Optimal liquidation with market parameter shift: a forward approach (with S. Nadtochiy and T. Zariphopoulou) Haoran Wang Ph.D. candidate University of Texas at Austin ICERM June, 2017
Problem Setup and Solution Review of Classical Approach Almgren-Chriss model Price under temporary and permanent impact P t = P 0 + σw t + γ(x t X 0 ) }{{} permanent impact + λẋ t }{{} temporary impact Inventory process: X t = x t 0 ξ sds, with X T = 0 T liquidation time, 0 < T < T chosen at t = 0 Parameters σ, γ, λ also assessed at t = 0
Problem Setup and Solution Review of Classical Approach Optimal liquidation problem (Schied et al.) Controlled state processes: inventory and revenue t Xt ξ := x ξ s ds 0 t t Rt ξ := r + σ Xs ξ dw s λ ξs 2 ds 0 0 Value function [ ] V (x, r, 0; T ) := sup E e Rξ T X ξ 0 = x, Rξ 0 = r ξ with the liquidation terminal constraint { v(0, r, T ; T ) = e r, v(x, r, T ; T ) =, for x 0
Problem Setup and Solution Review of Classical Approach Solution under CARA utility Value function (σ = 1) ( V (x, r, 0; T ) = exp r + λ ( T )) 2 x 2 coth 2λ Optimal trading rate for 0 t T ξt = V x(xt, Rt, t; T ) 2λV r (Xt, Rt, t; T ) = 1 ( T t ) Xt coth 2λ 2λ Optimal inventory process X t = ( ) x sinh T t 2λ sinh ( T 2λ ) ; X T = 0 Model commitment: temporary price impact parameter λ pre-chosen at t = 0, unrealistic!
Problem Setup and Solution Review of Classical Approach Empirical studies point out to non-constant λ Liquidity profile not constant; intraday and intraweek patterns Small- and medium-capitalization stocks have liquidity profiles harder to predict/model (for longer horizons) Extreme events, e.g., Flash Crash (May 6, 2010); may need adaptive trading objectives that respond to what actually occurs as time enfolds How can we then accommodate dynamic model change?
Problem Setup and Solution Review of Classical Approach Classical approach Inflexibility w.r.t. model revision (λ; σ, γ) t = 0 pre-chosen model DPP U T (x, r) T backward construction
Problem Setup and Solution Review of Classical Approach Model commitment Dynamic Programming Principle V t,t ( ) = sup A [t,t ] E (U T (X T ) F t ) = sup A [t,s] E (V s,t (X s ) F t ) A two-period discrete time example Model choice at t = 0 : 0 λ 1 T 2 λ 2 T Value functions : V 0 ( ; λ 1, λ 2 ) V T ( ; λ 2 ) U T ( ) 2 Dynamics/controls : R 0 ξ 1 ( ;λ 1,λ 2 ) R ξ 1 T 2 ξ 2 ( ;λ 2) R ξ 1,ξ 2 T
Problem Setup and Solution Review of Classical Approach Is model revision viable after we start? At t = T 2, suppose that model revision yields a new temporary price impact parameter ˆλ 2 F T 2 The initially chosen λ 2 is not anymore valid Two consequences 1. Time-inconsistency for the second period given the new model ( ξ2 ( ; λ 2 ) arg max E (U T ξ 2 X ξ2 T ) F T, ˆλ ) 2 2 2. ξ1 is still affected since λ 2 is already embedded in [0, T 2 ]; ξ1 would not have been optimal under the new model ( ( ξ1 ( ; λ 1, λ 2 ) arg max E V T ξ 1 2 X ξ1 T 2 ; ˆλ 2 ) F0 )
Problem Setup and Solution Review of Classical Approach How do we then incorporate model revision, time-consistency, optimality and liquidation? Shall we commit at t = 0 to a specific model for many periods ahead? If yes, how do we then incorporate learning? Recall that the revised ˆλ 2 is F T -mble and not F 0 -mble. 2 If we proceed incrementally forward in time, how do we then define optimality? What to do if we want to remain time-consistent? What are the consequences on (perfect) liquidation?
Forward performance approach (Musiela & Zariphopoulou, 2003,...) Allows for dynamic model revision as market evolves It is built to maintain time-consistency Forward performance and portfolio processes track the market up to current time Existing forward results mainly assume continuous model revision and continuous preference revision
Liquidation under forward criteria Revision of price impact parameter λ may be done continuously or discretely Discrete revision is more realistic and practical When revision is done discretely, the λ is kept piece-wise constant λ i t i t i+1 λ i F ti Shall we then align the frequency at which the forward performance process is built with the frequency the model is revised? After all, this is the also the case in the classical setting! Model chosen at t = 0, utility U T F 0
Predictable forward performance criteria (Angoshtari, Zariphopoulou, Zhou, 2017) A family of random functions and random times {U n, τ n } n 0 is a predictable forward performance process if 1. U 0 is a deterministic utility function with τ 0 = 0, and U n U(F τn 1 ), τ n F τn 1, n 1, where U(F τn 1 ) is the set of F τn 1 -measurable utility functions 2. For any admissible wealth process (X t ), U n 1 (X τn 1 ) E (U n (X τn ) ) F τn 1, for n 1 3. There exists an admissible wealth process (X t ), such that U n 1 (Xτ n 1 ) = E (U n (Xτ n ) ) F τn 1, for n 1
Optimal liquidation, dynamic model revision, and forward approach Assess λ 1 for a small period ahead / confidence time, say for (0, τ 1 ] Choose initial criterion at t = 0, solve forward problem in (0, τ 1 ] U 0 (x, r, 0) given= find U 1 (x, r, τ 1 ; λ 1 ) Assess λ 2 for the next liquidation period, (τ 1, τ 2 ] U 1 (x, r, τ 1 ; λ 1 ) given= find U 2 (x, r, τ 2 ; λ 1, λ 2 ) Continue this procedure Therefore, one incorporates learning as the market evolves while maintaining time-consistency by pasting bit-by-bit single inverse liquidation problems
Backward and forward approach M [0,T ] U T Backward 0 T............ DPP M (τ1,τ2] M [0,τ1] U 0 0 τ 1 τ 2 T Forward
Solving the first inverse liquidation problem λ 1 t = 0 U(r, x, 0) τ 1 T Find U(r, x, τ 1 ) F 0, s.t. ( ) U(r, x, 0) = sup E U(R τ1, X τ1, τ 1 ) F 0, λ 1 dx t = ξ t dt, dr t = λ 1 ξ 2 t dt + X t dw t What is a reasonable choice for the initial U(r, x, 0)? Perhaps U(r, x, 0)? = V (r, x, 0; T, λ 1 ), but the admissible class of U(r, x, 0) is actually bigger
Main result: solution of the first problem Suppose at time τ 0 = 0, the initial utility is U 0 (r, x) = e r+g(x). Furthermore, assume that there exist positive constants a b > 0, such that for any x > 0, the function g(x) satisfies g (x) a and g (x) x b Then, the forward problem is well posed in the sense that an optimal admissible liquidation strategy exists for all time t, 0 t < T g (λ), where T g (λ) := { 2λ min tanh 1 ( b 1 ), coth 1 ( a 1 )} 2λ 2λ (tanh 1 (1) = coth 1 (1) = )
Classical existing result is a special case of the single-period forward problem Choose U 0 (r, x) = V (r, x, 0; T ) or U 0 (r, x) = V (r, x, 0; ), then the two problems reconcile. In particular, T g (λ) = liquidation time = T The backward case is a special case of the forward by properly choosing the initial condition
General forward solution For each model revision period [τ n, τ n+1 ], solve a single-period forward problem with random coefficients λ n+1 F τn τ n U τn F τn 1 τ n+1 U τn+1 F τn Serious issues with ill-posedness Need to solve (period-by-period) a random ill-posed HJB U t (x, r, t; ω) + 1 ( 2 x 2 U rr (x, r, t; ω) min λ n+1 (ω)u r (x, r, t; ω)ξ 2 ξ ) +U x (x, r, t; ω)ξ = 0, τ n t < τ n + 1 Inverse problem: given U(x, r, τ n ; ω) F τn 1, find U(x, r, τ n+1 ; ω) F τn
Solution of the forward problem The predictable forward performance process at τ n depends on the realized market model up to τ n, rather than model dynamics beyond τ n as in the classical case U(r, x, τ n ) = e r+αn(λ 1,,λ n)x 2 F τn 1 The quadratic coefficients α ns can be computed recursively forward in time rather than through backward recursion as in the classical case ( ) ( 2λn 1 ) ( ) α n = α 2(τn τ n 1 cosh n 1 ) + α n 1 2 2(τn τ sinh n 1 ) 2λn 1 4 2λn 1 2λn 1 [ ( ) ( )] 2, τn τ cosh 2λn 1 n 1 2 τn τ α n 1 n 1 2λn 1 with α 1 (λ 1 ) = λ1 2 coth ( 2λ1 τ 1 ) λ n 1 sinh
Reconcile with the classical results If λ 1 = λ 2 = λ, i.e., constant price impact profile, then (τ n ) n 0 is deterministic and τ n T, as n U(r, x, τ n ) = V (r, x, τ n ; T ), for n 0 Forward optimal strategy identical to the classical one In particular, liquidation completes exactly at T
When do we fully liquidate under model revision? Do we liquidate exactly at T? Earlier or later than T? How do we balance accurate model revision with (preassigned) liquidation time T? The classical liquidation policy is inaccurate; because the initial model has by now changed!
Trade-off Classical case Perfect liquidation at T wrong model wrong value function Forward case Liquidation time may be close to T (depending on model fluctuations) accurately revised model preservation of optimality
Forward liquidation and classical liquidation time T Depending on the fluctuations of price impact, we may liquidate before or after T, or even at T One of the results λ n τ n 1 X τ n = 0 τ n T Early liquidation occurs if n-th period occurs before T and 2α n 1 (λ 1,, λ n 1 ) > 2λ n a.s. i.e., full liquidation can be achieved if price impact is small
Classical approach requires full model specification at t = 0 Forward approach is flexible to track the market, revising trading targets (e.g. trading horizon and volume) as market evolves Single-horizon and multiple-horizon formulations generalize existing optimal liquidation results under CARA utility Robust convergence results to the continuous time case
References R. ALMGREN (2012): Optimal Trading with Stochastic Liquidity and Volatility, SIAM Journal on Financial Mathematics 3, 163-181. R. ALMGREN, and N. CHRISS (2000): Optimal Execution of Portfolio Transactions, Risk 3, 5-39. R. ALMGREN, and J. LORENZ (2007): Adaptive Arrival Price, Institutional Investor Journals: Algorithmic Trading III: Precision, Control, Execution. B. ANGOSHTARI, T. ZARIPHOPOULOU, and X. ZHOU (2017): Predictable Investment Preferences in a Binomial Model, submitted. M. MUSIELA and T.ZARIPHOPOULOU (2010): Portfolio choice under space-time monotone performance criteria, SIAM Journal on Financial Mathematics. 1, 326-365.
References A. SCHIED, and T. SCHÖNEBORN (2007): Optimal Portfolio Liquidation for CARA Investors, working paper. A. SCHIED, and T. SCHÖNEBORN (2009): Risk Aversion and the Dynamics of Optimal Liquidation Strategies in Illiquid Markets, Finance. Stoch. 13, 181-204. A. SCHIED, T. SCHÖNEBORN, and M. TEHRANCHI (2010): Optimal Basket Liquidation for CARA Investors is Deterministic, Applied Mathematical Finance. 17, 471-489.
Thank you.