (8)
with boundary conditions
vn+1i (0, 0) = 0, (9)
vn+1i (0, y) = Ei
sup
0 6 ζ 6 ySτ1S0
vnIτ1
(
ζ, ySτ1
S0− ζ
)
(10)
vn+1i (x , y) 6 C(x + y)p (11)
![Page 23: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/23.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Stochastic control representation of v n
Consider the stopping times
θ0 = 0,
θn+1 = inft > θn s.t. ∆Nt 6= 0 or ∆N
It−,Itt 6= 0
,
i.e. θn is the n-th time where we have either a change of regime ora trading time.
Proposition
Given v0, . . . , vn, (8)-(11) admits a unique viscosity solution.
Furthermore, we have for n > 0,
vni (x , y) = sup(ζ,c)∈Ai (x ,y)
E
[∫ θn
0e−ρtU(ct)dt
].
![Page 24: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/24.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Convergence result
Proposition
vn ր v.
For some C > 0, 0 < δ < 1,
vi (x , y)− vni (x , y) 6 C (x + y)pδn.
![Page 25: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/25.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Going back to U(c) = cp
p: vni (x , y) =
rp
pϕni (z).
ϕn+1i solves the following BVP :
(ρ− qii + λi − pbiz +1
2p(1− p)σ2
i z2)ϕn+1
i − z(1− z)(bi − z(1− p)σ2i )(ϕ
n+1i )′
−
1
2z2(1− z)2σ2
i (ϕn+1i )′′ − (1− p)
(
ϕn+1i −
z
p(ϕn+1
i )′)−
p1−p
=∑
j 6=i
qij
[
(1− zγij)pϕ
nj
(z(1− γij)
1− zγij
)
]
+ λi supπ∈[0,1]
ϕni (π),
(ρ− qii + λi )ϕn+1i
(0)− (1− p)ϕn+1i
(0)−
p1−p =
∑
j 6=i
qijϕnj (0) + λi sup
π∈[0,1]ϕni (π),
(ρ− qii + λi − pbi +1
2p(1− p)σ2
i )ϕn+1i
(1) =∑
j 6=i
qij (1− γij )pϕn
j (1) + λi supπ∈[0,1]
ϕni (π).
![Page 26: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/26.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Numerical Results : Single-regime case
ρ = 0.2, b = 0.4, σ = 1.Cost of liquidity P(r) : v(r + P(r)) = vM(r).Comparison with (Pham, Tankov 08) : similar model, the differenceis that the investor only observes the stock price at the tradingtimes, so that the consumption process is piecewise-deterministic.
λ Discrete observation Continuous observation
1 0.275 0.1535 0.121 0.01640 0.054 0.001
Table: Cost of liquidity P(1) as a function of λ.
![Page 27: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/27.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Value function ϕ(z) for different values of λ
1.65
1.7
1.75
1.8
1.85
1.9
1.95
2
2.05
0 0.2 0.4 0.6 0.8 1
λ = 1λ = 3
λ = 5λ = 10
Mertonz∗
![Page 28: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/28.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Optimal consumption rate c∗(z) for different values of λ
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.2 0.4 0.6 0.8 1
λ = 1λ = 3
λ = 5λ = 10
Merton
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The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Two regimes
Parameters :
b1 = 0.4, σ1 = 1,
b2 = 0.1, σ2 = 0.7,
γ12 = γ21 = 0,
q12 = q21 = 1.
Regime 1 is ”better” than regime 2 : b1σ1
> b2σ2.
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The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
1.4
1.45
1.5
1.55
1.6
1.65
1.7
1.75
1.8
1.85
0 0.2 0.4 0.6 0.8 1
ϕ1(z)
0 0.2 0.4 0.6 0.8 1
ϕ2(z)
(λ1, λ2) = (1, 1)(λ1, λ2) = (10, 1)
(λ1, λ2) = (1, 10)(λ1, λ2) = (10, 10)
Merton
![Page 31: Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q ijv n j x,y(1−γ ij) +λ i sup −y 6ζ 6x vn i (x−ζ,y+ζ) (8) withboundaryconditions](https://reader033.vdocuments.us/reader033/viewer/2022050512/5f9cf1d3c39d217870745587/html5/thumbnails/31.jpg)
The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Numerical results : cost of liquidity
λ P1(1) P2(1)
1 0.153 0.01910 0.004 0.000
Table: Cost of liquidity Pi (1) for the single-regime case.
(λ1, λ2) P1(1) P2(1)
(1,1) 0.111 0.089(10,1) 0.039 0.034(1,10) 0.051 0.041(10,10) 0.009 0.009
Table: Cost of liquidity Pi (1) as a function of (λ1, λ2).
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The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results
Conclusion
Future work : incomplete observation, the agent only observes thestock price at the trading times and must infer the liquidity regime.v(~π, x , y), where πi = P(I0 = i).