Liquidity costs and market impact for derivatives

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1 Liquidity costs and market impact for derivatives F. Abergel, G. Loeper Statistical modeling, financial data analysis and applications, Istituto Veneto di Scienze Lettere ed Arti. Abergel, G. Loeper Statistical modeling, financial data analysis and applications, Istituto Veneto di Scienze Lettere ed Arti Liquidity costs and market impact 1/20

2 Liquidity costs and market impact Liquidity costs and market impact The two main problems that faces the large trader Liquidity costs: the extra price one has to pay, due to the finiteness of available liquidity at the best possible price [4, Cetin, Jarrow, Protter][11, Roch] Market impact: the feedback on the asset dynamics of the large trader s strategy Because of finite liquidity, there always is an instantaneous market impact - the virtual impact in [13, Weber, Rosenow]. As several empirical works show, [3, Almgren et al][13, Weber, Rosenow], a relaxation phenomenon takes place, and the virtual impact becomes permanent. Main objective of this work: a tractable joint modelling of liquidity costs and market impact applied to option pricing and hedging Liquidity costs and market impact 2/20

3 Liquidity costs and market impact Liquidity costs and market impact for derivatives What about derivatives? There is a rather long history of modelling transaction costs, liquidity costs and market impact in the context of derivatives [6, Lamberton, Pham, Schweizer][4, Cetin et al][9, Abergel, Millot] model transaction costs or liquidity costs Early works [5, Frey, Stremme][12, Schonbucher, Wilmott][10, Schweizer, Platen] and more recent ones [7, Liu, Yong][11, Roch] [8, Loeper][2, Almgren, Li] model market impact There is no unified theory for option replication with liquidity costs and market impact The various models that have been proposed often lead to ill-posed pricing and hedging equation Liquidity costs and market impact 3/20

4 Liquidity costs and market impact Liquidity costs and market impact Main modelling assumptions Our goal is to derive a macroscopic model allowing for simplified features of the order book and liquidity supply and demand fine structure Liquidity costs are described by a simple, stationary order book, characterized by its shape around the best price Permanent market impact is measured by a numerical parameter γ as in [13, Weber, Rosenow] γ = 0 means no permanent impact: the price goes back to its original value once the transaction is completed γ = 1 means no relaxation: the price remains at its final value after the transaction is completed Liquidity costs and market impact 4/20

5 Liquidity costs and market impact Main result Using a replication (complete market) or risk-minimization (incomplete market) approach, we derive a pricing and hedging PDE Our main result is Main result The range of parameter for which the pricing equation is well-posed is 2 3 γ 1 A comparison with the literature is interesting [4, Cetin et al][9, Abergel, Millot][11, Roch]: γ = 0 [7, Liu, Yong]: γ = 1 2 [8, Loeper]: γ = 1 Liquidity costs and market impact 5/20

6 Basic modelling assumptions Order book, liquidity costs and impact A deterministic order book profile profile is considered around the price Ŝ t of the asset S at a given time t before the option position is delta-hedged M(x) x µ(u)du represents the number of shares 0 available up to level x We use log prices to define costs and impact, in order to avoid inconsistencies Denote by κ the function M 1. For simplicity κ(ɛ) λɛ the virtual market impact of a transaction of size ɛ is I virtual (ɛ) = Ŝ t (e λɛ 1) the permanent impact is measured by γ R: I permanent (ɛ) = Ŝ t (e γλɛ 1) the cost of the transaction is C(ɛ) = Ŝ t (e λɛ 1) λ Liquidity costs and market impact 6/20

7 Discrete time setting The observed price dynamics Observed price dynamics The dynamics of the observed price is described sequentially First, the price changes under the action of the "market" Ŝ k S k 1 + Ŝ k S k 1 e M k + A k Then, the hedger re-hedges her position and Ŝ k becomes S k = S k 1 e M k + A k e γλ(δ k δ k 1 ) (1) Liquidity costs and market impact 7/20

8 Discrete time setting Cost process Cost process The incremental cost C k of re-hedging at time t k is now studied. A (not necessarily self-financing) strategy consists in buying δ k δ k 1 shares of the asset rebalancing the cash account from β k 1 to β k Introducing the value process one has V k = β k + δ k S k β k + δ k Ŝ k (1 + γκ(δ k δ k 1 )) Incremental cost C k = (V k V k 1 ) δ k 1 (S k S k 1 )+S k ( eλ(δ k δ k 1 ) 1 λe γλ(δ k δ k 1 ) (δ k δ k 1 )). Abergel, G. Loeper Statistical modeling, financial data analysis and applications, Istituto Veneto di Scienze Lettere ed Arti Liquidity costs and market impact 8/20

9 Discrete time setting Optimality conditions Optimality conditions Hedging is implemented via local-risk minimization. There are two (pseudo-)optimality conditions for V k 1 and δ k 1 Optimality in discrete time E( C k F k 1 ) = 0 E(( C k )(S k S k 1 + S k g (δ k δ k 1 )) F k 1 ) = 0 Introducing the supply price process S as in [6, Lamberton et al], [4, Cetin et al] S 0 = S 0, Sk Sk 1 = S k e λγ(δ k δ k 1 ) (γ+(1 γ)e λ(δ k δ k 1 ) ) S k 1 the orthogonality condition can be rewritten as E(( C k )( Sk Sk 1 ) F k 1 ) = 0 Liquidity costs and market impact 9/20

10 Continuous time setting Observed price The observed price in continuous time The continuous-time equivalent of (1) is ds t = S t (dx t + da t + γκ (0)dδ t ) This modelling implies a strategy-dependent volatility of the observed price leading to fully non-linear pricing equation in the markovian setting Modified Volatility Consider a hedging strategy δ which is a function of time and the observed price S at time t: δ t δ(s t, t). Then, the observed price dynamics (2) can be rewritten as (1 γκ δ (0)S t S )ds t = dx t + da t S t Liquidity costs and market impact 10/20

11 Continuous time setting Optimality conditions Cost process and optimality conditions Pseudo-optimal solutions can be characterized Pseudo-optimality in continuous time The cost process of an admissible hedging strategy (δ, V) is given by C t t 0 (dv u δds u S ug (0)d < δ, δ > u ) A strategy is (pseudo-)optimal iff it satisfies the two conditions C is a martingale C is orthogonal to the supply price process S, with d St = ds t + S t (g (0)dδ t g(3) (0)d < δ, δ > t ) Liquidity costs and market impact 11/20

12 The complete market case Pricing and hedging equation In the complete market case, the optimality conditions imply Perfect replication V S = δ and the pricing equation Generalized BS equation V t (1 2γ)κ (0)S 2 V S 2 (1 γκ (0)S 2 V S 2 ) 2 σ2s2 2 2 V S 2 = 0 Liquidity costs and market impact 12/20

13 The complete market case Well posedness of the generalized Black and Scholes equation A sharp result Dependence on the resilience parameter γ The non-linear backward partial differential operator V V t (1 2γ)κ (0)S 2 V S 2 (1 γκ (0)S 2 V S 2 ) 2 σ2s2 2 2 V S 2 = 0 is unconditionally parabolic iff 2 3 γ 1 The requirement is that the function p F(p) = p(1 + (1 2γ)p) (1 γp) 2. (2) be monotonically increasing Liquidity costs and market impact 13/20

14 The complete market case Well-posedness and perfect replication As a consequence of the structure condition on γ, the following theorem holds true [1, Abergel, Loeper] Main result Every european-style contingent claim with payoff Φ satisfying the terminal constraint sup(s 2 Φ S R + S ) < 1 2 γκ (0) (3) can be perfectly replicated via a δ-hedging strategy given by the unique, smooth away from T, solution to the generalized Black-Scholes equation This result can be extended to the multi-asset, complete market case Liquidity costs and market impact 14/20

15 Incomplete markets Stochastic volatility The general theory becomes much more complicated: one has to deal with systems on fully nonlinear, coupled partial differential equations... The "simple" case γ = 1, ρ = 0 can be worked out easily δ = V (4) S and V t σ 2 S 2 + 2(1 κ (0)S( 2 V S 2 )) 2 V S V 2 2 σ 2 Σ2 (5) + 1 κ (0)SΣ 2 2 (1 κ (0)S( 2 V )) ( 2 V σ S )2 + L 1 V = 0 S 2 leading to a positive conclusion Full impact, no correlation The pricing equation (5) is of parabolic type Liquidity costs and market impact 15/20

16 Conclusions and perspectives Conclusions and perspectives A simple model incorporating both liquidity costs and market impact has been presented The well-posedness of the pricing and hedging equation is related to the level of permanent market impact Sharp bounds on the parameter γ are obtained the case γ > 1 leads to arbitrage with round-trip trades the case γ < 2 3 leads to arbitrage for a suitable option portfolio The case of (realistic) payoffs not satisfying the constraint leads to interesting discussions Empirical measurements of γ give values very close to Liquidity costs and market impact 16/20

17 References References I F. Abergel and G. Loeper. Pricing and hedging contingent claims with liquidity costs and market impact. Available at SSRN , R. Almgren and T. M. Li. A fully-dynamic closed-form solution for δ-hedging with market impact R. Almgren, C. Thum, E. Hauptmann, and H. Li. Direct estimation of equity market impact. working paper. Liquidity costs and market impact 17/20

18 References References II U. Cetin, R. Jarrow, and P. Protter. Liquidity risk and arbitrage pricing theory. Finance and Stochastics, 8: , R. Frey and A. Stremme. Market volatility and feedback effects from dynamic hedging. Mathematical Finance, 7(4): , D. Lamberton, H. Pham, and M. Schweizer. Local risk-minimization under transaction costs. Mathematics of Operations Research, 23: , Liquidity costs and market impact 18/20

19 References References III H. Liu and J. M. Yong. Option pricing with an illiquid underlying asset market. Journal of Economic Dynamics and Control, 29: , G. Loeper. Option pricing with market impact and non-linear black and scholes pde s N. Millot and F. Abergel. Non quadratic local risk-minimization for hedging contingent claims in the presence of transaction costs. Available at SSRN , Liquidity costs and market impact 19/20

20 References References IV E. Platen and M. Schweizer. On feedback effects from hedging derivatives. Mathematical Finance, 8(1):67 84, A. Roch. Liquidity risk, volatility and financial bubbles. PhD Thesis. P. J. Schönbucher and P. Wilmott. The feedback effect of hedging in illiquid markets. SIAM J. Appl. Maths, 61(1): , P. Weber and B. Rosenow. Order book approach to price impact. Quantitative Finance, 5(4): , Liquidity costs and market impact 20/20

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