Transaction Costs Made Tractable

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


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.


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?


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.


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.


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


Welfare, Liquidity Premium, Trading
Theorem
Trading 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


Gap
Theorem
• 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).


Trading Volume

Theorem
• 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


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 ∗


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 .
¯ ¯


Trading Boundaries v. Spread
0.75

0.70

0.65

0.60

0.55

0.50
0.00 0.02 0.04 0.06 0.08 0.10
µ = 8%, σ = 16%, γ = 5. Zero discount rate for consumption.


Liquidity Premium v. Spread
0.012

0.010

0.008

0.006

0.004

0.002

0.01 0.02 0.03 0.04 0.05
µ = 8%, σ = 16%. γ = 5, 1, 0.5.


Liquidity Premium v. Risk Aversion
0.010

0.008

0.006

0.004

0.002

0.000
0 2 4 6 8 10
µ = 8%, σ = 16%. ε = 0.01%, 0.1%, 1%, 10%.


Share Turnover v. Risk Aversion
0.5

0.4

0.3

0.2

0.1

0.0
0 2 4 6 8 10
µ = 8%, σ = 16%, γ = 5. ε = 0.01%, 0.1%, 1%, 10%.


Wealth Turnover v. Risk Aversion
0.5

0.4

0.3

0.2

0.1

0.0
0 2 4 6 8 10
µ = 8%, σ = 16%, γ = 5. ε = 0.01%, 0.1%, 1%, 10%.


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.


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


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


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.


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.


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.


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.


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


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.


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


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.


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.


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.


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.


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.


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.


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 )


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


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


Veriﬁcation

Theorem
˜
The shadow payoff XT of π =
˜ 1 µ˜
+ (1 − γ) σ w and the shadow
˜
γ σ2
˜ σ
˜
· µ
˜ y
discount 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 ˆ
where 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
˜


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 .


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.


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 .


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.

Transaction Costs Made Tractable

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