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A trade execution model under a composite dynamic coherent risk measure
Operations Research Letters 43 (2015) 52–58
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Operations Research Letters journal homepage: www.elsevier.com/locate/orl
A trade execution model under a composite dynamic coherent risk measure Qihang Lin a,∗ , Xi Chen b , Javier Peña c a
Tippie College of Business, The University of Iowa, Iowa City, IA 52245, United States
b
Stern School of Business, New York University, New York, NY 10012, United States
c
Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA 15213, United States
article
info
Article history: Received 12 February 2014 Received in revised form 3 November 2014 Accepted 8 November 2014 Available online 18 November 2014
abstract We investigate the trade execution problem where a risky asset must be sold before a deadline with a control on the transaction cost. The asset price is modeled as a discrete random walk perturbed by price impacts. We show that the optimal trading strategy minimizing a dynamic coherent risk of the transaction cost is time-consistent and deterministic. We obtain a closed-form expression for the optimal strategy and evaluate its numerical performance over real data. © 2014 Elsevier B.V. All rights reserved.
Keywords: Dynamic coherent risk measures Trade execution Risk-averse Markov decision process
1. Introduction One of the tasks in algorithmic trading is to execute the transaction of a large volume of an asset with a balance between transaction cost and risk. The sources leading to transaction cost include commissions, exchange fees, taxes, bid–ask spread and price impact, among which the last one typically dominates the others. Trading a large volume of an asset within a short period of time generally leads to an unbearable transaction cost due to the large adverse price movement it induces. A better execution strategy may break down a large order into several smaller ones and spread them over time to mitigate the price impact. Since the transaction cost is a random variable, a risk measure needs to be specified before comparing different execution strategies. The optimal execution with respect to a risk measure is the one minimizing the value of the risk measure of transaction cost. Besides optimality, another important property of an execution strategy is time-consistency. Roughly speaking, a strategy is time-consistent if it minimizes the risk value from any intermediate time to the deadline conditioning on the historical sample path. The time-consistency of an optimal trade execution strategy depends on the risk measure it minimizes. Bertsimas and Lo [9] consider a risk-neutral execution strategy by minimizing the expectation, the simplest risk measure, of transaction cost. The analytical form of the optimal execution is derived from the Bellman
∗
Corresponding author. E-mail address: [email protected] (Q. Lin).
http://dx.doi.org/10.1016/j.orl.2014.11.005 0167-6377/© 2014 Elsevier B.V. All rights reserved.
equation of dynamic programming, which guarantees its timeconsistency. In their seminal work, Almgren and Chriss [4] consider a deterministic risk-averse strategy under a mean–variance risk measure. Their strategy is time-consistent [17] but not optimal [5] because it is obtained after restricting the strategy to be deterministic. An adaptive risk-averse strategy under a mean–variance risk has been proposed by Lorenz and Almgren [18]. This strategy is both time-consistent and optimal but can only be computed approximately via a cost-to-go function on a set of grid points. Given the computational challenges and the dichotomy between optimality and time-consistency in the existing approaches, one of our main goals is to obtain a risk averse execution strategy under some risk measure that is computationally tractable, globally optimal, and time-consistent. The risk measure we utilize in this paper is a composite dynamic coherent risk measure that belongs to the general family of dynamic risk measures [8,12,13,19,22,21]. A composite dynamic coherent risk measure for trade execution is appropriate for a variety of reasons. First, a composite dynamic coherent risk measure evaluates the risk of the transaction cost at each intermediate stage, providing richer information to the decision maker. Second, a composite dynamic coherent risk measure satisfies the coherent properties [7] so it can better reflect the rationale of a risk-averse decision maker in a multistage setting. Third, the time-consistency of a composite dynamic coherent risk measure ensures that it can be minimized using the dynamic programming principle [21], which guarantees the time-consistency of the corresponding optimal execution strategy. In this paper we measure the transaction cost of the classical trade execution model by Almgren and Chriss [4] using a composite
Q. Lin et al. / Operations Research Letters 43 (2015) 52–58
dynamic coherent risk measure. We prove that the optimal strategy is time-consistent and deterministic. Furthermore, we show that the impact of the choice of a composite dynamic coherent risk measure on the optimal strategy is only through a real value ρ(−ξ ) where ρ is a coherent risk measure used to build the dynamic risk measure and ξ is the source of the randomness in the trade execution model [4]. This observation allows us to reformulate the trade execution problem as a quadratic program over a simplex with a closed-form solution. Under a continuous-time framework, Schied et al. [23] and Gatheral et al. [15] show respectively that, when the risk measure is the expected constant absolute risk aversion utility or the time-averaged value-at-risk, the resulting optimal strategy is also deterministic and time-consistent. However, depending on the parameters of the problem, their strategies may involve both buying and selling activities as well as holding short positions at intermediate times, which violate some regulatory rules in agency trading. By contrast, our strategy only sells during the liquidation. In [2], Alfonsi et al. consider a risk neutral strategy in a market with a limit order book of a general shape and also obtain a deterministic optimal strategy with a closed-form solution. The deterministic nature of the policy mentioned above and our policy are related to the assumption of deterministic market liquidity. During the review process of this paper, the unpublished work by Selivanov and Urusov [24] was brought to our attention. The setting and results in [24] are similar to ours and were independently developed. However, their work and the details of the relevant proofs remain unpublished. 2. Almgren and Chriss’s trade execution model We now briefly review the classical model for trade execution by Almgren and Chriss [4]. We focus on the strategy for liquidating (selling) an asset for simplicity. All of our results can be applied to the case of buying an asset. Consider a probability space (Ω , F , P ) on which there exist i.i.d. random variables ξ1 , ξ2 , . . . , ξN −1 . Let F0 = {Ω , ∅}, Fk = σ (ξ1 , . . . , ξk ) and ξ[k] = (ξ1 , . . . , ξk ) for k = 1, 2, . . . , N − 1. Let x0 ∈ R+ represent the initial volume of a risky asset that has to be sold within a finite time period [0, T ]. The total volume is partitioned into N smaller blocks n1 , n2 , . . . , nN , where nk is to be sold within a time interval [tk−1 , tk ]. Assume tk = kτ with τ = NT for k = 1, . . . , N. We call ∆ = (n1 , . . . , nN ) the trading strategy and denote the trading trajectory associated to ∆ by (x1 , . . . , xN ), where xk represents the remaining unsold asset at time tk . The relationship between nk ’s and xk ’s is given by xk = xk−1 − nk ,
k = 1, . . . , N .
(1)
Let Sk be the market price of the asset that can be observed by the trader at time tk for k = 0, . . . , N − 1. We assume S0 is a deterministic constant in R+ . Following [4], the dynamic of Sk is modeled as Sk = Sk−1 + ξk − g · nk ,
k = 1, . . . , N − 1.
(2)
The linear term gnk in (2) measures the permanent price impact incurred by selling nk . Due to the limited liquidity of the market during [tk−1 , tk ], the prices at which the asset nk is sold are not consistently the prices observed at tk−1 , i.e., Sk−1 . The actual average transaction price during [tk−1 , tk ] is denoted by Sk , which is related to Sk−1 as nk Sk = Sk−1 − h · , τ nk
k = 1, . . . , N .
(3)
The linear term h τ in (2) represents the temporary impact on the n price as a linear function of the trading speed τk . The assumption of the linearity of the price impacts is supported by results
53
from [16] where the authors show that only linear impact functions prevent the opportunity of quasi-arbitrage. In the rest of the paper, we make the following assumption.
≥ g. Assumption 1. 2h τ This assumption states that, each unit sold during a period of length τ generates a permanent impact that is at most twice the temporary impact. The temporary impact is due to the temporary unbalanced demand and supply while the permanent impact is due to the change of the (publicly believed) fundamental value of the asset caused by the signal revealed from a transaction. This assumption will typically hold for an asset whose fundamental value is consistent such as the stock of a well-established and stable company. The empirical evidence in [6] also justifies this assumption when the trade duration τ is short. At time tk−1 , the trader determines nk based on the history (x[k−1] , S[k−1] ), where x[k−1] = (x0 , . . . , xk−1 ) and S[k−1] = (S0 , . . . , Sk−1 ). Hence, Sk−1 , Sk , nk and xk are random variables in (Ω , Fk−1 , P ). To completely liquidate the asset by T , we have to N enforce the condition xN = 0, or equivalently, k=1 nk = x0 . We also assume buying is not allowed, i.e., xk−1 ≥ xk , or equivalently, nk ≥ 0 for k = 1, . . . , N. Given a trading strategy ∆, the total cost (negative revenue) N the trader obtains during [0, T ] is k=1 −Sk nk . Bertsimas and Lo [9] consider a risk neutral strategy by minimizing the expected N cost, i.e., E( k=1 − Sk nk ). Almgren et al. [4,5,3] consider a risk averse strategy by minimizing the mean-variation of the cost, i.e., N N E( k=1 − Sk nk ) + λ Var( k=1 − Sk nk ) with λ > 0. In contrast to these previous approaches, we consider a risk averse strategy by minimizing the composite dynamic coherent risk of the cost. 3. Composite dynamic coherent risk measure The general definition of dynamic risk measure is given in [19,1, 10]. The composite dynamic coherent risk measure used here is the special dynamic risk measure introduced in [21,25], which can be constructed as a nested composition of conditional risk mappings. p p We define respectively Lp = Lp (Ω , F , P ) and Lk = Lk (Ω , Fk , P ) as the sets of random variables on (Ω , F , P ) and (Ω , Fk , P ) with finite pth moment for p ∈ [1, ∞) for k = 0, . . . , N − 1. We p assume that for each Z ∈ Lk , there exists a measurable function on k R such that Z = Z (ξ1 , . . . , ξk ). Definition 1. A mapping ρ : Lp → R is called a coherent risk measure on Lp if it satisfies the following properties [7] for all X and Y in Lp : A1. Convexity: ρ(λX + (1 − λ)Y ) ≤ λρ(X ) + (1 − λ)ρ(Y ), ∀λ ∈ [0, 1]; A2. Monotonicity: ρ(X ) ≤ ρ(Y ) if X ≤ Y ; A3. Translation Equivalence: ρ(X + Y ) = X + ρ(Y ) if X is a constant; A4. Positive Homogeneity: ρ(β X ) = βρ(X ), ∀β ≥ 0. A coherent risk measure ρ is law invariant if ρ(X ) = ρ(Y ) for all X and Y in Lp with the same distribution. One commonly used law invariant coherent risk measure is conditional value at risk (CVaR) [20]:
CVaRα (X ) = inf r + r ∈R
1 1−α
E(X − r )+ ,
where α ∈ [0, 1) is a risk aversion parameter. Note that
|CVaRα (X )| ≤ CVaRα (|X |) ≤ p
E|X | 1−α
< +∞
(4)
for all X ∈ L . Based on a coherent risk measure, we can construct the corresponding conditional risk mapping.
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Q. Lin et al. / Operations Research Letters 43 (2015) 52–58
Definition 2. Given a law invariant coherent risk measure ρ on Lp p and Z = Z (ξ1 , ξ2 , . . . , ξk+1 ) ∈ Lk+1 , define Z ′ : Rk → R as
Z ′ (θ1 , θ2 , . . . , θk ) = ρ( Z (θ1 , θ2 , . . . , θk , ξk+1 )), for all (θ1 , θ2 , . . . , θk ) ∈ Rk and k = 0, . . . , N − 2. The conditional p p risk mapping induced by ρ at time tk is the mapping ρk : Lk+1 → Lk defined by
ρk (Z ) = Z ′ (ξ1 , ξ2 , . . . , ξk ),
∀Z ∈
p Lk+1
.
E|ρk (Z )| ≤ Eξ[k]
≤
Z (ξ1 , . . . , ξk , ξk+1 )| Eξk+1 |
p
(1 − α)p
(5)
The trade execution problem is to sell x0 by time T with a minimum composite dynamic coherent risk at t0 , i.e.,
s.t.
nk = x 0 ,
− Sk nk · · ·
.
k=1
− S k nk
=
N −1
− Sk nk + ρN −2 − SN nN .
k=1
N
− S k nk
(6) nk ≥ 0;
k=1
To have a well-defined objective function in (6), we make the following assumption. Assumption 2. The distribution of ξk ’s is such that p Lk−1 for k = 1, . . . , N for all feasible solutions of (6).
k
i=1 −Si ni ∈
By (2) and (3) and the constraint 0 ≤ nk ≤ x0 , one can show that this assumption is satisfied, for example, when ξk ’s are normally distributed. 4. Convex optimization formulation Proposition 1. Let Vk : R+ × R → R for k = 0, . . . , N − 1 be x and defined by VN −1 (x, S ) := − S − hx τ
hn Vk (x, S ) := min − S − n 0≤n≤x τ
(8)
Suppose n1 , . . . , nN −1 have been determined. To sell the asset completely by T , the only feasible and optimal decision at time tN −1 is nN = xN −1 , i.e., n∗N (xN −1 , SN −1 ) = xN −1 . Thus,
hxN −1 xN −1 = VN −1 (xN −1 , SN −1 ). − S N nN = − S N − 1 − τ Substituting this equality into (8), by property A2 of ρ and the definitions of ρk ’s as well as Bellman’s principle of optimality, we have
(n1 ,...,nN )
− Sk nk
nk is a random variable on (Ω , Fk−1 , P ).
N
= − S1 n1 + ρ0 − S2 n2 + ρ1 · · · − S N − 2 nN − 2 + ρN −3 − SN −1 nN −1 + ρN −2 − S N nN ··· .
min
k=1 N
= ρ0 ρ1 · · · ρN −2
− S k nk
k=1
ρk,N −1 (Z ) = ρk (ρk+1 (· · · ρN −2 (Z ) · · ·)).
(n1 ,...,nN )
ρ N −2
N
ρ0,N −1
< +∞
Definition 3. Given a law invariant coherent risk measure ρ on Lp , the composite dynamic coherent risk measure induced by ρ at time p p tk is the mapping ρk,N −1 : LN −1 → Lk defined by
N
The composite dynamic coherent risk measure is a nested composition of conditional risk mappings.
ρ0,N −1
Applying this argument repeatedly, we can show
p
min
k=1
for all Z ∈ Lk+1 for k = 0, . . . , N − 2. The conditional risk mapping above is a special case of the general conditional risk mapping defined in [22,21].
Since k=1 − Sk nk does not depend on ξN −1 , according to property A3 of ρ and the definition of ρN −2 , we have
p
1−α
Eξ[k+1] | Z (ξ1 , . . . , ξk , ξk+1 )|p
ρ0,N −1
N
N −1
p
k=1
Remark 1. We assume that ρ ensures ρk (Z ) ∈ Lk for all Z ∈ Lk+1 . This assumption is satisfied, for example, when ρ = CVaRα . In fact, by (4), we have p
Proof. Our proof is an adaptation of the general argument by Ruszczynski [21, Theorem 2] to the specific dynamic risk measure we defined. The dynamic programming equations (7) yield the time consistency of the optimal strategy. Recall that
ρ0,N −1
N
− S k nk
k=1
ρ0 · · · − SN −2 nN −2 + ρN −3 − S N − 1 nN − 1 (n1 ,...,nN −1 ) + ρN −2 VN −1 (xN −1 , SN −1 ) · · · = min ρ0 · · · − SN −2 nN −2
=
min
(n1 ,...,nN −2 )
+ ρN −3
min
0≤nN −1 ≤xN −2
− SN −1 nN −1
+ ρN −2 VN −1 (xN −1 , SN −1 ) · · · .
(9)
By (1)–(3), we have
− SN −1 nN −1 + ρN −2 VN −1 (xN −1 , SN −1 ) hnN −1 nN − 1 = − S N −2 − τ + ρN −2 VN −1 (xN −2 − nN −1 , SN −2 + ξN −1 − gnN −1 ) .
(10)
Note that VN −2 (x, S ) = min0≤n≤x φ(x, S , n), where
+ ρ Vk+1 (x − n, S + ξk+1 − gn)
(7)
for k = 0, . . . , N − 2. Then the optimal value of (6) equals V0 (x0 , S0 ). Moreover, the optimal trading strategy ∆∗ = (n∗1 , . . . , n∗N ) for (6) is Markovian, where n∗N (xN −1 , SN −1 ) = xN −1 and n∗k+1 (xk , Sk ) is the optimal solution of (7) with x = xk and S = Sk for k = 0, . . . , N − 2.
φ(x, S , n) := − S −
hn
τ
n + ρ VN −1 (x − n, S + ξN −1 − gn) .
Since nN −1 , xN −2 and SN −2 do not depend on ξN −1 , by the definition of ρN −2 , (10) is equal to φ(xN −2 , SN −2 , nN −1 ) when (10) is understood as a function of (xN −2 , SN −2 , nN −1 ). Although the decision nN −1 depends on the history (x[N −2] , S[N −2] ), it suffices to solve the inner problem in (9) over the real interval [0, xN −2 ] because the
Q. Lin et al. / Operations Research Letters 43 (2015) 52–58
objective function and the constraint both depend on the history only via (xN −2 , SN −2 ). Therefore, the optimal value of the innermost minimization in (9) equals min
0≤n≤xN −2
φ(xN −2 , SN −2 , n) = VN −2 (xN −2 , SN −2 )
for each fixed value of xN −2 and SN −2 . Moreover, the optimal decision n∗N −1 depends on (xN −2 , SN −2 ) as n∗N −1 (xN −2 , SN −2 ) = arg min φ(xN −2 , SN −2 , n). 0≤n≤xN −2
Therefore, we can further represent (9) as
min
(n1 ,...,nN )
=
ρ0,N −1
N
− Sk nk
k=1
S N − 2 nN − 2 ρ0 · · · − (n1 ,...,nN −2 ) + ρ N −3 VN −2 ( x N −2 , S N −2 ) · · · . min
We proceed with the similar argument as above for k = N − 3, . . . , 0 and obtain the conclusion of the proposition. Using the particular structure of problem (6), we can refine (7) and show that the optimal n∗k+1 (xk , Sk ) only depends on xk but not on Sk and thus is deterministic. Theorem 1. The function Vk (x, S ) satisfies Vk (x, S ) = fk (x) − Sx
(11)
for k = 0, . . . , N − 1, where fN −1 (x) := τ and
fk (x) := min
0≤n≤x
hn2
τ
(12)
for k = 0, . . . , N − 2. The optimal trading strategy ∆∗ = (n∗1 , . . . , n∗N ) is deterministic where n∗k+1 is the optimal solution of (12) with x = xk for k = 0, . . . , N − 2 and n∗N = xN −1 . Proof. We prove (11) by backward induction. When k = N − 1, VN −1 (x, S ) = − S − hx x = fN −1 (x) − Sx. Suppose (11) holds for τ k + 1 with k ≤ N − 2, we want to prove that it also holds for k. By the induction hypothesis, we have
= fk+1 (x − n) − (S + ξk+1 − gn)(x − n).
x0 −
k=1 N
nk = x 0 ,
k
ni
i =1
(14)
nk ≥ 0, k = 1, . . . , N ,
k=1
N
k=1
n2k + g
N
k=1
nk x 0 −
k
i=1
ni .
ρ (−ξ1 ) Nk=1 x0 − ki=1 ni is a linear function of ∆. Because k N x0 − domain, f (∆) can be i=1 ni = i=k+1 ni on the feasible N N N h 2 written as f (∆) = τ k=1 nk + g k=1 nk i=k+1 ni which is a − g IN ×N + gJN . Here, quadratic function of ∆ whose Hessian is 2h τ IN ×N is an N × N identity matrix and JN is an N × N all-one matrix. By Assumption 1, 2h − g IN ×N + gJN is positive semidefinite and τ so (14) is a convex problem.
5. The optimal liquidation strategy
Theorem 2. Let a =
ρ (Vk+1 (x − n, S + ξk+1 − gn))
= fk (x) − Sx.
f (∆) + ρ (−ξ1 )
We are now ready to present the main result of this paper, namely a closed-form solution to (14).
Using properties A3 and A4, we obtain
hn min − S − n + ρ (Vk+1 (x − n, S + ξk+1 − gn)) 0≤n≤x τ 2 hn = min + fk+1 (x − n) + gn(x − n) 0≤n≤x τ + ρ(−ξk+1 )(x − n) − Sx
min
(n1 ,...,nN )
N
5.1. A closed form for the optimal strategy
Vk+1 (x − n, S + ξk+1 − gn)
Therefore, by (7), we can write Vk (x, S ) as
Corollary 1. The optimal trading strategy ∆∗ = (n∗1 , . . . , n∗N ) is the solution of
Problem (14) can be solved easily if its objective function is convex over its feasible domain which is a simplex. We note that
= fk+1 (x − n) − (S − gn)(x − n) + ρ (−ξk+1 ) (x − n).
Note that the value ρ(−ξk+1 ) in (12) is the same for k = 0, . . . , N − 2 because ρ is law invariant and ξk ’s are i.i.d. It can be seen from (12) that the composite dynamic coherent risk measure impacts the optimal strategy only through the value ρ(−ξk+1 ). This is due to the particular price dynamic (2) we consider and the fact that ξk ’s are i.i.d. We can view (12) as dynamic programming equations, where the term ρ(−ξk+1 )(x − n) is interpreted as a penalty the trader pay at stage k for the risk of the unsold asset x−n. Similar to the policies in [23,15,2], our policy is deterministic partially because the price impact coefficients h and g (market liquidity) are deterministic. If h and g were stochastic, the optimal strategy would be stochastic [14]. The optimal trading strategy ∆∗ can be solved sequentially from (12). However, we can also solve ∆∗ from a single optimization problem by applying (12) recursively:
where f (∆) = τh
+ fk+1 (x − n) + gn(x − n)
+ ρ(−ξk+1 )(x − n)
Therefore, we have shown that (11) holds for k. By Proposition 1, the optimal strategy n∗k+1 solves the minimization in (7) with x = xk and S = Sk . According to the derivation above, n∗k+1 only depends on xk and can be computed sequentially by solving (12) and thus is deterministic.
s.t
hx2
55
(13)
N ∗ ≡ min N ,
β 2h
τ −g
−1 +
√
with β = ρ(−ξ1 ) and
1 + 8x0 /|a|
2
.
(15)
If β ≥ 0, the optimal liquidation starts at stage 0 and finishes at stage N ∗ . If β < 0, the optimal liquidation starts at stage N − N ∗ and finishes at stage N. The optimal trading strategy and the optimal trading trajectory are given as:
x (N ∗ − 2k + 1)a 0 + N∗ 2 + n∗k = ∗ ( 2N − N − 2k + 1)a x 0 + ∗ N
2
+
if β ≥ 0 (16) if β < 0
56
Q. Lin et al. / Operations Research Letters 43 (2015) 52–58
and
∗ (N ∗ − k)ka (N − k)x0 − if β ≥ 0 N∗ 2 + x∗k = (N − k)x0 (N − k)(N − N ∗ − k)a − ∧ x0 N∗ 2 if β < 0
(17)
where (x)+ = max{x, 0} and x ∧ y = min{x, y} for k = 1, . . . , N. Proof. Because of the equality x0 − optimization problem (14) becomes min
(n1 ,...,nN )
s.t
N h
τ k=1 N
n2k
+g
N k=1
nk = x0 ,
nk
N
k
i=1
ni =
ni
+β
i=k+1
N
i=k+1
N N k=1 i=k+1
ni , the
ni (18)
nk ≥ 0, k = 1, . . . , N .
intrinsic time horizon N ∗ . By (15), N ∗ will move towards 0 as β increases and the trader sells faster. In the limit where β = +∞, we have N ∗ = 0, n∗1 = x0 , and nk = 0 for k = 2, . . . , N. On the other hand, if µ >√0 then the sign of β depends on the trade-off between the risk τ σ CVaRα (N (0, 1)) and the profit τ µ from holding the asset one more period. If the latter is larger, β becomes negative and Theorem 2, implies that the optimal strategy should finish at stage N but begin at an intrinsic starting time N − N ∗ . As β decreases, N − N ∗ will move towards N and the trader will postpone selling. In the limit where β = −∞, we have N − N ∗ = N , n∗N = x0 and nk = 0, for k = 1, . . . , N − 1. We also note that the intrinsic time horizon N ∗ is an increasing function of x0 when β > 0. If x0 is small, we have N ∗ < N. In other words, even though the trader has N stages to sell the asset, he may only need to trade in the first few stages and do nothing in the remaining stages. By contrast, the mean–variance strategy [4] does not have this kind of sparsity property. Instead, the strategy in [4] typically ends with a ‘‘small tail’’ (see Fig. 1).
k=1
By Assumption 1, (18) is a convex quadratic program with a simplex constraint. Let λ and uk be the dual variables associated with the equality and inequality constraints respectively. We can then write down the corresponding KKT conditions [11] as follows: 2h
τ
nk + g
N
N
ni + (k − 1)β − λ − uk = 0,
i=1,i̸=k
nk = x 0 ,
(19) nk uk = 0 ,
nk ≥ 0, uk ≥ 0,
k=1
for k = 1, . . . , N. We next show that the solution to these KKT conditions is given by (16). − g )nk + gx0 + (k − 1)β − Observe that (19) is equivalent to ( 2h τ
λ−uk = 0 because Nk=1 nk = x0 . By the complementary slackness condition nk uk = 0 it follows that uk = (−λ+ gx0 +(k − 1)β)+ and λ − gx0 − (k − 1)β . (20) nk (λ) = 2h −g τ + N To finish, we only need to find the value of λ such that k=1 nk (λ) = x0 . Note that nk (λ) Nin (20) is a non-decreasing piecewise linear function of λ. So is k=1 nk (λ). By some elementary calculations, N we can show that k=1 nk (λ) = x0 holds when λ = λ∗ with x0 2h a(N ∗ − 1) 2h − g + gx0 if β ≥ 0 N∗ τ − g + 2 τ ∗ λ = x0 2h a(2N − N ∗ − 1) 2h − g + − g + gx0 ∗ τ 2 τ N if β < 0, where N ∗ is defined as (15). The optimal trading strategy (16) is obtained by substituting λ∗ into (20). 5.2. Discussion By (15) and (16), the optimal strategy and starting or finishing time depend on the parameter β = ρ(−ξ1 ). Assume that ξk ’s follow a normal distribution N (τ µ, τ σ 2 ) and ρ = CVaRα . By properties A3 andA4 of CVaR, we have β = CVaRα (−ξ1 ) =
√ √ τ σ CVaRα −ξ√1 τ+τσ µ − τ µ = τ σ CVaRα (N (0, 1)) − τ µ. Thus β integrates three factors: volatility σ , trend µ and risk aversion level α . If µ ≤ 0, slow trading will lead to both a high risk and a significant loss of wealth. Since β ≥ 0 when µ ≤ 0, Theorem 2 implies that the optimal strategy starts at stage 0 and finishes at an
6. Numerical experiments We summarize some numerical experiments to show different characteristics of the trading strategies under the dynamic risk measure. Furthermore, we compare the transaction cost and risk under different strategies using real data. We choose ρ = CVaRα in all experiments. 6.1. Trading behaviors under different risk measures In Fig. 1, we compare our trading strategy with Almgren and Chriss’s mean–variance strategy [4]. In all experiments, we first search a risk aversion parameter λ in Almgren and Chriss’s strategy so that their strategy and ours have the same expected transaction cost. We consider ten stages (N = 10) with an initial holding of x0 = 10 000 shares of a stock whose price changes ξk ’s are i.i.d. from N (τ µ, τ σ 2 ) with σ = 1 and τ = 390 = 39. We set up the 10 impact parameters as h = 0.1 and g = 0. In the left plot of Fig. 1, we set µ = 0 and use different levels of risk aversion with α = 0.1, 0.5 and 0.9. In the right plot of Fig. 1, we set α = 0.5 and use different price trends with µ = 0.3, 0 and −0.3. The comparisons results are consistent with our discussion in Section 5.2 for all cases. For example, when µ = 0.3 so that β = −6.7, our strategy will only trade in the last few stages to benefit from the increase of the price. When µ = −0.3 so that β = 16.6, our strategy tries to liquidate the asset early to reduce the risk and wealth loss. 6.2. Experiments with limit order book data We evaluate the performance of our trade execution strategy over real limit order book data from New York Stock Exchange (NYSE). There are five stocks in the sample of data: IBM, JNJ, TWC, WMT and XOM. Our data sample contains every limit order for these five stocks that were submitted to NYSE in July, 2010, which covers 20 trading days. The information of each order given in the data includes submission time, price, volume, side (bid or ask) and order type (limit order, executed order or cancellation order). We can use limit orders to reconstruct the limit order book at every millisecond in July, 2010, with which we can easily evaluate the market price Sk and the actual transaction price Sk associated to each trading size nk at any time. We compare performances of the risk neutral strategy from [9] (Ber and Lo) and the mean–variance-based strategy [4] (Almg and Chr) to our strategy (Lin et al.) by liquidating the five stocks mentioned above. On every trading date in July 2010, we use these three strategies to sell an initial volume x0 randomly chosen between 1% and 2% of the average daily trading volume. The values of x0 for each stock are: IBM:146380, JNJ:173220, TWC:54120, WMT:80247 and XOM:404200.
Q. Lin et al. / Operations Research Letters 43 (2015) 52–58
57
Fig. 1. The trading strategies ∆ = (n1 , n2 , . . . , nN ) that minimize composite dynamic risk (CDR) or mean–variance risk (MV) under different risk aversions α (left) and different price trends µ (right).
Table 1 Comparisons of trading cost ($) in holdout samples. The best result is highlighted in bold font. Strategy IBM JNJ TWC WMT XOM
Mean cost of all samples
Mean cost of the worst 30% samples
Ber and Lo
Alm and Chr
Lin et al.
Ber and Lo
Alm and Chr
Lin et al.
−73,430
−73,310
−72,876
126,439 674,461 241,223 446,871 1,082,796
126,511 666,362 216,381 446,262 1,032,115
125,822 665,577 215,616 445,958 1,026,350
201,064 66,232 196,340 194,713
199,527 58,036 199,514 207,901
200,522 58,583 199,679 207,581
The whole selling program starts at 9:30 AM and x0 must be completely sold by 4:00 PM. The T = 390 minutes in a trading day, we partition x0 into (n1 , . . . , nN ) with N = 78 so that nk is traded in the k-th five-minute interval (τ = 5) in a trading date. We choose α = 0.01 and g = 0 for our strategy and search the λ value in the mean–variance-based strategy [4] so that they give the same theoretical expected transaction cost. We assume ξk ’s are i.i.d. from N (τ µ, τ σ 2 ) where µ and σ 2 can be estimated based on the sequence {ξk = Sk − Sk−1 }k generated from the data. To estimate h, we first randomly generate a sample of nk and sell them at time tk−1 for all k within each trading day. Using the limit order book at time tk−1 , we can obtain the market prices Sk−1 as well as the actual transaction prices Sk corresponding to nk . Using the sample {(Sk−1 , Sk , nk )}k , we estimate h by fitting the linear regression model Sk−1 − Sk = τh nk = hvk according to (3). In our experiments, we utilize a ‘‘moving window’’ procedure with the window length equal to ten. In particular, we first estimate h, µ and σ using the {(Sk−1 − Sk , nk )} sample and {ξk = Sk − Sk−1 } sample from the last ten days as described above, and then implement the trading strategies in the eleventh day using the estimated parameters as inputs. The trading cost is defined N S0 x0 − k=1 Sk nk , where S0 is the initial price at 9:30 AM. After that, we move the window one day forward and so on. Since there are 20 trading days in July 2010, this moving window procedure will generate 10 daily transaction costs as holdout samples. In Table 1, we compute the mean of all and of 30% worst cases of the holdout samples from each strategy. According to Table 1, our dynamic risk averse strategy outperforms the other two on average in the worst 30% cases over these five stocks. However, the other two strategies can generate smaller mean cost than ours. This is because mean cost is not the objective function in our optimization model. Our strategy shows its merit to risk averse traders who are sensitive to large losses in the worst scenarios.
Acknowledgments We thank two anonymous reviewers for their careful reading and constructive feedback. Their input allowed us to improve our paper substantially. References [1] B. Acciaio, I. Penner, Dynamic convex risk measures, in: Advanced Mathematical Methods for Finance, Springer-Verlag, Berlin, Heidelberg, 2011, pp. 1–34. (Chapter 1). [2] A. Alfonsi, A. Fruth, A. Schied, Optimal execution strategies in limit order books with general shape functions, Quant. Finance 10 (2010) 143–157. [3] R. Almgren, Optimal execution with nonlinear impact functions and tradingenhanced risk, Appl. Math. Finance 10 (2003) 1–18. [4] R. Almgren, N. Chriss, Optimal execution of portfolio transactions, J. Risk 3 (2) (2000) 854–861. [5] R. Almgren, J. Lorenz, Adaptive arrival price, Algorithmic Trading III: Precision, Control, Execution 2007 (1) (2007) 59–66. [6] R. Almgren, C. Thum, E. Hauptmann, H. Li, Equity market impact, Risk (2005). [7] P. Artzner, F. Delbaen, J.-M. Eber, D. Heath, Coherent measures of risk, Math. Finance 9 (3) (1999) 203–228. [8] P. Artzner, F. Delbaen, J.-M. Eber, D. Heath, H. Ku, Coherent multiperiod risk adjusted values and Bellman’s principle, Ann. Oper. Res. 152 (1) (2007) 5–22. [9] D. Bertsimas, A.W. Lo, Optimal contraol of execution costs, J. Financ. Markets 1 (1998) 1–50. [10] J. Bion-Nadal, Dynamic risk measures: time consistency and risk measures from BMO martingales, Finance Stoch. 12 (2008) 219–244. [11] S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, 2003. [12] P. Cheridito, F. Delbaen, M. Kupper, Dynamic monetary risk measures for bounded discrete-time processes, Electron. J. Probab. 11 (3) (2006). [13] M. Frittelli, G. Scandolo, Risk measures and capital requirements for processes, Math. Finance 16 (4) (2006) 589–612. [14] A. Fruth, T. Schöneborn, M. Urusov, Optimal trade execution and price manipulation in order books with time-varying liquidity, Math. Finance 24 (4) (2014) 651–695. [15] J. Gatheral, A. Schied, Optimal trade execution under geometric Brownian motion in the Almgren and Chriss framework, Int. J. Theor. Appl. Finance 14 (3) (2011) 353–368.
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[16] G. Huberman, W. Stanzl, Price manipulation and quasi-arbitrage, Econometrica 72 (4) (2004) 1247–1275. [17] G. Huberman, W. Stanzl, Optimal liquidity trading, Rev. Finance 9 (2) (2005) 165–200. [18] J. Lorenz, R. Almgren, Mean–variance optimal adaptive execution, Appl. Math. Finance 18 (2011) 395–422. [19] F. Riedel, Dynamic coherent risk measures, Stochastic Process. Appl. 112 (2004) 185–200. [20] R.T. Rockafellar, S.P. Uryasev, Conditional value-at-risk for general loss distributions, J. Bank. Finance 26 (2002) 1443–1471.
[21] A. Ruszczynski, Risk-averse dynamic programming for Markov decision processes, Math. Program. B 125 (2) (2010) 235–261. [22] A. Ruszczynski, A. Shapiro, Conditional risk mappings, Math. Oper. Res. 31 (3) (2006) 544–561. [23] A. Schied, T. Schöneborn, M. Tehranchi, Optimal basket liquidation for CARA investors is deterministic, Appl. Math. Finance 17 (2010) 471–489. [24] A. Selivanov, M. Urusov, Optimal portfolio liquidation with dynamic coherent risk, http://www.mat.univie.ac.at/~anstap10/slides/Urusov_2.pdf. [25] A. Shapiro, Time consistency of dynamic risk measures, Tech. Rep., School of Industrial & Systems Engineering, Georgia Institute of Technology, 2012.
Contents lists available at ScienceDirect
Operations Research Letters journal homepage: www.elsevier.com/locate/orl
A trade execution model under a composite dynamic coherent risk measure Qihang Lin a,∗ , Xi Chen b , Javier Peña c a
Tippie College of Business, The University of Iowa, Iowa City, IA 52245, United States
b
Stern School of Business, New York University, New York, NY 10012, United States
c
Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA 15213, United States
article
info
Article history: Received 12 February 2014 Received in revised form 3 November 2014 Accepted 8 November 2014 Available online 18 November 2014
abstract We investigate the trade execution problem where a risky asset must be sold before a deadline with a control on the transaction cost. The asset price is modeled as a discrete random walk perturbed by price impacts. We show that the optimal trading strategy minimizing a dynamic coherent risk of the transaction cost is time-consistent and deterministic. We obtain a closed-form expression for the optimal strategy and evaluate its numerical performance over real data. © 2014 Elsevier B.V. All rights reserved.
Keywords: Dynamic coherent risk measures Trade execution Risk-averse Markov decision process
1. Introduction One of the tasks in algorithmic trading is to execute the transaction of a large volume of an asset with a balance between transaction cost and risk. The sources leading to transaction cost include commissions, exchange fees, taxes, bid–ask spread and price impact, among which the last one typically dominates the others. Trading a large volume of an asset within a short period of time generally leads to an unbearable transaction cost due to the large adverse price movement it induces. A better execution strategy may break down a large order into several smaller ones and spread them over time to mitigate the price impact. Since the transaction cost is a random variable, a risk measure needs to be specified before comparing different execution strategies. The optimal execution with respect to a risk measure is the one minimizing the value of the risk measure of transaction cost. Besides optimality, another important property of an execution strategy is time-consistency. Roughly speaking, a strategy is time-consistent if it minimizes the risk value from any intermediate time to the deadline conditioning on the historical sample path. The time-consistency of an optimal trade execution strategy depends on the risk measure it minimizes. Bertsimas and Lo [9] consider a risk-neutral execution strategy by minimizing the expectation, the simplest risk measure, of transaction cost. The analytical form of the optimal execution is derived from the Bellman
∗
Corresponding author. E-mail address: [email protected] (Q. Lin).
http://dx.doi.org/10.1016/j.orl.2014.11.005 0167-6377/© 2014 Elsevier B.V. All rights reserved.
equation of dynamic programming, which guarantees its timeconsistency. In their seminal work, Almgren and Chriss [4] consider a deterministic risk-averse strategy under a mean–variance risk measure. Their strategy is time-consistent [17] but not optimal [5] because it is obtained after restricting the strategy to be deterministic. An adaptive risk-averse strategy under a mean–variance risk has been proposed by Lorenz and Almgren [18]. This strategy is both time-consistent and optimal but can only be computed approximately via a cost-to-go function on a set of grid points. Given the computational challenges and the dichotomy between optimality and time-consistency in the existing approaches, one of our main goals is to obtain a risk averse execution strategy under some risk measure that is computationally tractable, globally optimal, and time-consistent. The risk measure we utilize in this paper is a composite dynamic coherent risk measure that belongs to the general family of dynamic risk measures [8,12,13,19,22,21]. A composite dynamic coherent risk measure for trade execution is appropriate for a variety of reasons. First, a composite dynamic coherent risk measure evaluates the risk of the transaction cost at each intermediate stage, providing richer information to the decision maker. Second, a composite dynamic coherent risk measure satisfies the coherent properties [7] so it can better reflect the rationale of a risk-averse decision maker in a multistage setting. Third, the time-consistency of a composite dynamic coherent risk measure ensures that it can be minimized using the dynamic programming principle [21], which guarantees the time-consistency of the corresponding optimal execution strategy. In this paper we measure the transaction cost of the classical trade execution model by Almgren and Chriss [4] using a composite
Q. Lin et al. / Operations Research Letters 43 (2015) 52–58
dynamic coherent risk measure. We prove that the optimal strategy is time-consistent and deterministic. Furthermore, we show that the impact of the choice of a composite dynamic coherent risk measure on the optimal strategy is only through a real value ρ(−ξ ) where ρ is a coherent risk measure used to build the dynamic risk measure and ξ is the source of the randomness in the trade execution model [4]. This observation allows us to reformulate the trade execution problem as a quadratic program over a simplex with a closed-form solution. Under a continuous-time framework, Schied et al. [23] and Gatheral et al. [15] show respectively that, when the risk measure is the expected constant absolute risk aversion utility or the time-averaged value-at-risk, the resulting optimal strategy is also deterministic and time-consistent. However, depending on the parameters of the problem, their strategies may involve both buying and selling activities as well as holding short positions at intermediate times, which violate some regulatory rules in agency trading. By contrast, our strategy only sells during the liquidation. In [2], Alfonsi et al. consider a risk neutral strategy in a market with a limit order book of a general shape and also obtain a deterministic optimal strategy with a closed-form solution. The deterministic nature of the policy mentioned above and our policy are related to the assumption of deterministic market liquidity. During the review process of this paper, the unpublished work by Selivanov and Urusov [24] was brought to our attention. The setting and results in [24] are similar to ours and were independently developed. However, their work and the details of the relevant proofs remain unpublished. 2. Almgren and Chriss’s trade execution model We now briefly review the classical model for trade execution by Almgren and Chriss [4]. We focus on the strategy for liquidating (selling) an asset for simplicity. All of our results can be applied to the case of buying an asset. Consider a probability space (Ω , F , P ) on which there exist i.i.d. random variables ξ1 , ξ2 , . . . , ξN −1 . Let F0 = {Ω , ∅}, Fk = σ (ξ1 , . . . , ξk ) and ξ[k] = (ξ1 , . . . , ξk ) for k = 1, 2, . . . , N − 1. Let x0 ∈ R+ represent the initial volume of a risky asset that has to be sold within a finite time period [0, T ]. The total volume is partitioned into N smaller blocks n1 , n2 , . . . , nN , where nk is to be sold within a time interval [tk−1 , tk ]. Assume tk = kτ with τ = NT for k = 1, . . . , N. We call ∆ = (n1 , . . . , nN ) the trading strategy and denote the trading trajectory associated to ∆ by (x1 , . . . , xN ), where xk represents the remaining unsold asset at time tk . The relationship between nk ’s and xk ’s is given by xk = xk−1 − nk ,
k = 1, . . . , N .
(1)
Let Sk be the market price of the asset that can be observed by the trader at time tk for k = 0, . . . , N − 1. We assume S0 is a deterministic constant in R+ . Following [4], the dynamic of Sk is modeled as Sk = Sk−1 + ξk − g · nk ,
k = 1, . . . , N − 1.
(2)
The linear term gnk in (2) measures the permanent price impact incurred by selling nk . Due to the limited liquidity of the market during [tk−1 , tk ], the prices at which the asset nk is sold are not consistently the prices observed at tk−1 , i.e., Sk−1 . The actual average transaction price during [tk−1 , tk ] is denoted by Sk , which is related to Sk−1 as nk Sk = Sk−1 − h · , τ nk
k = 1, . . . , N .
(3)
The linear term h τ in (2) represents the temporary impact on the n price as a linear function of the trading speed τk . The assumption of the linearity of the price impacts is supported by results
53
from [16] where the authors show that only linear impact functions prevent the opportunity of quasi-arbitrage. In the rest of the paper, we make the following assumption.
≥ g. Assumption 1. 2h τ This assumption states that, each unit sold during a period of length τ generates a permanent impact that is at most twice the temporary impact. The temporary impact is due to the temporary unbalanced demand and supply while the permanent impact is due to the change of the (publicly believed) fundamental value of the asset caused by the signal revealed from a transaction. This assumption will typically hold for an asset whose fundamental value is consistent such as the stock of a well-established and stable company. The empirical evidence in [6] also justifies this assumption when the trade duration τ is short. At time tk−1 , the trader determines nk based on the history (x[k−1] , S[k−1] ), where x[k−1] = (x0 , . . . , xk−1 ) and S[k−1] = (S0 , . . . , Sk−1 ). Hence, Sk−1 , Sk , nk and xk are random variables in (Ω , Fk−1 , P ). To completely liquidate the asset by T , we have to N enforce the condition xN = 0, or equivalently, k=1 nk = x0 . We also assume buying is not allowed, i.e., xk−1 ≥ xk , or equivalently, nk ≥ 0 for k = 1, . . . , N. Given a trading strategy ∆, the total cost (negative revenue) N the trader obtains during [0, T ] is k=1 −Sk nk . Bertsimas and Lo [9] consider a risk neutral strategy by minimizing the expected N cost, i.e., E( k=1 − Sk nk ). Almgren et al. [4,5,3] consider a risk averse strategy by minimizing the mean-variation of the cost, i.e., N N E( k=1 − Sk nk ) + λ Var( k=1 − Sk nk ) with λ > 0. In contrast to these previous approaches, we consider a risk averse strategy by minimizing the composite dynamic coherent risk of the cost. 3. Composite dynamic coherent risk measure The general definition of dynamic risk measure is given in [19,1, 10]. The composite dynamic coherent risk measure used here is the special dynamic risk measure introduced in [21,25], which can be constructed as a nested composition of conditional risk mappings. p p We define respectively Lp = Lp (Ω , F , P ) and Lk = Lk (Ω , Fk , P ) as the sets of random variables on (Ω , F , P ) and (Ω , Fk , P ) with finite pth moment for p ∈ [1, ∞) for k = 0, . . . , N − 1. We p assume that for each Z ∈ Lk , there exists a measurable function on k R such that Z = Z (ξ1 , . . . , ξk ). Definition 1. A mapping ρ : Lp → R is called a coherent risk measure on Lp if it satisfies the following properties [7] for all X and Y in Lp : A1. Convexity: ρ(λX + (1 − λ)Y ) ≤ λρ(X ) + (1 − λ)ρ(Y ), ∀λ ∈ [0, 1]; A2. Monotonicity: ρ(X ) ≤ ρ(Y ) if X ≤ Y ; A3. Translation Equivalence: ρ(X + Y ) = X + ρ(Y ) if X is a constant; A4. Positive Homogeneity: ρ(β X ) = βρ(X ), ∀β ≥ 0. A coherent risk measure ρ is law invariant if ρ(X ) = ρ(Y ) for all X and Y in Lp with the same distribution. One commonly used law invariant coherent risk measure is conditional value at risk (CVaR) [20]:
CVaRα (X ) = inf r + r ∈R
1 1−α
E(X − r )+ ,
where α ∈ [0, 1) is a risk aversion parameter. Note that
|CVaRα (X )| ≤ CVaRα (|X |) ≤ p
E|X | 1−α
< +∞
(4)
for all X ∈ L . Based on a coherent risk measure, we can construct the corresponding conditional risk mapping.
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Q. Lin et al. / Operations Research Letters 43 (2015) 52–58
Definition 2. Given a law invariant coherent risk measure ρ on Lp p and Z = Z (ξ1 , ξ2 , . . . , ξk+1 ) ∈ Lk+1 , define Z ′ : Rk → R as
Z ′ (θ1 , θ2 , . . . , θk ) = ρ( Z (θ1 , θ2 , . . . , θk , ξk+1 )), for all (θ1 , θ2 , . . . , θk ) ∈ Rk and k = 0, . . . , N − 2. The conditional p p risk mapping induced by ρ at time tk is the mapping ρk : Lk+1 → Lk defined by
ρk (Z ) = Z ′ (ξ1 , ξ2 , . . . , ξk ),
∀Z ∈
p Lk+1
.
E|ρk (Z )| ≤ Eξ[k]
≤
Z (ξ1 , . . . , ξk , ξk+1 )| Eξk+1 |
p
(1 − α)p
(5)
The trade execution problem is to sell x0 by time T with a minimum composite dynamic coherent risk at t0 , i.e.,
s.t.
nk = x 0 ,
− Sk nk · · ·
.
k=1
− S k nk
=
N −1
− Sk nk + ρN −2 − SN nN .
k=1
N
− S k nk
(6) nk ≥ 0;
k=1
To have a well-defined objective function in (6), we make the following assumption. Assumption 2. The distribution of ξk ’s is such that p Lk−1 for k = 1, . . . , N for all feasible solutions of (6).
k
i=1 −Si ni ∈
By (2) and (3) and the constraint 0 ≤ nk ≤ x0 , one can show that this assumption is satisfied, for example, when ξk ’s are normally distributed. 4. Convex optimization formulation Proposition 1. Let Vk : R+ × R → R for k = 0, . . . , N − 1 be x and defined by VN −1 (x, S ) := − S − hx τ
hn Vk (x, S ) := min − S − n 0≤n≤x τ
(8)
Suppose n1 , . . . , nN −1 have been determined. To sell the asset completely by T , the only feasible and optimal decision at time tN −1 is nN = xN −1 , i.e., n∗N (xN −1 , SN −1 ) = xN −1 . Thus,
hxN −1 xN −1 = VN −1 (xN −1 , SN −1 ). − S N nN = − S N − 1 − τ Substituting this equality into (8), by property A2 of ρ and the definitions of ρk ’s as well as Bellman’s principle of optimality, we have
(n1 ,...,nN )
− Sk nk
nk is a random variable on (Ω , Fk−1 , P ).
N
= − S1 n1 + ρ0 − S2 n2 + ρ1 · · · − S N − 2 nN − 2 + ρN −3 − SN −1 nN −1 + ρN −2 − S N nN ··· .
min
k=1 N
= ρ0 ρ1 · · · ρN −2
− S k nk
k=1
ρk,N −1 (Z ) = ρk (ρk+1 (· · · ρN −2 (Z ) · · ·)).
(n1 ,...,nN )
ρ N −2
N
ρ0,N −1
< +∞
Definition 3. Given a law invariant coherent risk measure ρ on Lp , the composite dynamic coherent risk measure induced by ρ at time p p tk is the mapping ρk,N −1 : LN −1 → Lk defined by
N
The composite dynamic coherent risk measure is a nested composition of conditional risk mappings.
ρ0,N −1
Applying this argument repeatedly, we can show
p
min
k=1
for all Z ∈ Lk+1 for k = 0, . . . , N − 2. The conditional risk mapping above is a special case of the general conditional risk mapping defined in [22,21].
Since k=1 − Sk nk does not depend on ξN −1 , according to property A3 of ρ and the definition of ρN −2 , we have
p
1−α
Eξ[k+1] | Z (ξ1 , . . . , ξk , ξk+1 )|p
ρ0,N −1
N
N −1
p
k=1
Remark 1. We assume that ρ ensures ρk (Z ) ∈ Lk for all Z ∈ Lk+1 . This assumption is satisfied, for example, when ρ = CVaRα . In fact, by (4), we have p
Proof. Our proof is an adaptation of the general argument by Ruszczynski [21, Theorem 2] to the specific dynamic risk measure we defined. The dynamic programming equations (7) yield the time consistency of the optimal strategy. Recall that
ρ0,N −1
N
− S k nk
k=1
ρ0 · · · − SN −2 nN −2 + ρN −3 − S N − 1 nN − 1 (n1 ,...,nN −1 ) + ρN −2 VN −1 (xN −1 , SN −1 ) · · · = min ρ0 · · · − SN −2 nN −2
=
min
(n1 ,...,nN −2 )
+ ρN −3
min
0≤nN −1 ≤xN −2
− SN −1 nN −1
+ ρN −2 VN −1 (xN −1 , SN −1 ) · · · .
(9)
By (1)–(3), we have
− SN −1 nN −1 + ρN −2 VN −1 (xN −1 , SN −1 ) hnN −1 nN − 1 = − S N −2 − τ + ρN −2 VN −1 (xN −2 − nN −1 , SN −2 + ξN −1 − gnN −1 ) .
(10)
Note that VN −2 (x, S ) = min0≤n≤x φ(x, S , n), where
+ ρ Vk+1 (x − n, S + ξk+1 − gn)
(7)
for k = 0, . . . , N − 2. Then the optimal value of (6) equals V0 (x0 , S0 ). Moreover, the optimal trading strategy ∆∗ = (n∗1 , . . . , n∗N ) for (6) is Markovian, where n∗N (xN −1 , SN −1 ) = xN −1 and n∗k+1 (xk , Sk ) is the optimal solution of (7) with x = xk and S = Sk for k = 0, . . . , N − 2.
φ(x, S , n) := − S −
hn
τ
n + ρ VN −1 (x − n, S + ξN −1 − gn) .
Since nN −1 , xN −2 and SN −2 do not depend on ξN −1 , by the definition of ρN −2 , (10) is equal to φ(xN −2 , SN −2 , nN −1 ) when (10) is understood as a function of (xN −2 , SN −2 , nN −1 ). Although the decision nN −1 depends on the history (x[N −2] , S[N −2] ), it suffices to solve the inner problem in (9) over the real interval [0, xN −2 ] because the
Q. Lin et al. / Operations Research Letters 43 (2015) 52–58
objective function and the constraint both depend on the history only via (xN −2 , SN −2 ). Therefore, the optimal value of the innermost minimization in (9) equals min
0≤n≤xN −2
φ(xN −2 , SN −2 , n) = VN −2 (xN −2 , SN −2 )
for each fixed value of xN −2 and SN −2 . Moreover, the optimal decision n∗N −1 depends on (xN −2 , SN −2 ) as n∗N −1 (xN −2 , SN −2 ) = arg min φ(xN −2 , SN −2 , n). 0≤n≤xN −2
Therefore, we can further represent (9) as
min
(n1 ,...,nN )
=
ρ0,N −1
N
− Sk nk
k=1
S N − 2 nN − 2 ρ0 · · · − (n1 ,...,nN −2 ) + ρ N −3 VN −2 ( x N −2 , S N −2 ) · · · . min
We proceed with the similar argument as above for k = N − 3, . . . , 0 and obtain the conclusion of the proposition. Using the particular structure of problem (6), we can refine (7) and show that the optimal n∗k+1 (xk , Sk ) only depends on xk but not on Sk and thus is deterministic. Theorem 1. The function Vk (x, S ) satisfies Vk (x, S ) = fk (x) − Sx
(11)
for k = 0, . . . , N − 1, where fN −1 (x) := τ and
fk (x) := min
0≤n≤x
hn2
τ
(12)
for k = 0, . . . , N − 2. The optimal trading strategy ∆∗ = (n∗1 , . . . , n∗N ) is deterministic where n∗k+1 is the optimal solution of (12) with x = xk for k = 0, . . . , N − 2 and n∗N = xN −1 . Proof. We prove (11) by backward induction. When k = N − 1, VN −1 (x, S ) = − S − hx x = fN −1 (x) − Sx. Suppose (11) holds for τ k + 1 with k ≤ N − 2, we want to prove that it also holds for k. By the induction hypothesis, we have
= fk+1 (x − n) − (S + ξk+1 − gn)(x − n).
x0 −
k=1 N
nk = x 0 ,
k
ni
i =1
(14)
nk ≥ 0, k = 1, . . . , N ,
k=1
N
k=1
n2k + g
N
k=1
nk x 0 −
k
i=1
ni .
ρ (−ξ1 ) Nk=1 x0 − ki=1 ni is a linear function of ∆. Because k N x0 − domain, f (∆) can be i=1 ni = i=k+1 ni on the feasible N N N h 2 written as f (∆) = τ k=1 nk + g k=1 nk i=k+1 ni which is a − g IN ×N + gJN . Here, quadratic function of ∆ whose Hessian is 2h τ IN ×N is an N × N identity matrix and JN is an N × N all-one matrix. By Assumption 1, 2h − g IN ×N + gJN is positive semidefinite and τ so (14) is a convex problem.
5. The optimal liquidation strategy
Theorem 2. Let a =
ρ (Vk+1 (x − n, S + ξk+1 − gn))
= fk (x) − Sx.
f (∆) + ρ (−ξ1 )
We are now ready to present the main result of this paper, namely a closed-form solution to (14).
Using properties A3 and A4, we obtain
hn min − S − n + ρ (Vk+1 (x − n, S + ξk+1 − gn)) 0≤n≤x τ 2 hn = min + fk+1 (x − n) + gn(x − n) 0≤n≤x τ + ρ(−ξk+1 )(x − n) − Sx
min
(n1 ,...,nN )
N
5.1. A closed form for the optimal strategy
Vk+1 (x − n, S + ξk+1 − gn)
Therefore, by (7), we can write Vk (x, S ) as
Corollary 1. The optimal trading strategy ∆∗ = (n∗1 , . . . , n∗N ) is the solution of
Problem (14) can be solved easily if its objective function is convex over its feasible domain which is a simplex. We note that
= fk+1 (x − n) − (S − gn)(x − n) + ρ (−ξk+1 ) (x − n).
Note that the value ρ(−ξk+1 ) in (12) is the same for k = 0, . . . , N − 2 because ρ is law invariant and ξk ’s are i.i.d. It can be seen from (12) that the composite dynamic coherent risk measure impacts the optimal strategy only through the value ρ(−ξk+1 ). This is due to the particular price dynamic (2) we consider and the fact that ξk ’s are i.i.d. We can view (12) as dynamic programming equations, where the term ρ(−ξk+1 )(x − n) is interpreted as a penalty the trader pay at stage k for the risk of the unsold asset x−n. Similar to the policies in [23,15,2], our policy is deterministic partially because the price impact coefficients h and g (market liquidity) are deterministic. If h and g were stochastic, the optimal strategy would be stochastic [14]. The optimal trading strategy ∆∗ can be solved sequentially from (12). However, we can also solve ∆∗ from a single optimization problem by applying (12) recursively:
where f (∆) = τh
+ fk+1 (x − n) + gn(x − n)
+ ρ(−ξk+1 )(x − n)
Therefore, we have shown that (11) holds for k. By Proposition 1, the optimal strategy n∗k+1 solves the minimization in (7) with x = xk and S = Sk . According to the derivation above, n∗k+1 only depends on xk and can be computed sequentially by solving (12) and thus is deterministic.
s.t
hx2
55
(13)
N ∗ ≡ min N ,
β 2h
τ −g
−1 +
√
with β = ρ(−ξ1 ) and
1 + 8x0 /|a|
2
.
(15)
If β ≥ 0, the optimal liquidation starts at stage 0 and finishes at stage N ∗ . If β < 0, the optimal liquidation starts at stage N − N ∗ and finishes at stage N. The optimal trading strategy and the optimal trading trajectory are given as:
x (N ∗ − 2k + 1)a 0 + N∗ 2 + n∗k = ∗ ( 2N − N − 2k + 1)a x 0 + ∗ N
2
+
if β ≥ 0 (16) if β < 0
56
Q. Lin et al. / Operations Research Letters 43 (2015) 52–58
and
∗ (N ∗ − k)ka (N − k)x0 − if β ≥ 0 N∗ 2 + x∗k = (N − k)x0 (N − k)(N − N ∗ − k)a − ∧ x0 N∗ 2 if β < 0
(17)
where (x)+ = max{x, 0} and x ∧ y = min{x, y} for k = 1, . . . , N. Proof. Because of the equality x0 − optimization problem (14) becomes min
(n1 ,...,nN )
s.t
N h
τ k=1 N
n2k
+g
N k=1
nk = x0 ,
nk
N
k
i=1
ni =
ni
+β
i=k+1
N
i=k+1
N N k=1 i=k+1
ni , the
ni (18)
nk ≥ 0, k = 1, . . . , N .
intrinsic time horizon N ∗ . By (15), N ∗ will move towards 0 as β increases and the trader sells faster. In the limit where β = +∞, we have N ∗ = 0, n∗1 = x0 , and nk = 0 for k = 2, . . . , N. On the other hand, if µ >√0 then the sign of β depends on the trade-off between the risk τ σ CVaRα (N (0, 1)) and the profit τ µ from holding the asset one more period. If the latter is larger, β becomes negative and Theorem 2, implies that the optimal strategy should finish at stage N but begin at an intrinsic starting time N − N ∗ . As β decreases, N − N ∗ will move towards N and the trader will postpone selling. In the limit where β = −∞, we have N − N ∗ = N , n∗N = x0 and nk = 0, for k = 1, . . . , N − 1. We also note that the intrinsic time horizon N ∗ is an increasing function of x0 when β > 0. If x0 is small, we have N ∗ < N. In other words, even though the trader has N stages to sell the asset, he may only need to trade in the first few stages and do nothing in the remaining stages. By contrast, the mean–variance strategy [4] does not have this kind of sparsity property. Instead, the strategy in [4] typically ends with a ‘‘small tail’’ (see Fig. 1).
k=1
By Assumption 1, (18) is a convex quadratic program with a simplex constraint. Let λ and uk be the dual variables associated with the equality and inequality constraints respectively. We can then write down the corresponding KKT conditions [11] as follows: 2h
τ
nk + g
N
N
ni + (k − 1)β − λ − uk = 0,
i=1,i̸=k
nk = x 0 ,
(19) nk uk = 0 ,
nk ≥ 0, uk ≥ 0,
k=1
for k = 1, . . . , N. We next show that the solution to these KKT conditions is given by (16). − g )nk + gx0 + (k − 1)β − Observe that (19) is equivalent to ( 2h τ
λ−uk = 0 because Nk=1 nk = x0 . By the complementary slackness condition nk uk = 0 it follows that uk = (−λ+ gx0 +(k − 1)β)+ and λ − gx0 − (k − 1)β . (20) nk (λ) = 2h −g τ + N To finish, we only need to find the value of λ such that k=1 nk (λ) = x0 . Note that nk (λ) Nin (20) is a non-decreasing piecewise linear function of λ. So is k=1 nk (λ). By some elementary calculations, N we can show that k=1 nk (λ) = x0 holds when λ = λ∗ with x0 2h a(N ∗ − 1) 2h − g + gx0 if β ≥ 0 N∗ τ − g + 2 τ ∗ λ = x0 2h a(2N − N ∗ − 1) 2h − g + − g + gx0 ∗ τ 2 τ N if β < 0, where N ∗ is defined as (15). The optimal trading strategy (16) is obtained by substituting λ∗ into (20). 5.2. Discussion By (15) and (16), the optimal strategy and starting or finishing time depend on the parameter β = ρ(−ξ1 ). Assume that ξk ’s follow a normal distribution N (τ µ, τ σ 2 ) and ρ = CVaRα . By properties A3 andA4 of CVaR, we have β = CVaRα (−ξ1 ) =
√ √ τ σ CVaRα −ξ√1 τ+τσ µ − τ µ = τ σ CVaRα (N (0, 1)) − τ µ. Thus β integrates three factors: volatility σ , trend µ and risk aversion level α . If µ ≤ 0, slow trading will lead to both a high risk and a significant loss of wealth. Since β ≥ 0 when µ ≤ 0, Theorem 2 implies that the optimal strategy starts at stage 0 and finishes at an
6. Numerical experiments We summarize some numerical experiments to show different characteristics of the trading strategies under the dynamic risk measure. Furthermore, we compare the transaction cost and risk under different strategies using real data. We choose ρ = CVaRα in all experiments. 6.1. Trading behaviors under different risk measures In Fig. 1, we compare our trading strategy with Almgren and Chriss’s mean–variance strategy [4]. In all experiments, we first search a risk aversion parameter λ in Almgren and Chriss’s strategy so that their strategy and ours have the same expected transaction cost. We consider ten stages (N = 10) with an initial holding of x0 = 10 000 shares of a stock whose price changes ξk ’s are i.i.d. from N (τ µ, τ σ 2 ) with σ = 1 and τ = 390 = 39. We set up the 10 impact parameters as h = 0.1 and g = 0. In the left plot of Fig. 1, we set µ = 0 and use different levels of risk aversion with α = 0.1, 0.5 and 0.9. In the right plot of Fig. 1, we set α = 0.5 and use different price trends with µ = 0.3, 0 and −0.3. The comparisons results are consistent with our discussion in Section 5.2 for all cases. For example, when µ = 0.3 so that β = −6.7, our strategy will only trade in the last few stages to benefit from the increase of the price. When µ = −0.3 so that β = 16.6, our strategy tries to liquidate the asset early to reduce the risk and wealth loss. 6.2. Experiments with limit order book data We evaluate the performance of our trade execution strategy over real limit order book data from New York Stock Exchange (NYSE). There are five stocks in the sample of data: IBM, JNJ, TWC, WMT and XOM. Our data sample contains every limit order for these five stocks that were submitted to NYSE in July, 2010, which covers 20 trading days. The information of each order given in the data includes submission time, price, volume, side (bid or ask) and order type (limit order, executed order or cancellation order). We can use limit orders to reconstruct the limit order book at every millisecond in July, 2010, with which we can easily evaluate the market price Sk and the actual transaction price Sk associated to each trading size nk at any time. We compare performances of the risk neutral strategy from [9] (Ber and Lo) and the mean–variance-based strategy [4] (Almg and Chr) to our strategy (Lin et al.) by liquidating the five stocks mentioned above. On every trading date in July 2010, we use these three strategies to sell an initial volume x0 randomly chosen between 1% and 2% of the average daily trading volume. The values of x0 for each stock are: IBM:146380, JNJ:173220, TWC:54120, WMT:80247 and XOM:404200.
Q. Lin et al. / Operations Research Letters 43 (2015) 52–58
57
Fig. 1. The trading strategies ∆ = (n1 , n2 , . . . , nN ) that minimize composite dynamic risk (CDR) or mean–variance risk (MV) under different risk aversions α (left) and different price trends µ (right).
Table 1 Comparisons of trading cost ($) in holdout samples. The best result is highlighted in bold font. Strategy IBM JNJ TWC WMT XOM
Mean cost of all samples
Mean cost of the worst 30% samples
Ber and Lo
Alm and Chr
Lin et al.
Ber and Lo
Alm and Chr
Lin et al.
−73,430
−73,310
−72,876
126,439 674,461 241,223 446,871 1,082,796
126,511 666,362 216,381 446,262 1,032,115
125,822 665,577 215,616 445,958 1,026,350
201,064 66,232 196,340 194,713
199,527 58,036 199,514 207,901
200,522 58,583 199,679 207,581
The whole selling program starts at 9:30 AM and x0 must be completely sold by 4:00 PM. The T = 390 minutes in a trading day, we partition x0 into (n1 , . . . , nN ) with N = 78 so that nk is traded in the k-th five-minute interval (τ = 5) in a trading date. We choose α = 0.01 and g = 0 for our strategy and search the λ value in the mean–variance-based strategy [4] so that they give the same theoretical expected transaction cost. We assume ξk ’s are i.i.d. from N (τ µ, τ σ 2 ) where µ and σ 2 can be estimated based on the sequence {ξk = Sk − Sk−1 }k generated from the data. To estimate h, we first randomly generate a sample of nk and sell them at time tk−1 for all k within each trading day. Using the limit order book at time tk−1 , we can obtain the market prices Sk−1 as well as the actual transaction prices Sk corresponding to nk . Using the sample {(Sk−1 , Sk , nk )}k , we estimate h by fitting the linear regression model Sk−1 − Sk = τh nk = hvk according to (3). In our experiments, we utilize a ‘‘moving window’’ procedure with the window length equal to ten. In particular, we first estimate h, µ and σ using the {(Sk−1 − Sk , nk )} sample and {ξk = Sk − Sk−1 } sample from the last ten days as described above, and then implement the trading strategies in the eleventh day using the estimated parameters as inputs. The trading cost is defined N S0 x0 − k=1 Sk nk , where S0 is the initial price at 9:30 AM. After that, we move the window one day forward and so on. Since there are 20 trading days in July 2010, this moving window procedure will generate 10 daily transaction costs as holdout samples. In Table 1, we compute the mean of all and of 30% worst cases of the holdout samples from each strategy. According to Table 1, our dynamic risk averse strategy outperforms the other two on average in the worst 30% cases over these five stocks. However, the other two strategies can generate smaller mean cost than ours. This is because mean cost is not the objective function in our optimization model. Our strategy shows its merit to risk averse traders who are sensitive to large losses in the worst scenarios.
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