Smooth solutions to portfolio liquidation problems under - PowerPoint PPT Presentation

Smooth solutions to portfolio liquidation problems under price-sensitive market impact Ulrich Horst 1 Humboldt-Universit¨ at zu Berlin Department of Mathematics & School of Business and Economics August 29, 2013 1 Based on Joint work with P. Graewe and E. S´ er´ e

Outline • Portfolio liquidation/accquisition under market impact • liquidation with active orders • liquidation with active and passive orders • Markovian Control Problem (with P. Graewe and E. S´ er´ e) • An HJB equation with singular terminal value • Existence of short-time solutions • Verification argument • Non-Markovian Control Problem (with P. Graewe and J. Qiu) • A BSPDE with singular terminal value • Existence of solutions • Verification argument • Conclusion

Portfolio Liquidation

Portfolio Liquidation • Traditional financial market models assume that investors can buy sell arbitrary amounts at given prices • This neglects market impact : large transactions (1%-3% of ADV, or more) move prices in an unfavorable direction

Portfolio Liquidation • Economists have long studied models of optimal block trading • Their focus is often on informational asymmetries • Stealth trading: split large blocks into a series of smaller ones • Mathematicians identified this topic only more recently • Their focus is often on ‘structural models’ (algorithmic trading) • Models of optimal portfolio liquidation give rise to novel stochastic control problems: • (‘Liquidation’) constraint on the terminal state • Value functions with singular terminal value • PDEs, BSDEs, BSPDEs, .... with singular terminal values

Portfolio Liquidation • Almost all trading nowadays takes place in limit order markets. • Limit order book: list of prices and available liquidity • Limited liquidity available at each price level • There are (essentially) two types of orders one can submit: • active orders submitted for immediate execution • passive orders submitted for future execution • We allow active and passive orders; price sensitive impact • Markovian model: PDE with singular terminal condition • non-Markovian model: BSPDE with singular terminal condition

Liquidation with active orders Consider an order to sell X > 0 shares by time T > 0: • ξ t rate of trading (control) � t • X t = X − ξ s ds remaining position (controlled state) 0 • S t market/benchmark price (uncontrolled state) The optimal liquidation problem is of the form �� T � min f ( ξ t , S t , X t ) dt s.t. X T − = 0 ( ξ t ) E 0 The liquidation constraint results in a singularity of the value function: � + ∞ for X � = 0 t → T − V ( t , S , X ) = lim 0 for X = 0

Benchmark: linear temporary impact For some martingale ( S t ), the transaction price is given by � S t = S t − ηξ t ( η = market impact factor) . The liquidity costs are then defined as C = book value − revenue � T � T � T � ηξ 2 = S 0 X − S t ξ t dt = − X t dS t + t dt 0 0 0 and the expected liquidity costs are � T ηξ 2 E [ C ] = t dt . 0 Usually, one minimizes expected liquidation + risk costs.

Literature review • Almgren & Chriss (2000): mean-variance, S t BM � T ηξ 2 t + λσ 2 X 2 t dt − → min 0 • Gatheral & Schied (2011): time-averaged VaR, S t GBM �� T � ηξ 2 t + λ S t X t dt − → min E 0 • Ankirchner & Kruse (2012): similar but dS t = σ ( S t ) dW t �� T � ηξ 2 t + λ ( S t ) X 2 E t dt − → min 0 • and many others ....

Markovian Models

Liquidation with active and passive orders Modeling the impact of active orders is comparably simple; the impact of passive orders is harder to model: • how does the market react to passive order placement? • using active and passive orders simultaneously may lead to market manipulation • .... To overcome this problem, we assume that passive orders are placed in a dark pool : • passive orders are not openly displayed • executed only when matching liquidity becomes available • if executed, then at prices coming from some primary venue Dark trading: reduced trading costs vs. execution uncertainty .

Liquidation with active and passive orders We allow for active and passive orders: • active order placements: ( ξ t ) t ∈ [0 , T ) • passive order placements: ( ν t ) t ∈ [0 , T ) For X 0 = X the portfolio dynamics is given by dX t = − ξ t dt − ν t d π t X T − = 0 a . s . with Our value function is given by V ( T , S , X ) �� T � η ( S t ) | ξ t | p + γ ( S t ) | ν t | p + λ ( S t ) | X t | p dt = inf ( ξ,ν ) ∈ A ( T , X ) E 0 where the coefficients η, σ, γ, λ are nice enough and p > 1.

Remark (Power-structure of cost function) Kratz (2012) and H & Naujokat (2013) consider the cost function �� T � η | ξ t | 2 + γ | ν t | 1 + λ | X t | 2 dt . E 0 In this case, no passive orders are used after first execution. This property does not carry over to price-sensitive impact factors. We thus consider �� T � η ( S t ) | ξ t | p + γ ( S t ) | ν t | p + λ ( S t ) | X t | p dt . E 0

Theorem (Structure of the Value Function) The value function is of the form (‘power-utility’) V ( T , S , X ) = v ( T , S ) | X | p and the optimal controls are: t = v ( T − t , S t ) β v ( T − t , S t ) β ξ ∗ ν ∗ X t , t = γ ( S t ) β + v ( T − t , S t ) β X t , η ( S t ) β 1 where β := p − 1 > 0 and the “inflator” v solves the PDE � � v T = 1 1 γ ( S ) v βη ( S ) β v β +1 − θ 2 σ 2 ( S ) v SS + λ ( S ) − v − . ( γ ( S ) β + v β ) 1 /β � �� � F ( S , v )

Boundary condition for v The final position when following ξ ∗ and ν ∗ is � � ∆ π t � =0 � T � � � v ( T − t , S t ) β v ( T − t , S t ) β X exp − dt 1 − . γ ( S t ) β + v ( T − t , S t ) β η ( S t ) β 0 0 ≤ t < T • To ensure X ∗ T − = 0 one needs v ( T − t , S ) β − → ∞ as t → T (uniformly in S ). η ( S ) β • Through a-priori estimates one shows that v ( T , S ) ∼ η ( S ) as T → 0 uniformly in S . 1 T β If η ≡ const , no passive orders, then this holds automatically.

Theorem (PDE for v) After a change of variables, the inflator v is the unique classical solution of v t = 1 2 ∆ v − 1 2 σ ′ ( x ) ∇ v + F ( x , v ) such that v ( t , x ) → 0 as t → 0 uniformly in x. This solution satisfies: v ( t , x ) ∼ η ( x ) as t → 0 uniformly in x. 1 t β

Remark • The operator A = 1 2 ∆ − 1 2 σ ′ ( x ) ∇ generates an analytic (yet not strongly continuous) semigroup e tA in C ( R ) and a priori bounds give that any short-time solution extends to a global solution. • For the short-time solution, we express the asymptotics in terms of an equation: v ( t , x ) = η ( x ) + ‘correction’ 1 t β

Existence of a short-time solution Our ansatz is to additively separate the “leading singular term”: v ( t , x ) = η ( x ) + u ( t , x ) u ( t , x ) ∈ O ( t 2 ) as t → 0 uniformly in x β +1 , 1 1 t t β Results in an evolution equation in C ( R ) for the correction term: u ′ ( t ) = Au + f ( t , u ( t )) , u (0) ≡ 0 , with the singular nonlinearity of the form: � u ( t ) � k ∞ � f ( t , u ( t )) = . . . . . . . . . . t η k =2 Remark We move the singularity from the terminal condition into the non-linearity in such a way that it causes no harm.

Existence of a short-time solution The contraction argument giving a short-time solution by a fixed point of the operator � t e ( t − s ) A f ( s , u ( s )) ds Γ( u )( t ) = 0 is then carried out in the space E = { u ∈ C ([0 , δ ]; C ( R )) : � u � E < ∞} where � t − 2 u ( t ) } � u � E = sup t ∈ (0 ,δ ] Theorem (Existence of solutions) The operator Γ has a fixed point for all sufficiently small t ∈ [0 , T ] .

Lemma It is enough to consider only strategies that yield monotone portfolio processes. For such strategies � | p � v ( T − t , S t ) | X ξ,ν E − → 0 as t → T . t Theorem (Value Function) The value function for our control problem is V ( T , S , X ) = v ( T , X ) | X | p .

Non-Markovian Models

Probability space Consider a probability space (Ω , ¯ F , { ¯ F t } t ≥ 0 , P ) with { ¯ F t } t ≥ 0 being generated by three mutually independent processes: • m -dimensional Brownian motion W ; • m -dimensional Brownian motion B ; • stationary Poisson point process J on Z ⊂ R l with • finite characteristic measure : µ ( dz ); • counting measure π ( dt , dz ) on R + × Z ; and • { ˜ π ([0 , t ] × A ) } t ≥ 0 a martingale where ˜ π ([0 , t ] × A := π ([0 , t ] × A ) − t µ ( A ) . • The filtration generated by W is denoted F .

The control problem • The controlled process is � t � t � x t = x − ξ s ds − ρ s ( z ) π ( dz , ds ); x T − = 0 0 0 Z the set of admissible strategies is the set of all pairs ( ξ, ρ ) ∈ L 2 F (0 , T ) × L 4 F (0 , T ; L 2 ( Z )) with x T − = 0 a . s . ¯ ¯ • The uncontrolled factors follow the dynamics � t � t � t y t = y + b s ( y s , ω ) ds + σ s ( y s , ω ) dB s + ¯ σ s ( y s , ω ) dW s 0 0 0 where the processes b ( y , · ) , σ ( y ; · ) , ¯ σ ( y , · ) are F -adapted.