Illiquidity, Position Limits, and Optimal Investment for Mutual - PowerPoint PPT Presentation

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds Min Dai National University of Singapore Joint with Hanqing Jin Hong Liu Oxford University Washington University in St. Louis Workshop on Stochastic Analysis & Finance, 29 Jun-3th July, City University of Hong Kong Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 1/21

Introduction: Motivation • Mutual funds are often restricted to allocate certain percentages of fund assets to certain securities ( Almazan, Brown, Carlson, and Chapman 2004, Clarke, de Silva, and Thorley 2002 ). ◦ funds prevented from shorting selling and/or buying-on-margin ◦ a small cap fund may set a lower bound on its holdings of small cap stocks. • Mutual funds can also face significant illiquidity in trading securities ( Chalmers, Edelen, and Kadlec 1999, Delib and Varma 2002 ) • The coexistence of position limits and asset illiquidity and the interactions among them are important for the optimal trading strategy of a mutual fund. Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 2/21

Motivation (continued) • The existing literature ignores the coexistence of position limits and asset illiquidity and the interactions among them. ◦ Portfolio selection with position limits: Fleming and Zariphopoulou (1991), Cuoco (1997), Cuoco and Liu (2000) . ◦ Portfolio selection with transaction costs: Constantinides (1986), Davis and Norman (1990), Shreve and Soner (1994), Liu and Loewenstein (2002), Øksendal and Sulem (2002), Liu (2004), Dai and Yi (2009), Dai et al. (2009) . Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 3/21

The Model • An investor with a finite horizon T ∈ (0 , ∞ ) maximizes his CRRA utility from terminal non-liquidated wealth: u ( W ) = W 1 − γ − 1 , γ > 0 . 1 − γ • Two assets: 1 liquid stock, and 1 illiquid stock. • The liquid stock price S Lt evolves as dS Lt S Lt = µ L dt + σ L dB Lt , where µ L and σ L > 0 are both constants and B Lt is a one-dimensional Brownian motion. Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 4/21

The Illiquid Stock • The investor can buy the illiquid stock at the ask price S A It = (1 + θ ) S It and sell the stock at the bid price S B It = (1 − α ) S It , where θ ≥ 0 and 0 ≤ α < 1 represent the proportional transaction cost rates and S It follows the process dS It S It = µ I dt + σ I dB It , where µ I and σ I > 0 are both constants and B It is another one-dimensional Brownian motion that has a correlation of ρ with B Lt with | ρ | < 1 . Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 5/21

Position Limits • Let x t and y t be the dollar amount invested in the liquid stock and the illiquid stock respectively. • The investor is subject to the following position limits: b ≤ y t W t ≤ ¯ b, ∀ t ≥ 0 , (1) where W t = x t + y t is the non-liquidated wealth process. Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 6/21

The Dynamic Budget Constraint When α + θ > 0 , we have dx t = µ L x t dt + σ L x t dB Lt − (1 + θ ) dI t + (1 − α ) dD t , (2) dy t = µ I y t dt + σ I y t dB It + dI t − dD t , (3) where the processes D and I represent the cumulative dollar amount of sales and purchases of the illiquid stock, respectively. D and I are nondecreasing and right continuous adapted processes with D (0) = I (0) = 0 . Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 7/21

The Solvency Region • Let Θ( x 0 , y 0 ) denote the set of admissible trading strategies ( D, I ) such that (1), (2), (3), and ˆ W t ≥ 0 , ∀ t ≥ 0 , where ˆ W t = x t + (1 − α ) y + t − (1 + θ ) y − t is the time t wealth after liquidation. x _ x=-(1+ )y θ x=z y BUY Solvency Line No-Transaction x=z y _ S y E L L 0 Solvency Line x=-(1- )y α Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 8/21

The Investor’s Problem and HJB Equation • The investor’s problem is then sup E [ u ( W T )] , ( D,I ) ∈ Θ( x 0 ,y 0 ) which is a singular stochastic control problem with state constraints. • HJB Equation max { V t + LV, (1 − α ) V x − V y , − (1 + θ ) V x + V y } = 0 , with the boundary conditions y y x + y = ¯ (1 − α ) V x − V y = 0 on b, (1 + θ ) V x − V y = 0 on x + y = b, and the terminal condition V ( x, y, T ) = ( x + y ) 1 − γ − 1 , where 1 − γ LV = 1 I y 2 V yy + 1 2 σ 2 2 σ 2 L x 2 V xx + ρσ I σ L xyV xy + µ I yV y + µ L xV x Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 9/21

Analytical Results: The No Transaction Cost Case Theorem 1. Suppose that α = θ = 0 . Then the optimal trading policy is given by  ¯ ≥ ¯ if π M b b I   < ¯ π ∗ π M if b < π M , π ∗ L = 1 − π ∗ I = b I I I if π M  b ≤ b  I where = µ I − µ L + γσ L ( σ L − ρσ I ) π M I γ ( σ 2 L + σ 2 I − 2 ρσ L σ I ) is the optimal fraction of wealth invested in illiquid stock in the unconstraint case. Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 10/21

The Transaction Cost Case: Change of Variables � � x y 1 ( x + y ) 1 − γ V V ( x, y, t ) = x + y , x + y , t − 1 − γ 1 ( x + y ) 1 − γ V (1 − π, π, t ) − = 1 − γ 1 ( x + y ) 1 − γ ϕ ( π, t ) − ≡ 1 − γ , where y π = x + y ∈ ( α − 1 , ∞ ) . Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 11/21

A Reduced Equation It follows max { ϕ t + L 1 ϕ, − (1 − απ ) ϕ π − α (1 − γ ) ϕ, (1 + θπ ) ϕ π − θ (1 − γ ) ϕ } = 0 , with the boundary conditions − (1 − απ ) ϕ π − α (1 − γ ) ϕ = 0 on π = ¯ b, (1 + θπ ) ϕ π − θ (1 − γ ) ϕ = 0 on π = b, and the terminal condition 1 ϕ ( π, T ) = 1 − γ , where � � L 1 ϕ = 1 β 3 + β 2 π − 1 2 β 1 π 2 (1 − π ) 2 ϕ ππ + ( β 2 − γβ 1 π ) π (1 − π ) ϕ π + (1 − γ ) 2 γβ 1 π 2 ϕ. Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 12/21

A Further Transformation Let 1 w = 1 − γ log [(1 − γ ) ϕ ] . Then  � � α θ max w t + L 2 w, − 1 − απ − w π , w π − = 0  1+ θπ    1 − απ on π = ¯ α w π = − b,  θ w π = 1+ θπ on π = b,     w ( π, T ) = 0  − 1 θ , 1 � � in × [0 , T ) , where α L 2 w = 1 +( β 2 − γβ 1 π ) π (1 − π ) w π + β 3 + β 2 π − 1 2 β 1 π 2 (1 − π ) 2 � w ππ + (1 − γ ) w 2 2 γβ 1 π 2 . � π Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 13/21

An Equivalent Standard Variational Inequality Denote v = w π . Since ∂ 1 2 β 1 π 2 (1 − π ) 2 v ππ + [ β 1 + β 2 − (2 + γ ) β 1 π ] π (1 − π ) v π ∆ L v = ∂π ( L 2 w ) = + [ β 2 (1 − 2 π ) − γβ 1 π (2 − 3 π )] v + (1 − γ ) β 1 π (1 − π ) v [(1 − 2 π ) v + π (1 − π ) v π ] + β 2 − γβ 1 π Using the technique developed by Dai and Yi (2009), we can show  α θ v t + L v = 0 if − 1 − απ < v < 1+ θπ ,    α v t + L v ≤ 0 if v = − 1 − απ ,     θ  v t + L v ≥ 0 if v = 1+ θπ ,  1 − απ on π = ¯ α v = − b,    θ v = 1+ θπ on π = b,      v ( π, T ) = 0 ,  − 1 θ , 1 � � in × [0 , T ) . α Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 14/21

The Case with Transaction Costs and Constraints The following verification theorem shows the existence and the uniqueness of the optimal trading strategy. It also ensures the smoothness of the value function except for a set of measure zero. Theorem 3. (i) The HJB equation admits a unique viscosity solution, and the value function is the viscosity solution. (ii) The value function is C 2 , 2 , 1 in { ( x, y, t ) : x + (1 − α ) y + − (1 + θ ) y − > 0 , b < y/ ( x + y ) < ¯ b, 0 ≤ t < T } \ ( { y = 0 } ) . Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 15/21

The Transaction Cost Case Without Constraints Proposition 2. Assume − 1 α + 1 < π M < 1 θ + 1 . We have ∀ t ∈ [0 , T ] , I 1. for the sell boundary, there exists t < T such that π M 1 I α = π I ( s ) ≥ π I ( t ) ≥ I ) , for any t and all s > t ; 1 − α (1 − π M 2. for the buy boundary, there exists t < T such that π M − 1 I θ = π I ( s ) ≤ π I ( t ) ≤ I ) , for any t and all s > t. 1 + θ (1 − π M Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 16/21

The Case with Transaction Costs and Constraints Proposition 4. We have 1. for the sell boundary, there exists t b < T such that π M � � � � b = π c I ( s ; b, b ) ≥ π c I I ( t ; b, b ) ≥ max min I ) , b , b , for any t and s > t b ; 1 − α (1 − π M 2. for the buy boundary, there exists t b < T such that π M � � � � I ≥ π c I ( t ; b, b ) ≥ π c min max I ) , b , b I ( s ; b, b ) = b, for any t and s > t b . 1 + θ (1 − π M 3. both π c I ( t ; b, b ) and π c I ( t ; b, b ) are increasing in b and b for all t ∈ [0 , T ] ; Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 17/21