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4 results

Hydrodynamics of Markets

Hydrodynamics of Markets
  • Hidden Links between Physics and Finance
  • Alexander Lipton
  • Published online:
    26 November 2024
    Print publication:
    02 January 2025
    • Element
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      • Open access
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  • An intriguing link between a wide range of problems occurring in physics and financial engineering is presented. These problems include the evolution of small perturbations of linear flows in hydrodynamics, the movements of particles in random fields described by the Kolmogorov and Klein-Kramers equations, the Ornstein-Uhlenbeck and Feller processes, and their generalizations. They are reduced to affine differential and pseudo-differential equations and solved in a unified way by using Kelvin waves and developing a comprehensive math framework for calculating transition probabilities and expectations. Kelvin waves are instrumental for studying the well-known Black-Scholes, Heston, and Stein-Stein models and more complex path-dependent volatility models, as well as the pricing of Asian options, volatility and variance swaps, bonds, and bond options. Kelvin waves help to solve several cutting-edge problems, including hedging the impermanent loss of Automated Market Makers for cryptocurrency trading. This title is also available as Open Access on Cambridge Core.

  • 1 - An Overview of Numerical Weather Prediction

  • Eugenia Kalnay, University of Maryland, College Park, Safa Mote, University of Maryland, College Park, Cheng Da, University of Maryland, College Park
  • Book:
    Earth System Modeling, Data Assimilation and Predictability
    Published online:
    21 March 2025
    Print publication:
    31 October 2024, pp 1-34

  • Summary

    Bjerknes pointed out for the first time that the evolution of the atmosphere is governed by a complete set of seven equations with seven unknowns. If we know both the equations and the initial conditions with sufficient accuracy, we can predict the weather. In this chapter, we first introduce the continuous equations that govern the atmosphere. From these equations, we then discuss several fundamental wave oscillations existing in the atmosphere and their filtering approximations. For comparison, we then introduce the primitive equations for the oceans and discuss the Kelvin and Equatorially trapped waves, a special kind of waves that can appear in both the atmosphere and the ocean.

  • 2 - The Continuous Equations

  • Eugenia Kalnay, University of Maryland, College Park, Safa Mote, University of Maryland, College Park, Cheng Da, University of Maryland, College Park
  • Book:
    Earth System Modeling, Data Assimilation and Predictability
    Published online:
    21 March 2025
    Print publication:
    31 October 2024, pp 35-75

  • Summary

    Bjerknes pointed out for the first time that the evolution of the atmosphere is governed by a complete set of seven equations with seven unknowns. If we know both the equations and the initial conditions with sufficient accuracy, we can predict the weather. In this chapter, we first introduce the continuous equations that govern the atmosphere. From these equations, we then discuss several fundamental wave oscillations existing in the atmosphere and their filtering approximations. For comparison, we then introduce the primitive equations for the oceans and discuss the Kelvin and Equatorially trapped waves, a special kind of waves that can appear in both the atmosphere and the ocean.

  • 5 - Tidal Wave Propagation

  • Theo Gerkema, Royal Netherlands Institute for Sea Research
  • Book:
    An Introduction to Tides
    Published online:
    08 June 2019
    Print publication:
    20 June 2019, pp 87-121

  • Summary

    The propagation of waves at tidal frequencies is studied analytically for simple configurations involving a wall, a channel, semi-enclosed basins, or a continental slope with adjacent shelf sea. The equations of motions are presented, and are simplified using the linear and hydrostatic approximations. The fundamental wave types (Poincaré and Kelvin waves) are derived. The appearance of amphidromic points is explained. A detailed analysis is provided of the solution of the Taylor problem in the case of perfect reflection. The parameter space is explored for modified Kelvin waves in the presence of a shelf sea. As a special case, the double Kelvin wave is obtained.