Model Results Heuristics This Talk • Inputs • Price exogenous. Geometric Brownian Motion. • Constant relative risk aversion and long horizon. • Nonlinear price impact: trading rate one-percent highers means impact α -percent higher. • Outputs • Optimal trading policy and welfare. • High liquidity asymptotics. • Linear impact and bid-ask spreads as extreme cases. • Focus is on temporary price impact: • No permanent impact as in Huberman and Stanzl (2004) • No transient impact as in Obizhaeva and Wang (2006) or Gatheral (2010).
Model Results Heuristics This Talk • Inputs • Price exogenous. Geometric Brownian Motion. • Constant relative risk aversion and long horizon. • Nonlinear price impact: trading rate one-percent highers means impact α -percent higher. • Outputs • Optimal trading policy and welfare. • High liquidity asymptotics. • Linear impact and bid-ask spreads as extreme cases. • Focus is on temporary price impact: • No permanent impact as in Huberman and Stanzl (2004) • No transient impact as in Obizhaeva and Wang (2006) or Gatheral (2010).
Model Results Heuristics This Talk • Inputs • Price exogenous. Geometric Brownian Motion. • Constant relative risk aversion and long horizon. • Nonlinear price impact: trading rate one-percent highers means impact α -percent higher. • Outputs • Optimal trading policy and welfare. • High liquidity asymptotics. • Linear impact and bid-ask spreads as extreme cases. • Focus is on temporary price impact: • No permanent impact as in Huberman and Stanzl (2004) • No transient impact as in Obizhaeva and Wang (2006) or Gatheral (2010).
Model Results Heuristics This Talk • Inputs • Price exogenous. Geometric Brownian Motion. • Constant relative risk aversion and long horizon. • Nonlinear price impact: trading rate one-percent highers means impact α -percent higher. • Outputs • Optimal trading policy and welfare. • High liquidity asymptotics. • Linear impact and bid-ask spreads as extreme cases. • Focus is on temporary price impact: • No permanent impact as in Huberman and Stanzl (2004) • No transient impact as in Obizhaeva and Wang (2006) or Gatheral (2010).
Model Results Heuristics This Talk • Inputs • Price exogenous. Geometric Brownian Motion. • Constant relative risk aversion and long horizon. • Nonlinear price impact: trading rate one-percent highers means impact α -percent higher. • Outputs • Optimal trading policy and welfare. • High liquidity asymptotics. • Linear impact and bid-ask spreads as extreme cases. • Focus is on temporary price impact: • No permanent impact as in Huberman and Stanzl (2004) • No transient impact as in Obizhaeva and Wang (2006) or Gatheral (2010).
Model Results Heuristics This Talk • Inputs • Price exogenous. Geometric Brownian Motion. • Constant relative risk aversion and long horizon. • Nonlinear price impact: trading rate one-percent highers means impact α -percent higher. • Outputs • Optimal trading policy and welfare. • High liquidity asymptotics. • Linear impact and bid-ask spreads as extreme cases. • Focus is on temporary price impact: • No permanent impact as in Huberman and Stanzl (2004) • No transient impact as in Obizhaeva and Wang (2006) or Gatheral (2010).
Model Results Heuristics This Talk • Inputs • Price exogenous. Geometric Brownian Motion. • Constant relative risk aversion and long horizon. • Nonlinear price impact: trading rate one-percent highers means impact α -percent higher. • Outputs • Optimal trading policy and welfare. • High liquidity asymptotics. • Linear impact and bid-ask spreads as extreme cases. • Focus is on temporary price impact: • No permanent impact as in Huberman and Stanzl (2004) • No transient impact as in Obizhaeva and Wang (2006) or Gatheral (2010).
Model Results Heuristics Market • Brownian Motion ( W t ) t ≥ 0 with natural filtration ( F t ) t ≥ 0 . • Best quoted price of risky asset. Price for an infinitesimal trade. dS t = µ dt + σ dW t S t • Trade ∆ θ shares over time interval ∆ t . Order filled at price α � � � ∆ θ t S t � ˜ sgn ( ˙ � � S t (∆ θ ) := S t 1 + λ θ ) � � ∆ tX t � � where X t 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. • Impact of dollar trade S t ∆ θ declines as large investor’s wealth increases. • Makes model scale-invariant. Doubling wealth, and all subsequent trades, doubles final payoff exactly.
Model Results Heuristics Market • Brownian Motion ( W t ) t ≥ 0 with natural filtration ( F t ) t ≥ 0 . • Best quoted price of risky asset. Price for an infinitesimal trade. dS t = µ dt + σ dW t S t • Trade ∆ θ shares over time interval ∆ t . Order filled at price α � � � ∆ θ t S t � ˜ sgn ( ˙ � � S t (∆ θ ) := S t 1 + λ θ ) � � ∆ tX t � � where X t 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. • Impact of dollar trade S t ∆ θ declines as large investor’s wealth increases. • Makes model scale-invariant. Doubling wealth, and all subsequent trades, doubles final payoff exactly.
Model Results Heuristics Market • Brownian Motion ( W t ) t ≥ 0 with natural filtration ( F t ) t ≥ 0 . • Best quoted price of risky asset. Price for an infinitesimal trade. dS t = µ dt + σ dW t S t • Trade ∆ θ shares over time interval ∆ t . Order filled at price α � � � ∆ θ t S t � ˜ sgn ( ˙ � � S t (∆ θ ) := S t 1 + λ θ ) � � ∆ tX t � � where X t 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. • Impact of dollar trade S t ∆ θ declines as large investor’s wealth increases. • Makes model scale-invariant. Doubling wealth, and all subsequent trades, doubles final payoff exactly.
Model Results Heuristics Market • Brownian Motion ( W t ) t ≥ 0 with natural filtration ( F t ) t ≥ 0 . • Best quoted price of risky asset. Price for an infinitesimal trade. dS t = µ dt + σ dW t S t • Trade ∆ θ shares over time interval ∆ t . Order filled at price α � � � ∆ θ t S t � ˜ sgn ( ˙ � � S t (∆ θ ) := S t 1 + λ θ ) � � ∆ tX t � � where X t 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. • Impact of dollar trade S t ∆ θ declines as large investor’s wealth increases. • Makes model scale-invariant. Doubling wealth, and all subsequent trades, doubles final payoff exactly.
Model Results Heuristics Market • Brownian Motion ( W t ) t ≥ 0 with natural filtration ( F t ) t ≥ 0 . • Best quoted price of risky asset. Price for an infinitesimal trade. dS t = µ dt + σ dW t S t • Trade ∆ θ shares over time interval ∆ t . Order filled at price α � � � ∆ θ t S t � ˜ sgn ( ˙ � � S t (∆ θ ) := S t 1 + λ θ ) � � ∆ tX t � � where X t 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. • Impact of dollar trade S t ∆ θ declines as large investor’s wealth increases. • Makes model scale-invariant. Doubling wealth, and all subsequent trades, doubles final payoff exactly.
Model Results Heuristics Market • Brownian Motion ( W t ) t ≥ 0 with natural filtration ( F t ) t ≥ 0 . • Best quoted price of risky asset. Price for an infinitesimal trade. dS t = µ dt + σ dW t S t • Trade ∆ θ shares over time interval ∆ t . Order filled at price α � � � ∆ θ t S t � ˜ sgn ( ˙ � � S t (∆ θ ) := S t 1 + λ θ ) � � ∆ tX t � � where X t 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. • Impact of dollar trade S t ∆ θ declines as large investor’s wealth increases. • Makes model scale-invariant. Doubling wealth, and all subsequent trades, doubles final payoff exactly.
Model Results Heuristics Market • Brownian Motion ( W t ) t ≥ 0 with natural filtration ( F t ) t ≥ 0 . • Best quoted price of risky asset. Price for an infinitesimal trade. dS t = µ dt + σ dW t S t • Trade ∆ θ shares over time interval ∆ t . Order filled at price α � � � ∆ θ t S t � ˜ sgn ( ˙ � � S t (∆ θ ) := S t 1 + λ θ ) � � ∆ tX t � � where X t 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. • Impact of dollar trade S t ∆ θ declines as large investor’s wealth increases. • Makes model scale-invariant. Doubling wealth, and all subsequent trades, doubles final payoff exactly.
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) S t (∆ θ ) := S t + λ ∆ θ ˜ ∆ 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. � 1 + λ ∆ θ � ˜ S t (∆ θ ) := S t θ t ∆ t • Problematic. Infinite price impact with cash position.
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) S t (∆ θ ) := S t + λ ∆ θ ˜ ∆ 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. � 1 + λ ∆ θ � ˜ S t (∆ θ ) := S t θ t ∆ t • Problematic. Infinite price impact with cash position.
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) S t (∆ θ ) := S t + λ ∆ θ ˜ ∆ 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. � 1 + λ ∆ θ � ˜ S t (∆ θ ) := S t θ t ∆ t • Problematic. Infinite price impact with cash position.
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) S t (∆ θ ) := S t + λ ∆ θ ˜ ∆ 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. � 1 + λ ∆ θ � ˜ S t (∆ θ ) := S t θ t ∆ t • Problematic. Infinite price impact with cash position.
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) S t (∆ θ ) := S t + λ ∆ θ ˜ ∆ 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. � 1 + λ ∆ θ � ˜ S t (∆ θ ) := S t θ t ∆ t • Problematic. Infinite price impact with cash position.
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) S t (∆ θ ) := S t + λ ∆ θ ˜ ∆ 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. � 1 + λ ∆ θ � ˜ S t (∆ θ ) := S t θ t ∆ t • Problematic. Infinite price impact with cash position.
Model Results Heuristics Wealth and Portfolio • Continuous time: cash position α � 1 + α � � � � � � � ˙ S t ˙ ˙ θ t S t sgn ( ˙ θ t θ t S t dC t = − S t 1 + λ θ ) d θ t = − X t + λ X t dt � � � � X t X t � � � � ˙ θ t S t • Trading volume as wealth turnover u t := X t . Amount traded in unit of time, as fraction of wealth. • Dynamics for wealth X t := θ t S t + C t and risky portfolio weight Y t := θ t S t X t dX t = Y t ( µ dt + σ dW t ) − λ | u t | 1 + α dt X t dY t = ( Y t ( 1 − Y t )( µ − Y t σ 2 ) + ( u t + λ Y t | u t | 1 + α )) dt + σ Y t ( 1 − Y t ) dW t • Illiquidity... • ...reduces portfolio return ( − λ u 1 + α ). t Turnover effect quadratic: quantities times price impact. • ...increases risky weight ( λ Y t u 1 + α ). Buy: pay more cash. Sell: get less. t Turnover effect linear in risky weight Y t . Vanishes for cash position.
Model Results Heuristics Wealth and Portfolio • Continuous time: cash position α � 1 + α � � � � � � � ˙ S t ˙ ˙ θ t S t sgn ( ˙ θ t θ t S t dC t = − S t 1 + λ θ ) d θ t = − X t + λ X t dt � � � � X t X t � � � � ˙ θ t S t • Trading volume as wealth turnover u t := X t . Amount traded in unit of time, as fraction of wealth. • Dynamics for wealth X t := θ t S t + C t and risky portfolio weight Y t := θ t S t X t dX t = Y t ( µ dt + σ dW t ) − λ | u t | 1 + α dt X t dY t = ( Y t ( 1 − Y t )( µ − Y t σ 2 ) + ( u t + λ Y t | u t | 1 + α )) dt + σ Y t ( 1 − Y t ) dW t • Illiquidity... • ...reduces portfolio return ( − λ u 1 + α ). t Turnover effect quadratic: quantities times price impact. • ...increases risky weight ( λ Y t u 1 + α ). Buy: pay more cash. Sell: get less. t Turnover effect linear in risky weight Y t . Vanishes for cash position.
Model Results Heuristics Wealth and Portfolio • Continuous time: cash position α � 1 + α � � � � � � � ˙ S t ˙ ˙ θ t S t sgn ( ˙ θ t θ t S t dC t = − S t 1 + λ θ ) d θ t = − X t + λ X t dt � � � � X t X t � � � � ˙ θ t S t • Trading volume as wealth turnover u t := X t . Amount traded in unit of time, as fraction of wealth. • Dynamics for wealth X t := θ t S t + C t and risky portfolio weight Y t := θ t S t X t dX t = Y t ( µ dt + σ dW t ) − λ | u t | 1 + α dt X t dY t = ( Y t ( 1 − Y t )( µ − Y t σ 2 ) + ( u t + λ Y t | u t | 1 + α )) dt + σ Y t ( 1 − Y t ) dW t • Illiquidity... • ...reduces portfolio return ( − λ u 1 + α ). t Turnover effect quadratic: quantities times price impact. • ...increases risky weight ( λ Y t u 1 + α ). Buy: pay more cash. Sell: get less. t Turnover effect linear in risky weight Y t . Vanishes for cash position.
Model Results Heuristics Wealth and Portfolio • Continuous time: cash position α � 1 + α � � � � � � � ˙ S t ˙ ˙ θ t S t sgn ( ˙ θ t θ t S t dC t = − S t 1 + λ θ ) d θ t = − X t + λ X t dt � � � � X t X t � � � � ˙ θ t S t • Trading volume as wealth turnover u t := X t . Amount traded in unit of time, as fraction of wealth. • Dynamics for wealth X t := θ t S t + C t and risky portfolio weight Y t := θ t S t X t dX t = Y t ( µ dt + σ dW t ) − λ | u t | 1 + α dt X t dY t = ( Y t ( 1 − Y t )( µ − Y t σ 2 ) + ( u t + λ Y t | u t | 1 + α )) dt + σ Y t ( 1 − Y t ) dW t • Illiquidity... • ...reduces portfolio return ( − λ u 1 + α ). t Turnover effect quadratic: quantities times price impact. • ...increases risky weight ( λ Y t u 1 + α ). Buy: pay more cash. Sell: get less. t Turnover effect linear in risky weight Y t . Vanishes for cash position.
Model Results Heuristics Wealth and Portfolio • Continuous time: cash position α � 1 + α � � � � � � � ˙ S t ˙ ˙ θ t S t sgn ( ˙ θ t θ t S t dC t = − S t 1 + λ θ ) d θ t = − X t + λ X t dt � � � � X t X t � � � � ˙ θ t S t • Trading volume as wealth turnover u t := X t . Amount traded in unit of time, as fraction of wealth. • Dynamics for wealth X t := θ t S t + C t and risky portfolio weight Y t := θ t S t X t dX t = Y t ( µ dt + σ dW t ) − λ | u t | 1 + α dt X t dY t = ( Y t ( 1 − Y t )( µ − Y t σ 2 ) + ( u t + λ Y t | u t | 1 + α )) dt + σ Y t ( 1 − Y t ) dW t • Illiquidity... • ...reduces portfolio return ( − λ u 1 + α ). t Turnover effect quadratic: quantities times price impact. • ...increases risky weight ( λ Y t u 1 + α ). Buy: pay more cash. Sell: get less. t Turnover effect linear in risky weight Y t . Vanishes for cash position.
Model Results Heuristics Wealth and Portfolio • Continuous time: cash position α � 1 + α � � � � � � � ˙ S t ˙ ˙ θ t S t sgn ( ˙ θ t θ t S t dC t = − S t 1 + λ θ ) d θ t = − X t + λ X t dt � � � � X t X t � � � � ˙ θ t S t • Trading volume as wealth turnover u t := X t . Amount traded in unit of time, as fraction of wealth. • Dynamics for wealth X t := θ t S t + C t and risky portfolio weight Y t := θ t S t X t dX t = Y t ( µ dt + σ dW t ) − λ | u t | 1 + α dt X t dY t = ( Y t ( 1 − Y t )( µ − Y t σ 2 ) + ( u t + λ Y t | u t | 1 + α )) dt + σ Y t ( 1 − Y t ) dW t • Illiquidity... • ...reduces portfolio return ( − λ u 1 + α ). t Turnover effect quadratic: quantities times price impact. • ...increases risky weight ( λ Y t u 1 + α ). Buy: pay more cash. Sell: get less. t Turnover effect linear in risky weight Y t . Vanishes for cash position.
Model Results Heuristics Admissible Strategies Definition Admissible strategy: process ( u t ) t ≥ 0 , adapted to F t , such that system dX t = Y t ( µ dt + σ dW t ) − λ | u t | 1 + α dt X t dY t = ( Y t ( 1 − Y t )( µ − Y t σ 2 ) + ( u t + λ Y t | u t | 1 + α )) dt + σ Y t ( 1 − Y t ) dW t has unique solution satisfying X t ≥ 0 a.s. for all t ≥ 0. • Contrast to models without frictions or with transaction costs: control variable is not risky weight Y t , but its “rate of change” u t . • Portfolio weight Y t is now a state variable . • Illiquid vs. perfectly liquid market. Steering a ship vs. driving a race car. µ • Frictionless solution Y t = γσ 2 unfeasible. A still ship in stormy sea.
Model Results Heuristics Admissible Strategies Definition Admissible strategy: process ( u t ) t ≥ 0 , adapted to F t , such that system dX t = Y t ( µ dt + σ dW t ) − λ | u t | 1 + α dt X t dY t = ( Y t ( 1 − Y t )( µ − Y t σ 2 ) + ( u t + λ Y t | u t | 1 + α )) dt + σ Y t ( 1 − Y t ) dW t has unique solution satisfying X t ≥ 0 a.s. for all t ≥ 0. • Contrast to models without frictions or with transaction costs: control variable is not risky weight Y t , but its “rate of change” u t . • Portfolio weight Y t is now a state variable . • Illiquid vs. perfectly liquid market. Steering a ship vs. driving a race car. µ • Frictionless solution Y t = γσ 2 unfeasible. A still ship in stormy sea.
Model Results Heuristics Admissible Strategies Definition Admissible strategy: process ( u t ) t ≥ 0 , adapted to F t , such that system dX t = Y t ( µ dt + σ dW t ) − λ | u t | 1 + α dt X t dY t = ( Y t ( 1 − Y t )( µ − Y t σ 2 ) + ( u t + λ Y t | u t | 1 + α )) dt + σ Y t ( 1 − Y t ) dW t has unique solution satisfying X t ≥ 0 a.s. for all t ≥ 0. • Contrast to models without frictions or with transaction costs: control variable is not risky weight Y t , but its “rate of change” u t . • Portfolio weight Y t is now a state variable . • Illiquid vs. perfectly liquid market. Steering a ship vs. driving a race car. µ • Frictionless solution Y t = γσ 2 unfeasible. A still ship in stormy sea.
Model Results Heuristics Admissible Strategies Definition Admissible strategy: process ( u t ) t ≥ 0 , adapted to F t , such that system dX t = Y t ( µ dt + σ dW t ) − λ | u t | 1 + α dt X t dY t = ( Y t ( 1 − Y t )( µ − Y t σ 2 ) + ( u t + λ Y t | u t | 1 + α )) dt + σ Y t ( 1 − Y t ) dW t has unique solution satisfying X t ≥ 0 a.s. for all t ≥ 0. • Contrast to models without frictions or with transaction costs: control variable is not risky weight Y t , but its “rate of change” u t . • Portfolio weight Y t is now a state variable . • Illiquid vs. perfectly liquid market. Steering a ship vs. driving a race car. µ • Frictionless solution Y t = γσ 2 unfeasible. A still ship in stormy sea.
Model Results Heuristics Admissible Strategies Definition Admissible strategy: process ( u t ) t ≥ 0 , adapted to F t , such that system dX t = Y t ( µ dt + σ dW t ) − λ | u t | 1 + α dt X t dY t = ( Y t ( 1 − Y t )( µ − Y t σ 2 ) + ( u t + λ Y t | u t | 1 + α )) dt + σ Y t ( 1 − Y t ) dW t has unique solution satisfying X t ≥ 0 a.s. for all t ≥ 0. • Contrast to models without frictions or with transaction costs: control variable is not risky weight Y t , but its “rate of change” u t . • Portfolio weight Y t is now a state variable . • Illiquid vs. perfectly liquid market. Steering a ship vs. driving a race car. µ • Frictionless solution Y t = γσ 2 unfeasible. A still ship in stormy sea.
Model Results Heuristics Objective • Investor with relative risk aversion γ . • Maximize equivalent safe rate, i.e., power utility over long horizon: 1 1 � � X 1 − γ 1 − γ max lim T log E T u T →∞ • Tradeoff between speed and impact. • Optimal policy and welfare. • Implied trading volume. • Dependence on parameters. • Asymptotics for small λ . • Comparison with linear impact and transaction costs.
Model Results Heuristics Objective • Investor with relative risk aversion γ . • Maximize equivalent safe rate, i.e., power utility over long horizon: 1 1 � � X 1 − γ 1 − γ max lim T log E T u T →∞ • Tradeoff between speed and impact. • Optimal policy and welfare. • Implied trading volume. • Dependence on parameters. • Asymptotics for small λ . • Comparison with linear impact and transaction costs.
Model Results Heuristics Objective • Investor with relative risk aversion γ . • Maximize equivalent safe rate, i.e., power utility over long horizon: 1 1 � � X 1 − γ 1 − γ max lim T log E T u T →∞ • Tradeoff between speed and impact. • Optimal policy and welfare. • Implied trading volume. • Dependence on parameters. • Asymptotics for small λ . • Comparison with linear impact and transaction costs.
Model Results Heuristics Objective • Investor with relative risk aversion γ . • Maximize equivalent safe rate, i.e., power utility over long horizon: 1 1 � � X 1 − γ 1 − γ max lim T log E T u T →∞ • Tradeoff between speed and impact. • Optimal policy and welfare. • Implied trading volume. • Dependence on parameters. • Asymptotics for small λ . • Comparison with linear impact and transaction costs.
Model Results Heuristics Objective • Investor with relative risk aversion γ . • Maximize equivalent safe rate, i.e., power utility over long horizon: 1 1 � � X 1 − γ 1 − γ max lim T log E T u T →∞ • Tradeoff between speed and impact. • Optimal policy and welfare. • Implied trading volume. • Dependence on parameters. • Asymptotics for small λ . • Comparison with linear impact and transaction costs.
Model Results Heuristics Objective • Investor with relative risk aversion γ . • Maximize equivalent safe rate, i.e., power utility over long horizon: 1 1 � � X 1 − γ 1 − γ max lim T log E T u T →∞ • Tradeoff between speed and impact. • Optimal policy and welfare. • Implied trading volume. • Dependence on parameters. • Asymptotics for small λ . • Comparison with linear impact and transaction costs.
Model Results Heuristics Objective • Investor with relative risk aversion γ . • Maximize equivalent safe rate, i.e., power utility over long horizon: 1 1 � � X 1 − γ 1 − γ max lim T log E T u T →∞ • Tradeoff between speed and impact. • Optimal policy and welfare. • Implied trading volume. • Dependence on parameters. • Asymptotics for small λ . • Comparison with linear impact and transaction costs.
Model Results Heuristics Objective • Investor with relative risk aversion γ . • Maximize equivalent safe rate, i.e., power utility over long horizon: 1 1 � � X 1 − γ 1 − γ max lim T log E T u T →∞ • Tradeoff between speed and impact. • Optimal policy and welfare. • Implied trading volume. • Dependence on parameters. • Asymptotics for small λ . • Comparison with linear impact and transaction costs.
Model Results Heuristics Verification Theorem µ If γσ 2 ∈ ( 0 , 1 ) , then the optimal wealth turnover and equivalent safe rate are: 1 /α � � q ( y ) ˆ EsR γ (ˆ u ( y ) = sgn ( q ( y )) u ) = β � � ( α + 1 ) λ ( 1 − yq ( y )) � � µ 2 where β ∈ ( 0 , 2 γσ 2 ) and q : [ 0 , 1 ] �→ R are the unique pair that solves the ODE β + µ y − γ σ 2 2 y 2 + y ( 1 − y )( µ − γσ 2 y ) q − ˆ α + 1 ( 1 − yq ) 1 /α λ − 1 /α + σ 2 α | q | α 2 y 2 ( 1 − y ) 2 ( q ′ + ( 1 − γ ) q 2 ) = 0 + ( α + 1 ) 1 + 1 /α α α + 1 , α + 1 � � ( 1 − q ( 1 )) 1 /α λ − 1 /α = ˆ β − µ + γ σ 2 1 1 α ˆ | q ( 1 ) | α + 1 ( α + 1 ) α + 1 α α q ( 0 ) = λ β α + 1 ( α + 1 ) 1 + 1 /α 2 • License to solve an ODE of Abel type. Function q and scalar β not explicit. • Asymptotic expansion for λ near zero?
Model Results Heuristics Verification Theorem µ If γσ 2 ∈ ( 0 , 1 ) , then the optimal wealth turnover and equivalent safe rate are: 1 /α � � q ( y ) ˆ EsR γ (ˆ u ( y ) = sgn ( q ( y )) u ) = β � � ( α + 1 ) λ ( 1 − yq ( y )) � � µ 2 where β ∈ ( 0 , 2 γσ 2 ) and q : [ 0 , 1 ] �→ R are the unique pair that solves the ODE β + µ y − γ σ 2 2 y 2 + y ( 1 − y )( µ − γσ 2 y ) q − ˆ α + 1 ( 1 − yq ) 1 /α λ − 1 /α + σ 2 α | q | α 2 y 2 ( 1 − y ) 2 ( q ′ + ( 1 − γ ) q 2 ) = 0 + ( α + 1 ) 1 + 1 /α α α + 1 , α + 1 � � ( 1 − q ( 1 )) 1 /α λ − 1 /α = ˆ β − µ + γ σ 2 1 1 α ˆ | q ( 1 ) | α + 1 ( α + 1 ) α + 1 α α q ( 0 ) = λ β α + 1 ( α + 1 ) 1 + 1 /α 2 • License to solve an ODE of Abel type. Function q and scalar β not explicit. • Asymptotic expansion for λ near zero?
Model Results Heuristics Verification Theorem µ If γσ 2 ∈ ( 0 , 1 ) , then the optimal wealth turnover and equivalent safe rate are: 1 /α � � q ( y ) ˆ EsR γ (ˆ u ( y ) = sgn ( q ( y )) u ) = β � � ( α + 1 ) λ ( 1 − yq ( y )) � � µ 2 where β ∈ ( 0 , 2 γσ 2 ) and q : [ 0 , 1 ] �→ R are the unique pair that solves the ODE β + µ y − γ σ 2 2 y 2 + y ( 1 − y )( µ − γσ 2 y ) q − ˆ α + 1 ( 1 − yq ) 1 /α λ − 1 /α + σ 2 α | q | α 2 y 2 ( 1 − y ) 2 ( q ′ + ( 1 − γ ) q 2 ) = 0 + ( α + 1 ) 1 + 1 /α α α + 1 , α + 1 � � ( 1 − q ( 1 )) 1 /α λ − 1 /α = ˆ β − µ + γ σ 2 1 1 α ˆ | q ( 1 ) | α + 1 ( α + 1 ) α + 1 α α q ( 0 ) = λ β α + 1 ( α + 1 ) 1 + 1 /α 2 • License to solve an ODE of Abel type. Function q and scalar β not explicit. • Asymptotic expansion for λ near zero?
Model Results Heuristics Trading Rate ( µ = 8 % , σ = 16 % , λ = 0 . 1 % , γ = 5) Trading rate (vertical) against current risky weight (horizontal) for α = 1 / 3 (red) and α = 2 / 3 (blue). Dashed lines are no-trade boundaries ( α = 0).
Model Results Heuristics Asymptotics Theorem c α and s α unique pair that solves | s α ( z ) | s ′ ( z ) = z 2 − c − α ( α + 1 ) − ( 1 + 1 /α ) | s ( z ) | 1 + 1 /α α α + 1 = ( α + 1 ) α − lim α + 1 2 α z →±∞ | z | � α + 1 �� � 3 � 1 / 2 α + 3 α σ 2 � − γ ¯ Y 4 ( 1 − ¯ Y ) 4 2 l α α + 1 Set l α := , A α = , B α = l . α 2 γσ 2 Asymptotic optimal strategy and welfare: 1 /α � � α + 3 ( y − ¯ 1 s α ( λ − Y ) / A α ) � � y − ¯ ˆ � � u ( y ) = − sgn Y � � � B α ( α + 1 ) � � � µ 2 2 2 α + 3 + o ( λ α + 3 ) EsR γ (ˆ u ) = 2 γσ 2 − c α l α λ • Implications?
Model Results Heuristics Asymptotics Theorem c α and s α unique pair that solves | s α ( z ) | s ′ ( z ) = z 2 − c − α ( α + 1 ) − ( 1 + 1 /α ) | s ( z ) | 1 + 1 /α α α + 1 = ( α + 1 ) α − lim α + 1 2 α z →±∞ | z | � α + 1 �� � 3 � 1 / 2 α + 3 α σ 2 � − γ ¯ Y 4 ( 1 − ¯ Y ) 4 2 l α α + 1 Set l α := , A α = , B α = l . α 2 γσ 2 Asymptotic optimal strategy and welfare: 1 /α � � α + 3 ( y − ¯ 1 s α ( λ − Y ) / A α ) � � y − ¯ ˆ � � u ( y ) = − sgn Y � � � B α ( α + 1 ) � � � µ 2 2 2 α + 3 + o ( λ α + 3 ) EsR γ (ˆ u ) = 2 γσ 2 − c α l α λ • Implications?
Model Results Heuristics Trading Policy • Trade towards ¯ Y . Buy for y < ¯ Y , sell for y > ¯ Y . • Trade faster if market deeper. Higher volume in more liquid markets. • Trade slower than with linear impact near target. Faster away from target. With linear impact trading rate proportional to displacement | y − ¯ Y | . • As α ↓ 0, trading rate: vanishes inside no-trade region explodes to ±∞ outside region.
Model Results Heuristics Trading Policy • Trade towards ¯ Y . Buy for y < ¯ Y , sell for y > ¯ Y . • Trade faster if market deeper. Higher volume in more liquid markets. • Trade slower than with linear impact near target. Faster away from target. With linear impact trading rate proportional to displacement | y − ¯ Y | . • As α ↓ 0, trading rate: vanishes inside no-trade region explodes to ±∞ outside region.
Model Results Heuristics Trading Policy • Trade towards ¯ Y . Buy for y < ¯ Y , sell for y > ¯ Y . • Trade faster if market deeper. Higher volume in more liquid markets. • Trade slower than with linear impact near target. Faster away from target. With linear impact trading rate proportional to displacement | y − ¯ Y | . • As α ↓ 0, trading rate: vanishes inside no-trade region explodes to ±∞ outside region.
Model Results Heuristics Trading Policy • Trade towards ¯ Y . Buy for y < ¯ Y , sell for y > ¯ Y . • Trade faster if market deeper. Higher volume in more liquid markets. • Trade slower than with linear impact near target. Faster away from target. With linear impact trading rate proportional to displacement | y − ¯ Y | . • As α ↓ 0, trading rate: vanishes inside no-trade region explodes to ±∞ outside region.
Model Results Heuristics Long-term weight ( µ = 8 % , σ = 16 % , γ = 5) Density (vertical) of the long-term density of rescaled risky weight Z 0 (horizontal) for α = 1 / 3 (red) and α = 2 / 3 (blue). Dashed line is uniform density ( α → 0).
Model Results Heuristics Universal Constant c α c α (vertical) against α (horizontal).
Model Results Heuristics Linear Impact ( α = 1) • Solution to s ′ ( z ) = z 2 − c − α ( α + 1 ) − ( 1 + 1 /α ) | s ( z ) | 1 + 1 /α is c 1 = 2 and s 1 ( z ) = − 2 z . • Optimal policy and welfare: � γ 2 λ ( ¯ ˆ u ( y ) = σ Y − y ) + O ( 1 ) µ 2 � γ Y ) 2 λ 1 / 2 + O ( λ ) Y 2 ( 1 − ¯ ¯ EsR γ (ˆ 2 γσ 2 − σ 3 u ) = 2
Model Results Heuristics Linear Impact ( α = 1) • Solution to s ′ ( z ) = z 2 − c − α ( α + 1 ) − ( 1 + 1 /α ) | s ( z ) | 1 + 1 /α is c 1 = 2 and s 1 ( z ) = − 2 z . • Optimal policy and welfare: � γ 2 λ ( ¯ ˆ u ( y ) = σ Y − y ) + O ( 1 ) µ 2 � γ Y ) 2 λ 1 / 2 + O ( λ ) Y 2 ( 1 − ¯ ¯ EsR γ (ˆ 2 γσ 2 − σ 3 u ) = 2
Model Results Heuristics Transaction Costs ( α ↓ 0) • Solution to s ′ ( z ) = z 2 − c − α ( α + 1 ) − ( 1 + 1 /α ) | s ( z ) | 1 + 1 /α converges to c 0 = ( 3 / 2 ) 2 / 3 and z ∈ ( −∞ , −√ c 0 ] , 1 , z ∈ ( −√ c 0 , √ c 0 ) , z 3 / 3 − c 0 z , s 0 ( z ) := lim α → 0 s α ( z ) = z ∈ [ √ c 0 , + ∞ ) . − 1 , • Optimal policy and welfare: � 3 � 1 / 3 µ Y 2 ( 1 − ¯ ¯ Y ) 2 ε 1 / 3 Y ± = γσ 2 ± 4 γ � 3 � 2 / 3 2 γσ 2 − γσ 2 µ 2 Y 2 ( 1 − ¯ ¯ Y ) 2 ε 2 / 3 EsR γ (ˆ u ) = 2 4 γ • Compare to transaction cost model (Gerhold et al., 2014).
Model Results Heuristics Transaction Costs ( α ↓ 0) • Solution to s ′ ( z ) = z 2 − c − α ( α + 1 ) − ( 1 + 1 /α ) | s ( z ) | 1 + 1 /α converges to c 0 = ( 3 / 2 ) 2 / 3 and z ∈ ( −∞ , −√ c 0 ] , 1 , z ∈ ( −√ c 0 , √ c 0 ) , z 3 / 3 − c 0 z , s 0 ( z ) := lim α → 0 s α ( z ) = z ∈ [ √ c 0 , + ∞ ) . − 1 , • Optimal policy and welfare: � 3 � 1 / 3 µ Y 2 ( 1 − ¯ ¯ Y ) 2 ε 1 / 3 Y ± = γσ 2 ± 4 γ � 3 � 2 / 3 2 γσ 2 − γσ 2 µ 2 Y 2 ( 1 − ¯ ¯ Y ) 2 ε 2 / 3 EsR γ (ˆ u ) = 2 4 γ • Compare to transaction cost model (Gerhold et al., 2014).
Model Results Heuristics Transaction Costs ( α ↓ 0) • Solution to s ′ ( z ) = z 2 − c − α ( α + 1 ) − ( 1 + 1 /α ) | s ( z ) | 1 + 1 /α converges to c 0 = ( 3 / 2 ) 2 / 3 and z ∈ ( −∞ , −√ c 0 ] , 1 , z ∈ ( −√ c 0 , √ c 0 ) , z 3 / 3 − c 0 z , s 0 ( z ) := lim α → 0 s α ( z ) = z ∈ [ √ c 0 , + ∞ ) . − 1 , • Optimal policy and welfare: � 3 � 1 / 3 µ Y 2 ( 1 − ¯ ¯ Y ) 2 ε 1 / 3 Y ± = γσ 2 ± 4 γ � 3 � 2 / 3 2 γσ 2 − γσ 2 µ 2 Y 2 ( 1 − ¯ ¯ Y ) 2 ε 2 / 3 EsR γ (ˆ u ) = 2 4 γ • Compare to transaction cost model (Gerhold et al., 2014).
Model Results Heuristics Trading Volume and Welfare • Expected Trading Volume � T 1 �� � 1 � 3 α + 3 σ 2 1 1 γ ¯ Y 4 ( 1 − ¯ λ − α + 3 + o ( λ − | ˆ Y ) 4 | ET | := lim u λ ( Y t ) | dt = K α α + 3 2 T T →∞ 0 • Define welfare loss as decrease in equivalent safe rate due to friction: µ 2 2 γσ 2 − EsR γ (ˆ LoS = u ) • Zero loss if no trading necessary, i.e. ¯ Y ∈ { 0 , 1 } . • Universal relation: LoS = N α λ | ET | 1 + α where constant N α depends only on α . • Linear effect with transaction costs (price, not quantity). Superlinear effect with liquidity (price times quantity).
Model Results Heuristics Trading Volume and Welfare • Expected Trading Volume � T 1 �� � 1 � 3 α + 3 σ 2 1 1 γ ¯ Y 4 ( 1 − ¯ λ − α + 3 + o ( λ − | ˆ Y ) 4 | ET | := lim u λ ( Y t ) | dt = K α α + 3 2 T T →∞ 0 • Define welfare loss as decrease in equivalent safe rate due to friction: µ 2 2 γσ 2 − EsR γ (ˆ LoS = u ) • Zero loss if no trading necessary, i.e. ¯ Y ∈ { 0 , 1 } . • Universal relation: LoS = N α λ | ET | 1 + α where constant N α depends only on α . • Linear effect with transaction costs (price, not quantity). Superlinear effect with liquidity (price times quantity).
Model Results Heuristics Trading Volume and Welfare • Expected Trading Volume � T 1 �� � 1 � 3 α + 3 σ 2 1 1 γ ¯ Y 4 ( 1 − ¯ λ − α + 3 + o ( λ − | ˆ Y ) 4 | ET | := lim u λ ( Y t ) | dt = K α α + 3 2 T T →∞ 0 • Define welfare loss as decrease in equivalent safe rate due to friction: µ 2 2 γσ 2 − EsR γ (ˆ LoS = u ) • Zero loss if no trading necessary, i.e. ¯ Y ∈ { 0 , 1 } . • Universal relation: LoS = N α λ | ET | 1 + α where constant N α depends only on α . • Linear effect with transaction costs (price, not quantity). Superlinear effect with liquidity (price times quantity).
Model Results Heuristics Trading Volume and Welfare • Expected Trading Volume � T 1 �� � 1 � 3 α + 3 σ 2 1 1 γ ¯ Y 4 ( 1 − ¯ λ − α + 3 + o ( λ − | ˆ Y ) 4 | ET | := lim u λ ( Y t ) | dt = K α α + 3 2 T T →∞ 0 • Define welfare loss as decrease in equivalent safe rate due to friction: µ 2 2 γσ 2 − EsR γ (ˆ LoS = u ) • Zero loss if no trading necessary, i.e. ¯ Y ∈ { 0 , 1 } . • Universal relation: LoS = N α λ | ET | 1 + α where constant N α depends only on α . • Linear effect with transaction costs (price, not quantity). Superlinear effect with liquidity (price times quantity).
Model Results Heuristics Trading Volume and Welfare • Expected Trading Volume � T 1 �� � 1 � 3 α + 3 σ 2 1 1 γ ¯ Y 4 ( 1 − ¯ λ − α + 3 + o ( λ − | ˆ Y ) 4 | ET | := lim u λ ( Y t ) | dt = K α α + 3 2 T T →∞ 0 • Define welfare loss as decrease in equivalent safe rate due to friction: µ 2 2 γσ 2 − EsR γ (ˆ LoS = u ) • Zero loss if no trading necessary, i.e. ¯ Y ∈ { 0 , 1 } . • Universal relation: LoS = N α λ | ET | 1 + α where constant N α depends only on α . • Linear effect with transaction costs (price, not quantity). Superlinear effect with liquidity (price times quantity).
Model Results Heuristics Neither a Borrower nor a Shorter Be Theorem µ If γσ 2 ≤ 0 , then Y t = 0 and ˆ u = 0 for all t optimal. Equivalent safe rate zero. µ u = 0 for all t optimal. Equivalent safe rate µ − γ γσ 2 ≥ 1 , then Y t = 1 and ˆ 2 σ 2 . If • 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.
Model Results Heuristics Neither a Borrower nor a Shorter Be Theorem µ If γσ 2 ≤ 0 , then Y t = 0 and ˆ u = 0 for all t optimal. Equivalent safe rate zero. µ u = 0 for all t optimal. Equivalent safe rate µ − γ γσ 2 ≥ 1 , then Y t = 1 and ˆ 2 σ 2 . If • 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.
Model Results Heuristics Neither a Borrower nor a Shorter Be Theorem µ If γσ 2 ≤ 0 , then Y t = 0 and ˆ u = 0 for all t optimal. Equivalent safe rate zero. µ u = 0 for all t optimal. Equivalent safe rate µ − γ γσ 2 ≥ 1 , then Y t = 1 and ˆ 2 σ 2 . If • 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.
Model Results Heuristics Neither a Borrower nor a Shorter Be Theorem µ If γσ 2 ≤ 0 , then Y t = 0 and ˆ u = 0 for all t optimal. Equivalent safe rate zero. µ u = 0 for all t optimal. Equivalent safe rate µ − γ γσ 2 ≥ 1 , then Y t = 1 and ˆ 2 σ 2 . If • 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.
Model Results Heuristics Neither a Borrower nor a Shorter Be Theorem µ If γσ 2 ≤ 0 , then Y t = 0 and ˆ u = 0 for all t optimal. Equivalent safe rate zero. µ u = 0 for all t optimal. Equivalent safe rate µ − γ γσ 2 ≥ 1 , then Y t = 1 and ˆ 2 σ 2 . If • 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.
Model Results Heuristics Neither a Borrower nor a Shorter Be Theorem µ If γσ 2 ≤ 0 , then Y t = 0 and ˆ u = 0 for all t optimal. Equivalent safe rate zero. µ u = 0 for all t optimal. Equivalent safe rate µ − γ γσ 2 ≥ 1 , then Y t = 1 and ˆ 2 σ 2 . If • 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.
Model Results Heuristics Neither a Borrower nor a Shorter Be Theorem µ If γσ 2 ≤ 0 , then Y t = 0 and ˆ u = 0 for all t optimal. Equivalent safe rate zero. µ u = 0 for all t optimal. Equivalent safe rate µ − γ γσ 2 ≥ 1 , then Y t = 1 and ˆ 2 σ 2 . If • 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.
Model Results Heuristics Neither a Borrower nor a Shorter Be Theorem µ If γσ 2 ≤ 0 , then Y t = 0 and ˆ u = 0 for all t optimal. Equivalent safe rate zero. µ u = 0 for all t optimal. Equivalent safe rate µ − γ γσ 2 ≥ 1 , then Y t = 1 and ˆ 2 σ 2 . If • 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.
Model Results Heuristics Neither a Borrower nor a Shorter Be Theorem µ If γσ 2 ≤ 0 , then Y t = 0 and ˆ u = 0 for all t optimal. Equivalent safe rate zero. µ u = 0 for all t optimal. Equivalent safe rate µ − γ γσ 2 ≥ 1 , then Y t = 1 and ˆ 2 σ 2 . If • 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.
Model Results Heuristics Control Argument • Value function v depends on (1) current wealth X t , (2) current risky weight Y t , and (3) calendar time t . dv ( t , X t , Y t ) = v t dt + v x dX t + v y dY t + v xx 2 d � X � t + v yy 2 d � Y � t + v xy d � X , Y � t = v t dt + v x ( µ X t Y t − λ X t | u t | α + 1 ) dt + v x X t Y t σ dW t + v y ( Y t ( 1 − Y t )( µ − Y t σ 2 ) + u t + λ Y t | u t | α + 1 ) dt + v y Y t ( 1 − Y t ) σ dW t � σ 2 t + σ 2 � t ( 1 − Y t ) 2 + σ 2 v xy X t Y 2 2 v xx X 2 t Y 2 2 v yy Y 2 + t ( 1 − Y t ) dt , • Maximize drift over u , and set result equal to zero: v t + y ( 1 − y )( µ − σ 2 y ) v y + µ xyv x + σ 2 y 2 � x 2 v xx + ( 1 − y ) 2 v yy + 2 x ( 1 − y ) v xy � 2 − λ x | u | α + 1 v x + v y u + λ y | u | α + 1 �� � � + max = 0 . u
Model Results Heuristics Control Argument • Value function v depends on (1) current wealth X t , (2) current risky weight Y t , and (3) calendar time t . dv ( t , X t , Y t ) = v t dt + v x dX t + v y dY t + v xx 2 d � X � t + v yy 2 d � Y � t + v xy d � X , Y � t = v t dt + v x ( µ X t Y t − λ X t | u t | α + 1 ) dt + v x X t Y t σ dW t + v y ( Y t ( 1 − Y t )( µ − Y t σ 2 ) + u t + λ Y t | u t | α + 1 ) dt + v y Y t ( 1 − Y t ) σ dW t � σ 2 t + σ 2 � t ( 1 − Y t ) 2 + σ 2 v xy X t Y 2 2 v xx X 2 t Y 2 2 v yy Y 2 + t ( 1 − Y t ) dt , • Maximize drift over u , and set result equal to zero: v t + y ( 1 − y )( µ − σ 2 y ) v y + µ xyv x + σ 2 y 2 � x 2 v xx + ( 1 − y ) 2 v yy + 2 x ( 1 − y ) v xy � 2 − λ x | u | α + 1 v x + v y u + λ y | u | α + 1 �� � � + max = 0 . u
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. � y q ( z ) dz ) reduces HJB equation • Substitution v ( t , x , y ) = x 1 − γ 1 − γ e ( 1 − γ )( β ( T − t )+ 2 y 2 + qy ( 1 − y )( µ − γσ 2 y ) + σ 2 2 y 2 ( 1 − y ) 2 ( q ′ + ( 1 − γ ) q 2 ) − β + µ y − γ σ 2 − λ | u | α + 1 + ( u + λ y | u | α + 1 ) q � � + max = 0 , u 1 /α � � q ( y ) • Maximum for | u ( y ) | = . � � ( α + 1 ) λ ( 1 − yq ( y )) � � • Plugging yields − β + µ y − γ σ 2 2 y 2 + y ( 1 − y )( µ − γσ 2 y ) q α + 1 ( 1 − yq ) 1 /α λ − 1 /α + σ 2 α | q | α 2 y 2 ( 1 − y ) 2 ( q ′ + ( 1 − γ ) q 2 ) = 0 . + ( α + 1 ) 1 + 1 /α µ 2 µ • β = 2 γσ 2 , q = 0, y = γσ 2 corresponds to Merton solution. • Classical model as a singular limit.
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. � y q ( z ) dz ) reduces HJB equation • Substitution v ( t , x , y ) = x 1 − γ 1 − γ e ( 1 − γ )( β ( T − t )+ 2 y 2 + qy ( 1 − y )( µ − γσ 2 y ) + σ 2 2 y 2 ( 1 − y ) 2 ( q ′ + ( 1 − γ ) q 2 ) − β + µ y − γ σ 2 − λ | u | α + 1 + ( u + λ y | u | α + 1 ) q � � + max = 0 , u 1 /α � � q ( y ) • Maximum for | u ( y ) | = . � � ( α + 1 ) λ ( 1 − yq ( y )) � � • Plugging yields − β + µ y − γ σ 2 2 y 2 + y ( 1 − y )( µ − γσ 2 y ) q α + 1 ( 1 − yq ) 1 /α λ − 1 /α + σ 2 α | q | α 2 y 2 ( 1 − y ) 2 ( q ′ + ( 1 − γ ) q 2 ) = 0 . + ( α + 1 ) 1 + 1 /α µ 2 µ • β = 2 γσ 2 , q = 0, y = γσ 2 corresponds to Merton solution. • Classical model as a singular limit.
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. � y q ( z ) dz ) reduces HJB equation • Substitution v ( t , x , y ) = x 1 − γ 1 − γ e ( 1 − γ )( β ( T − t )+ 2 y 2 + qy ( 1 − y )( µ − γσ 2 y ) + σ 2 2 y 2 ( 1 − y ) 2 ( q ′ + ( 1 − γ ) q 2 ) − β + µ y − γ σ 2 − λ | u | α + 1 + ( u + λ y | u | α + 1 ) q � � + max = 0 , u 1 /α � � q ( y ) • Maximum for | u ( y ) | = . � � ( α + 1 ) λ ( 1 − yq ( y )) � � • Plugging yields − β + µ y − γ σ 2 2 y 2 + y ( 1 − y )( µ − γσ 2 y ) q α + 1 ( 1 − yq ) 1 /α λ − 1 /α + σ 2 α | q | α 2 y 2 ( 1 − y ) 2 ( q ′ + ( 1 − γ ) q 2 ) = 0 . + ( α + 1 ) 1 + 1 /α µ 2 µ • β = 2 γσ 2 , q = 0, y = γσ 2 corresponds to Merton solution. • Classical model as a singular limit.
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. � y q ( z ) dz ) reduces HJB equation • Substitution v ( t , x , y ) = x 1 − γ 1 − γ e ( 1 − γ )( β ( T − t )+ 2 y 2 + qy ( 1 − y )( µ − γσ 2 y ) + σ 2 2 y 2 ( 1 − y ) 2 ( q ′ + ( 1 − γ ) q 2 ) − β + µ y − γ σ 2 − λ | u | α + 1 + ( u + λ y | u | α + 1 ) q � � + max = 0 , u 1 /α � � q ( y ) • Maximum for | u ( y ) | = . � � ( α + 1 ) λ ( 1 − yq ( y )) � � • Plugging yields − β + µ y − γ σ 2 2 y 2 + y ( 1 − y )( µ − γσ 2 y ) q α + 1 ( 1 − yq ) 1 /α λ − 1 /α + σ 2 α | q | α 2 y 2 ( 1 − y ) 2 ( q ′ + ( 1 − γ ) q 2 ) = 0 . + ( α + 1 ) 1 + 1 /α µ 2 µ • β = 2 γσ 2 , q = 0, y = γσ 2 corresponds to Merton solution. • Classical model as a singular limit.
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. � y q ( z ) dz ) reduces HJB equation • Substitution v ( t , x , y ) = x 1 − γ 1 − γ e ( 1 − γ )( β ( T − t )+ 2 y 2 + qy ( 1 − y )( µ − γσ 2 y ) + σ 2 2 y 2 ( 1 − y ) 2 ( q ′ + ( 1 − γ ) q 2 ) − β + µ y − γ σ 2 − λ | u | α + 1 + ( u + λ y | u | α + 1 ) q � � + max = 0 , u 1 /α � � q ( y ) • Maximum for | u ( y ) | = . � � ( α + 1 ) λ ( 1 − yq ( y )) � � • Plugging yields − β + µ y − γ σ 2 2 y 2 + y ( 1 − y )( µ − γσ 2 y ) q α + 1 ( 1 − yq ) 1 /α λ − 1 /α + σ 2 α | q | α 2 y 2 ( 1 − y ) 2 ( q ′ + ( 1 − γ ) q 2 ) = 0 . + ( α + 1 ) 1 + 1 /α µ 2 µ • β = 2 γσ 2 , q = 0, y = γσ 2 corresponds to Merton solution. • Classical model as a singular limit.
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. � y q ( z ) dz ) reduces HJB equation • Substitution v ( t , x , y ) = x 1 − γ 1 − γ e ( 1 − γ )( β ( T − t )+ 2 y 2 + qy ( 1 − y )( µ − γσ 2 y ) + σ 2 2 y 2 ( 1 − y ) 2 ( q ′ + ( 1 − γ ) q 2 ) − β + µ y − γ σ 2 − λ | u | α + 1 + ( u + λ y | u | α + 1 ) q � � + max = 0 , u 1 /α � � q ( y ) • Maximum for | u ( y ) | = . � � ( α + 1 ) λ ( 1 − yq ( y )) � � • Plugging yields − β + µ y − γ σ 2 2 y 2 + y ( 1 − y )( µ − γσ 2 y ) q α + 1 ( 1 − yq ) 1 /α λ − 1 /α + σ 2 α | q | α 2 y 2 ( 1 − y ) 2 ( q ′ + ( 1 − γ ) q 2 ) = 0 . + ( α + 1 ) 1 + 1 /α µ 2 µ • β = 2 γσ 2 , q = 0, y = γσ 2 corresponds to Merton solution. • Classical model as a singular limit.
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. � y q ( z ) dz ) reduces HJB equation • Substitution v ( t , x , y ) = x 1 − γ 1 − γ e ( 1 − γ )( β ( T − t )+ 2 y 2 + qy ( 1 − y )( µ − γσ 2 y ) + σ 2 2 y 2 ( 1 − y ) 2 ( q ′ + ( 1 − γ ) q 2 ) − β + µ y − γ σ 2 − λ | u | α + 1 + ( u + λ y | u | α + 1 ) q � � + max = 0 , u 1 /α � � q ( y ) • Maximum for | u ( y ) | = . � � ( α + 1 ) λ ( 1 − yq ( y )) � � • Plugging yields − β + µ y − γ σ 2 2 y 2 + y ( 1 − y )( µ − γσ 2 y ) q α + 1 ( 1 − yq ) 1 /α λ − 1 /α + σ 2 α | q | α 2 y 2 ( 1 − y ) 2 ( q ′ + ( 1 − γ ) q 2 ) = 0 . + ( α + 1 ) 1 + 1 /α µ 2 µ • β = 2 γσ 2 , q = 0, y = γσ 2 corresponds to Merton solution. • Classical model as a singular limit.
Model Results Heuristics Asymptotics away from Target • Guess that q ( y ) → 0 as λ ↓ 0. Limit equation: γσ 2 α Y − y ) 2 = lim α + 1 2 ( ¯ α + 1 ( α + 1 ) − 1 /α | q | α λ − 1 /α . λ → 0 µ 2 • Expand equivalent safe rate as β = 2 γσ 2 − c ( λ ) • Function c represents welfare impact of illiquidity. • Expand function as q ( y ) = q ( 1 ) ( y ) λ 1 / 2 + o ( λ 1 / 2 ) . • Plug expansion in HJB equation q 2 − β + µ y − γ σ 2 4 λ ( 1 − yq ) + σ 2 2 y 2 + y ( 1 − y )( µ − γσ 2 y ) q + 2 y 2 ( 1 − y ) 2 ( q ′ +( 1 − γ ) q 2 ) = 0 • which suggests asymptotic approximation α γσ 2 � α + 1 � α + 1 2 α 1 1 | ¯ α + 1 sgn ( ¯ q ( 1 ) ( y ) = λ α + 1 ( α + 1 ) Y − y | Y − y ) . α + 1 α 2 • Derivative explodes at target ¯ Y . Need different expansion.
Model Results Heuristics Asymptotics away from Target • Guess that q ( y ) → 0 as λ ↓ 0. Limit equation: γσ 2 α Y − y ) 2 = lim α + 1 2 ( ¯ α + 1 ( α + 1 ) − 1 /α | q | α λ − 1 /α . λ → 0 µ 2 • Expand equivalent safe rate as β = 2 γσ 2 − c ( λ ) • Function c represents welfare impact of illiquidity. • Expand function as q ( y ) = q ( 1 ) ( y ) λ 1 / 2 + o ( λ 1 / 2 ) . • Plug expansion in HJB equation q 2 − β + µ y − γ σ 2 4 λ ( 1 − yq ) + σ 2 2 y 2 + y ( 1 − y )( µ − γσ 2 y ) q + 2 y 2 ( 1 − y ) 2 ( q ′ +( 1 − γ ) q 2 ) = 0 • which suggests asymptotic approximation α γσ 2 � α + 1 � α + 1 2 α 1 1 | ¯ α + 1 sgn ( ¯ q ( 1 ) ( y ) = λ α + 1 ( α + 1 ) Y − y | Y − y ) . α + 1 α 2 • Derivative explodes at target ¯ Y . Need different expansion.
Model Results Heuristics Asymptotics away from Target • Guess that q ( y ) → 0 as λ ↓ 0. Limit equation: γσ 2 α Y − y ) 2 = lim α + 1 2 ( ¯ α + 1 ( α + 1 ) − 1 /α | q | α λ − 1 /α . λ → 0 µ 2 • Expand equivalent safe rate as β = 2 γσ 2 − c ( λ ) • Function c represents welfare impact of illiquidity. • Expand function as q ( y ) = q ( 1 ) ( y ) λ 1 / 2 + o ( λ 1 / 2 ) . • Plug expansion in HJB equation q 2 − β + µ y − γ σ 2 4 λ ( 1 − yq ) + σ 2 2 y 2 + y ( 1 − y )( µ − γσ 2 y ) q + 2 y 2 ( 1 − y ) 2 ( q ′ +( 1 − γ ) q 2 ) = 0 • which suggests asymptotic approximation α γσ 2 � α + 1 � α + 1 2 α 1 1 | ¯ α + 1 sgn ( ¯ q ( 1 ) ( y ) = λ α + 1 ( α + 1 ) Y − y | Y − y ) . α + 1 α 2 • Derivative explodes at target ¯ Y . Need different expansion.
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