Application of the kinetic theory to reveal the non-Gaussianity in finance

Tokyo Tech, Kiyoshi Kanazawa

10th March, 2017YITP

Collaborators

Takumi SueshigeTokyo Tech

Hideki TakayasuSony CSL, Tokyo Tech

Misako TakayasuTokyo Tech

To appear on arXiv soon…

Brownian motion in finance: Price fluctuation in FX markets

Market : Stock, Foreign Exchange (FX)

Ideal markets: Price movements ≒ pure random walkReal markets : Deviation from pure random walk for short time

Recently, detailed electronic FX data is available

Focus: hierarchies in FX markets by data analysis and theory

Price

Time

Market

Review: Rule of FX markets

Double Auction1. In advance, FX traders quote their price for ask or bid.

2. When a bid and an ask price match, there is a transaction

3. The transacted price is recorded and displayed

Ask: ¥100/$

Bid: ¥90/$

Ask: ¥90/$

¥90/$

Price

Time

Market

Hierarchy in the FX markets: microscopic – mesoscopic – macroscopic hierarchies

reducePrice

Time

Macroscopic price dynamics

Mainly focused in previous studies

• Hierarchy in FX markets: 1. Microscopic : dynamics of Individual traders2. Mesoscopic : distribution of order-books3. Macroscopic: diffusion of price

• Previous studies: mesoscopic or macroscopic analysis→because microscopic data is confidential…

Mesoscopic: Order-Book distributions

Order-book = current situation bid & ask prices1. Time of new order-submission

2. Cancel time or execution time

Statistical laws for the order-book(e.g., J.-P. Bouchaud q-fin 2002, Y. Yura PRL2014)

Volume Bid order-book Ask order-book

Price

10:20 Cancellation 10:15 New ask

Best bid Best ask

Genders of Ordersbid (buy) or ask (sell)

Order-book distribution in real time

Green：New OrderRed ：Annihilation

Bid Dist.

Ask Dist.

Question: Is it possible to systematically understand the market from microscopic dynamics

Motivation: micro data analysis & corresponding micro theory

1. Data analysis : Direct observation of traders’ microscopic dynamics

2. Theory : Agent model and the corresponding statistical mechanics

1. Data analysis of anonymized traders’ IDs2. Application of the techniques in the molecular kinetic theory

Idea for Solution

reducePrice

Time

Macroscopic price dynamics

Previous studiesToday’s talk

Solution 1: Anonymized traders’ IDs

Trajectoriesof the best bid and ask

for a single trader

1. Submission, cancellation, execution information for all traders

2. Traders’ ID is also accessible with anonymized bank codes

1. Traders’ Strategies can be directly revealed from anonymized traders’ IDs2. New microscopic agent model can be proposed on the basis of this data

Solution 2: “Kinetic” theory for traders

Standard methods in the kinetic theory

1. Liouville equation (Newton equation)

2. BBGKY Hierarchy

3. Boltzmann equation

Application of this technique for FX trader systems

Agent model based on the data analysis

“Kinetic” theory for financial systems1. “Liouville” equation2. “BBGKY” Hierarchy3. “Boltzmann” equation

Mathematical analogy

Review of the kinetic theory0th Step: Newton’s Equation

Complete description of N body systems by Newton

𝑁𝑁 ∼ 1023 →Too many particle to calculate!

= Point in the phase space

1st step: Liouville equation

Master equation for the N-body dist.

Mathematically equivalent to Newton’s equation

Exact and closed, but cannot be solved…

𝑃𝑃𝑁𝑁 �⃗�𝑥1,𝑝𝑝1; … ; �⃗�𝑥𝑁𝑁,𝑝𝑝𝑁𝑁= 𝑁𝑁-body dist.

2nd step: Bogolioubov-Born-Green-Kirkwood-Yvon hierarchy

Reduction of the N-body to the 1- & 2-body dists.𝑃𝑃𝑁𝑁 ⟹ 𝑃𝑃1,𝑃𝑃2

Exact, but not closed…

Reduction

3nd Step: Boltzmann equation

Exact proof exists for Hamiltonian dynamics(In the dilute limit＝the Boltzmann-Grad limit)

Can be solved: the solution is the Maxwell-Boltzmann dist.

𝑓𝑓 �⃗�𝑣 =𝑚𝑚2𝜋𝜋𝜋𝜋

3/2𝑒𝑒−𝑚𝑚𝑣𝑣2/2𝑇𝑇

Assumption of the molecular chaos

Theoretical idea: from the molecular to the trader kinetic theory

Molecular kinetic theory

1. Newton’s equation

2. Liouville equation

3. BBGKY hierarchy

4. Boltzmann equation

Trader “kinetic” theory

1. Stochastic equation

2. “Liouville” equation

3. “BBGKY” hierarchy

4. Mean-field equation

Goal of this talk: Direct revelation of FX market microstructure via both data analysis and theory

Data Analysis:1. Dynamical characterization of

individual traders

2. Trend-following strategy

Microscopic theory:1. Trader model based on

microscopic empirical law

2. “Kinetic” theory to derive a mean field equation

3. From micro to meso & macro

Trajectories of single trader

Empirical Analysis of Individual TradersTrend-following Strategy on the microscopic level

Typical trajectory of High Frequency Traders (HFT)

• High Frequency Trader (HFT) obeying algorithms• Time-precision is millisecond; mean interval is a few sec. to 1 min.• Definition: traders who submit once a min. in the week average

(134 among 1015)

Typical trajectory of HFT

2) spread≒const.

price

ask

bid

1) Two-sided quotesas a market-maker(liquidity provider)

Tight-coupling for bid and ask

Time series of spread & spread distribution

Time series of spread

Fluctuation around constant 𝐿𝐿𝑖𝑖 ≃ 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐.

𝐿𝐿𝑖𝑖: Spread of 𝑖𝑖th trader

Spread distribution 𝜌𝜌𝐿𝐿

𝛾𝛾-distribution (for one week)

𝜌𝜌𝐿𝐿 =𝐿𝐿3e−𝐿𝐿/𝐿𝐿∗

6𝐿𝐿∗4

Spread

Time

Typical trajectory of HFT

3) Trend-following：Correlation betwee 𝛥𝛥𝑧𝑧𝑖𝑖 & 𝛥𝛥𝑝𝑝

𝛥𝛥𝑃𝑃

𝛥𝛥𝑧𝑧

Trend-following of individual traders

Δ𝑧𝑧𝑖𝑖 = Movement of quoted price for 𝑖𝑖th trader (based on best mid-price)

Δ𝑝𝑝 = Previous price movement

The same strategy for Top 20 HFTs Δ𝑧𝑧𝑖𝑖𝑐𝑐𝑖𝑖∗

≃ tanhΔ𝑝𝑝Δ𝑝𝑝𝑖𝑖∗

𝑐𝑐𝑖𝑖∗ & Δ𝑝𝑝𝑖𝑖∗: characteristic constants.

∼ sgn Δ𝑝𝑝 (|Δ𝑝𝑝| → ∞)

Microscopic theory based on the empirical laws for individual traders“KINETIC” THEORY FOR TRADERS

Model: trend-following random walks

Market-maker: traders who quote both bid and ask simultaneously

𝑖𝑖th trader: bid price 𝑏𝑏𝑖𝑖ask price 𝑎𝑎𝑖𝑖buy-sell spread 𝐿𝐿𝑖𝑖

Buy-sell spread 𝐿𝐿𝑖𝑖 depends on the trader: spread dist. 𝜌𝜌(𝐿𝐿)

Identical trader

𝑏𝑏𝑖𝑖 𝑎𝑎𝑖𝑖

Mid price: 𝑧𝑧𝑖𝑖 ≡𝑎𝑎𝑖𝑖+𝑏𝑏𝑖𝑖2

𝐿𝐿𝑖𝑖

Model dynamics

mid price 𝑧𝑧𝑖𝑖 ≡ (𝑎𝑎𝑖𝑖 + 𝑏𝑏𝑖𝑖)/2 (Pricemovement 𝛥𝛥𝑝𝑝 & constants 𝑐𝑐,Δ𝑝𝑝∗)𝑑𝑑𝑧𝑧𝑖𝑖𝑑𝑑𝑐𝑐 = 𝑐𝑐tanh Δ𝑝𝑝/Δ𝑝𝑝∗ + 𝜎𝜎𝜂𝜂𝑖𝑖 , 𝜂𝜂𝑖𝑖 = White Gaussian noise

Transaction rule

Requotation 𝑏𝑏𝑏𝑖𝑖 = 𝑏𝑏𝑖𝑖 − 𝐿𝐿𝑖𝑖/2𝑎𝑎𝑏𝑗𝑗 = 𝑎𝑎𝑗𝑗 + 𝐿𝐿𝑗𝑗/2

𝑧𝑧𝑏𝑖𝑖 = 𝑝𝑝𝑖𝑖 − 𝐿𝐿𝑖𝑖/2𝑧𝑧𝑏𝑗𝑗 = 𝑝𝑝𝑗𝑗 + 𝐿𝐿𝑗𝑗/2⟺

𝑏𝑏𝑖𝑖 = 𝑎𝑎𝑗𝑗 ⟺ 𝑧𝑧𝑖𝑖 − 𝑧𝑧𝑗𝑗 = (𝐿𝐿𝑖𝑖 + 𝐿𝐿𝑗𝑗)/2

0. Stochastic Differential Equations

Trader number: 𝑵𝑵

Spread dist. : 𝜌𝜌(𝐿𝐿)

Transaction rule

Dist. 𝜌𝜌(𝐿𝐿)

𝑧𝑧𝑏𝑖𝑖 = 𝑧𝑧𝑖𝑖 − 𝐿𝐿𝑖𝑖/2𝑧𝑧𝑏𝑗𝑗 = 𝑧𝑧𝑗𝑗 + 𝐿𝐿𝑗𝑗/2𝑧𝑧𝑖𝑖 − 𝑧𝑧𝑗𝑗 =

𝐿𝐿𝑖𝑖 + 𝐿𝐿𝑗𝑗2

Asymptotic solution for 𝑵𝑵 → ∞via BBGKY equation

𝑑𝑑𝑧𝑧𝑖𝑖𝑑𝑑𝑐𝑐 = 𝑐𝑐tanh Δ𝑝𝑝/Δ𝑝𝑝∗ + 𝜎𝜎𝜂𝜂𝑖𝑖𝑅𝑅 + 𝜂𝜂𝑖𝑖𝐸𝐸

Requotation

𝐿𝐿𝑖𝑖

𝐿𝐿𝑗𝑗

Volume

Bid order-book Ask order-book

Price

1. “Liouville” equation for the trader model

Master equation for the N-body distribution

Corresponding to “Liouville” equation in the kinetic theory

𝑃𝑃𝑁𝑁 𝑟𝑟1, … , 𝑟𝑟𝑁𝑁 = 𝑁𝑁-body dist.̅𝑧𝑧 ≡ �

𝑖𝑖

𝑁𝑁�̂�𝑧𝑁𝑁

= Center of mass

�̂�𝑟𝑛𝑛 ≡ �̂�𝑧𝑛𝑛 − ̅𝑧𝑧 = relative position

2. “BBGKY” hierarchy for the trader model:Conditional distributions on spreads

Exact equation, but not closed…

As a next step, an approximation is necessary…

= 1-body dist. for 1 trader with spread 𝐿𝐿

= 2-body dist. for 2 traders with spread 𝐿𝐿 & 𝐿𝐿𝑏

Corresponding to the collision integral

3. Boltzmann-like equation for one-body dist.

Mean-field equation based on the “molecular chaos.”

Closed equation for 1-body dist. 𝜙𝜙𝐿𝐿(𝑟𝑟)

𝜙𝜙𝐿𝐿∗ 𝑟𝑟 ≡ lim𝑁𝑁→∞

𝜙𝜙𝐿𝐿 𝑟𝑟 = 𝐿𝐿/2 − 𝑟𝑟 /(𝐿𝐿/2)2 ( 𝑟𝑟 ≤ 𝐿𝐿/2)0 ( 𝑟𝑟 > 𝐿𝐿/2)

Tent function

Average order-book distribution

𝜌𝜌 𝐿𝐿 = 𝛿𝛿(𝐿𝐿 − 𝐿𝐿∗) 𝜌𝜌 𝐿𝐿 =𝛿𝛿 𝐿𝐿 − 𝐿𝐿∗ + 𝛿𝛿 𝐿𝐿 − 2𝐿𝐿∗

2

𝑓𝑓𝐴𝐴 𝑟𝑟 = 𝑓𝑓𝐵𝐵 −𝑟𝑟 = �−𝐿𝐿min

𝐿𝐿max𝑑𝑑𝐿𝐿𝜌𝜌 𝐿𝐿 𝜙𝜙𝐿𝐿∗(𝑟𝑟 − 𝐿𝐿)

Average order-book given by the 𝜌𝜌 𝐿𝐿 -weighed superposition of tent functions

Mesoscopic hierarchy: Average order-book profile

Solution (Mean field):

Numerical validation (𝑁𝑁 = 1000)

𝜌𝜌 𝐿𝐿 =𝐿𝐿3𝑒𝑒−𝐿𝐿/𝐿𝐿∗

6𝐿𝐿∗4 ⟹ 𝑓𝑓𝐴𝐴 𝑟𝑟 =4

3𝐿𝐿∗ 𝑒𝑒−3𝑟𝑟2𝐿𝐿∗ 2 +

𝑟𝑟𝐿𝐿∗ sinh

𝑟𝑟2𝐿𝐿∗ −

𝑟𝑟2𝐿𝐿∗ 𝑒𝑒

− 𝑟𝑟2𝐿𝐿∗

𝑓𝑓𝐴𝐴 𝑟𝑟

𝑟𝑟

Agreement with data without fitting parameters

Macroscopic hierarchy:Exponential price movement distribution 𝑃𝑃(Δ𝑃𝑃)

Solution (phenomenological): Exponential tail 𝑃𝑃(≥ |Δ𝑝𝑝|)

𝑃𝑃 ≥ |𝛥𝛥𝑃𝑃| ∼ 𝑒𝑒−3 Δ𝑃𝑃2Δ𝑧𝑧∗ , Δ𝑧𝑧∗ ≡ 𝑐𝑐 𝜏𝜏 , 𝜏𝜏 ≡

3𝑁𝑁𝜎𝜎2

Numerical validation (𝑁𝑁 = 1000)

𝑃𝑃 ≥ |Δ𝑃𝑃|

Δ𝑃𝑃

Exponential dists. in data

Note #1: Scaling for price movements

Exponential dists. Every two-hours→Decay length 𝜅𝜅 varies over time

Single master curve when scaled for horizontal and vertical axises

𝑃𝑃2h ≥ Δ𝑝𝑝 ; 𝜅𝜅 ∼ e− Δ𝑝𝑝 𝜅𝜅 → �𝑃𝑃2h ≥ Δ �𝑝𝑝 ∼ 𝑒𝑒−|Δ �𝑝𝑝|

Note #2: Relation to power-law dist.

Power-law for price movements? (𝛼𝛼 = 3.6 ± 0.13)𝑃𝑃w ≥ Δ𝑝𝑝 ∼ Δ𝑝𝑝 −𝛼𝛼

Power-law for one-week = Superposition of exponential dist.

Indeed, decay length 𝜅𝜅 obeys powar-law (𝑚𝑚 = 3.5 ± 0.13)𝑄𝑄 ≥ 𝜅𝜅 ∼ 𝜅𝜅−𝑚𝑚 → 𝑃𝑃w Δ𝑝𝑝 = �𝑑𝑑𝜅𝜅𝑄𝑄 𝜅𝜅 𝑃𝑃2ℎ ≥ Δ𝑝𝑝 ;𝜅𝜅 ∼ Δ𝑝𝑝 −𝑚𝑚

Note #3: Candidates of origin of power-law?(Variation of numbers of traders?)

Why does decay length 𝜅𝜅 distribute according to power-law?

Related to intraday activity of market?(Non-stationary because traders are sleeping in midnight…)→𝜅𝜅 becomes longest just after opening of the market (Sunday midnight in London)

Trader 𝑁𝑁 vs. decay length 𝜅𝜅 (Spearman correlation 𝜌𝜌 = 0.3）Trader 𝑁𝑁 vs. mean movement |Δ𝑝𝑝| (Spearman correlation 𝜌𝜌 = 0.7）

𝑃𝑃(Δ �𝑝𝑝) ∼ 𝑒𝑒−Δ �𝑝𝑝

Significance of our study:From microscopic to macroscopic laws systematically

𝑓𝑓𝐴𝐴 𝑟𝑟 =4𝑒𝑒−

3𝑟𝑟2𝐿𝐿∗

3𝐿𝐿∗ 2 +𝑟𝑟𝐿𝐿∗ sinh

𝑟𝑟2𝐿𝐿∗ −

𝑟𝑟𝑒𝑒−𝑟𝑟2𝐿𝐿∗

2𝐿𝐿∗

𝑑𝑑𝑧𝑧𝑖𝑖𝑑𝑑𝑐𝑐

= 𝑐𝑐tanhΔ𝑝𝑝Δ𝑝𝑝∗

+𝜎𝜎𝜂𝜂𝑖𝑖𝑅𝑅 + 𝜂𝜂𝑖𝑖𝐸𝐸

Summary of this presentation

Dynamics of individual traders:1. Two-sided quote with stable spread

2. Trend-following

Trend-following random walk model

Asymptotic analysis for 𝑁𝑁 → ∞(based on the kinetic formulation)

From micro to meso & macro:1. Meso: average order-book

2. Macro: exponential price movement dist.

Power-law as superposition of exponential

reduce

𝑃𝑃(Δ �𝑝𝑝) ∼ 𝑒𝑒−Δ �𝑝𝑝𝑓𝑓𝐴𝐴 𝑟𝑟 =4𝑒𝑒−

3𝑟𝑟2𝐿𝐿∗

3𝐿𝐿∗ 2 +𝑟𝑟𝐿𝐿∗ sinh

𝑟𝑟2𝐿𝐿∗ −

𝑟𝑟𝑒𝑒−𝑟𝑟2𝐿𝐿∗

2𝐿𝐿∗

𝑑𝑑𝑧𝑧𝑖𝑖𝑑𝑑𝑐𝑐

= 𝑐𝑐tanhΔ𝑝𝑝Δ𝑝𝑝∗

+𝜎𝜎𝜂𝜂𝑖𝑖𝑅𝑅 + 𝜂𝜂𝑖𝑖𝐸𝐸

Thank you for your attention!