Math in finance or vice versa

Quantum Mechanics and Market Observables

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Quantum mechanics has quite powerful notion of observables. They are just properties of system state that can be determined by measurement process. Let’s recall main principles of quantum mechanics and how they could be applied to markets.

Quantization principle

Observables in quantum mechanics are represented by operators on a Hilbert space. However, not all mathematical operators are suitable for representing observables. Since the values of observables have to represent real quantities, like coordinate and momentum in physics, price and volume in markets, the operator associated to a quantum observable must be a Hermitian operator. An operator \hat{O} said to be Hermitian when \hat{O}=\left(\hat{O}^* \right)^T.

Hermitian operator could be represented by it’s spectrum or eigenvalues and eigenvectors:
\hat{O} = \sum_j {o_j \left| o_j \right> \left< o_j \right|}
where o_j is eigenvalue and \left| o_j \right> is eigenvector.

Price representation is subject of different regulations. Notably, price decimalization is one of things had influenced markets last time. But price itself remains discrete but theoretically infinite. Therefore, quantum operator of fair price must be defined on infinite-dimensional Hilbert space and must have discrete spectrum.
Non-injective operators have only discrete spectrum. Operator is said to be injective if and only if for all \left| a \right> and \left| b \right> from operator domain, if \hat{O} \left| a \right> = \hat{O} \left| b \right> then \left| a \right> = \left| b \right> This property should be used to construct fair price operator.

The following observables are present at market:

  • Buy/sell distribution by price, i.e. volume by price – \hat{P}
  • Fair price of asset – \hat{F}
  • Implied volatility – \hat{V}
  • Statistical algorithm

    Given that a quantum system is completely defined by a vector \left| \psi \right> (Eq. (2.15)), the probability of having a determinate measurement result – an eigenvalue o_k of the measured observable \hat{O} – is given by
    p(o_k,\psi) = \left| \left< o_k | \psi \right> \right|^2

    What it gives us is a method for establishing the link between order book properties and observable. Thus order book provides buy/sell distribution by price \hat{P}.

    Simultaneous measurability

    The necessary and sufficient condition for two observables \hat{A} and \hat{B}  to be simultaneously measurable with arbitrary precision is that they commute:
    \left[\hat{A},\hat{B}\right] = \hat{A}\hat{B} - \hat{B}\hat{A} = 0

    As order book is visible for any market participant and fair price and implied volatility are not measurable, then their respective observables must not commute:
    \left[\hat{P},\hat{F}\right] \neq 0
    \left[\hat{P},\hat{V}\right] \neq 0

    Written by fxpaul

    January 15, 2011 at 10:53

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