Mean-field and n -agent games for optimal investment under relative - PowerPoint PPT Presentation

Mean-field and n -agent games for optimal investment under relative performance criteria WCMF 2017 Seattle Thaleia Zariphopoulou UT-Austin Oxford-Man Institute, Oxford

Portfolio management under competition and asset specialization

References • Mean-field and n -agent games for optimal investment under relative performance criteria (with Dan Lacker) • Relative forward performance criteria: passive and competitive cases, under asset specialization and diversification (with Tianran Geng)

Competition among fund manangers in mutual and hedge funds Chevalier and Ellison (1997) Sirri and Tufano (1998) Agarwal, Daniel and Naik (2004) Ding, Getmansky, Liang and Wermers (2007) Goriaev et al. (2003) Li and Tiwari (2006) Gallaher, Kaniel and Starks(2006) Brown, Goetzmann and Park(2001) Kempf and Ruenzi (2008) Basak and Makarov (2013, 2016) Espinoza and Touzi (2013), .... Career advancement motives, seeking higher money inflows from their clients, preferential compensation contracts,... Only two managers, mainly discrete-time models, criteria involving risk neutrality, relative performance with respect to an absolute benchmark or a critical threshhold, constraints on the managers’ risk aversion parameters, ...

Asset specialization for fund managers Brennan (1975) Merton (1987) Coval and Moskowitz (1999) Karperczyk, Sialm and Zheng (2005) van Nieuwerburgh and Veldkamp (2009, 2010) Uppal and Wang (2003) Boyle, Garlappi, Uppal and Wang (2012) Liu (2012) Mitton and Vorkink (2007), ... Familiarity, learning cost reduction, ambiguity aversion, solvency requirements, trading costs and constraints, liquidation risks, informational frictions, ....

The n -player game

The competition setting n fund managers • common investment horizon [0 , T ] • common riskless asset (bond) • asset specialization • individual stock S i , i = 1 , ..., n dS i t = µ i dt + ν i dW i t + σ i dB t , S i t µ i > 0 , σ i ≥ 0 , and ν i ≥ 0 , σ i + ν i > 0 � B t , W 1 � ( � W n t , . . . , W n t ) t ∈ [0 ,T ] := t ∈ [0 ,T ] is an ( n + 1) -dim. BM t • B common noise and W i an indiosyncratic noise

Special case Single stock • Coefficients ( µ i , σ i ) = ( µ, σ ) , and ν i = 0 , i = 1 , ..., n • All stocks are identical • Managers invest in identical markets • Managers differ only in their risk preferences and personal competition concerns

Policies and wealth processes i th fund manager, i = 1 , ..., n • Uses self-financing portfolios π i (other usual admissibility conds) • Trades in [0 , T ] • Has wealth process X i dX i t = π i t ( µ i dt + ν i dW i t + σ i dB t ) • W i : indiosyncratic noise • B : common noise

Utility under competition • Utility function U i : R 2 → R depends on both her individual wealth x , and the average wealth of all investors , m , � � − 1 U i ( x, m ) := − exp ( x − θ i m ) δ i • δ i > 0 is the personal risk tolerance • θ i ∈ [0 , 1] as the personal social comparison parameter • θ i = 0 means no relative concerns • Both δ i , θ i are unitless quantities

Expected utility under competition • Fund managers choose admissible strategies π 1 t , . . . , π n t , t ∈ [0 , T ] • The payoff for investor i is given by � � ��� � − 1 T − θ i ¯ J i ( π 1 , . . . , π n ) := E X i − exp X T δ i • Average wealth of the managers’ population n � X T = 1 ¯ X k T n k =1 • Alternatively, � � ��� � − 1 J i ( π 1 , . . . , π n ) = E (1 − θ i ) X i T + θ i ( X i T − ¯ − exp X T ) δ i • X i T : personal, absolute wealth T − ¯ • X i X T : personal, relative to the population, wealth

Nash equilibrium

Nash equilibrium • A vector ( π 1 , ∗ , . . . , π n, ∗ ) of admissible strategies is a Nash equilibrium if, for all admissible π i ∈ A and i = 1 , . . . , n , J i ( π 1 , ∗ , . . . , π i, ∗ , . . . , π n, ∗ ) ≥ J i ( π 1 , ∗ , . . . , π i − 1 , ∗ , π i , π i +1 , ∗ , . . . , π n, ∗ ) • A constant Nash equilibrium is one in which, for each i , π i, ∗ is constant in time, i.e., π i, ∗ = π i, ∗ 0 , for all t ∈ [0 , T ] t • A constant Nash equilibrium is thus a vector π ∗ = ( π 1 , ∗ , . . . , π n, ∗ ) ∈ R n

Construction of Nash equilibria

Main result • δ i > 0 , θ i ∈ [0 , 1] • µ i > 0 , σ i ≥ 0 , ν i ≥ 0 , and σ i + ν i > 0 • Define the constants � n � n σ 2 µ i σ i ϕ n := 1 ψ n := 1 i =1 δ i and i =1 θ i i n σ 2 i + ν 2 n σ 2 i + ν 2 i (1 − θ i /n ) i (1 − θ i /n ) Nash equilibria • If ψ n < 1 , there exists a unique constant equilibrium, given by µ i σ i ϕ n π i, ∗ = δ i i (1 − θ i /n ) + θ i σ 2 i + ν 2 σ 2 i + ν 2 i (1 − θ i /n ) 1 − ψ n • If ψ n = 1 , there is no constant equilibrium

Main steps in the proof • Fix i and assume that all other k th agents, k � = i, follow constant investment strategies, α k ∈ R • Competitor’s wealth X k t , � � X k t = x k µ k t + ν k W k 0 + α k t + σ k B t • Competitors’ aggregate wealth � Y t := 1 X k t n k � = i • The i th fund manager solves the optimization problem   ��� � �� � � � − 1 1 − θ i 0 , Y 0 = 1 X i � � X 0 = x i x k  − exp  E sup T − θ i Y T 0 δ i n n π ∈A k � = i with dX t = π t ( µ i dt + ν i dW i t + σ i dB t ) , ναdW k dY t = � µαdt + � t + � σαdB t � � � µα := 1 να := 1 σα := 1 µ k α k , � ν k α k and σ k α k � � n n n k � = i k � = i k � = i

Connection with indifference valuation   ��� � �� � � � − 1 1 − θ i 0 , Y 0 = 1 X i � � X 0 = x i x k  − exp  sup E T − θ i Y T 0 δ i n n π ∈A k � = i The i th fund manager → writer of liability G ( Y T ) := θ i 1 − θ i /n Y T , � � Risk aversion γ i := 1 1 − θ i δ i n Thus, the above supremum is equal to v ( X 0 , Y 0 , 0) , with v ( x, y, t ) solving the HJB eqn � 1 � 2( σ 2 i + ν 2 i ) π 2 v xx + π ( µ i v x + σ i � v t + max σαv xy ) π ∈ R � � +1 σα 2 + 1 � ( να ) 2 � v yy + � µαv y = 0 , 2 n � for ( x, y, t ) ∈ R × R × [0 , T ] , and � ( να ) 2 := 1 k � = i ν 2 k α 2 k , n � �� � �� − 1 1 − θ i v ( x, y, T ) = − e − γ i ( x − G ( y )) = − exp x − θ i y δ i n

Candidate Nash equilibria • The i th agent’s optimal feedback control µ i v x ( x, y, t ) σ i � σαv xy ( x, y, t ) π i, ∗ ( x, y, t ) := − i ) v xx ( x, y, t ) − ( σ 2 i + ν 2 ( σ 2 i + ν 2 i ) v xx ( x, y, t ) • The HJB equation admits separable solutions v ( x, y, t ) = − e − γ i x F ( y, t ) • It then turns out that the optimal policy is of the form δ i µ i θ i σ i π i, ∗ = i )(1 − θ i /n ) + i )(1 − θ i /n ) � σα ( σ 2 i + ν 2 ( σ 2 i + ν 2

Construction of Nash equilibria • For a candidate portfolio vector ( α 1 , . . . , α n ) to be a Nash equilibrium, we need π i, ∗ = α i , i = 1 , ..., n δ i µ i θ i σ i a i = i )(1 − θ i /n ) + i )(1 − θ i /n ) � σα ( σ 2 i + ν 2 ( σ 2 i + ν 2 • Set n � σα := 1 σα + 1 σ k α k = � nσ i α i n k =1 • Then, we must have θ i σ 2 δ i µ i + σ i θ i σα α i = π i, ∗ = i i )(1 − θ i /n ) − i )(1 − θ i /n ) α i , ( σ 2 i + ν 2 n ( σ 2 i + ν 2 and δ i µ i σ i θ i a i = i (1 − θ i /n ) + i (1 − θ i /n ) σα σ 2 i + ν 2 σ 2 i + ν 2

Construction of Nash equilibria (cont.) δ i µ i σ i θ i a i = i (1 − θ i /n ) + i (1 − θ i /n ) σα σ 2 i + ν 2 σ 2 i + ν 2 n � σα := 1 σα + 1 σ k α k = � nσ i α i n k =1 Multiplying both sides by σ i and then averaging over i = 1 , . . . , n , gives σα = ϕ n + ψ n σα � n � n σ 2 σ i µ i ϕ n := 1 ψ n := 1 i =1 δ i and i =1 θ i i n σ 2 i + ν 2 n σ 2 i + ν 2 i (1 − θ i /n ) i (1 − θ i /n ) • Existence • Uniqueness

Existence of Nash equilibria δ i µ i σ i θ i a i = i (1 − θ i /n ) + i (1 − θ i /n ) σα σ 2 i + ν 2 σ 2 i + ν 2 σα = ϕ n + ψ n σα � n � n σ 2 ϕ n := 1 σ i µ i ψ n := 1 i =1 δ i and i =1 θ i i n σ 2 i + ν 2 i (1 − θ i /n ) n σ 2 i + ν 2 i (1 − θ i /n ) • If ψ n < 1 , then σα = ϕ n / (1 − ψ n ) , and Nash equilibrium exists • If ψ n = 1 and ϕ n > 0 , eqn has no solution; no constant equilibria exist • If ψ n = 1 and ϕ n = 0 , eqn has infinitely many solutions, but this case not feasible

Uniqueness of smooth solutions to the HJB equation Recall that the candidate Nash equilibria were constructed from the smooth solutions of the HJB eqn � 1 � 2( σ 2 i + ν 2 i ) π 2 v xx + π ( µ i v x + σ i � v t + max σαv xy ) π ∈ R � � +1 σα 2 + 1 � ( να ) 2 � v yy + � µαv y = 0 , 2 n � �� � �� − 1 1 − θ i v ( x, y, T ) = − e − γ i ( x − G ( y )) = − exp x − θ i y δ i n This equation has a unique smooth solution that is strictly concave and strictly increasing in x (Duffie et al. (1996), Musiela and Z. (2002))