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A problem of portfolio/consumption choice in a liquidity risk model with random trading times

Huyˆ en PHAM ∗ Special Semester on Stochastics with Emphasis on Finance, Kick-off workshop, Linz, September 8-12, 2008

∗University Paris 7 and Institut Universitaire de France

based on joint papers with: Peter TANKOV, University Paris 7 Fausto GOZZI, Alessandra CRETAROLA, Luiss University, Roma

1

• 0. Introduction
• Liquidity risk: one of the most significant risk factors in financial economy
• In general terms, illiquidity: trading restriction

◮ (il)liquidity measures affected by:

• volume: size of traded position
• price: costs caused by trading the position
• time: point in time when one has to trade the position

2

• A market liquidity modeling with random trading times:

⋆ Illiquid asset prices are quoted and observed only at random arrival times:

• discrete nature of financial data: tick-by-tick stock prices
• exogenous random times ←

→ arrivals of buy/sell orders in illiquid markets,

• r instances at which large trade occurs or market maker updates his quotes

in reaction to new information (e.g. publication of the results of a hedge fund) → Such a context is largely considered in econometrics of high-frequency data for the estimation of the jump times intensity and/or volatility: e.g. Rogers and Zane (98, 02), Frey and Runggaldier (01), Cvitanic, Liptser, Ro- zovskii (06), Ait-Sahalia, Mykland, Zhang (04), Barndorff-Nielsen and Shep- hard (06), Ait-Sahalia and Jacod (07, ...), Woerner (07),

3

• Our liquidity risk model for portfolio selection:

⋆ Asset prices are observed only at random arrival times ⋆ Discrete trading are possible only at these random times ⋆ The investor may consume continuously from its cash holding (or distributes dividends to shareholders)

◮ Problem of optimal portfolio/consumption choice ◮ Cost of such liquidity effect with respect to the perfect liquid market (e.g.

Merton model)

4

• 1. Model and Problem formulation
• Stock price S is observed and traded only at exogenous random times

(τk)k≥0 with τ0 = 0 < τ1 < . . . < τk < . . .

• The investor may consume continuously from the bank account between

two trading dates.

◮ Observation continuous filtration: Gc = (Gt)t≥0,

Gt = σ{(τk, Sτk) : τk ≤ t}

◮ Observation discrete filtration: Gd

= (Gτk)k≥0,

5

• Control policy: mixed discrete/continuous time process (α, c):

⋆ α = (αk)k≥1 is a real-valued Gd-predictable process: αk represents the amount of stock invested for the period (τk−1, τk] after

• bserving the stock price Sτk−1 at time τk−1

⋆ c = (ct)t≥0 is a nonnegative Gc-adapted process: ct represents the consumption rate at time t based on the observation of random arrival times and stock prices until t.

6

• Wealth discrete process: starting from an initial capital x ≥ 0, and given

a strategy (α, c), the wealth Xx

k of the investor at time τk is:

Xx

k

= x −

τk

ctdt +

k

• i=1

αi Sτi − Sτi−1 Sτi−1 , k ≥ 1,

7

• Wealth discrete process: starting from an initial capital x ≥ 0, and given

a strategy (α, c), the wealth Xx

k of the investor at time τk is:

Xx

k

= x −

τk

ctdt +

k

• i=1

αi Zi, k ≥ 1, where Zk = Sτk − Sτk−1 Sτk−1 is the observed return process valued in (−1, ∞).

◮ Admissible control policy: given x ≥ 0, we say that (α, c) is admissible,

(α, c) ∈ A(x) if: Xx

k

≥ 0, a.s. ∀k ≥ 1.

8

• Optimal portfolio/consumption problem:

◮ Utility function U : R+ → R, C1, increasing, concave, U(0) = 0, satisfying

the Inada conditions U′(0) = ∞, U′(∞) = 0, and the growth condition U(w) ≤ K1wγ, γ ∈ [0, 1).

◮ Value function:

v(x) = sup

(α,c)∈A(x)

E

∞

e−ρtU(ct)dt

• ,

x ≥ 0. → Mixed discrete/continuous stochastic control problem → Not completely standard in the literature on control

9

• Conditions on (τk, Zk):

(H1) {τk}k≥1 is the increasing sequence of jump times of a Poisson process with intensity λ. (H2) (i) “L´ evy property” on the return process: Conditionally on the interarrival time τk − τk−1 = t, Zk is independent from {τi, Zi}i<k and has a distribution p(t, dz). (ii) “Arbitrage property”: the support of p(t, dz) is either

• an interval of interior (−z, ¯

z), z ∈ (0, 1] and ¯ z ∈ (0, ∞],

• or is finite equal to {−z, . . . , ¯

z}, z ∈ (0, 1] and ¯ z ∈ (0, ∞). (H3) There exist some κ, b ∈ R+ s.t.

|z|p(t, dz) ≤ κebt, ∀ t ≥ 0

(H4) Continuity of the measure p(t, dz): limt→t0

w(z)p(t, dz) = w(z)p(t0, dz), t0 ≥ 0,

for all measurable functions w with linear growth condition

10

• Example:

S extracted from a Black-Scholes model: dSt = bStdt + σStdWt. Then p(t, dz) is the distribution of Z(t) = exp

• b − σ2

2

• t + σWt
• − 1,

with support (−1, ∞).

11

Mixed discrete/continuous stochastic control problem v(x) = sup

(α,c)∈A(x)

E

∞

e−ρtU(ct)dt

• ,

x ≥ 0. αk+1 Gτk-measurable, Gτk = σ{(τi, Zi), i ≤ k}, k ∈ N ct Gt-measurable, Gt = σ{(τk, Zk), τk ≤ t}, t ∈ R+ Xx

k

= x −

τk

ctdt +

k

• i=1

αi Zi, ≥ 0, ← → A(x), k ∈ N∗.

12

• 2. Dynamic programming and first-order coupled nonlinear IPDE

Relation on the value function by considering two consecutive trading dates: stationarity of the problem → between τ0 = 0 and τ1: v(x) = sup

(α,c)∈A(x)

E

τ1

e−ρtU(ct)dt + e−ρτ1v(Xx

1)

• 13

• 2. Dynamic programming and first-order coupled nonlinear IPDE

Relation on the value function by considering two consecutive trading dates: stationarity of the problem → between τ0 = 0 and τ1: v(x) = sup

(α,c)∈A(x)

E

τ1

e−ρtU(ct)dt + e−ρτ1v(Xx

1)

• =

sup

(a,c)∈Ad(x)

E

τ1

e−ρtU(ct)dt + e−ρτ1v(x −

τ1

ctdt + aZ1)

• ,

Ad(x): pair of deterministic constants a and nonnegative processes c = (ct)t≥0 s.t.: x −

τ1

0 ctdt + aZ1 ≥ 0 a.s. , i.e. by (H2)(ii)

14

• 2. Dynamic programming and first-order coupled nonlinear IPDE

Relation on the value function by considering two consecutive trading dates: stationarity of the problem → between τ0 = 0 and τ1: v(x) = sup

(α,c)∈A(x)

E

τ1

e−ρtU(ct)dt + e−ρτ1v(Xx

1)

• =

sup

(a,c)∈Ad(x)

E

τ1

e−ρtU(ct)dt + e−ρτ1v(x −

τ1

ctdt + aZ1)

• ,

Ad(x): pair of deterministic constants a and nonnegative processes c = (ct)t≥0 s.t.: x −

τ1

0 ctdt + aZ1 ≥ 0 a.s. , i.e. by (H2)(ii)

−x ¯ z ≤ a ≤ x z x −

t

0 cudu ≥ ℓ(a),

∀t ≥ 0 : c ∈ Ca(x), where ℓ(a) = max(az, −a¯ z).

15

◮ Split into two (coupled) optimization problems:

• Fix a. Optimal consumption problem over c:

ˆ v(0, x, a) := sup

c∈Ca(x)

E

τ1

e−ρtU(ct)dt + e−ρτ1v(Y x

τ1 + aZ1)

• ,

with wealth Y x

s

= x −

s

0 cudu

≥ ℓ(a). → Given an investment a in the stock at τk and hold until the next trading date τk+1, ˆ v(., a) is the value function for an optimal consumption problem between τk and τk+1, with wealth Y , and reward utility v(Yτk+1 + aZk+1) at τk+1.

• Extremum (scalar) problem on portfolio:

v(x) = sup

a ∈ [−x/¯ z, x/z]

ˆ v(0, x, a).

16

Under conditions (H1) and (H2)(i) on (τ1, Z1), the value function ˆ v for the

• ptimal consumption problem is expressed as:

ˆ v(0, x, a) = sup

c∈Ca(x)

∞

e−(ρ+λ)s

• U(cs) + λ
• v(Y x

s + az)p(s, dz)

• ds.

→ Deterministic control problem on infinite horizon,

17

Under conditions (H1) and (H2)(i) on (τ1, Z1), the value function ˆ v for the

• ptimal consumption problem is expressed as:

ˆ v(t, x, a) = sup

c∈Ca(t,x)

∞

t

e−(ρ+λ)(s−t)

• U(cs) + λ
• v(Y t,x

s

+ az)p(s, dz)

• ds.

→ Deterministic control problem on infinite horizon, but nonstationary when p(t, dz) depends on t

18

Under conditions (H1) and (H2)(i) on (τ1, Z1), the value function ˆ v for the

• ptimal consumption problem is expressed as:

ˆ v(t, x, a) = sup

c∈Ca(t,x)

∞

t

e−(ρ+λ)(s−t)

• U(cs) + λ
• v(Y t,x

s

+ az)p(s, dz)

• ds.

→ Deterministic control problem on infinite horizon, but nonstationary when p(t, dz) depends on t → We can write the Hamilton Jacobi (HJ) equation for ˆ v. (ρ + λ)ˆ v − ∂ˆ v ∂t − ˜ U

∂ˆ

v ∂x

• − λ
• v(x + az)p(t, dz)

= 0, t ≥ 0, x ≥ ℓ(a), ˜ U(p) = supc≥0[U(c) − cp] convex conjugate of U.

19

◮ Dynamic programming −

→ system of coupled nonlinear IPDE: (ρ + λ)ˆ v − ∂ˆ v ∂t − ˜ U

∂ˆ

v ∂x

• − λ
• v(x + az)p(t, dz)

= 0, t ≥ 0, x ≥ ℓ(a), v(x) = Hˆ v(x) := sup

a ∈ [−x/¯ z, x/z]

ˆ v(0, x, a), x ≥ 0,

20

◮ Dynamic programming −

→ system of coupled nonlinear IPDE: (ρ + λ)ˆ v − ∂ˆ v ∂t − ˜ U

∂ˆ

v ∂x

• − λ
• v(x + az)p(t, dz)

= 0, t ≥ 0, x ≥ ℓ(a), v(x) = Hˆ v(x) := sup

a ∈ [−x/¯ z, x/z]

ˆ v(0, x, a), x ≥ 0, Compare with the HJB equation for the Merton problem (λ = ∞): ρv − ˜ U

∂v

∂x

• − sup

a∈R

• abx∂v

∂x + 1 2a2x2σ2∂2v ∂x2

• =

21

Boundary conditions on v, ˆ v

• Proposition. Under (H1)-(H2)-(H3)-(H4), suppose that ρ satisfies

(Hρ) ρ > bγ + λ(κγ zγ − 1). Then, v and ˆ v are continuous on R+ and D and satisfy: (G1) Growth condition: there exists some positive constant K s.t. ˆ v(t, x, a) ≤ K(ebtx)γ, ∀t ≥ 0, x ≥ ℓ(a), v(x) ≤ Kxγ, ∀x ≥ 0. (B1) Boundary data ← → nonnegative wealth constraint: ˆ v(t, x, a) = λ

∞

t

e−(ρ+λ)(s−t)

• v(x + az)p(s, dz)ds,

at x = ℓ(a) (Coupled Dirichlet condition!)

22

• 3. PDE characterization: viscosity approach

Viscosity solutions for the coupled IPDE: (ρ + λ) ˆ w − ∂ ˆ w ∂t − ˜ U

∂ ˆ

w ∂x

• − λ
• w(x + az)p(t, dz)

= 0, t ≥ 0, x > ℓ(a), w(x) = H ˆ w(x) = sup

a ∈ [−x/¯ z, x/z]

ˆ w(0, x, a), x ∈ R+, Definition. A pair of functions (w, ˆ w) ∈ C+(R+) × C+(D) is a viscosity supersolution to the above IPDE if: (i) w ≥ H ˆ w and (ii) for all a ∈ R, (¯ t, ¯ x) ∈ R+ × (ℓ(a), ∞), (ρ + λ) ˆ w(¯ t, ¯ x, a) − ∂ϕ ∂t (¯ t, ¯ x) − ˜ U

∂ϕ

∂x(¯ t, ¯ x)

• − λ
• w(¯

x + az)p(¯ t, dz) ≥ 0, for any test function ϕ ∈ C1(R+ × (ℓ(a), ∞)), which is a local minimum of ( ˆ w(., ., a) − ϕ).

23

• 3. PDE characterization: viscosity approach

Viscosity solutions for the coupled IPDE: (ρ + λ) ˆ w − ∂ ˆ w ∂t − ˜ U

∂ ˆ

w ∂x

• − λ
• w(x + az)p(t, dz)

= 0, t ≥ 0, x > ℓ(a), w(x) = H ˆ w(x) = sup

a ∈ [−x/¯ z, x/z]

ˆ w(0, x, a), x ∈ R+, Definition. A pair of functions (w, ˆ w) ∈ C+(R+) × C+(D) is a viscosity subsolution to the above IPDE if: (i) w ≤ H ˆ w and (ii) for all a ∈ R, (¯ t, ¯ x) ∈ R+ × (ℓ(a), ∞), (ρ + λ) ˆ w(¯ t, ¯ x, a) − ∂ϕ ∂t (¯ t, ¯ x) − ˜ U

∂ϕ

∂x(¯ t, ¯ x)

• − λ
• w(¯

x + az)p(¯ t, dz) ≤ 0, for any test function ϕ ∈ C1(R+ × (ℓ(a), ∞)), which is a local maximum of ( ˆ w(., ., a) − ϕ).

24

• Theorem. Under (H1)-(H2)-(H3)-(H4), and

(Hρ) ρ > bγ The pair of value functions (v, ˆ v) is the unique viscosity solution to the IPDE: (ρ + λ)ˆ v − ∂ˆ v ∂t − ˜ U

∂ˆ

v ∂x

• − λ
• v(x + az)p(t, dz)

= 0, t ≥ 0, x > ℓ(a), v(x) = Hˆ v(x) = sup

a ∈ [−x/¯ z, x/z]

ˆ v(0, x, a), x ∈ R+, satisfying the boundary conditions (G1)-(B1).

25

• 4. Regularity results and optimal strategies
• What about the optimal strategies (α, c)?

→ Usually obtained by a classical verification theorem → Need to go beyond viscosity solutions, and to get some smoothness pro- perties on the value functions ˆ v!

26

Assumptions: (H5) U is strictly concave. (H6) p(t, dz) admits a density f(t, z) continuously differentiable

• Theorem. The value function ˆ

v(., a) is C1 on (0, ∞) × (ℓ(a), ∞), for all a ∈ R. Moreover, v is C1 on (0, ∞). The proof relies on arguments from nonsmooth analysis, (semi)concavity and convex Hamiltonians + viscosity solutions (see e.g. book by Bardi, Capuzzo- Dolcetta).

27

Existence and characterization of optimal strategies by a verification theorem

• Feedback trading portfolio from the scalar maximum problem:

ˆ α(x) ∈ arg max

a ∈ [−x/¯ z, x/z] ˆ

v(0, x, a) → α∗

k+1

= ˆ α(Xx

k),

k ≥ 0. (1)

• Feedback consumption from the deterministic control problem:

ˆ c(t, x, a) ∈ arg max

c≥0

• U(c) − c ∂ˆ

v ∂x(t, x, a)

• = I

∂ˆ

v ∂x(t, x, a)

• →

c∗

t

= ˆ c(t − τk, Y

τk,Xx

k

t

, α∗

k+1),

τk < t ≤ τk+1. (2) where I = (U′)−1, and Y

τk,Xx

k

t

= Xx

k −

t

τk

c∗

sds,

is the wealth between two trading dates τk, τk+1.

28

• 5. Numerical resolution and experiments

Coupling IPDE: (ρ + λ)ˆ v − ∂ˆ v ∂t − ˜ U

∂ˆ

v ∂x

• − λ
• v(x + az)p(t, dz)

= 0, t ≥ 0, x > ℓ(a), ˆ v(t, x, a) = λ

∞

t

e−(ρ+λ)(s−t)

• v(x + az)p(s, dz)ds,

t ≥ 0, x = ℓ(a). v(x) = Hˆ v(x) := sup

a ∈ [−x/¯ z, x/z]

ˆ v(0, x, a), x ≥ 0, − → Theoretical and numerical difficulties

29

A numerical decoupling algorithm Decoupling system with the following iterative procedure:

◮ start from v0:

the value function of the consumption problem without trading: v0(x) = sup

c∈C(x)

∞

e−ρtU(ct)dt, C(x): set of nonnegative functions c = (ct)t s.t. x −

t

0 csds ≥ 0 for all t ≥ 0.

← → ρv0 − ˜ U

∂v0

∂x

• =

0, x > 0, v0(0+) = 0.

30

◮ Step n → n + 1: first-order PDE

(ρ + λ)ˆ vn+1 − ∂ˆ vn+1 ∂t − ˜ U

∂ˆ

vn+1 ∂x

• − λ
• vn(x + az)p(t, dz)

= 0, t ≥ 0, x > ℓ(a) ˆ vn+1(t, x, a) = λ

∞

t

e−(ρ+λ)(s−t)

• vn(x + az)p(s, dz)ds,

t ≥ 0, x = ℓ(a). vn+1(x) = Hˆ vn+1(x) := sup

a ∈ [−x/¯ z, x/z]

ˆ vn+1(0, x, a), x ≥ 0,

31

Convergence of the algorithm

• Theorem. Under (H1)-(H2)-(H3)-(H4), and

(Hρ) ρ > bγ the sequence of functions (ˆ vn, vn)n≥1 converges increasingly and uniformly on any compact subset of D and R+ to (ˆ v, v). More precisely, for any compact set F and G of D and R+: 0 ≤ sup

F

(ˆ v − ˆ vn) ≤ CF δn 0 ≤ sup

G

(v − vn) ≤ CG δn, for some positive constants CF and CG, and where under (Hρ) δ := λ ρ − bγ + λ < 1

32

Numerical illustrations

• (Zk) extracted from a BS model
• Jump times (τk) of intensity λ ↔ “illiquidity” of the market
• U(x) = xγ/γ, γ = 0.5.
• ρ = 0.2, b = 0.4, σ = 1:

◮ value function without trading: v0(x) = K0xγ, and K0 = 3.16 ◮ value function of Merton problem: vM(x) = KMxγ, and KM = 4.08

and the amount invested in stock is proportional to wealth: αM

t

= 0.8Xt.

33

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 1 2 3 4 5 6 7 8 9 10 v0 4 iterations 10 iterations 50 iterations Merton 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1 iteration 5 iterations 50 iterations Merton

Left : Convergence of the iterative algorithm for computing the value function with λ = 1. Right : Convergence of the iterative algorithm for computing the optimal investment policy (the amount to invest in stock as a function of the total wealth at the trading date). 34

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 1 2 3 4 5 6 7 8 9 10 v0 lambda=1 lambda=5 lambda=40 Merton 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 lambda=1 lambda=5 lambda=40 Merton

Behavior of the value function in an illiquid market (left) and of the optimal investment policy (right) for different values of the Poisson parameter λ. 35

Cost of illiquidity: π = π(x) s.t. v(x + π(x)) = vM(x). λ 0 (No trading) 1 5 40 π(1) 0.6671 0.2749 0.1214 0.0539

36

Profile of consumption between two trading dates

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

• Exp. bound

Optimal wealth 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Consumption corresponding to Y0 Optimal consumption

Left: typical profile of the optimal wealth process Yt, and of the wealth process Y 0 corresponding to an investor, which only consumes and has a zero reward utility at a random time. Right: the corresponding consumption strategies. In the presence of investment opportunities, the agent first consumes slowly but if the investment opportunity does not appear, the agent eventually “gets disappointed” and starts to consume fast. 37

• 6. Conclusion
• A (simple) model of liquidity risk with random trading times:

◮ Theoretical and numerical study, and quantitative cost of illiquidity

• Further questions and development:

◮ Expansion in 1/λ of the value function and optimal strategy around the

Merton problem (λ = ∞)

38

◮ More general and realistic models, e.g.:

• Cox processes for jump times: λ = λ(θt)
• Models with unobservable volatility
• Models with microstructure noise on the price

39