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 ( x S 1 instead of xs ) 2q x }{{} average price per share
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,... : where S n = S n γ n 1 i=0 x i Payout at time n: S n+ = S n (κ + γ)x n, ( x n S n κ + γ ) 2 x n 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) ER N+ + λvarr N+ Xdet Optimal strategy is of the form min X n = C 1 e Kn C 2 e Kn ( ) (A) + (B) (C) Konishi and Makimoto (2001) ER N+ + λ VarR N+ Xdet Optimal strategy is again of the form ( ) (A) (B) (C) + min
It would be more interesting to optimize over X rather than over X det Almgren and Lorenz (2007) ( ) is no longer optimal (A) (C):? ER N+ + λvarr N+ X min Schied, Schöneborn, and Tehranchi (2010) For U(x) = e αx, EU(R N+ ) X max Optimal strategy is deterministic (cf. with Schied and Schöneborn (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 Föllmer 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 ) (X i ), where X n = X 0 n 1 i=0 x i
Problem Settings Setting 1 For a strategy x = (x i ) N i=0 define the cashflow F x by ( Fn x = x n S n γ ) n 1 i=0 x i κ+γ 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 ( Gn x = x n 1 S n 1 + ξn 2 γ ) n 2 i=0 x i κ+γ 2 x n 1, The problem: ρ 0 (G x ) min over x X n = 1,..., N + 1.
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 x i = X ( ) 0 N N + 1 + a 2 i, i = 0,..., N, x i = 0, i = N + 1,..., N, where N = N ( ceil 1 + 1 + 8X 0 /a 2 1 ) 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, Schöneborn, 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, Schöneborn, 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), Schöneborn (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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