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


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.


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).


Outline

1 The illiquid market model

2 HJB equations and characterization of the solution

3 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 ).


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

).


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 ,


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.


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+.


Outline





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).


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+.


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)


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.


Outline





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 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)


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).


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.


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)


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)


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

].


Convergence result

Proposition

vn ր v.

For some C > 0, 0 < δ < 1,

vi (x , y)− vni (x , y) 6 C (x + y)pδn.


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).


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.


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∗


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


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.


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


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).


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).

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

Documents

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