Optimal Portfolio Liquidation with Dynamic Coherent Risk Andrey Selivanov 1 Mikhail Urusov 2 1 Moscow State University and Gazprom Export 2 Ulm University Analysis, Stochastics, and Applications. A Conference in Honour of Walter Schachermayer – Vienna University, July 12–16, 2010
Outline Optimal Portfolio Liquidation Dynamic Risk Main Result
Outline Optimal Portfolio Liquidation Dynamic Risk Main Result
A trader sells x > 0 shares of a stock in an illiquid market. In selling the price falls from S − to S + = S − − 1 q x . The trader gets the payout � � S − − 1 x 2 q x � �� � average price per share instead of xS −
OPL How to sell optimally X 0 shares until time N ? X 0 , N are specified by a client, X 0 is very big Time horizon is usually short A strategy is a sequence x = ( x i ) N i = 0 , where all x i ≥ 0 and � N i = 0 x i = X 0 x i means the number of shares to sell at time i , i = 0 , . . . , N X (resp., X det ) denotes the set of adapted (resp., deterministic) strategies
Model for unaffected price A random walk ( S n ) (short time horizon) Model for price impact A block-shaped limit order book with infinite resilience Optimization problem Minimize a certain dynamic coherent risk measure
Model for price impact Linear permanent and temporary impacts with the coefficients γ ≥ 0 resp. κ > 0 Selling x k ≥ 0 shares at times k , k = 0 , 1 , . . . : S n + = � � S n − − ( κ + γ ) x n , S n − = S n − γ � n − 1 where � i = 0 x i Payout at time n : � � S n − − κ + γ � x n x n 2 Cf. with Bertsimas and Lo (1998), Almgren and Chriss (2001) LOB with finite resilience: Obizhaeva and Wang (2005), Alfonsi, Fruth, and Schied (2010)
Notation X n := X 0 − � n − 1 i = 0 x i , n = 1 , . . . , N + 1, the number of shares remaining at hand at time n − . Note that X N + 1 = 0 ( x i ) ← → ( X i ) Properties of strategies desirable for practitioners (A) Dynamic consistency (B) Presence of an intrinsic time horizon N ∗ such that N ∗ < N for small X 0 , N ∗ = N for large X 0 , N ∗ is increasing as a function of X 0 (C) Relative selling speed decreasing in the position size: x 0 X 0 decreases as a function of X 0
Notation R N + revenue from the liquidation Almgren and Chriss (2001) − E R N + + λ Var R N + − − → min X det Optimal strategy is of the form X n = C 1 e − Kn − C 2 e Kn ( ∗ ) (A) + (B) − (C) − Konishi and Makimoto (2001) � − E R N + + λ Var R N + − − → min X det Optimal strategy is again of the form ( ∗ ) (A) − (B) − (C) +
It would be more interesting to optimize over X rather than over X det Almgren and Lorenz (2007) − E R N + + λ Var R N + − − − → min X ( ∗ ) is no longer optimal (A)–(C): ? oneborn, and Tehranchi (2010) For U ( x ) = − e − α x , Schied, Sch¨ E U ( R N + ) − − − → max X Optimal strategy is deterministic (cf. with Schied and Sch¨ oneborn (2009)) If ( S n ) is a Gaussian random walk, then the optimal strategy is the Almgren–Chriss one with λ = α/ 2 (A) + (B) − (C) −
Outline Optimal Portfolio Liquidation Dynamic Risk Main Result
Static Risk (Ω , F , P ) R : Ω → R P&L of a bank How to measure risk of R ? Artzner, Delbaen, Eber, and Heath (1997, 1999): Coherent risk measures F¨ ollmer and Schied (2002), Frittelli and Rosazza Gianin (2002): Convex risk measures Notation ρ ( R ) a law invariant coherent risk measure ρ ( Law R ) := ρ ( R ) � E.g. CV @ R λ ( R ) = − E ( R | R ≤ q λ ( R )) (modulo a technicality), where q λ ( R ) is λ -quantile of R
Dynamizing ρ (Ω , F , ( F n ) N n = 0 , P ) Cashflow F = ( F n ) N n = 0 : an adapted process F n means P&L of a bank at time n Need to define dynamic risk ρ ( F ) ρ ( F ) = ( ρ n ( F )) N n = 0 an adapted process ρ n ( F ) ≡ ρ ( F n , . . . , F N ) means the risk of the remaining part ( F n , . . . , F N ) of the cashflow measured at time n Define inductively: ρ N ( F ) = − F N , � � ρ n ( F ) = − F n + � ρ Law [ − ρ n + 1 ( F ) |F n ] , n = N − 1 , . . . , 0 Cf. with Riedel (2004), Cheridito and Kupper (2006), Cherny (2009)
Outline Optimal Portfolio Liquidation Dynamic Risk Main Result
Inputs X 0 > 0 a large number of shares to sell until time N S n = S 0 + � n i = 1 ξ i , where ( ξ i ) iid F n = σ ( ξ 1 , . . . , ξ n ) , where F 0 = triv A strategy is an ( F n ) -adapted sequence x = ( x i ) N i = 0 , where all x i ≥ 0 and � N i = 0 x i = X 0 X (resp., X det ) denotes the set of all (resp., deterministic) strategies → ( X i ) , where X n = X 0 − � n − 1 ( x i ) ← i = 0 x i
Problem Settings i = 0 define the cashflow F x by Setting 1 For a strategy x = ( x i ) N � � S n − γ � n − 1 F x i = 0 x i − κ + γ n = x n 2 x n , n = 0 , . . . , N . The problem: ρ 0 ( F x ) − → min over x ∈ X Setting 2 For a strategy x define G x by G x 0 = 0 and � � 2 − γ � n − 2 S n − 1 + ξ n i = 0 x i − κ + γ G x n = x n − 1 2 x n − 1 , n = 1 , . . . , N + 1 . The problem: ρ 0 ( G x ) − → min over x ∈ X
Main Result Standing assumption 0 < � ρ ( Law ξ ) < ∞ Set a := � ρ ( Law ξ ) /κ , so a > 0 Theorem Optimal strategy is the same in both settings. Moreover, it is deterministic and given by the formulas � N ∗ � X 0 i = 0 , . . . , N ∗ , x i = N ∗ + 1 + a 2 − i , i = N ∗ + 1 , . . . , N , x i = 0 , where � � � ceil − 1 + 1 + 8 X 0 / a N ∗ = N ∧ − 1 2 with ceil y denoting the minimal integer d such that y ≤ d
Discussion If we maximized over X det rather than over X , then the optimizer would be the same in both settings. This is not clear a priori when we maximize over X The proof consists of two parts: first we prove that optimizing over X does not do a better job, than optimizing over X det , and then perform just a deterministic optimization Cf. with Alfonsi, Fruth, and Schied (2010), Schied, Sch¨ oneborn, and Tehranchi (2010), where the optimal strategies are also deterministic Why is the optimal strategy deterministic? Because here liquidity ( κ ) is deterministic Cf. with Fruth, Sch¨ oneborn, and Urusov (2010), where stochastic liquidity leads to stochastic optimal strategies
Remarks ◮ (A) + (B) + (C) + (recall “ + − − ” for the Almgren–Chriss strategy) ◮ ( X n ) parabola vs. X n = C 1 e − Kn − C 2 e Kn (Almgren–Chriss is now a benchmark for practitioners) ◮ Setting N = ∞ (time horizon is not specified by the client) we get a strategy with a purely intrinsic time horizon N ∗ . Cf. with Almgren (2003), Sch¨ oneborn (2008) ◮ a ↑ leads to a quicker liquidation in the beginning = ⇒ reasonable dependence of the liquidation strategy on volatility risk ( � ρ ( Law ξ ) ) and on liquidity risk ( κ )
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Possible Generalizations ◮ More general price impact? Optimal strategies are again deterministic ◮ Convex risk measure ρ ? Optimal strategies are again deterministic, however, different in Settings 1 and 2 Typically (A) + (B) − Also (C) − in an example with entropic risk measure, which was worked out explicitly
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