Optimal Portfolio Liquidation with Dynamic Coherent Risk Andrey - - PowerPoint PPT Presentation

Optimal Portfolio Liquidation with Dynamic Coherent Risk

Andrey Selivanov1 Mikhail Urusov2

1Moscow State University and Gazprom Export 2Ulm 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

2q x

• average price per share

instead of xS−

OPL How to sell optimally X0 shares until time N? X0, N are specified by a client, X0 is very big Time horizon is usually short A strategy is a sequence x = (xi)N

i=0, where all xi ≥ 0 and

N

i=0 xi = X0

xi means the number of shares to sell at time i, i = 0, . . . , N X (resp., Xdet) denotes the set of adapted (resp., deterministic) strategies

Model for unaffected price A random walk (Sn) (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 xk ≥ 0 shares at times k, k = 0, 1, . . . :

• Sn+ =

Sn− − (κ + γ)xn, where Sn− = Sn − γ n−1

i=0 xi

Payout at time n: xn

• Sn− − κ + γ

2 xn

• Cf. with Bertsimas and Lo (1998), Almgren and Chriss (2001)

LOB with finite resilience: Obizhaeva and Wang (2005), Alfonsi, Fruth, and Schied (2010)

Notation Xn := X0 − n−1

i=0 xi, n = 1, . . . , N + 1, the number of

shares remaining at hand at time n−. Note that XN+1 = 0 (xi) ← → (Xi) Properties of strategies desirable for practitioners (A) Dynamic consistency (B) Presence of an intrinsic time horizon N∗ such that N∗ < N for small X0, N∗ = N for large X0, N∗ is increasing as a function of X0 (C) Relative selling speed decreasing in the position size:

x0 X0 decreases as a function of X0

Notation RN+ revenue from the liquidation Almgren and Chriss (2001) −ERN+ + λVarRN+ − − →

Xdet

min Optimal strategy is of the form Xn = C1e−Kn − C2eKn (∗) (A) + (B) − (C) − Konishi and Makimoto (2001) −ERN+ + λ

• VarRN+ −

− →

Xdet

min Optimal strategy is again of the form (∗) (A) − (B) − (C) +

It would be more interesting to optimize over X rather than

• ver Xdet

Almgren and Lorenz (2007) −ERN+ + λVarRN+ − − − →

X

min (∗) is no longer optimal (A)–(C):

?

Schied, Sch¨

• neborn, and Tehranchi (2010) For U(x) = −e−αx,

EU(RN+) − − − →

X

max Optimal strategy is deterministic (cf. with Schied and Sch¨

• neborn (2009))

If (Sn) 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¨

• llmer 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@Rλ(R) = −E(R|R ≤ qλ(R)) (modulo a technicality), where qλ(R) is λ-quantile of R

Dynamizing ρ

(Ω, F, (Fn)N

n=0, P)

Cashflow F = (Fn)N

n=0: an adapted process

Fn means P&L of a bank at time n Need to define dynamic risk ρ(F) ρ(F) = (ρn(F))N

n=0 an adapted process

ρn(F) ≡ ρ(Fn, . . . , FN) means the risk of the remaining part (Fn, . . . , FN) of the cashflow measured at time n Define inductively: ρN(F) = −FN, ρn(F) = −Fn + ρ

• Law[−ρn+1(F)|Fn]
• ,

n = N − 1, . . . , 0

• Cf. with Riedel (2004), Cheridito and Kupper (2006), Cherny

(2009)

Outline

Optimal Portfolio Liquidation Dynamic Risk Main Result

Inputs

X0 > 0 a large number of shares to sell until time N Sn = S0 + n

i=1 ξi, where (ξi) iid

Fn = σ(ξ1, . . . , ξn), where F0 = triv A strategy is an (Fn)-adapted sequence x = (xi)N

i=0, where all

xi ≥ 0 and N

i=0 xi = X0

X (resp., Xdet) denotes the set of all (resp., deterministic) strategies (xi) ← → (Xi), where Xn = X0 − n−1

i=0 xi

Problem Settings

Setting 1 For a strategy x = (xi)N

i=0 define the cashflow F x by

F x

n = xn

• Sn − γ n−1

i=0 xi − κ+γ 2 xn

• ,

n = 0, . . . , N. The problem: ρ0(F x) − → min over x ∈ X Setting 2 For a strategy x define Gx by Gx

0 = 0 and

Gx

n = xn−1

• Sn−1 + ξn

2 − γ n−2 i=0 xi − κ+γ 2 xn−1

• ,

n = 1, . . . , N + 1. The problem: ρ0(Gx) − → 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 xi = X0 N∗ + 1 + a N∗ 2 − i

• ,

i = 0, . . . , N∗, xi = 0, i = N∗ + 1, . . . , N, where N∗ = N ∧

• ceil −1 +
• 1 + 8X0/a

2 − 1

• with ceil y denoting the minimal integer d such that y ≤ d

Discussion

If we maximized over Xdet rather than over X, then the

• ptimizer 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

• ver X does not do a better job, than optimizing over Xdet, and

then perform just a deterministic optimization

• Cf. with Alfonsi, Fruth, and Schied (2010),

Schied, Sch¨

• neborn, 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¨
• neborn, and Urusov (2010), where

stochastic liquidity leads to stochastic optimal strategies

Remarks

◮ (A) +

(B) + (C) + (recall “+ − −” for the Almgren–Chriss strategy)

◮ (Xn) parabola vs. Xn = C1e−Kn − C2eKn

(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¨
• neborn (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)

Thank you for your attention!

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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