Download - 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 FinanceUniversity 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

• Janecek and Shreve (2004):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

St= (r + µ)dt + σdWt

• Bid price (1− ε)St . ε is the spread.• Investor with power utility U(x) = x1−γ/(1− γ).• Maximize certainty equivalent rate (Dumas and Luciano, 1991):

maxπ

limT→∞

1T

log E[X 1−γ

T

] 11−γ


Welfare, Liquidity Premium, TradingTheoremTrading the risky asset with transaction costs is equivalent to:• investing all wealth at hypothetical safe certainty equivalent rate

CeR = r +µ2 − λ2

2γσ2

• trading a hypothetical asset, at no transaction costs, with samevolatility σ, but expected return decreased by the liquidity premium

LiP = µ−√µ2 − λ2.

• Optimal to keep risky weight within buy and sell boundaries(evaluated at buy and sell prices respectively)

π− =µ− λγσ2 , π+ =

µ+ λ

γσ2 ,


GapTheorem

• λ identified as unique value for which solution of Cauchy problem

w ′(x) + (1− γ)w(x)2 +

(2µσ2 − 1

)w(x)− γ

(µ− λγσ2

)(µ+ λ

γσ2

)= 0

w(0) =µ− λγσ2 ,

satisfies the terminal value condition:

w(log(u(λ)/l(λ))) = µ+λγσ2 , where u(λ)

l(λ) = 11−ε

(µ+λ)(µ−λ−γσ2)(µ−λ)(µ+λ−γσ2)

.

• Asymptotic expansion:

λ = γσ2(

34γπ∗

2 (1− π∗)2)1/3

ε1/3 + O(ε).


Trading Volume

Theorem

• Share turnover (shares traded d ||ϕ||t divided by shares held |ϕt |).

ShT = limT→∞

1T

∫ T0

d‖ϕ‖t|ϕt | = σ2

2

(2µσ2 − 1

)(1−π−

(u/l)2µσ2−1

−1− 1−π+

(u/l)1− 2µ

σ2 −1

).

• Wealth turnover, (wealth traded divided by the wealth held):

WeT = limT→∞

1T

(∫ T0

(1−ε)St dϕ↓t

ϕ0t S0

t +ϕt (1−ε)St+∫ T

0St dϕ

↑t

ϕ0t S0

t +ϕt St

)= σ2

2

(2µσ2 − 1

)(π−(1−π−)

(u/l)2µσ2−1

−1− π+(1−π+)

(u/l)1− 2µ

σ2 −1

).


Asymptotics

π± = π∗ ±(

34γπ2∗ (1− π∗)2

)1/3

ε1/3 + O(ε).

CeR = r +µ2

2γσ2 −γσ2

2

(3

4γπ2∗ (1− π∗)2

)2/3

ε2/3 + O(ε4/3).

LiP =µ

2π2∗

(3

4γπ2∗ (1− π∗)2

)2/3

ε2/3 + O(ε4/3).

ShT =σ2

2(1− π∗)2π∗

(3

4γπ2∗ (1− π∗)2

)−1/3

ε−1/3 + O(ε1/3)

WeT =γσ2

3

(3

4γπ2∗(1− π∗)2

)2/3

ε−1/3 + O(ε).


Implications

• λ/σ2 depends on mean-variance ratio µ = µ/σ2. 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

proportional to business time∫ t

0 σ2s ds. Trading strategy invariant.

• All results extend to St such that:

dSt

St= (r + µσt )dt + σtdWt

with σt independent of Wt and ergodic (limT→∞1T

∫ T0 σ2

t dt = σ2).• Same formulas hold,

replacing µ/σ2 with µ, and residual factor σ2 with σ2.


Trading Boundaries v. Spread

0.00 0.02 0.04 0.06 0.08 0.100.50

0.55

0.60

0.65

0.70

0.75

µ = 8%, σ = 16%, γ = 5. Zero discount rate for consumption.


Liquidity Premium v. Spread

0.01 0.02 0.03 0.04 0.05

0.002

0.004

0.006

0.008

0.010

0.012

µ = 8%, σ = 16%. γ = 5,1,0.5.


Liquidity Premium v. Risk Aversion

0 2 4 6 8 100.000

0.002

0.004

0.006

0.008

0.010

µ = 8%, σ = 16%. ε = 0.01%,0.1%,1%,10%.


Share Turnover v. Risk Aversion

0 2 4 6 8 100.0

0.1

0.2

0.3

0.4

0.5

µ = 8%, σ = 16%, γ = 5. ε = 0.01%,0.1%,1%,10%.


Wealth Turnover v. Risk Aversion

0 2 4 6 8 100.0

0.1

0.2

0.3

0.4

0.5

µ = 8%, σ = 16%, γ = 5. ε = 0.01%,0.1%,1%,10%.


Welfare, Volume, and Spread

• Liquidity premium and share turnover:

LiPShT

=34ε+ O(ε5/3)

• Certainty equivalent rate and wealth turnover:

(r + µ2

γσ2 )− CeR

WeT=

34ε+ O(ε5/3).

• 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

t units of safe asset. ϕt shares of risky asset at time t .

• ϕt = ϕ↑t − ϕ↓t . Shares bought ϕ↑t minus shares sold ϕ↓t .

• Self-financing condition:

dϕ0t = − St

S0t

dϕ↑ + (1− ε)St

S0t

dϕ↓t

• X 0t = ϕ0

t S0t ,Xt = ϕtSt safe and risky wealth, at ask price St .

dX 0t =rX 0

t dt − Stdϕ↑t + (1− ε)Stdϕ

↓t ,

dXt =(µ+ r)Xtdt + σXtdWt + Stdϕ↑t − Stdϕ↓


Control Argument

• V (t , x , y) value function. Depends on time, and on asset positions.• By Itô’s formula:

dV (t ,X 0t ,Xt ) = Vtdt + VxdX 0

t + VydXt +12

Vyyd〈X ,X 〉t

=

(Vt + rX 0

t Vx + (µ+ r)XtVy +σ2

2X 2

t Vyy

)dt

+ St (Vy − Vx )dϕ↑t + St ((1− ε)Vx − Vy )dϕ↓t + σXtdWt

• V (t ,X 0t ,Xt ) supermartingale for any ϕ.

• ϕ↑, ϕ↓ increasing, hence Vy − Vx ≤ 0 and (1− ε)Vx − Vy ≤ 0

1 ≤ Vx

Vy≤ 1

1− ε


No Trade Region

• When 1 ≤ VxVy≤ 1

1−ε does not bind, drift is zero:

Vt + rX 0t Vx + (µ+ r)XtVy +

σ2

2X 2

t Vyy = 0 if 1 <Vx

Vy<

11− ε

.

• This is the no-trade region.• Ansatz: value function homogeneous in wealth.

Grows exponentially with the horizon.

V (t ,X 0t ,Xt ) = (X 0

t )1−γv(Xt/X 0t )e−(1−γ)(β+r)t

• Set z = y/x . For 1 + z < (1−γ)v(z)v ′(z) < 1

1−ε + z, HJB equation isσ2

2z2v ′′(z) + µzv ′(z)− (1− γ)βv(z) = 0

• Linear second order ODE. But β unknown.


Smooth Pasting

• Suppose 1 + z < (1−γ)v(z)v ′(z) < 1

1−ε + z same as l ≤ z ≤ u.

• For l < u to be found. Free boundary problem:

σ2

2z2v ′′(z) + µzv ′(z)− (1− γ)βv(z) = 0 if l < z < u,

(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:

−σ2

2 (1− γ)γ l2(1+l)2 v + µ(1− γ) l

1+l v − (1− γ)βv = 0.

• Setting π− = l/(1 + l), and factoring out (1− γ)v :

−γσ2

2π2− + µπ− − β = 0.

• π− risky weight on buy boundary, using ask price.• Same argument for u. Other solution to quadratic equation is:

π+ = u(1−ε)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

π± =µ

γσ2 ±√µ2 − 2βγσ2

γσ2 .

• Set β = (µ2 − λ2)/2γσ2. β = µ2/2γσ2 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±λγσ2 .


Symmetric Trading Boundaries

• Trading boundaries symmetric around frictionless weight µ/γσ2.• 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:

v(z) = e(1−γ)∫ log(z/l(λ)) w(y)dy , i.e. w(y) = l(λ)ey v ′(l(λ)ey )

(1−γ)v(l(λ)ey )

• Then linear decond order ODE becomes first order Riccati ODE

w ′(x) + (1− γ)w(x)2 +(

2µσ2 − 1

)w(x)− γ

(µ−λγσ2

)(µ+λγσ2

)= 0

w(0) =µ− λγσ2

w(log(u(λ)/l(λ))) =µ+ λ

γσ2

where u(λ)l(λ) = 1

1−επ+(1−π−)π−(1−π+) = 1

1−ε(µ+λ)(µ−λ−γσ2)(µ−λ)(µ+λ−γσ2)

.

• For each λ, initial value problem has solution w(λ, ·).• λ identified by second boundary w(λ, log(u(λ)/l(λ))) = µ+λ

γσ2 .


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 withtransaction 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

eYtg(eYt )

eYt = (Xt/X 0t )/l ratio between risky and safe positions at mid-price

S, and centered at the buying boundary

l =π−

1− π−=

(µ− λ)

γσ2 − (µ− λ).

• 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/S0) 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− σ2/2)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/lg′(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:

dSt/St = (µ(Yt ) + r)dt + σ(Yt )dWt ,

where

µ(y) =µg′(ey )ey + σ2

2 g′′(ey )e2y

g(ey ), and σ(y) =

σg′(ey )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:

dXt = r Xtdt + πt µ(Yt )Xtdt + πt σ(Yt )XtdWt .

• Setting Vt = V (t , Xt ,Yt ), Itô’s formula yields for dVt :

(Vt + r Xt Vx + µπt Xt Vx + σ2

2 π2t X 2

t Vxx +(µ−σ2

2 )Vy + σ2

2 Vyy +σσπt Xt Vxy )dt

+ Vy (dLt − dUt ) + (σπt Xt Vx + σVy )dWt ,

• V supermartingale for any strategy, martingale for optimal strategy.• HJB equation:

supπ(Vt +rxVx +µπxVx + σ2

2 π2x2Vxx +(µ−σ2

2 )Vy +σ2

2 Vyy +σσπxVxy ) = 0

with Neumann boundary conditions

Vy (0) = Vy (log(u/l)) = 0.


Homogeneity• Homogeneous value function V (t , x , y) = x1−γ v(t , y) implies:

πt =1γ

(µ

σ2 +σ

σ

vy

v

).

• Plugging equality back into the HJB equation:

vt + (1− γ)r v +

(µ− σ2

2

)vy +

σ2

2vyy +

1− γ2γ

(µ

σ2 + σvy

v

)2

v = 0.

• Certainty equivalent rate β = (µ2 − λ2)/(2γσ2) for shadow marketmust be the same as for transaction cost market. Set

v(t , y) = e−(1−γ)(β+r)te(1−γ)∫ y w(z)dz ,

• Since vy/v = (1− γ)w , equation reduces to Riccati ODE

w ′ + (1− γ)w2 +(

2µσ2 − 1

)w − 2β

σ2 + 1γσ2

(µσ + σ(1− γ)w

)2= 0

with boundary conditions w(0) = w(log(u/l)) = 0.


Matching HJB Equations• Shadow market value function

Vt = e−(1−γ)(β+r)t X 1−γt e(1−γ)

∫ y w(z)dz

must coincide with transaction cost value function:

Vt = e−(1−γ)(β+r)t (X 0t )1−γe(1−γ)

∫ y w(z)dz

• X 0 safe position, X shadow wealth. Related by

Xt

X 0t

=ϕ0

t S0t + ϕt St

ϕ0t S0

t= 1 + g(eYt )l = φ(Yt ).

• Condition V = V implies that

0 = log (1 + g(ey )l) +

∫ y(w(z)− w(z))dz,

• Which in turn means that

w(y) = w(y)− g′(ey )ey l1 + g(ey )l

= w(y)− φ′(y)

φ(y).


Shadow price ODE

• Plug w(y) into ODE for w , use ODE for w , and simplify. Result:((1− γ)w(y) +

µ(y)

σσ(y)− g′(ey )ey l

1 + g(ey )l

)2

= 0.

• Plug µ(y) and σ(y) to obtain (ugly) ODE for g:

g′′(ey )ey

g′(ey )− 2g′(ey )ey l

1 + g(ey )l+

2µσ2 + 2(1− γ)w(y) = 0.

• Substitution k(y) = 1+g(ey )lg′(ey )ey l makes ODE linear:

k ′(y) = k(y)

(2µσ2 − 1 + 2(1− γ)w(y)

)− 1.


Explicit Solutions• First, solve ODE for w(x , λ). Solution (for positive discriminant):

w(λ, x) =a(λ) tan[tan−1(b(λ)

a(λ) ) + a(λ)x ] + ( µσ2 − 1

2)

γ − 1,

where

a(λ) =

√(γ − 1)µ

2−λ2

γσ4 −(

12 −

µσ2

)2,b(λ) = 1

2 −µσ2 + (γ − 1)µ−λ

γσ2 .

• Plug expression into ODE for k . Solution:

k(y) = cos2 [tan−1 (ba

)+ ay

] (−1

a tan[tan−1 (b

a

)+ay

]+ b

a2 + (a2+b2)γσ2

a2(µ−λ)

).

• Plug into g(ey ) =(

11−ε + 1

l

)exp

(∫ y0

1k(x)dx

)− 1

l , which yields:

g(y) = 11−ε

(1 + γσ2

µ−λ

(1

1+µ−λγσ2

(b

b2+a2−a

a2+b2 tan[tan−1( ba )+ay]

) − 1

))


Verification

TheoremThe shadow payoff XT of π = 1

γ

(µσ2 + (1− γ)σσ w

)and the shadow

discount factor MT = E(−∫ ·

0µσdWt )T satisfy (with q(y) =

∫ y w(z)dz):

E[X 1−γ

T

]=e(1−γ)βT E

[e(1−γ)(q(Y0)−q(YT ))

],

E[M

1− 1γ

T

]γ=e(1−γ)βT E

[e( 1

γ−1)(q(Y0)−q(YT ))

]γ.

where E [·] is the expectation with respect to the myopic probability P:

dPdP

= exp

(∫ T

0

(− µσ

+ σπ

)dWt −

12

∫ T

0

(− µσ

+ σπ

)2

dt

).


First Bound (1)• µ, σ, π, w functions of Yt . Argument omitted for brevity.• For first bound, write shadow wealth X as:

X 1−γT = exp

((1− γ)

∫ T0

(µπ − σ2

2 π2)

dt + (1− γ)∫ T

0 σπdWt

).

• Hence:

X 1−γT =dP

dP exp(∫ T

0

((1− γ)

(µπ − σ2

2 π2)

+ 12

(− µσ + σπ

)2)

dt)

× exp(∫ T

0

((1− γ)σπ −

(− µσ + σπ

))dWt

).

• Plug π = 1γ

(µσ2 + (1− γ)σσ w

). Second integrand is −(1− γ)σw .

• First integrand is 12µ2

σ2 + γ σ2

2 π2 − γµπ, which equals to

(1− γ)2 σ2

2 w2 + 1−γ2γ

(µσ + σ(1− γ)w

)2.


First Bound (2)• In summary,

(XT

)1−γequals to:

dPdP exp

((1− γ)

∫ T0

((1− γ)σ

2

2 w2 + 12γ

(µσ + σ(1− γ)w

)2)

dt)

× exp(−(1− γ)

∫ T0 σwdWt

).

• By Itô’s formula, and boundary conditions w(0) = w(log(u/l)) = 0,

q(YT )− q(Y0) =∫ T

0 w(Yt )dYt + 12

∫ T0 w ′(Yt )d〈Y ,Y 〉t + w(0)LT − w(u/l)UT

=∫ T

0

((µ− σ2

2

)w + σ2

2 w ′)

dt +∫ T

0 σwdWt .

• Use identity to replace∫ T

0 σwdWt , and X 1−γT equals to:

dPdP exp

((1− γ)

∫ T0 (β) dt)

)× exp (−(1− γ)(q(YT )− q(Y0))) .

as σ2

2 w ′ + (1− γ)σ2

2 w2 +(µ− σ2

2

)w + 1

2γ

(µσ + σ(1− γ)w

)2= β.

• First bound follows.


Second Bound• Argument for second bound similar.• Discount factor MT = E(−

∫ ·0µσdW )T , myopic probability P satisfy:

M1− 1

γ

T = exp(

1−γγ

∫ T0

µσdW + 1−γ

2γ

∫ T0

µ2

σ2 dt),

dPdP

= exp(

1−γγ

∫ T0

(µσ + σw

)dWt − (1−γ)2

2γ2

∫ T0

(µσ + σw

)2dt).

• Hence, M1− 1

γ

T equals to:

dPdP exp

(−1−γ

γ

∫ T0 σwdWt + 1−γ

γ

∫ T0

12

(µ2

σ2 + 1−γγ

(µσ + σw

)2)

dt).

• Note µ2

σ2 + 1−γγ

(µσ + σw

)2= (1− γ)σ2w2 + 1

γ

(µσ + σ(1− γ)w

)2

• Plug∫ T

0 σwdWt = q(YT )− q(Y0)−∫ T

0

((µ− σ2

2

)w + σ2

2 w ′)

dt

• HJB equation yields M1− 1

γ

T = dPdP e

1−γγβT− 1−γ

γ(q(YT )−q(Y0)).


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.

Download - Transaction Costs Made Tractable

Top Related