Imperfect Competition Among Liquidity Providers

Transcription

1 Imperfect Competition Among Liquidity Providers Angad Singh Caltech November 2018 Angad Singh (Caltech) November / 22

2 Table of Contents 1 Motivation & Related Work 2 The Model 3 Equilibrium Characterization 4 Equilibrium Analysis 5 Summary Angad Singh (Caltech) November / 22

3 Table of Contents 1 Motivation & Related Work 2 The Model 3 Equilibrium Characterization 4 Equilibrium Analysis 5 Summary Angad Singh (Caltech) November / 22

4 Motivation Angad Singh (Caltech) November / 22

5 Motivation Angad Singh (Caltech) November / 22

6 Motivation LO specifies a price and a quantity Angad Singh (Caltech) November / 22

7 Motivation LO specifies a price and a quantity MO specifies only a quantity Angad Singh (Caltech) November / 22

8 Motivation LO specifies a price and a quantity MO specifies only a quantity Trade means MO executes against LO Angad Singh (Caltech) November / 22

9 Motivation LO specifies a price and a quantity price elastic demand MO specifies only a quantity Trade means MO executes against LO Angad Singh (Caltech) November / 22

10 Motivation LO specifies a price and a quantity price elastic demand MO specifies only a quantity price inelastic demand Trade means MO executes against LO Angad Singh (Caltech) November / 22

11 Related Work Angad Singh (Caltech) November / 22

12 Related Work Grossman-Miller (1985) Angad Singh (Caltech) November / 22

13 Related Work Grossman-Miller (1985) Kyle (1989) Angad Singh (Caltech) November / 22

14 Related Work Grossman-Miller (1985) Kyle (1989) Almgren & Chriss (2001), Cartea & Jaimungal Angad Singh (Caltech) November / 22

15 Related Work Grossman-Miller (1985) Kyle (1989) Almgren & Chriss (2001), Cartea & Jaimungal Garleanu & Pedersen (2016) Angad Singh (Caltech) November / 22

16 Related Work Grossman-Miller (1985) Kyle (1989) Almgren & Chriss (2001), Cartea & Jaimungal Garleanu & Pedersen (2016) Bouchard, Fukasawa, Herdegen & Muhle-Karbe (2018) Angad Singh (Caltech) November / 22

17 Related Work Grossman-Miller (1985) Kyle (1989) Almgren & Chriss (2001), Cartea & Jaimungal Garleanu & Pedersen (2016) Bouchard, Fukasawa, Herdegen & Muhle-Karbe (2018) Rostek & Wretka (2015) Angad Singh (Caltech) November / 22

18 Related Work Grossman-Miller (1985) Kyle (1989) Almgren & Chriss (2001), Cartea & Jaimungal Garleanu & Pedersen (2016) Bouchard, Fukasawa, Herdegen & Muhle-Karbe (2018) Rostek & Wretka (2015) Sannikov & Skrzypacz (2016) Angad Singh (Caltech) November / 22

19 Related Work Grossman-Miller (1985) Kyle (1989) Almgren & Chriss (2001), Cartea & Jaimungal Garleanu & Pedersen (2016) Bouchard, Fukasawa, Herdegen & Muhle-Karbe (2018) Rostek & Wretka (2015) Sannikov & Skrzypacz (2016) Ma, Wang & Zhang (2015) Angad Singh (Caltech) November / 22

20 Table of Contents 1 Motivation & Related Work 2 The Model 3 Equilibrium Characterization 4 Equilibrium Analysis 5 Summary Angad Singh (Caltech) November / 22

21 The Model Angad Singh (Caltech) November / 22

22 The Model Single asset in zero net supply Angad Singh (Caltech) November / 22

23 The Model Single asset in zero net supply Smooth trading on an infinite horizon Angad Singh (Caltech) November / 22

24 The Model Single asset in zero net supply Smooth trading on an infinite horizon Two types of traders: Angad Singh (Caltech) November / 22

25 The Model Single asset in zero net supply Smooth trading on an infinite horizon Two types of traders: N Market Makers (MMs) Angad Singh (Caltech) November / 22

26 The Model Single asset in zero net supply Smooth trading on an infinite horizon Two types of traders: N Market Makers (MMs) Liquidity Traders (LTs) Angad Singh (Caltech) November / 22

27 The Model Single asset in zero net supply Smooth trading on an infinite horizon Two types of traders: N Market Makers (MMs) Liquidity Traders (LTs) Trades occur at a uniform price p t to be determined endogenously Angad Singh (Caltech) November / 22

28 The Model Angad Singh (Caltech) November / 22

29 The Model LTs inventory is S t, where ds t = F t dt df t = ψf t dt + σ F db F t, ψ > 0 Angad Singh (Caltech) November / 22

30 The Model LTs inventory is S t, where ds t = F t dt df t = ψf t dt + σ F dbt F, ψ > 0 MMs inventories evolve according to dx n t = q n t dt Angad Singh (Caltech) November / 22

31 The Model LTs inventory is S t, where ds t = F t dt df t = ψf t dt + σ F dbt F, ψ > 0 MMs inventories evolve according to dx n t = q n t dt where q n t will be determined endogenously Angad Singh (Caltech) November / 22

32 The Model LTs inventory is S t, where ds t = F t dt df t = ψf t dt + σ F dbt F, ψ > 0 MMs inventories evolve according to dx n t = q n t dt where q n t will be determined endogenously MMs cash evolves according to dc n t = q n t p t dt Angad Singh (Caltech) November / 22

33 MM Objectives Angad Singh (Caltech) November / 22

34 MM Objectives MMs valuation of the stock is exogenously given as D t where dd t = µdt + σ D db D t and {B D t } is independent of {B F t } Angad Singh (Caltech) November / 22

35 MM Objectives MMs valuation of the stock is exogenously given as D t where dd t = µdt + σ D db D t and {B D t } is independent of {B F t } MMs marked-to-market wealth is W n t = C n t + X n t D t Angad Singh (Caltech) November / 22

36 MM Objectives MMs valuation of the stock is exogenously given as D t where dd t = µdt + σ D db D t and {B D t } is independent of {B F t } MMs marked-to-market wealth is MMs want to maximize W n t = C n t + X n t D t Angad Singh (Caltech) November / 22

37 MM Objectives MMs valuation of the stock is exogenously given as D t where dd t = µdt + σ D db D t and {B D t } is independent of {B F t } MMs marked-to-market wealth is W n t = C n t + X n t D t MMs want to maximize [ E e ρt( dwt n γ ) ] 2 d W n t 0 Angad Singh (Caltech) November / 22

38 MM Objectives MMs valuation of the stock is exogenously given as D t where dd t = µdt + σ D db D t and {B D t } is independent of {B F t } MMs marked-to-market wealth is W n t = C n t + X n t D t MMs want to maximize [ E e ρt( dwt n γ ) ] 0 2 d W n t = [ E e ρt( qt n (p t D t ) + µxt n γσ2 D 2 (X t n ) 2) ] dt 0 Angad Singh (Caltech) November / 22

39 The Trading Mechanism Angad Singh (Caltech) November / 22

40 The Trading Mechanism At time t each MM submits a demand schedule of the form α n t β n t q n t = p t Angad Singh (Caltech) November / 22

41 The Trading Mechanism At time t each MM submits a demand schedule of the form α n t β n t q n t = p t p t and q n t are then given implicitly by the N + 1 equations α n t β n t q n t = p t q 1 t + + q N t = F t Angad Singh (Caltech) November / 22

42 The Trading Mechanism At time t each MM submits a demand schedule of the form α n t β n t q n t = p t p t and q n t are then given implicitly by the N + 1 equations α n t β n t q n t = p t q 1 t + + q N t = F t Price and trades are set in Nash equilibrium Angad Singh (Caltech) November / 22

43 Equilibrium Concept Angad Singh (Caltech) November / 22

44 Equilibrium Concept A profile α 1 t, β 1 t,, α N t, β N t is admissible if Angad Singh (Caltech) November / 22

45 Equilibrium Concept A profile αt 1, βt 1,, αt N, βt N 1 βt n > 0 a.s. is admissible if Angad Singh (Caltech) November / 22

46 Equilibrium Concept A profile αt 1, βt 1,, αt N, βt N 1 βt n > 0 a.s. is admissible if 2 E[e ρt (X n t ) 2 ] 0 as t Angad Singh (Caltech) November / 22

47 Equilibrium Concept A profile αt 1, βt 1,, αt N, βt N 1 βt n > 0 a.s. is admissible if 2 E[e ρt (Xt n ) 2 ] 0 as t ) 3 αt n, βt n σ ({D s } 0 s t, {p s } 0 s<t, {Xs n } 0 s t, {qs n } 0 s<t, S 0 Angad Singh (Caltech) November / 22

48 Equilibrium Concept A profile αt 1, βt 1,, αt N, βt N 1 βt n > 0 a.s. is admissible if 2 E[e ρt (Xt n ) 2 ] 0 as t ) 3 αt n, βt n σ ({D s } 0 s t, {p s } 0 s<t, {Xs n } 0 s t, {qs n } 0 s<t, S 0 A profile is linear and symmetric if a < 0, e > 0, and b, c, ξ R s.t. α n t = ax n t + bd t + cs t + ξ β n t = e Angad Singh (Caltech) November / 22

49 Equilibrium Concept A profile αt 1, βt 1,, αt N, βt N 1 βt n > 0 a.s. is admissible if 2 E[e ρt (Xt n ) 2 ] 0 as t ) 3 αt n, βt n σ ({D s } 0 s t, {p s } 0 s<t, {Xs n } 0 s t, {qs n } 0 s<t, S 0 A profile is linear and symmetric if a < 0, e > 0, and b, c, ξ R s.t. α n t = ax n t + bd t + cs t + ξ β n t = e Will focus on linear symmetric Nash equilibrium Angad Singh (Caltech) November / 22

50 Table of Contents 1 Motivation & Related Work 2 The Model 3 Equilibrium Characterization 4 Equilibrium Analysis 5 Summary Angad Singh (Caltech) November / 22

51 Explicit Equilibrium Theorem Angad Singh (Caltech) November / 22

52 Explicit Equilibrium Theorem If N 3 then there is a unique linear symmetric Nash equilibrium. Angad Singh (Caltech) November / 22

53 Explicit Equilibrium Theorem If N 3 then there is a unique linear symmetric Nash equilibrium. In equilibrium the price is p t = D t + µ ρ 1 γ ρ N σ2 D S t γ N 1 σ 2 ( D 1 N N 2 ρ + ψ δ + 1 ) F t ρ Angad Singh (Caltech) November / 22

54 Explicit Equilibrium Theorem If N 3 then there is a unique linear symmetric Nash equilibrium. In equilibrium the price is p t = D t + µ ρ 1 γ ρ N σ2 D S t γ N 1 σ 2 ( D 1 N N 2 ρ + ψ δ + 1 ) F t ρ and trading rates are ( qt n = κ X n t S t N ) + 1 N F t Angad Singh (Caltech) November / 22

55 Explicit Equilibrium Theorem If N 3 then there is a unique linear symmetric Nash equilibrium. In equilibrium the price is p t = D t + µ ρ 1 γ ρ N σ2 D S t γ N 1 σ 2 ( D 1 N N 2 ρ + ψ δ + 1 ) F t ρ and trading rates are where ( qt n = κ X n t S t N ) + 1 N F t κ = ρ(n 2) ρ + ψ ρ + δ δ = ρ 2 + 2ρ(ρ + ψ)(n 2) Angad Singh (Caltech) November / 22

56 Sketch of Proof Angad Singh (Caltech) November / 22

57 Sketch of Proof On equilibrium ˆp t = a N S t + bd t + cs t + ξ e N F t ˆq t n = a ( ˆX t n S ) t + 1 e N N F t Angad Singh (Caltech) November / 22

58 Sketch of Proof On equilibrium Off equilibrium p t = ˆp t = a N S t + bd t + cs t + ξ e N F t ˆq t n = a ( ˆX t n S ) t + 1 e N N F t a N 1 (S t X t ) + bd t + cs t + ξ e N 1 (F t q t ) p t = α t β t q t Angad Singh (Caltech) November / 22

59 Sketch of Proof Optimize directly over {q t } Angad Singh (Caltech) November / 22

60 Sketch of Proof Optimize directly over {q t } Almgren-Chriss type problem Angad Singh (Caltech) November / 22

61 Sketch of Proof Optimize directly over {q t } Almgren-Chriss type problem p t = a N 1 S t + bd t + cs t + ξ e N 1 F t a N 1 X t + e N 1 q t Angad Singh (Caltech) November / 22

62 Sketch of Proof Optimize directly over {q t } Almgren-Chriss type problem p t = a N 1 S t + bd t + cs t + ξ = M t + ΛX t + λq t e N 1 F t a N 1 X t + e N 1 q t Angad Singh (Caltech) November / 22

63 Sketch of Proof Optimize directly over {q t } Almgren-Chriss type problem p t = a N 1 S t + bd t + cs t + ξ = M t + ΛX t + λq t e N 1 F t Dynamic Programming, HJB, Feedback Controller a N 1 X t + e N 1 q t Angad Singh (Caltech) November / 22

64 Sketch of Proof Optimize directly over {q t } Almgren-Chriss type problem p t = a N 1 S t + bd t + cs t + ξ = M t + ΛX t + λq t e N 1 F t Dynamic Programming, HJB, Feedback Controller [ 2e ( a ) N 1 ˆq = V x N 1 + c s a N 1 X t + a x + (b 1)d + ξ e N 1 e N 1 q t ] N 1 f Angad Singh (Caltech) November / 22

65 Table of Contents 1 Motivation & Related Work 2 The Model 3 Equilibrium Characterization 4 Equilibrium Analysis 5 Summary Angad Singh (Caltech) November / 22

66 Interpreting the Price Angad Singh (Caltech) November / 22

67 Interpreting the Price p t = D t + µ ρ 1 γ ρ N σ2 D S t γ N 1 σ 2 ( D 1 N N 2 ρ + ψ δ + 1 ) F t ρ Angad Singh (Caltech) November / 22

68 Interpreting the Price p t = D t + µ ρ 1 γ ρ N σ2 D S t γ N 1 σ 2 ( D 1 N N 2 ρ + ψ δ + 1 ) F t ρ First term is MMs current asset valuation Angad Singh (Caltech) November / 22

69 Interpreting the Price p t = D t + µ ρ 1 γ ρ N σ2 D S t γ N 1 σ 2 ( D 1 N N 2 ρ + ψ δ + 1 ) F t ρ First term is MMs current asset valuation Second term is a premium for expected valuation growth Angad Singh (Caltech) November / 22

70 Interpreting the Price p t = D t + µ ρ 1 γ ρ N σ2 D S t γ N 1 σ 2 ( D 1 N N 2 ρ + ψ δ + 1 ) F t ρ First term is MMs current asset valuation Second term is a premium for expected valuation growth Third term is a discount to compensate MMs for bearing risk Angad Singh (Caltech) November / 22

71 Interpreting the Price p t = D t + µ ρ 1 γ ρ N σ2 D S t γ N 1 σ 2 ( D 1 N N 2 ρ + ψ δ + 1 ) F t ρ First term is MMs current asset valuation Second term is a premium for expected valuation growth Third term is a discount to compensate MMs for bearing risk Fourth term captures liquidity Angad Singh (Caltech) November / 22

72 Interpreting the Price p t = D t + µ ρ 1 γ ρ N σ2 D S t γ N 1 σ 2 ( D 1 N N 2 ρ + ψ δ + 1 ) F t ρ First term is MMs current asset valuation Second term is a premium for expected valuation growth Third term is a discount to compensate MMs for bearing risk Fourth term captures liquidity Definition N 1 σ 2 D ( Price Impact := γ 1 N N 2 ρ+ψ δ + 1 ρ 1 Liquidity := Price Impact ) Angad Singh (Caltech) November / 22

73 Comparative Statics for Liquidity Proposition Angad Singh (Caltech) November / 22

74 Comparative Statics for Liquidity Proposition Liquidity < 0. Liquidity is decreasing in market makers risk aversion. 1 γ Angad Singh (Caltech) November / 22

75 Comparative Statics for Liquidity Proposition Liquidity < 0. Liquidity is decreasing in market makers risk aversion. 1 γ 2 σ D Liquidity < 0. Liquidity is decreasing in fundamental volatility. Angad Singh (Caltech) November / 22

76 Comparative Statics for Liquidity Proposition Liquidity < 0. Liquidity is decreasing in market makers risk aversion. 1 γ 2 σ D Liquidity < 0. Liquidity is decreasing in fundamental volatility. 3 If γ N is held fixed then N Liquidity > 0. Liquidity is increasing in market maker competition. Angad Singh (Caltech) November / 22

77 Comparative Statics for Liquidity Proposition Liquidity < 0. Liquidity is decreasing in market makers risk aversion. 1 γ 2 σ D Liquidity < 0. Liquidity is decreasing in fundamental volatility. 3 If γ N is held fixed then N Liquidity > 0. Liquidity is increasing in market maker competition. Liquidity > 0. Liquidity is decreasing in order flow risk. 4 ψ Angad Singh (Caltech) November / 22

78 Limiting Cases Angad Singh (Caltech) November / 22

79 Limiting Cases Price Impact 0 as ψ Angad Singh (Caltech) November / 22

80 Limiting Cases Price Impact 0 as ψ Price Impact γ 0 ρ σ 2 D ρ+ψ as N with γ N = γ 0 fixed Angad Singh (Caltech) November / 22

81 Limiting Cases Price Impact 0 as ψ Price Impact γ 0 ρ σ 2 D ρ+ψ as N with γ N = γ 0 fixed In both cases the price is asymptotic to D t + µ [ ] ρ E t e ρ(t t) γ 0 σd 2 S T dt t Angad Singh (Caltech) November / 22

82 Interpreting the Trading Rates Angad Singh (Caltech) November / 22

83 Interpreting the Trading Rates ( qt n = κ X n t S t N ) + 1 N F t Angad Singh (Caltech) November / 22

84 Interpreting the Trading Rates ( qt n = κ X n t S t N ) + 1 N F t MMs aggregate time t inventory is S t Angad Singh (Caltech) November / 22

85 Interpreting the Trading Rates ( qt n = κ X n t S t N ) + 1 N F t MMs aggregate time t inventory is S t The efficient allocation is for each MM to hold St N Angad Singh (Caltech) November / 22

86 Interpreting the Trading Rates ( qt n = κ X n t S t N ) + 1 N F t MMs aggregate time t inventory is S t The efficient allocation is for each MM to hold St N Second term says that new supply shocks are absorbed efficiently Angad Singh (Caltech) November / 22

87 Interpreting the Trading Rates ( qt n = κ X n t S t N ) + 1 N F t MMs aggregate time t inventory is S t The efficient allocation is for each MM to hold St N Second term says that new supply shocks are absorbed efficiently First term says that existing misallocations move towards efficiency Angad Singh (Caltech) November / 22

88 Interpreting the Trading Rates ( qt n = κ X n t S t N ) + 1 N F t MMs aggregate time t inventory is S t The efficient allocation is for each MM to hold St N Second term says that new supply shocks are absorbed efficiently First term says that existing misallocations move towards efficiency Xt n = S t N + e κt( X0 n S ) 0 N Angad Singh (Caltech) November / 22

89 Interpreting the Trading Rates ( qt n = κ X n t S t N ) + 1 N F t MMs aggregate time t inventory is S t The efficient allocation is for each MM to hold St N Second term says that new supply shocks are absorbed efficiently First term says that existing misallocations move towards efficiency Xt n = S t N + e κt( X0 n S ) 0 N Definition Rate of Convergence to Efficiency := κ := ρ(n 2) ρ+ψ ρ+δ Angad Singh (Caltech) November / 22

90 Table of Contents 1 Motivation & Related Work 2 The Model 3 Equilibrium Characterization 4 Equilibrium Analysis 5 Summary Angad Singh (Caltech) November / 22

91 Summary Angad Singh (Caltech) November / 22

92 Summary Stochastic differential game to compete for order flow Angad Singh (Caltech) November / 22

93 Summary Stochastic differential game to compete for order flow Endogenous liquidity level driven by Angad Singh (Caltech) November / 22

94 Summary Stochastic differential game to compete for order flow Endogenous liquidity level driven by Imperfect competition Angad Singh (Caltech) November / 22

95 Summary Stochastic differential game to compete for order flow Endogenous liquidity level driven by Imperfect competition Order flow risk Angad Singh (Caltech) November / 22

96 Summary Stochastic differential game to compete for order flow Endogenous liquidity level driven by Imperfect competition Order flow risk Risk discount today is the present value of future risk discounts Angad Singh (Caltech) November / 22

97 Summary Stochastic differential game to compete for order flow Endogenous liquidity level driven by Imperfect competition Order flow risk Risk discount today is the present value of future risk discounts Risk reallocation among liquidity providers can be slow Angad Singh (Caltech) November / 22