Utility maimization problem with transaction costs - Shadow price approach Jin Hyuk Choi joint work with Mihai Sîrbu and Gordan Žitković Department of Mathematics University of Teas at Austin Carnegie Mellon University Pittsburgh, Sep 12 th, 2011 1 / 25
Outline Problem settings and related work Merton problem Shadow price approach Heuristic derivation of the free boundary ODE Our result (for power utility) Value function < Shadow price process Eplicit condition for market parameters Eistence of C 2 solution to the free boundary ODE Sketch of the proof Eistence of C 2 solution Construction and Verification of the Shadow process. 2 / 25
Market Model : Davis and Norman ( 90), Shreve and Soner ( 94) Stock : ds t = S t (µdt + σdw t ), µ>0, σ> 0. Bond : B 1. Transaction costs : λ> 0, λ (0, 1), Portfolio : ϕ 0 t ϕ t c t Bid S t (1 + λ)s t, Ask S t (1 λ)s t : # of shares of B : # of shares of S. : consumption rate Initial position :(ϕ 0 0,ϕ 0)=(η B,η S ) Admissibility : (ϕ 0, ϕ, c) A (S,λ, λ) iff Self-financing dϕ 0 t = S t dϕ t S tdϕ t c tdt No-bankrupcy ϕ 0 t + S t ϕ+ t S t ϕ t 0 where ϕ = ϕ t ϕ t is the pathwide minimal decomposition of ϕ into a difference of two non-decreasing processes. (ϕ t = L t,ϕ t = M t in Shreve and Soner ( 94)) 3 / 25
Investor : Davis and Norman ( 90), Shreve and Soner ( 94) { ln, p = 0 Utility functions (CRRA) : U p () = p, for > 0. p, p (, 1) \{0} Investor s goal : to maimize epected utility by consumption u(s,λ, λ) where δ> 0 is a discount factor. sup (ϕ 0,ϕ,c) A (S,λ,λ) E [ e δt U p (c t )dt ], 0 4 / 25
Related Work - no transaction cost, Melton( 71) Proof : There eists an eplicit solution to the HJB equation, we can write down optimal strategy and do the verification. 2(1 p)σ Remark : For 0 < p < 1, u(s, 0, 0) < µ< 2 δ p. 5 / 25
Related Work - transaction cost Magill and Constantinides ( 76) : Optimal behavior (heuristic) Davis and Norman ( 90) : Analytic proof, several technical conditions. Shreve and Soner ( 94) : Rigorous proof, only assumed u(s,λ, λ) <. 6 / 25
Shadow price approach Motivation : In the frictionless market, we can use duality. Can we construct a stock price process which somehow absorbs transaction costs? (S,λ, λ) ( S, 0, 0). Consistent price processes : S { S : S t S t S t, t 0} Shadow price process : We call S S a shadow price if Observations : u(s,λ, λ) = u( S, 0, 0) < For S S, u(s,λ, λ) u( S, 0, 0) u(s,λ, λ) = u( S, 0, 0) = inf u( S, 0, 0) S S 7 / 25
Related work - Construction of Shadow price Kallsen and Muhle-Karbe ( 10) showed that if p = 0 (log utility) and µ<σ 2, the shadow price process can be constructed. They derived a free boundary ODE based on the following observation - the optimal strategy does not trade when S t < S t < S t. In log utility case (p=0), their methods work, becasue the eplicit epression for the optimal strategy eists. But their methods only work for p = 0 (log utility). How can we do for p 0? 8 / 25
Our approach - derivation of free boundary problem If the Shadow price process S eists, u( S, 0, 0) = inf u( S, 0, 0) S S Let s parametrize S S with an Ito-process Y : S t = e Y t S t, dy t = µ t dt + σ t dw t, with y Y t y, y ln (1 λ), y ln (1 + λ) In the complete market, by conve duality, u( S, 0, 0) = (η B+η S S 0 ) p p where θ t µ+ µ s+σ σ s + 1 2 σ2 s σ s +σ. (E [ E ( θ W) 0 p p 1 t e δ p 1 t dt ]) 1 p, 9 / 25
Our approach - derivation of free boundary problem u( S, 0, 0) = inf S S = inf Y = inf y [y,y] = inf y [y,y] where u( S, 0, 0) u( S, 0, 0) inf µ, σ u( S, 0, 0) (η B +η S e y S 0 ) p p w(y) 1 p w(y) inf µ, σ We set d P = E ( p 1 pθ W)dP, then, w(y) = inf µ, σ E P [ 0 E [ E ( θ W) 0 p t e δ p 1 t e 2(1 p) 2 p p 1 t 0 θ2 udu dt ]. e δ p 1 t dt ]. We can write down the HJB-equation for this optimal stochastic control. 10 / 25
Our approach - derivation of free boundary problem HJB equation : For y (y, y), inf µ, σ ( ( p 2(1 p) 2 θ 2 δ 1 p )w(y) + ( µ + p 1 p σθ)w (y) + 1 2 σw (y) + 1 ) = 0 Boundary condition : At y and y, turn off the diffusion, which correspond to w (y) = w (y) = Order reduction : Set w(y) = g(w (y)) for g :[, ] R,( = w (y), = w (y)), inf µ, σ ( p ( θ 2 δ 2(1 p) 2 1 p )g() + ( µ + 1 p σθ) p 1 + 2 σ g () + 1) = 0, with g () = g () = 0, (boundary conditions from g () = I () = g () d = y y. w (I()), where I() = (w ) 1 ().) 11 / 25
Free boundary ODE Finally, after simple change of variable, we obtain a free boundary problem : g () = L(, g()) on [, ] g () = g g () = 0, () d = ±(y y) where L(, y) = 2µ+δσ 2 2 +2δ y (δ 1)(2δ y (2µ σ 2 )), p = 0 (1 p) 3 σ 2 2 +2q(1+µ)y 2pδy ( 2, )y+2pδy 2 p 0 (1 p)(2+2µ+(p 2 1)σ 2 )+ 2q+(q(2µ σ 2 ) 2δ) 12 / 25
Main result Theorem. The followings are equivalent : 1. u(s,λ, λ) <. 2. The Shadow price process eists. g () = L(, g()) on [, ] 3.,, g C 2 ([, ]) with g () = g g () = 0, () d = ± ln ( ) 1+λ, 1 λ where L(, y) = 2µ+δσ 2 2 +2δ y (δ 1)(2δ y (2µ σ 2 )), p = 0 (1 p) 3 σ 2 2 +2q(1+µ)y 2pδy 2 (1 p)(2+2µ+(p 2 1)σ 2 )+(2q+(q(2µ σ 2 ) 2δ))y+2pδy 2, p 0. 13 / 25
Main result 4. If 0 < p < 1 and 2(1 p)σ 2 δ p µ< δ p + (1 p)σ2 2, ln ( 1+λ 1 λ) > C(µ, σ, δ, p). k 1 where C(µ, σ, δ, p) = 2q(pµ 2qδσ 2 ) p(2δ 2qµ+qσ 2 2p 2qδσ 2 ) h 1 (k) 2qµ+(2δ+q(2µ σ2 ))k+ h 2 (k) 2qµ+(2δ+q(2µ σ2 ))k k1 0, q p(1 p) ( kh 1 (k) k h 1 (k) kh 2 (k) k h 2 (k)) dk = Eplicit! 4δ 2 k 2 +q 2 (σ 2 k 2µ(1+k)) 2 4qδ(2µk(1+k)+σ 2 (k 2 (2p 2 1)+2(1 p) 2 4qk)) 4pδ(1+k) 4δ 2 k 2 +q 2 (σ 2 k 2µ(1+k)) 2 4qδ(2µk(1+k)+σ 2 (k 2 (2p 2 1)+2(1 p) 2 4qk)) 4pδ(1+k) 14 / 25
Main result Remarks : We get a direct proof to the original control problem, by constructing the shadow price process. We get an eplicit necessary and sufficient condition for finiteness of the value function, this was only known for two bonds market. δ C(µ, 0, δ, p) = ln( δ pµ ) < ln ( ) 1+λ 1 λ = δ> pµ, 1 1 λ 1+λ coinside with the result in Shreve & Soner ( 94) for the two bonds market. Sensitivity analysis for small transaction cost possible. 15 / 25
2(1 p)σ Solution of the free boundary ODE (µ < 2 δ p ) Pick α, evolve g α until it meets blue line (at β α ). Then, g α(α) = g α(β α ) = 0. α lim α 1 βα α βα α g α() d is continous g α() d = 0, lim α 0 βα α g α() d = =, g () d = y y 16 / 25
2(1 p)σ Solution of the free boundary ODE (µ < 2 δ p ), such that Need to show g C 2. g () d = y y (by patching curves at 0). 17 / 25
2(1 p)σ Solution of the free boundary ODE ( 2 δ p µ< δ p + (1 p)σ2 2 ) α lim α 0 βα α βα α g α() d is continous g α() d =, lim α βα If y y > C(µ, σ, δ, p), then, α g α() d = C(µ, σ, δ, p) g () d = y y 18 / 25
Solution of the free boundary ODE (µ > δ p + (1 p)σ2 2 ) g α does NOT hit the blue line, there is no solution. Remark : In this case, we can eplicity write down the trading/consumption strategy which leads to the infinite value function. 19 / 25
Backward Construction of Shadow price (p = 0) Let s define f C 2 ([, ]) and 0 as min [,] f () y + g (t) dt t ( g() + ln (η B +η S e f () S 0 )+ln δ 1 δ ) at 0 Let {X t } t, {Φ t } t be the solution of the following Skorokhod SDE t t 2δg(X X t = 0 + δx s ds + s )+(σ 2 2µ)X s σ dw s + Φ t, 0 0 where Φ t is the "instantaneous inward reflection" at the boundary,. Shadow price : e f (X t) S t will be a Shadow price process. 20 / 25
Verification (p = 0) Let s define Shadow price : Ŝ t e f (X t) S t Optimal wealth : ˆV t (η B + η S Ŝ 0 )E ( 0 δx t dŝ s Ŝ s 0 δds) t Optimal strategy : ( ˆϕ 0 t, ˆϕ t, ĉ t ) ( (1 δx t ) ˆV t, δx t ˆV t Ŝ t,δ ˆV t ) with ( ˆϕ 0 0, ˆϕ 0) = (η B,η S ) Check ( ˆϕ 0, ˆϕ, ĉ) is optimal for (Ŝ, frictionless) market. d(ln ˆϕ t ) = 1 X t dφ t = d ˆϕ t = 1 {Ŝt =S t } d ˆϕ t 1 {Ŝ t =S t } d ˆϕ t. Thus ( ˆϕ 0, ˆϕ, ĉ) is also optimal for (S, transaction costs) market. u(s,λ, λ) = u(ŝ, 0, 0), conclude that Ŝ is the Shadow price process! 21 / 25
Sensitivity analysis (currently working on this) λ ln (1 + λ) = g () d C(g() g()) = Ch3 + o(h 3 ) = u(s, 0, 0) u(s, 0,λ) = h 2 + O(h 3 ) = Cλ 2 3 + O(λ), Size of the no-trade region : Cλ 1 3 + O(λ 2 3 ) By using the series epension of g, we may find a fractional tailor epansion for any order. 22 / 25
Summary We get a direct proof to the original control problem, by constructing the shadow price process. We get an eplicit necessary and sufficient condition for finiteness of the value function. We may do sensitivity analysis for small transaction cost, any order. We may etract more financial interpretations from the structure of the ODE. 23 / 25
References A. Skorokhod. Stochastic equations for diffusion processes in a bounded region. Theory of Probability and its Applications, 6:264-274, 1961. J. Kallsen and J. Muhle-Karbe. On using shadow prices in portfolio optimization with transaction costs. Annals of Applied Probability, 20:1341âĂŞ1358, 2010. M. Davis and A. Norman. Portfolio selection with transaction costs. Mathematics of Operations Research, 15:676-713, 1990. S. Shreve and M. Soner. Optimal investment and consumption with transaction costs. The Annals of Applied Probability, 4:609-692, 1994. 24 / 25
Thank you! 25 / 25