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Efficient Markets and Contingent Claims Valuation: An Information Theoretic Approach.

Jussi Lindgren1

  • 1Department of Mathematics and Systems Analysis, Aalto University, 02150 Espoo, Finland.

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Summary

This study links derivative pricing to information theory and stochastic control. It reveals that financial markets, when viewed as information processors, align with Black-Scholes pricing and thermodynamic principles.

Keywords:
Black–Scholes equationBurgers equationentropyfinancial marketsfree energyinformation theoryoptions pricingstochastic optimal control

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Area of Science:

  • Financial mathematics
  • Stochastic calculus
  • Information theory

Background:

  • Derivative pricing models are crucial in financial markets.
  • Existing models often lack a unified theoretical framework connecting finance, control theory, and information theory.
  • Understanding market dynamics through an information-processing lens is an emerging area.

Purpose of the Study:

  • To develop a novel framework for derivative pricing using stochastic optimal control and information theory.
  • To demonstrate the connection between the Hamilton-Jacobi-Bellman equation and the Black-Scholes pricing model.
  • To explore the thermodynamic equilibrium of financial markets.

Main Methods:

  • Formulating derivative pricing within stochastic optimal control theory.
  • Applying information theory to model financial markets as information processing systems.
  • Deriving the Black-Scholes equation from a linearized Hamilton-Jacobi-Bellman equation.
  • Investigating market drift using optimal transport equations and the backwards Burgers equation.
  • Analyzing market behavior in terms of thermodynamic equilibrium, free energy, and entropy.

Main Results:

  • The pricing of derivative securities can be unified under stochastic optimal control and information theory.
  • The linearized Hamilton-Jacobi-Bellman equation corresponds to the Black-Scholes pricing equation.
  • A Hamiltonian for financial markets leads to an optimal transport equation for market drift.
  • Market drift follows a backwards Burgers equation within this framework.
  • The financial market tends towards a thermodynamical equilibrium, minimizing free energy and maximizing entropy.

Conclusions:

  • This research provides a new theoretical foundation for derivative pricing by integrating stochastic control and information theory.
  • The Black-Scholes model emerges naturally from this unified framework, supported by thermodynamic principles.
  • The findings suggest that financial markets exhibit behaviors analogous to physical systems reaching equilibrium.