The Self-financing Condition: Remembering the Limit Order Book

Transcription

1 The Self-financing Condition: Remembering the Limit Order Book R. Carmona, K. Webster Bendheim Center for Finance ORFE, Princeton University November 6, 2013

2 Structural relationships? From LOB Models to Low Frequency Models Understand transition from discrete to continuous models Incorporate market microstructure into low-frequency models. Differentiate between liquidity takers and providers Identify self-financing conditions

3 Structural relationships? From LOB Models to Low Frequency Models Understand transition from discrete to continuous models Incorporate market microstructure into low-frequency models. Differentiate between liquidity takers and providers Identify self-financing conditions Searching for Answers in the Data Nasdaq ITCH data includes all limit and market orders. Perfect reconstruction of visible limit order book. Example: KO (Coca Cola) on 18/04/13.

4 Midprice Conventions Trade clock, n = 1,...N corresponds to the times t 1 <... < t N at which trades occur Notation: n x = x n+1 x n. p n : mid-price just before the trade at time t n (i.e. p tn )

5 Goal: All trades happen at the best bid or best ask First: remove trades against hidden orders Check the result Next: remove trades with special deals

6 Bid-Ask Spread Convention, Notation, Assumption All trades (100%) happen at the best bid or best ask s n : bid-ask spread just before the trade. s n n p.

7 Aggregate inventory Conventions and comments Inventory of the aggregate liquidity provider L n : Inventory just before the trade. n L < 0 means that a market order bought at the ask.

8 Price Impact Empirical Fact n L n p 0 holds 99.1% of the time. Prices move in favor of market orders

9 Statistical Test of the Hypothesis Assume mid-price p and inventory L are Itô processes { dp t = µ t dt + σ t dw t dl t = b t dt + l t dw t with d[w, W ] t = ρ t dt. Let p N and L N be discrete samplings Define { C N t = Nt 1 n=1 n p N n L N V N t Then = N Nt 2 n=1 p N n = p n/n and L N n = L n/n ( ( n p N n+1 L N) 2 + n p N n L N n+1 p N n+1 L N ) ( ) Ct N [p, L] t L N(0, 1) N 1 Vt N Confidence intervals for [W, W ] t and test of H 0 : t [0, 1], ρ t > 0

10 Tests Results Stock proba reject nb wrong trades total nb trades percent false MSFT KO BA GPS GE CS CPB BCS JNJ UPS CLX T DELL XOM CAT COF AAPL PG GOOG HSY WFC DTV BBY MT GM CL MA KSU GIS

11 Constant Correlation (99% conf. int.) Stock correlation conf int LB conf int UB MSFT KO BA GPS GE CS CPB BCS JNJ UPS CLX T DELL XOM CAT COF AAPL PG GOOG HSY WFC DTV BBY MT GM CL MA KSU GIS

12 (Integrated) Quadratic Covariations

13 Cash Account Convention and Comment K n : cash holdings just before the trade. Self-financing by construction: changes in cash are the amounts exchanged during trades. No more, no less

14 Wealth Definition X n = L n p n + K n Accounting rule: wealth is the value of the inventory marked to the mid-price plus the cash holdings.

15 Self-financing equations Possible wealth dynamics n X = L n n p (1) n X = L n n p + s n 2 nl (2) n X = L n n p + s n 2 nl + n p n L (3)

16 Self-financing equations Possible wealth dynamics n X = L n n p (1) n X = L n n p + s n 2 nl (2) n X = L n n p + s n 2 nl + n p n L (3) Corresponding relationships for self-financing cash n K = p n+1 n L (1) n K = p n+1 n L + s n 2 nl (2) n K = p n n L + s n 2 nl (3)

17 Comparing the three wealth equations Comments True wealth coincides with (3). Difference between (1) and (2) (transaction costs) is large. Difference between (2) and (3) (price impact) cannot be neglected

18 The continuous limit What are the issues? 3 self-financing wealth conditions to choose from. Choice of assumptions on p and L: jumps? finite variation? Bid-ask spread: fixed? Vanishing?

19 The continuous limit What are the issues? 3 self-financing wealth conditions to choose from. Choice of assumptions on p and L: jumps? finite variation? Bid-ask spread: fixed? Vanishing? Informally Want (3) to include transaction costs and price impact = p n/n and L N n = L n/n from p t and L t continuous Itô processes sampled at 1/N, 2/N,..., 1 (trade clock) p N n So n p N = O(1/ N) and n L N = O(1/ N) as N We will also want s N n = O(1/ N)

20 Continuous setup Continuous data Assume { dp t = µ t dt + σ t dw t dl t = b t dt + l t dw t for some µ t, b t, σ t > 0, l t > 0, adapted, and [W, W ] t = t 0 ρ s ds for some ρ t [ 1, 1] and assume s t is continuous and adapted

21 Continuous setup Continuous data Assume { dp t = µ t dt + σ t dw t dl t = b t dt + l t dw t for some µ t, b t, σ t > 0, l t > 0, adapted, and [W, W ] t = t and assume s t is continuous and adapted Discretization choice 0 ρ s ds for some ρ t [ 1, 1] p N n = p n/n ; L N n = L n/n and s N n = 1 N s n/n

22 Proposed discrete equations Wealth dynamics n X N = L N n n p N + s n/n 2 N nl N + n p N n L N Price impact constraint n p N n L N 0 If we want the discretization to mimic the micro structure of a LOB

23 Back to the diffusion limit Wealth dynamics dx t = L t dp t + s tl t 2π dt + d[l, p] t Price impact constraint A necessary condition for an inventory obtained by limit orders is: d[l, p] t 0

24 Applications: I. Hedging Model assumptions Price model: dp t = µ(t, p t )dt + σ(t, p t )dw t Inventory model: Spread model: (empirical studies) dl t = b t dt l t dw t s t = 2πλσ t, with λ > 1/2 No dividend or interest rate. NB: Note that λ = 1 implies dx t = L t dp t (frictionless case)

25 Applications: I. Hedging Model assumptions Price model: dp t = µ(t, p t )dt + σ(t, p t )dw t Inventory model: Spread model: (empirical studies) dl t = b t dt l t dw t s t = 2πλσ t, with λ > 1/2 No dividend or interest rate. NB: Note that λ = 1 implies dx t = L t dp t (frictionless case) Objective Given the model for p and s, find L such that X hedges a European option with payoff f (p T ).

26 Replication argument Markovian setup, so price of the option given by function v(t, p). Itô s formula: d(x t v(t, p t )) = (L t t ) dp t (Θ t Γ tσ 2 (t, p t ))dt + s t l t dt + d[p, L] t 2π

27 Replication argument Markovian setup, so price of the option given by function v(t, p). Itô s formula: d(x t v(t, p t )) = (L t t ) dp t (Θ t Γ tσ 2 (t, p t ))dt + s t l t dt + d[p, L] t 2π Matching Itô decompositions L t = t which also implies l t = Γ t σ(t, p t )

28 Final Solution Delta hedging { L t = t l t = Γ t σ(t, p t ) Only negative Gamma options can be replicated via limit orders!

29 Final Solution Delta hedging { L t = t l t = Γ t σ(t, p t ) Only negative Gamma options can be replicated via limit orders! Pricing PDE ( t v(t, p) + λ 1 ) σ 2 (t, p) 2 2 pv(t, p) = 0 local volatility multiplied by a factor of 2λ 1

30 Applications: II. Market Making Problem (Spirit of Avellaneda - Stoikov) How should the market maker choose the spread s and the volume L? Model assumptions Price model: p N n = p [n/n] for dp t = µ(t, p t )dt + σ(t, p t )dw t Inventory model: Market Maker chooses spread s n only Almgren-Chriss = n L agg = λ n p

31 Thinning Argument By thinning n L = ɛ n+1 n L agg = λɛ n+1 n p with ɛ n+1 = { 1 if market order hits me at time n 0 otherwise P{ɛ n+1 = 1 F n } = g 2 (s n ) with Predictable quadratic variation and co-variation: n 1 [ L, L] n = λ 2 g 2 (s k )E [ k p 2 ] F k k=1 n 1 [ L, p] n = λ g 2 (s k )E [ k p 2 ] Fk k=1

32 Continuum Limit Control Problem { dp t = µ t dt + σ t dw t dl t = λg 2 (s t )µ t dt + λg(s t )σ t dw t with d[w, W ] t = t 0 g(s u)du. In particular dl t = λg 2 (s t )dp t + λg(s t ) 1 g 2 (s t )σ t dwt with W t independent of W t. Our self-financing condition reads: T T X T = L T p T p t dl t λ 2π σ t s t g(s t )dt So a risk neutral Market Maket will want to solve sup EX T. s t [0,s max ],t [0,T )

33 Solution by Stochastic Maximum Principle Hamiltonian H(s, L, Y, Z, Z ) = λg 2 (s) [(Y t p t) µ t + λg(s)σ tz] + λσt 2π sg(s) + λσ tg(s) 1 g 2 (s)z Adjoint equation solved by dy t = L H(t, L t, Y t, Z t, Z t )dt + Z tdw t + Z t dw t Y t = E [ p T F t], Z t = 0, Z t by martingale representation Maximizing Hamiltonian over Optimal Path with ŝ t = arg s [0,smax ]F t (s) F t(s) = sg(s) α tg 2 (s) and α t = E [ p T p t F t] µt σ t + Z t OK as F t is continuous and F t(s max) = 0

34 Geometric Brownian Motion Mid-Price If g(s) = 1 s, s max = 1 and p satisfies dp t = µp t dt + σp t dw t then the adjoint processes become Y t = p t e µ(t t) ; and the optimal spread s t = 1 + α t 2 + α t µ(t t) Z t = σp t e with If µ = 0, α t = p t ( µ σ ( ) e µ(t t) µ(t 1 + σe t)) 1 s t = σp t

35 A version of the Merton problem Assume Investor s control variable. We have dp t = µp t dt + σp t dw t f t = L tp t X t = F t + t 0 b u dw u dl t = dfv t + f t dx t f tx t p t pt 2 dp t + X t df t p t ( ) ft L t = dfv t + p t = dfv t + [ (f 2 t f tx t p 2 t dp t + b tx t dw t p t f t )σ + b t ] X t p t dw t where FV denotes a generic finite variation process.

36 Wealth in our Merton problem dx t = [ µf t (λ 1) σ [ σ(f 2 t f t ) + b t ]] Xt dt + f t X t σdw t If agent uses log-utility function, needs to maximize: E[log X T ] = T 0 [ E (σ(λ 1) + 12 ) ] σ2 ft 2 + (µ σ(λ 1)) f t σ (λ 1) b t dt over all processes f t = F t + t 0 b udw u.

37 Conclusions Standard self-financing equations miss certain features of high-frequency microstructure.

38 Conclusions Standard self-financing equations miss certain features of high-frequency microstructure. We propose an equation which takes into account: transaction costs price impact differentiates between limit orders and market orders

39 Conclusions Standard self-financing equations miss certain features of high-frequency microstructure. We propose an equation which takes into account: transaction costs price impact differentiates between limit orders and market orders Same level of complexity

40 Conclusions Standard self-financing equations miss certain features of high-frequency microstructure. We propose an equation which takes into account: transaction costs price impact differentiates between limit orders and market orders Same level of complexity Similar equation for market orders

41 Conclusions Standard self-financing equations miss certain features of high-frequency microstructure. We propose an equation which takes into account: transaction costs price impact differentiates between limit orders and market orders Same level of complexity Similar equation for market orders Fits data well (tested on 30 large cap stocks).

42 Conclusions Standard self-financing equations miss certain features of high-frequency microstructure. We propose an equation which takes into account: transaction costs price impact differentiates between limit orders and market orders Same level of complexity Similar equation for market orders Fits data well (tested on 30 large cap stocks). Generalizes to a full LOB

43 Conclusions Standard self-financing equations miss certain features of high-frequency microstructure. We propose an equation which takes into account: transaction costs price impact differentiates between limit orders and market orders Same level of complexity Similar equation for market orders Fits data well (tested on 30 large cap stocks). Generalizes to a full LOB Needs more attention Relate ρ t 0 to the queuing systems in LOBs Which limit order strategies produce a given inventory L t?...