Path Integrals and Financial Markets

The following sections are included:

• Fluctuation Properties of Financial Assets

  • Harmonic Approximation to Fluctuations

  • Lévy Distributions

  • Truncated Lévy Distributions

  • Student or Tsallis Distribution

  • Asymmetric Truncated Lévy Distributions

  • Boltzmann Distribution

  • Meixner Distributions

  • Generalized Hyperbolic Distributions

  • Lévy-Khintchine Formula

  • Debye-Waller Factor for Non-Gaussian Fluctuations

  • Path Integral for Non-Gaussian Distribution

  • Semigroup Property of Asset Distributions

  • Time Evolution of Distribution

  • Fokker-Planck-Type Equation

• Martingales

  • Gaussian Martingales

  • Non-Gaussian Martingales

• Origin of Heavy Tails

  • Pair of Stochastic Differential Equations

  • Fokker-Planck Equation

  • Solution of Fokker-Planck Equation

  • Pure x-Distribution

  • Long-Time Behavior

  • Tail Behavior for all Times

  • Path Integral Calculation

  • Natural Martingales

• Spectral Decomposition of Power Behaviors

• Option Pricing

  • Black-Scholes Option Pricing Model

  • Evolution Equations of Portfolios with Options

  • Option Pricing for Gaussian Fluctuations

  • Option Pricing for Boltzmann Distribution

  • Option Pricing for General Non-Gaussian Fluctuations

  • Option Pricing for Fluctuating Variance

  • Perturbation Expansion and Smile

• Appendix 20A Large-x Behavior of Truncated Lévy Distribution

• Appendix 20B Gaussian Weight

• Appendix 20C Comparison with Dow-Jones Data

• Notes and References