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UT Austin - Portugal Lectures on Portfolio Choice
1. Long Run Investment: Opportunities and Frictions
Paolo Guasoni
Boston University and Dublin City University
July 2-6, 2012
CoLab Mathematics Summer School
2. Outline
• Long Run and Stochastic Investment Opportunities
• High-water Marks and Hedge Fund Fees
• Transaction Costs
• Price Impact
• Open Problems
3. Long Run and Stochastic Investment Opportunities
One
Long Run and Stochastic Investment Opportunities
Based on
Portfolios and Risk Premia for the Long Run
with Scott Robertson
4. Long Run and Stochastic Investment Opportunities
Independent Returns?
• Higher yields tend to be followed by higher long-term returns.
• Should not happen if returns independent!
5. Long Run and Stochastic Investment Opportunities
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.
6. Long Run and Stochastic Investment Opportunities
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
7. Long Run and Stochastic Investment Opportunities
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.
8. Long Run and Stochastic Investment Opportunities
(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.
9. Long Run and Stochastic Investment Opportunities
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
10. Long Run and Stochastic Investment Opportunities
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.
11. Long Run and Stochastic Investment Opportunities
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.
12. Long Run and Stochastic Investment Opportunities
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.
13. Long Run and Stochastic Investment Opportunities
Solution Strategy
• Duality Bound.
• Stationary HJB equation and finite-horizon bounds.
• Criteria for long-run optimality.
14. Long Run and Stochastic Investment Opportunities
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.
15. Long Run and Stochastic Investment Opportunities
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.
16. Long Run and Stochastic Investment Opportunities
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 →∞
17. Long Run and Stochastic Investment Opportunities
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?
18. Long Run and Stochastic Investment Opportunities
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?
19. Long Run and Stochastic Investment Opportunities
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.
20. Long Run and Stochastic Investment Opportunities
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.
21. Long Run and Stochastic Investment Opportunities
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?
22. Long Run and Stochastic Investment Opportunities
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.
23. Long Run and Stochastic Investment Opportunities
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.
24. Long Run and Stochastic Investment Opportunities
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 .
25. Long Run and Stochastic Investment Opportunities
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.
26. Long Run and Stochastic Investment Opportunities
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
27. Long Run and Stochastic Investment Opportunities
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
28. Long Run and Stochastic Investment Opportunities
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.
29. Long Run and Stochastic Investment Opportunities
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.
30. Long Run and Stochastic Investment Opportunities
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
31. Long Run and Stochastic Investment Opportunities
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
32. Long Run and Stochastic Investment Opportunities
Proof of wealth bound (2)
• 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
(µ + pΥ w) Σσ ρdBt
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
33. Long Run and Stochastic Investment Opportunities
Proof of wealth bound (3)
• 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 ).
34. Long Run and Stochastic Investment Opportunities
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
(µ + pΥ w) Σσ ρdBt
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
35. Long Run and Stochastic Investment Opportunities
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.
36. Long Run and Stochastic Investment Opportunities
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
37. Long Run and Stochastic Investment Opportunities
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.
38. Long Run and Stochastic Investment Opportunities
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.
39. Long Run and Stochastic Investment Opportunities
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.
40. High-water Marks and Hedge Fund Fees
Two
High-water Marks and Hedge Fund Fees
Based on
The Incentives of Hedge Fund Fees and High-Water Marks
with Jan Obłoj
41. High-water Marks and Hedge Fund Fees
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.
42. High-water Marks and Hedge Fund Fees
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.
43. High-water Marks and Hedge Fund Fees
High-Water Marks
2.5
2.0
1.5
1.0
0.5
20 40 60 80 100
44. High-water Marks and Hedge Fund Fees
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.
45. High-water Marks and Hedge Fund Fees
Questions
• Portfolio:
Effect of fees and risk-aversion?
• Welfare:
Effect on investors and managers?
• High-Water Mark Contracts:
consistent with any investor’s objective?
46. High-water Marks and Hedge Fund Fees
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.
47. High-water Marks and Hedge Fund Fees
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.
48. High-water Marks and Hedge Fund Fees
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
49. High-water Marks and Hedge Fund Fees
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.
50. High-water Marks and Hedge Fund Fees
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.
51. High-water Marks and Hedge Fund Fees
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.
52. High-water Marks and Hedge Fund Fees
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.
53. High-water Marks and Hedge Fund Fees
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
54. High-water Marks and Hedge Fund Fees
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 )
55. High-water Marks and Hedge Fund Fees
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).
56. High-water Marks and Hedge Fund Fees
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).
57. High-water Marks and Hedge Fund Fees
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 .
58. High-water Marks and Hedge Fund Fees
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) .
59. High-water Marks and Hedge Fund Fees
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∗
60. Transaction Costs
Three
Transaction Costs
Based on
Transaction Costs, Trading Volume, and the Liquidity Premium
with Stefan Gerhold, Johannes Muhle-Karbe, and Walter
Schachermayer
61. 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
62. 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
63. 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
64. 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.
65. 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.
66. 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.
67. 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.
68. 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
.
69. 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.
70. 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
.
72. 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.
73. 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
74. 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.
75. 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].
76. 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
˜
77. 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 .
78. 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.
80. 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 .
81. 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