The Micro-Price Sasha Stoikov Cornell University Jim Gatheral @ - PowerPoint PPT Presentation

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion The Micro-Price Sasha Stoikov Cornell University Jim Gatheral @ NYU

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion High frequency traders (HFT) • HFTs are good: • Optimal order splitting • Pairs trading / statistical arbitrage • Market making / liquidity provision • Latency arbitrage • Sentiment analysis of news • HFTs are evil: • The flash crash • Front running • Market manipulation and spoofing

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion HFTs care about the imbalance Figure: Buy and sell volume conditional on (pre-trade) Imbalance

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion The mid-price • The mid-price M = P b + P a 2 • P b is the best bid price • P a is the best ask price • Not a martingale (Bid-ask bounce) • Low frequency signal • Doesn’t use volume at the best bid and ask prices.

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion The weighted mid-price • The weighted mid-price M w = IP a + (1 − I ) P b Q b • The imbalance I = Q b + Q a • Q b is the bid size and Q a is the ask size. • Gatheral and Oomen (2009) • Not a martingale • Noisy • Counter-intuitive examples

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion The weighted mid-price example • Assume P b = $32 . 17, Q b = 9, P a = $31 . 18, Q a = 1 • Assume the second best ask is $31.19 and the second best ask size is 27 • M w = $32 . 179 = 0 . 1 · 32 . 17 + 0 . 9 · 32 . 18 • Order of size 1 at P a = $31 . 18 cancels • New M w = $32 . 1725 = 0 . 25 · 32 . 17 + 0 . 75 · 32 . 19 • The ‘fair’ price just moved down after an ask order canceled?

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion Features of the Micro-Price • P micro = F ( M t , I t , S t ) t • Markov • Martingale • Computationally fast • Better short term price predictions

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion Outline • General definition • Toy models 1 micro-price = mid price 2 micro-price = weighted mid price • A discrete Markov model • Data analysis • Conclusion

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion Micro-price definition Define P micro i →∞ P i = lim t t where the approximating sequence of martingale prices is given by P i t = E [ M τ i |F t ] • F t is the information contained in the order book at time t . • τ 1 , ..., τ n are (random) times when the mid-price M t changes

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion Assumptions Assumption The information in the order book is given by the 3 dimensional Markov process F t = ( M t , I t , S t ) where M t = 1 2 ( P b t + P a t ) is the Q b mid-price S t = 1 2 ( P a t − P b t ) is the bid-ask spread I t = t t is the Q b t + Q a imbalance at the top of the order book. Assumption The dynamics of ( M t , I t , S t ) is independent of the level M t , i.e. E [ M τ 1 − M t | M t , I t , S t ] � g 1 ( I t , S t )

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion Main result Theorem Given Assumptions 1 and Assumption 2, the i-th approximation to the micro-price can be written as i � P i g k ( I t , S t ) t = M t + k =1 where g 1 ( I t , S t ) = E [ M τ 1 − M t | I t , S t ] and g i +1 ( I t , S t ) = E g i ( I τ 1 , S τ 1 ) | I t , S t � � , ∀ j ≥ 0 can be computed recursively.

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion 3 examples 1 Mid-price independent of imbalance 2 Brownian imbalance 3 Discrete-time, finite state space Interesting questions: • Does the micro-price converge? • What does it converge to? • Is the micro-price between the bid and the ask? • Is it sensible for large tick and small tick stocks?

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion First example If • M s − M t is independent of I t for all s > t • M t is a continuous time random walk. The jumps are binomial and symmetric, i.e. M τ i +1 − M τ i takes values in ( − 1 , 1), have up and down probabilities of 0.5. • The spread S t = 1 then P micro = M t t

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion Second example If • The process I t is a Brownian motion on the interval [0 , 1]. • Let τ down = inf { s > t : I s = 0 } and τ up = inf { s > t : I s = 1 } and τ 1 = min( τ up , τ down ) • When I t is absorbed to 1, the mid-price jumps up with probability 0.5 or bounces back with probability 0.5. • When I t is absorbed to 0, the mid-price jumps down with probability 0.5 or bounces back with probability 0.5. • The spread S t = 1 then = M t + I t − 1 P micro t 2

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion Assumptions • The time step is now discrete with t ∈ Z + , • The imbalance I t takes discrete values 1 ≤ i I ≤ n , • The spread S t takes discrete values 1 ≤ i S ≤ m • The mid-price changes M t +1 − M t takes integer values in K = { k | 0 < | k | ≤ 2 m } . • Define the state X t = ( I t , S t ) with discrete values 1 ≤ i ≤ nm

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion Computing g 1 The first step approximation to the micro-price g 1 ( i ) = E [ M τ 1 − M t | X t = i ] � = k · P ( M τ 1 − M t = k | X t = i ) k ∈ K � � = k · P ( M τ 1 − M t = k ∧ τ 1 − t = s | X t = i ) s k ∈ K

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion The transition probability matrix T 1 Then we define an absorbing Markov chain completely identified by the transition probability matrix T 1 in canonical form: � Q R 1 � T 1 = 0 I • Q is nm × nm matrix • R 1 is nm × 4 m matrix • I is the 4 m × 4 m matrix

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion Computing g 1 Absorbing states R 1 ik := P ( M t +1 − M t = k | X t = i ) Transient states Q ij := P ( M t +1 − M t = 0 ∧ X t +1 = j | X t = i ) Note that R 1 is an nm × 4 m matrix and Q is an nm × nm matrix. � − 1 R 1 k � � g 1 ( i ) = Q s − 1 R 1 � � k = 1 − Q s � T � where k = − 2 m , − 2 m + 1 , . . . , − 1 , 1 , . . . 2 m − 1 , 2 m

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion Computing g i +1 Define a new matrix of absorbing states R 2 ik := P ( M t +1 − M t � = 0 ∧ I t +1 = k | I t = i ) Once again applying standard techniques for discrete time Markov processes with absorbing states g i = � − 1 R 2 g i � � g i +1 ( i ) = Q s − 1 R 2 � � 1 − Q s

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion Checking that the micro-price converges � − 1 R 2 . � Define B := 1 − Q Theorem If B has strictly positive entries and lim k →∞ B k = W where W is the unique stationary distribution and W g 1 = 0 , then the limit i →∞ p i t = p micro lim t converges.

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion A spectral decomposition for the micro-price Perron-Frobenius decomposition nm � exp( λ i ) B i g 1 p micro i →∞ p i = lim t = M t + t i =2 where λ i are the eigenvalues of B and B i are matrices formed from normalized left and right eigenvectors of B .

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion The data Bid and ask quotes for Bank of America (BAC) and Chevron (CVX), for the month of March 2011. Figure: Spread histograms for BAC and CVX. BAC is a typical large tick stock and CVX is a typical small tick stock.

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion The in-sample estimation • Estimate transition probabilities Q , R 1 and R 2 • Compute g 1 = � − 1 R 1 k . This function is symmetrized � 1 − Q to ensure that g 1 ( i I , i S ) = 1 − g 1 ( n − i I , i S ). � − 1 R 2 . This function is symmetrized to � • Compute B = 1 − Q ensure that B ( i I , i S ) , ( j I , j S ) = B ( n − i I , i S ) , ( n − j I , j S ) . Note that the g 1 = 0 and that the symmetrizing procedure ensures that B ˙ micro-price converges as guaranteed by Theorem 2. • Perform a spectral decomposition of B in terms of eigenvalues λ i and matrices B i • Compute the micro-price adjustment: nm G ∗ = p micro − M = � exp( λ i ) B i g 1 i =2

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion In-sample results Figure: G ∗ = p micro − M t as a function of I and S t

Introduction General Framework Toy models Discrete Markov model Data Analysis Conclusion Out of sample validation part 1 • Compute averages of M t +60 − M t grouped by I t and S t for 3 out of sample days • Compare to G ∗ obtained from the first day or March.