title: Musings on Mathematical Finance author: Keith A. Lewis institution: KALX, LLC email: [email protected] classoption: fleqn abstract: Waiting for Heisenberg, Schrodinger, and von Neumann ...
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Much of current academic teaching and research has been criticized for its lack of relevance, that is, of immediate practical impact. ... The trouble is caused, however, not by an inadequate selection of targets, but rather by our inability to hit squarely on them, ... by the palpable inadequacy of the scientific means with which they try to solve them. ... The weak and all too slowly growing empirical foundations clearly cannot support the proliferating superstructure of pure, or should I say, speculative economic theory.... By the time it comes to interpretations of the substantive conclusions, the assumptions on which the model has been based are easily forgotten. But it is precisely the empirical validity of these assumptions on which the usefulness of the entire exercise depends. ... A natural Darwinian feedback operating through selection of academic personnel contributes greatly to the perpetuation of this state of affairs. -- Василий Васильевич Леонтьев
The Nobel Prize winning work of Fischer Black, Myron Scholes, and Robert Merton had a profound influence on the application of Mathematical Finance to the valuation of financial instruments. They showed the probability of future prices were irrelevant to valuing derivatives.
real-world
hedge - risk managment - How's my hedging?
Prize said value, but value comes from hedge.
In Physics when a theory does not fit observation, it is time to come up with a new theory that does. In Mathematical Finance the problems to solve are related to Economics and less tractable.
UV catastrophy and hydrogen atom Leontiev won a Nobel prize for his model pioneering work
In physics, when a theory does not match observation it is time to come up with a new theory. This happened at the end of the 19th century when physicists were getting a bit smug about the success of Newtow's theory of gravity and Maxwell's theory of electromagnatism. Some attribute "there is nothing left but to carry measurement to a few more decimal places" to Lord Kelvin. However there were two nagging problems: the ultraviolet catastrophy and the spectrum of hydrogen atoms.
Max Plank could fit observed black-body radiation by assuming photons were emitted in discrete packages. Rydberg noted the frequency of radiation from hydrogen atoms was proportional to differences of the reciprocals of squares of integers. The Bohr model of the hydrogen atom assumed electrons orbited at distances equal to integer multiples of the Plack constant divided by 2π and this agreed with Rydberg's empirical formula. What could not be explained was how electons can orbit a nucleus as if it were a planet without emitting radiation. Maxwell's theory said an accelerating electron must do that.
Heisenberg and Schrodinger stepped up to provide the initial theories of quantum mechanics. Heisenberg's original paper involved a clumsy reinvention of matrix multiplication (later ... by Jordan) while Schrodinger took a more principled approach based on the classical equations of motion due to Lagrange and Hamilton. Eventually John von Neumann reconciled these by showing both were identical to a theory involving unbounded self-adjoint operators on Hilbert spaces.
Mathematical Finance is still at its 19th century fin de siècle stage. We have had our Nobel prize winners ... But we are still waiting for our Heisenberg and Schrodinger. We will probably never have another von Neumann.
Shortly after Black, Scholes, and Merton \cite{BSM73} Merton \cite{Mer76} used the reflection principal of Brownian motion to derive a closed form solution for the value of a barrier option. When used to value a barrier option that knocks in the second time the barrier is touched it gives the wrong anser. The theory says it should have the same value as a barrier option that knocks in the first time it is touched. In fact, the theory says a barrier option that knocks in the millionth time the barrier is touched also has the same value.
This is a mathematical artifact of using geometric Brownian motion as the model of stock prices and the unrealistic assumption that continuous time hedging is possible.
https://www.nobelprize.org/prizes/economic-sciences/1997/press-release/
for a new method to determine the value of derivatives.
Black, Merton and Scholes thus laid the foundation for the rapid growth of markets for derivatives in the last ten years. Their method has more general applicability, however, and has created new areas of research – inside as well as outside of financial economics. A similar method may be used to value insurance contracts and guarantees, or the flexibility of physical investment projects.
Turned back of the envelope into math.
value only because hedge is perfect in their moddel.
Quesney (tableau) and Walras equilibrium theory.
Gosplan.