Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q...
Transcript of Investment-consumption problem in illiquid ... - uni-jena.de · U(c)−c ∂v i ∂x = X j6= i q...
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
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.
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).
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
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
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 ).
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
).
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 ,
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.
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+.
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
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).
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+.
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)
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.
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
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)
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)
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).
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.
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)
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)
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
].
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.
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).
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.
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∗
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
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.
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
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).
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).