quantum coupled-wave model of price formation

Quantum coupled-wave model of price formationPhysics of Financial Markets Series

Jack Sarkissian, PhDAlgostox Trading

[email protected]© Algostox Trading LLC. All rights reserved.

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2Market maker’s problem

• Market makers (MM) provide liquidity and profit from bid-ask spread• When MM’s quote is matched the MM forms a position (inventory)• Inventory carries market risk (residual risk)• MMs try to close the position at profit as soon as possible• Understanding spread behavior is crucial to MMs• Will allow to price securities when liquidity is essential• Will allow to manage risks of illiquid securities

© Algostox Trading LLC. All rights reserved.

3How does price form?

Price formation There is a spectrum of attainable prices Price is localized between bid and ask levels Single value is selected from spectrum at the time of transaction Each transaction is an elementary act of price measurement

Common points with two-level quantum systems Interaction and transfer between price levels Bandgap between bid and ask and noncrossing terms (Wigner-

von-Neumann theorem) Fluctuation of coupling coefficients

This points at possibility to apply quantum framework to price formation

Ask size Price Bid size100 27.87 1000 27.90 100 27.95 300 28.15 100 28.20

27.83 100 27.82 400 27.80 200 27.79 1200 27.78 100

Order book


4Quantum framework for price

Security state is described by probability amplitude:

States are eigenfunctions and prices are eigenvalues of the price operator:

For two-level system and

Price operator fluctuates in time:


5Spread as a bandgap

and Ask size Price Bid size100 27.87 1000 27.90 100 27.95 300 28.15 100 28.20

27.83 100 27.82 400 27.80 200 27.79 1200 27.78 100

Order book


6Price simulation in coupled-wave modelSpread parametrization,

,

, , and .

Coupled-wave model

, , and

or

0

20

40

60

80

100

120

140

160

180

0 20 40 60 80 100

pric

e

time

Price simulation in coupled-wave model

ask bid close


7Data sources for the model

Order book: NBBO Order book: effective bid/ask

OHLC bar time series


8Calibration to bid-ask spread

AAPL AMZN

GOOG GAZP

Relative spread (bp)

Ticker ξ1 κ0 κ1

AAPL 1.1 2.3 1.6AMZN 3.2 4.8 2.7GOOG 4.1 3.3 2.1INTC 4.2 5.5 0.7MSFT 2.6 3.2 0.16GAZP (MoEx) 0.2 3.5 1.7


9Calibration to bar data

AAPL AMZN

GOOG GAZP

High-low (relative)

Ticker ξ1 κ0 κ1

AAPL 1.75% 1.31% 0.51%AMZN 1.53% 1.49% 0.45%GOOG 1.24% 1.02% 0.34%INTC 1.39% 1.11% 0.31%MSFT 1.31% 1.10% 0.39%GAZP (MoEx) 1.73% 1.30% 0.35%


10Risk management

Coupled-wave model also allows to manage risk of illiquid securities

Risk sources:• Uncertainty of mid-price• Uncertainty of spread / high-low bar size


11Conclusions

• Every financial transaction is an elementary act of price measurement

• Price is an eigenvalue of price operator with fluctuating matrix elements

• Bid and ask prices behave like quantum chaotic quantities

• Model can be calibrated to simulate bid-ask, effective bid/ask, or OHLC prices

• Provides capabilities for pricing illiquid securities and managing their risks


12Notice

Please cite as: J. Sarkissian, “Quantum coupled-wave model of price formation (Physics of Financial Markets Series)”, released by Algostox Trading, June 1, 2016. Available from www.algostox.com.

Copyright © Algostox Trading LLC. All rights reserved. Federal law prohibits the unauthorized reproduction, distribution or exhibition of the materials. Violations of copyright law will be prosecuted.

These materials are released for informational purposes only. No representation or warranty, express or implied, is made by Algostox Trading LLC as to the accuracy, completeness, or fitness for any particular purpose of enclosed materials. Under no circumstances shall Algostox Trading LLC have any liability to any other person or any entity for (a) any loss, damage or other injury in whole or in part caused by, resulting from or relating to, any error (negligent or otherwise), of Algostox Trading LLC in connection with the compilation, analysis, interpretation, communication, publication or delivery of enclosed materials, or (b) any direct, indirect, special, consequential, incidental or compensatory damages whatsoever (including, without limitation, lost profits), in either case caused by reliance upon or otherwise resulting from or relating to the use of (including the inability to use) enclosed materials. The enclosed materials do not constitute or imply any financial advice. No recommendation is being made to buy or sell any securities.


quantum coupled-wave model of price formation

Economy & Finance