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

Transcription

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

2 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

3 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, or 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, Rozovskii (06), Ait-Sahalia, Mykland, Zhang (04), Barndorff-Nielsen and Shephard (06), Ait-Sahalia and Jacod (07,...), Woerner (07), 3

4 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

5 1. Model and Problem formulation Stock price S is observed and traded only at exogenous random times (τ k ) k 0 with τ 0 = 0 < τ 1 <... < τ k <... The investor may consume continuously from the bank account between two trading dates. Observation continuous filtration: G c = (G t ) t 0, G t = σ{(τ k, S τk ) : τ k t} Observation discrete filtration: G d = (G τk ) k 0, 5

6 Control policy: mixed discrete/continuous time process (α, c): α = (α k ) k 1 is a real-valued G d -predictable process: α k represents the amount of stock invested for the period (τ k 1, τ k ] after observing the stock price S τk 1 at time τ k 1 c = (c t ) t 0 is a nonnegative G c -adapted process: c t represents the consumption rate at time t based on the observation of random arrival times and stock prices until t. 6

7 Wealth discrete process: starting from an initial capital x 0, and given a strategy (α, c), the wealth X x k of the investor at time τ k is: X x k = x τk 0 c tdt + k i=1 α i S τi S τi 1 S τi 1, k 1, 7

8 Wealth discrete process: starting from an initial capital x 0, and given a strategy (α, c), the wealth X x k of the investor at time τ k is: where X x k = x τk 0 c tdt + k i=1 α i Z i, k 1, Z k = S τ k S τk 1 S τk 1 is the observed return process valued in ( 1, ). Admissible control policy: (α, c) A(x) if: given x 0, we say that (α, c) is admissible, X x k 0, a.s. k 1. 8

9 Optimal portfolio/consumption problem: Utility function U : R + R, C 1, increasing, concave, U(0) = 0, satisfying the Inada conditions U (0) =, U ( ) = 0, and the growth condition Value function: U(w) K 1 w γ, γ [0, 1). v(x) = sup E (α,c) A(x) [ 0 e ρt U(c t )dt ], x 0. Mixed discrete/continuous stochastic control problem Not completely standard in the literature on control 9

10 Conditions on (τ k, Z k ): (H1) (H2) {τ k } k 1 is the increasing sequence of jump times of a Poisson process with intensity λ. (i) Lévy property on the return process: Conditionally on the interarrival time τ k τ k 1 = t, Z k is independent from {τ i, Z i } 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) κe bt, t 0 (H4) Continuity of the measure p(t, dz): lim t t0 w(z)p(t, dz) = w(z)p(t0, dz), t 0 0, for all measurable functions w with linear growth condition 10

11 Example: S extracted from a Black-Scholes model: ds t = bs t dt + σs t dw t. Then p(t, dz) is the distribution of with support ( 1, ). Z(t) = exp [( b σ2 2 ) t + σw t ] 1, 11

12 Mixed discrete/continuous stochastic control problem [ ] v(x) = sup E (α,c) A(x) 0 e ρt U(c t )dt, x 0. α k+1 G τk -measurable, G τk = σ{(τ i, Z i ), i k}, k N c t G t -measurable, G t = σ{(τ k, Z k ), τ k t}, t R + X x k = x τk 0 c tdt + k i=1 α i Z i, 0, A(x), k N. 12

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 τ 0 = 0 and τ 1 : v(x) = sup E (α,c) A(x) [ τ1 0 e ρt U(c t )dt + e ρτ 1v(X x 1 ) ] 13

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 τ 0 = 0 and τ 1 : v(x) = sup E (α,c) A(x) [ τ1 0 e ρt U(c t )dt + e ρτ 1v(X x 1 ) [ τ1 = sup E (a,c) A d (x) 0 e ρt U(c t )dt + e ρτ 1v(x τ1 ] 0 c tdt + az 1 ) A d (x): pair of deterministic constants a and nonnegative processes c = (c t ) t 0 s.t.: x τ 1 0 c tdt + az 1 0 a.s., i.e. by (H2)(ii) ], 14

15 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 τ 0 = 0 and τ 1 : v(x) = sup E (α,c) A(x) [ τ1 0 e ρt U(c t )dt + e ρτ 1v(X x 1 ) [ τ1 = sup E (a,c) A d (x) 0 e ρt U(c t )dt + e ρτ 1v(x τ1 ] 0 c tdt + az 1 ) A d (x): pair of deterministic constants a and nonnegative processes c = (c t ) t 0 s.t.: x τ 1 0 c tdt + az 1 0 a.s., i.e. by (H2)(ii) ], x t where l(a) = max(az, a z). x z a x z 0 c udu l(a), t 0 : c C a (x), 15

16 Split into two (coupled) optimization problems: Fix a. Optimal consumption problem over c: with wealth ˆv(0, x, a) := sup E c C a (x) Y x s [ τ1 0 e ρt U(c t )dt + e ρτ 1v(Y x τ 1 + az 1 ) s = x c udu l(a). 0 Given an investment a in the stock at τ k and hold until the next trading date τ k+1, ˆv(., a) is the value function for an optimal consumption problem between τ k and τ k+1, with wealth Y, and reward utility v(y τk+1 + az k+1 ) at τ k+1. ], Extremum (scalar) problem on portfolio: v(x) = sup a [ x/ z, x/z] ˆv(0, x, a). 16

17 Under conditions (H1) and (H2)(i) on (τ 1, Z 1 ), the value function ˆv for the optimal consumption problem is expressed as: [ ˆv(0, x, a) = sup c C a (x) 0 e (ρ+λ)s U(c s ) + λ Deterministic control problem on infinite horizon, v(y x s + az)p(s, dz) ] ds. 17

18 Under conditions (H1) and (H2)(i) on (τ 1, Z 1 ), the value function ˆv for the optimal consumption problem is expressed as: ˆv(t, x, a) = sup c C a (t,x) t e (ρ+λ)(s t) [U(c s ) + λ v(y t,x s Deterministic control problem on infinite horizon, ] + az)p(s, dz) ds. but nonstationary when p(t, dz) depends on t 18

19 Under conditions (H1) and (H2)(i) on (τ 1, Z 1 ), the value function ˆv for the optimal consumption problem is expressed as: ˆv(t, x, a) = sup c C a (t,x) t e (ρ+λ)(s t) [U(c s ) + λ v(y t,x s Deterministic control problem on infinite horizon, ] + az)p(s, dz) ds. but nonstationary when p(t, dz) depends on t We can write the Hamilton Jacobi (HJ) equation for ˆv. (ρ + λ)ˆv ˆv ( ) ˆv t Ũ λ v(x + az)p(t, dz) = 0, t 0, x l(a), x Ũ(p) = sup c 0 [U(c) cp] convex conjugate of U. 19

20 Dynamic programming system of coupled nonlinear IPDE: (ρ + λ)ˆv ˆv t Ũ ( ˆv x ) λ v(x + az)p(t, dz) = 0, t 0, x l(a), v(x) = Hˆv(x) := sup a [ x/ z, x/z] ˆv(0, x, a), x 0, 20

21 Dynamic programming system of coupled nonlinear IPDE: (ρ + λ)ˆv ˆv t Ũ ( ˆv x ) λ v(x + az)p(t, dz) = 0, t 0, x l(a), v(x) = Hˆv(x) := sup a [ x/ z, x/z] ˆv(0, x, a), x 0, Compare with the HJB equation for the Merton problem (λ = ): ρv Ũ ( v x ) sup a R [ abx v x a2 x 2 σ 2 2 v x 2 ] = 0 21

22 Boundary conditions on v, ˆv Proposition. Under (H1)-(H2)-(H3)-(H4), suppose that ρ satisfies (Hρ) ρ > bγ + λ( κγ z γ 1). Then, v and ˆv are continuous on R + and D and satisfy: (G1) Growth condition: there exists some positive constant K s.t. ˆv(t, x, a) K(e bt x) γ, t 0, x l(a), v(x) Kx γ, x 0. (B1) Boundary data nonnegative wealth constraint: ˆv(t, x, a) = λ t (Coupled Dirichlet condition!) e (ρ+λ)(s t) v(x + az)p(s, dz)ds, at x = l(a) 22

23 3. PDE characterization: viscosity approach Viscosity solutions for the coupled IPDE: (ρ + λ)ŵ ŵ ( ) ŵ t Ũ λ x w(x + az)p(t, dz) = 0, t 0, x > l(a), w(x) = Hŵ(x) = ŵ(0, x, a), x R +, Definition. sup a [ x/ z, x/z] A pair of functions (w, ŵ) C + (R + ) C + (D) is a viscosity supersolution to the above IPDE if: (i) w Hŵ and (ii) for all a R, ( t, x) R + (l(a), ), (ρ + λ)ŵ( t, x, a) ϕ ( ) ϕ t ( t, x) Ũ x ( t, x) λ w( x + az)p( t, dz) 0, for any test function ϕ C 1 (R + (l(a), )), which is a local minimum of (ŵ(.,., a) ϕ). 23

24 3. PDE characterization: viscosity approach Viscosity solutions for the coupled IPDE: (ρ + λ)ŵ ŵ ( ) ŵ t Ũ λ x w(x + az)p(t, dz) = 0, t 0, x > l(a), w(x) = Hŵ(x) = ŵ(0, x, a), x R +, Definition. sup a [ x/ z, x/z] A pair of functions (w, ŵ) C + (R + ) C + (D) is a viscosity subsolution to the above IPDE if: (i) w Hŵ and (ii) for all a R, ( t, x) R + (l(a), ), (ρ + λ)ŵ( t, x, a) ϕ ( ) ϕ t ( t, x) Ũ x ( t, x) λ w( x + az)p( t, dz) 0, for any test function ϕ C 1 (R + (l(a), )), which is a local maximum of (ŵ(.,., a) ϕ). 24

25 Theorem. Under (H1)-(H2)-(H3)-(H4), and (Hρ) ρ > bγ The pair of value functions (v, ˆv) is the unique viscosity solution to the IPDE: (ρ + λ)ˆv ˆv ( ) ˆv t Ũ λ x v(x) = Hˆv(x) = v(x + az)p(t, dz) = 0, t 0, x > l(a), sup a [ x/ z, x/z] satisfying the boundary conditions (G1)-(B1). ˆv(0, x, a), x R +, 25

26 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 properties on the value functions ˆv! 26

27 Assumptions: (H5) U is strictly concave. (H6) p(t, dz) admits a density f(t, z) continuously differentiable Theorem. The value function ˆv(., a) is C 1 on (0, ) (l(a), ), for all a R. Moreover, v is C 1 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

28 Existence and characterization of optimal strategies by a verification theorem - Feedback trading portfolio from the scalar maximum problem: ˆα(x) arg max ˆv(0, x, a) a [ x/ z, x/z] α k+1 = ˆα(Xx k ), k 0. (1) - Feedback consumption from the deterministic control problem: [ ĉ(t, x, a) arg max U(c) c ˆv ] ( ˆv (t, x, a) = I (t, x, a) c 0 x x c t = ĉ(t τ k, Y τ k,x x k t, α k+1 ), τ k < t τ k+1. (2) where I = (U ) 1, and Y τ k,x x k t = X x k t τ k c sds, is the wealth between two trading dates τ k, τ k+1. ) 28

29 5. Numerical resolution and experiments Coupling IPDE: (ρ + λ)ˆv ˆv ( ˆv t Ũ x ˆv(t, x, a) = λ t ) λ v(x + az)p(t, dz) = 0, t 0, x > l(a), e (ρ+λ)(s t) v(x + az)p(s, dz)ds, t 0, x = l(a). v(x) = Hˆv(x) := sup a [ x/ z, x/z] ˆv(0, x, a), x 0, Theoretical and numerical difficulties 29

30 A numerical decoupling algorithm Decoupling system with the following iterative procedure: start from v 0 : trading: the value function of the consumption problem without v 0 (x) = sup c C(x) 0 e ρt U(c t )dt, C(x): set of nonnegative functions c = (c t ) t s.t. x t 0 c s ds 0 for all t 0. ( ) v0 ρv 0 Ũ x = 0, x > 0, v 0 (0 + ) = 0. 30

31 Step n n + 1: first-order PDE (ρ + λ)ˆv n+1 ˆv n+1 Ũ t ˆv n+1 (t, x, a) = λ t ( ) ˆvn+1 x λ v n (x + az)p(t, dz) = 0, t 0, x > l(a) e (ρ+λ)(s t) v n (x + az)p(s, dz)ds, t 0, x = l(a). v n+1 (x) = Hˆv n+1 (x) := sup a [ x/ z, x/z] ˆv n+1 (0, x, a), x 0, 31

32 Convergence of the algorithm Theorem. Under (H1)-(H2)-(H3)-(H4), and (Hρ) ρ > bγ the sequence of functions (ˆv n, v n ) n 1 converges increasingly and uniformly on any compact subset of D and R + to (ˆv, v). More precisely, for any compact set F and G of D and R + : 0 sup(ˆv ˆv n ) C F δ n F 0 sup G for some positive constants C F (v v n ) C G δ n, and C G, and where under (Hρ) δ := λ ρ bγ + λ < 1 32

33 Numerical illustrations (Z k ) extracted from a BS model Jump times (τ k ) of intensity λ illiquidity of the market U(x) = x γ /γ, γ = 0.5. ρ = 0.2, b = 0.4, σ = 1: value function without trading: v 0 (x) = K 0 x γ, and K 0 = 3.16 value function of Merton problem: v M (x) = K M x γ, and K M = 4.08 and the amount invested in stock is proportional to wealth: α M t = 0.8X t. 33

34 v0 4 iterations 10 iterations 50 iterations Merton 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

35 v0 lambda=1 lambda=5 lambda=40 Merton 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

36 Cost of illiquidity: π = π(x) s.t. v(x + π(x)) = v M (x). λ 0 (No trading) π(1)

37 Profile of consumption between two trading dates Exp. bound Optimal wealth Consumption corresponding to Y0 Optimal consumption Left: typical profile of the optimal wealth process Y t, 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

38 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

39 More general and realistic models, e.g.: Cox processes for jump times: λ = λ(θ t ) Models with unobservable volatility Models with microstructure noise on the price 39