Optimal liquidation in an Almgren-Chriss type model with L´ evy processes and finite time horizons Junwei Xu joint work with Arne Løkka Department of Mathematics London School of Economics and Political Science 3rd Young Researchers Meeting in Probability, Numerics and Finance Xu (LSE) Optimal liquidation June 29 - July 1, 2016 1 / 24
What is an optimal liquidation problem? We consider an investor who aims to sell a large amount of shares. Since the trading volume by this investor is large, due to a lack of liquidity, the market price will drop during selling, potentially resulting in huge execution costs. The investor seeks to find an optimal strategy, maximising the final cash subject to some optimisation criterion. Xu (LSE) Optimal liquidation June 29 - July 1, 2016 2 / 24
The Almgen-Chriss framework For t ≥ 0, let Y t ≥ 0 be the position in a stock and ξ t ∈ R be the execution speed. So with Y 0 = y , � t Y t = y + ξ s ds . 0 The observed market price of this stock at time t is S u S t = + α ( Y t − Y 0 ) + F ( ξ t ) , t ���� � �� � � �� � unaffected price permanent impact temporary impact where α ≥ 0 is the coefficient of permanent impact and F : R → R is the temporary impact function which is increasing and satisfies F (0) = 0. Xu (LSE) Optimal liquidation June 29 - July 1, 2016 3 / 24
Our model: L´ evy unaffected price Let (Ω , F , ( F t ) , P ) be a complete filtered probability space satisfying the usual conditions, which supports a one dimensional, non-trivial, F -adapted L´ evy process L . Our unaffected price is given by S u t = s + L t � x ˜ = s + µ t + σ W t + N ( t , dx ) . R Xu (LSE) Optimal liquidation June 29 - July 1, 2016 4 / 24
Our model: Admissible strategies For any time horizon T ∈ (0 , ∞ ) and any initial stock position y ≥ 0, any admissible liquidation strategy Y ∈ A ( T , y ) satisfies � Y is adapted and absolutely continuous; � Y 0 = y , Y T = 0 and Y t ≥ 0 for t ∈ [0 , T ]. A D ( T , y ) is the set of all deterministic admissible strategies. Xu (LSE) Optimal liquidation June 29 - July 1, 2016 5 / 24
Our model: General temporary impact function The temporary impact function F : R → R satisfies � F ∈ C ( R ) ∩ C 1 ( R \ { 0 } ); � F (0) = 0; � the function x �→ xF ( x ) is strictly convex on R ; � some other technical conditions Xu (LSE) Optimal liquidation June 29 - July 1, 2016 6 / 24
Cash position Stock price at time t ≥ 0 is given by S t = s + L t + α ( Y t − Y 0 ) + F ( ξ t ) . Let C be the process of cash position with C 0 = c . Then for any Y ∈ A ( T , y ), the total cash at time T is given by � T C T = c − S t dY t 0 � T � T = c + sy − 1 2 α y 2 + Y t − dL t − ξ t F ( ξ t ) dt a . s . 0 0 Xu (LSE) Optimal liquidation June 29 - July 1, 2016 7 / 24
The large investor’s optimisation problem Suppose the investor has a constant absolute risk aversion, and wants to maximise the expected utility of final cash. Therefore, his problem is given by � � sup E − exp( − AC T ) Y ∈A ( T , y ) where A > 0 is the risk aversion. This problem is equivalent to � � �� � T � T inf exp − AY t − dL t + A ξ t F ( ξ t ) dt . Y ∈A ( T , y ) E 0 0 Xu (LSE) Optimal liquidation June 29 - July 1, 2016 8 / 24
Problem simplification � Define the function κ A : [0 , ¯ δ A ) → R by κ A ( x ) = κ ( − Ax ) , where A > 0 is a constant, κ is the cumulant generating function of L 1 . � Therefore, �� t � t � M t = exp − AY u − dL u − κ A ( Y u ) du , t ∈ [0 , T ] 0 0 is a martingale. Xu (LSE) Optimal liquidation June 29 - July 1, 2016 9 / 24
� If we assume that y ∈ [0 , ¯ δ A ), then based on the idea in Schied, Sch¨ oneborn and Tehranchi (2010) , we calculate that � � � T � T �� Y ∈A ( T , y ) E inf exp − AY t − dL t + A ξ t F ( ξ t ) dt 0 0 � � � � T � T = inf exp − AY t − dL t − κ A ( Y t ) dt × Y ∈A ( T , y ) E 0 0 �� T �� × exp κ A ( Y t ) + A ξ t F ( ξ t ) dt 0 � �� T �� � = inf exp κ A ( Y t ) + A ξ t F ( ξ t ) dt E Y ∈A ( T , y ) 0 � �� T �� � = inf E exp κ A ( Y t ) + A ξ t F ( ξ t ) dt Y ∈A D ( T , y ) 0 �� T � � � = Y ∈A D ( T , y ) exp inf κ A ( Y t ) + A ξ t F ( ξ t ) dt 0 � � � T ∼ inf κ A ( Y t ) + A ξ t F ( ξ t ) dt , , Y ∈A D ( T , y ) 0 Xu (LSE) Optimal liquidation June 29 - July 1, 2016 10 / 24
Possible approaches Simplified problem: � � � T inf κ A ( Y t ) + A ξ t F ( ξ t ) dt , Y ∈A D ( T , y ) 0 where ( T , y ) ∈ (0 , ∞ ) × [0 , ¯ δ A ) and dY t = ξ t dt . � The Euler-Lagrange equation: A ( Y t ) − d κ ′ dt [ AF ( ξ t ) + A ξ t F ′ ( ξ t )] = 0 with Y 0 = y and Y T = 0. But F ′ is not differentiable! � The Beltrami identity: κ A ( Y t ) − A ξ 2 t F ′ ( ξ t ) ≡ const with Y 0 = y and Y T = 0. It is equivalent to the Euler-Lagrange equation if F ′ is differentiable. Xu (LSE) Optimal liquidation June 29 - July 1, 2016 11 / 24
A general optimisation problem � φ : R × R → R is proper and lower semi-continuous. � D := D 1 × D 2 := { ( x , ˙ x ) | φ ( x , ˙ x ) < ∞} has non-empty interior. � φ ∈ C 1 (int( D )). � For any x ∈ D 1 , φ ( x , · ) is strictly convex on D 2 . Consider the problem � b � � inf φ x ( t ) , ˙ x ( t ) dt , (1) x ( · ) a where any admissible x ( · ) satisfies that � x ( · ) is absolutely continuous; � x ( a ) = x a and x ( b ) = x b with x a , x b ∈ D 1 ; � �� � b � � φ � dt < ∞ . x ( t ) , ˙ x ( t ) � a Xu (LSE) Optimal liquidation June 29 - July 1, 2016 12 / 24
Necessary and sufficient conditions for the optimiser For any x , ˙ x ∈ R and p ∈ R , write H ( x , ˙ x , p ) = ˙ xp − φ ( x , ˙ x ) , and define the Hamiltonian to be H ( x , p ) = sup H ( x , ˙ x , p ) . x ∈ R ˙ Xu (LSE) Optimal liquidation June 29 - July 1, 2016 13 / 24
Theorem 1 (necessary) x ( · ) be an optimal admissible function for (1) and ˙ Let ˆ ˆ x ( · ) be the derivative of x ( · ) . Suppose ˆ ˆ x ( · ) is Lipschitz continuous with associated Lipschitz constant belonging to the interior of D 2 . We also suppose that ˆ x ( · ) takes values in the interior of D 1 . Then there exists a function p : [ a , b ] → R satisfying � � x ( t ) , ˙ p ( t ) = − H x ˙ ˆ ˆ x ( t ) , p ( t ) a.e. t ∈ [ a , b ] , , and � � � � x ( t ) , ˙ H ˆ x ( t ) , p ( t ) ˆ = max x ∈ R H x ( t ) , ˙ ˆ x ( t ) , p ( t ) , a.e. t ∈ [ a , b ] . ˙ If in addition that � � � � x ( t ) , ˙ H x x ( t ) , p ( t ) ˆ = H x ˆ x ( t ) , p ( t ) ˆ , a.e. t ∈ [ a , b ] , then the optimal function ˆ x ( · ) satisfies � � � � x ( t ) , ˙ − ˙ x ( t ) , ˙ φ ˆ x ( t ) ˆ ˆ x ( t ) φ ˙ ˆ x ( t ) ˆ ≡ K , a.e. t ∈ [ a , b ] , x where K is some constant. Xu (LSE) Optimal liquidation June 29 - July 1, 2016 14 / 24
Theorem 2 (sufficient) Suppose for any x ∈ D 1 and any p ∈ { φ ˙ x ( x , ˙ x ) | ˙ x ∈ D 2 } , H ( · , p ) is concave. Let � � x ( t ) , ˙ x ( · ) be an admissible function such that t �→ φ ˙ ˆ ˆ ˆ x ( t ) is absolutely x continuous on [ a , b ] . Suppose � � � � x ( t ) , ˙ − ˙ x ( t ) , ˙ φ ˆ ˆ x ( t ) x ( t ) φ ˙ ˆ ˆ ˆ x ( t ) ≡ K , for all t ∈ [ a , b ]; x and when ˙ ˆ x ( · ) = 0 a.e. on some interval contained in [ a , b ] , we have � � H x x ( · ) , p ( · ) ˆ = 0 on the same interval. Then such ˆ x ( · ) is optimal for problem (1). Xu (LSE) Optimal liquidation June 29 - July 1, 2016 15 / 24
The optimal liquidation strategy � Recall the simplified optimal liquidation problem: ( T , y ) ∈ (0 , ∞ ) × [0 , ¯ δ A ), � � � T inf κ A ( Y t ) + A ξ t F ( ξ t ) dt . Y ∈A D ( T , y ) 0 � Consider the Beltrami identity K T , y + κ A ( Y t ) = A ξ 2 t F ′ ( ξ t ) . Define the continuous functions G − : [0 , ∞ ) → ( −∞ , 0] and G + : [0 , ∞ ) → [0 , ∞ ) to be the inverses of x �→ x 2 F ′ ( x ) restricted on intervals ( −∞ , 0] and [0 , ∞ ), respectively. Xu (LSE) Optimal liquidation June 29 - July 1, 2016 16 / 24
� Given any ( T , y ) ∈ (0 , ∞ ) × [0 , ¯ δ A ), the unique optimal strategy satisfies either � ˆ � � K T , y + κ A � d ˆ Y T , y Y T , y = ˆ ξ T , y t t = G − t dt A or � ˆ � � K T , y + κ A � d ˆ Y T , y Y T , y t = ˆ ξ T , y t = G + ✶ [0 ,τ y ) ( t ) t dt A � ˆ � � K T , y + κ A � Y T , y t + G − ✶ [ τ y , T ] ( t ) , A where τ y = inf { t ≥ 0 | ˆ Y T , y = κ − 1 A ( − K T , y ) } . t � K T , y is uniquely determined by T and y . Xu (LSE) Optimal liquidation June 29 - July 1, 2016 17 / 24
Figure: An illustration of optimal liquidation trajectories for µ ≤ 0. Xu (LSE) Optimal liquidation June 29 - July 1, 2016 18 / 24
Figure: An illustration of optimal liquidation trajectories for µ > 0. Xu (LSE) Optimal liquidation June 29 - July 1, 2016 19 / 24
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