Dynamic Trading Volume

Model Results Heuristics

Dynamic Trading Volume

Paolo Guasoni1,2 Marko Weber2,3

Boston University1

Dublin City University2

Scuola Normale Superiore3

Probability, Control and Finance
A Conference in Honor of Ioannis Karatzas
Columbia University, June 5th , 2012


Outline

• Motivation:
Liquidity, Market Depth, Trading Volume.
• Model:
Finite Depth. Constant investment opportunities and risk aversion.
• Results:
Optimal policy and welfare. Trading volume dynamics.


Price and Volume

• Geometric Brownian Motion: basic stochastic process for price.
Since Samuelson, Black and Scholes, Merton.
• Basic stochastic process for volume?


Unsettling Answers

• Volume: rate of change in total quantities traded.
• Portfolio models, exogenous prices. After Merton (1969, 1973).
Continuous rebalancing. Quantities are diffusions.
Volume infinite.
• Equilibrium models, endogenous prices. After Lucas (1973).
Representative agent holds risky asset at all times.
Volume zero.
• Multiple agents models. After Milgrom and Stokey (1982).
Prices change, portfolios don’t.
Volume zero.
• All these models: no frictions.
• Transaction costs, exogenous prices. After Constantinides (1986).
Volume finite as time-average.
Either zero (no trade region) or infinite (trading boundaries).


This Talk

• Question:
if price is geometric Brownian Motion, what is the process for volume?
• Inputs
• Price exogenous. Geometric Brownian Motion.
• Representative agent.
Constant relative risk aversion and long horizon.
• Friction. Execution price linear in volume.
• Outputs
• Stochastic process for trading volume.
• Optimal trading policy and welfare.
• Small friction asymptotics explicit.


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.


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.


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

˙t ˙ S ˙
dCt = −St 1 + λ θXSt dθt = −St θt + λ Xtt θt2 dt
t

˙
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.


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.


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.


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?


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?


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.


Turnover (µ = 8%, σ = 16%, λ = 10−3 , γ = 5)
4

2

0.2 0.4 0.6 0.8 1.0

2

4

• Solution to ODE almost linear. Asymptotics accurate.


Turnover (µ = 8%, σ = 16%, λ = 10−3 , γ = 10)
4

2

0.2 0.4 0.6 0.8 1.0

2

4

• Hiher risk aversion: lower target and narrower interval.


Effect of Illiquidity (µ = 8%, σ = 16%, γ = 5)
2

1

0.55 0.60 0.65 0.70 0.75

1

2
• λ = 10−4 (solid), 10−3 (long), 10−2 , (short), and 10−1 (dotted).


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


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
1950-1949 10% 12% 3.855 2.554
1960-1949 10% 15% 4.365 3.064
1970-1949 13% 20% 4.107 2.806
1980-1949 15% 63% 5.699 4.398
1990-1949 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).


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 ≈ π (1 − π∗ ) 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).


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.
• Phenomenon disappears with exponential utility.


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


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.


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 ).


Issues

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


Verification
Lemma
y
Let q solve the HJB equation, and define Q(y ) = q(z)dz. There exists a
ˆ
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

and equality holds for the optimal strategy.

• Solution of HJB equation yields asymptotic upper bound for any strategy.
• Upper bound reached for optimal strategy.
• Valid for any β, for corresponding Q.
• Idea: pick largest β ∗ to make Q disappear in the long run.
• A priori bounds:
µ2
β∗ < (frictionless solution)
2γσ 2
γ 2
max 0, µ − σ <β ∗ (all in safe or risky asset)
2


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!


Explosion with Leverage

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

dYt = Yt (1 − Yt )(µdt − Yt σ 2 dt + σdWt ) + ut dt + λYt ut2 dt

explodes in ﬁnite time with positive probability.

• Feller’s criterion for explosions.
• No strategy admissible if it begins with levered or negative position.


Conclusion

• Finite market depth. Execution price linear in volume as wealth turnover.
• Representative agent with constant relative risk aversion.
• Base price geometric Brownian Motion.
• Trade towards frictionless portfolio.
• Dynamics for trading volume.
• Do not lever an illiquid asset!


Happy Birthday Yannis!

Dynamic Trading Volume

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