Optimal Execution Beyond Optimal Liquidation

Transcription

1 Optimal Execution Beyond Optimal Liquidation Olivier Guéant Université Paris-Diderot Market Microstructure, Confronting Many Viewpoints. December 2014 This work has been supported by the Research Initiative Modélisation des marchés financiers à haute fréquence financed by HSBC France.

2 Introduction Outline Towards block trade pricing Almgren-Chriss model revisited Block trade pricing Option hedging (and pricing) Introduction The model Examples Accelerated share repurchase Introduction and notations The model Numerics and Examples

3 Introduction Outline Towards block trade pricing Almgren-Chriss model revisited Block trade pricing Option hedging (and pricing) Introduction The model Examples Accelerated share repurchase Introduction and notations The model Numerics and Examples

4 Introduction - Optimal Liquidation Basic question: How to liquidate a portfolio with q 0 shares? Classical trade-off Liquidating fast is costly. Execution costs and market impact. But if one liquidates too slowly...

5 ... the price may go down while we are liquidating and we would have been better executing faster. Need to find an optimal trading schedule.

6 Literature Classical literature on optimal liquidation Models à la Almgren-Chriss. Models with transient market impact e.g. Obizhaeva-Wang, Alfonsi-Schied,... Models with limit orders, iceberg orders or dark pools (risk of not being executed). Literature mainly focused on the optimal way to buy/sell a given number of shares (IS orders). Remark: A few papers also on POV orders, VWAP orders, Target close orders,...

7 But optimal execution is more than optimal liquidation! Risk trades Block trade pricing: what should be the price to buy/sell q 0 shares (as a block)? Guaranteed VWAP: what should be the premium to obtain the VWAP for q 0 shares? Optimal execution is everywhere Portfolio choice. Option hedging (and pricing). Complex share buyback programs Accelerated Share Repurchase.

8 Introduction Outline Towards block trade pricing Almgren-Chriss model revisited Block trade pricing Option hedging (and pricing) Introduction The model Examples Accelerated share repurchase Introduction and notations The model Numerics and Examples

9 Introduction Outline Towards block trade pricing Almgren-Chriss model revisited Block trade pricing Option hedging (and pricing) Introduction The model Examples Accelerated share repurchase Introduction and notations The model Numerics and Examples

10 Almgren-Chriss model revisited We consider the liquidation of q 0 shares. Framework in continuous time with 4 variables Time: t Number of shares: q t = q 0 t 0 v sds Price: ds t = σdw t kv t dt Cash: dx t = v t S t dt V t L ( vt V t ) dt where (V t ) t is the market volume curve, assumed to be deterministic. A very general class of functions is admissible for L. In practice: L(ρ) = η ρ 1+φ + ψ ρ

11 Optimization problems Optimization problem Types of orders IS order: B T = S 0 sup E [ exp( γ(x T q 0 B T ))] (v t) t A Target Close order: B T = S T VWAP order: B T = T 0 VtStdt T 0 Vtdt

12 Optimization problems Admissible strategies can be with/without participation constraints: A without = { (v t ) t [0,T ] prog mes, ˆ T 0 ˆ } T v t dt L, v t dt = q 0 0 { A with = (v t ) t [0,T ] prog mes, v t ρ m V t, ˆ T 0 v t dt = q 0 }

13 Classical results Deterministic strategies are optimal. Hamiltonian equations (case of an IS order): { ṗ(t) = γσ 2 q(t) q(t) = V t H (p(t)) q(0) = q 0, q(t ) = 0, where H(p) = sup ρ ρm ρp L(ρ). Generalization in high dimension Numerical methods (difficult in high dimension, with bid-ask spread and participation constraints)

14 Introduction Outline Towards block trade pricing Almgren-Chriss model revisited Block trade pricing Option hedging (and pricing) Introduction The model Examples Accelerated share repurchase Introduction and notations The model Numerics and Examples

15 Question: what should be the price for a block of q 0 shares? Indifference price P(T, q 0, S 0 ): sup E [ exp( γx T )] = exp( γp(t, q 0, S 0 )) (v t) t A with or without constraints. Using results on IS orders we find: where θ T (t, q) = P(T, q 0, S 0 ) = q 0 S 0 k 2 q2 0 θ T (0, q 0 ) ˆ T ( inf V s L q W 1,1 q,0 (t,t ) t ( q (s) V s ) + 1 ) 2 γσ2 q 2 (s) ds

16 What happens when T +? (Liquidation with no time constraint) Theorem If V t = V, then: ˆ ( ) q γσ lim θ T (t, q) = θ (q) := H 1 2 T + 0 2V x 2 dx, where H 1 is the inverse of H : p R + sup ρ ρm ρp L(ρ). Block trade pricing formula P(q, S) = qs k 2 q2 ˆ q 0 ( ) γσ H 1 2 2V x 2 dx We call qs P(q, S) a risk-liquidity premium/discount.

17 Block trade pricing formula If L(ρ) = η ρ 1+φ + ψ ρ and without participation constraints: where l(q) = k 2 q2 + ψq + η 1 1+φ P(q, S) = qs l(q) φ φ 1+φ (1 + φ) φ ( γσ 2 2V ) φ 1+φ q 1+3φ 1+φ is the risk-liquidity discount/premium in this particular case. This type of premium/discount gives a price to liquidity: it can be used in many problems as a penalization function.... but γ need to be chosen!

18 Choice of the risk aversion parameter Hard to answer a question about our own risk aversion Easier to decide on a participation rate POV orders A POV = sup E [ exp( γ(x T q 0 S 0 ))] (v t) t A { } (v t ) t R+, ρ > 0, t 0, v t = ρv t 1 { t 0 ρvsds q 0} If L(ρ) = η ρ 1+φ + ψ ρ and V t = V : ρ = ( γσ 2 6ηφ q 2 0 V ) 1 1+φ This formula can be inverted to choose γ!

19 References Block trade pricing*: Optimal execution and block trade pricing: a general framework, to app. in AMF Numerical methods: A convex duality method for optimal liquidation with participation constraints, submitted (with J.-M. Lasry and J. Pu). POV orders*: Execution and block trade pricing with optimal constant rate of participation, JMF, 2014 Guaranteed VWAP: VWAP execution and guaranteed VWAP, SIFIN, 2014 (with G. Royer)

20 Introduction Outline Towards block trade pricing Almgren-Chriss model revisited Block trade pricing Option hedging (and pricing) Introduction The model Examples Accelerated share repurchase Introduction and notations The model Numerics and Examples

21 Introduction Outline Towards block trade pricing Almgren-Chriss model revisited Block trade pricing Option hedging (and pricing) Introduction The model Examples Accelerated share repurchase Introduction and notations The model Numerics and Examples

22 Introduction - Option pricing / hedging Classical framework for option pricing: Black-Scholes and extensions frictionless market, price-taker agent Sometimes super-replication + transaction costs but... Issues Not suited for options on illiquid assets Not suited when nominal is large Not suited when Γ is too large No difference between physical and cash settlement Here: Option pricing and hedging with execution costs and market impact, joint work with J. Pu

23 Optimal execution and options Other routes Transaction costs (fixed or proportional), Supply curve approach (Çetin-Jarrow-Protter (2004), Çetin-Soner-Touzi (2010)). A few papers with some form of market impact (Lasry-Lions, Abergel-Loeper, Bouchard-Loeper) Recently, optimal execution met option pricing: L. C. Rogers, S. Singh, The cost of illiquidity and its effects on hedging. Mathematical Finance, 20(4), , T. M. Li, R. Almgren, Option hedging with smooth market impact, 2014

24 Introduction Outline Towards block trade pricing Almgren-Chriss model revisited Block trade pricing Option hedging (and pricing) Introduction The model Examples Accelerated share repurchase Introduction and notations The model Numerics and Examples

25 Call Option Call option on a stock with: Strike K Maturity T Nominal N (in shares) N matters because the introduction of execution costs and market impact makes the problem a non-linear one. We consider that we have sold this call option with physical settlement.

26 Notations Model without permanent market impact for the sake of simplicity. Framework in continuous time with 4 variables Time: t Number of shares: q t = q 0 + t 0 v sds Price: ds t = σdw t ( ) Cash: dx t = v t S t dt V t L vt V t dt Remarks: q 0 is important here! V t can be set to 0 at night! Additional features (in the paper) Interest rate r Drift µ Permanent market impact k

27 Payoff Case 1 the option is exercised: The trader has whatever is on his cash account X T The trader receives KN The trader buys (N q T ) shares and deliver N shares The payoff in that case is: X }{{} T + }{{} KN ((N q T )S T + l(n q T )) }{{} cash account payment of the client cost of buying N q T shares

28 Payoff Case 2 the option is not exercised: The trader has whatever is on his cash account X T. The trader liquidates the q T shares remaining in his portfolio. The payoff in that case is: X }{{} T + q T S T l(q T ) }{{} cash account gain of selling the q T shares Payoff X T + q T S T + 1 ST K (N(K S T ) l(n q T )) 1 ST <K l(q T ) In case of cash settlement, one just needs to replace l(n q T ) by l(q T ).

29 Optimization Problem Optimization Problem The bank maximizes its expected utility: where Y T = X T + q T S T sup E [ exp ( γy T )], v A +1 ST K (N(K S T ) l(n q T )) 1 ST <K l(q T )

30 HJB Equation The HJB equation associated to this stochastic optimal control problem is: HJB equation 0 = t u 1 { ( ( ) ) } v 2 σ2 SSu 2 sup v q u + vs L V t x u v R Vt with terminal condition: u(t, x, q, S) = exp ( ( γ x + qs 1 S<K l(q)) )) +1 S K (N(K S) l(n q))

31 Change of Variables We use the following change of variables: Definition We introduce θ by: u(t, x, q, S) = exp ( γ (x + qs θ(t, q, S))) Indifference Price θ(0, q 0, S 0 ) can be interpreted as the indifference price of the following contract: We write the call with the client We give q 0 S 0 to the client in cash The client gives us q 0 shares

32 PDE for θ The PDE satisfied by θ is the following: PDE t θ 1 2 σ2 2 SSθ 1 2 γσ2 ( S θ q) 2 + V t H( q θ) = 0 where H is as above H(p) = sup ρ ρ m {pρ L(ρ)}. Terminal Condition θ(t, q, S) = 1 S K (N(S K) + l(n q)) + 1 S<K l(q)

33 PDE Interpretation of the PDE: t θ 1 2 σ2 SSθ 2 1 }{{} 2 γσ2 ( S θ q) 2 + V t H( q θ) }{{}}{{} Execution costs Black-Scholes PDE "Mishedge" = 0 An optimal control is formally given by: Optimal Control v (t, q, S) = H ( q θ(t, q, S)) Remark: This PDE is not an HJB equation. θ is rather the value function of a player in a zero-sum differential game.

34 Introduction Outline Towards block trade pricing Almgren-Chriss model revisited Block trade pricing Option hedging (and pricing) Introduction The model Examples Accelerated share repurchase Introduction and notations The model Numerics and Examples

35 Reference Scenario S 0 = K = 45 σ = 0.6 day 1/2 ( 21% annual volatility) T = 63 days V = shares day 1 N = shares L(ρ) = η ρ 1+φ with η = 0.1 shares 1 day 1 and φ = 0.75 That corresponds to 9 bp for a participation rate of 30% and 13 bp for a participation rate of 50% γ = l corresponds to liquidation with POV when at rate 50%

36 Reference Scenario Price Price Strike Time Figure : Reference Scenario - Stock Price

37 Reference Scenario 2 numerical methods: a tree method and a finite difference scheme. We see that we do not mean-revert around the usual. q/n Tree PDE Bachelier Delta Time Figure : Reference Scenario - Strategy

38 Reference Scenario Model/Method Bachelier Tree-Based approach PDE approach Price Table : Prices of the call option for the two numerical methods. We see the difference between the classical model and our model.

39 Importance of Initial Position q/n Initial Portfolio with No Share Initial Portfolio with Bachelier Delta Time Figure : Optimal portfolio when q 0 = 0, when a participation limit of 50% is imposed

40 Execution Costs q/n Bachelier Delta eta = 0.01 eta = 0.05 eta = 0.1 eta = Time Figure : Optimal portfolio for different values of η

41 Execution Costs When η increases: The trajectories are smoother They are closer to the position 0.5N to avoid round trips When η 0, we obtain the limiting case of -Hedging. The prices are given by: η (Bachelier) Price of the call Prices are higher when η increases.

42 Price risk and risk aversion First risk (binary/digital): the trader will have to deliver either N shares or none. Being averse to this risk encourages the trader to stay close to a neutral portfolio with q = 0.5N. Second risk: price at which shares are bought/sold. Being averse to price risk encourages the trader to have a portfolio that evolves in the same direction as the price, as it is the case in the Bachelier model.

43 Price risk and risk aversion q/n gamma = 2e 7 gamma = 5e 8 gamma = 2e 8 gamma = 1e Time Figure : Optimal portfolio for different values of γ - 1

44 Price risk and risk aversion q/n gamma = 2e 7 gamma = 1e 6 gamma = 2e 6 gamma = 5e Time Figure : Optimal portfolio for different values of γ - 2

45 Price risk and risk aversion The two effects are important. In terms of price however, there is monotonicity: γ Price of the call Table : Prices of the call option for different values of γ. Prices are increasing with γ. Prices also increasing with σ.

46 Introduction Outline Towards block trade pricing Almgren-Chriss model revisited Block trade pricing Option hedging (and pricing) Introduction The model Examples Accelerated share repurchase Introduction and notations The model Numerics and Examples

47 Introduction Outline Towards block trade pricing Almgren-Chriss model revisited Block trade pricing Option hedging (and pricing) Introduction The model Examples Accelerated share repurchase Introduction and notations The model Numerics and Examples

48 Introduction We presented a way to use optimal execution models for option hedging. There exist execution contracts with options inside the contracts... Nature of the problem Accelerated share repurchase contracts An optimal execution problem An optimal stopping problem. An option pricing and hedging problem with asian payoff.

49 What is an ASR? ASR contracts are used by firms to buy back shares instead of paying dividends! Why not simply buying shares on markets?... to commit to the decision of a share repurchase program! Many repurchase programs are slowed down, postponed, or cancelled after announcement (because of unexpected shocks on prices for instance). ASR contracts are mainly of two kinds: with fixed number of shares / with fixed notional.

50 ASR Contracts (fixed number of shares Q) A bank is asked by a firm to repurchase a number Q > 0 of the firm s own shares. At time t = 0, the bank borrows Q shares from shareholders and delivers them to the firm. At time t = 0, the firm pays an amount F to the bank. The bank buys back shares on the market to give them back to initial shareholders. The bank chooses a date τ among a set of dates in [0, T ], to exercise the option. At the exercise date τ, the firm pays QA τ F to the bank, where A τ is the average price between 0 and τ.

51 Introduction Outline Towards block trade pricing Almgren-Chriss model revisited Block trade pricing Option hedging (and pricing) Introduction The model Examples Accelerated share repurchase Introduction and notations The model Numerics and Examples

52 Setup of the model (fixed number of shares Q) Model in discrete time: δt = 1 day, n = 0 corresponds to t = 0 and T = Nδt is the horizon of the ASR contract. Time + 4 variables Number of shares: q n+1 = q n + v n δt, q 0 = 0 Price (daily VWAP): S n+1 = S n + σ δtɛ n+1 Average price: A n = 1 n nk=1 S k Cash spent: X n+1 = X n + v n S n+1 δt + L ( vn V n+1 ) V n+1 δt

53 Exercice dates Setup of the model (continued) N {1,..., N 1} is the set of possible exercise times before expiry (usually, N = {n 0,..., N 1}). The exercise time n is a stopping time taking value in N {N}. After the option is exercised At time n, Q q n shares remain to be bought. The pure optimal execution problem after time n is replaced by a proxy: (Q q n )S n + l(q q n ), where l is a penalty function.

54 Objective function We consider an expected utility framework: Maximization problem sup E [ exp ( γ (QA n X n (Q q n )S n l(q q n )))] (v,n ) A

55 Value function The value function u n (x, q, S, A) of the problem can be written as: ( ( exp γ (Q(A S) X + qs θ n q, S A σ δt ))). and the optimal control will only depend on n, q and the spread Z = S A σ δt.

56 Bellman equation for θ n Proposition (Bellman for θ n ) for n = N: θ n (q, Z) = l(q q), for n N : θ n (q, Z) = min { θn,n+1 (q, Z), l(q q) }, for n / N : θ n (q, Z) = θ n,n+1 (q, Z), where θ n,n+1 is equal to: ( [ ( ( 1 inf v R γ log E exp γ σ (( ) n δt n + 1 Q q ɛ n+1 Q ) n + 1 Z ( ) ( ) v n ))]) + L V n+1 δt + θ n+1 q + vδt, V n+1 n + 1 (Z + ɛ n+1).

57 Analysis of θ n Our change of variables can be interpreted easily since θ n (q, Z) is equal to: ( [ ( ( 1 inf (v,n ) γ log E exp γ σ ( n 1 ( ) j δt n j=n Q q j ɛ j+1 (1 n ) ) n QZ }{{}}{{} Z term risk term n ( ) ))]) 1 vj + L V j+1 δt + l(q q n ). V j=n j+1 }{{}}{{} liquidity and risk term after exercise liquidity term before exercise

58 Analysis of θ n The previous formula helps to understand the effects at stake: The risk term The risk term measures the risk associated to a deviation from a straight-line strategy. If the bank buys Q shares evenly until a given exercise date (or until T ), then the risk is indeed perfectly hedged. But to benefit from the option contract, the bank will not follow this strategy. The Z-term If the price goes down, then there is an incentive to exercise to benefit from the difference between A and S but this incentive depends on q (see below).

59 Analysis of θ n The l term Before time n, the execution process is partially hedged (this is the risk term) After time n, the execution process is not hedged (the risk is in the l-term). Hence, there is an incentive to delay exercise if we have still a large number of shares to buy. The consequence is that when S goes down the bank should accelerate the execution (buying) process, but not too much (because of execution costs).

60 Introduction Outline Towards block trade pricing Almgren-Chriss model revisited Block trade pricing Option hedging (and pricing) Introduction The model Examples Accelerated share repurchase Introduction and notations The model Numerics and Examples

61 Numerical scheme We consider a pentanomial tree model for innovations (ɛ n ) n 1 : ɛ n = +2 with probability with probability with probability with probability with probability 1 12 This model for ɛ n is chosen to match the first four moments of the standard normal distribution, i.e. we have: [ ] [ ] [ ] E [ɛ n ] = 0, E ɛ 2 n = 1, E ɛ 3 n = 0, E ɛ 4 n = 3.

62 Numerical scheme Each node of the tree corresponds to a couple (n, Z) and we associate a grid for q to each node. The tree is not recombinant in the classical sense. However nz n + n(n 1) is an integer between 0 and 2n(n 1). Hence the tree has a number of nodes that is a cubic function of N.

63 Reference case S 0 = 45 σ = 0.6 day 1/2, which corresponds to an annual volatility approximately equal to 21%. T = 63 trading days V = stocks day 1 Q = stocks L(ρ) = η ρ 1+φ with η = 0.1 stock 1 day 1 and φ = 0.75 γ = l(q) corresponds to execution at participation rate 25% after the exercise date. The set of possible exercise dates is N = [22, 62] N.

64 Price trajectory and optimal strategy I Inventory q (in millions of shares) Optimal Strategy S A Price Time Figure : Optimal Strategy when price goes up.

65 Price trajectory and optimal strategy I In that case: Exercise at terminal time. Minimizing execution costs by trading almost in straight line. When S decreases, acceleration of the buying process When S increases, the buying process slows down or even turns into a selling process (for hedging purposes).

66 Price trajectory and optimal strategy II Inventory q (in millions of shares) Optimal Strategy S A Price Time Figure : Optimal Strategy when price goes down.

67 Price trajectory and optimal strategy II In that case: Exercise almost as soon as possible (to benefit from A S) As S is below A, acceleration of the buying process to buy a lot before exercising.

68 Price trajectory and optimal strategy III Inventory q (in millions of shares) Optimal Strategy S A Price Time Figure : Optimal Strategy when price oscillates.

69 Questions?