In a market with price-impact proportional to a power of the order flow, we derive optimal trading policies and their implied welfare for long-term investors with constant relative risk aversion, who trade one safe asset and one risky asset that follows geometric Brownian motion. These quantities admit asymptotic explicit formulas up to a structural constant that depends only on the price-impact exponent. Trading rates are finite as with linear impact, but they are lower near the target portfolio, and higher away from the target. The model nests the square-root impact law and, as extreme cases, linear impact and proportional transaction costs.
1. Model Results Heuristics
Nonlinear Price Impact and Portfolio Choice
Paolo Guasoni1,2
Marko Weber2,3
Boston University1
Dublin City University2
Scuola Normale Superiore3
Financial Mathematics Seminar
Princeton University, February 26th
, 2015
2. Model Results Heuristics
Outline
ā¢ Motivation:
Optimal Rebalancing and Execution.
ā¢ Model:
Nonlinear Price Impact.
Constant investment opportunities and risk aversion.
ā¢ Results:
Optimal policy and welfare. Implications.
3. Model Results Heuristics
Price Impact and Market Frictions
ā¢ Classical theory: no price impact.
Same price for any quantity bought or sold.
Merton (1969) and many others.
ā¢ Bid-ask spread: constant (proportional) āimpactā.
Price depends only on sign of trade.
Constantinides (1985), Davis and Norman (1990), and extensions.
ā¢ Price linear in trading rate.
Asymmetric information equilibria (Kyle, 1985), (Back, 1992).
Quadratic transaction costs (Garleanu and Pedersen, 2013)
ā¢ Price nonlinear in trading rate.
Square-root rule: Loeb (1983), BARRA (1997), Grinold and Kahn (2000).
Empirical evidence: Hasbrouck and Seppi (2001), Plerou et al. (2002),
Lillo et al. (2003), Almgren et al. (2005).
ā¢ Literature on nonlinear impact focuses on optimal execution.
Portfolio choice?
4. Model Results Heuristics
Portfolio Choice with Frictions
ā¢ With constant investment opportunities and constant relative risk aversion:
ā¢ Classical theory: hold portfolio weights constant at Merton target.
ā¢ Proportional bid-ask spreads:
hold portfolio weight within buy and sell boundaries (no-trade region).
ā¢ Linear impact:
trading rate proportional to distance from target.
ā¢ Rebalancing rule for nonlinear impact?
5. Model Results Heuristics
This Talk
ā¢ Inputs
ā¢ Price exogenous. Geometric Brownian Motion.
ā¢ Constant relative risk aversion and long horizon.
ā¢ Nonlinear price impact:
trading rate one-percent highers means impact Ī±-percent higher.
ā¢ Outputs
ā¢ Optimal trading policy and welfare.
ā¢ High liquidity asymptotics.
ā¢ Linear impact and bid-ask spreads as extreme cases.
ā¢ Focus is on temporary price impact:
ā¢ No permanent impact as in Huberman and Stanzl (2004)
ā¢ No transient impact as in Obizhaeva and Wang (2006) or Gatheral (2010).
6. Model Results Heuristics
Market
ā¢ Brownian Motion (Wt )tā„0 with natural ļ¬ltration (Ft )tā„0.
ā¢ Best quoted price of risky asset. Price for an inļ¬nitesimal trade.
dSt
St
= Āµdt + ĻdWt
ā¢ Trade āĪø shares over time interval āt. Order ļ¬lled at price
ĖSt (āĪø) := St 1 + Ī»
St āĪøt
Xt āt
Ī±
sgn( ĖĪø)
where Xt is investorās wealth.
ā¢ Ī» measures illiquidity. 1/Ī» market depth. Like Kyleās (1985) lambda.
ā¢ Price worse for larger quantity |āĪø| or shorter execution time āt.
Price linear in quantity, inversely proportional to execution time.
ā¢ Impact of dollar trade St āĪø declines as large investorās wealth increases.
ā¢ Makes model scale-invariant.
Doubling wealth, and all subsequent trades, doubles ļ¬nal payoff exactly.
7. Model Results Heuristics
Alternatives?
ā¢ Alternatives: quantities āĪø, or share turnover āĪø/Īø. Consequences?
ā¢ Quantities (āĪø):
Bertsimas and Lo (1998), Almgren and Chriss (2000), Schied and
Shoneborn (2009), Garleanu and Pedersen (2011)
ĖSt (āĪø) := St + Ī»
āĪø
āt
ā¢ Price impact independent of price. Not invariant to stock splits!
ā¢ Suitable for short horizons (liquidation) or mean-variance criteria.
ā¢ Share turnover:
Stationary measure of trading volume (Lo and Wang, 2000). Observable.
ĖSt (āĪø) := St 1 + Ī»
āĪø
Īøt āt
ā¢ Problematic. Inļ¬nite price impact with cash position.
8. Model Results Heuristics
Wealth and Portfolio
ā¢ Continuous time: cash position
dCt = āSt 1 + Ī»
ĖĪøt St
Xt
Ī±
sgn( ĖĪø) dĪøt = ā St
ĖĪøt
Xt
+ Ī»
ĖĪøt St
Xt
1+Ī±
Xt dt
ā¢ Trading volume as wealth turnover ut :=
ĖĪøt St
Xt
.
Amount traded in unit of time, as fraction of wealth.
ā¢ Dynamics for wealth Xt := Īøt St + Ct and risky portfolio weight Yt := Īøt St
Xt
dXt
Xt
= Yt (Āµdt + ĻdWt ) ā Ī»|ut |1+Ī±
dt
dYt = (Yt (1 ā Yt )(Āµ ā Yt Ļ2
) + (ut + Ī»Yt |ut |1+Ī±
))dt + ĻYt (1 ā Yt )dWt
ā¢ Illiquidity...
ā¢ ...reduces portfolio return (āĪ»u1+Ī±
t ).
Turnover effect quadratic: quantities times price impact.
ā¢ ...increases risky weight (Ī»Yt u1+Ī±
t ). Buy: pay more cash. Sell: get less.
Turnover effect linear in risky weight Yt . Vanishes for cash position.
9. Model Results Heuristics
Admissible Strategies
Deļ¬nition
Admissible strategy: process (ut )tā„0, adapted to Ft , such that system
dXt
Xt
= Yt (Āµdt + ĻdWt ) ā Ī»|ut |1+Ī±
dt
dYt = (Yt (1 ā Yt )(Āµ ā Yt Ļ2
) + (ut + Ī»Yt |ut |1+Ī±
))dt + ĻYt (1 ā Yt )dWt
has unique solution satisfying Xt ā„ 0 a.s. for all t ā„ 0.
ā¢ Contrast to models without frictions or with transaction costs:
control variable is not risky weight Yt , but its ārate of changeā ut .
ā¢ Portfolio weight Yt is now a state variable.
ā¢ Illiquid vs. perfectly liquid market.
Steering a ship vs. driving a race car.
ā¢ Frictionless solution Yt = Āµ
Ī³Ļ2 unfeasible. A still ship in stormy sea.
10. Model Results Heuristics
Objective
ā¢ Investor with relative risk aversion Ī³.
ā¢ Maximize equivalent safe rate, i.e., power utility over long horizon:
max
u
lim
Tāā
1
T
log E X1āĪ³
T
1
1āĪ³
ā¢ Tradeoff between speed and impact.
ā¢ Optimal policy and welfare.
ā¢ Implied trading volume.
ā¢ Dependence on parameters.
ā¢ Asymptotics for small Ī».
ā¢ Comparison with linear impact and transaction costs.
11. Model Results Heuristics
Veriļ¬cation
Theorem
If Āµ
Ī³Ļ2 ā (0, 1), then the optimal wealth turnover and equivalent safe rate are:
Ėu(y) = q(y)
(Ī±+1)Ī»(1āyq(y))
1/Ī±
sgn(q(y)) EsRĪ³(Ėu) = Ī²
where Ī² ā (0, Āµ2
2Ī³Ļ2 ) and q : [0, 1] ā R are the unique pair that solves the ODE
ā ĖĪ² + Āµy ā Ī³
Ļ2
2
y2
+ y(1 ā y)(Āµ ā Ī³Ļ2
y)q
+
Ī±
(Ī± + 1)1+1/Ī±
|q|
Ī±+1
Ī±
(1 ā yq)1/Ī±
Ī»ā1/Ī±
+
Ļ2
2
y2
(1 ā y)2
(q + (1 ā Ī³)q2
) = 0
q(0) = Ī»
1
Ī±+1 (Ī± + 1)
1
Ī±+1 Ī±+1
Ī±
ĖĪ²
Ī±
Ī±+1
, Ī±
(Ī±+1)1+1/Ī±
|q(1)|
Ī±+1
Ī±
(1āq(1))1/Ī± Ī»ā1/Ī±
= ĖĪ² ā Āµ + Ī³ Ļ2
2
ā¢ License to solve an ODE of Abel type. Function q and scalar Ī² not explicit.
ā¢ Asymptotic expansion for Ī» near zero?
14. Model Results Heuristics
Trading Policy
ā¢ Trade towards ĀÆY. Buy for y < ĀÆY, sell for y > ĀÆY.
ā¢ Trade faster if market deeper. Higher volume in more liquid markets.
ā¢ Trade slower than with linear impact near target. Faster away from target.
With linear impact trading rate proportional to displacement |y ā ĀÆY|.
ā¢ As Ī± ā 0, trading rate:
vanishes inside no-trade region
explodes to Ā±ā outside region.
15. Model Results Heuristics
Welfare
ā¢ Welfare cost of friction:
cĪ±
Ļ2
2
3
Ī³ ĀÆY4
(1 ā ĀÆY)4
Ī±+1
Ī±+3
Ī»
2
Ī±+3
ā¢ Last factor accounts for effect of illiquidity parameter.
ā¢ Middle factor reļ¬ects volatility of portfolio weight.
ā¢ Constant cĪ± depends on Ī± alone. No explicit expression for general Ī±.
ā¢ Exponents 2/(Ī± + 3) and (Ī± + 1)/(Ī± + 3) sum to one. Geometric average.
17. Model Results Heuristics
Portfolio Dynamics
Proposition
Rescaled portfolio weight ZĪ»
s := Ī»ā 1
Ī±+3 (YĪ»2/(Ī±+3)s ā ĀÆY) converges weakly to the
process Z0
s , deļ¬ned by
dZ0
s = vĪ±(Z0
s )ds + ĀÆY(1 ā ĀÆY)ĻdWs
ā¢ āNonlinearā stationary process. Ornstein-Uhlenbeck for linear impact.
ā¢ No explicit expression for drift ā even asymptotically.
ā¢ Long-term distribution?
18. Model Results Heuristics
Long-term weight (Āµ = 8%, Ļ = 16%, Ī³ = 5)
0.4 0.2 0.0 0.2 0.4
2
4
6
8
Density (vertical) of the long-term density of rescaled risky weight Z0
(horizontal) for Ī± = 1/8, 1/4, 1/2, 1. Dashed line is uniform density (Ī± ā 0).
19. Model Results Heuristics
Linear Impact (Ī± = 1)
ā¢ Solution to
s (z) = z2
ā c ā Ī±(Ī± + 1)ā(1+1/Ī±)
|s(z)|1+1/Ī±
is c1 = 2 and s1(z) = ā2z.
ā¢ Optimal policy and welfare:
Ėu(y) = Ļ
Ī³
2Ī»
( ĀÆY ā y) + O(1)
EsRĪ³(Ėu) =
Āµ2
2Ī³Ļ2
ā Ļ3 Ī³
2
ĀÆY2
(1 ā ĀÆY)2
Ī»1/2
+ O(Ī»)
20. Model Results Heuristics
Transaction Costs (Ī± ā 0)
ā¢ Solution to
s (z) = z2
ā c ā Ī±(Ī± + 1)ā(1+1/Ī±)
|s(z)|1+1/Ī±
converges to c0 = (3/2)2/3
and
s0(z) := lim
Ī±ā0
sĪ±(z) =
ļ£±
ļ£“ļ£²
ļ£“ļ£³
1, z ā (āā, ā
ā
c0],
z3
/3 ā c0z, z ā (ā
ā
c0,
ā
c0),
ā1, z ā [
ā
c0, +ā).
ā¢ Optimal policy and welfare:
YĀ± =
Āµ
Ī³Ļ2
Ā±
3
4Ī³
ĀÆY2
(1 ā ĀÆY)2
1/3
Īµ1/3
EsRĪ³(Ėu) =
Āµ2
2Ī³Ļ2
ā
Ī³Ļ2
2
3
4Ī³
ĀÆY2
(1 ā ĀÆY)2
2/3
Īµ2/3
ā¢ Compare to transaction cost model (Gerhold et al., 2014).
21. Model Results Heuristics
Trading Volume and Welfare
ā¢ Expected Trading Volume
|ET| := lim
Tāā
1
T
T
0
|ĖuĪ»(Yt )|dt = KĪ±
Ļ2
2
3
Ī³ ĀÆY4
(1 ā ĀÆY)4
1
Ī±+3
Ī»ā 1
Ī±+3 +o(Ī»ā 1
Ī±+3
ā¢ Deļ¬ne welfare loss as decrease in equivalent safe rate due to friction:
LoS =
Āµ2
2Ī³Ļ2
ā EsRĪ³(Ėu)
ā¢ Zero loss if no trading necessary, i.e. ĀÆY ā {0, 1}.
ā¢ Universal relation:
LoS = NĪ±Ī» |ET|
1+Ī±
where constant NĪ± depends only on Ī±.
ā¢ Linear effect with transaction costs (price, not quantity).
Superlinear effect with liquidity (price times quantity).
22. Model Results Heuristics
Hacking the Model (Ī± > 1)
0.61 0.62 0.63 0.64 0.65
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
ā¢ Empirically improbable. Theoretically possible.
ā¢ Trading rates below one cheap. Above one expensive.
ā¢ As Ī± ā ā, trade at rate close to one. Compare to Longstaff (2001).
23. Model Results Heuristics
Neither a Borrower nor a Shorter Be
Theorem
If Āµ
Ī³Ļ2 ā¤ 0, then Yt = 0 and Ėu = 0 for all t optimal. Equivalent safe rate zero.
If Āµ
Ī³Ļ2 ā„ 1, then Yt = 1 and Ėu = 0 for all t optimal. Equivalent safe rate Āµ ā Ī³
2 Ļ2
.
ā¢ If Merton investor shorts, keep all wealth in safe asset, but do not short.
ā¢ If Merton investor levers, keep all wealth in risky asset, but do not lever.
ā¢ Portfolio choice for a risk-neutral investor!
ā¢ Corner solutions. But without constraints?
ā¢ Intuition: the constraint is that wealth must stay positive.
ā¢ Positive wealth does not preclude borrowing with block trading,
as in frictionless models and with transaction costs.
ā¢ Block trading unfeasible with price impact proportional to turnover.
Even in the limit.
ā¢ Phenomenon disappears with exponential utility.
24. Model Results Heuristics
Control Argument
ā¢ Value function v depends on (1) current wealth Xt , (2) current risky weight
Yt , and (3) calendar time t.
dv(t, Xt , Yt ) = vt dt + vx dXt + vy dYt +
vxx
2
d X t +
vyy
2
d Y t + vxy d X, Y t
= vt dt + vx (ĀµXt Yt ā Ī»Xt |ut |Ī±+1
)dt + vx Xt Yt ĻdWt
+ vy (Yt (1 ā Yt )(Āµ ā Yt Ļ2
) + ut + Ī»Yt |ut |Ī±+1
)dt + vy Yt (1 ā Yt )ĻdWt
+
Ļ2
2
vxx X2
t Y2
t +
Ļ2
2
vyy Y2
t (1 ā Yt )2
+ Ļ2
vxy Xt Y2
t (1 ā Yt ) dt,
ā¢ Maximize drift over u, and set result equal to zero:
vt +y(1āy)(ĀµāĻ2
y)vy +Āµxyvx +
Ļ2
y2
2
x2
vxx + (1 ā y)2
vyy + 2x(1 ā y)vxy
+ max
u
āĪ»x|u|Ī±+1
vx + vy u + Ī»y|u|Ī±+1
= 0.
25. Model Results Heuristics
Homogeneity and Long-Run
ā¢ Homogeneity in wealth v(t, x, y) = x1āĪ³
v(t, 1, y).
ā¢ Guess long-term growth at equivalent safe rate Ī², to be found.
ā¢ Substitution v(t, x, y) = x1āĪ³
1āĪ³ e(1āĪ³)(Ī²(Tāt)+ y
q(z)dz)
reduces HJB equation
āĪ² + Āµy ā Ī³ Ļ2
2 y2
+ qy(1 ā y)(Āµ ā Ī³Ļ2
y) + Ļ2
2 y2
(1 ā y)2
(q + (1 ā Ī³)q2
)
+ max
u
āĪ»|u|Ī±+1
+ (u + Ī»y|u|Ī±+1
)q = 0,
ā¢ Maximum for |u(y)| = q(y)
(Ī±+1)Ī»(1āyq(y))
1/Ī±
.
ā¢ Plugging yields
āĪ² + Āµy ā Ī³
Ļ2
2
y2
+ y(1 ā y)(Āµ ā Ī³Ļ2
y)q
+
Ī±
(Ī± + 1)1+1/Ī±
|q|
Ī±+1
Ī±
(1 ā yq)1/Ī±
Ī»ā1/Ī±
+
Ļ2
2
y2
(1 ā y)2
(q + (1 ā Ī³)q2
) = 0.
ā¢ Ī² = Āµ2
2Ī³Ļ2 , q = 0, y = Āµ
Ī³Ļ2 corresponds to Merton solution.
ā¢ Classical model as a singular limit.
26. Model Results Heuristics
Asymptotics away from Target
ā¢ Guess that q(y) ā 0 as Ī» ā 0. Limit equation:
Ī³Ļ2
2
( ĀÆY ā y)2
= lim
Ī»ā0
Ī±
Ī± + 1
(Ī± + 1)ā1/Ī±
|q|
Ī±+1
Ī± Ī»ā1/Ī±
.
ā¢ Expand equivalent safe rate as Ī² = Āµ2
2Ī³Ļ2 ā c(Ī»)
ā¢ Function c represents welfare impact of illiquidity.
ā¢ Plug expansion in HJB equation
āĪ²+ĀµyāĪ³ Ļ2
2 y2
+y(1āy)(ĀµāĪ³Ļ2
y)q+ q2
4Ī»(1āyq) +Ļ2
2 y2
(1āy)2
(q +(1āĪ³)q2
) =
ā¢ which suggests asymptotic approximation
q(1)
(y) = Ī»
1
Ī±+1 (Ī± + 1)
1
Ī±+1
Ī± + 1
Ī±
Ī³Ļ2
2
Ī±
Ī±+1
| ĀÆY ā y|
2Ī±
Ī±+1 sgn( ĀÆY ā y).
ā¢ Derivative explodes at target ĀÆY. Need different expansion.
28. Model Results Heuristics
Issues
ā¢ How to make argument rigorous?
ā¢ Heuristics yield ODE, but no boundary conditions!
ā¢ Relation between ODE and optimization problem?
29. Model Results Heuristics
Veriļ¬cation
Lemma
Let q solve the HJB equation, and deļ¬ne Q(y) =
y
q(z)dz. There exists a
probability ĖP, equivalent to P, such that the terminal wealth XT of any
admissible strategy satisļ¬es:
E[X1āĪ³
T ]
1
1āĪ³ ā¤ eĪ²T+Q(y)
EĖP[eā(1āĪ³)Q(YT )
]
1
1āĪ³ ,
and equality holds for the optimal strategy.
ā¢ Solution of HJB equation yields asymptotic upper bound for any strategy.
ā¢ Upper bound reached for optimal strategy.
ā¢ Valid for any Ī², for corresponding Q.
ā¢ Idea: pick largest Ī²ā
to make Q disappear in the long run.
ā¢ A priori bounds:
Ī²ā
<
Āµ2
2Ī³Ļ2
(frictionless solution)
max 0, Āµ ā
Ī³
2
Ļ2
<Ī²ā
(all in safe or risky asset)
30. Model Results Heuristics
Existence
Theorem
Assume 0 < Āµ
Ī³Ļ2 < 1. There exists Ī²ā
such that HJB equation has solution
q(y) with positive ļ¬nite limit in 0 and negative ļ¬nite limit in 1.
ā¢ for Ī² > 0, there exists a unique solution q0,Ī²(y) to HJB equation with
positive ļ¬nite limit in 0.
ā¢ for Ī² > Āµ ā Ī³Ļ2
2 , there exists a unique solution q1,Ī²(y) to HJB equation
with negative ļ¬nite limit in 1.
ā¢ there exists Ī²u such that q0,Ī²u
(y) > q1,Ī²u
(y) for some y;
ā¢ there exists Ī²l such that q0,Ī²l
(y) < q1,Ī²l
(y) for some y;
ā¢ by continuity and boundedness, there exists Ī²ā
ā (Ī²l , Ī²u) such that
q0,Ī²ā (y) = q1,Ī²ā (y).
ā¢ Boundary conditions are natural!
31. Model Results Heuristics
Explosion with Leverage
Lemma
If Yt that satisļ¬es Y0 ā (1, +ā) and
dYt = Yt (1 ā Yt )(Āµdt ā Yt Ļ2
dt + ĻdWt ) + ut dt + Ī»Yt |ut |1+Ī±
dt
explodes in ļ¬nite time with positive probability.
Lemma
Let Ļ be the exploding time of Yt . Then wealth XĻ = 0 a.s on {Ļ < +ā}.
ā¢ Fellerās criterion for explosions.
ā¢ No strategy admissible if it begins with levered or negative position.
32. Model Results Heuristics
Conclusion
ā¢ Finite market depth. Execution price power of wealth turnover.
ā¢ Large investor with constant relative risk aversion.
ā¢ Base price geometric Brownian Motion.
ā¢ Halfway between linear impact and bid-ask spreads.
ā¢ Trade towards frictionless portfolio.
ā¢ Do not lever an illiquid asset!