Transaction Costs Made Tractable

1,262 views

Published on

Published in: Economy & Finance, Business
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
1,262
On SlideShare
0
From Embeds
0
Number of Embeds
29
Actions
Shares
0
Downloads
12
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

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.

×