UT Austin - Portugal Lectures on Portfolio Choice

Long Run Investment: Opportunities and Frictions

Paolo Guasoni

Boston University and Dublin City University

July 2-6, 2012
CoLab 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 on
Portfolios and Risk Premia for the Long Run
with Scott Robertson


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:
π
max E[U(XT )]
π

• Easy for logarithmic utility U(x) = log x.
Myopic portfolio π = Σ−1 µ optimal. Numeraire argument.
• Portfolio does not depend on horizon (even random!), and on the
dynamics 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) = x 1−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
t
• Safe asset St0 = exp 0
r (Ys )ds , d risky assets, k state variables.

dSti
=r (Yt )dt + dRti 1≤i ≤d
Sti
n
dRti =µi (Yt )dt + σij (Yt )dZtj 1≤i ≤d
j=1
k
dYti =bi (Yt )dt + aij (Yt )dWtj 1≤i ≤k
j=1

d Zi, Wj 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 ∈ C γ (E, R), b ∈ C 1,γ (E, Rk ), µ ∈ C 1,γ (E, Rn ), A ∈ C 2,γ (E, Rk ×k ),
Σ ∈ C 2,γ (E, Rn×n ) and Υ ∈ C 2,γ (E, Rn×k ). The symmetric matrices A and Σ
¯
are 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 =Yt dt + σdZt
dYt = − λYt dt + dWt

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

dRt =νYt dt + Yt dZt
dYt =κ(θ − Yt )dt + a Yt dWt

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


(In)Completeness

• Υ Σ−1 Υ: covariance of hedgeable state shocks:
Measures degree of market completeness.
• A = Υ Σ−1 Υ: 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−1 Υ/a2 = ρ ρ.
Equivalent to R 2 of regression of state shocks on returns.


Well Posedness
Assumption
There exists unique solution P (r ,y ) ) r ∈Rn ,y ∈E
to martingale problem:
n+k n+k
1 ˜ ∂2 ˜ ∂ ˜ Σ Υ ˜ µ
L= Ai,j (x) + bi (x) A= b=
2 ∂xi ∂xj ∂xi Υ A b
i,j=1 i=1

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

Deﬁnition
(P x )x∈Rn ×E on (Ω, B) solves martingale problem if, for all x ∈ Rn × E:
• P x (X0 = x) = 1
• P x (Xt ∈ Rn × E, ∀t ≥ 0) = 1
t
• f (Xt ) − f (X0 ) − 0
(Lf )(Xu )du is P x -martingale for all f ∈ C0 (Rn × E)
2


Trading and Payoffs

Deﬁnition
Trading strategy: process (πti )1≤i≤d , adapted to Ft = Bt+ , the right-continuous
t≥0
envelope of the ﬁltration generated by (R, Y ), and R-integrable.

• Investor trades without frictions. Wealth dynamics:

dXtπ
= r (Yt )dt + πt dRt
Xtπ

• In particular, Xtπ ≥ 0 a.s. for all t.
Admissibility implied by R-integrability.


Equivalent Safe Rate
• Maximizing power utility E (XT )1−γ /(1 − γ) equivalent to maximizing the
π
1
certainty equivalent E (XT )1−γ
π 1−γ
.
• Observation: in most models of interest, wealth grows exponentially with
the horizon. And so does the certainty equivalent.
• Example: with r , µ, Σ constant, the certainty equivalent is exactly
1
exp (r + 2γ µ Σ−1 µ)T . Only total Sharpe ratio matters.
• Intuition: an investor with a long horizon should try to maximize the rate at
which the certainty equivalent grows:

1 1
β = max lim inf log E (XT )1−γ
π 1−γ

π T →∞ T
• 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:

1 π
β = max lim inf − log E e−αXT
π T →∞ αT
• 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 Factors
Deﬁnition
Stochastic discount factor: strictly positive adapted M = (Mt )t≥0 , such that:
y
EP Mt Sti Fs = Ms Ss
i
for all 0 ≤ s ≤ t, 0 ≤ i ≤ d

Martingale measure: probability Q, such that Q|Ft and P y |Ft equivalent for all
t ∈ [0, ∞), and discounted prices S i /S 0 Q-martingales for 1 ≤ i ≤ d.

• Martingale measures and stochastic discount factors related by:
dQ t
= exp 0 r (Ys )ds Mt
dP y Ft
• Local martingale property: all stochastic discount factors satisfy
t · ·
Mtη = exp − 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
π η
• For any payoff X = and any discount factor M = MT , E[XM] ≤ x.
XT
Because XM is a local martingale.
• Duality bound for power utility:
γ
1
1−γ
E X 1−γ 1−γ
≤ xE M 1−1/γ

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

1 x 1 M M
− log E e−αX ≤ + E log
α E[M] α E[M] 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 Optimality
Deﬁnition (Power Utility)
An admissible portfolio π is long run optimal if it solves:
1 1
max lim inf log E (XT )1−γ 1−γ
π
π T →∞ T

The risk premia η are long run optimal if they solve:
γ
1 η 1−γ
min lim sup log E (MT )1−1/γ
η T →∞ T

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:
1 π η η
max lim inf − log E e−αXT 1
min lim sup T E MT log MT
π T →∞ T η T →∞


HJB Equation
• V (t, x, y ) depends on time t, wealth x, and state variable y .
• Itô’s formula:
1
dV (t, Xt , Yt ) = Vt dt + Vx dXt + Vy dYt + (Vxx d X t + Vxy d X , Y t + Vyy d Y t )
2
• Vector notation. Vy , Vxy k -vectors. Vyy k × k matrix.
• Wealth dynamics:

dXt = (r + πt µt )Xt dt + Xt πt σt dZt

• Drift reduces to:

1 x2
Vt + xVx r + Vy b + tr(Vyy A) + xπ (µVx + ΥVxy ) + Vxx π Σπ
2 2
• Maximizing over π, the optimal value is:

Vx −1 Vxy −1
π=− Σ µ− Σ Υ
xVxx xVxx
• Second term is new. Interpretation?


Intertemporal Hedging
V V
• π = − xVx Σ−1 µ − Σ−1 Υ xVxy
xx xx

• 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 + 2 π Aπ = − 1 b A−1 b).
1
2
HJB equation becomes:

1 1 Σ−1
Vt + xVx r + Vy b + tr(Vyy A) − (µVx + ΥVxy ) (µVx + ΥVxy ) = 0
2 2 Vxx
• 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 ) = x 1−γ
1−γ v (t, y ).

x 1−γ
Vt = vt Vx = x −γ v Vxx = −γx −γ−1 v
1−γ
x 1−γ x 1−γ
Vxy = x −γ vy Vy = vy Vyy = vyy
1−γ 1−γ
• Optimal portfolio becomes:

1 −1 1 vy
π= Σ µ + Σ−1 Υ
γ γ v
• Plugging in, HJB equation becomes:

1 1−γ
vt + (1 − γ) r + µ Σ−1 µ v + b + Υ Σ−1 µ vy
2γ γ
1 1 − γ vy Υ Σ−1 Υvy
+ tr(vyy A) + =0
2 γ 2v
• 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:
1 1−γ
−β + r + µ Σ−1 µ + b + Υ Σ−1 µ wy
2γ γ
1 1−γ 1−γ
+ tr(wyy A) + wy A− Υ Σ−1 Υ wy = 0
2 2 γ
• And the optimal portfolio:
1 −1 1
π= Σ µ+ 1− γ Σ−1 Υwy
γ
• 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 =νYt dt + Yt dZt
dYt =κ(θ − Yt )dt + ε Yt dWt
• Substitute values in stationary HJB equation:

ν2 1−γ
−β + r + y + κ(θ − y ) + ρενy wy
2γ γ
ε2 ε2 y 2 1−γ 2
+ wyy + (1 − γ) wy 1 − ρ =0
2 2 γ
• Try a linear guess w = λy . Set constant and linear terms to zero.
• System of equations in β and λ:
−β + r + κθλ =0
2
1−γ 2 1−γ ν2
λ2 (1 − γ) 1− ρ +λ νρ − κ + =0
2 γ γ 2γ
• 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.
1
π= (ν + βρε)
γ
• 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:
θν 2 1 νρ
β= 1− 1− ε + O(ε2 )
2γ γ κ

• Correction term changes sign as γ crosses 1.


Martingale Measure
• Many martingale measures. With incomplete market, local martingale
condition does not identify a single measure.
• For any arbitrary k -valued ηt , the process:
· ·
Mt = E − (µ Σ−1 + η Υ Σ−1 )σdZ + η adW
0 0 t

is a local martingale such that MR is also a local martingale.
π
• Recall that MT = yU (XT ).
• If local martingale M is a martingale, it deﬁnes a stochastic discount factor.
1 −1
• π= γΣ (µ + Υ(1 − γ)wy ) yields:
T T
µ− 1 π Σπ)dt−γ
U (XT ) = (XT )−γ =x −γ e−γ
π π 0
(π 2 0
π σdZt

T
+(1−γ)wy Υ )Σ−1 σdZt + T
=e− 0
(µ 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 =σd Zt
˜
dYt =(b − Υ Σ−1 µ + (A − Υ Σ−1 Υ)η)dt + ad Wt

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

˜
dRt =σd Zt
˜
dYt = b − Υ Σ−1 µ + (A − Υ Σ−1 Υ)(1 − γ)wy dt + ad Wt

• 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:

1 1 1
−β + µ Σ−1 µ + b − Υ Σ−1 µ wy + tr(wyy A)− wy A − Υ Σ−1 Υ wy = 0
2 2 2
• And the optimal portfolio:
1 −1
Σ µ − Σ−1 Υwy
xπ =
α
• 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−1 µ − (A − Υ Σ−1 Υ)wy dt + ad Wt


HJB Equation

Assumption
w ∈ C 2 (E, R) and β ∈ R solve the ergodic HJB equation:

1 ¯ 1−γ 1 ¯
r+ µ Σµ + w A − (1 −)Υ ΣΥ w+
2γ 2 γ
1 ¯ 1
w b − (1 − )Υ Σµ + tr AD 2 w = β
γ 2

• 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 Probability
Assumption
ˆ
There exists unique solution (P r ,y )r ∈Rn ,y ∈Rk to to martingale problem

n+k n+k
ˆ 1 ˜ ∂2 ˆ ∂
L= Ai,j (x) + bi (x)
2 ∂xi ∂xj ∂xi
i,j=1 i=1
1
γ (µ + (1 − γ)Υ w)
ˆ
b=
b − (1 − 1 ¯ 1 ¯
Σµ + A − (1 − γ )Υ ΣΥ (1 − γ) w
γ )Υ

ˆ
• Under P, the diffusion has dynamics:

1 ˆ
dRt = γ (µ + (1 − γ)Υ w) dt + σd Zt
1 ¯ 1 ¯ ˆ
dYt = b − (1 − γ )Υ Σµ + (A − (1 − γ )Υ ΣΥ)(1 − γ) w dt + ad Wt

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


Finite horizon bounds

Theorem
Under previous assumptions:

1¯
π = Σ (µ + (1 − γ)Υ w) , η = (1 − γ) w
γ

satisfy the equalities:
1 1
y y 1−γ
EP (XT )1−γ
π 1−γ
= eβT +w(y ) EP e−(1−γ)w(YT )
ˆ
γ γ
γ−1 1−γ 1−γ 1−γ
y y
η
EP (MT ) γ = eβT +w(y ) EP e−
ˆ
γ w(YT )

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

1 y 1−γ
lim sup log EP e− γ w(YT ) = 0
ˆ
T →∞ T
1 y
lim sup log EP e−(1−γ)w(YT ) = 0
ˆ
T →∞ T


Proof of wealth bound (1)
ˆ
dP
dP |Ft , which equals to E(M), where:
• Z = ρW + ρB. Define Dt =
¯
t
Mt = ¯ ¯
−qΥ Σµ + A − qΥ ΣΥ v (a )−1 dWt
0
t
− ¯ ¯
q Σµ + ΣΥ v σ ρdBt
¯
0

• For the portfolio bound, it suffices to show that:
p
(XT ) = ep(βT +w(y )−w(YT )) DT
π

π 1
which is the same as log XT − p log DT = βT + w(y ) − w(YT ).
• The first term on the left-hand side is:
T T
π 1
log XT = r +π µ− π Σπ dt + π σdZt
0 2 0


• Set π = 1 ¯ π
1−p Σ (µ + pΥ w). log XT becomes:

T 1−2p ¯ p2 ¯ 1 p
2
¯
0
r+ 2(1−p) µ Σµ − (1−p)2
µ ΣΥ w − 2 (1−p)2 w Υ ΣΥ w dt
1 T ¯ 1 T ¯ ¯
+ 1−p 0
(µ + pΥ w) ΣσρdWt − 1−p 0
(µ + pΥ w) Σσ ρdBt

• Similarly, log DT /p becomes:

T p ¯ p ¯ p p(2−p) ¯
0
− 2(1−p)2 µ Σµ − (1−p)2
µ ΣΥ w − 2 w A+ (1−p)2
Υ ΣΥ w dt+
T 1 ¯ 1 T ¯ ¯
0
w a+ 1−p (µ + pΥ w) Σσρ dWt + 1−p 0

π
• Subtracting yields for log XT − log DT /p

T 1 ¯ p ¯ p p ¯
0
r+ 2(1−p) µ Σµ + 1−p µ ΣΥ w + 2 w A+ 1−p Υ ΣΥ w dt
T
− 0
w adWt



• Now, Itô’s formula allows to substitute:
T T T
1
− w adWt = w(y ) − w(YT ) + w bdt + tr(AD 2 w)dt
0 0 2 0

• The resulting dt term matches the one in the HJB equation.
π
• log XT − 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:

1 η 1 1
p−1 log MT − p log DT = 1−p (βT + w(y ) − w(YT ))

1 η
• The term p−1 log MT equals to:

1 T 1 ¯ p2 ¯
1−p 0
r + 2 µ Σµ + 2 w A − Υ ΣΥ w dt+
1 T ¯ 1 T ¯ ¯
p−1 0
p w a − (µ + pΥ w) Σσρ dWt + 1−p 0

1 1 η 1
• Subtracting p log DT yields for p−1 log MT − p log DT :

1 T 1 ¯ p ¯ p p ¯
1−p 0
r+ 2(1−p) µ Σµ + 1−p µ ΣΥ w + 2 w A+ 1−p Υ ΣΥ w dt
1 T
− 1−p 0
w adWt


Proof of martingale bound (2)

T
• Replacing again 0
w adWt with Itô’s formula yields:

1 T 1 ¯ p ¯
1−p 0
(r + 2(1−p) µ Σµ + ( 1−p µ ΣΥ + b ) w+
1 p p ¯
2 tr(AD 2 w) + 2 w A+ 1−p Υ ΣΥ w)dt
1
+ 1−p (w(y ) − w(YT ))

• And the integral equals 1
1−p βT by the HJB equation.


Exponential Utility
Theorem
If r = 0 and w solves equation:
1 ¯ 1 ¯ ¯ 1
µ Σµ − w A − Υ ΣΥ w+ w b − Υ Σµ + tr AD 2 w = β
2 2 2
and myopic dynamics is well posed:
ˆ
dRt =σd Zt
¯ ¯ ˆ
dYt = b − Υ Σµ − (A − Υ ΣΥ) w dt + ad Wt

Then for the portfolio and risk premia (π, η) given by:
1
xπ = Σ−1 µ − Σ−1 Υ w η=− w
α
ﬁnite-horizon bounds hold as:
1 y π 1 y
− log EP e−α(XT −x) =βT + log EP ew(y )−w(YT )
ˆ
α α
1 y η 1 y
EP [M log M η ] =βT + EP [w(y ) − w(YT )]
2 α ˆ


Long-Run Optimality

Theorem
If, in addition to the assumptions for ﬁnite-horizon bounds,
ˆ
• the random variables (Yt )t≥0 are P y -tight in E for each y ∈ E;
• supy ∈E (1 − γ)F (y ) < +∞, where F ∈ C(E, R) is deﬁned as:

µ Σ−1 µ (1−γ)2
 r −β+
2γ − 2γ w Υ Σ−1 Υ w e−(1−γ)w γ>1
F = µ Σ−1 µ (1−γ)2 − 1−γ w
 r −β+
2γ − 2 w A − Υ Σ−1 Υ w e γ 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:

1 1 y η 1 1−p y
0 ≤ lim inf log EP (MT )q log EP [(XT )p ]
− π
T →∞ p T T
1 y η 1−p 1 y
≤ lim sup log EP (MT )q − lim inf log EP [(XT )p ]
π
T →∞ pT T →∞ pT

1−p y 1 1 y
= lim sup log EP e− 1−p v (YT ) − lim inf
ˆ log EP e−v (YT )
ˆ
T →∞ pT T →∞ pT

• For p < 0 enough to show lower bound

1 y 1
lim inf log EP exp − 1−p v (YT )
ˆ ≥0
T →∞ T
and upper bound:
1 y
lim supT →∞ T log EP [exp (−v (YT ))] ≤ 0
ˆ

• Lower bound follows from tightness.


Proof of Long-Run Optimality (2)
• For upper bound, set:
1
Lf = f b − qΥ Σ−1 µ + A − qΥ Σ−1 Υ v + 2 tr AD 2 f
• Then, for α ∈ R, the HJB equation implies that L (eαv ) equals to:
1
αeαv v b − qΥ Σ−1 µ + A − qΥ Σ−1 Υ v + 2 tr AD 2 v + 1 α v A v
2
= αeαv 1
2 v (1 + α)A − qΥ Σ−1 Υ v + pβ − pr + q µ Σ−1 µ
2
• Set α = −1, to obtain that:
q q
L e−v = e−v
v Υ Σ−1 Υ v − λ + pr − µ Σ−1 µ
2 2
• The boundedness hypothesis on F allows to conclude that:
y
EP e−v (YT ) ≤ e−v (y ) + (K ∨ 0) T
ˆ

whence
1 y
log EP e−v (YT ) ≤ 0
ˆ lim sup
T →∞ T
• 0 < p < 1 similar. Reverse inequalities for upper and lower bounds.
1
Use α = − 1−p for upper bound.

High-water Marks and Hedge Fund Fees

Two

Based on
The Incentives of Hedge Fund Fees and High-Water Marks
with Jan Obłoj


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 Fees
0 100 100 100 0
1 110 108 108 2
2 100 100 108 2
3 118 116 116 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

2.5

2.0

1.5

1.0

0.5

20 40 60 80 100


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:

1 µ
π=
γ ∗ σ2
γ ∗ =(1 − α)γ + α
γ =Manager’s Risk Aversion
α =Performance Fee (e.g. 20%)

• Manager behaves as if owned fund, but were more myopic (γ ∗ weighted
average 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:

1 µ
π=
γ ∗ σ2


Price Dynamics

dSt
= (r + µ)dt + σdWt (Risky Asset)
St
dSt α ∗
dXt = (r − ϕ)Xt dt + Xt πt St − rdt − 1−α dXt (Fund)
α
dFt = rFt dt + ϕXt dt + dX ∗ (Fees)
1−α t
Xt∗ = max Xs (High-Water Mark)
0≤s≤t

• 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
dSt α
dXt = Xt πt − ϕXt dt − dX ∗
St 1−α t
• Yes. Pathwise construction.

Proposition
∗
The unique solution is Xt = eRt −αRt , where:
t t
σ2 2
Rt = µπs − π − ϕ ds + σ πs dWs
0 2 s 0

is the cumulative log return.


Fund Value Explicit

Lemma
Let Y be a continuous process, and α > 0.
Then Yt + 1−α Yt∗ = Rt if and only if Yt = Rt − αRt∗ .
α

Proof.
Follows from:

α α 1
Rt∗ = sup Ys + sup Yu = Yt∗ + Yt∗ = Y∗
s≤t 1 − α u≤s 1−α 1−α t

• Apply Lemma to cumulative log return.


Long Horizon

• The manager chooses the portfolio π which maximizes expected power
utility from fees at a long horizon.
• Maximizes the long-run objective:

1 p
max lim log E[FT ] = β
π T →∞ pT
• Dumas and Luciano (1991), Grossman and Vila (1992), Grossman and
Zhou (1993), Cvitanic and Karatzas (1995). Risk-Sensitive Control:
Bielecki and Pliska (1999) and many others.
2
1 µ
• β=r+ γ 2σ 2 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:
α
Ft = (X ∗ − X0 )
∗
1−α t
• Thus, the manager’s objective is equivalent to:

1 ∗
max lim log E[(XT )p ]
π T →∞ pT

• Finite-horizon value function:

1
V (x, z, t) = sup E[XT p |Xt = x, Xt∗ = z]
∗
π p
1
dV (Xt , Xt∗ , t) = Vt dt + Vx dXt + Vxx d X t + Vz dXt∗
2
2
= Vt dt + Vz − α
1−α Vx dXt∗ + Vx Xt (πt µ − ϕ)dt + Vxx σ πt2 Xt2 dt
2


Dynamic Programming
• Hamilton-Jacobi-Bellman equation:
 2
Vt + supπ xVx (πµ − ϕ) + Vxx σ π 2 x 2 ) x <z

 2

α
Vz = 1−α Vx x =z

p
V = z /p

 x =0

V = z p /p t =T


• Maximize in π, and use homogeneity
V (x, z, t) = z p /pV (x/z, 1, t) = z p /pu(x/z, 1, t).
 2
ut − ϕxux − µ22 ux = 0 x ∈ (0, 1)
2σ uxx



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−pβt w(x), forgetting the terminal
condition: 2
µ2 wx
−pβw − ϕxwx − 2σ2 wxx = 0 for x < 1
wx (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.
1−α µ2
• w(x) = x p(1−α) , for β = (1−α)γ+α 2σ 2 − ϕ(1 − α).


Veriﬁcation

Theorem
µ2 1 1
If ϕ − r < 2σ 2
min γ∗ , γ∗2 , then for any portfolio π:

1 π p 1 µ2
lim log E (FT ) ≤ max (1 − α) + r − ϕ ,r
T ↑∞ pT γ ∗ 2σ 2
2
α 1 µ
Under the nondegeneracy condition ϕ + 1−α r < γ∗ 2σ 2 , the unique optimal
µ
solution is π = γ1∗ σ2 .
ˆ

• 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:

T σ2 2 T ˜
RT = − π dt
0 2 t
+ 0
σπt d Wt

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

∗ ∗ µ ˜ µ2
E[(XT )p ] = E[ep(1−α)RT ] = EQ ep(1−α)RT e σ WT − 2σ2 T
π

• For δ > 1, Hölder’s inequality:
δ−1
δ µ ˜ µ2 δ
µ2
σ WT − 2σ 2
µ ˜ 1 T
∗ ∗
σ WT − 2σ 2 T
p(1−α)RT δp(1−α)RT δ δ−1
EQ e e ≤ EQ e EQ e

1 µ2
• Second term exponential normal moment. Just e δ−1 2σ2 T .


Upper Bound (2)
∗
• Estimate EQ eδp(1−α)RT .
• 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:

∗ ∗ 1
EQ eδp(1−α)RT ≤ EQ eδp(1−α)R∞ =
1 − δp(1 − α)

• In summary, for 1 < δ < 1
p(1−α) :

1 π p 1 µ2
lim log E (FT ) ≤
T ↑∞ pT p(δ − 1) 2σ 2

• Thesis follows as δ → 1
p(1−α) .


High-Water Marks and Drawdowns
• Imagine fund’s assets Xt and manager’s fees Ft in the same account
Ct = Xt + Ft .
dSt
dCt = (Ct − Ft )πt
St
• Fees Ft proportional to high-water mark Xt∗ :
α
Ft = (X ∗ − X0 )
∗
1−α t
• Account increase dCt∗ as fund increase plus fees increase:
t t
α 1
Ct∗ − C0 =
∗ ∗
(dXs + dFs ) = ∗
+ 1 dXs = (X ∗ − X0 )
0 0 1−α 1−α t
• Obvious bound Ct ≥ Ft yields:

Ct ≥ α(Ct∗ − X0 )
• X0 negligible as t ↑ ∞. Approximate drawdown constraint.

Ct ≥ αCt∗

Transaction Costs

Three

Transaction Costs
Based on
Transaction Costs, Trading Volume, and the Liquidity Premium
with Stefan Gerhold, Johannes Muhle-Karbe, and Walter
Schachermayer

Transaction Costs

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 equivalent safe rate:
1
1 1−γ 1−γ
max lim log E XT
π T →∞ T

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 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 dϕ↑ + (1 − ε) 0 dϕ↓
t
St0 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

Transaction Costs

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

Transaction Costs

No Trade Region

Vx 1
• When 1 ≤ Vy ≤ 1−ε does not bind, drift is zero:

σ2 2 Vx 1
Vt + rXt0 Vx + (µ + r )Xt Vy + X Vyy = 0 if 1 < < .
2 t Vy 1−ε

• This is the no-trade region.
• Long-run guess:

V (t, Xt0 , Xt ) = (Xt0 )1−γ v (Xt /Xt0 )e−(1−γ)(β+r )t
(1−γ)v (z) 1
• Set z = y /x. For 1 + z < v (z) < 1−ε + z, HJB equation is
2
σ 2
z v (z) + µzv (z) − (1 − γ)βv (z) = 0
2
• Linear second order ODE. But β unknown.

Transaction Costs

Smooth Pasting
(1−γ)v (z) 1
• Suppose 1 + z < v (z) < 1−ε + z same as l ≤ z ≤ u.
• 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.

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:
2 2
l 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.

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

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.
µ±λ
• With this notation, buy and sell boundaries are π± = γσ 2
.

Transaction Costs

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.

Transaction Costs

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 second 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
2
u(λ) 1 π+ (1−π− ) 1 (µ+λ)(µ−λ−γσ )
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, ·).
µ+λ
• λ identiﬁed by second boundary w(λ, log(u(λ)/l(λ))) = γσ 2
.

Transaction Costs

Explicit Solutions

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

a(λ) tan[tan−1 ( b(λ) ) + a(λ)x] + ( σ2 − 2 )
a(λ)
µ 1
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

• 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, Xt0 − δ St , Xt + δSt ) ≤ V (t, Xt0 , Xt )
˜
• Expanding for small δ, −δVx St + δVy St ≤ 0
˜
• Must hold for δ positive and negative. Guess for St :

˜ Vy
St = St
Vx
log(Xt /lXt0 )
• Plugging guess V (t, Xt0 , Xt ) = e−(1−γ)(r +β)t (Xt0 )1−γ e(1−γ) 0
w(y )dy
,

˜ w(Yt )
St = St
leYt (1
− w(Yt ))
• Shadow portfolio weight is precisely w:
w(Yt )
˜
ϕt St 1−w(Yt )
πt =
˜ = = w(Yt ),
˜
ϕ0 St0 + ϕt St 1 w(Yt )
+ 1−w(Yt )
t

Transaction Costs

Shadow Value Function
˜ ˜
• In terms of shadow wealth Xt = ϕ0 St0 + ϕt St , the value function becomes:
t

1−γ
Xt0 Yt
V (t, Xt , Yt ) = V (t, Xt0 , Xt ) = e−(1−γ)(r +β)t Xt1−γ
˜ ˜ ˜ e(1−γ) 0
w(y )dy
.
˜
Xt
˜ ˜
• Note that Xt0 /Xt = 1 − w(Yt ) by deﬁnition of St , and rewrite as:

1−γ
Xt0 Yt
w(y )dy Yt
e(1−γ) 0 = exp (1 − γ) log(1 − w(Yt )) + 0
w(y )dy
˜
Xt
Yt w (y )
= (1 − w(0))γ−1 exp (1 − γ) 0
w(y ) − 1−w(y ) dy .

˜
• Set w = w − w
1−w to obtain
Yt
V (t, Xt , Yt ) = e−(1−γ)(r +β)t Xt1−γ e(1−γ)
˜ ˜ ˜ 0
˜
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 − σ 2 /2)dt + σdWt + dLt − dUt , Y0 ∈ [0, log(u/l)],

Lemma
˜ w(Y )
The dynamics of S = S leY (1−w(Y )) is given by

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

where µ(·) and σ (·) are defined as
˜ ˜

σ 2 w (y ) w (y ) σw (y )
µ(y ) =
˜ − (1 − γ)w(y ) , σ (y ) =
˜ .
w(y )(1 − w(y )) 1 − w(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

Theorem
˜
The shadow payoff XT of the portfolio π = w(Yt ) and the shadow discount
˜
· µ
˜ y
˜
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 ˜ ˜
E MT =e(1−γ)βT E e( γ −1)(q (Y0 )−q (YT ))
ˆ .

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
˜

Transaction Costs

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

˜ 1−γ = exp (1 − γ) T σ2 2
˜ T
XT 0
µπ −
˜˜ 2 π
˜ dt + (1 − γ) 0
σ π dWt .
˜˜

• Hence:

ˆ T 2
˜ 1−γ = d P exp σ2 2
˜ 1 µ
˜
XT dP 0
(1 − γ) µπ −
˜˜ 2 π
˜ + 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
˜
˜ ˜
1µ˜2 2
• First integrand is 2 σ2
˜
+ γ σ π 2 − γ µπ , which equals to
˜
2 ˜ ˜˜
2 2
1−γ µ
˜
(1 − γ)2 σ w 2 +
2
˜ 2γ σ
˜
˜
+ σ(1 − γ)w .

Transaction Costs

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/l)UT
T σ2 2
σ 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
σ2 ˜ 2
σ2 1 µ
˜
as 2 w + (1 − γ) σ w 2 + µ −
2
˜ 2
˜
w+ 2γ σ
˜
˜
+ σ(1 − γ)w = β.
• First bound follows.

Transaction Costs

Second Bound
• Argument for second bound similar.
· µ
˜ ˆ
• Discount factor MT = E(− ˜ dW )T ,
0 σ
myopic probability P satisfy:
1
1− γ 1−γ T µ
˜ 1−γ T µ2
˜
MT = exp γ ˜ dW
0 σ
+ 2γ 0 σ2
˜
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
µ2
˜ 1−γ µ
˜ 1 µ
˜
• Note σ2
˜
+ γ σ
˜
˜
+ σw = (1 − γ)σ 2 w 2 +
˜ γ σ
˜
˜
+ σ(1 − γ)w
T T σ2 σ2 ˜
• Plug 0
˜ ˜ ˜
σ wdWt = q (YT ) − q (Y0 ) − 0
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 .

Transaction Costs

Buy and Sell at Boundaries
Lemma
˜ ˜
For 0 < µ/γσ 2 = 1, the number of shares ϕt = w(Yt )Xt /St follows:

dϕt µ−λ µ+λ
= 1− dLt − 1 − dUt .
ϕt γσ 2 γσ 2

˜
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)σ 2 w (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 w (Yt )
= d(Lt − Ut ).
ϕt w(Yt )
• Since Lt and Ut only increase (decrease for µ/γσ 2 > 1) on {Yt = 0} and
{Yt = log(u/l)}, claim follows from boundary conditions for w and w .

Transaction Costs

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

µ2 − λ2
ESR = r +
2γσ 2
• 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)

µ−λ µ+λ
π− = , π+ = ,
γσ 2 γσ 2

Transaction Costs

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:
2
µ+λ u(λ) 1 (µ+λ)(µ−λ−γσ )
w(log(u(λ)/l(λ))) = γσ 2
, where l(λ) = 1−ε (µ−λ)(µ+λ−γσ 2 ) .

• Asymptotic expansion:

1/3
3 2
λ = γσ 2 4γ π∗
2
(1 − π∗ ) ε1/3 + O(ε).

Transaction Costs

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+
ShTu = lim |ϕt | = σ2
−1 2µ − 2µ .
T →∞ T 0 2
(u/l) σ2
−1
−1
1−
(u/l) σ2 −1

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

1 T (1−ε)St dϕ↓ T St dϕ↑
WeTu = 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

Transaction Costs

Asymptotics

1/3
3 2 2
π± = π∗ ± π (1 − π∗ ) ε1/3 + O(ε).
4γ ∗
2/3
µ2 γσ 2 3 2 2
ESR = r + − π (1 − π∗ ) ε2/3 + O(ε4/3 ).
2γσ 2 2 4γ ∗
2/3
µ 3 2 2
LiPr = π (1 − π∗ ) ε2/3 + O(ε4/3 ).
2π∗2 4γ ∗
−1/3
σ2 3 2 2
ShTu = (1 − π∗ )2 π∗ π (1 − π∗ ) ε−1/3 + O(ε1/3 )
2 4γ ∗
2/3
γσ 2 3 2
WeTu = π (1 − π∗ )2 ε−1/3 + O(ε).
3 4γ ∗

Transaction Costs

Welfare, Volume, and Spread

• Liquidity premium and share turnover:

LiPr 3
= ε + O(ε5/3 )
ShTu 4
• Equivalent safe rate and wealth turnover:

µ2
(r + γσ 2
) − ESR 3
= ε + O(ε5/3 ).
WeTu 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.

Price Impact

Four

Price Impact
Based on
Dynamic 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
= µdt + σdWt
St
• Trade ∆θ shares over time interval ∆t. Order filled at price

˜ St ∆θ
St (∆θ) := St 1+λ
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), Schied
and Shoneborn (2009), Garleanu and Pedersen (2011)

˜ ∆θ
St (∆θ) := St + λ
∆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.

˜ ∆θ
St (∆θ) := St 1+λ
θt ∆t

• Problematic. Inﬁnite price impact with cash position.

Price Impact

Wealth and Portfolio
˜ ˙ ˙t
• Continuous trading: execution price St (θt ) = St 1 + λ θXSt , cash position
t

˙t ˙ ˙
dCt = −St 1 + λ θXSt dθt = −St θt + λ Stt θt2 dt
t X

˙
θt St
• Trading volume as wealth turnover ut := . Xt
Amount traded in unit of time, as fraction of wealth.
θt St
• Dynamics for wealth Xt := θt St + Ct and risky portfolio weight Yt := Xt

dXt
= Yt (µdt + σdWt ) − λut2 dt
Xt
dYt = (Yt (1 − Yt )(µ − Yt σ 2 ) + (ut + λYt ut2 ))dt + σYt (1 − Yt )dWt

• Illiquidity...
• ...reduces portfolio return (−λut2 ).
Turnover effect quadratic: quantities times price impact.
• ...increases risky weight (λYt ut2 ). Buy: pay more cash. Sell: get less cash.
Turnover effect linear in risky weight Yt . Vanishes for cash position.

Price Impact

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

dXt
= Yt (µdt + σdWt ) − λut2 dt
Xt
dYt = (Yt (1 − Yt )(µ − Yt σ 2 ) + (ut + λYt ut2 ))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.
µ
• Frictionless solution Yt = γσ 2
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:
1
1 1−γ 1−γ
max lim log E XT
u T →∞ T

• 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 weight
Yt , and (3) calendar time t.
˙
θt St
• Evolution for ﬁxed trading strategy u = Xt :

dv (Xt , Yt , t) =vt dt + vx (µXt Yt − λXt ut2 )dt + vx Xt Yt σdWt
+vy (Yt (1 − Yt )(µ − Yt σ 2 ) + ut + λYt ut2 )dt + vy Yt (1 − Yt )σdWt
σ2 σ2
+ vxx Xt2 Yt2 + vyy Yt2 (1 − Yt )2 + σ 2 vxy Xt Yt2 (1 − Yt ) dt
2 2

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

vt + max vx µxy − λxu 2 + vy y (1 − y )(µ − σ 2 y ) + u + λyu 2
u
σ2 y 2
+ vxx x 2 + vyy (1 − y )2 + 2vxy x(1 − y ) =0
2

Price Impact

Homogeneity and Long-Run
• Homogeneity in wealth v (t, x, y ) = x 1−γ v (t, 1, y ).
• Guess long-term growth at equivalent safe rate β, to be found.
• Substitution v (t, x, y ) = x 1−γ (1−γ)(β(T −t)+ y
q(z)dz)
1−γ e reduces HJB equation

2
−β + maxu µy − γ σ y 2 − λu 2 + q(y (1 − y )(µ − γσ 2 y ) + u + λyu 2 )
2
2
+ σ y 2 (1 − y )2 (q + (1 − γ)q 2 ) = 0.
2

q(y )
• Maximum for u(y ) = 2λ(1−yq(y )) .
• Plugging yields
2 2 2
q
µy −γ σ y 2 +y (1−y )(µ−γσ 2 y )q+ 4λ(1−yq) + σ y 2 (1−y )2 (q +(1−γ)q 2 ) = β
2 2

µ2 µ
• β= 2γσ 2
, q = 0, y = γσ 2
corresponds to Merton solution.
• Classical model as a singular limit.

Price Impact

Asymptotics

µ2
• Expand equivalent safe rate as β = 2γσ 2
− 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
2 2 2
q
−β+µy −γ σ y 2 +y (1−y )(µ−γσ 2 y )q+ 4λ(1−yq) + σ y 2 (1−y )2 (q +(1−γ)q 2 ) =
2 2

• Yields
q (1) (y ) = σ −1 (γ/2)−1/2 (µ − γσ 2 y )
q(y )
• Turnover follows through u(y ) = 2λ(1−yq(y )) .
¯
• Plug q (1) (y ) back in equation, and set y = Y .
¯ ¯
• Yields welfare loss c(λ) = σ 3 (γ/2)−1/2 Y 2 (1 − Y )2 λ1/2 + 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
Theorem
µ
If γσ 2
∈ (0, 1), then the optimal wealth turnover and equivalent safe rate are:

1 q(y )
ˆ
u (y ) = ˆ
ESRγ (u ) = β
2λ 1 − yq(y )
2
µ
where β ∈ (0, 2γσ2 ) and q : [0, 1] → R are the unique pair that solves the ODE

2 2 2
q
−β+µy −γ σ y 2 +y (1−y )(µ−γσ 2 y )q+ 4λ(1−yq) + σ y 2 (1−y )2 (q +(1−γ)q 2 ) = 0
2 2

√
and q(0) = 2 λβ, q(1) = λd − λd(λd − 2), where d = −γσ 2 − 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
Theorem
µ
Assume 0 < γσ2 < 1. There exists β ∗ such that HJB equation has solution
q(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 with
√
positive finite limit in 0 (and the limit is 2 λβ);
γσ 2
• for β > µ − 2 ,
there exists a unique solution q1,β (y ) to HJB equation
with negative finite limit in 1 (and the limit is λd − λd(λd − 2), where
γσ 2
d := 2(µ − 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)
Lemma
y ˆ
Define Q(y ) = q(z)dz. There exists a probability P, equivalent to P, such
that the terminal wealth XT of any admissible strategy satisfies:
1−γ 1 1
E[XT ] 1−γ ≤ eβT +Q(y ) EP [e−(1−γ)Q(YT ) ] 1−γ
ˆ (1)

and equality holds for the optimal strategy.
ˆ
• Define probability P as:
ˆ
dP T (1−γ)2 Ys (1+q(Ys )(1−Ys ))2 σ 2
2
T
dP
= exp 0
− 2
ds + 0
(1 − γ)Ys (1 + q(Ys )(1 − Ys ))σdWs

• It suffices to check that
1 ˆ
dP
log XT − 1−γ
log dP
≤ βT − Q(YT ) + Q(y ) .
• By self-financing condition, and Itô formula:
T Yt2 (1−Yt )2 σ 2
Q(YT ) − Q(y ) = 0
q(Yt )(Yt (1 − Yt )(µ − Yt σ 2 ) + ut + λYt ut2 ) + q (Yt ) 2
dt
T
+ q(Yt )Yt (1 − Yt )σdWt
0

Price Impact

Finite Horizon Bound (2)
• Also,
T T
Yt2 σ 2
log XT = Yt µ − λut2 − dt + Yt σdWt .
0 2 0

1 ˆ
• Plugging equalities into log XT − 1−γ log d P , and using HJB equation:
dP

T Yt2 σ 2 σ 2 (1−γ)Yt2 (1+q(Yt )(1−Yt ))2 T
0
µYt − λut2 − 2
+ 2
dt − 0
q(Yt )Yt (1 − Yt )σdWt ≤
T 2 σ2
≤ 0
β − q(Yt )(Yt (1 − Yt )(µ − Yt σ ) + ut + λYt ut2 ) + 2
q (Yt )Yt2 (1 − Yt )2 dt
T
− q(Yt )Yt (1 − Yt )σdWt .
0

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

2
2 2 (1−γ)y 2 (1+q(y )(1−y ))2 2
y µ−λu 2 − y 2 + σ
σ
2
≤ β−q(y )(y (1−y )(µ−y σ 2 )+u+λyu 2 )−q (y ) y (1−y
2

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

Price Impact

Asymptotics

Theorem
¯
Y = µ
∈ (0, 1). Asymptotic expansions for turnover and equivalent safe rate:
γσ 2

γ ¯
ˆ
u (y ) = σ (Y − y ) + O(1)
2λ
µ2 γ ¯2 ¯
ˆ
ESRγ (u ) = 2
− σ3 Y (1 − Y )2 λ1/2 + O(λ)
2γσ 2

Long-term average of (unsigned) turnover
1 T ¯ ¯ 1/4
|ET| = lim |u (Yt )|dt = π −1/2 σ 3/2 Y (1 − Y ) (γ/2) λ−1/4 + O(λ1/2 )
ˆ
T →∞ T 0

• Implications?

Price Impact

Turnover

• Turnover:
γ ¯
ˆ
u (y ) ∼ σ (Y − y )
2λ
¯ ¯ ¯
• 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:

γ 2 ¯2 ¯ γ ¯ ¯
ˆ
d ut = σ (σ Y (1 − Y )(1 − γ) − ut )dt − σ 2
ˆ Y (1 − Y )dWt
2λ 2λ
• In the following sense:

Theorem
ˆ
The process ut has asymptotic moments:
T
1 ¯ ¯
ET := lim u (Yt )dt = σ 2 Y 2 (1 − Y )(1 − γ) + o(1) ,
ˆ
T →∞ T 0
1 T 1 ¯ ¯
VT := lim (u (Yt ) − ET)2 dt = σ 3 Y 2 (1 − Y )2 (γ/2)1/2 λ1/2 + o(λ1/2 ) ,
ˆ
T →∞ T 0 2
1 ¯ ¯
QT := lim E[ u (Y ) T ] = σ 4 Y 2 (1 − Y )2 (γ/2)λ−1 + o(λ−1 )
ˆ
T →∞ T

Price Impact

How big is λ ?
• Implied share turnover:
T 1/4
lim 1 | u(Yt ) |dt = π −1/2 σ 3/2 (1 − Y ) (γ/2)
¯ λ−1/4
T →∞ T 0 Yt

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

Period Volatility Share − log10 λ
Turnover high low
1926-1929 20% 125% 6.164 4.863
1930-1939 30% 39% 3.082 1.781
1940-1949 14% 12% 3.085 1.784
1950-1959 10% 12% 3.855 2.554
1960-1969 10% 15% 4.365 3.064
1970-1979 13% 20% 4.107 2.806
1980-1989 15% 63% 5.699 4.398
1990-1999 13% 95% 6.821 5.520
2000-2009 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:

µ2 γ ¯2 ¯
LoS = − ESRγ (u ) ∼ σ 3
ˆ Y (1 − Y )2 λ1/2
2γσ 2 2
¯
• Zero loss if no trading necessary, i.e. Y ∈ {0, 1}.
• Universal relation:
2
LoS = πλ |ET|
Welfare loss equals squared turnover times liquidity, times π.
• Compare to proportional transaction costs ε:
2/3
µ2 γσ 2 3 ¯2 ¯ 2
LoS = − ESRγ ≈ Y 1−Y ε2/3
2γσ 2 2 4γ
• Universal relation:
3
LoS = ε |ET|
4
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

Theorem
µ
If γσ 2
ˆ
≤ 0, then Yt = 0 and u = 0 for all t optimal. Equivalent safe rate zero.
µ
If γσ 2
≥ 1, then Yt = 1 and u = 0 for all t optimal. Equivalent safe rate µ − γ σ 2 .
ˆ 2

• 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 Leverage
Theorem
If Yt that satisfies Y0 ∈ (1, +∞) and

dYt = Yt (1 − Yt )(µdt − Yt σ 2 dt + σdWt ) + ut (1 + λYt ut )dt

explodes in finite time with positive probability.
1
• For y ∈ [1, +∞), drift bounded below by µ(y ) := y (1 − y )(µ − y σ 2 ) − 4λ .
˜
• Comparison principle: enough to check explosion with lower bound drift.
x y µ(z)
˜
• Scale function s(x) = c
exp −2 c σ 2 (z)
dz dy finite at both 1 and ∞.
1
• For y ∼ 1, µ(y ) ∼
˜ σ (y ) ∼ σ (1 − y )2 .
− 4λ and
˜ 2 2

For y ∼ ∞, µ(y ) ∼ σ y and σ 2 (y ) ∼ σ 2 y 4 .
˜ 2 3
˜
• Feller’s test for explosions: check that next integral finite at z = ∞.
y 2µ(s)
 ˜

z x
2˜(s)
µ x exp c σ 2 (s)
ds
exp − ds  dy  dx
c c σ 2 (s) c σ 2 (y )

• 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)].

T 2 T
Yt µ−λut2 − σ Yt2 dt+σ Yt dWt
XT = xe 0 2 0
.
τ τ σ2 2 τ
• If 0
Yt2 dt = ∞, then − Y dt
0 2 t
+ 0
Yt σdWt = −∞ by LLN.
τ ¯
dP τ σ2 2 τ
• If 0
Yt2 dt < ∞, deﬁne measure dP = exp − Y dt
0 2 t
+ 0
Yt σdWt
τ
• In both cases, check that 0
Yt µ − λut2 dt = −∞.
St
• 0 < (ω) < Xt (ω) < M(ω) on [0, τ (ω)], hence:
T T
T 2
limT →τ 0
Yt µ − λut2 dt < lim Mµ θt dt − λ θt2 dt
˙
T →τ 0 0
T T
θt dt
= lim θt2 dt
˙ Mµ lim 0
T
−λ 2
= −∞.
T →τ 0 T →τ ˙
θ2 dt
0 t

Price Impact

Optimality

µ
• Check that Yt ≡ 1 optimal if γσ 2
> 1.
• By Itô’s formula,
T T
1−γ
XT = x 1−γ e(1−γ) 0 (Yt µ− 1 Yt2 σ2 −λut2 )dt+(1−γ)
2 0
Yt σdWt
.

• and hence, for g(y , u) = y µ − 1 y 2 γσ 2 − λu 2 ,
2

1−γ T
E[XT ] = x 1−γ EP e
ˆ 0
(1−γ)g(Yt ,ut )dt
,

ˆ
dP T T
where dP = exp{ 0
− 1 Yt2 (1 − γ)2 σ 2 dt +
2 0
Yt (1 − γ)σdWt }.
• g(y , u) on [0, 1] × R has maximum g(1, 0) = µ − 1 γσ 2 at (1, 0).
2
• Since Yt ≡ 1 and ut ≡ 0 is admissible, it is also optimal.

UT Austin - Portugal Lectures on Portfolio Choice

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UT Austin - Portugal Lectures on Portfolio Choice