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How are the notions of randomness, i.e., stochastic processes, linked to theories in physics and what have they got to do with options pricing in economics?
How did the prevailing world view change from 1900 to 1905?
What connects the mathematicians Bachelier, Markov, Kolmogorov, Ito to the physicists Langevin, Fokker, Planck, Einstein and the economists Black, Scholes, Merton?

The Setting

  • Science up to 1900 was in essence the study of solutions of differential equations (Newton's heritage);
  • Was very successful, e.g., Maxwell's equations: four differential equations describing everything about (classical) electromagnetism;
  • Prevailing world view:
    • Deterministic universe;
    • Initial conditions plus the solution of differential equation yield certain prediction of the future.

Three Pillars

By the end of the 20th Century, it became clear that there are (at least?) two additional aspects needed in a completer understanding of reality:
  • Inherent randomness: statistical evaluations of sets of outcomes of single observations/experiments;
    • Quantum mechanics (Planck 1900; Einstein 1905) contains a fundamental element of randomness;
    • In chaos theory (e.g., Mandelbrot 1963) non-linear dynamics leads to a sensitivity to initial conditions which renders even simple differential equations essentially unpredictable;
  • Complex systems (e.g., Wolfram 1983), i.e., self-organization and emergent behavior, best understood as outcomes of simple rules.

Stochastic Processes

  • Systems which evolve probabilistically in time;
  • Described by a time-dependent random variable;
  • The probability density function describes the distribution of the measurements at time t;
  • Prototype: The Markov process.
For a Markov process, only the present state of the system influences its future evolution: there is no long-term memory. Examples:
  • Wiener process or Einstein-Wiener process or Brownian motion:
    • Introduced by Bachelier in 1900;
    • Continuous (in t and the sample path)
    • Increments are independent and drawn from a Gaussian normal distribution;
  • Random walk:
    • Discrete steps (jumps), continuous in t;
    • Is a Wiener process in the limit of the step size going to zero.
To summarize, there are three possible characteristics:
  1. Jumps (in sample path);
  2. Drift (of the probability density function);
  3. Diffusion (widening of the probability density function).
Probability distribution function showing drift and diffusion: Probability distribution function with drift and diffusion But how to deal with stochastic processes?

The Micro View

Einstein:
  • Presented a theory of Brownian motion in 1905;
  • New paradigm: stochastic modeling of natural phenomena; statistics as intrinsic part of the time evolution of system;
  • Mean-square displacement of Brownian particle proportional to time;
  • Equation for the Brownian particle similar to a diffusion (differential) equation.
Langevin:
  • Presented a new derivation of Einstein's results in 1908;
  • First stochastic differential equation, i.e., a differential equation of a "rapidly and irregularly fluctuating random force" (today called a random variable)
  • Solutions of differential equation are random functions.
However, no formal mathematical grounding until 1942, when Ito developed stochastic calculus:
  • Langevin's equations interpreted as Ito stochastic differential equations using Ito integrals;
  • Ito integral defined to deal with non-differentiable sample paths of random functions;
  • Ito lemma (generalized integration rule) used to solve stochastic differential equations.
Note:
  • The Markov process is a solution to a simple stochastic differential equation;
  • The celebrated Black-Scholes option pricing formula is a stochastic differential equation employing Brownian motion.

The Fokker-Planck Equation: Moving To The Macro View

  • The Langevin equation describes the evolution of the position of a single "stochastic particle";
  • The Fokker-Planck equation describes the behavior of a large population of of "stochastic particles";
    • Formally: The Fokker-Planck equation gives the time evolution of the probability density function of the system as a function of time;
  • Results can be derived more directly using the Fokker-Planck equation than using the corresponding stochastic differential equation;
  • The theory of Markov processes can be developed from this macro point of view.

The Historical Context

Bachelier

  • Developed a theory of Brownian motion (Einstein-Wiener process) in 1900 (five years before Einstein, and long before Wiener);
  • Was the first person to use a stochastic process to model financial systems;
  • Essentially his contribution was forgotten until the late 1950s;
  • Black, Scholes and Merton's publication in 1973 finally gave Brownian motion the break-through in finance.

Planck

  • Founder of quantum theory;
  • 1900 theory of black-body radiation;
  • Central assumption: electromagnetic energy is quantized, E = h v;
  • In 1914 Fokker derives an equation on Brownian motion which Planck proves;
  • Applies the Fokker-Planck equation as quantum mechanical equation, which turns out to be wrong;
  • In 1931 Kolmogorov presented two fundamental equations on Markov processes;
  • It was later realized, that one of them was actually equivalent to the Fokker-Planck equation.

Einstein

1905 "Annus Mirabilis" publications. Fundamental paradigm shifts in the understanding of reality:
  • Photoelectric effect:
    • Explained by giving Planck's (theoretical) notion of energy quanta a physical reality (photons),
    • Further establishing quantum theory,
    • Winning him the Nobel Prize;
  • Brownian motion:
    • First stochastic modeling of natural phenomena,
    • The experimental verification of the theory established the existence of atoms, which had been heavily debate at the time,
    • Einstein's most frequently cited paper, in the fields of biology, chemistry, earth and environmental sciences, life sciences, engineering;
  • Special theory of relativity: the relative speeds of the observers' reference frames determines the passage of time;
  • Equivalence of energy and mass (follows from special relativity): E = m c^2.
Einstein was working at the Patent Office in Bern at the time and submitted his Ph.D. to the University of Zurich in July 1905. Later Work:
  • 1915: general theory of relativity, explaining gravity in terms of the geometry (curvature) of space-time;
    • Planck also made contributions to general relativity;
  • Although having helped in founding quantum mechanics, he fundamentally opposed its probabilistic implications: "God does not throw dice";
  • Dreams of a unified field theory:
    • Spend his last 30 years or so trying to (unsuccessfully) extend the general theory of relativity to unite it with electromagnetism;
    • Kaluza and Klein elegantly managed to do this in 1921 by developing general relativity in five space-time dimensions;
    • Today there is still no empirically validated theory able to explain gravity and the (quantum) Standard Model of particle physics, despite intense theoretical research (string/M-theory, loop quantum gravity);
    • In fact, one of the main goals of the LHC at CERN (officially operational on the 21st of October 2008) is to find hints of such a unified theory (supersymmetric particles, higher dimensions of space).
This was originally posted on Wednesday, September 3, 2008 in my tech blog...
 

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