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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
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Model Results Heuristics Price and Volume • Geometric Brownian Motion: basic stochastic process for price. Since Samuelson, Black and Scholes, Merton. • Basic stochastic process for volume?
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Model Results Heuristics 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).
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Model Results Heuristics 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.
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Model Results Heuristics 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.
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Model Results Heuristics 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.
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Model Results Heuristics 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.
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Model Results Heuristics Admissible StrategiesDeﬁnitionAdmissible 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 )dWthas 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.
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Model Results Heuristics 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.
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Model Results Heuristics VeriﬁcationTheorem µIf 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 µwhere 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?
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Model Results Heuristics AsymptoticsTheorem¯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 2Long-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?
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Model Results Heuristics 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.
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Model Results Heuristics 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 ˆThe 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
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Model Results Heuristics 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).
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Model Results Heuristics 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).
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Model Results Heuristics Neither a Borrower nor a Shorter BeTheorem µIf γσ 2 ˆ ≤ 0, then Yt = 0 and u = 0 for all t optimal. Equivalent safe rate zero. µIf 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.
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Model Results Heuristics 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
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Model Results Heuristics 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.
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Model Results Heuristics 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 ).
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Model Results Heuristics Issues • How to make argument rigorous? • Heuristics yield ODE, but no boundary conditions! • Relation between ODE and optimization problem?
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Model Results Heuristics VeriﬁcationLemma yLet q solve the HJB equation, and deﬁne Q(y ) = q(z)dz. There exists a ˆprobability P, equivalent to P, such that the terminal wealth XT of anyadmissible strategy satisﬁes: 1−γ 1 1 E[XT ] 1−γ ≤ eβT +Q(y ) EP [e−(1−γ)Q(YT ) ] 1−γ , ˆand 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
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Model Results Heuristics ExistenceTheorem µAssume 0 < γσ2 < 1. There exists β ∗ such that HJB equation has solutionq(y ) with positive ﬁnite limit in 0 and negative ﬁnite limit in 1. • for β > 0, there exists a unique solution q0,β (y ) to HJB equation 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!
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Model Results Heuristics Explosion with LeverageTheoremIf Yt that satisﬁes Y0 ∈ (1, +∞) and dYt = Yt (1 − Yt )(µdt − Yt σ 2 dt + σdWt ) + ut dt + λYt ut2 dtexplodes in ﬁnite time with positive probability. • Feller’s criterion for explosions. • No strategy admissible if it begins with levered or negative position.
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Model Results Heuristics 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!
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Model Results Heuristics Happy Birthday Yannis!
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