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Model                 Results                       Heuristics            Method             Transaction Costs Made Tracta...
Model                    Results                Heuristics      Method                                   Outline   • Motiv...
Model                  Results                 Heuristics           Method                           Transaction Costs   •...
Model                     Results                 Heuristics               Method                                    Liter...
Model                   Results                 Heuristics            Method                                  This Paper  ...
Model                  Results                   Heuristics      Method                                      Model   • Saf...
Model                    Results                   Heuristics           Method                Welfare, Liquidity Premium, ...
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Model                 Results             Heuristics          Method                Trading Boundaries v. Spread0.750.700....
Model                 Results           Heuristics          Method                Liquidity Premium v. Spread0.0120.0100.0...
Model               Results             Heuristics       Method            Liquidity Premium v. Risk Aversion0.0100.0080.0...
Model               Results              Heuristics       Method             Share Turnover v. Risk Aversion0.50.40.30.20....
Model               Results              Heuristics       Method             Wealth Turnover v. Risk Aversion0.50.40.30.20...
Model                 Results                          Heuristics      Method                 Welfare, Volume, and Spread ...
Model                   Results                 Heuristics                Method                            Wealth Dynamic...
Model                     Results                  Heuristics               Method                              Control Ar...
Model                    Results                     Heuristics              Method                              No Trade ...
Model                 Results                Heuristics                   Method                            Smooth Pasting...
Model                  Results                 Heuristics   Method                         Solution Procedure   • Unknown:...
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Model                    Results                 Heuristics             Method                               Shadow Market...
Model                    Results                   Heuristics             Method                           Shadow Price Fo...
Model                    Results               Heuristics                Method               Shadow Price at Trading Boun...
Model                    Results                  Heuristics                 Method                                   Shad...
Model                      Results                   Heuristics              Method                          Shadow HJB Eq...
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Model                    Results                    Heuristics                Method                       Matching HJB Eq...
Model                  Results                     Heuristics              Method                         Shadow price ODE...
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Transaction Costs Made Tractable

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Transaction Costs Made Tractable

  1. 1. Model Results Heuristics Method Transaction Costs Made Tractable Paolo Guasoni Stefan Gerhold Johannes Muhle-Karbe Walter Schachermayer Boston University and Dublin City University Stochastic Analysis in Insurance and Finance University of Michigan at Ann Arbor, May 17th , 2011
  2. 2. Model Results Heuristics Method Outline • Motivation: Trading Bounds, Liquidity Premia, and Trading Volume. • Model: Constant investment opportunities and risk aversion. • Results: Explicit formulas. Asymptotics. • Method: Shadow Price and long-run optimality.
  3. 3. Model Results Heuristics Method Transaction Costs • Classical portfolio choice: 1 Constant ratio of risky and safe assets. 2 Sharpe ratio alone determines discount factor. 3 Continuous rebalancing and infinite trading volume. • Transaction costs: 1 Variation in risky/safe ratio. Tradeoff between higher tracking error and higher costs. 2 Liquidity premium. Trading costs equivalent to lower expected return. 3 Finite trading volume. Understand dependence on model parameters. • Tractability?
  4. 4. Model Results Heuristics Method Literature • Magill and Constantinides (1976): the no-trade region. • Constantinides (1986): no-trade region large, but liquidity premium small. • Davis and Norman (1990): rigorous solution. Algorithm for trading boundaries. • Taksar, Klass, and Assaf (1998), Dumas and Luciano (1991): long-run control argument. Numerical solution. • Shreve and Soner (1994): Viscosity solution. Utility impact of ε transaction cost of order ε2/3 • Janeˇ ek and Shreve (2004): c Trading boundaries of order ε1/3 . Asymptotic expansion. • Kallsen and Muhle-Karbe (2010), Gerhold, Muhle-Karbe and Schachermayer (2010): Logarithmic solution with shadow price. Asymptotics.
  5. 5. Model Results Heuristics Method This Paper • Long-run portfolio choice. No consumption. • Constant relative risk aversion γ. • Explicit formulas for: 1 Trading boundaries. 2 Certainty equivalent rate (expected utility). 3 Trading volume (relative turnover). 4 Liquidity premium. In terms of gap parameter. • Expansion for gap yields asymptotics for all other quantities. Of any order. • Shadow price solution. Long-run verification theorem. • Shadow price also explicit.
  6. 6. Model Results Heuristics Method Model • Safe rate r . • Ask (buying) price of risky asset: dSt = (r + µ)dt + σdWt St • Bid price (1 − ε)St . ε is the spread. • Investor with power utility U(x) = x 1−γ /(1 − γ). • Maximize certainty equivalent rate (Dumas and Luciano, 1991): 1 1 1−γ 1−γ max lim log E XT π T →∞ T
  7. 7. Model Results Heuristics Method Welfare, Liquidity Premium, TradingTheoremTrading the risky asset with transaction costs is equivalent to: • investing all wealth at hypothetical safe certainty equivalent rate µ2 − λ 2 CeR = r + 2γσ 2 • trading a hypothetical asset, at no transaction costs, with same volatility σ, but expected return decreased by the liquidity premium LiP = µ − µ2 − λ 2 . • Optimal to keep risky weight within buy and sell boundaries (evaluated at buy and sell prices respectively) µ−λ µ+λ π− = , π+ = , γσ 2 γσ 2
  8. 8. Model Results Heuristics Method GapTheorem • λ identified as unique value for which solution of Cauchy problem 2µ µ−λ µ+λ w (x) + (1 − γ)w(x)2 + − 1 w(x) − γ =0 σ2 γσ 2 γσ 2 µ−λ w(0) = , γσ 2 satisfies the terminal value condition: µ+λ u(λ) 1 (µ+λ)(µ−λ−γσ 2 ) w(log(u(λ)/l(λ))) = γσ 2 , where l(λ) = 1−ε (µ−λ)(µ+λ−γσ 2 ) . • Asymptotic expansion: 1/3 λ = γσ 2 3 2 4γ π∗ (1 − π ∗ )2 ε1/3 + O(ε).
  9. 9. Model Results Heuristics Method Trading VolumeTheorem • Share turnover (shares traded d||ϕ||t divided by shares held |ϕt |). 1 T d ϕ t σ2 2µ 1−π− 1−π+ ShT = lim 0 |ϕt | = σ2 −1 2µ − 2µ . T →∞ T 2 −1 (u/l) σ2 −1 (u/l) 1− σ2 −1 • Wealth turnover, (wealth traded divided by the wealth held): 1 T (1−ε)St dϕ↓ T St dϕ↑ WeT = lim t 0 ϕ0 St0 +ϕt (1−ε)St + t 0 ϕ0 St0 +ϕt St T →∞ T t t σ2 2µ π− (1−π− ) π+ (1−π+ ) = 2 σ2 −1 2µ −1 − 1− 2µ . (u/l) σ2 −1 (u/l) σ2 −1
  10. 10. Model Results Heuristics Method Asymptotics 1/3 3 2 π± = π∗ ± π (1 − π∗ )2 ε1/3 + O(ε). 4γ ∗ 2/3 µ2 γσ 2 3 2 CeR = r + − π (1 − π∗ )2 ε2/3 + O(ε4/3 ). 2γσ 2 2 4γ ∗ 2/3 µ 3 2 LiP = π (1 − π∗ )2 ε2/3 + O(ε4/3 ). 2π∗2 4γ ∗ −1/3 σ2 3 2 ShT = (1 − π∗ )2 π∗ π (1 − π∗ )2 ε−1/3 + O(ε1/3 ) 2 4γ ∗ 2/3 γσ 2 3 2 WeT = π (1 − π∗ )2 ε−1/3 + O(ε). 3 4γ ∗
  11. 11. Model Results Heuristics Method Implications • λ/σ 2 depends on mean-variance ratio µ = µ/σ 2 . Only. ¯ • Trading boundaries depend only on µ. ¯ • Certainty equivalent, liquidity premium, volume per unit variance depend only on µ. ¯ • Interpretation: certainty equivalent, liquidity premium, volume t 2 proportional to business time 0 σs ds. Trading strategy invariant. • All results extend to St such that: dSt = (r + µσt )dt + σt dWt ¯ St 1 T with σt independent of Wt and ergodic (limT →∞ T 0 σt2 dt = σ 2 ). ¯ • Same formulas hold, replacing µ/σ 2 with µ, and residual factor σ 2 with σ 2 . ¯ ¯
  12. 12. Model Results Heuristics Method Trading Boundaries v. Spread0.750.700.650.600.550.50 0.00 0.02 0.04 0.06 0.08 0.10µ = 8%, σ = 16%, γ = 5. Zero discount rate for consumption.
  13. 13. Model Results Heuristics Method Liquidity Premium v. Spread0.0120.0100.0080.0060.0040.002 0.01 0.02 0.03 0.04 0.05µ = 8%, σ = 16%. γ = 5, 1, 0.5.
  14. 14. Model Results Heuristics Method Liquidity Premium v. Risk Aversion0.0100.0080.0060.0040.0020.000 0 2 4 6 8 10µ = 8%, σ = 16%. ε = 0.01%, 0.1%, 1%, 10%.
  15. 15. Model Results Heuristics Method Share Turnover v. Risk Aversion0.50.40.30.20.10.0 0 2 4 6 8 10µ = 8%, σ = 16%, γ = 5. ε = 0.01%, 0.1%, 1%, 10%.
  16. 16. Model Results Heuristics Method Wealth Turnover v. Risk Aversion0.50.40.30.20.10.0 0 2 4 6 8 10µ = 8%, σ = 16%, γ = 5. ε = 0.01%, 0.1%, 1%, 10%.
  17. 17. Model Results Heuristics Method Welfare, Volume, and Spread • Liquidity premium and share turnover: LiP 3 = ε + O(ε5/3 ) ShT 4 • Certainty equivalent rate and wealth turnover: µ2 (r + γσ 2 ) − CeR 3 = ε + O(ε5/3 ). WeT 4 • Two relations, one meaning. • Welfare effect proportional to spread, holding volume constant. • For same welfare, spread and volume inversely proportional. • Relations independent of market and preference parameters. • 3/4 universal constant.
  18. 18. Model Results Heuristics Method Wealth Dynamics • Number of shares must have a.s. locally finite variation. • Otherwise infinite costs in finite time. • Strategy: predictable process (ϕ0 , ϕ) of finite variation. • ϕ0 units of safe asset. ϕt shares of risky asset at time t. t • ϕt = ϕ↑ − ϕ↓ . Shares bought ϕ↑ minus shares sold ϕ↓ . t t t t • Self-financing condition: St St dϕ0 = − t 0 dϕ↑ + (1 − ε) 0 dϕ↓ t St St • Xt0 = ϕ0 St0 , Xt = ϕt St safe and risky wealth, at ask price St . t dXt0 =rXt0 dt − St dϕ↑ + (1 − ε)St dϕ↓ , t t dXt =(µ + r )Xt dt + σXt dWt + St dϕ↑ − St dϕ↓ t
  19. 19. Model Results Heuristics Method Control Argument • V (t, x, y ) value function. Depends on time, and on asset positions. • By Itô’s formula: 1 dV (t, Xt0 , Xt ) = Vt dt + Vx dXt0 + Vy dXt + Vyy d X , X t 2 σ2 2 = Vt + rXt0 Vx + (µ + r )Xt Vy + X Vyy dt 2 t + St (Vy − Vx )dϕ↑ + St ((1 − ε)Vx − Vy )dϕ↓ + σXt dWt t t • V (t, Xt0 , Xt ) supermartingale for any ϕ. • ϕ↑ , ϕ↓ increasing, hence Vy − Vx ≤ 0 and (1 − ε)Vx − Vy ≤ 0 Vx 1 1≤ ≤ Vy 1−ε
  20. 20. Model Results Heuristics Method No Trade Region V 1 • When 1 ≤ Vx ≤ 1−ε does not bind, drift is zero: y σ2 2 Vx 1 Vt + rXt0 Vx + (µ + r )Xt Vy + Xt Vyy = 0 if 1 < < . 2 Vy 1−ε • This is the no-trade region. • Ansatz: value function homogeneous in wealth. Grows exponentially with the horizon. V (t, Xt0 , Xt ) = (Xt0 )1−γ v (Xt /Xt0 )e−(1−γ)(β+r )t • Set z = y /x. For 1 + z < (1−γ)v (z) < 1−ε + z, HJB equation is v (z) 1 σ2 2 z v (z) + µzv (z) − (1 − γ)βv (z) = 0 2 • Linear second order ODE. But β unknown.
  21. 21. Model Results Heuristics Method Smooth Pasting • Suppose 1 + z < (1−γ)v (z) < 1−ε + z same as l ≤ z ≤ u. v (z) 1 • For l < u to be found. Free boundary problem: σ2 2 z v (z) + µzv (z) − (1 − γ)βv (z) = 0 if l < z < u, 2 (1 + l)v (l) − (1 − γ)v (l) = 0, (1/(1 − ε) + u)v (u) − (1 − γ)v (u) = 0. • Conditions not enough to find solution. Matched for any l, u. • Smooth pasting conditions. • Differentiate boundary conditions with respect to l and u: (1 + l)v (l) + γv (l) = 0, (1/(1 − ε) + u)v (u) + γv (u) = 0.
  22. 22. Model Results Heuristics Method Solution Procedure • Unknown: trading boundaries l, u and rate β. • Strategy: find l, u in terms of β. • Free bounday problem becomes fixed boundary problem. • Find unique β that solves this problem.
  23. 23. Model Results Heuristics Method Trading Boundaries • Plug smooth-pasting into boundary, and result into ODE. Obtain: 2 l 2 l − σ (1 − γ)γ (1+l)2 v + µ(1 − γ) 1+l v − (1 − γ)βv = 0. 2 • Setting π− = l/(1 + l), and factoring out (1 − γ)v : γσ 2 2 − π + µπ− − β = 0. 2 − • π− risky weight on buy boundary, using ask price. • Same argument for u. Other solution to quadratic equation is: u(1−ε) π+ = 1+u(1−ε) , • π+ risky weight on sell boundary, using bid price.
  24. 24. Model Results Heuristics Method Gap • Optimal policy: buy when “ask" weight falls below π− , sell when “bid" weight rises above π+ . Do nothing in between. • π− and π+ solve same quadratic equation. Related to β via µ µ2 − 2βγσ 2 π± = ± . γσ 2 γσ 2 • Set β = (µ2 − λ2 )/2γσ 2 . β = µ2 /2γσ 2 without transaction costs. • Investor indifferent between trading with transaction costs asset with volatility σ and excess return µ, and... • ...trading hypothetical frictionless asset, with excess return µ2 − λ2 and same volatility σ. • µ− µ2 − λ2 is liquidity premium. • With this notation, buy and sell boundaries are π± = µ±λ . γσ 2
  25. 25. Model Results Heuristics Method Symmetric Trading Boundaries • Trading boundaries symmetric around frictionless weight µ/γσ 2 . • Each boundary corresponds to classical solution, in which expected return is increased or decreased by the gap λ. • With l(λ), u(λ) identified by π± in terms of λ, it remains to find λ. • This is where the trouble is.
  26. 26. Model Results Heuristics Method First Order ODE • Use substitution: log(z/l(λ)) w(y )dy , l(λ)ey v (l(λ)ey ) v (z) = e(1−γ) i.e. w(y ) = (1−γ)v (l(λ)ey ) • Then linear decond order ODE becomes first order Riccati ODE 2µ µ−λ µ+λ w (x) + (1 − γ)w(x)2 + σ2 − 1 w(x) − γ γσ 2 γσ 2 =0 µ−λ w(0) = γσ 2 µ+λ w(log(u(λ)/l(λ))) = γσ 2 u(λ) 1 π+ (1−π− ) 1 (µ+λ)(µ−λ−γσ 2 ) where l(λ) = 1−ε π− (1−π+ ) = 1−ε (µ−λ)(µ+λ−γσ 2 ) . • For each λ, initial value problem has solution w(λ, ·). • λ identified by second boundary w(λ, log(u(λ)/l(λ))) = µ+λ . γσ 2
  27. 27. Model Results Heuristics Method Shadow Market • Find shadow price to make argument rigorous. ˜ • Hypothetical price S of frictionless risky asset, such that trading in ˜ S withut transaction costs is equivalent to trading in S with transaction costs. For optimal policy. • For all other policies, shadow market is better. • Use frictionless theory to show that candidate optimal policy is optimal in shadow market. • Then it is optimal also in transaction costs market.
  28. 28. Model Results Heuristics Method Shadow Price Form • Look for a shadow price of the form ˜ St St = Y g(eYt ) e t eYt = (Xt /Xt0 )/l ratio between risky and safe positions at mid-price S, and centered at the buying boundary π− (µ − λ) l= = 2 − (µ − λ) . 1 − π− γσ • Idea: risky/safe ratio is the state variable of the shadow market. • Shadow price has stochastic investment opportunities. • Numbers of units ϕ0 and ϕ remain constant inside no-trade region. • Y = log(ϕ/lϕ0 ) + log(S/S 0 ) follows Brownian motion with drift.
  29. 29. Model Results Heuristics Method Shadow Price at Trading Boundaries • Y must remain in [0, log(u/l)], so Y reflected at boundaries: dYt = (µ − σ 2 /2)dt + σdWt + dLt − dUt , for L, U that only increase when {Yt = 0} and {Yt = log(u/l)}. • g : [1, u/l] → [1, (1 − ε)u/l] satisfies conditions g(1) =1 g(u/l) =(1 − ε)u/l g (1) =1 g (u/l) =1 − ε. ˜ • Boundary conditions: S equals bid and the ask at boundaries. ˜ • Smooth-pasting: diffusion of S/S zero at boundaries.
  30. 30. Model Results Heuristics Method Shadow Price ˜ • Itô’s formula and conditions on g imply that S satisfies: ˜ ˜ d St /St = (˜(Yt ) + r )dt + σ (Yt )dWt , µ ˜ where 2 µg (ey )ey + σ g (ey )e2y 2 σg (ey )ey µ(y ) = ˜ , and σ (y ) = ˜ . g(ey ) g(ey ) ˜ • Local time terms vanish in the dynamics of S. • How to find function g? • First derive the HJB equation for generic g. • Then, compare HJB equation to that for transaction cost problem. • Value function must be the same. Matching the two HJB equations identifies the function g.
  31. 31. Model Results Heuristics Method Shadow HJB Equation • Shadow wealth process of policy π is: ˜ ˜ ˜ ˜ ˜ ˜ ˜˜ ˜ d Xt = r Xt dt + πt µ(Yt )Xt dt + πt σ (Yt )Xt dWt . ˜ ˜ ˜ ˜ • Setting Vt = V (t, Xt , Yt ), Itô’s formula yields for d Vt : ˜ ˜ ˜ ˜2 2 2 ˜˜ ˜ ˜ 2 ˜ ˜ ˜ 2 ˜ 2 ˜ σ˜ ˜ ˜ (Vt + r Xt Vx + µπt Xt Vx + σ πt2 Xt2 Vxx +(µ−σ )Vy + σ Vyy +σ˜ πt Xt Vxy )d ˜ σ˜ ˜ ˜ ˜ + Vy (dLt − dUt ) + (˜ πt Xt Vx + σ Vy )dWt , ˜ • V supermartingale for any strategy, martingale for optimal strategy. • HJB equation: ˜ ˜ µ˜ ˜ ˜2 ˜ ˜ 2 ˜ 2 ˜ σ˜ ˜ supπ (Vt +rx Vx +˜π x Vx + σ π 2 x 2 Vxx +(µ− σ )Vy + σ Vyy +σ˜ π x Vxy ) = 0 ˜ 2 2 2 with Neumann boundary conditions ˜ ˜ Vy (0) = Vy (log(u/l)) = 0.
  32. 32. Model Results Heuristics Method Homogeneity ˜ • Homogeneous value function V (t, x, y ) = x 1−γ v (t, y ) implies: ˜ 1 µ˜ ˜ σ vy πt = ˜ 2 + . γ σ ˜ ˜ ˜ σ v • Plugging equality back into the HJB equation: 2 σ2 σ2 1−γ µ˜ ˜ vy ˜ ˜ vt + (1 − γ)r v + µ − ˜ vy + ˜ vyy + +σ ˜ v = 0. 2 2 2γ σ ˜ 2 ˜ v • Certainty equivalent rate β = (µ2 − λ2 )/(2γσ 2 ) for shadow market must be the same as for transaction cost market. Set y ˜ v (t, y ) = e−(1−γ)(β+r )t e(1−γ) ˜ w(z)dz , ˜ ˜ ˜ • Since vy /v = (1 − γ)w, equation reduces to Riccati ODE 2 2µ 2β 1 µ ˜ w + (1 − γ)w 2 + ˜ ˜ σ2 ˜ −1 w − σ2 + γσ 2 σ ˜ ˜ + σ(1 − γ)w =0 ˜ ˜ with boundary conditions w(0) = w(log(u/l)) = 0.
  33. 33. Model Results Heuristics Method Matching HJB Equations • Shadow market value function y Vt = e−(1−γ)(β+r )t Xt1−γ e(1−γ) ˜ ˜ ˜ w(z)dz must coincide with transaction cost value function: y Vt = e−(1−γ)(β+r )t (Xt0 )1−γ e(1−γ) w(z)dz ˜ • X 0 safe position, X shadow wealth. Related by ˜ Xt ˜ ϕ0 S 0 + ϕt St = t t0 0 = 1 + g(eYt )l = φ(Yt ). Xt0 ϕt St ˜ • Condition V = V implies that y 0 = log (1 + g(ey )l) + ˜ (w(z) − w(z))dz, • Which in turn means that g (ey )ey l φ (y ) ˜ w(y ) = w(y ) − y )l = w(y ) − . 1 + g(e φ(y )
  34. 34. Model Results Heuristics Method Shadow price ODE ˜ ˜ • Plug w(y ) into ODE for w, use ODE for w, and simplify. Result: 2 µ(y ) ˜ g (ey )ey l (1 − γ)w(y ) + − = 0. σ˜ (y ) 1 + g(ey )l σ • Plug µ(y ) and σ (y ) to obtain (ugly) ODE for g: ˜ ˜ g (ey )ey 2g (ey )ey l 2µ y) − y )l + 2 + 2(1 − γ)w(y ) = 0. g (e 1 + g(e σ 1+g(e )ly • Substitution k (y ) = g (ey )ey l makes ODE linear: 2µ k (y ) = k (y ) − 1 + 2(1 − γ)w(y ) − 1. σ2
  35. 35. Model Results Heuristics Method Explicit Solutions • First, solve ODE for w(x, λ). Solution (for positive discriminant): a(λ) tan[tan−1 ( b(λ) ) + a(λ)x] + ( σ2 − 1 ) a(λ) µ 2 w(λ, x) = , γ−1 where 2 2 2 −λ a(λ) = (γ − 1) µγσ4 − 1 2 − µ σ2 , b(λ) = 1 2 − µ σ2 + (γ − 1) µ−λ . γσ 2 • Plug expression into ODE for k . Solution: 2 2 2 k (y ) = cos2 tan−1 b a + ay − a tan tan−1 1 b a +ay + a2 + (aa2 (µ−λ) b +b )γσ 1 1 y 1 • Plug into g(ey ) = 1−ε + l exp 0 k (x) dx − 1 , which yields: l 1 γσ 2 1 g(y ) = 1−ε 1+ µ−λ −1 1+ µ−λ 2 b − 2a 2 tan[tan−1 ( b )+ay ] γσ b2 +a2 a +b a
  36. 36. Model Results Heuristics Method VerificationTheorem ˜The shadow payoff XT of π = ˜ 1 µ˜ + (1 − γ) σ w and the shadow ˜ γ σ2 ˜ σ ˜ · µ ˜ ydiscount factor MT = E(− 0 σ dWt )T ˜ ˜ satisfy (with q (y ) = ˜ w(z)dz): ˜ 1−γ =e(1−γ)βT E e(1−γ)(q (Y0 )−q (YT )) , E XT ˆ ˜ ˜ 1 γ γ 1− γ 1 ˜ ˜ ( γ −1)(q (Y0 )−q (YT )) E MT ˆ =e(1−γ)βT E e . ˆ ˆwhere E[·] is the expectation with respect to the myopic probability P: ˆ T T 2 dP µ ˜ 1 µ ˜ = exp − + σ π dWt − ˜˜ − + σπ ˜˜ dt . dP 0 σ ˜ 2 0 σ ˜
  37. 37. Model Results Heuristics Method First Bound (1) ˜ ˜ ˜ ˜ • µ, σ , π , w functions of Yt . Argument omitted for brevity. ˜ • For first bound, write shadow wealth X as: ˜ 1−γ T σ2 2 ˜ T XT = exp (1 − γ) 0 µπ − ˜˜ 2 π ˜ dt + (1 − γ) 0 σ π dWt . ˜˜ • Hence: ˆ 2 ˜ 1−γ T σ2 2 µ ˜ XT = d P exp dP 0 (1 − γ) µπ − ˜˜ ˜ 2 π ˜ + 1 2 − σ + σπ ˜ ˜˜ dt T µ ˜ × exp 0 (1 − γ)˜ π − − σ + σ π σ˜ ˜ ˜˜ dWt . 1 µ˜ • Plug π = γ ˜ σ2 ˜ + (1 − γ) σ w . Second integrand is −(1 − γ)σ w. σ ˜ ˜ ˜ µ ˜ 2 2 • First integrand is 1 σ2 + γ σ π 2 − γ µπ , which equals to 2˜ ˜ 2 ˜ ˜˜ 2 2 1−γ µ ˜ (1 − γ)2 σ w 2 + 2 ˜ 2γ σ ˜ ˜ + σ(1 − γ)w .
  38. 38. Model Results Heuristics Method First Bound (2) 1−γ ˜ • In summary, XT equals to: ˆ T 2 2 dP 1 µ ˜ dP exp (1 − γ) 0 (1 − γ) σ w 2 + 2 ˜ 2γ σ ˜ ˜ + σ(1 − γ)w dt T × exp −(1 − γ) 0 ˜ σ wdWt . ˜ ˜ • By Itô’s formula, and boundary conditions w(0) = w(log(u/l)) = 0, T 1 T ˜ ˜ q (YT ) − q (Y0 ) = 0 ˜ w(Yt )dYt + 2 0 ˜ w (Yt )d Y , Y t ˜ ˜ + w(0)LT − w(u T σ2 ˜ σ2 ˜ T ˜ = 0 µ− 2 w+ 2 w dt + 0 σ wdWt . T ˜ 1−γ equals to: ˜ • Use identity to replace 0 σ wdWt , and XT ˆ dP T dP exp (1 − γ) 0 ˜ ˜ (β) dt) × exp (−(1 − γ)(q (YT ) − q (Y0 ))) . 2 2 2 σ2 1 µ ˜ as σ w + (1 − γ) σ w 2 + µ − 2 ˜ 2 ˜ 2 ˜ w+ 2γ σ ˜ ˜ + σ(1 − γ)w = β. • First bound follows.
  39. 39. Model Results Heuristics Method Second Bound • Argument for second bound similar. · µ ˜ ˆ • Discount factor MT = E(− 0 σ dW )T , myopic probability P satisfy: ˜ 1 1− γ 1−γ T µ ˜ 1−γ T µ2 ˜ MT = exp γ 0 σ dW ˜ + 2γ 0 σ 2 dt ˜ , ˆ dP 1−γ T µ ˜ (1−γ)2 T µ ˜ 2 = exp γ 0 σ ˜ ˜ + σ w dWt − 2γ 2 0 σ ˜ ˜ + σw dt . dP 1 1− γ • Hence, MT equals to: 2 ˆ T T 1 µ2 dP dP exp − 1−γ γ 0 ˜ σ wdWt + 1−γ γ 0 2 ˜ σ2 ˜ + 1−γ γ µ ˜ σ ˜ ˜ + σw dt . 2 2 2 • Note σ2 + 1−γ µ ˜ ˜ γ µ ˜ σ ˜ ˜ + σw = (1 − γ)σ 2 w 2 + ˜ 1 γ µ ˜ σ ˜ ˜ + σ(1 − γ)w T ˜ ˜ ˜ T • Plug 0 σ wdWt = q (YT ) − q (Y0 ) − 0 σ2 ˜ σ2 ˜ µ− 2 w+ 2 w dt 1 1− γ ˆ dP 1−γ βT − 1−γ (q (YT )−q (Y0 )) ˜ ˜ • HJB equation yields MT = dP e γ γ .
  40. 40. Model Results Heuristics Method Conclusion • Portfolio choice with transaction costs. • Constant risk aversion and long horizon. • Formulas for trading boundaries, certainty equivalent rate, liquidity premium and trading volume. All in terms of gap parameter. • Gap identified as solution of scalar equation. • Expansion for gap yield asymptotics for all quantities. • Verification by shadow price. • Shadow price also explicit.

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