Microprice slides

Introduction

The Micro-Price

Data Analysis

Price Prediction with Big Data Analytics Sasha Stoikov Cornell Financial Engineering Manhattan

May 11, 2018

Conclusion

Introduction

The Micro-Price

Data Analysis

Financial markets have always been Big Data

For US equities 1 2

Full order book data is 1TB/day Level I quotes data is 100GB/day • For each timestamp we have Ft = σ(Ptb , Pta , Qtb , Qta ) • Ptb bid price • Pta ask price • Qtb bid size • Qta ask size

Conclusion

Introduction

The Micro-Price

Data Analysis

What can we do with all this Big Data?

• Compute the fair price • Should be a martingale

Pt∗ = E [PT∗ |Ft ], ∀t < T • This is useful for: • Optimal timing of market orders (Stoikov and Waeber 2016) • Market-making (Avellaneda and Stoikov 2008) • High frequency volatility, correlation and cointegration (Ait-Sahalia 2011)

Conclusion

Introduction

The Micro-Price

Data Analysis

Why not use the mid price?

The mid price Mt =

Ptb + Pta 2

â&#x20AC;˘ Ignores sizes Qtb and Qta â&#x20AC;˘ Not a martingale, i.e.

Mt 6= E [MT |It ] where It = is the imbalance

Qtb Qtb + Qta

Conclusion

Introduction

The Micro-Price

Data Analysis

The imbalance matters

Figure: Buy and sell volume conditional on (pre-trade) Imbalance

Conclusion

Introduction

The Micro-Price

Data Analysis

The main objective To define the micro-price Pt∗ = Mt + g ∗ (It , St ) as a function of • The midprice: Mt = 12 (Ptb + Pta ) • The spread: St = 12 (Pta − Ptb ) b

• The imbalance: It = bQt a Qt +Qt

in such a way that it is a martingale Pt∗ = E [PT∗ |Ft ]

Conclusion

Introduction

The Micro-Price

Data Analysis

Conclusion

Micro-price definition

• Consider the i-th mid price prediction

Pti = E [Mτi |Ft ] where τ1 , ..., τn are random times when the mid-price changes • Define the micro-price:

Ptmicro = lim Pti i→∞

Introduction

The Micro-Price

Data Analysis

Main result

Theorem The i-th mid-price prediction can be written as Pti = Mt +

i X

g k (It , St )

k=1

where g 1 (I , S) = E [Mτ1 − Mt |It = I , St = S] and

g i+1 (I , S) = E g i (Iτ1 , Sτ1 )|It = I , St = S , ∀i ≥ 0 can be computed recursively.

Conclusion

Introduction

The Micro-Price

Data Analysis

Conclusion

Discretizing the state space

• Define the state Xt = (It , St ) with discrete values 1 ≤ i ≤ nm • The first step approximation to the micro-price

G 1 (i) =E [Mτ1 − Mt |Xt = i] XX = k · P(Mτ1 − Mt = k ∧ τ1 − t = s|Xt = i) k∈K

s

Introduction

The Micro-Price

Data Analysis

Computing G 1 Absorbing states Rik := P(Mt+1 − Mt = k|Xt = i) Transient states Qij := P(Mt+1 − Mt = 0 ∧ Xt+1 = j|Xt = i) Note that R is an nm × 4 matrix and Q is an nm × nm matrix.

1

G =

∞ X

−1 Q s−1 R K = 1 − Q RK

s=1

where K = [−0.01

− 0.005

0.005

0.01]T

Conclusion

Introduction

The Micro-Price

Data Analysis

Computing G i+1

Define a new matrix of absorbing states Rij2 := P(Mt+1 − Mt 6= 0 ∧ It+1 = j|It = i) Therefore G i+1 =

∞ X s=1

−1 Q s−1 T G i = 1 − Q TG i := BG i

Conclusion

Introduction

The Micro-Price

Data Analysis

Checking that the micro-price converges

B := 1 − Q

−1

T.

Theorem If B ∗ = limk→∞ B k and B ∗ G 1 = 0, then the limit lim Pti = Ptmicro

i→∞

converges.

Conclusion

Introduction

The Micro-Price

Data Analysis

The data Bid and ask quotes for Bank of America (BAC) and Chevron (CVX), for the month of March 2011.

Figure: BAC = large tick

vs.

CVX = small tick

Conclusion

Introduction

The Micro-Price

Data Analysis

Estimation

• Symmetrize the data. For every tuple (It , St , It+1 , St+1 , dM)

there is a corresponding symmetric tuple (1 − It , St , 1 − It+1 , St+1 , −dM) • Estimate transition probabilities Q, R and T • Compute G 1 = 1 − Q • Compute B = 1 − Q

−1

−1

RK .

T.

• Compute the micro-price adjustment: ∗

G =P

micro

1

−M =G +

∞ X i=1

Bi G 1

Conclusion

Introduction

The Micro-Price

Data Analysis

BAC and CVX mid price adjustments

Figure: G ∗ = P micro − M for BAC and CVX

Conclusion

Introduction

The Micro-Price

Data Analysis

Comparing micro-price to empirical mid-price changes

• Sample the mid price every T seconds • Compute the empirical change in mid prices

E [Mt+T − Mt |St , It ] • Compare to the micro-price adjustment G ∗

Conclusion

Introduction

The Micro-Price

Data Analysis

BAC Microprice vs. empirical mid price changes

Figure: G â&#x2C6;&#x2014; = Ptmicro â&#x2C6;&#x2019; Mt vs. empirical price changes for BAC

Conclusion

Introduction

The Micro-Price

Data Analysis

CVX Microprice vs. empirical mid price changes

Conclusion

Introduction

The Micro-Price

Data Analysis

Micro-price properties

1

Itâ&#x20AC;&#x2122;s the expected mid-price in the distant future

2

Converges if B â&#x2C6;&#x2014; G 1 = 0

3

Good predictor of future mid prices for big-tick/small-tick microstructures

4

Horizon independent

5

Lives between the bid and the ask

Conclusion

Introduction

The Micro-Price

Data Analysis

What can you do with the micro-price?

• Optimal timing of market orders • Market-making and posting of limit orders • High frequency volatility estimation • High frequency correlation and cointegration

Conclusion

Introduction

The Micro-Price

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Data Analysis

Conclusion