Forecasting prices from level-i quotes in the presence of hidden liquidity

Transcription

1 Forecasting prices from level-i quotes in the presence of hidden liquidity S. Stoikov, M. Avellaneda and J. Reed December 5, 2011

2 Background Automated or computerized trading Accounts for 70% of equity trades taking place in the US U.S. Securities and Exchange Commission (SEC) authorized electronic exchanges in 1998 Archipelago-Arca-NYSE, Island-Instinet-Inet-NASDAQ, BATS, CME, Tokyo stock exchange, Eurex, London stock exchange

3 Background Automated or computerized trading Accounts for 70% of equity trades taking place in the US U.S. Securities and Exchange Commission (SEC) authorized electronic exchanges in 1998 Archipelago-Arca-NYSE, Island-Instinet-Inet-NASDAQ, BATS, CME, Tokyo stock exchange, Eurex, London stock exchange Algorithmic trading Brokers executing client transactions Optimally splitting of client orders

4 Background Automated or computerized trading Accounts for 70% of equity trades taking place in the US U.S. Securities and Exchange Commission (SEC) authorized electronic exchanges in 1998 Archipelago-Arca-NYSE, Island-Instinet-Inet-NASDAQ, BATS, CME, Tokyo stock exchange, Eurex, London stock exchange Algorithmic trading Brokers executing client transactions Optimally splitting of client orders High frequency trading Computerized trading strategies characterized by extremely short position-holding periods Market-making Flash crash!

5 Market in the 90s

6 Market today This is often referred to as the order book

7 A simplified view of the trading world Agent Type of decision Data Mutual/hedge fund Investment Daily close prices Banks, brokers Order splitting 5 min prices Algorithms, HFT Market vs. limit, order routing Level I trades and quotes Electronic market Order matching, messenging Level II trades and quotes

8 Literature Price impact and optimal execution Almgren and Chriss (2000) Schied and Schoneborn (2007) Bouchaud (2009)

9 Literature Price impact and optimal execution Almgren and Chriss (2000) Schied and Schoneborn (2007) Bouchaud (2009) Market microstructure and the information content of the order book Hasbrouck (1993) Parlour and Seppi (2008) Hellstroem and Simonsen (2009) Cao, Hansch and Wang (2009)

10 Literature Price impact and optimal execution Almgren and Chriss (2000) Schied and Schoneborn (2007) Bouchaud (2009) Market microstructure and the information content of the order book Hasbrouck (1993) Parlour and Seppi (2008) Hellstroem and Simonsen (2009) Cao, Hansch and Wang (2009) Limit order book models, zero-intelligence Smith, Farmer, Gillemot, and Krishnamurthy (2003) Cont, Stoikov and Talreja (2010) Cont, De Larrard (2011)

11 Model objectives Making short term price predictions 1 Given the best bid/ask quotes 2 Given statistics on the arrival rates of orders 3 Given a single hidden liquidity parameter

12 Model objectives Making short term price predictions 1 Given the best bid/ask quotes 2 Given statistics on the arrival rates of orders 3 Given a single hidden liquidity parameter Improving the micro-price or fair price ( ) ( ) q ask q bid p micro = p bid + p ask q ask + q bid q ask + q bid

13 Model objectives Making short term price predictions 1 Given the best bid/ask quotes 2 Given statistics on the arrival rates of orders 3 Given a single hidden liquidity parameter Improving the micro-price or fair price ( ) ( ) q ask q bid p micro = p bid + p ask q ask + q bid q ask + q bid Comparing the quality of various exchanges

14 Model objectives Making short term price predictions 1 Given the best bid/ask quotes 2 Given statistics on the arrival rates of orders 3 Given a single hidden liquidity parameter Improving the micro-price or fair price ( ) ( ) q ask q bid p micro = p bid + p ask q ask + q bid q ask + q bid Comparing the quality of various exchanges Estimating hidden liquidity

15 Outline 1 The discrete model A queuing model for level 1 quotes The probability of an upward move in price

16 Outline 1 The discrete model A queuing model for level 1 quotes The probability of an upward move in price 2 The diffusion limit Diffusion approximation Hidden liquidity and boundary conditions Closed form solution

17 Outline 1 The discrete model A queuing model for level 1 quotes The probability of an upward move in price 2 The diffusion limit Diffusion approximation Hidden liquidity and boundary conditions Closed form solution 3 Data analysis Trades and quotes (TAQ) data Estimating hidden liquidity

18 Outline 1 The discrete model A queuing model for level 1 quotes The probability of an upward move in price 2 The diffusion limit Diffusion approximation Hidden liquidity and boundary conditions Closed form solution 3 Data analysis Trades and quotes (TAQ) data Estimating hidden liquidity 4 Conclusion

19 Modeling Level I quotes Assume the bid-ask spread is 1 tick One of the following must happen first: 1 The ask queue is depleted and the price moves up. 2 The bid queue is depleted and the price moves down.

20 A continuous-time Markov chain Let (X t, Y t ) be the bid and ask sizes. Changes in the bid and ask sizes occur at exponential times with rates: λ = arrival rate of orders at the ask (bid) µ = departure rate of orders at the ask (bid) η = rate of simultaneous arrival at the bid (ask) and departure at the ask (bid) h = minimum order size

21 Infinitesimal means and variances E [X t+ t X t X t, Y t ] = h (λ µ) t + o( t) E [Y t+ t Y t X t, Y t ] = h (λ µ) t + o( t) E [ (X t+ t X t ) 2 X t, Y t ] = h 2 (λ + µ + 2η) t + o( t) E [ (Y t+ t Y t ) 2 X t, Y t ] = h 2 (λ + µ + 2η) t + o( t) E [(X t+ t X t )(Y t+ t Y t ) X t, Y t ] = h 2 (2η) t + o( t). If λ = µ, drifts and the variances of the queue sizes are given by m X = m Y = 0 σ 2 X = σ 2 Y = 2h 2 (λ + η) ρ = η λ + η

22 The probability of an upward move in price τ X is the first time the bid size hits zero τ Y is the first time the ask size hits zero The probability that the price moves up before it moves down Prob.{ P > 0 X t, Y t } = Prob.{τ Y < τ X X t, Y t } = p(x t, Y t ) This probability may be computed using Laplace transform methods (see Cont. et al. (2010)) Here we will look at the diffusion limit.

23 Continuous limit Assume that the average queue sizes are much larger than the minimum size < X >=< Y > h Assume that the frequency of orders per unit time is high, λ, η 1. Define the coarse-grained variables x = X / < X >, y = Y / < Y >, σ 2 = 2h2 (λ + η) < X > 2,

24 Continuous limit Assume that the average queue sizes are much larger than the minimum size < X >=< Y > h Assume that the frequency of orders per unit time is high, λ, η 1. Define the coarse-grained variables x = X / < X >, y = Y / < Y >, σ 2 = 2h2 (λ + η) < X > 2, The process (x t, y t ) can be approximated by the diffusion dx t = σdw t dy t = σdz t E (dwdz) = ρdt,

25 The diffusion limit

26 The partial differential equation Let u(x, y) = P(τ y < τ x x t = x, y t = y) be the probability that the next price move is up, given the bid and ask sizes.

27 The partial differential equation Let u(x, y) = P(τ y < τ x x t = x, y t = y) be the probability that the next price move is up, given the bid and ask sizes. It solves the following PDE: σ 2 (u xx + 2ρu xy + u yy ) = 0, x > 0, y > 0,

28 The partial differential equation Let u(x, y) = P(τ y < τ x x t = x, y t = y) be the probability that the next price move is up, given the bid and ask sizes. It solves the following PDE: σ 2 (u xx + 2ρu xy + u yy ) = 0, x > 0, y > 0, Boundary conditions u(0, y) = 0, for y > 0, u(x, 0) = 1, for x > 0. The price moves as soon as x t or y t hit zero

29 Hidden liquidity Empirically, the probability of the price going up when the ask size is small does not tend to zero.

30 Hidden liquidity Empirically, the probability of the price going up when the ask size is small does not tend to zero. Orders on other exchanges prevent the price from moving up (REG NMS)

31 Hidden liquidity Empirically, the probability of the price going up when the ask size is small does not tend to zero. Orders on other exchanges prevent the price from moving up (REG NMS) Hidden or iceberg orders

32 Boundary condition We model a fixed hidden liquidity H

33 Boundary condition We model a fixed hidden liquidity H This translates in σ 2 (p xx + 2ρp xy + p yy ) = 0, x > H, y > H, with the boundary condition p( H, y) = 0, for y > H, p(x, H) = 1, for x > H.

34 Boundary condition We model a fixed hidden liquidity H This translates in σ 2 (p xx + 2ρp xy + p yy ) = 0, x > H, y > H, with the boundary condition p( H, y) = 0, for y > H, p(x, H) = 1, for x > H.

35 Boundary condition We model a fixed hidden liquidity H This translates in σ 2 (p xx + 2ρp xy + p yy ) = 0, x > H, y > H, with the boundary condition p( H, y) = 0, for y > H, p(x, H) = 1, for x > H. In other words we can solve the problem with boundary conditions at zero and use the relation p(x, y; H) = u(x + H, y + H)

36 Solution Theorem The probability of an upward move in the mid price is given by p(x, y; H) = u(x + H, y + H), (1) where u(x, y) = 1 Arctan 1 2 Arctan ( 1+ρ y x 1 ρ y+x ( ) 1+ρ 1 ρ ). (2)

37 Uncorrelated queues (ρ = 0) Problem p xx + p yy = 0, x > H, y > H, and p( H, y) = 0, for y > H, p(x, H) = 1, for x > H.

38 Uncorrelated queues (ρ = 0) Problem p xx + p yy = 0, x > H, y > H, and p( H, y) = 0, for y > H, p(x, H) = 1, for x > H. Solution p(x, y; H) = 2 ( ) x + H π Arctan. y + H

39 Perfectly negatively correlated queues (ρ = 1) Problem p xx 2p xy + p yy = 0, x > H, y > H, and p( H, y) = 0, for y > H, p(x, H) = 1, for x > H.

40 Perfectly negatively correlated queues (ρ = 1) Problem p xx 2p xy + p yy = 0, x > H, y > H, and p( H, y) = 0, for y > H, p(x, H) = 1, for x > H. Solution p(x, y; H) = x + H x + y + 2H.

41 The data Best bid and ask quotes for tickers QQQQ, XLF, JPM, and AAPL, over the first five trading days in 2010

42 The data Best bid and ask quotes for tickers QQQQ, XLF, JPM, and AAPL, over the first five trading days in 2010 All four tickers are traded on various exchanges (NASDAQ, NYSE and BATS)

43 The data Best bid and ask quotes for tickers QQQQ, XLF, JPM, and AAPL, over the first five trading days in 2010 All four tickers are traded on various exchanges (NASDAQ, NYSE and BATS) Using the perfectly negatively correlated queues model, i.e. p(x, y; H) = x + H x + y + 2H we obtain the implied hidden size for each ticker and exchange.

44 Data sample Obtained from the consolidated quotes of the NYSE-TAQ database, provided by WRDS symbol date time bid ask bsize asize exchange QQQQ :30: T QQQQ :30: T QQQQ :30: T QQQQ :30: P QQQQ :30: P QQQQ :30: P

45 Summary statistics Ticker Exchange num qt qt/sec spread bsize+asize price XLF NASDAQ 0.7M XLF NYSE 0.4M XLF BATS 0.4M QQQQ NASDAQ 2.7M QQQQ NYSE 4.0M QQQQ BATS 1.6M JPM NASDAQ 1.2M JPM NYSE 0.7M JPM BATS 0.6M AAPL NASDAQ 1.3M AAPL NYSE 0.4M AAPL BATS 0.6M Table: Summary statistics

46 Estimation procedure 1 We filter the data set by exchange and ticker

47 Estimation procedure 1 We filter the data set by exchange and ticker 2 We bucket the bid and ask sizes in deciles

48 Estimation procedure 1 We filter the data set by exchange and ticker 2 We bucket the bid and ask sizes in deciles 3 For each bucket (i, j), we compute the empirical probability that the price goes up u ij.

49 Estimation procedure 1 We filter the data set by exchange and ticker 2 We bucket the bid and ask sizes in deciles 3 For each bucket (i, j), we compute the empirical probability that the price goes up u ij. 4 We count the number of occurrences of the (i, j) bucket, and denote this distribution d ij.

50 Estimation procedure 1 We filter the data set by exchange and ticker 2 We bucket the bid and ask sizes in deciles 3 For each bucket (i, j), we compute the empirical probability that the price goes up u ij. 4 We count the number of occurrences of the (i, j) bucket, and denote this distribution d ij. 5 We minimize least squares for the negatively correlated queues model, i.e. [ 10 ( min u ij i + H ) 2 d ij] H i + j + 2H i,j=1 and obtain an implied hidden liquidity H for each exchange.

51 Empirical probability (XLF on NASDAQ) decile

52 Model probabilities (XLF on NASDAQ) decile

53 Results Ticker NASDAQ NYSE BATS XLF QQQQ JPM AAPL s = AAPL s = AAPL s = Table: Implied hidden liquidity across tickers and exchanges

54 Future research Level 2 data, predictions on longer time scales Bid ask spreads greater than 1 High frequency volatility estimation Optimal execution with limit and market orders More general dynamics for the bid and ask processes