Volatility and Liquidity Trading

Volatility and Liquidity Trading OxfordPrinceton Volatility and Liquidity Trading Motivation Market Impact Modeling Liquidity Trading Game René Carmona and Z. Joseph Yang Bendheim Center for Finance & Dept. of ORFE, Princeton University Summary References March 28th, 2009 Oxford-Princeton Volatility and Liquidity Trading Outline Volatility and Liquidity Trading OxfordPrinceton 1 Motivation 2 Market Impact Modeling The Permanent Component is Necessarily Linear Time-Homogeneity is Obtained By Subordinating in Volume-Time 3 Formulation of the Liquidity Trading Game The Stochastic Optimal Control Problem for Each Player The Nash-Equilibrium for the Liquidity Trading Game Numerical Analysis of the NE 4 Summary and Ongoing Work Motivation Market Impact Modeling Liquidity Trading Game Summary References Oxford-Princeton Volatility and Liquidity Trading The Liquidity Nature of the Financial Markets Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game Summary References Liquidity Risk Asset Liquidity / Market Liquidity inventory approach/information-asymmetry approach empirical data modeling approach optimal control theory approach game theoretic approach Funding Liquidity margin requirement shift and illiquidity spiral capital structure analysis etc. Market Risk Credit Risk Oxford-Princeton Volatility and Liquidity Trading The Liquidity Nature of the Financial Markets Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game Summary References Liquidity Risk Asset Liquidity / Market Liquidity inventory approach/information-asymmetry approach empirical data modeling approach optimal control theory approach game theoretic approach Funding Liquidity margin requirement shift and illiquidity spiral capital structure analysis etc. Market Risk Credit Risk Oxford-Princeton Volatility and Liquidity Trading The Liquidity Nature of the Financial Markets Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game Summary References Liquidity Risk Asset Liquidity / Market Liquidity inventory approach/information-asymmetry approach empirical data modeling approach optimal control theory approach game theoretic approach Funding Liquidity margin requirement shift and illiquidity spiral capital structure analysis etc. Market Risk Credit Risk Oxford-Princeton Volatility and Liquidity Trading The Liquidity Nature of the Financial Markets Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game Summary References Liquidity Risk Asset Liquidity / Market Liquidity inventory approach/information-asymmetry approach empirical data modeling approach optimal control theory approach game theoretic approach Funding Liquidity margin requirement shift and illiquidity spiral capital structure analysis etc. Market Risk Credit Risk Oxford-Princeton Volatility and Liquidity Trading The Liquidity Nature of the Financial Markets Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game Summary References Liquidity Risk Asset Liquidity / Market Liquidity inventory approach/information-asymmetry approach empirical data modeling approach optimal control theory approach game theoretic approach Funding Liquidity margin requirement shift and illiquidity spiral capital structure analysis etc. Market Risk Credit Risk Oxford-Princeton Volatility and Liquidity Trading The Liquidity Nature of the Financial Markets Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game Summary References Liquidity Risk Asset Liquidity / Market Liquidity inventory approach/information-asymmetry approach empirical data modeling approach optimal control theory approach game theoretic approach Funding Liquidity margin requirement shift and illiquidity spiral capital structure analysis etc. Market Risk Credit Risk Oxford-Princeton Volatility and Liquidity Trading The Liquidity Nature of the Financial Markets Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game Summary References Liquidity Risk Asset Liquidity / Market Liquidity inventory approach/information-asymmetry approach empirical data modeling approach optimal control theory approach game theoretic approach Funding Liquidity margin requirement shift and illiquidity spiral capital structure analysis etc. Market Risk Credit Risk Oxford-Princeton Volatility and Liquidity Trading The Liquidity Nature of the Financial Markets Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game Summary References Liquidity Risk Asset Liquidity / Market Liquidity inventory approach/information-asymmetry approach empirical data modeling approach optimal control theory approach game theoretic approach Funding Liquidity margin requirement shift and illiquidity spiral capital structure analysis etc. Market Risk Credit Risk Oxford-Princeton Volatility and Liquidity Trading Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game Summary The crucial role of understanding the Liquidity Nature of a financial market, for both market participants and regulators alike Black Monday in 1987 LTCM and sovereign bond crisis in 1998 Riot of subprime credit products in 2007 the collapse of Amaranth in 2006, and so on References Oxford-Princeton Volatility and Liquidity Trading Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game Summary The crucial role of understanding the Liquidity Nature of a financial market, for both market participants and regulators alike Black Monday in 1987 LTCM and sovereign bond crisis in 1998 Riot of subprime credit products in 2007 the collapse of Amaranth in 2006, and so on References Oxford-Princeton Volatility and Liquidity Trading Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game Summary The crucial role of understanding the Liquidity Nature of a financial market, for both market participants and regulators alike Black Monday in 1987 LTCM and sovereign bond crisis in 1998 Riot of subprime credit products in 2007 the collapse of Amaranth in 2006, and so on References Oxford-Princeton Volatility and Liquidity Trading Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game Summary The crucial role of understanding the Liquidity Nature of a financial market, for both market participants and regulators alike Black Monday in 1987 LTCM and sovereign bond crisis in 1998 Riot of subprime credit products in 2007 the collapse of Amaranth in 2006, and so on References Oxford-Princeton Volatility and Liquidity Trading Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game Summary The crucial role of understanding the Liquidity Nature of a financial market, for both market participants and regulators alike Black Monday in 1987 LTCM and sovereign bond crisis in 1998 Riot of subprime credit products in 2007 the collapse of Amaranth in 2006, and so on References Oxford-Princeton Volatility and Liquidity Trading Volatility and Liquidity Trading OxfordPrinceton Motivation Intension/activity of buying and selling does affect the market price of an asset Trading tactics inspired by the liquidity rationale and trend of the market Market Impact Modeling optimal execution for a single player Liquidity Trading Game strategic play between multiple players Summary References Bertsimas & Lo, 98; Almgren & Chriss, 01; Almgren 03 Brunnermeier & Pedersen, 05 Carlin, Lobo, Viswanathan, 07 Schoneborn & Schied, 07 Oxford-Princeton Volatility and Liquidity Trading Volatility and Liquidity Trading OxfordPrinceton Motivation Intension/activity of buying and selling does affect the market price of an asset Trading tactics inspired by the liquidity rationale and trend of the market Market Impact Modeling optimal execution for a single player Liquidity Trading Game strategic play between multiple players Summary References Bertsimas & Lo, 98; Almgren & Chriss, 01; Almgren 03 Brunnermeier & Pedersen, 05 Carlin, Lobo, Viswanathan, 07 Schoneborn & Schied, 07 Oxford-Princeton Volatility and Liquidity Trading Volatility and Liquidity Trading OxfordPrinceton Motivation Intension/activity of buying and selling does affect the market price of an asset Trading tactics inspired by the liquidity rationale and trend of the market Market Impact Modeling optimal execution for a single player Liquidity Trading Game strategic play between multiple players Summary References Bertsimas & Lo, 98; Almgren & Chriss, 01; Almgren 03 Brunnermeier & Pedersen, 05 Carlin, Lobo, Viswanathan, 07 Schoneborn & Schied, 07 Oxford-Princeton Volatility and Liquidity Trading Volatility and Liquidity Trading OxfordPrinceton Motivation Intension/activity of buying and selling does affect the market price of an asset Trading tactics inspired by the liquidity rationale and trend of the market Market Impact Modeling optimal execution for a single player Liquidity Trading Game strategic play between multiple players Summary References Bertsimas & Lo, 98; Almgren & Chriss, 01; Almgren 03 Brunnermeier & Pedersen, 05 Carlin, Lobo, Viswanathan, 07 Schoneborn & Schied, 07 Oxford-Princeton Volatility and Liquidity Trading Volatility and Liquidity Trading OxfordPrinceton Motivation Intension/activity of buying and selling does affect the market price of an asset Trading tactics inspired by the liquidity rationale and trend of the market Market Impact Modeling optimal execution for a single player Liquidity Trading Game strategic play between multiple players Summary References Bertsimas & Lo, 98; Almgren & Chriss, 01; Almgren 03 Brunnermeier & Pedersen, 05 Carlin, Lobo, Viswanathan, 07 Schoneborn & Schied, 07 Oxford-Princeton Volatility and Liquidity Trading A quick overview of previous models Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game Summary References Trading in continuous-time → differential game permanent component and temporary component of market impact Nash-equilibrium of the game, either predation or providing liquidity only open-loop strategies are allowed for each player essentially deterministic optimal control and volatility never plays a role Oxford-Princeton Volatility and Liquidity Trading A quick overview of previous models Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game Summary References Trading in continuous-time → differential game permanent component and temporary component of market impact Nash-equilibrium of the game, either predation or providing liquidity only open-loop strategies are allowed for each player essentially deterministic optimal control and volatility never plays a role Oxford-Princeton Volatility and Liquidity Trading A quick overview of previous models Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game Summary References Trading in continuous-time → differential game permanent component and temporary component of market impact Nash-equilibrium of the game, either predation or providing liquidity only open-loop strategies are allowed for each player essentially deterministic optimal control and volatility never plays a role Oxford-Princeton Volatility and Liquidity Trading A quick overview of previous models Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game Summary References Trading in continuous-time → differential game permanent component and temporary component of market impact Nash-equilibrium of the game, either predation or providing liquidity only open-loop strategies are allowed for each player essentially deterministic optimal control and volatility never plays a role Oxford-Princeton Volatility and Liquidity Trading A quick overview of previous models Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game Summary References Trading in continuous-time → differential game permanent component and temporary component of market impact Nash-equilibrium of the game, either predation or providing liquidity only open-loop strategies are allowed for each player essentially deterministic optimal control and volatility never plays a role Oxford-Princeton Volatility and Liquidity Trading A quick overview of previous models Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game Summary References Trading in continuous-time → differential game permanent component and temporary component of market impact Nash-equilibrium of the game, either predation or providing liquidity only open-loop strategies are allowed for each player essentially deterministic optimal control and volatility never plays a role Oxford-Princeton Volatility and Liquidity Trading A quick overview of previous models Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game Summary References Trading in continuous-time → differential game permanent component and temporary component of market impact Nash-equilibrium of the game, either predation or providing liquidity only open-loop strategies are allowed for each player essentially deterministic optimal control and volatility never plays a role Oxford-Princeton Volatility and Liquidity Trading Market Impact Modeling Volatility and Liquidity Trading OxfordPrinceton Rt X(t) = 0 ξ(s) ds, where ξ(t) is the trading intensity in continuous-time. market (mid-quote) price Motivation Market Impact Modeling Permanent Component is Necessarily Linear Subordinating in VolumeTime Liquidity Trading Game dP (t) = µ(t, · · · )dt + f (ξ(t))dt + σ(t, · · · )dW (t) COST = Z t P̃ (s)ξ(s) ds = 0 Z (1) t (P (s) + g(ξ(s)))ξ(s) ds 0 where f (·) and g(·) are the so-called permanent component function and temporary component function. Summary References Oxford-Princeton Volatility and Liquidity Trading Market Impact Modeling Volatility and Liquidity Trading OxfordPrinceton Rt X(t) = 0 ξ(s) ds, where ξ(t) is the trading intensity in continuous-time. market (mid-quote) price Motivation Market Impact Modeling Permanent Component is Necessarily Linear Subordinating in VolumeTime Liquidity Trading Game dP (t) = µ(t, · · · )dt + f (ξ(t))dt + σ(t, · · · )dW (t) COST = Z t P̃ (s)ξ(s) ds = 0 Z (1) t (P (s) + g(ξ(s)))ξ(s) ds 0 where f (·) and g(·) are the so-called permanent component function and temporary component function. Summary References Oxford-Princeton Volatility and Liquidity Trading Market Impact Modeling Volatility and Liquidity Trading OxfordPrinceton Rt X(t) = 0 ξ(s) ds, where ξ(t) is the trading intensity in continuous-time. market (mid-quote) price Motivation Market Impact Modeling Permanent Component is Necessarily Linear Subordinating in VolumeTime Liquidity Trading Game dP (t) = µ(t, · · · )dt + f (ξ(t))dt + σ(t, · · · )dW (t) COST = Z t P̃ (s)ξ(s) ds = 0 Z (1) t (P (s) + g(ξ(s)))ξ(s) ds 0 where f (·) and g(·) are the so-called permanent component function and temporary component function. Summary References Oxford-Princeton Volatility and Liquidity Trading Market Impact Modeling Volatility and Liquidity Trading OxfordPrinceton Rt X(t) = 0 ξ(s) ds, where ξ(t) is the trading intensity in continuous-time. market (mid-quote) price Motivation Market Impact Modeling Permanent Component is Necessarily Linear Subordinating in VolumeTime Liquidity Trading Game dP (t) = µ(t, · · · )dt + f (ξ(t))dt + σ(t, · · · )dW (t) COST = Z t P̃ (s)ξ(s) ds = 0 Z (1) t (P (s) + g(ξ(s)))ξ(s) ds 0 where f (·) and g(·) are the so-called permanent component function and temporary component function. Summary References Oxford-Princeton Volatility and Liquidity Trading Market Impact Modeling Volatility and Liquidity Trading OxfordPrinceton Rt X(t) = 0 ξ(s) ds, where ξ(t) is the trading intensity in continuous-time. market (mid-quote) price Motivation Market Impact Modeling Permanent Component is Necessarily Linear Subordinating in VolumeTime Liquidity Trading Game dP (t) = µ(t, · · · )dt + f (ξ(t))dt + σ(t, · · · )dW (t) COST = Z t P̃ (s)ξ(s) ds = 0 Z (1) t (P (s) + g(ξ(s)))ξ(s) ds 0 where f (·) and g(·) are the so-called permanent component function and temporary component function. Summary References Oxford-Princeton Volatility and Liquidity Trading Market Impact Modeling Volatility and Liquidity Trading OxfordPrinceton Rt X(t) = 0 ξ(s) ds, where ξ(t) is the trading intensity in continuous-time. market (mid-quote) price Motivation Market Impact Modeling Permanent Component is Necessarily Linear Subordinating in VolumeTime Liquidity Trading Game dP (t) = µ(t, · · · )dt + f (ξ(t))dt + σ(t, · · · )dW (t) COST = Z t P̃ (s)ξ(s) ds = 0 Z (1) t (P (s) + g(ξ(s)))ξ(s) ds 0 where f (·) and g(·) are the so-called permanent component function and temporary component function. Summary References Oxford-Princeton Volatility and Liquidity Trading Market Impact Modeling Volatility and Liquidity Trading OxfordPrinceton Rt X(t) = 0 ξ(s) ds, where ξ(t) is the trading intensity in continuous-time. market (mid-quote) price Motivation Market Impact Modeling Permanent Component is Necessarily Linear Subordinating in VolumeTime Liquidity Trading Game dP (t) = µ(t, · · · )dt + f (ξ(t))dt + σ(t, · · · )dW (t) COST = Z t P̃ (s)ξ(s) ds = 0 Z (1) t (P (s) + g(ξ(s)))ξ(s) ds 0 where f (·) and g(·) are the so-called permanent component function and temporary component function. Summary References Oxford-Princeton Volatility and Liquidity Trading Market Impact Modeling Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Permanent Component is Necessarily Linear Subordinating in VolumeTime Call a trading scheme a clean-hand trade if the strategy RT {ξ(t)}t∈[0,T ] satisfies 0 = 0 ξ(t) dt = X(T ) − X(0), and the integral process X(t) is bounded.  Z Π=E − 0 T  P̃ (t)ξ(t) dt ≤ 0 Liquidity Trading Game Summary References Oxford-Princeton Volatility and Liquidity Trading (2) Market Impact Modeling Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Permanent Component is Necessarily Linear Subordinating in VolumeTime Call a trading scheme a clean-hand trade if the strategy RT {ξ(t)}t∈[0,T ] satisfies 0 = 0 ξ(t) dt = X(T ) − X(0), and the integral process X(t) is bounded.  Z Π=E − 0 T  P̃ (t)ξ(t) dt ≤ 0 Liquidity Trading Game Summary References Oxford-Princeton Volatility and Liquidity Trading (2) Market Impact Modeling Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Permanent Component is Necessarily Linear Subordinating in VolumeTime Call a trading scheme a clean-hand trade if the strategy RT {ξ(t)}t∈[0,T ] satisfies 0 = 0 ξ(t) dt = X(T ) − X(0), and the integral process X(t) is bounded.  Z Π=E − 0 T  P̃ (t)ξ(t) dt ≤ 0 Liquidity Trading Game Summary References Oxford-Princeton Volatility and Liquidity Trading (2) Market Impact Modeling Volatility and Liquidity Trading " Z Π = E − T " Z = E − T (P (t) + g(ξ(t)))ξ(t) dt 0 OxfordPrinceton Motivation Market Impact Modeling " T − Z T g(ξ(t))ξ(t) dt 0 Summary References +E = Z 0 g(ξ(t))ξ(t) dt # T X(t) (f (ξ(t))dt + σ(t, P (t))dW (t)) # X(t)f (ξ(t)) dt − "Z T (3) 0 T Z T 0 = E − P (t)X(t)|0 + Permanent Component is Necessarily Linear Subordinating in VolumeTime Liquidity Trading Game P (t) dX(t) − 0 Z # Z 0 T g(ξ(t))ξ(t) dt # X(t)σ(t, P (t))dW (t) 0 Oxford-Princeton Volatility and Liquidity Trading Market Impact Modeling Volatility and Liquidity Trading " Z Π = E − T " Z = E − T (P (t) + g(ξ(t)))ξ(t) dt 0 OxfordPrinceton Motivation Market Impact Modeling " T − Z T g(ξ(t))ξ(t) dt 0 Summary References +E = Z 0 g(ξ(t))ξ(t) dt # T X(t) (f (ξ(t))dt + σ(t, P (t))dW (t)) # X(t)f (ξ(t)) dt − "Z T (3) 0 T Z T 0 = E − P (t)X(t)|0 + Permanent Component is Necessarily Linear Subordinating in VolumeTime Liquidity Trading Game P (t) dX(t) − 0 Z # Z 0 T g(ξ(t))ξ(t) dt # X(t)σ(t, P (t))dW (t) 0 Oxford-Princeton Volatility and Liquidity Trading Market Impact Modeling Volatility and Liquidity Trading OxfordPrinceton Π = Z T 2 ξtf (ξ) dt + 0 Motivation − Market Impact Modeling Permanent Component is Necessarily Linear Subordinating in VolumeTime Liquidity Trading Game Z T 2 Z g(ξ)ξ dt − 0 T ξ(T − t)f (−ξ) dt T 2 Z T g(−ξ)(−ξ) dt T 2 T2 T ξ (f (ξ) + f (−ξ)) + ξ (g(−ξ) − g(ξ)) 8 2 ≤ 0 = ⇒ f (−ξ) = −f (ξ), for any ξ ∈ R and g(ξ) ≥ g(−ξ), for any ξ > 0 Summary References Oxford-Princeton Volatility and Liquidity Trading (4) Market Impact Modeling Volatility and Liquidity Trading OxfordPrinceton Π = Z T 2 ξtf (ξ) dt + 0 Motivation − Market Impact Modeling Permanent Component is Necessarily Linear Subordinating in VolumeTime Liquidity Trading Game Z T 2 Z g(ξ)ξ dt − 0 T ξ(T − t)f (−ξ) dt T 2 Z T g(−ξ)(−ξ) dt T 2 T2 T ξ (f (ξ) + f (−ξ)) + ξ (g(−ξ) − g(ξ)) 8 2 ≤ 0 = ⇒ f (−ξ) = −f (ξ), for any ξ ∈ R and g(ξ) ≥ g(−ξ), for any ξ > 0 Summary References Oxford-Princeton Volatility and Liquidity Trading (4) Market Impact Modeling Volatility and Liquidity Trading OxfordPrinceton Π = Z T 2 ξtf (ξ) dt + 0 Motivation − Market Impact Modeling Permanent Component is Necessarily Linear Subordinating in VolumeTime Liquidity Trading Game Z T 2 Z g(ξ)ξ dt − 0 T ξ(T − t)f (−ξ) dt T 2 Z T g(−ξ)(−ξ) dt T 2 T2 T ξ (f (ξ) + f (−ξ)) + ξ (g(−ξ) − g(ξ)) 8 2 ≤ 0 = ⇒ f (−ξ) = −f (ξ), for any ξ ∈ R and g(ξ) ≥ g(−ξ), for any ξ > 0 Summary References Oxford-Princeton Volatility and Liquidity Trading (4) Market Impact Modeling Volatility and Liquidity Trading Π = OxfordPrinceton Permanent Component is Necessarily Linear Subordinating in VolumeTime Liquidity Trading Game Summary References ξ2 ξ1 +ξ2 = = ≤ T ξ1 tf (ξ1 ) dt + 0 − Motivation Market Impact Modeling Z Z ξ2 ξ1 +ξ2 Z T g(ξ1 )ξ1 dt − 0 T ξ2 ξ1 +ξ2 Z ξ2 (T − t)f (−ξ2 ) dt T T ξ2 ξ1 +ξ2 g(−ξ2 )(−ξ2 ) dt T 1 ξ22 T 2 1 ξ12 T 2 · + ξ f (−ξ ) · 2 2 2 (ξ1 + ξ2 )2 2 (ξ1 + ξ2 )2 ξ1 T ξ2 T + g(−ξ2 )ξ2 −g(ξ1 )ξ1 ξ1 + ξ2 ξ1 + ξ2 2 T ξ1 ξ2 ξ1 ξ2 (ξ2 f (ξ1 ) − ξ1 f (ξ2 )) + T (g(−ξ2 ) − g(ξ1 )) 2 (ξ1 + ξ2 )2 ξ1 + ξ2 0 ξ1 f (ξ1 ) ⇒ ξ2 f (ξ1 ) − ξ1 f (ξ2 ) = 0, for any ξ1 , ξ2 ∈ R namely, there ∃γ, s.t. f (ξ) = γξ, for any ξ ∈ R Oxford-Princeton Volatility and Liquidity Trading Market Impact Modeling Volatility and Liquidity Trading Π = OxfordPrinceton Permanent Component is Necessarily Linear Subordinating in VolumeTime Liquidity Trading Game Summary References ξ2 ξ1 +ξ2 = = ≤ T ξ1 tf (ξ1 ) dt + 0 − Motivation Market Impact Modeling Z Z ξ2 ξ1 +ξ2 Z T g(ξ1 )ξ1 dt − 0 T ξ2 ξ1 +ξ2 Z ξ2 (T − t)f (−ξ2 ) dt T T ξ2 ξ1 +ξ2 g(−ξ2 )(−ξ2 ) dt T 1 ξ22 T 2 1 ξ12 T 2 · + ξ f (−ξ ) · 2 2 2 (ξ1 + ξ2 )2 2 (ξ1 + ξ2 )2 ξ1 T ξ2 T + g(−ξ2 )ξ2 −g(ξ1 )ξ1 ξ1 + ξ2 ξ1 + ξ2 2 T ξ1 ξ2 ξ1 ξ2 (ξ2 f (ξ1 ) − ξ1 f (ξ2 )) + T (g(−ξ2 ) − g(ξ1 )) 2 (ξ1 + ξ2 )2 ξ1 + ξ2 0 ξ1 f (ξ1 ) ⇒ ξ2 f (ξ1 ) − ξ1 f (ξ2 ) = 0, for any ξ1 , ξ2 ∈ R namely, there ∃γ, s.t. f (ξ) = γξ, for any ξ ∈ R Oxford-Princeton Volatility and Liquidity Trading Market Impact Modeling Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Permanent Component is Necessarily Linear Subordinating in VolumeTime This analytical argument is supported by empirical studies such as Almgren et al. (05) conducted on large-scale datasets traded at NYSE. Time-homogeneity can be obtained by rescaling real time using the intra-day volume up to that moment, so-called volume time. E.g., a VWAP execution in real time is essentially a constant-intensity trading trajectory in volume time Liquidity Trading Game Summary References Oxford-Princeton Volatility and Liquidity Trading Market Impact Modeling Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Permanent Component is Necessarily Linear Subordinating in VolumeTime This analytical argument is supported by empirical studies such as Almgren et al. (05) conducted on large-scale datasets traded at NYSE. Time-homogeneity can be obtained by rescaling real time using the intra-day volume up to that moment, so-called volume time. E.g., a VWAP execution in real time is essentially a constant-intensity trading trajectory in volume time Liquidity Trading Game Summary References Oxford-Princeton Volatility and Liquidity Trading The Story of a Liquidity Trading Game Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis A distressed trader with maximum inventory x0 , constrained by an exogenous time horizon [0, T ] Objective, to generate cash as much as possible Only allowed to monotonely sell (liquidate), not allowed to buy back at any moment during [0, T ] A perfectly solvent trader, sophisticated and aggressive Able to buy or sell at any moment The only constraint is to be clean-hand by T̄ ≫ T . Each player looks to closed-loop optimal control strategies, aiming to utilize the updates/feedback of market evolution to refine her control. ⇒ Subgame-Perfect. Summary References Oxford-Princeton Volatility and Liquidity Trading The Story of a Liquidity Trading Game Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis A distressed trader with maximum inventory x0 , constrained by an exogenous time horizon [0, T ] Objective, to generate cash as much as possible Only allowed to monotonely sell (liquidate), not allowed to buy back at any moment during [0, T ] A perfectly solvent trader, sophisticated and aggressive Able to buy or sell at any moment The only constraint is to be clean-hand by T̄ ≫ T . Each player looks to closed-loop optimal control strategies, aiming to utilize the updates/feedback of market evolution to refine her control. ⇒ Subgame-Perfect. Summary References Oxford-Princeton Volatility and Liquidity Trading The Story of a Liquidity Trading Game Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis A distressed trader with maximum inventory x0 , constrained by an exogenous time horizon [0, T ] Objective, to generate cash as much as possible Only allowed to monotonely sell (liquidate), not allowed to buy back at any moment during [0, T ] A perfectly solvent trader, sophisticated and aggressive Able to buy or sell at any moment The only constraint is to be clean-hand by T̄ ≫ T . Each player looks to closed-loop optimal control strategies, aiming to utilize the updates/feedback of market evolution to refine her control. ⇒ Subgame-Perfect. Summary References Oxford-Princeton Volatility and Liquidity Trading Setting up the Liquidity Trading Game Model Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis Summary An illiquid asset with permanent component coef γ, temporary component coef λ, intra-day volatility σ, in common knowledge to the two players dZ(t) = (γξ(t) + γη(t))dt + σdW (t) CL control strategies for the two players φ(t, x, y, z) and ψ(t, x, y, z) Given the CL strategy of the opponent, each player solves her optimal control problem Agreeing at the Nash-equilibrium of this game, when nobody has incentive to deviate. References Oxford-Princeton Volatility and Liquidity Trading Setting up the Liquidity Trading Game Model Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis Summary An illiquid asset with permanent component coef γ, temporary component coef λ, intra-day volatility σ, in common knowledge to the two players dZ(t) = (γξ(t) + γη(t))dt + σdW (t) CL control strategies for the two players φ(t, x, y, z) and ψ(t, x, y, z) Given the CL strategy of the opponent, each player solves her optimal control problem Agreeing at the Nash-equilibrium of this game, when nobody has incentive to deviate. References Oxford-Princeton Volatility and Liquidity Trading Setting up the Liquidity Trading Game Model Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis Summary An illiquid asset with permanent component coef γ, temporary component coef λ, intra-day volatility σ, in common knowledge to the two players dZ(t) = (γξ(t) + γη(t))dt + σdW (t) CL control strategies for the two players φ(t, x, y, z) and ψ(t, x, y, z) Given the CL strategy of the opponent, each player solves her optimal control problem Agreeing at the Nash-equilibrium of this game, when nobody has incentive to deviate. References Oxford-Princeton Volatility and Liquidity Trading Setting up the Liquidity Trading Game Model Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis Summary An illiquid asset with permanent component coef γ, temporary component coef λ, intra-day volatility σ, in common knowledge to the two players dZ(t) = (γξ(t) + γη(t))dt + σdW (t) CL control strategies for the two players φ(t, x, y, z) and ψ(t, x, y, z) Given the CL strategy of the opponent, each player solves her optimal control problem Agreeing at the Nash-equilibrium of this game, when nobody has incentive to deviate. References Oxford-Princeton Volatility and Liquidity Trading Setting up the Liquidity Trading Game Model Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis Summary An illiquid asset with permanent component coef γ, temporary component coef λ, intra-day volatility σ, in common knowledge to the two players dZ(t) = (γξ(t) + γη(t))dt + σdW (t) CL control strategies for the two players φ(t, x, y, z) and ψ(t, x, y, z) Given the CL strategy of the opponent, each player solves her optimal control problem Agreeing at the Nash-equilibrium of this game, when nobody has incentive to deviate. References Oxford-Princeton Volatility and Liquidity Trading The Stochastic Optimal Control Problem Volatility and Liquidity Trading OxfordPrinceton Given the CL strategy ψ(· · · ) of the 2nd player, the optimal control problem for the 1st player U (t, x, y, z) = min E ξ(·)∈A Motivation hR T t Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis (Z(s) + λ(ξ(s) + ψ(s, X(s), Y (s), Z(s)))) # ·ξ(s) ds X(t) = x Y (t) = y Z(t) = z where dZ(t) = (γξ(t) + γψ(t, X(t), Y (t), Z(t)))dt + σdW (t) dX(t) = ξ(t)dt dY (t) = ψ(t, X(t), Y (t), Z(t))dt Summary References Oxford-Princeton Volatility and Liquidity Trading The Stochastic Optimal Control Problem Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis Summary References The HJB equation for player 1  Ut + min λξ 2 + λψ(t, x, y, z)ξ + zξ + ξUx + ψ(t, x, y, z)Uy 1 +(γξ + γψ(t, x, y, z))Uz + σ 2 Uzz |ξ ≤ 0 } = 0 2 1 −Ut = ψ(t, x, y, z)(Uy + γUz ) + σ 2 Uzz 2 1 − [(z + λψ(t, x, y, z) + Ux + γUz )+ ]2 4λ for t ∈ [0, T ], x ∈ [0, x0 ], y ∈ R, z ∈ R+ , and where φ(t, x, y, z) = − 1 (z + λψ(t, x, y, z) + Ux + γUz )+ 2λ Oxford-Princeton Volatility and Liquidity Trading The Stochastic Optimal Control Problem Volatility and Liquidity Trading OxfordPrinceton Given the CL strategy φ(· · · ) of the 1st player, the optimal control problem for the 2nd player V (t, x, y, z) = min E η(·)∈A Motivation hR T t Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis (Z(s) + λ(φ(s, X(s), Y (s), Z(s)) + η(s))) # ·η(s) ds X(t) = x Y (t) = y Z(t) = z where dZ(t) = (γφ(t, X(t), Y (t), Z(t)) + γη(t))dt + σdW (t) dX(t) = φ(t, X(t), Y (t), Z(t))dt dY (t) = η(t)dt Summary References Oxford-Princeton Volatility and Liquidity Trading The Stochastic Optimal Control Problem Volatility and Liquidity Trading OxfordPrinceton The HJB equation for player 2 1 (z + λφ(t, x, y, z) + Vy + γVz )2 4λ 1 + φ(t, x, y, z)(Vx + γVz ) + σ 2 Vzz = 0 2 Vt − Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis for t ∈ [0, T ], x ∈ [0, x0 ], y ∈ R, z ∈ R+ , and where ψ(t, x, y, z) = − 1 (z + λφ(t, x, y, z) + Vy + γVz ) 2λ Summary References Oxford-Princeton Volatility and Liquidity Trading Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis Clean up the entanglement of φ(· · · ) and ψ(· · · ), we get 1 (z + 2Ux − Vy + 2γUz − γVz )+ 3λ 1 = − (δ(t, x, y, z))+ 3λ  − 1 (z − Ux + 2Vy − γUz + 2γVz )    3λ if δ(t, x, y, z) > 0 ψ(t, x, y, z) = 1 − (z + V  2λ y + γVz )   if δ(t, x, y, z) ≤ 0 φ(t, x, y, z) = − where δ(t, x, y, z) := z + 2Ux − Vy + 2γUz − γVz . Summary References Oxford-Princeton Volatility and Liquidity Trading The Nash-Equilibrium of This Game Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis Summary References Define δ(t, x, y, z) := z + 2Ux − Vy + 2γUz − γVz and δ ∗ (t, x, y, z) := z + 2Vy − Ux + 2γVz − γUz where δ(t, x, y, z) > 0,  1 ∗ −Ut = − 3λ δ (t, x, y, z)(Uy + γUz )    1 − 9λ (δ(t, x, y, z))2 + 21 σ 2 Uzz 1 δ(t, x, y, z)(Vx + γVz ) −Vt = − 3λ    1 − 9λ (δ ∗ (t, x, y, z))2 + 12 σ 2 Vzz where δ(t, x, y, z) ≤ 0,  1 (z + Vy + γVz )(Uy + γUz ) + 21 σ 2 Uzz −Ut = − 2λ 1 (z + Vy + γVz )2 + 21 σ 2 Vzz −Vt = − 4λ Oxford-Princeton Volatility and Liquidity Trading Numerical Analysis of the NE Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis In order to obtain stable numerical solutions, let us induce some viscosity condiments to the numerical scheme when solving the PDE system. For the higher-order partials, instead of σ 2 Uzz consider σ 2 Uzz + 21 ǫσ 2 Uxx + ǫσ 2 Uyy and let ǫ → 0 Key verification: the numerical solution obtained is not sensitive at all to the choice of ǫ Summary References Oxford-Princeton Volatility and Liquidity Trading Numerical Analysis of the NE Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis A trial example: the after-story of the liquidity trading game The HJB equation for the 2nd player during the sequel period [T, T̄ ]  Vt + min λη 2 + (z + Vy + γVz )η |η ≤ 0 1 1 + ǫσ 2 Vyy + σ 2 Vzz = 0 2 2 −Vt = − 2 1 1 1  (z + Vy + γVz )+ + σ 2 Vzz + ǫσ 2 Vyy 4λ 2 2 Summary References Oxford-Princeton Volatility and Liquidity Trading Numerical Analysis of the NE Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis A trial example: the after-story of the liquidity trading game The HJB equation for the 2nd player during the sequel period [T, T̄ ]  Vt + min λη 2 + (z + Vy + γVz )η |η ≤ 0 1 1 + ǫσ 2 Vyy + σ 2 Vzz = 0 2 2 −Vt = − 2 1 1 1  (z + Vy + γVz )+ + σ 2 Vzz + ǫσ 2 Vyy 2 4λ 2 Summary References Oxford-Princeton Volatility and Liquidity Trading Numerical Analysis of the NE Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis A trial example: the after-story of the liquidity trading game The HJB equation for the 2nd player during the sequel period [T, T̄ ]  Vt + min λη 2 + (z + Vy + γVz )η |η ≤ 0 1 1 + ǫσ 2 Vyy + σ 2 Vzz = 0 2 2 −Vt = − 2 1 1  1 (z + Vy + γVz )+ + σ 2 Vzz + ǫσ 2 Vyy 4λ 2 2 Summary References Oxford-Princeton Volatility and Liquidity Trading Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis Summary References Figure: The numerical solution almost does not depend on the choice of coefficient for the artificial viscosity term. Here, ǫ = 2.5 ∗ 10−3 Oxford-Princeton Volatility and Liquidity Trading Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis Summary References Figure: The numerical solution almost does not depend on the choice of coefficient for the artificial viscosity term. Here, ǫ = 1.0 ∗ 10−4 Oxford-Princeton Volatility and Liquidity Trading Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis Summary References Figure: The numerical solution almost does not depend on the choice of coefficient for the artificial viscosity term Oxford-Princeton Volatility and Liquidity Trading Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis Summary References Figure: The numerical solution almost does not depend on the choice of coefficient for the artificial viscosity term Oxford-Princeton Volatility and Liquidity Trading Volatility Does Enter the Picture and Make A Difference Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis Summary References Figure: The terminal value function for the predator under different volatility levels Oxford-Princeton Volatility and Liquidity Trading Volatility Does Enter the Picture and Make A Difference Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game The Stochastic Optimal Control Problem The NashEquilibrium Numerical Analysis Summary References Figure: The terminal value function for the predator under different volatility levels Oxford-Princeton Volatility and Liquidity Trading Summary and Ongoing Work Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Liquidity Trading Game Summary References Strategic interplay is an important source for Market Liquidity behaviors Adopt a reasonable market impact model, and think in volume time Closed-Loop strategies guarantee subgame perfectness, and usher volatility into the picture Numerical analysis of the NE of such a liquidity trading game Oxford-Princeton Volatility and Liquidity Trading References Volatility and Liquidity Trading T. Başar and G. Olsder (1999). Dynamic noncooperative game theory. SIAM Classics in Applied Mathematics. OxfordPrinceton M. Brunnermeier and L. Pedersen (2005). Predatory trading. Journal of Finance, 60(4):1825–1863. Motivation B. Carlin, et al. (2007). Episodic Liquidity Crises: Cooperative and Predatory Trading. Journal of Finance, 62(5):2235–2274. Market Impact Modeling Liquidity Trading Game Summary References E. Dockner, et al. (2000). Differential Games in Economics and Management Science. Cambridge University Press. W.H. Fleming and H.M. Soner (2006). Controlled Markov Processes and Viscosity Solutions (2nd Ed). Spinger. T. Schoneborn and A. Schied (2008). Liquidation in the face of adversity: stealth vs. sunshine trading, predatory trading vs. liquidity provision. Working paper. J.W. Thomas (1995). Numerical Partial Differential Equations. Springer. Oxford-Princeton Volatility and Liquidity Trading The End Volatility and Liquidity Trading OxfordPrinceton Motivation Market Impact Modeling Thank You Liquidity Trading Game Summary References Oxford-Princeton Volatility and Liquidity Trading