ut austin - portugal lectures on portfolio choice

Long Run Investment: Opportunities and Frictions

Paolo Guasoni

Boston University and Dublin City University

July 2-6, 2012CoLab Mathematics Summer School

Outline

• Long Run and Stochastic Investment Opportunities• High-water Marks and Hedge Fund Fees• Transaction Costs• Price Impact• Open Problems

Long Run and Stochastic Investment Opportunities

One


Based onPortfolios and Risk Premia for the Long Run

with Scott Robertson

http://dx.doi.org/10.1214/11-AAP767


Independent Returns?

• Higher yields tend to be followed by higher long-term returns.• Should not happen if returns independent!


Utility Maximization• Basic portfolio choice problem: maximize utility from terminal wealth:

maxπ

E [U(XπT )]

• Easy for logarithmic utility U(x) = log x .Myopic portfolio π = Σ−1µ optimal. Numeraire argument.

• Portfolio does not depend on horizon (even random!), and on thedynamics of the the state variable, but only its current value.

• But logarithmic utility leads to counterfactual predictions.And implies that unhedgeable risk premia η are all zero.

• Power utility U(x) = x1−γ/(1− γ) is more flexible.Portfolio no longer myopic. Risk premia η nonzero, and depend on γ.

• Power utility far less tractable.Joint dependence on horizon and state variable dynamics.

• Explicit solutions few and cumbersome.• Goal: keep dependence from state variable dynamics, lose from horizon.• Tool: assume long horizon.


Asset Prices and State Variables• Safe asset S0

t = exp(∫ t

0 r(Ys)ds)

, d risky assets, k state variables.

dSit

Sit

=r(Yt )dt + dR it 1 ≤ i ≤ d

dR it =µi (Yt )dt +

n∑j=1

σij (Yt )dZ jt 1 ≤ i ≤ d

dY it =bi (Yt )dt +

k∑j=1

aij (Yt )dW jt 1 ≤ i ≤ k

d〈Z i ,W j〉t =ρij (Yt )dt 1 ≤ i ≤ d ,1 ≤ j ≤ k

• Z , W Brownian Motions.• Σ(y) = (σσ′)(y), Υ(y) = (σρa′)(y), A(y) = (aa′)(y).

Assumption

r ∈ Cγ(E ,R), b ∈ C1,γ(E ,Rk ), µ ∈ C1,γ(E ,Rn), A ∈ C2,γ(E ,Rk×k ),Σ ∈ C2,γ(E ,Rn×n) and Υ ∈ C2,γ(E ,Rn×k ). The symmetric matrices A and Σare strictly positive definite for all y ∈ E. Set Σ = Σ−1


State Variables

• Investment opportunities:safe rate r , excess returns µ, volatilities σ, and correlations ρ.

• State variables: anything on which investment opportunities depend.• Example with predictable returns:

dRt =Ytdt + σdZt

dYt =− λYtdt + dWt

• State variable is expected return. Oscillates around zero.• Example with stochastic volatility:

dRt =νYtdt +√

YtdZt

dYt =κ(θ − Yt )dt + a√

YtdWt

• State variable is squared volatility. Oscillates around positive value.• State variables are generally stationary processes.


(In)Completeness

• Υ′Σ−1Υ: covariance of hedgeable state shocks:Measures degree of market completeness.

• A = Υ′Σ−1Υ: complete market.State variables perfectly hedgeable, hence replicable.

• Υ = 0: fully incomplete market.State shocks orthogonal to returns.

• Otherwise state variable partially hedgeable.• One state: Υ′Σ−1Υ/a2 = ρ′ρ.

Equivalent to R2 of regression of state shocks on returns.


Well Posedness

Assumption

There exists unique solution(P(r ,y))

)r∈Rn,y∈E to martingale problem:

L =12

n+k∑i,j=1

Ai,j (x)∂2

∂xi∂xj+

n+k∑i=1

bi (x)∂

∂xiA =

(Σ ΥΥ′ A

)b =

(µb

)

• Ω = C([0,∞),Rn+k ) with uniform convergence on compacts.• B Borel σ-algebra, (Bt )t≥0 natural filtration.

Definition(Px )x∈Rn×Eon (Ω,B) solves martingale problem if, for all x ∈ Rn × E :• Px (X0 = x) = 1• Px (Xt ∈ Rn × E ,∀t ≥ 0) = 1

• f (Xt )− f (X0)−∫ t

0 (Lf )(Xu)du is Px -martingale for all f ∈ C20 (Rn × E)


Trading and Payoffs

DefinitionTrading strategy: process (πi

t )1≤i≤dt≥0 , adapted to Ft = Bt+, the right-continuous

envelope of the filtration generated by (R,Y ), and R-integrable.

• Investor trades without frictions. Wealth dynamics:

dXπt

Xπt

= r(Yt )dt + π′t dRt

• In particular, Xπt ≥ 0 a.s. for all t .

Admissibility implied by R-integrability.


Equivalent Safe Rate• Maximizing power utility E

[(Xπ

T )1−γ] /(1− γ) equivalent to maximizing the

certainty equivalent E[(Xπ

T )1−γ] 11−γ .

• Observation: in most models of interest, wealth grows exponentially withthe horizon. And so does the certainty equivalent.

• Example: with r , µ,Σ constant, the certainty equivalent is exactlyexp

((r + 1

2γµ′Σ−1µ)T

). Only total Sharpe ratio matters.

• Intuition: an investor with a long horizon should try to maximize the rate atwhich the certainty equivalent grows:

β = maxπ

lim infT→∞

1T

log E[(Xπ

T )1−γ] 11−γ

• Imagine a “dream” market, without risky assets, but only a safe rate ρ.• If β < ρ, an investor with long enough horizon prefers the dream.• If β > ρ, he prefers to wake up.• At β = ρ, his dream comes true.


Equivalent Annuity

• Exponential utility U(x) = −e−αx leads to a similar, but distinct idea.• Suppose the safe rate is zero.• Then optimal wealth typically grows linearly with the horizon, and so does

the certainty equivalent.• Then it makes sense to consider the equivalent annuity:

β = maxπ

lim infT→∞

− 1αT

log E[e−αXπT

]• The dream market now does not offer a higher safe rate, but instead a

stream of fixed payments, at rate ρ. The safe rate remains zero.• The investor is indifferent between dream and reality for β = ρ.• For positive safe rate, use definition with discounted quantities.• Undiscounted equivalent annuity always infinite with positive safe rate.


Solution Strategy

• Duality Bound.• Stationary HJB equation and finite-horizon bounds.• Criteria for long-run optimality.


Stochastic Discount FactorsDefinitionStochastic discount factor: strictly positive adapted M = (Mt )t≥0, such that:

EyP

[MtSi

t

∣∣Fs]

= MsSis for all 0 ≤ s ≤ t ,0 ≤ i ≤ d

Martingale measure: probability Q, such that Q|Ft and Py |Ft equivalent for allt ∈ [0,∞), and discounted prices Si/S0 Q-martingales for 1 ≤ i ≤ d .

• Martingale measures and stochastic discount factors related by:

dQdPy

∣∣Ft

= exp(∫ t

0 r(Ys)ds)

Mt

• Local martingale property: all stochastic discount factors satisfy

Mηt = exp

(−∫ t

0 rdt)E(−∫ ·

0(µ′Σ−1 + η′Υ′Σ−1)σdZ +∫ ·

0 η′adW

)t

for some adapted, Rk -valued process η.• η represents the vector of unhedgeable risk premia.• Intuitively, the Sharpe ratios of shocks orthogonal to dR.


Duality Bound• For any payoff X = Xπ

T and any discount factor M = MηT , E [XM] ≤ x .

Because XM is a local martingale.• Duality bound for power utility:

E[X 1−γ] 1

1−γ ≤ xE[M1−1/γ

] γ1−γ

• Proof: exercise with Hölder’s inequality.• Duality bound for exponential utility:

−1α

log E[e−αX ] ≤ x

E [M]+

1α

E[

ME [M]

logM

E [M]

]• Proof: Jensen inequality under risk-neutral densities.• Both bounds true for any X and for any M.

Pass to sup over X and inf over M.• Note how α disappears from the right-hand side.• Both bounds in terms of certainty equivalents.• As T →∞, bounds for equivalent safe rate and annuity follow.


Long Run OptimalityDefinition (Power Utility)

An admissible portfolio π is long run optimal if it solves:

maxπ

lim infT→∞

1T

log E[(Xπ

T )1−γ] 11−γ

The risk premia η are long run optimal if they solve:

minη

lim supT→∞

1T

log E[(Mη

T )1−1/γ] γ

1−γ

Pair (π, η) long run optimal if both conditions hold, and limits coincide.

• Easier to show that (π, η) long run optimal together.• Each η is an upper bound for all π and vice versa.

Definition (Exponential Utility)

Portfolio π and risk premia η long run optimal if they solve:

maxπ

lim infT→∞

− 1T

log E[e−αXπT

]minη

lim supT→∞

1T E[Mη

T log MηT

]


HJB Equation• V (t , x , y) depends on time t , wealth x , and state variable y .• Itô’s formula:

dV (t ,Xt ,Yt ) = Vtdt + Vx dXt + Vy dYt +12

(Vxx d〈X 〉t + Vxy d〈X ,Y 〉t + Vyy d〈Y 〉t )

• Vector notation. Vy ,Vxy k -vectors. Vyy k × k matrix.• Wealth dynamics:

dXt = (r + π′tµt )Xtdt + Xtπ′tσtdZt

• Drift reduces to:

Vt + xVx r + Vy b +12

tr(Vyy A) + xπ′ (µVx + ΥVxy ) +x2

2Vxxπ

′Σπ

• Maximizing over π, the optimal value is:

π = − Vx

xVxxΣ−1µ−

Vxy

xVxxΣ−1Υ

• Second term is new. Interpretation?


Intertemporal Hedging

• π = − VxxVxx

Σ−1µ− Σ−1ΥVxyxVxx

• First term: optimal portfolio if state variable frozen at current value.• Myopic solution, because state variable will change.• Second term hedges shifts in state variables.• If risk premia covary with Y , investors may want to use a portfolio which

covaries with Y to control its changes.• But to reduce or increase such changes? Depends on preferences.• When does second term vanish?• Certainly if Υ = 0. Then no portfolio covaries with Y .

Even if you want to hedge, you cannot do it.• Also if Vy = 0, like for constant r , µ and σ.

But then state variable is irrelevant.• Any other cases?


HJB Equation

• Maximize over π, recalling max(π′b + 12π′Aπ = − 1

2 b′A−1b).HJB equation becomes:

Vt + xVx r + Vy b +12

tr(Vyy A)− 12

(µVx + ΥVxy )′Σ−1

Vxx(µVx + ΥVxy ) = 0

• Nonlinear PDE in k + 2 dimensions. A nightmare even for k = 1.• Need to reduce dimension.• Power utility eliminates wealth x by homogeneity.


Homogeneity• For power utility, V (t , x , y) = x1−γ

1−γ v(t , y).

Vt =x1−γ

1− γvt Vx = x−γv Vxx = −γx−γ−1v

Vxy = x−γvy Vy =x1−γ

1− γvy Vyy =

x1−γ

1− γvyy

• Optimal portfolio becomes:

π =1γ

Σ−1µ+1γ

Σ−1Υvy

v

• Plugging in, HJB equation becomes:

vt + (1− γ)

(r +

12γµ′Σ−1µ

)v +

(b +

1− γγ

Υ′Σ−1µ

)vy

+12

tr(vyy A) +1− γγ

v ′y Υ′Σ−1Υvy

2v= 0

• Nonlinear PDE in k + 1 variables. Still hard to deal with.


Long Run Asymptotics• For a long horizon, use the guess v(t , y) = e(1−γ)(β(T−t)+w(y)).• It will never satisfy the boundary condition. But will be close enough.• Here β is the equivalent safe, to be found.• We traded a function v(t , y) for a function w(y), plus a scalar β.• The HJB equation becomes:(

−β + r +1

2γµ′Σ−1µ

)+

(b +

1− γγ

Υ′Σ−1µ

)wy

+12

tr(wyy A) +1− γ

2w ′y

(A− 1− γ

γΥ′Σ−1Υ

)wy = 0

• And the optimal portfolio:

π =1γ

Σ−1µ+(

1− 1γ

)Σ−1Υwy

• Stationary portfolio. Depends on state variable, not horizon.• HJB equation involved gradient wy and Hessian wyy , but not w .• With one state, first-order ODE.• Optimality? Accuracy? Boundary conditions?


Example• Stochastic volatility model:

dRt =νYtdt +√

YtdZt

dYt =κ(θ − Yt )dt + ε√

YtdWt

• Substitute values in stationary HJB equation:(−β + r +

ν2

2γy)

+

(κ(θ − y) +

1− γγ

ρενy)

wy

+ε2

2wyy + (1− γ)

ε2y2

w2y

(1− 1− γ

γρ2)

= 0

• Try a linear guess w = λy . Set constant and linear terms to zero.• System of equations in β and λ:

−β + r + κθλ =0

λ2(1− γ)ε2

2

(1− 1− γ

γρ2)

+ λ

(1− γγ

ενρ− κ)

+ν2

2γ=0

• Second equation quadratic, but only larger solution acceptable.Need to pick largest possible β.


Example (continued)• Optimal portfolio is constant, but not the usual constant.

π =1γ

(ν + βρε)

• Hedging component depends on various model parameters.• Hedging is zero if ρ = 0 or ε = 0.• ρ = 0: hedging impossible. Returns do not covary with state variable.• ε = 0: hedging unnecessary. State variable deterministic.• Hedging zero also if β = 0, which implies logarithmic utility.• Logarithmic investor does not hedge, even if possible.• Lives every day as if it were the last one.• Equivalent safe rate:

β =θν2

2γ

(1−

(1− 1

γ

)νρ

κε

)+ O(ε2)

• Correction term changes sign as γ crosses 1.


Martingale Measure

• Many martingale measures. With incomplete market, local martingalecondition does not identify a single measure.

• For any arbitrary k -valued ηt , the process:

Mt = E(−∫ ·

0(µ′Σ−1 + η′Υ′Σ−1)σdZ +

∫ ·0η′adW

)t

is a local martingale such that MR is also a local martingale.• Recall that MT = yU ′(Xπ

T ).• If local martingale M is a martingale, it defines a stochastic discount factor.• π = 1

γΣ−1(µ+ Υ(1− γ)wy ) yields:

U ′(XπT ) = (Xπ

T )−γ =x−γe−γ∫ T

0 (π′µ− 12π′Σπ)dt−γ

∫ T0 π′σdZt

=e−∫ T

0 (µ′+(1−γ)wy Υ′)Σ−1σdZt +∫ T

0 (... )dt

• Mathing the two expressions, we guess η = (1− γ)wy .


Risk Neutral Dynamics

• To find dynamics of R and Y under Q, recall Girsanov Theorem.• If Mt has previous representation, dynamics under Q is:

dRt =σdZt

dYt =(b −Υ′Σ−1µ+ (A−Υ′Σ−1Υ)η)dt + adWt

• Since η = (1− γ)wy , it follows that:

dRt =σdZt

dYt =(b −Υ′Σ−1µ+ (A−Υ′Σ−1Υ)(1− γ)wy

)dt + adWt

• Formula for (long-run) risk neutral measure for a given risk aversion.• For γ = 1 (log utility) boils down to minimal martingale measure.• Need to find w to obtain explicit solution.• And need to check that above martingale problem has global solution.


Exponential Utility• Instead of homogeneity, recall that wealth factors out of value function.• Long-run guess: V (x , y , t) = e−αx+αβt+w(y).β is now equivalent annuity.

• Set r = 0, otherwise safe rate wipes out all other effects.• The HJB equation becomes:(

−β +12µ′Σ−1µ

)+(

b −Υ′Σ−1µ)

wy +12

tr(wyy A)−12

w ′y(

A−Υ′Σ−1Υ)

wy = 0

• And the optimal portfolio:

xπ =1α

Σ−1µ− Σ−1Υwy

• Rule of thumb to obtain exponential HJB equation:write power HJB equation in terms of w = γw , then send γ ↑ ∞.Then remove the˜

• Exponential utility like power utility with “∞” relative risk aversion.• Risk-neutral dynamics is minimal entropy martingale measure:

dYt =(b −Υ′Σ−1µ− (A−Υ′Σ−1Υ)wy

)dt + adWt


HJB Equation

Assumption

w ∈ C2 (E ,R) and β ∈ R solve the ergodic HJB equation:

r +1

2γµ′Σµ+

1− γ2∇w ′

(A− (1− 1

γ)Υ′ΣΥ

)∇w+

∇w ′(

b − (1− 1γ

)Υ′Σµ

)+

12

tr(AD2w

)= β

• Solution must be guessed one way or another.• PDE becomes ODE for a single state variable• PDE becomes linear for logarithmic utility (γ = 1).• Must find both w and β


Myopic ProbabilityAssumption

There exists unique solution (Pr ,y )r∈Rn,y∈Rk to to martingale problem

L =12

n+k∑i,j=1

Ai,j (x)∂2

∂xi∂xj+

n+k∑i=1

bi (x)∂

∂xi

b =

( 1γ (µ+ (1− γ)Υ∇w)

b − (1− 1γ )Υ′Σµ+

(A− (1− 1

γ )Υ′ΣΥ)

(1− γ)∇w

)

• Under P, the diffusion has dynamics:

dRt = 1γ (µ+ (1− γ)Υ∇w) dt + σdZt

dYt =(

b − (1− 1γ )Υ′Σµ+ (A− (1− 1

γ )Υ′ΣΥ)(1− γ)∇w)

dt + adWt

• Same optimal portfolio as a logarthmic investor living under P.


Finite horizon bounds

TheoremUnder previous assumptions:

π =1γ

Σ (µ+ (1− γ)Υ∇w) , η = (1− γ)∇w

satisfy the equalities:

EyP

[(Xπ

T )1−γ] 11−γ = eβT +w(y)Ey

P

[e−(1−γ)w(YT )

] 11−γ

EyP

[(Mη

T )γ−1γ

] γ1−γ

= eβT +w(y)EyP

[e−

1−γγ w(YT )

] γ1−γ

• Bounds are almost the same. Differ in Lp norm• Long run optimality if expectations grow less than exponentially.


Path to Long Run solution

• Find candidate pair w , β that solves HJB equation.• Different β lead to to different solutions w .• Must find w corresponding to the lowest β that has a solution.• Using w , check that myopic probabiity is well defined.

Y does not explode under dynamics of P.• Then finite horizon bounds hold.• To obtain long run optimality, show that:

lim supT→∞

1T

log EyP

[e−

1−γγ w(YT )

]= 0

lim supT→∞

1T

log EyP

[e−(1−γ)w(YT )

]= 0


Proof of wealth bound (1)

• Z = ρW + ρB. Define Dt = dPdP |Ft , which equals to E(M), where:

Mt =

∫ t

0

(−qΥ′Σµ+

(A− qΥ′ΣΥ

)∇v)′

(a′)−1dWt

−∫ t

0q(Σµ+ ΣΥ∇v

)′σρdBt

• For the portfolio bound, it suffices to show that:

(XπT )p = ep(βT +w(y)−w(YT ))DT

which is the same as log XπT −

1p log DT = βT + w(y)− w(YT ).

• The first term on the left-hand side is:

log XπT =

∫ T

0

(r + π′µ− 1

2π′Σπ

)dt +

∫ T

0π′σdZt


Proof of wealth bound (2)• Set π = 1

1−p Σ (µ+ pΥ∇w). log XπT becomes:

∫ T0

(r + 1−2p

2(1−p)µ′Σµ− p2

(1−p)2µ′ΣΥ∇w − 1

2p2

(1−p)2∇w ′Υ′ΣΥ∇w)

dt

+ 11−p

∫ T0 (µ+ pΥ∇w)′ ΣσρdWt − 1

1−p

∫ T0 (µ+ pΥ∇w)′ ΣσρdBt

• Similarly, log DT/p becomes:∫ T0

(− p

2(1−p)2µ′Σµ− p

(1−p)2µ′ΣΥ∇w − p

2∇w ′(

A + p(2−p)(1−p)2 Υ′ΣΥ

)∇w

)dt+∫ T

0

(∇w ′a + 1

1−p (µ+ pΥ∇w)′ Σσρ)

dWt + 11−p

∫ T0 (µ+ pΥ∇w) ΣσρdBt

• Subtracting yields for log XπT − log DT/p∫ T

0

(r + 1

2(1−p)µ′Σµ+ p

1−pµ′ΣΥ∇w + p

2∇w ′(

A + p1−p Υ′ΣΥ

)∇w

)dt

−∫ T

0 ∇w ′adWt


Proof of wealth bound (3)

• Now, Itô’s formula allows to substitute:

−∫ T

0∇w ′adWt = w(y)− w(YT ) +

∫ T

0∇w ′bdt +

12

∫ T

0tr(AD2w)dt

• The resulting dt term matches the one in the HJB equation.• log Xπ

T − log DT/p equals to βT + w(y)− w(YT ).


Proof of martingale bound (1)

• For the discount factor bound, it suffices to show that:

1p−1 log Mη

T −1p log DT = 1

1−p (βT + w(y)− w(YT ))

• The term 1p−1 log Mη

T equals to:

11−p

∫ T0

(r + 1

2µ′Σµ+ p2

2 ∇w ′(A−Υ′ΣΥ

)∇w

)dt+

1p−1

∫ T0

(p∇w ′a− (µ+ pΥ∇w)′ Σσρ

)dWt + 1

1−p

∫ T0 (µ+ pΥ∇w) ΣσρdBt

• Subtracting 1p log DT yields for 1

p−1 log MηT −

1p log DT :

11−p

∫ T0

(r + 1

2(1−p)µ′Σµ+ p

1−pµ′ΣΥ∇w + p

2∇w ′(


)∇w

)dt

− 11−p

∫ T0 ∇w ′adWt


Proof of martingale bound (2)

• Replacing again∫ T

0 ∇w ′adWt with Itô’s formula yields:

11−p

∫ T0 (r + 1

2(1−p)µ′Σµ+ ( p

1−pµ′ΣΥ + b′)∇w+

12 tr(AD2w) + p

2∇w ′(


)∇w)dt

+ 11−p (w(y)− w(YT ))

• And the integral equals 11−pβT by the HJB equation.


Exponential UtilityTheoremIf r = 0 and w solves equation:

12µ′Σµ− 1

2∇w ′

(A−Υ′ΣΥ

)∇w +∇w ′

(b −Υ′Σµ

)+

12

tr(AD2w

)= β

and myopic dynamics is well posed:dRt =σdZt

dYt =(b −Υ′Σµ− (A−Υ′ΣΥ)∇w

)dt + adWt

Then for the portfolio and risk premia (π, η) given by:

xπ =1α

Σ−1µ− Σ−1Υ∇w η = −∇w

finite-horizon bounds hold as:

−1α

log EyP

[e−α(XπT −x)

]=βT +

1α

log EyP

[ew(y)−w(YT )

]12

EyP [Mη log Mη] =βT +

1α

EyP

[w(y)− w(YT )]

• Myopic probability is minimal entropy martingale measure!


Long-Run Optimality

TheoremIf, in addition to the assumptions for finite-horizon bounds,• the random variables (Yt )t≥0 are Py -tight in E for each y ∈ E;• supy∈E (1− γ)F (y) < +∞, where F ∈ C(E ,R) is defined as:

F =

(

r − β + µ′Σ−1µ2γ − (1−γ)2

2γ ∇w ′Υ′Σ−1Υ∇w)

e−(1−γ)w γ > 1(r − β + µ′Σ−1µ

2γ − (1−γ)2

2 ∇w ′(A−Υ′Σ−1Υ

)∇w

)e−

1−γγ w γ < 1

Then long-run optimality holds.

• Straightforward to check, once w is known.• Tightness checked with some moment condition.• Does not require transition kernel for Y under any probability.


Proof of Long-Run Optimality (1)• By the duality bound:

0 ≤ lim infT→∞

1p

(1T

log EyP

[(Mη

T )q]1−p − 1T

log EyP [(Xπ

T )p]

)≤ lim sup

T→∞

1pT

log EyP

[(Mη

T )q]1−p − lim infT→∞

1pT

log EyP [(Xπ

T )p]

= lim supT→∞

1− ppT

log EyP

[e−

11−p v(YT )

]− lim inf

T→∞

1pT

log EyP

[e−v(YT )

]• For p < 0 enough to show lower bound

lim infT→∞

1T

log EyP

[exp

(− 1

1−p v(YT ))]≥ 0

and upper bound:

lim supT→∞1T log Ey

P[exp (−v(YT ))] ≤ 0

• Lower bound follows from tightness.


Proof of Long-Run Optimality (2)• For upper bound, set:

Lf = ∇f ′(b − qΥ′Σ−1µ+

(A− qΥ′Σ−1Υ

)∇v)

+ 12 tr(AD2f

)• Then, for α ∈ R, the HJB equation implies that L (eαv ) equals to:

αeαv(∇v ′

(b − qΥ′Σ−1µ+

(A− qΥ′Σ−1Υ

)∇v)

+ 12 tr(AD2v

)+ 1

2α∇v ′A∇v)

= αeαv( 1

2∇v ′((1 + α)A− qΥ′Σ−1Υ

)∇v + pβ − pr + q

2µ′Σ−1µ

)• Set α = −1, to obtain that:

L(e−v) = e−v

(q2∇v ′Υ′Σ−1Υ∇v − λ+ pr − q

2µ′Σ−1µ

)• The boundedness hypothesis on F allows to conclude that:

EyP

[e−v(YT )

]≤ e−v(y) + (K ∨ 0) T

whencelim sup

T→∞

1T

log EyP

[e−v(YT )

]≤ 0

• 0 < p < 1 similar. Reverse inequalities for upper and lower bounds.Use α = − 1

1−p for upper bound.

High-water Marks and Hedge Fund Fees

Two


Based onThe Incentives of Hedge Fund Fees and High-Water Marks

with Jan Obłoj

http://ssrn.com/abstract=1784429


Two and Twenty

• Hedge Funds Managers receive two types of fees.• Regular fees, like Mutual Funds.• Unlike Mutual Funds, Performance Fees.• Regular fees:

a fraction ϕ of assets under management. 2% typical.• Performance fees:

a fraction α of trading profits. 20% typical.• High-Water Marks:

Performance fees paid after losses recovered.


High-Water Marks

Time Gross Net High-Water Mark Fees0 100 100 100 01 110 108 108 22 100 100 108 23 118 116 116 4

• Fund assets grow from 100 to 110.The manager is paid 2, leaving 108 to the fund.

• Fund drops from 108 to 100.No fees paid, nor past fees reimbursed.

• Fund recovers from 100 to 118.Fees paid only on increase from 108 to 118.Manager receives 2.


High-Water Marks

20 40 60 80 100

0.5

1.0

1.5

2.0

2.5


Risk Shifting?

• Manager shares investors’ profits, not losses.Does manager take more risk to increase profits?

• Option Pricing Intuition:Manager has a call option on the fund value.Option value increases with volatility. More risk is better.

• Static, Complete Market Fallacy:Manager has multiple call options.

• High-Water Mark: future strikes depend on past actions.• Option unhedgeable: cannot short (your!) hedge fund.


Questions

• Portfolio:Effect of fees and risk-aversion?

• Welfare:Effect on investors and managers?

• High-Water Mark Contracts:consistent with any investor’s objective?


Answers

• Goetzmann, Ingersoll and Ross (2003):Risk-neutral value of management contract (future fees).Exogenous portfolio and fund flows.

• High-Water Mark contract worth 10% to 20% of fund.• Panageas and Westerfield (2009):

Exogenous risky and risk-free asset.Optimal portfolio for a risk-neutral manager.Fees cannot be invested in fund.

• Constant risky/risk-free ratio optimal.Merton proportion does not depend on fee size.Same solution for manager with Hindy-Huang utility.


This Model

• Manager with Power Utility and Long Horizon.Exogenous risky and risk-free asset.Fees cannot be invested in fund.

• Optimal Portfolio:

π =1γ∗

µ

σ2

γ∗ =(1− α)γ + α

γ =Manager’s Risk Aversionα =Performance Fee (e.g. 20%)

• Manager behaves as if owned fund, but were more myopic (γ∗ weightedaverage of γ and 1).

• Performance fees α matter. Regular fees ϕ don’t.


Three Problems, One Solution

• Power utility, long horizon. No regular fees.1 Manager maximizes utility of performance fees.

Risk Aversion γ.2 Investor maximizes utility of wealth. Pays no fees.

Risk Aversion γ∗ = (1− α)γ + α.3 Investor maximizes utility of wealth. Pays no fees.

Risk Aversion γ. Maximum Drawdown 1− α.

• Same optimal portfolio:

π =1γ∗

µ

σ2


Price Dynamics

dSt

St= (r + µ)dt + σdWt (Risky Asset)

dXt = (r − ϕ)Xtdt + Xtπt

(dStSt− rdt

)− α

1−αdX ∗t (Fund)

dFt = rFtdt + ϕXtdt +α

1− αdX ∗t (Fees)

X ∗t = max0≤s≤t

Xs (High-Water Mark)

• One safe and one risky asset.• Gain split into α for the manager and 1− α for the fund.• Performance fee is α/(1− α) of fund increase.


Dynamics Well Posed?

• Problem: fund value implicit.Find solution Xt for

dXt = XtπtdSt

St− ϕXtdt − α

1− αdX ∗t

• Yes. Pathwise construction.

Proposition

The unique solution is Xt = eRt−αR∗t , where:

Rt =

∫ t

0

(µπs −

σ2

2π2

s − ϕ)

ds + σ

∫ t

0πsdWs

is the cumulative log return.


Fund Value Explicit

LemmaLet Y be a continuous process, and α > 0.Then Yt + α

1−αY ∗t = Rt if and only if Yt = Rt − αR∗t .

Proof.Follows from:

R∗t = sups≤t

(Ys +

α

1− αsupu≤s

Yu

)= Y ∗t +

α

1− αY ∗t =

11− α

Y ∗t

• Apply Lemma to cumulative log return.


Long Horizon

• The manager chooses the portfolio π which maximizes expected powerutility from fees at a long horizon.

• Maximizes the long-run objective:

maxπ

limT→∞

1pT

log E [F pT ] = β

• Dumas and Luciano (1991), Grossman and Vila (1992), Grossman andZhou (1993), Cvitanic and Karatzas (1995). Risk-Sensitive Control:Bielecki and Pliska (1999) and many others.

• β = r + 1γµ2

2σ2 for Merton problem with risk-aversion γ = 1− p.


Solving It• Set r = 0 and ϕ = 0 to simplify notation.• Cumulative fees are a fraction of the increase in the fund:

Ft =α

1− α(X ∗t − X ∗0 )

• Thus, the manager’s objective is equivalent to:

maxπ

limT→∞

1pT

log E [(X ∗T )p]

• Finite-horizon value function:

V (x , z, t) = supπ

1p

E [X ∗Tp|Xt = x ,X ∗t = z]

dV (Xt ,X ∗t , t) = Vtdt + VxdXt +12

Vxxd〈X 〉t + VzdX ∗t

= Vtdt +(

Vz − α1−αVx

)dX ∗t +

(VxXt (πtµ− ϕ)dt + Vxx

σ2

2 π2t X 2

t

)dt


Dynamic Programming

• Hamilton-Jacobi-Bellman equation:Vt + supπ

(xVx (πµ− ϕ) + Vxx

σ2

2 π2x2)

)x < z

Vz = α1−αVx x = z

V = zp/p x = 0V = zp/p t = T

• Maximize in π, and use homogeneityV (x , z, t) = zp/pV (x/z,1, t) = zp/pu(x/z,1, t).

ut − ϕxux − µ2

2σ2u2

xuxx

= 0 x ∈ (0,1)

ux (1, t) = p(1− α)u(1, t) t ∈ (0,T )

u(x ,T ) = 1 x ∈ (0,1)

u(0, t) = 1 t ∈ (0,T )


Long-Run Heuristics

• Long-run limit.Guess a solution of the form u(t , x) = ce−pβtw(x), forgetting the terminalcondition:

−pβw − ϕxwx − µ2

2σ2w2

xwxx

= 0 for x < 1wx (1) = p(1− α)w(1)

• This equation is time-homogeneous, but β is unknown.• Any β with a solution w is an upper bound on the rate λ.• Candidate long-run value function:

the solution w with the lowest β.

• w(x) = xp(1−α), for β = 1−α(1−α)γ+α

µ2

2σ2 − ϕ(1− α).


Verification

TheoremIf ϕ− r < µ2

2σ2 min

1γ∗, 1γ2∗

, then for any portfolio π:

limT↑∞

1pT

log E[(Fπ

T )p] ≤ max

(1− α)

(1γ∗

µ2

2σ2 + r − ϕ), r

Under the nondegeneracy condition ϕ+ α1−α r < 1

γ∗

µ2

2σ2 , the unique optimalsolution is π = 1

γ∗µσ2 .

• Martingale argument. No HJB equation needed.• Show upper bound for any portfolio π (delicate).• Check equality for guessed solution (easy).


Upper Bound (1)• Take p > 0 (p < 0 symmetric).• For any portfolio π:

RT = −∫ T

0σ2

2 π2t dt +

∫ T0 σπtdWt

• Wt = Wt + µ/σt risk-neutral Brownian Motion• Explicit representation:

E [(XπT )p] = E [ep(1−α)R∗T ] = EQ

[ep(1−α)R∗T e

µσ WT− µ2

2σ2 T]

• For δ > 1, Hölder’s inequality:

EQ

[ep(1−α)R∗T e

µσ WT− µ2

2σ2 T]≤ EQ

[eδp(1−α)R∗T

] 1δ EQ

[e

δδ−1

(µσ WT− µ2

2σ2 T)] δ−1

δ

• Second term exponential normal moment. Just e1δ−1

µ2

2σ2 T .


Upper Bound (2)

• Estimate EQ[eδp(1−α)R∗T

].

• Mt = eRt strictly positive continuous local martingale.Converges to zero as t ↑ ∞.

• Fact:inverse of lifetime supremum (M∗∞)−1 uniform on [0,1].

• Thus, for δp(1− α) < 1:

EQ

[eδp(1−α)R∗T

]≤ EQ

[eδp(1−α)R∗∞

]=

11− δp(1− α)

• In summary, for 1 < δ < 1p(1−α) :

limT↑∞

1pT

log E[(Fπ

T )p] ≤ 1p(δ − 1)

µ2

2σ2

• Thesis follows as δ → 1p(1−α) .


High-Water Marks and Drawdowns• Imagine fund’s assets Xt and manager’s fees Ft in the same account

Ct = Xt + Ft .

dCt = (Ct − Ft )πtdSt

St

• Fees Ft proportional to high-water mark X ∗t :

Ft =α

1− α(X ∗t − X ∗0 )

• Account increase dC∗t as fund increase plus fees increase:

C∗t − C∗0 =

∫ t

0(dX ∗s + dFs) =

∫ t

0

(α

1− α+ 1)

dX ∗s =1

1− α(X ∗t − X0)

• Obvious bound Ct ≥ Ft yields:

Ct ≥ α(C∗t − X0)

• X0 negligible as t ↑ ∞. Approximate drawdown constraint.

Ct ≥ αC∗t

Transaction Costs

Three

Transaction Costs

Based onTransaction Costs, Trading Volume, and the Liquidity Premium

with Stefan Gerhold, Johannes Muhle-Karbe, and WalterSchachermayer

http://ssrn.com/abstract=1905077

Transaction Costs

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 equivalent safe rate:

maxπ

limT→∞

1T

log E[X 1−γ

T

] 11−γ

Transaction Costs

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↓

Transaction Costs

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 + Vy dXt +12

Vyy d〈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

Transaction Costs

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.• Long-run guess:

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 is

σ2

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

• Linear second order ODE. But β unknown.

Transaction Costs

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.

Transaction Costs

Solution Procedure

• Unknown: trading boundaries l ,u and rate β.• Strategy: find l ,u in terms of β.• Free boundary problem becomes fixed boundary problem.• Find unique β that solves this problem.

Transaction Costs

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.

Transaction Costs

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 .

Transaction Costs

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.

Transaction Costs

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 second 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 .

Transaction Costs

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 .

• Similar expressions for zero and negaive discriminants.

Transaction Costs

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.

Transaction Costs

Shadow Price as Marginal Rate of Substitution• Imagine trading is now frictionless, but price is St .• Intuition: the value function must not increase by trading δ ∈ R shares.• In value function, risky position valued at ask price St . Thus

V (t ,X 0t − δSt ,Xt + δSt ) ≤ V (t ,X 0

t ,Xt )

• Expanding for small δ, −δVx St + δVy St ≤ 0• Must hold for δ positive and negative. Guess for St :

St =Vy

VxSt

• Plugging guess V (t ,X 0t ,Xt ) = e−(1−γ)(r+β)t (X 0

t )1−γe(1−γ)∫ log(Xt/lX0

t )

0 w(y)dy ,

St =w(Yt )

leYt (1− w(Yt ))St

• Shadow portfolio weight is precisely w :

πt =ϕt St

ϕ0t S0

t + ϕt St=

w(Yt )1−w(Yt )

1 + w(Yt )1−w(Yt )

= w(Yt ),

Transaction Costs

Shadow Value Function• In terms of shadow wealth Xt = ϕ0

t S0t + ϕt St , the value function becomes:

V (t , Xt ,Yt ) = V (t ,X 0t ,Xt ) = e−(1−γ)(r+β)t X 1−γ

t

(X 0

t

Xt

)1−γ

e(1−γ)∫ Yt

0 w(y)dy .

• Note that X 0t /Xt = 1− w(Yt ) by definition of St , and rewrite as:

(X 0

t

Xt

)1−γ

e(1−γ)∫ Yt

0 w(y)dy = exp(

(1− γ)[log(1− w(Yt )) +

∫ Yt

0 w(y)dy])

= (1− w(0))γ−1 exp(

(1− γ)∫ Yt

0

(w(y)− w ′(y)

1−w(y)

)dy).

• Set w = w − w ′1−w to obtain

V (t , Xt ,Yt ) = e−(1−γ)(r+β)t X 1−γt e(1−γ)

∫ Yt0 w(y)dy (1− w(0))γ−1.

• We are now ready for verification.

Transaction Costs

Construction of Shadow Price

• So far, a string of guesses. Now construction.• Define Yt as reflected diffusion on [0, log(u/l)]:

dYt = (µ− σ2/2)dt + σdWt + dLt − dUt , Y0 ∈ [0, log(u/l)],

LemmaThe dynamics of S = S w(Y )

leY (1−w(Y ))is given by

dS(Yt )/S(Yt ) = (µ(Yt ) + r) dt + σ(Yt )dWt ,

where µ(·) and σ(·) are defined as

µ(y) =σ2w ′(y)

w(y)(1− w(y))

(w ′(y)

1− w(y)− (1− γ)w(y)

), σ(y) =

σw ′ (y)

w(y)(1− w(y)).

And the process S takes values within the bid-ask spread [(1− ε)S,S].

Transaction Costs

Finite-horizon Bound

TheoremThe shadow payoff XT of the portfolio π = w(Yt ) and the shadow discountfactor 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

).

Transaction Costs

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.

Transaction Costs

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.

Transaction Costs

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

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

M1− 1

γ

T = exp(

1−γγ

∫ T0µσdW + 1−γ

2γ

∫ T0µ2

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−γγ βT− 1−γ

γ (q(YT )−q(Y0)).

Transaction Costs

Buy and Sell at BoundariesLemmaFor 0 < µ/γσ2 6= 1, the number of shares ϕt = w(Yt )Xt/St follows:

dϕt

ϕt=

(1− µ− λ

γσ2

)dLt −

(1− µ+ λ

γσ2

)dUt .

Thus, ϕt increases only when Yt = 0, that is, when St equals the ask price,and decreases only when Yt = log(u/l), that is, when St equals the bid price.

• Itô’s formula and the ODE yield

dw(Yt ) = −(1− γ)σ2w ′(Yt )w(Yt )dt + σw ′(Yt )dWt + w ′(Yt )(dLt − dUt ).

• Integrate ϕt = w(Yt )Xt/St by parts, insert dynamics of w(Yt ), Xt , St ,

dϕt

ϕt=

w ′(Yt )

w(Yt )d(Lt − Ut ).

• Since Lt and Ut only increase (decrease for µ/γσ2 > 1) on Yt = 0 andYt = log(u/l), claim follows from boundary conditions for w and w ′.

Transaction Costs

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

ESR = r +µ2 − λ2

2γσ2

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

LiPr = 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 ,

Transaction Costs

Gap

Theorem

• 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).

Transaction Costs

Trading Volume

Theorem

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

ShTu = 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):

WeTu = 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

).

Transaction Costs

Asymptotics

π± = π∗ ±(

34γπ2∗ (1− π∗)2

)1/3

ε1/3 + O(ε).

ESR = r +µ2

2γσ2 −γσ2

2

(3

4γπ2∗ (1− π∗)2

)2/3

ε2/3 + O(ε4/3).

LiPr =µ

2π2∗

(3

4γπ2∗ (1− π∗)2

)2/3

ε2/3 + O(ε4/3).

ShTu =σ2

2(1− π∗)2π∗

(3

4γπ2∗ (1− π∗)2

)−1/3

ε−1/3 + O(ε1/3)

WeTu =γσ2

3

(3

4γπ2∗(1− π∗)2

)2/3

ε−1/3 + O(ε).

Transaction Costs

Welfare, Volume, and Spread

• Liquidity premium and share turnover:

LiPrShTu

=34ε+ O(ε5/3)

• Equivalent safe rate and wealth turnover:

(r + µ2

γσ2 )− ESR

WeTu=

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.

Price Impact

Four

Price Impact

Based onDynamic Trading Volume

with Marko Weber

Price Impact

Market• Brownian Motion (Wt )t≥0 with natural filtration (Ft )t≥0.• Best quoted price of risky asset. Price for an infinitesimal trade.

dSt

St= µdt + σdWt

• Trade ∆θ shares over time interval ∆t . Order filled at price

St (∆θ) := St

(1 + λ

St ∆θ

Xt ∆t

)where Xt is investor’s wealth.

• λ measures illiquidity. 1/λ market depth. Like Kyle’s (1985) lambda.• Price worse for larger quantity |∆θ| or shorter execution time ∆t .

Price linear in quantity, inversely proportional to execution time.• Same amount St ∆θ has lower impact if investor’s wealth larger.• Makes model scale-invariant.

Doubling wealth, and all subsequent trades, doubles final payoff exactly.

Price Impact

Alternatives?

• Alternatives: quantities ∆θ, or share turnover ∆θ/θ. Consequences?• Quantities (∆θ):

Kyle (1985), Bertsimas and Lo (1998), Almgren and Chriss (2000), Schiedand Shoneborn (2009), Garleanu and Pedersen (2011)

St (∆θ) := St + λ∆θ

∆t

• Price impact independent of price. Not invariant to stock splits!• Suitable for short horizons (liquidation) or mean-variance criteria.• Share turnover:

Stationary measure of trading volume (Lo and Wang, 2000). Observable.

St (∆θ) := St

(1 + λ

∆θ

θt ∆t

)• Problematic. Infinite price impact with cash position.

Price Impact

Wealth and Portfolio• Continuous trading: execution price St (θt ) = St

(1 + λ θt St

Xt

), cash position

dCt = −St

(1 + λ θt St

Xt

)dθt = −St

(θt + λSt

Xtθ2

t

)dt

• Trading volume as wealth turnover ut := θt StXt

.Amount traded in unit of time, as fraction of wealth.

• Dynamics for wealth Xt := θtSt + Ct and risky portfolio weight Yt := θt StXt

dXt

Xt= Yt (µdt + σdWt )− λu2

t dt

dYt = (Yt (1− Yt )(µ− Ytσ2) + (ut + λYtu2

t ))dt + σYt (1− Yt )dWt

• Illiquidity...• ...reduces portfolio return (−λu2

t ).Turnover effect quadratic: quantities times price impact.

• ...increases risky weight (λYtu2t ). Buy: pay more cash. Sell: get less cash.

Turnover effect linear in risky weight Yt . Vanishes for cash position.

Price Impact

Admissible Strategies

DefinitionAdmissible strategy: process (ut )t≥0, adapted to Ft , such that system

dXt

Xt= Yt (µdt + σdWt )− λu2

t dt

dYt = (Yt (1− Yt )(µ− Ytσ2) + (ut + λYtu2

t ))dt + σYt (1− Yt )dWt

has unique solution satisfying Xt ≥ 0 a.s. for all t ≥ 0.

• Contrast to models without frictions or with transaction costs:control variable is not risky weight Yt , but its “rate of change” ut .

• Portfolio weight Yt is now a state variable.• Illiquid vs. perfectly liquid market.

Steering a ship vs. driving a race car.• Frictionless solution Yt = µ

γσ2 unfeasible. A still ship in stormy sea.

Price Impact

Objective

• Investor with relative risk aversion γ.• Maximize equivalent safe rate, i.e., power utility over long horizon:

maxu

limT→∞

1T

log E[X 1−γ

T

] 11−γ

• Tradeoff between speed and impact.• Optimal policy and welfare.• Implied trading volume.• Dependence on parameters.• Asymptotics for small λ.• Comparison with transaction costs.

Price Impact

Control Argument

• Value function v depends on (1) current wealth Xt , (2) current risky weightYt , and (3) calendar time t .

• Evolution for fixed trading strategy u = θt StXt

:

dv(Xt ,Yt , t) =vtdt + vx (µXtYt − λXtu2t )dt + vxXtYtσdWt

+vy (Yt (1− Yt )(µ− Ytσ2) + ut + λYtu2

t )dt + vy Yt (1− Yt )σdWt

+

(σ2

2vxxX 2

t Y 2t +

σ2

2vyy Y 2

t (1− Yt )2 + σ2vxy XtY 2

t (1− Yt )

)dt

• Maximize drift over u, and set result equal to zero:

vt + maxu

(vx(µxy − λxu2)+ vy

(y(1− y)(µ− σ2y) + u + λyu2)

+σ2y2

2(vxxx2 + vyy (1− y)2 + 2vxy x(1− y)

) )= 0

Price Impact

Homogeneity and Long-Run

• Homogeneity in wealth v(t , x , y) = x1−γv(t ,1, y).• Guess long-term growth at equivalent safe rate β, to be found.

• Substitution v(t , x , y) = x1−γ

1−γ e(1−γ)(β(T−t)+∫ y q(z)dz) reduces HJB equation

−β + maxu

((µy − γ σ

2

2 y2 − λu2)

+ q(y(1− y)(µ− γσ2y) + u + λyu2)

+σ2

2 y2(1− y)2(q′ + (1− γ)q2))

= 0.

• Maximum for u(y) = q(y)2λ(1−yq(y)) .

• Plugging yields

µy−γ σ2

2 y2+y(1−y)(µ−γσ2y)q+ q2

4λ(1−yq) + σ2

2 y2(1−y)2(q′+(1−γ)q2) = β

• β = µ2

2γσ2 , q = 0, y = µγσ2 corresponds to Merton solution.

• Classical model as a singular limit.

Price Impact

Asymptotics

• Expand equivalent safe rate as β = µ2

2γσ2 − c(λ)

• Function c represents welfare impact of illiquidity.• Expand function as q(y) = q(1)(y)λ1/2 + o(λ1/2).• Plug expansion in HJB equation

−β+µy−γ σ2

2 y2+y(1−y)(µ−γσ2y)q+ q2

4λ(1−yq) +σ2

2 y2(1−y)2(q′+(1−γ)q2) = 0

• Yieldsq(1)(y) = σ−1(γ/2)−1/2(µ− γσ2y)

• Turnover follows through u(y) = q(y)2λ(1−yq(y)) .

• Plug q(1)(y) back in equation, and set y = Y .• Yields welfare loss c(λ) = σ3(γ/2)−1/2Y 2(1− Y )2λ1/2 + o(λ1/2).

Price Impact

Issues

• How to make argument rigorous?• Heuristics yield ODE, but no boundary conditions!• Relation between ODE and optimization problem?

Price Impact

Verification

TheoremIf µγσ2 ∈ (0,1), then the optimal wealth turnover and equivalent safe rate are:

u(y) =1

2λq(y)

1− yq(y)ESRγ(u) = β

where β ∈ (0, µ2

2γσ2 ) and q : [0,1] 7→ R are the unique pair that solves the ODE

−β+µy−γ σ2

2 y2+y(1−y)(µ−γσ2y)q+ q2

4λ(1−yq) + σ2

2 y2(1−y)2(q′+(1−γ)q2) = 0

and q(0) = 2√λβ, q(1) = λd −

√λd(λd − 2), where d = −γσ2 − 2β + 2µ.

• A license to solve an ODE, of Abel type.• Function q and scalar β not explicit.• Asymptotic expansion for λ near zero?

Price Impact

Existence

TheoremAssume 0 < µ

γσ2 < 1. There exists β∗ such that HJB equation has solutionq(y) with positive finite limit in 0 and negative finite limit in 1.

• for β > 0, there exists a unique solution q0,β(y) to HJB equation withpositive finite limit in 0 (and the limit is 2

√λβ);

• for β > µ− γσ2

2 , there exists a unique solution q1,β(y) to HJB equationwith negative finite limit in 1 (and the limit is λd −

√λd(λd − 2), where

d := 2(µ− γσ2

2 − β));• there exists βu such that q0,βu (y) > q1,βu (y) for some y ;• there exists βl such that q0,βl (y) < q1,βl (y) for some y ;• by continuity and boundedness, there exists β∗ ∈ (βl , βu) such that

q0,β∗(y) = q1,β∗(y).• Boundary conditions are natural!

Price Impact

Finite Horizon Bound (1)LemmaDefine Q(y) =

∫ y q(z)dz. There exists a probability P, equivalent to P, suchthat the terminal wealth XT of any admissible strategy satisfies:

E [X 1−γT ]

11−γ ≤ eβT +Q(y)EP [e−(1−γ)Q(YT )]

11−γ (1)

and equality holds for the optimal strategy.

• Define probability P as:dPdP = exp

∫ T0 −

(1−γ)2Y 2s (1+q(Ys)(1−Ys))2σ2

2 ds +∫ T

0 (1− γ)Ys(1 + q(Ys)(1− Ys))σdWs

.

• It suffices to check that

log XT − 11−γ

log dPdP ≤ βT −Q(YT ) + Q(y) .

• By self-financing condition, and Itô formula:

Q(YT )−Q(y) =∫ T

0

(q(Yt )(Yt (1− Yt )(µ− Ytσ

2) + ut + λYtu2t ) + q′(Yt )

Y 2t (1−Yt )2σ2

2

)dt

+

∫ T

0q(Yt )Yt (1− Yt )σdWt

Price Impact

Finite Horizon Bound (2)

• Also,

log XT =

∫ T

0

(Ytµ− λu2

t −Y 2

t σ2

2

)dt +

∫ T

0YtσdWt .

• Plugging equalities into log XT − 11−γ log dP

dP , and using HJB equation:

∫ T0 µYt − λu2

t −Y 2

t σ2

2 +σ2(1−γ)Y 2

t (1+q(Yt )(1−Yt ))2

2 dt −∫ T

0 q(Yt )Yt (1− Yt )σdWt ≤

≤∫ T

0 β −(

q(Yt )(Yt (1− Yt )(µ− Ytσ2) + ut + λYtu2

t ) + σ2

2 q′(Yt )Y 2t (1− Yt )

2)

dt

−∫ T

0q(Yt )Yt (1− Yt )σdWt .

• Stochastic integrals on both sides are equal. Enough to prove that:

yµ−λu2− y2σ2

2 +σ2(1−γ)y2(1+q(y)(1−y))2

2 ≤ β−q(y)(y(1−y)(µ−yσ2)+u+λyu2)−q′(y) y2(1−y)2σ2

2

• But this is just the HJB equation rearranged. Equality with optimal control.

Price Impact

Asymptotics

TheoremY = µ

γσ2 ∈ (0,1). Asymptotic expansions for turnover and equivalent safe rate:

u(y) = σ

√γ

2λ(Y − y) + O(1)

ESRγ(u) =µ2

2γσ2 − σ3√γ

2Y 2(1− Y )2λ1/2 + O(λ)

Long-term average of (unsigned) turnover

|ET| = limT→∞

1T

∫ T0 |u(Yt )|dt = π−1/2σ3/2Y (1− Y ) (γ/2)1/4

λ−1/4 + O(λ1/2)

• Implications?

Price Impact

Turnover

• Turnover:

u(y) ∼ σ√

γ

2λ(Y − y)

• Trade towards Y . Buy for y < Y , sell for y > Y .• Trade speed proportional to displacement |y − Y |.• Trade faster with more volatility. Volume typically increases with volatility.• Trade faster if market deeper. Higher volume in more liquid markets.• Trade faster if more risk averse. Reasonable, not obvious.

Dual role of risk aversion.• More risk aversion means less risky asset but more trading speed.

Price Impact

Trading Volume• Wealth turnover approximately Ornstein-Uhlenbeck:

dut = σ

√γ

2λ(σ2Y 2(1− Y )(1− γ)− ut )dt − σ2

√γ

2λY (1− Y )dWt

• In the following sense:

TheoremThe process ut has asymptotic moments:

ET := limT→∞

1T

∫ T

0u(Yt )dt = σ2Y 2(1− Y )(1− γ) + o(1) ,

VT := limT→∞

1T

∫ T

0(u(Yt )− ET)2dt =

12σ3Y 2(1− Y )2(γ/2)1/2λ1/2 + o(λ1/2) ,

QT := limT→∞

1T

E [〈u(Y )〉T ] = σ4Y 2(1− Y )2(γ/2)λ−1 + o(λ−1)

Price Impact

How big is λ ?• Implied share turnover:

limT→∞

1T

∫ T0 |

u(Yt )Yt|dt = π−1/2σ3/2(1− Y ) (γ/2)1/4

λ−1/4

• Match formula with observed share turnover. Bounds on γ, Y .

Period Volatility Share − log10 λTurnover high low

1926-1929 20% 125% 6.164 4.8631930-1939 30% 39% 3.082 1.7811940-1949 14% 12% 3.085 1.7841950-1959 10% 12% 3.855 2.5541960-1969 10% 15% 4.365 3.0641970-1979 13% 20% 4.107 2.8061980-1989 15% 63% 5.699 4.3981990-1999 13% 95% 6.821 5.5202000-2009 22% 199% 6.708 5.407

• γ = 1, Y = 1/2 (high), and γ = 10, Y = 0 (low).

Price Impact

Welfare• Define welfare loss as decrease in equivalent safe rate due to friction:

LoS =µ2

2γσ2 − ESRγ(u) ∼ σ3√γ

2Y 2(1− Y )2λ1/2

• Zero loss if no trading necessary, i.e. Y ∈ 0,1.• Universal relation:

LoS = πλ |ET|2

Welfare loss equals squared turnover times liquidity, times π.• Compare to proportional transaction costs ε:

LoS =µ2

2γσ2 − ESRγ ≈γσ2

2

(3

4γY 2 (1− Y

)2)2/3

ε2/3

• Universal relation:LoS =

34ε |ET|

Welfare loss equals turnover times spread, times constant 3/4.• Linear effect with transaction costs (price, not quantity).

Quadratic effect with liquidity (price times quantity).

Price Impact

Neither a Borrower nor a Shorter Be

TheoremIf µγσ2 ≤ 0, then Yt = 0 and u = 0 for all t optimal. Equivalent safe rate zero.

If µγσ2 ≥ 1, then Yt = 1 and u = 0 for all t optimal. Equivalent safe rate µ− γ

2σ2.

• If Merton investor shorts, keep all wealth in safe asset, but do not short.• If Merton investor levers, keep all wealth in risky asset, but do not lever.• Portfolio choice for a risk-neutral investor!• Corner solutions. But without constraints?• Intuition: the constraint is that wealth must stay positive.• Positive wealth does not preclude borrowing with block trading,

as in frictionless models and with transaction costs.• Block trading unfeasible with price impact proportional to turnover.

Even in the limit.

Price Impact

Explosion with LeverageTheoremIf Yt that satisfies Y0 ∈ (1,+∞) and

dYt = Yt (1− Yt )(µdt − Ytσ2dt + σdWt ) + ut (1 + λYtut )dt

explodes in finite time with positive probability.

• For y ∈ [1,+∞), drift bounded below by µ(y) := y(1− y)(µ− yσ2)− 14λ .

• Comparison principle: enough to check explosion with lower bound drift.• Scale function s(x) =

∫ xc exp

(−2∫ y

cµ(z)σ2(z)

dz)

dy finite at both 1 and∞.

• For y ∼ 1, µ(y) ∼ − 14λ and σ2(y) ∼ σ2(1− y)2.

For y ∼ ∞, µ(y) ∼ σ2y3 and σ2(y) ∼ σ2y4.• Feller’s test for explosions: check that next integral finite at z =∞.∫ z

cexp

(−∫ x

c

2µ(s)

σ2(s)ds)∫ x

c

exp(∫ y

c2µ(s)

σ2(s)ds)

σ2(y)dy

dx

• This shows that risky weight Y explodes. What about wealth X?

Price Impact

Bankruptcy

• τ explosion time for Yt . Show that Xτ (ω) = 0 on ω ∈ τ < +∞.• By contradiction, suppose Xt (ω) does not hit 0 on [0, τ(ω)].

XT = xe∫ T

0

(Ytµ−λu2

t −σ2

2 Y 2t

)dt+σ

∫ T0 Yt dWt .

• If∫ τ

0 Y 2t dt =∞, then −

∫ τ0σ2

2 Y 2t dt +

∫ τ0 YtσdWt = −∞ by LLN.

• If∫ τ

0 Y 2t dt <∞, define measure dP

dP = exp(−∫ τ

0σ2

2 Y 2t dt +

∫ τ0 YtσdWt

)• In both cases, check that

∫ τ0

(Ytµ− λu2

t)

dt = −∞.

• 0 < ε(ω) < StXt

(ω) < M(ω) on [0, τ(ω)], hence:

limT→τ

∫ T0

(Ytµ− λu2

t)

dt < limT→τ

Mµ

∫ T

0θtdt − λε2

∫ T

0θ2

t dt

=

(lim

T→τ

∫ T

0θ2

t dt)(

Mµ limT→τ

∫ T0 θtdt∫ T0 θ2

t dt− λε2

)= −∞.

Price Impact

Optimality

• Check that Yt ≡ 1 optimal if µγσ2 > 1.

• By Itô’s formula,

X 1−γT = x1−γe(1−γ)

∫ T0 (Ytµ− 1

2 Y 2t σ

2−λu2t )dt+(1−γ)

∫ T0 YtσdWt .

• and hence, for g(y ,u) = yµ− 12 y2γσ2 − λu2,

E [X 1−γT ] = x1−γEP

[e∫ T

0 (1−γ)g(Yt ,ut )dt],

where dPdP = exp

∫ T0 −

12 Y 2

t (1− γ)2σ2dt +∫ T

0 Yt (1− γ)σdWt.• g(y ,u) on [0,1]× R has maximum g(1,0) = µ− 1

2γσ2 at (1,0).

• Since Yt ≡ 1 and ut ≡ 0 is admissible, it is also optimal.

ut austin - portugal lectures on portfolio choice

Economy & Finance