A note on Almgren-Chriss optimal execution problem with geometric Brownian motion
AA note on Almgren-Chriss optimal execution problem withgeometric Brownian motion ∗ Bastien
Baldacci
CMAP, Ecole [email protected]
Benveniste
Ritter Alpha LP and [email protected] 25, 2020
Abstract
We solve explicitly the Almgren-Chriss optimal liquidation problem where the stock priceprocess follows a geometric Brownian motion. Our technique is to work in terms of cash andto use functional analysis tools. We show that this framework extends readily to the case of astochastic drift for the price process and the liquidation of a portfolio.
Keywords:
Almgren-Chriss, optimal liquidation, adjoint operator.
Optimal liquidation is a problem faced by a trader when he needs to liquidate a large number of shares.The trader faces a tradeoff between fast execution, reducing a risk related to price changes and slowexecution, allowing to avoid high trading costs. Since the seminal paper by Almgren and Chriss [2],various extensions of optimal liquidation problems have been studied, see for example [1, 3, 8]. Thecommon framework to address this issue introduced in [2] assumes the following: • the efficient price process follows an arithmetic Brownian motion (ABM), • permanent market impact is linear, • transaction costs are a linear function of the trading rate. ∗ The ideas presented in this paper do not necessarily reflect the views or practices at Ritter Alpha LP. This workbenefits from the financial support of the Chaires Analytics and Models for Regulation, Financial Risk and Financeand Sustainable Development. Bastien Baldacci gratefully acknowledge the financial support of the ERC Grant 679836Staqamof. The authors would like to thank Robert Almgren, Peter Carr, Jim Gatheral, Dylan Possamai and MathieuRosenbaum for fruitful discussions. In particular, Peter Carr deserves warm thanks as this article would not existwithout his inputs. a r X i v : . [ q -f i n . T R ] J un he execution of a large order is then formulated in discrete time as a tradeoff between expectedcosts and risk of the trading strategy, with variance as a risk measure. Under this framework, thereexists a unique optimal liquidation strategy, which is a deterministic function of time and the initialposition of the trader.Continuous versions of this problem have been considered, notably in [5, 6], where the author showsthe ill-posedness of the mean-variance framework leading to time-inconsistent solutions. To overcomethis issue, the authors suggest using alternative objective functions, particularly mean-quadratic vari-ation. Under this choice, the authors solve a two-dimensional Hamilton-Jacobi-Bellman equationnumerically. Moreover, in [8], the authors consider the optimal execution problem with CARA utilityobjective function. To the best of our knowledge, there is no closed-form solution to the continuousversion of the Almgren-Chriss framework with quadratic variation as a risk measure and geometricBrownian motion (GBM) assumption for the efficient price process. In [7], the authors solve a modi-fied version of the problem with GBM, accounting for the risk with a linear function of the trading rate.In this paper, we solve the optimal liquidation problem under the Almgren-Chriss framework in con-tinuous time, in the case where the efficient price process follows a GBM, and the risk measure isquadratic variation. Motivated by [4], we assume that trading costs are a quadratic function of theamount of cash, instead of shares. Thus, we reformulate the problem in terms of cash traded andderive in closed-form the optimal control of the trader liquidating his position. As the method isbased on the resolution of a system of ODEs, it does not suffer from the curse of dimensionality. Inparticular, we show how to extend this framework to the case of the liquidation of a portfolio of N ,possibly correlated, assets. It also enables us to treat the case where the return’s drift is a stochasticprocess without any BSDE methods with a singular condition.The paper is organized as follows. In Section 2, we describe the Almgren Chriss framework incontinuous time and reformulate the optimization problem in terms of cash. In Section 3, we obtaina closed-form solution of the Almgren Chriss framework with GBM for the efficient price process.Finally in Section 4, we present numerical applications under different market conditions. We define (Ω , F t ∈ [0 ,T ] , P ) a filtered probability space, on which all stochastic process are defined, anda trading horizon is T > We rapidly recall the well-known Almgren-Chriss problem in continuous time. We consider the issueof the liquidation of q ∈ R shares of a stock whose price at time t is defined by S t . The number ofshares hold by the trader is defined by an absolutely continuous measurable process q t := q − R t . q s d s where ( . q s ) s ∈ [0 ,T ] is the trading rate, controlled by the trader. The transaction price is˜ S t := S t + λ . q t + γ ( q t − q ) , λ, γ ∈ R + are constants related respectively to temporary and permanent price impact. Indeed,the term λ . q t is the impact of trading . q t shares at time t , whereas the term γ ( q t − q ) is the impactgenerated by the flow of transactions up to time t . In the original framework in discrete time, see[2], and in most of the extensions in continuous time, see [8] for example, the price process follows anABM. The number of shares hold by the trader satisfies the boundary condition q T = 0. Therefore,the cost of this strategy during the trading period is C ( . q ) := Z T ˜ S t q t d t. Aiming at remedying the time inconsistency of the optimal strategies in the pre-commitment mean-variance framework, inspired by [6], we replace the variance by the quadratic variation in the penalty.The optimal execution problem consists in the optimization of a mean-quadratic variation objectivefunction over the strategies ( . q t ) t ∈ [0 ,T ] ∈ A where A := (cid:26) ( . q t ) t ∈ [0 ,T ] , F t − measurable such that Z T . q s ds = q (cid:27) . The problem can be written as follows:sup v ∈A E h − C ( . q ) − κ hCi T i , with κ > hCi T := Z T q t d h S i t . The use of quadratic variation leads to time-consistent strategies. Moreover, in contrast to the vari-ance, quadratic variation takes into account the trajectory of liquidation. A direct integration byparts on C ( q ) gives C ( q ) = − q S − Z T q t d S t + λ Z T . q t d t + γ q . Therefore, the problem writes assup . q ∈A E (cid:20) Z T q t d S t − λ Z T . q t d t − κ Z T q t d h S i t (cid:21) . (2.1)When the price process follows an ABM, Problem (2.1) boils down to a simple calculus of variationsproblem, which has been solved, for example, in [8]. The case where the dynamics are given bya GBM is more intricate. In [7], the authors consider it analytically intractable when a quadraticvariation penalty is used. Moreover, in [6], the authors derive a numerical solution of (2.1) by solvingthe corresponding Hamilton-Jacobi-Bellman equation. Note that strategies under ABM assumptionare good proxies of the ones under GBM assumption in period of low volatility.3 .2 Reformulation in terms of cash We now reformulate the optimal execution problem in terms of cash. We emphasize that we treat thevery same problem as in (2.1), except that we modify the transaction costs such that the penalty for . q t becomes . q t S t .We assume that the price process follows a GBM:d S t = σS t d W t . Multiplying above and below by S t , we obtain that Z T q t d S t = Z T θ t d y t , where d y t := σ d W t is the return of the price process, and θ t := q t S t is the trader’s position expressedin dollars. Moreover, the quadratic variation penalty has the form κσ Z T θ t d t. Applying Ito’s formula, we derive that the cash position θ t := θ ut has the following dynamics: d θ ut = u t d t + θ ut d y t = u t d t + σθ ut dW t , (2.2)where u t = . q t S t is the trading’s rate in dollar at time t .Recall that in the classical Almgren-Chriss framework (2.1), trading costs are a quadratic function ofthe number of shares traded at time t defined by . x t (the second term in (2.1)). The only modificationwe make here is to assume that instantaneous costs are a quadratic function of the amount of cash.According to [4], working with dollar holdings and returns is more consistent with common practice.We define the set of admissible control processes ( u t ) t ∈ [0 ,T ] as A := (cid:26) ( u t ) t ∈ [0 ,T ] measurable, s.t Z T | u t | d t < + ∞ , θ uT = 0 (cid:27) . where the last condition ensures the complete liquidation of the trader’s position at terminal time T . Following the problem formulation in terms of cash instead of shares, we consider the followingmean-quadratic variation optimization problem:lim a → + ∞ sup u ∈A E (cid:20) Z T − (cid:18) λ u t + κσ θ ut ) (cid:19) d t − a θ uT ) (cid:21) . (2.3)The limit over a > θ uT = 0. Equation (2.3) can be seen asa classical linear-quadratic optimization problem, which is reduced to the resolution of a Riccati equa-tion in dimension one. However such equations are not well suited for multidimensional extensionsof this problem, that is to say the liquidation of a portfolio of N assets. Furthermore when addinga possibly non-Markovian drift ( α t ) t ∈ [0 ,T ] to the price process, one has to rely on BSDE methods to We write the superscript u since ( u t ) t ∈ [0 ,T ] is the control process. α t ) t ∈ [0 ,T ] without using the BSDE framework. Finally, an explicit solution can beobtained in the case of the liquidation of a portfolio of N possibly correlated assets.We solve in the next section Problem (2.3) under the dynamics (2.2) for the trader’s position. Wetreat the non-zero drift case in Section 5.1. Throughout this section, we work on the following functional space: H := (cid:26) ( v t ) t ∈ [0 ,T ] : E h Z T v t d t i < + ∞ (cid:27) , with its associated inner product and norm h u, v i t = E (cid:20) Z t u s v s d s (cid:21) , k u k = E (cid:20) Z t u s d s (cid:21) . We also define for all t ∈ [0 , T ] the exponential martingale M t := exp (cid:18) σW t − σ t (cid:19) and the associatedchange of measure d Q d P (cid:12)(cid:12)(cid:12)(cid:12) F T = M T . We begin with a lemma characterizing the trader’s position. Lemma 3.1.
The unique solution of (2.2) is given by Z t M t M − s u s d s. For all v ∈ H , we define the operator ( Kv ) t := Z t M t M − s v s d s. The adjoint process ( K ? v ) is equal for all s ∈ [0 , T ] to ( K ? v ) s := Z Ts E Q [ v t |F s ]d t. The proof is given in Appendix A.1 and relies on a straightforward application of Ito’s formula.Therefore the optimization problem (2.3) can be rewritten, with a fixed a >
0, assup u ∈A − λ || u || − κσ || Ku || − a Ku ) T . (3.1)5he problem is a supremum over a concave function of u , which is Gateaux-differentiable on H . Thusfirst order condition gives: κσ λ K ? Ku + u + aλ ( Ku ) T = 0 , (3.2)or equivalently κσ λ Z Ts Z t E Q [ M t M − τ u τ |F s ]d τ d t + u s + aλ M T Z T M − τ u τ d τ = 0 . (3.3)For all ( s, s ) ∈ [0 , T ] such that s ≥ s , we apply E Q [ ·|F s ] on both sides of (3.3). This leads to thefollowing technical lemma. Lemma 3.2.
We define v ( s ) := E Q [ u s |F s ] such that v ( s ) = u s , and assume that it is differentiablewith respect to s . We also set z ( t ) = e σ ( t − s ) θ s + Z ts e σ ( t − τ ) v ( τ )d τ, where we recall that θ s := ( Ku ) s .i) Equation (3.2) can be rewritten v ( s ) + κσ λ Z Ts z ( t )d t + aλ z ( T ) = 0 . ii) The couple ( v, z ) satisfies the following system of differential equations ( v ( s ) = κσ λ z ( s ) z ( s ) = σ z ( s ) + v ( s ) , with boundary conditions ( v ( T ) = − aλ z ( T ) z ( s ) = θ s . Thus, the control problem (2.3) is reduced to the resolution of a linear system of ODEs with constantcoefficients. We can now state our main theorem.
Theorem 3.3.
Consider the problem (2.3) , the optimal control is given explicitly for all time t ∈ [0 , T ] by u ?t = θ u ? t Γ( t ) , where Γ( · ) is a deterministic function of time defined in (A.1) and the optimal trader’s positionsatisfies θ u ? t = θ u ? exp (cid:16) Z t (Γ( s ) − σ s + σW t (cid:17) . See Appendix A.2 for well-definedness of the first order condition. It will be shown ex-post, by a direct verification argument, that v ( · ) is differentiable. aggressive in-the-money selling strategy, similar to [7], in the sensethat the trader liquidates faster when the stock price increases and conversely. This is illustratedin the following section. Moreover, the trader’s position is a geometric Brownian motion, so that italways stays positive, in contrast to [7]. As the function Γ( t ) → t → T −∞ superlinearly, we have θ ut → t → T We simulate one Brownian motion trajectory, and plot the corresponding stock price process, aswell as trading strategy ( u ?t ) t ∈ [0 ,T ] and trader’s cash position ( θ ?t ) t ∈ [0 ,T ] and in shares ( θ ?t /S t ) t ∈ [0 ,T ] fordifferent values of σ . We take a stock with initial price S = 100$ following a GBM without drift(whose trajectories for different values of σ are in Figure 5), a portfolio of 10 shares to liquidateover T = 20 days, with λ = κ = 0 .
2. In Figure 1, we see an increase of the cash position at thebeginning, which can be misleading but is only due to the initial increase of the stock price process.This is also represented in the trading strategy of Figure 2, where we see that the trader liquidates hisposition faster when the stock process has a higher volatility. Figures 3 and 4 show the position andthe trading strategy in terms of shares. We also compare in Figure 6 our trading strategy in sharesto the one in [7], which is defined as q ?t := (cid:16) T − tT (cid:17)(cid:18) q − κT Z t S u d u (cid:19) , (4.1)where q = θ S is the initial number of shares hold by the trader, and d S t = σS t d W t . The trader stillliquidates faster with a high volatility but his trading strategy, in this rather extreme regime, can gonegative. Figure 1: Evolution of the cash position with re-spect to time. Figure 2: Trading strategy in cash with respect totime. igure 3: Evolution of the share’s position withrespect to time. Figure 4: Trading strategy in shares with respectto time.Figure 5: Evolution of the stock price with respectto time. Figure 6: Evolution of the share’s position withrespect to time using (4.1). We now fix σ = 0 . κ = 0 .
2. The various cases of the impact of the transaction costs λ on thetrader’s behavior are represented in Figures 7,8,9 and 10. Obviously, the price process is insensitiveto a variation of λ . Moreover, the trading strategies in Figures 8 and 10 are decreasing functions of λ meaning that the trader liquidates his position using smaller sell orders when transactions costs arehigher. This is also shown in the trader’s position in Figures 7 and 9.8 igure 7: Evolution of the cash position with re-spect to time. Figure 8: Trading strategy in cash with respect totime.Figure 9: Evolution of the share’s position withrespect to time. Figure 10: Trading strategy in shares with respectto time. Finally, we set σ = 0 . , λ = 0 . κ . In Figures12 and 14, we see that a highly risk averse trader will liquidate faster than a low risk averse trader.This is shown in terms of his position in Figures 11 and 13. Figure 11: Evolution of the cash position with re-spect to time. Figure 12: Trading strategy in cash with respectto time. igure 13: Evolution of the share’s position withrespect to time. Figure 14: Trading strategy in shares with respectto time. We now show how to extend our framework to the case of a stochastic drift for the price process andthe liquidation of a portfolio of N assets. We now consider the case of a stochastic drift, that is we solvelim a → + ∞ sup u ∈A E (cid:20) Z T α t θ t − (cid:18) λ u t + κσ θ ut ) (cid:19) d t − a θ uT ) (cid:21) , where ( α t ) t ∈ [0 ,T ] is a stochastic drift of the price process. We consider a slight modification of theproblem where we neglect the part α t S t of the price’s drift. Therefore, we simplify the dynamics ofthe price process, and assume d θ ut = u t d t + σθ ut d W t . (5.1)The first-order condition associated to this optimization problem writes as κσ λ Z Ts Z t E Q [ M t M − τ u τ |F s ]d τ d t + u s + aλ M T Z T M − τ u τ d τ = 1 λ Z Ts E Q [ α t |F s ]d t. Then, the analogous of Equation (3.2) can be rewritten v ( s ) + κσ λ Z Ts z ( t )d t + aλ z ( T ) = 1 λ Z Ts E Q [ α t |F s ]d t. where the couple ( v, z ) satisfies the following system of differential equations ( v ( s ) = κσ λ z ( s ) − λ E Q [ α s |F s ] z ( s ) = σ z ( s ) + v ( s ) , It can be shown that, if there exists η > t | α t | < η , then the trading strategy derived in this sectionis arbitrary closed (as a function of η, T, σ ) to the optimal strategy without simplification of the drift. ( v ( T ) = − aλ z ( T ) z ( s ) = θ s . We finally obtain the following theorem:
Theorem 5.1.
The optimal control at any time t ∈ [0 , T ] is given by u ?t = θ u ? t Γ( t ) + ν ( t ) , where the optimal trader’s position is defined as θ u ? t = H t Z t H − s ν ( s )d s with d H t = Γ( t ) H t d t + σH t d W t , and Γ( · ) , ν ( · ) are deterministic functions defined in (A.1) . The term ν ( · ) is a linear function of both α t and E Q [ α T |F t ], representing the influence of the drift onthe optimal strategy. It is an increasing function of the drift α t meaning that we aim at liquidatingfaster our position when the stock price increases. Moreover, it is a decreasing function of E Q [ α T |F t ]:when the expected drift at the terminal time is high, the trader prefers to liquidate slower, waiting fora future stock price increase. As in the zero-drift case, we observe an aggressive in-the-money sellingstrategy. This model extends directly to the problem of optimal execution of a portfolio of N assets. We definethe return of the i -th asset as d y it = σ i d W it , where ( W , . . . , W N ) are Brownian motions with non singular covariance matrix Σ = ( σ i σ j ρ i,j ) ≤ i,j ≤ N , σ i > i -th asset and ρ i,j is the correlation between the i -th and the j -thBrownian motion. The cash position of the trader with respect to the i -th asset is defined byd θ u,it = u it d t + θ u,it d y it = u it d t + σ i θ u,it d W it , (5.2)where ( u it ) t ∈ [0 ,T ] is the trading rate on the i -th asset. Therefore the optimization problem (2.3) rewritesas lim a → + ∞ sup u ∈A E " Z T N X i =1 − λ u it ) − κ (cid:18) N X i =1 σ i ( θ u,it ) d t + N X i,j =1 i = j ρ i,j σ i σ j θ u,it θ u,jt d t (cid:19) − a N X i =1 ( θ u,iT ) , where A := (cid:26) ( u it ) t ∈ [0 ,T ] ,i ∈{ ,...,N } measurable, s.t for all i ∈ { , . . . , N } Z T | u it | d t < + ∞ , θ u,iT = 0 (cid:27) .
11e define ( K i u it ) t ∈ [0 ,T ] ,i ∈{ ,...,N } as the solution of the SDE (5.2):( K i u i ) t = M it Z t ( M is ) − u is d s, where d Q i d P (cid:12)(cid:12)(cid:12)(cid:12) F T = M iT := exp (cid:16) σ i W iT − ( σ i ) T (cid:17) . The adjoint operator is defined as( K ?i u i ) s = Z Ts E Q i [ u it |F s ]d t. For a fixed a >
0, the optimization problem rewritessup u ∈A − λ k u k − κ h Ku, Σ Ku i − a N X i =1 ( K i u i ) T . The first order condition gives the following system u i + κ λ (cid:16) σ i K ?i K i u i + N X j = i ρ i,j σ i σ j K ?i K j u j (cid:17) + aλ ( K i u i ) T = 0 , i = 1 , . . . , N, (5.3)or equivalently for all i = 1 , . . . , N , u is + κ λ (cid:18) σ i Z Ts Z t (cid:16) E Q i [ M it ( M iτ ) − u iτ |F s ]d τ d t + N X j = i ρ i,j σ i σ j E Q i [ M jt ( M jτ ) − u jτ |F s ]d τ d t (cid:17)(cid:19) + aλ ( K i u i ) T = 0 . For any s ≥ s , apply E Q i h · |F s i on both sides of the equations. Simple but tedious computationslead to E Q i h u is |F s i + κσ i λ (cid:18) Z Ts e σ i ( t − s ) ( K i u i ) s d t + Z Ts Z ts e σ i ( t − τ ) E Q i h u iτ |F s i d τ d t (cid:19) + κλ N X j = i ρ i,j σ i σ j (cid:18) Z Ts e σ i σ j ρ i,j ( t − s ) ( K j u j ) s + Z Ts Z ts e σ i σ j ρ i,j ( t − τ ) E Q j h u jτ |F s i d τ d t (cid:19) + aλ (cid:18) e σ i ( T − s ) ( K i u i ) s + Z Ts e σ i ( T − τ ) E Q i h u iτ |F s i d τ (cid:19) = 0 . By denoting for all i = 1 , . . . , N , v i ( s ) = E Q i h u is |F s i , and θ is = ( K i u i ) s the system becomes v i ( s ) + κσ i λ (cid:18) Z Ts e σ i ( t − s ) θ is d t + Z Ts Z ts e σ i ( t − τ ) v i ( τ )d τ d t (cid:19) + κλ N X j = i ρ i,j σ i σ j (cid:18) Z Ts e σ i σ j ρ i,j ( t − s ) θ js + Z Ts Z ts e σ i σ j ρ i,j ( t − τ ) v j ( τ )d τ d t (cid:19) + aλ (cid:18) e σ i ( T − s ) θ is + Z Ts e σ i ( T − τ ) v i ( τ )d τ (cid:19) = 0 . We define z i ( t ) := e σ i ( t − s ) θ is + Z ts e σ i ( t − τ ) v i ( τ )d τ,z i,j ( t ) := e σ i σ j ρ i,j ( t − s ) θ js + Z ts e σ i σ j ρ i,j ( t − τ ) v j ( τ )d τ, i = 1 , . . . , N : v i ( s ) + κσ i λ (cid:18) Z Ts z i ( t )d t (cid:19) + κλ N X j = i ρ i,j σ i σ j Z Ts z i,j ( t )d t + aλ z i ( T ) = 0 . Therefore first-order condition (5.3) is equivalent to the system of differential equations v i ( s ) − κσ i λ z i ( s ) − κλ P Nj = i ρ i,j σ i σ j z i,j ( s ) = 0 z i ( s ) = σ i z i ( s ) + v i ( s ) z i,j ( s ) = σ i σ j ρ i,j z i,j ( s ) + v j ( s ) , with initial conditions v i ( T ) = − aλ z i ( T ) z i ( s ) = θ is z i,j ( s ) = θ js . We obtain a system of linear differential equations with constant coefficients. Thus, by noting thatfor all i = 1 , . . . , N and s ∈ [0 , T ], v i ( s ) = u is , we obtain the controls u it for all t ∈ [0 , T ] and i = 1 , . . . , N by solving this system of ODEs. In this article, we present a way to solve the traditional Almgren-Chriss liquidation problem when theunderlying asset is driven by a GBM. By working in terms of cash and using functional analysis tools,we can provide the optimal control of the problem explicitly. We provide an extension to the case ofa GBM with stochastic drift and the liquidation of a portfolio of correlated assets. In particular, ourmethod does not suffer from the curse of dimensionality.
A Appendix
A.1 Proof of Lemma 3.1
An application of Ito’s formula givesd( Ku ) t = u t d t + σ ( Ku ) t d W t , hence solving (2.2). The adjoint of K is the operator K ? such that for all ( u, v ) ∈ A , h Ku, v i = h u, K ? v i . h Ku, v i = E (cid:20) Z T ( Ku ) t v t d t (cid:21) = E (cid:20) Z T Z t M t M − s u s v t d s d t (cid:21) = E (cid:20) Z T Z Ts M t M − s u s v t d t d s (cid:21) = E (cid:20) Z T u s ( K ? v ) s d s (cid:21) = h u, K ? v i , where ( K ? v ) s = R Ts M t M − s v t d t = R Ts E Q [ v t |F s ]d t . A.2 Gateaux differentiability in (3.1)
We define the map Ξ : H → R byΞ( u ) = − λ k u k − κσ k Ku k − a Ku ) . As K is a linear operator of u ∈ H , and λ, κ >
0, we deduce that Ξ is continuous, strictly concaveand Gateaux differentiable; with Gateaux derivative given, for any h ∈ H , byΞ( u )[ h ] = − λ h u, h i − κσ h Ku, Kh i − a ( Kh ) . By setting Ξ( u )[ h ] = 0, and using the definition of an adjoint operator, we obtain Equation (3.2). A.3 Proof of Lemma 3.2
By the fact that E Q (cid:20) Z Ts Z t E Q [ M t M − τ u τ |F s ]d τ d t (cid:12)(cid:12)(cid:12) F s (cid:21) = Z Ts Z t E Q [ M t M − τ u τ |F s ]d τ d t = Z Ts (cid:18) Z s M − τ u τ E Q [ M t |F s ]d τ + Z ts E Q [ M t M − τ u τ |F s ]d τ (cid:19) d t = Z Ts (cid:18) e σ ( t − s ) ( Ku ) s d t + Z ts E Q h E Q [ M t M − τ u τ |F s ] |F τ i d τ (cid:19) d t = Z Ts (cid:18) e σ ( t − s ) ( Ku ) s d t + Z ts e σ ( t − τ ) E Q [ u τ |F s ]d τ (cid:19) d t. Condition (3.3) can be rewritten κσ λ Z Ts e σ ( t − s ) ( Ku ) s d t + κσ λ Z Ts Z ts e σ ( t − τ ) E Q [ u τ |F s ]d τ d t + aλ e σ ( T − s ) θ s + aλ Z Ts e σ ( T − τ ) E Q [ u τ |F s ] + E Q [ u s |F s ] = 0 , which proves the first statement of the theorem. We obtain the second point by a straightforwardderivation of the functions z and v . 14 .4 Proof of Theorem 5.1 The solution of the ODE for z is given by z ( s ) = θ s e σ ( s − s ) + Z ss e σ ( s − u ) v ( u )d u, and we can rewrite v ( s ) = κσ λ ( θ s e σ ( s − s ) + Z ss e σ ( s − u ) v ( u )d u ) − λ E Q [ α s |F s ] . Multiplying by e − σ s on both sides we have e − σ s v ( s ) = kσ λ ( e − σ s θ s + Z ss e − σ u v ( u )d u ) − e − σ s λ E Q [ α s |F s ] , and defining w ( s ) = e − σ s v ( s ) we obtain w ( s ) = kσ λ ( e − σ s θ s + Z ss w ( u )d u ) − σ w ( s ) − e − σ s λ E Q [ α s |F s ] . We note y ( s ) = R ss w ( u )d u , satisfying the following differential equation y ( s ) = κσ λ y ( s ) − σ y ( s ) + κσ λ e − σ s θ s − e − σ s λ E Q [ α s |F s ] . Solving this ODE without second member, we have y ( s ) = C e γ s + C e γ s , where C , C ∈ R , γ = − σ −√ ∆2 , γ = − σ + √ ∆2 , ∆ = σ ( σ + 4 κλ ) >
0. A particular solution is givenby the function y ( s ) = − θ s e − σ s + e − σ s κσ E Q [ α s |F s ]. The general solution is therefore given by: y ( s ) = C e γ s + C e γ s − θ s e − σ s + e − σ s κσ E Q [ α s |F s ] . To find C , C we use the fact that y ( s ) = 0 and y ( T ) = w ( T ) = e − σ T v ( T ) = − e − σ T aλ z ( T ).Substituting the previous expression of y , and making a → + ∞ to ensure liquidation at terminaltime, we obtain C ∞ ( s ) = β ∞ ( s ) e γ T − σ s ( θ s − α s κσ ) + e γ s − σ T E Q [ α T |F s ] κσ ! ,C ∞ ( s ) = β ∞ ( s ) − e γ T − σ s ( θ s − α s κσ ) − e γ s − σ T E Q [ α T |F s ] κσ ! , where β ∞ ( s ) = e γ s γ T − e γ T + γ s >
0. Note that v ( s ) = u s = e σ s y ( s ), which gives u s = θ s Γ( s ) + ν ( s ) , s ) := β ∞ ( s )( e γ s + γ T γ − e γ T + γ s γ ) ,ν ( s ) = β ∞ ( s ) γ e ( γ + σ ) s (cid:18) − α s κσ e γ T − σ s + e γ s − σ T E Q [ α T |F s ] κσ (cid:19) + γ e ( γ + σ ) s (cid:18) α s κσ e γ T − σ s − e γ s − σ T E Q [ α T |F s ] κσ (cid:19) − α s κ . (A.1)Substituting this expression in (5.1), the trader’s position becomesd θ ut = ( ν ( t ) + Γ( t ) θ ut )d t + σθ ut d W t . Therefore, we have the optimal position defined by θ u ? t = H t Z t H − s ν ( s )d s, where d H t = Γ( t ) H t d t + σH t d W t . The optimal control is finally given explicitly at any time t ∈ [0 , T ]by u ?t = θ u ? t Γ( t ) + ν ( t ) . References [1] A. Alfonsi, A. Fruth, and A. Schied. Optimal execution strategies in limit order books with generalshape functions.
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