# Derivative Pricing Theory: Call, Put Options and “Black, Scholes'” Hedged Portfolio.Thought of the Day 152.0

Fischer Black and Myron Scholes revolutionized the pricing theory of options by showing how to hedge continuously the exposure on the short position of an option. Consider the writer of a call option on a risky asset. S/he is exposed to the risk of unlimited liability if the asset price rises above the strike price. To protect the writer’s short position in the call option, s/he should consider purchasing a certain amount of the underlying asset so that the loss in the short position in the call option is offset by the long position in the asset. In this way, the writer is adopting the hedging procedure. A hedged position combines an option with its underlying asset so as to achieve the goal that either the asset compensates the option against loss or otherwise. By adjusting the proportion of the underlying asset and option continuously in a portfolio, Black and Scholes demonstrated that investors can create a riskless hedging portfolio where the risk exposure associated with the stochastic asset price is eliminated. In an efficient market with no riskless arbitrage opportunity, a riskless portfolio must earn an expected rate of return equal to the riskless interest rate.

Black and Scholes made the following assumptions on the financial market.

1. Trading takes place continuously in time.
2. The riskless interest rate r is known and constant over time.
3. The asset pays no dividend.
4. There are no transaction costs in buying or selling the asset or the option, and no taxes.
5. The assets are perfectly divisible.
6. There are no penalties to short selling and the full use of proceeds is permitted.
7. There are no riskless arbitrage opportunities.

The stochastic process of the asset price St is assumed to follow the geometric Brownian motion

dSt/St = μ dt + σ dZt —– (1)

where μ is the expected rate of return, σ is the volatility and Zt is the standard Brownian process. Both μ and σ are assumed to be constant. Consider a portfolio that involves short selling of one unit of a call option and long holding of Δt units of the underlying asset. The portfolio value Π (St, t) at time t is given by

Π = −c + Δt St —– (2)

where c = c(St, t) denotes the call price. Note that Δt changes with time t, reflecting the dynamic nature of hedging. Since c is a stochastic function of St, we apply the Ito lemma to compute its differential as follows:

dc = ∂c/∂t dt + ∂c/∂St dSt + σ2/2 St2 ∂2c/∂St2 dt

such that

-dc + Δt dS= (-∂c/∂t – σ2/2 St2 ∂2c/∂St2)dt + (Δ– ∂c/∂St)dSt

= [-∂c/∂t – σ2/2 St2 ∂2c/∂St+ (Δ– ∂c/∂St)μSt]dt + (Δ– ∂c/∂St)σSdZt

The cumulative financial gain on the portfolio at time t is given by

G(Π (St, t )) = ∫0t -dc + ∫0t Δu dSu

= ∫0t [-∂c/∂u – σ2/2 Su22c/∂Su2 + (Δ– ∂c/∂Su)μSu]du + ∫0t (Δ– ∂c/∂Su)σSdZ—– (3)

The stochastic component of the portfolio gain stems from the last term, ∫0t (Δ– ∂c/∂Su)σSdZu. Suppose we adopt the dynamic hedging strategy by choosing Δu = ∂c/∂Su at all times u < t, then the financial gain becomes deterministic at all times. By virtue of no arbitrage, the financial gain should be the same as the gain from investing on the risk free asset with dynamic position whose value equals -c + Su∂c/∂Su. The deterministic gain from this dynamic position of riskless asset is given by

Mt = ∫0tr(-c + Su∂c/∂Su)du —– (4)

By equating these two deterministic gains, G(Π (St, t)) and Mt, we have

-∂c/∂u – σ2/2 Su22c/∂Su2 = r(-c + Su∂c/∂Su), 0 < u < t

which is satisfied for any asset price S if c(S, t) satisfies the equation

∂c/∂t + σ2/2 S22c/∂S+ rS∂c/∂S – rc = 0 —– (5)

This parabolic partial differential equation is called the Black–Scholes equation. Strangely, the parameter μ, which is the expected rate of return of the asset, does not appear in the equation.

To complete the formulation of the option pricing model, let’s prescribe the auxiliary condition. The terminal payoff at time T of the call with strike price X is translated into the following terminal condition:

c(S, T ) = max(S − X, 0) —– (6)

for the differential equation.

Since both the equation and the auxiliary condition do not contain ρ, one concludes that the call price does not depend on the actual expected rate of return of the asset price. The option pricing model involves five parameters: S, T, X, r and σ. Except for the volatility σ, all others are directly observable parameters. The independence of the pricing model on μ is related to the concept of risk neutrality. In a risk neutral world, investors do not demand extra returns above the riskless interest rate for bearing risks. This is in contrast to usual risk averse investors who would demand extra returns above r for risks borne in their investment portfolios. Apparently, the option is priced as if the rates of return on the underlying asset and the option are both equal to the riskless interest rate. This risk neutral valuation approach is viable if the risks from holding the underlying asset and option are hedgeable.

The governing equation for a put option can be derived similarly and the same Black–Scholes equation is obtained. Let V (S, t) denote the price of a derivative security with dependence on S and t, it can be shown that V is governed by

∂V/∂t + σ2/2 S22V/∂S+ rS∂V/∂S – rV = 0 —– (7)

The price of a particular derivative security is obtained by solving the Black–Scholes equation subject to an appropriate set of auxiliary conditions that model the corresponding contractual specifications in the derivative security.

The original derivation of the governing partial differential equation by Black and Scholes focuses on the financial notion of riskless hedging but misses the precise analysis of the dynamic change in the value of the hedged portfolio. The inconsistencies in their derivation stem from the assumption of keeping the number of units of the underlying asset in the hedged portfolio to be instantaneously constant. They take the differential change of portfolio value Π to be

dΠ =−dc + Δt dSt,

which misses the effect arising from the differential change in Δt. The ability to construct a perfectly hedged portfolio relies on the assumption of continuous trading and continuous asset price path. It has been commonly agreed that the assumed Geometric Brownian process of the asset price may not truly reflect the actual behavior of the asset price process. The asset price may exhibit jumps upon the arrival of a sudden news in the financial market. The interest rate is widely recognized to be fluctuating over time in an irregular manner rather than being constant. For an option on a risky asset, the interest rate appears only in the discount factor so that the assumption of constant/deterministic interest rate is quite acceptable for a short-lived option. The Black–Scholes pricing approach assumes continuous hedging at all times. In the real world of trading with transaction costs, this would lead to infinite transaction costs in the hedging procedure.

# Symmetrical – Asymmetrical Dialectics Within Catastrophical Dynamics. Thought of the Day 148.0

Catastrophe theory has been developed as a deterministic theory for systems that may respond to continuous changes in control variables by a discontinuous change from one equilibrium state to another. A key idea is that system under study is driven towards an equilibrium state. The behavior of the dynamical systems under study is completely determined by a so-called potential function, which depends on behavioral and control variables. The behavioral, or state variable describes the state of the system, while control variables determine the behavior of the system. The dynamics under catastrophe models can become extremely complex, and according to the classification theory of Thom, there are seven different families based on the number of control and dependent variables.

Let us suppose that the process yt evolves over t = 1,…, T as

dyt = -dV(yt; α, β)dt/dyt —– (1)

where V (yt; α, β) is the potential function describing the dynamics of the state variable ycontrolled by parameters α and β determining the system. When the right-hand side of (1) equals zero, −dV (yt; α, β)/dyt = 0, the system is in equilibrium. If the system is at a non-equilibrium point, it will move back to its equilibrium where the potential function takes the minimum values with respect to yt. While the concept of potential function is very general, i.e. it can be quadratic yielding equilibrium of a simple flat response surface, one of the most applied potential functions in behavioral sciences, a cusp potential function is defined as

−V(yt; α, β) = −1/4yt4 + 1/2βyt2 + αyt —– (2)

with equilibria at

-dV(yt; α, β)dt/dyt = -yt3 + βyt + α —– (3)

being equal to zero. The two dimensions of the control space, α and β, further depend on realizations from i = 1 . . . , n independent variables xi,t. Thus it is convenient to think about them as functions

αx = α01x1,t +…+ αnxn,t —– (4)

βx = β0 + β1x1,t +…+ βnxn,t —– (5)

The control functions αx and βx are called normal and splitting factors, or asymmetry and bifurcation factors, respectively and they determine the predicted values of yt given xi,t. This means that for each combination of values of independent variables there might be up to three predicted values of the state variable given by roots of

-dV(yt; αx, βx)dt/dyt = -yt3 + βyt + α = 0 —– (6)

This equation has one solution if

δx = 1/4αx2 − 1/27βx3 —– (7)

is greater than zero, δx > 0 and three solutions if δx < 0. This construction can serve as a statistic for bimodality, one of the catastrophe flags. The set of values for which the discriminant is equal to zero, δx = 0 is the bifurcation set which determines the set of singularity points in the system. In the case of three roots, the central root is called an “anti-prediction” and is least probable state of the system. Inside the bifurcation, when δx < 0, the surface predicts two possible values of the state variable which means that the state variable is bimodal in this case.

Most of the systems in behavioral sciences are subject to noise stemming from measurement errors or inherent stochastic nature of the system under study. Thus for a real-world applications, it is necessary to add non-deterministic behavior into the system. As catastrophe theory has primarily been developed to describe deterministic systems, it may not be obvious how to extend the theory to stochastic systems. An important bridge has been provided by the Itô stochastic differential equations to establish a link between the potential function of a deterministic catastrophe system and the stationary probability density function of the corresponding stochastic process. Adding a stochastic Gaussian white noise term to the system

dyt = -dV(yt; αx, βx)dt/dyt + σytdWt —– (8)

where -dV(yt; αx, βx)dt/dyt is the deterministic term, or drift function representing the equilibrium state of the cusp catastrophe, σyt is the diffusion function and Wt is a Wiener process. When the diffusion function is constant, σyt = σ, and the current measurement scale is not to be nonlinearly transformed, the stochastic potential function is proportional to deterministic potential function and probability distribution function corresponding to the solution from (8) converges to a probability distribution function of a limiting stationary stochastic process as dynamics of yt are assumed to be much faster than changes in xi,t. The probability density that describes the distribution of the system’s states at any t is then

fs(y|x) = ψ exp((−1/4)y4 + (βx/2)y2 + αxy)/σ —– (9)

The constant ψ normalizes the probability distribution function so its integral over the entire range equals to one. As bifurcation factor βx changes from negative to positive, the fs(y|x) changes its shape from unimodal to bimodal. On the other hand, αx causes asymmetry in fs(y|x).

# Knowledge Limited for Dummies….Didactics.

Bertrand Russell with Alfred North Whitehead, in the Principia Mathematica aimed to demonstrate that “all pure mathematics follows from purely logical premises and uses only concepts defined in logical terms.” Its goal was to provide a formalized logic for all mathematics, to develop the full structure of mathematics where every premise could be proved from a clear set of initial axioms.

Russell observed of the dense and demanding work, “I used to know of only six people who had read the later parts of the book. Three of those were Poles, subsequently (I believe) liquidated by Hitler. The other three were Texans, subsequently successfully assimilated.” The complex mathematical symbols of the manuscript required it to be written by hand, and its sheer size – when it was finally ready for the publisher, Russell had to hire a panel truck to send it off – made it impossible to copy. Russell recounted that “every time that I went out for a walk I used to be afraid that the house would catch fire and the manuscript get burnt up.”

Momentous though it was, the greatest achievement of Principia Mathematica was realized two decades after its completion when it provided the fodder for the metamathematical enterprises of an Austrian, Kurt Gödel. Although Gödel did face the risk of being liquidated by Hitler (therefore fleeing to the Institute of Advanced Studies at Princeton), he was neither a Pole nor a Texan. In 1931, he wrote a treatise entitled On Formally Undecidable Propositions of Principia Mathematica and Related Systems, which demonstrated that the goal Russell and Whitehead had so single-mindedly pursued was unattainable.

The flavor of Gödel’s basic argument can be captured in the contradictions contained in a schoolboy’s brainteaser. A sheet of paper has the words “The statement on the other side of this paper is true” written on one side and “The statement on the other side of this paper is false” on the reverse. The conflict isn’t resolvable. Or, even more trivially, a statement like; “This statement is unprovable.” You cannot prove the statement is true, because doing so would contradict it. If you prove the statement is false, then that means its converse is true – it is provable – which again is a contradiction.

The key point of contradiction for these two examples is that they are self-referential. This same sort of self-referentiality is the keystone of Gödel’s proof, where he uses statements that imbed other statements within them. This problem did not totally escape Russell and Whitehead. By the end of 1901, Russell had completed the first round of writing Principia Mathematica and thought he was in the homestretch, but was increasingly beset by these sorts of apparently simple-minded contradictions falling in the path of his goal. He wrote that “it seemed unworthy of a grown man to spend his time on such trivialities, but . . . trivial or not, the matter was a challenge.” Attempts to address the challenge extended the development of Principia Mathematica by nearly a decade.

Yet Russell and Whitehead had, after all that effort, missed the central point. Like granite outcroppings piercing through a bed of moss, these apparently trivial contradictions were rooted in the core of mathematics and logic, and were only the most readily manifest examples of a limit to our ability to structure formal mathematical systems. Just four years before Gödel had defined the limits of our ability to conquer the intellectual world of mathematics and logic with the publication of his Undecidability Theorem, the German physicist Werner Heisenberg’s celebrated Uncertainty Principle had delineated the limits of inquiry into the physical world, thereby undoing the efforts of another celebrated intellect, the great mathematician Pierre-Simon Laplace. In the early 1800s Laplace had worked extensively to demonstrate the purely mechanical and predictable nature of planetary motion. He later extended this theory to the interaction of molecules. In the Laplacean view, molecules are just as subject to the laws of physical mechanics as the planets are. In theory, if we knew the position and velocity of each molecule, we could trace its path as it interacted with other molecules, and trace the course of the physical universe at the most fundamental level. Laplace envisioned a world of ever more precise prediction, where the laws of physical mechanics would be able to forecast nature in increasing detail and ever further into the future, a world where “the phenomena of nature can be reduced in the last analysis to actions at a distance between molecule and molecule.”

What Gödel did to the work of Russell and Whitehead, Heisenberg did to Laplace’s concept of causality. The Uncertainty Principle, though broadly applied and draped in metaphysical context, is a well-defined and elegantly simple statement of physical reality – namely, the combined accuracy of a measurement of an electron’s location and its momentum cannot vary far from a fixed value. The reason for this, viewed from the standpoint of classical physics, is that accurately measuring the position of an electron requires illuminating the electron with light of a very short wavelength. But the shorter the wavelength the greater the amount of energy that hits the electron, and the greater the energy hitting the electron the greater the impact on its velocity.

What is true in the subatomic sphere ends up being true – though with rapidly diminishing significance – for the macroscopic. Nothing can be measured with complete precision as to both location and velocity because the act of measuring alters the physical properties. The idea that if we know the present we can calculate the future was proven invalid – not because of a shortcoming in our knowledge of mechanics, but because the premise that we can perfectly know the present was proven wrong. These limits to measurement imply limits to prediction. After all, if we cannot know even the present with complete certainty, we cannot unfailingly predict the future. It was with this in mind that Heisenberg, ecstatic about his yet-to-be-published paper, exclaimed, “I think I have refuted the law of causality.”

The epistemological extrapolation of Heisenberg’s work was that the root of the problem was man – or, more precisely, man’s examination of nature, which inevitably impacts the natural phenomena under examination so that the phenomena cannot be objectively understood. Heisenberg’s principle was not something that was inherent in nature; it came from man’s examination of nature, from man becoming part of the experiment. (So in a way the Uncertainty Principle, like Gödel’s Undecidability Proposition, rested on self-referentiality.) While it did not directly refute Einstein’s assertion against the statistical nature of the predictions of quantum mechanics that “God does not play dice with the universe,” it did show that if there were a law of causality in nature, no one but God would ever be able to apply it. The implications of Heisenberg’s Uncertainty Principle were recognized immediately, and it became a simple metaphor reaching beyond quantum mechanics to the broader world.

This metaphor extends neatly into the world of financial markets. In the purely mechanistic universe of classical physics, we could apply Newtonian laws to project the future course of nature, if only we knew the location and velocity of every particle. In the world of finance, the elementary particles are the financial assets. In a purely mechanistic financial world, if we knew the position each investor has in each asset and the ability and willingness of liquidity providers to take on those assets in the event of a forced liquidation, we would be able to understand the market’s vulnerability. We would have an early-warning system for crises. We would know which firms are subject to a liquidity cycle, and which events might trigger that cycle. We would know which markets are being overrun by speculative traders, and thereby anticipate tactical correlations and shifts in the financial habitat. The randomness of nature and economic cycles might remain beyond our grasp, but the primary cause of market crisis, and the part of market crisis that is of our own making, would be firmly in hand.

The first step toward the Laplacean goal of complete knowledge is the advocacy by certain financial market regulators to increase the transparency of positions. Politically, that would be a difficult sell – as would any kind of increase in regulatory control. Practically, it wouldn’t work. Just as the atomic world turned out to be more complex than Laplace conceived, the financial world may be similarly complex and not reducible to a simple causality. The problems with position disclosure are many. Some financial instruments are complex and difficult to price, so it is impossible to measure precisely the risk exposure. Similarly, in hedge positions a slight error in the transmission of one part, or asynchronous pricing of the various legs of the strategy, will grossly misstate the total exposure. Indeed, the problems and inaccuracies in using position information to assess risk are exemplified by the fact that major investment banking firms choose to use summary statistics rather than position-by-position analysis for their firmwide risk management despite having enormous resources and computational power at their disposal.

Perhaps more importantly, position transparency also has implications for the efficient functioning of the financial markets beyond the practical problems involved in its implementation. The problems in the examination of elementary particles in the financial world are the same as in the physical world: Beyond the inherent randomness and complexity of the systems, there are simply limits to what we can know. To say that we do not know something is as much a challenge as it is a statement of the state of our knowledge. If we do not know something, that presumes that either it is not worth knowing or it is something that will be studied and eventually revealed. It is the hubris of man that all things are discoverable. But for all the progress that has been made, perhaps even more exciting than the rolling back of the boundaries of our knowledge is the identification of realms that can never be explored. A sign in Einstein’s Princeton office read, “Not everything that counts can be counted, and not everything that can be counted counts.”

The behavioral analogue to the Uncertainty Principle is obvious. There are many psychological inhibitions that lead people to behave differently when they are observed than when they are not. For traders it is a simple matter of dollars and cents that will lead them to behave differently when their trades are open to scrutiny. Beneficial though it may be for the liquidity demander and the investor, for the liquidity supplier trans- parency is bad. The liquidity supplier does not intend to hold the position for a long time, like the typical liquidity demander might. Like a market maker, the liquidity supplier will come back to the market to sell off the position – ideally when there is another investor who needs liquidity on the other side of the market. If other traders know the liquidity supplier’s positions, they will logically infer that there is a good likelihood these positions shortly will be put into the market. The other traders will be loath to be the first ones on the other side of these trades, or will demand more of a price concession if they do trade, knowing the overhang that remains in the market.

This means that increased transparency will reduce the amount of liquidity provided for any given change in prices. This is by no means a hypothetical argument. Frequently, even in the most liquid markets, broker-dealer market makers (liquidity providers) use brokers to enter their market bids rather than entering the market directly in order to preserve their anonymity.

The more information we extract to divine the behavior of traders and the resulting implications for the markets, the more the traders will alter their behavior. The paradox is that to understand and anticipate market crises, we must know positions, but knowing and acting on positions will itself generate a feedback into the market. This feedback often will reduce liquidity, making our observations less valuable and possibly contributing to a market crisis. Or, in rare instances, the observer/feedback loop could be manipulated to amass fortunes.

One might argue that the physical limits of knowledge asserted by Heisenberg’s Uncertainty Principle are critical for subatomic physics, but perhaps they are really just a curiosity for those dwelling in the macroscopic realm of the financial markets. We cannot measure an electron precisely, but certainly we still can “kind of know” the present, and if so, then we should be able to “pretty much” predict the future. Causality might be approximate, but if we can get it right to within a few wavelengths of light, that still ought to do the trick. The mathematical system may be demonstrably incomplete, and the world might not be pinned down on the fringes, but for all practical purposes the world can be known.

Unfortunately, while “almost” might work for horseshoes and hand grenades, 30 years after Gödel and Heisenberg yet a third limitation of our knowledge was in the wings, a limitation that would close the door on any attempt to block out the implications of microscopic uncertainty on predictability in our macroscopic world. Based on observations made by Edward Lorenz in the early 1960s and popularized by the so-called butterfly effect – the fanciful notion that the beating wings of a butterfly could change the predictions of an otherwise perfect weather forecasting system – this limitation arises because in some important cases immeasurably small errors can compound over time to limit prediction in the larger scale. Half a century after the limits of measurement and thus of physical knowledge were demonstrated by Heisenberg in the world of quantum mechanics, Lorenz piled on a result that showed how microscopic errors could propagate to have a stultifying impact in nonlinear dynamic systems. This limitation could come into the forefront only with the dawning of the computer age, because it is manifested in the subtle errors of computational accuracy.

The essence of the butterfly effect is that small perturbations can have large repercussions in massive, random forces such as weather. Edward Lorenz was testing and tweaking a model of weather dynamics on a rudimentary vacuum-tube computer. The program was based on a small system of simultaneous equations, but seemed to provide an inkling into the variability of weather patterns. At one point in his work, Lorenz decided to examine in more detail one of the solutions he had generated. To save time, rather than starting the run over from the beginning, he picked some intermediate conditions that had been printed out by the computer and used those as the new starting point. The values he typed in were the same as the values held in the original simulation at that point, so the results the simulation generated from that point forward should have been the same as in the original; after all, the computer was doing exactly the same operations. What he found was that as the simulated weather pattern progressed, the results of the new run diverged, first very slightly and then more and more markedly, from those of the first run. After a point, the new path followed a course that appeared totally unrelated to the original one, even though they had started at the same place.

Lorenz at first thought there was a computer glitch, but as he investigated further, he discovered the basis of a limit to knowledge that rivaled that of Heisenberg and Gödel. The problem was that the numbers he had used to restart the simulation had been reentered based on his printout from the earlier run, and the printout rounded the values to three decimal places while the computer carried the values to six decimal places. This rounding, clearly insignificant at first, promulgated a slight error in the next-round results, and this error grew with each new iteration of the program as it moved the simulation of the weather forward in time. The error doubled every four simulated days, so that after a few months the solutions were going their own separate ways. The slightest of changes in the initial conditions had traced out a wholly different pattern of weather.

Intrigued by his chance observation, Lorenz wrote an article entitled “Deterministic Nonperiodic Flow,” which stated that “nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states.” Translation: Long-range weather forecasting is worthless. For his application in the narrow scientific discipline of weather prediction, this meant that no matter how precise the starting measurements of weather conditions, there was a limit after which the residual imprecision would lead to unpredictable results, so that “long-range forecasting of specific weather conditions would be impossible.” And since this occurred in a very simple laboratory model of weather dynamics, it could only be worse in the more complex equations that would be needed to properly reflect the weather. Lorenz discovered the principle that would emerge over time into the field of chaos theory, where a deterministic system generated with simple nonlinear dynamics unravels into an unrepeated and apparently random path.

The simplicity of the dynamic system Lorenz had used suggests a far-reaching result: Because we cannot measure without some error (harking back to Heisenberg), for many dynamic systems our forecast errors will grow to the point that even an approximation will be out of our hands. We can run a purely mechanistic system that is designed with well-defined and apparently well-behaved equations, and it will move over time in ways that cannot be predicted and, indeed, that appear to be random. This gets us to Santa Fe.

The principal conceptual thread running through the Santa Fe research asks how apparently simple systems, like that discovered by Lorenz, can produce rich and complex results. Its method of analysis in some respects runs in the opposite direction of the usual path of scientific inquiry. Rather than taking the complexity of the world and distilling simplifying truths from it, the Santa Fe Institute builds a virtual world governed by simple equations that when unleashed explode into results that generate unexpected levels of complexity.

In economics and finance, institute’s agenda was to create artificial markets with traders and investors who followed simple and reasonable rules of behavior and to see what would happen. Some of the traders built into the model were trend followers, others bought or sold based on the difference between the market price and perceived value, and yet others traded at random times in response to liquidity needs. The simulations then printed out the paths of prices for the various market instruments. Qualitatively, these paths displayed all the richness and variation we observe in actual markets, replete with occasional bubbles and crashes. The exercises did not produce positive results for predicting or explaining market behavior, but they did illustrate that it is not hard to create a market that looks on the surface an awful lot like a real one, and to do so with actors who are following very simple rules. The mantra is that simple systems can give rise to complex, even unpredictable dynamics, an interesting converse to the point that much of the complexity of our world can – with suitable assumptions – be made to appear simple, summarized with concise physical laws and equations.

The systems explored by Lorenz were deterministic. They were governed definitively and exclusively by a set of equations where the value in every period could be unambiguously and precisely determined based on the values of the previous period. And the systems were not very complex. By contrast, whatever the set of equations are that might be divined to govern the financial world, they are not simple and, furthermore, they are not deterministic. There are random shocks from political and economic events and from the shifting preferences and attitudes of the actors. If we cannot hope to know the course of the deterministic systems like fluid mechanics, then no level of detail will allow us to forecast the long-term course of the financial world, buffeted as it is by the vagaries of the economy and the whims of psychology.

# Econophysics: Financial White Noise Switch. Thought of the Day 115.0

What is the cause of large market fluctuation? Some economists blame irrationality behind the fat-tail distribution. Some economists observed that social psychology might create market fad and panic, which can be modeled by collective behavior in statistical mechanics. For example, the bi-modular distribution was discovered from empirical data in option prices. One possible mechanism of polarized behavior is collective action studied in physics and social psychology. Sudden regime switch or phase transition may occur between uni-modular and bi-modular distribution when field parameter changes across some threshold. The Ising model in equilibrium statistical mechanics was borrowed to study social psychology. Its phase transition from uni-modular to bi-modular distribution describes statistical features when a stable society turns into a divided society. The problem of the Ising model is that its key parameter, the social temperature, has no operational definition in social system. A better alternative parameter is the intensity of social interaction in collective action.

A difficult issue in business cycle theory is how to explain the recurrent feature of business cycles that is widely observed from macro and financial indexes. The problem is: business cycles are not strictly periodic and not truly random. Their correlations are not short like random walk and have multiple frequencies that changing over time. Therefore, all kinds of math models are tried in business cycle theory, including deterministic, stochastic, linear and nonlinear models. We outline economic models in terms of their base function, including white noise with short correlations, persistent cycles with long correlations, and color chaos model with erratic amplitude and narrow frequency band like biological clock.

The steady state of probability distribution function in the Ising Model of Collective Behavior with h = 0 (without central propaganda field). a. Uni-modular distribution with low social stress (k = 0). Moderate stable behavior with weak interaction and high social temperature. b. Marginal distribution at the phase transition with medium social stress (k = 2). Behavioral phase transition occurs between stable and unstable society induced by collective behavior. c. Bi-modular distribution with high social stress (k = 2.5). The society splits into two opposing groups under low social temperature and strong social interactions in unstable society.

Deterministic models are used by Keynesian economists for endogenous mechanism of business cycles, such as the case of the accelerator-multiplier model. The stochastic models are used by the Frisch model of noise-driven cycles that attributes external shocks as the driving force of business fluctuations. Since 1980s, the discovery of economic chaos and the application of statistical mechanics provide more advanced models for describing business cycles. Graphically,

The steady state of probability distribution function in socio-psychological model of collective choice. Here, “a” is the independent parameter; “b” is the interaction parameter. a Centered distribution with b < a (denoted by short dashed curve). It happens when independent decision rooted in individualistic orientation overcomes social pressure through mutual communication. b Horizontal flat distribution with b = a (denoted by long dashed line). Marginal case when individualistic orientation balances the social pressure. c Polarized distribution with b > a (denoted by solid line). It occurs when social pressure through mutual communication is stronger than independent judgment.

Numerical 1 autocorrelations from time series generated by random noise and harmonic wave. The solid line is white noise. The broken line is a sine wave with period P = 1.

Linear harmonic cycles with unique frequency are introduced in business cycle theory. The auto-correlations from harmonic cycle and white noise are shown in the above figure. Auto-correlation function from harmonic cycles is a cosine wave. The amplitude of cosine wave is slightly decayed because of limited data points in numerical experiment. Auto-correlations from a random series are an erratic series with rapid decade from one to residual fluctuations in numerical calculation. The auto-regressive (AR) model in discrete time is a combination of white noise term for simulating short-term auto-correlations from empirical data.

The deterministic model of chaos can be classified into white chaos and color chaos. White chaos is generated by nonlinear difference equation in discrete-time, such as one-dimensional logistic map and two-dimensional Henon map. Its autocorrelations and power spectra look like white noise. Its correlation dimension can be less than one. White noise model is simple in mathematical analysis but rarely used in empirical analysis, since it needs intrinsic time unit.

Color chaos is generated by nonlinear differential equations in continuous-time, such as three-dimensional Lorenz model and one-dimensional model with delay-differential model in biology and economics. Its autocorrelations looks like a decayed cosine wave, and its power spectra seem a combination of harmonic cycles and white noise. The correlation dimension is between one and two for 3D differential equations, and varying for delay-differential equation.

History shows the remarkable resilience of a market that experienced a series of wars and crises. The related issue is why the economy can recover from severe damage and out of equilibrium? Mathematically speaking, we may exam the regime stability under parameter change. One major weakness of the linear oscillator model is that the regime of periodic cycle is fragile or marginally stable under changing parameter. Only nonlinear oscillator model is capable of generating resilient cycles within a finite area under changing parameters. The typical example of linear models is the Samuelson model of multiplier-accelerator. Linear stochastic models have similar problem like linear deterministic models. For example, the so-called unit root solution occurs only at the borderline of the unit root. If a small parameter change leads to cross the unit circle, the stochastic solution will fall into damped (inside the unit circle) or explosive (outside the unit circle) solution.

# Appropriation of (Ir)reversibility of Noise Fluctuations to (Un)Facilitate Complexity

The logical depth is a suitable measure of subjective complexity for physical as well as mathematical objects. this, upon considering the effect of irreversibility, noise, and spatial symmetries of the equations of motion and initial conditions on the asymptotic depth-generating abilities of model systems.

“Self-organization” suggests a spontaneous increase of complexity occurring in a system with simple, generic (e.g. spatially homogeneous) initial conditions. The increase of complexity attending a computation, by contrast, is less remarkable because it occurs in response to special initial conditions. An important question, which would have interested Turing, is whether self-organization is an asymptotically qualitative phenomenon like phase transitions. In other words, are there physically reasonable models in which complexity, appropriately defined, not only increases, but increases without bound in the limit of infinite space and time? A positive answer to this question would not explain the natural history of our particular finite world, but would suggest that its quantitative complexity can legitimately be viewed as an approximation to a well-defined qualitative property of infinite systems. On the other hand, a negative answer would suggest that our world should be compared to chemical reaction-diffusion systems (e.g. Belousov-Zhabotinsky), which self-organize on a macroscopic, but still finite scale, or to hydrodynamic systems which self-organize on a scale determined by their boundary conditions.

The suitability of logical depth as a measure of physical complexity depends on the assumed ability (“physical Church’s thesis”) of Turing machines to simulate physical processes, and to do so with reasonable efficiency. Digital machines cannot of course integrate a continuous system’s equations of motion exactly, and even the notion of computability is not very robust in continuous systems, but for realistic physical systems, subject throughout their time development to finite perturbations (e.g. electromagnetic and gravitational) from an uncontrolled environment, it is plausible that a finite-precision digital calculation can approximate the motion to within the errors induced by these perturbations. Empirically, many systems have been found amenable to “master equation” treatments in which the dynamics is approximated as a sequence of stochastic transitions among coarse-grained microstates.

We concentrate arbitrarily on cellular automata, in the broad sense of discrete lattice models with finitely many states per site, which evolve according to a spatially homogeneous local transition rule that may be deterministic or stochastic, reversible or irreversible, and synchronous (discrete time) or asynchronous (continuous time, master equation). Such models cover the range from evidently computer-like (e.g. deterministic cellular automata) to evidently material-like (e.g. Ising models) with many gradations in between.

More of the favorable properties need to be invoked to obtain “self-organization,” i.e. nontrivial computation from a spatially homogeneous initial condition. A rather artificial system (a cellular automaton which is stochastic but noiseless, in the sense that it has the power to make purely deterministic as well as random decisions) undergoes this sort of self-organization. It does so by allowing the nucleation and growth of domains, within each of which a depth-producing computation begins. When two domains collide, one conquers the other, and uses the conquered territory to continue its own depth-producing computation (a computation constrained to finite space, of course, cannot continue for more than exponential time without repeating itself). To achieve the same sort of self-organization in a truly noisy system appears more difficult, partly because of the conflict between the need to encourage fluctuations that break the system’s translational symmetry, while suppressing fluctuations that introduce errors in the computation.

Irreversibility seems to facilitate complex behavior by giving noisy systems the generic ability to correct errors. Only a limited sort of error-correction is possible in microscopically reversible systems such as the canonical kinetic Ising model. Minority fluctuations in a low-temperature ferromagnetic Ising phase in zero field may be viewed as errors, and they are corrected spontaneously because of their potential energy cost. This error correcting ability would be lost in nonzero field, which breaks the symmetry between the two ferromagnetic phases, and even in zero field it gives the Ising system the ability to remember only one bit of information. This limitation of reversible systems is recognized in the Gibbs phase rule, which implies that under generic conditions of the external fields, a thermodynamic system will have a unique stable phase, all others being metastable. Even in reversible systems, it is not clear why the Gibbs phase rule enforces as much simplicity as it does, since one can design discrete Ising-type systems whose stable phase (ground state) at zero temperature simulates an aperiodic tiling of the plane, and can even get the aperiodic ground state to incorporate (at low density) the space-time history of a Turing machine computation. Even more remarkably, one can get the structure of the ground state to diagonalize away from all recursive sequences.

# String’s Depth of Burial

A string’s depth might be defined as the execution time of its minimal program.

The difficulty with this definition arises in cases where the minimal program is only a few bits smaller than some much faster program, such as a print program, to compute the same output x. In this case, slight changes in x may induce arbitrarily large changes in the run time of the minimal program, by changing which of the two competing programs is minimal. Analogous instability manifests itself in translating programs from one universal machine to another. This instability emphasizes the essential role of the quantity of buried redundancy, not as a measure of depth, but as a certifier of depth. In terms of the philosophy-of-science metaphor, an object whose minimal program is only a few bits smaller than its print program is like an observation that points to a nontrivial hypothesis, but with only a low level of statistical confidence.

To adequately characterize a finite string’s depth one must therefore consider the amount of buried redundancy as well as the depth of its burial. A string’s depth at significance level s might thus be defined as that amount of time complexity which is attested by s bits worth of buried redundancy. This characterization of depth may be formalized in several ways.

A string’s depth at significance level s be defined as the time required to compute the string by a program no more than s bits larger than the minimal program.

This definition solves the stability problem, but is unsatisfactory in the way it treats multiple programs of the same length. Intuitively, 2k distinct (n + k)-bit programs that compute same output ought to be accorded the same weight as one n-bit program; but, by the present definition, they would be given no more weight than one (n + k)-bit program.

A string’s depth at signicifcance level s depth might be defined as the time t required for the string’s time-bounded algorithmic probability Pt(x) to rise to within a factor 2−s of its asymptotic time-unbounded value P(x).

This formalizes the notion that for the string to have originated by an effective process of t steps or fewer is less plausible than for the first s tosses of a fair coin all to come up heads.

It is not known whether there exist strings that are deep, in other words, strings having no small fast programs, even though they have enough large fast programs to contribute a significant fraction of their algorithmic probability. Such strings might be called deterministically deep but probabilistically shallow, because their chance of being produced quickly in a probabilistic computation (e.g. one where the input bits of U are supplied by coin tossing) is significant compared to their chance of being produced slowly. The question of whether such strings exist is probably hard to answer because it does not relativize uniformly. Deterministic and probabilistic depths are not very different relative to a random coin-toss oracle A of the equality of random-oracle-relativized deterministic and probabilistic polynomial time complexity classes; but they can be very different relative to an oracle B deliberately designed to hide information from deterministic computations (this parallels Hunt’s proof that deterministic and probabilistic polynomial time are unequal relative to such an oracle).

(Depth of Finite Strings): Let x and w be strings and s a significance parameter. A string’s depth at significance level s, denoted Ds(x), will be defined as min{T(p) : (|p|−|p| < s)∧(U(p) = x)}, the least time required to compute it by a s-incompressible program. At any given significance level, a string will be called t-deep if its depth exceeds t, and t-shallow otherwise.

The difference between this definition and the previous one is rather subtle philosophically and not very great quantitatively. Philosophically, when each individual hypothesis for the rapid origin of x is implausible at the 2−s confidence level, then it requires only that a weighted average of all such hypotheses be implausible.

There exist constants c1 and c2 such that for any string x, if programs running in time ≤ t contribute a fraction between 2−s and 2−s+1 of the string’s total algorithmic probability, then x has depth at most t at significance level s + c1 and depth at least t at significance level s − min{H(s), H(t)} − c2.

Proof : The first part follows easily from the fact that any k-compressible self-delimiting program p is associated with a unique, k − O(1) bits shorter, program of the form “execute the result of executing p∗”. Therefore there exists a constant c1 such that if all t-fast programs for x were s + c1– compressible, the associated shorter programs would contribute more than the total algorithmic probability of x. The second part follows because, roughly, if fast programs contribute only a small fraction of the algorithmic probability of x, then the property of being a fast program for x is so unusual that no program having that property can be random. More precisely, the t-fast programs for x constitute a finite prefix set, a superset S of which can be computed by a program of size H(x) + min{H(t), H(s)} + O(1) bits. (Given x∗ and either t∗ or s∗, begin enumerating all self-delimiting programs that compute x, in order of increasing running time, and quit when either the running time exceeds t or the accumulated measure of programs so far enumerated exceeds 2−(H(x)−s)). Therefore there exists a constant c2 such that, every member of S, and thus every t-fast program for x, is compressible by at least s − min{H(s), H(t)} − O(1) bits.

The ability of universal machines to simulate one another efficiently implies a corresponding degree of machine-independence for depth: for any two efficiently universal machines of the sort considered here, there exists a constant c and a linear polynomial L such that for any t, strings whose (s+c)-significant depth is at least L(t) on one machine will have s-significant depth at least t on the other.

Depth of one string relative to another may be defined analogously, and represents the plausible time required to produce one string, x, from another, w.

(Relative Depth of Finite Strings): For any two strings w and x, the depth of x relative to w at significance level s, denoted Ds(x/w), will be defined as min{T(p, w) : (|p|−|(p/w)∗| < s)∧(U(p, w) = x)}, the least time required to compute x from w by a program that is s-incompressible relative to w.

Depth of a string relative to its length is a particularly useful notion, allowing us, as it were, to consider the triviality or nontriviality of the “content” of a string (i.e. its bit sequence), independent of its “form” (length). For example, although the infinite sequence 000… is intuitively trivial, its initial segment 0n is deep whenever n is deep. However, 0n is always shallow relative to n, as is, with high probability, a random string of length n.

In order to adequately represent the intuitive notion of stored mathematical work, it is necessary that depth obey a “slow growth” law, i.e. that fast deterministic processes be unable to transform a shallow object into a deep one, and that fast probabilistic processes be able to do so only with low probability.

(Slow Growth Law): Given any data string x and two significance parameters s2 > s1, a random program generated by coin tossing has probability less than 2−(s2−s1)+O(1) of transforming x into an excessively deep output, i.e. one whose s2-significant depth exceeds the s1-significant depth of x plus the run time of the transforming program plus O(1). More precisely, there exist positive constants c1, c2 such that for all strings x, and all pairs of significance parameters s2 > s1, the prefix set {q : Ds2(U(q, x)) > Ds1(x) + T(q, x) + c1} has measure less than 2−(s2−s1)+c2.

Proof: Let p be a s1-incompressible program which computes x in time Ds1(x), and let r be the restart prefix mentioned in the definition of the U machine. Let Q be the prefix set {q : Ds2(U(q, x)) > T(q, x) + Ds1(x) + c1}, where the constant c1 is sufficient to cover the time overhead of concatenation. For all q ∈ Q, the program rpq by definition computes some deep result U(q, x) in less time than that result’s own s2-significant depth, and so rpq must be compressible by s2 bits. The sum of the algorithmic probabilities of strings of the form rpq, where q ∈ Q, is therefore

Σq∈Q P(rpq)< Σq∈Q 2−|rpq| + s2 = 2−|r|−|p|+s2 μ(Q)

On the other hand, since the self-delimiting program p can be recovered from any string of the form rpq (by deleting r and executing the remainder pq until halting occurs, by which time exactly p will have been read), the algorithmic probability of p is at least as great (within a constant factor) as the sum of the algorithmic probabilities of the strings {rpq : q ∈ Q} considered above:

P(p) > μ(Q) · 2−|r|−|p|+s2−O(1)

Recalling the fact that minimal program size is equal within a constant factor to the −log of algorithmic probability, and the s1-incompressibility of p, we have P(p) < 2−(|p|−s1+O(1)), and therefore finally

μ(Q) < 2−(s2−s1)+O(1), which was to be demonstrated.

# Stephen Wolfram and Stochasticity of Financial Markets. Note Quote.

The most obvious feature of essentially all financial markets is the apparent randomness with which prices tend to fluctuate. Nevertheless, the very idea of chance in financial markets clashes with our intuitive sense of the processes regulating the market. All processes involved seem deterministic. Traders do not only follow hunches but act in accordance with specific rules, and even when they do appear to act on intuition, their decisions are not random but instead follow from the best of their knowledge of the internal and external state of the market. For example, traders copy other traders, or take the same decisions that have previously worked, sometimes reacting against information and sometimes acting in accordance with it. Furthermore, nowadays a greater percentage of the trading volume is handled algorithmically rather than by humans. Computing systems are used for entering trading orders, for deciding on aspects of an order such as the timing, price and quantity, all of which cannot but be algorithmic by definition.

Algorithmic however, does not necessarily mean predictable. Several types of irreducibility, from non-computability to intractability to unpredictability, are entailed in most non-trivial questions about financial markets.

whether the market generates its own randomness, starting from deterministic and purely algorithmic rules. Wolfram points out that the fact that apparent randomness seems to emerge even in very short timescales suggests that the randomness (or a source of it) that one sees in the market is likely to be the consequence of internal dynamics rather than of external factors. In economists’ jargon, prices are determined by endogenous effects peculiar to the inner workings of the markets themselves, rather than (solely) by the exogenous effects of outside events.

Wolfram points out that pure speculation, where trading occurs without the possibility of any significant external input, often leads to situations in which prices tend to show more, rather than less, random-looking fluctuations. He also suggests that there is no better way to find the causes of this apparent randomness than by performing an almost step-by-step simulation, with little chance of besting the time it takes for the phenomenon to unfold – the time scales of real world markets being simply too fast to beat. It is important to note that the intrinsic generation of complexity proves the stochastic notion to be a convenient assumption about the market, but not an inherent or essential one.

Economists may argue that the question is irrelevant for practical purposes. They are interested in decomposing time-series into a non-predictable and a presumably predictable signal in which they have an interest, what is traditionally called a trend. Whether one, both or none of the two signals is deterministic may be considered irrelevant as long as there is a part that is random-looking, hence most likely unpredictable and consequently worth leaving out.

What Wolfram’s simplified model show, based on simple rules, is that despite being so simple and completely deterministic, these models are capable of generating great complexity and exhibit (the lack of) patterns similar to the apparent randomness found in the price movements phenomenon in financial markets. Whether one can get the kind of crashes in which financial markets seem to cyclicly fall into depends on whether the generating rule is capable of producing them from time to time. Economists dispute whether crashes reflect the intrinsic instability of the market, or whether they are triggered by external events. Sudden large changes are Wolfram’s proposal for modeling market prices would have a simple program generate the randomness that occurs intrinsically. A plausible, if simple and idealized behavior is shown in the aggregate to produce intrinsically random behavior similar to that seen in price changes.

In the figure above, one can see that even in some of the simplest possible rule-based systems, structures emerge from a random-looking initial configuration with low information content. Trends and cycles are to be found amidst apparent randomness.

An example of a simple model of the market, where each cell of a cellular automaton corresponds to an entity buying or selling at each step. The behaviour of a given cell is determined by the behaviour of its two neighbors on the step before according to a rule. A rule like rule 90 is additive, hence reversible, which means that it does not destroy any information and has ‘memory’ unlike the random walk model. Yet, due to its random looking behaviour, it is not trivial shortcut the computation or foresee any successive step. There is some randomness in the initial condition of the cellular automaton rule that comes from outside the model, but the subsequent evolution of the system is fully deterministic.

internally generated suggesting large changes are more predictable – both in magnitude and in direction as the result of various interactions between agents. If Wolfram’s intrinsic randomness is what leads the market one may think one could then easily predict its behaviour if this were the case, but as suggested by Wolfram’s Principle of Computational Equivalence it is reasonable to expect that the overall collective behaviour of the market would look complicated to us, as if it were random, hence quite difficult to predict despite being or having a large deterministic component.

Wolfram’s Principle of Computational Irreducibility says that the only way to determine the answer to a computationally irreducible question is to perform the computation. According to Wolfram, it follows from his Principle of Computational Equivalence (PCE) that

almost all processes that are not obviously simple can be viewed as computations of equivalent sophistication: when a system reaches a threshold of computational sophistication often reached by non-trivial systems, the system will be computationally irreducible.

# Random Uniform Deviate, or Correlation Dimension

A widely used dimension algorithm in data analysis is the correlation dimension. Fix m, a positive integer, and r, a positive real number. Given a time-series of data u(1), u(2), …, u(N),from measurements equally spaced in time, form a sequence of vectors x(1), x(2),…, x(N- m + 1) in R’, defined by x(i) = [u(i), u(i+ 1),…,u(i+ m – 1)]. Next, define for each i, 1 ≤ i ≤ N – m + 1,

Cmi (r)= (number of j such that d[x(i), x(j)] ≤ r)/(N-m+1) ———- [1]

We must define d[x(i), x(j)] for vectors x(i) and x(j). We define

d[x(i), x(j)]= maxk = 1,2,…,m (|u(i+k-1) – u(j+k-1)j) ———- [2]

From the Cmi (r), define

Cm(r) = (N- m + i)-1 ∑(N – m + 1)i = 1 Cmi (r) ———- [3]

and define

βm = limr → 0 limn → ∞ log Cm(r)/log r ———- [4]

The assertion is that for m sufficiently large, βmis the correlation dimension. Such a limiting slope has been shown to exist for the commonly studied chaotic attractors. This procedure has frequently been applied to experimental data; investigators seek a “scaling range”of r values for which log Cm(r)/log r is nearly constant for large m, and they infer that this ratio is the correlation dimension. In some instances, investigators have concluded that this procedure establishes deterministic chaos.

The later conclusion is not necessarily correct: a converged, finite correlation dimension value does not guarantee that the defining process is deterministic. Consider the following stochastic process. Fix 0 ≤ p ≤1. Define Xj = α-l/2 sin(2πj/12) ∀ j,where α is specified below. Define Yj as a family of independent identicaly distributed (i.i.d.) real random variables, with uniform density on the interval [-√3, √3]. Define Zj = 1 with probability p, Zj = 0 with probability 1 – p.

α = (∑j = 112 sin2(2πj/12)/12 ———- [5]

and define MI Xj = (1- Zj) Xj + ZjYj. Intuitively, MI X(p) is generated by first ascertaining, for each j, whether the jth sample will be from the deterministic sine wave or from the random uniform deviate, with likelihood (1- p) of the former choice, then calculating either Xj or Yj. Increasing p marks a tendency towards greater system randomness. We now show that almost surely βmin [4] equals 0 ∀ m for the MI X(p) process, p ≠ 1. Fix m, define Kj = (12m)j- 12m, and define Nj = 1 if (MI Xk(j)+l,…, k(j)+m) = (X1,. . ., Xm), Nj = 0 otherwise. The Nj are i.i.d.random variables, with the expected value of Nj, E(Nj) ≥ (1- p)m. By the Strong Law of Large Numbers,

limN → ∞ ∑j = 1N Nj/N = E(N1) ≥ (1-p)m

Observe that (∑j = 1N Nj/12 mN)2 is a lower bound to Cm(r), since xk(i)+1,…., xk(j)+1 if Ni = Nj = 1. Thus for r ‹ 1

limN → ∞ sup log Cm(r)/log r ≤ (1/log r) limN → ∞ (∑j = 1N Nj/12 mN)2 ≤ log (1-p)2m/(12m)2/log r

Since, (1-p)2m/(12m)2 is independent of r, βm = limr → 0 limN → ∞ log Cm(r)/log r = 0. Since, βm ≠ 0 with probability 0 for each m, by countable additivity, ∀m, β= 0.

The MIX(p) process can be motivated by considering an autonomous unit that produces sinusoidal output, surrounded by a world of interacting processes that in ensemble produces output that resembles noise relative to the timing of the unit. The extent to which the surrounding world interacts with the unit could be controlled by a gateway between the two, with a larger gateway admitting greater apparent noise to compete with the sinusoidal signal. It is easy to show that, given a sequence Xj, a sequence of k = 1, 2,…, m i.i.d.Yj, defined by a density function and independent of the Xj, and Z= X+ Yj, then Zj has an infinite correlation dimension. It appears that correlation dimension distinguishes between correlated and uncorrelated successive iterates, with larger estimates of dimension corresponding to uncorrelated data. For a more complete interpretation of correlation dimension results, stochastic processes with correlated increments should be analyzed. Error estimates in dimension calculations are commonly seen. In statistics, one presumes a specified underlying stochastic distribution to estimate misclassification probabilities. Without knowing the form of a distribution, or if the system is deterministic or stochastic, one must be suspicious of error estimates. There often appears to be a desire to establish a noninteger dimension value, to give a fractal and chaotic interpretation to the result, but again, prior to a thorough study of the relationship between the geometric Hausdorff dimension and the time series formula labeled correlation dimension, it is speculation to draw conclusions from a noninteger correlation dimension value.

# Feynman Path Integrals, Trajectories and Copenhagen Interpretation. Note Quote.

As the trajectory exists by precept in the trajectory representation, there is no need for Copenhagen’s collapse of the wave function. The trajectory representation can describe an individual particle. On the other hand, Copenhagen describes an ensemble of particles while only rendering probabilities for individual particles.

The trajectory representation renders microstates of the Schrödinger’s wave function for the bound state problem. Each microstate by the equation

ψ = (2m)1/4cos(W/h ̄)/(W′)1/2[a − c2/(4b)]1/2

(aφ2 + bθ2 + cφθ)1/2/[a − c2/(4b)]1/2 cos[arctan(b(θ/φ) + c/2)/(ab − c2/4)1/2 = φ

is sufficient by itself to determine the Schrödinger’s wave function. Thus, the existence of microstates is a counter example refuting the Copenhagen assertion that the Schrödinger’s wave function be an exhaustive description of non-relativistic quantum phenomenon. The trajectory representation is deterministic. We can now identify a trajectory and its corresponding Schrödinger wave function with sub-barrier energy that tunnels through the barrier with certainty. Hence, tunneling with certainty is a counter example refuting Born’s postulate of the Copenhagen interpretation that attributes a probability amplitude to the Schrödinger’s wave function. As the trajectory representation is deterministic and does not need ψ, much less to assign a probability amplitude to it, the trajectory representation does not need a wave packet to describe or localize a particle. The equation of motion,

t − τ = ∂W/∂E, where t is the trajectory time, relative to its constant coordinate τ, and given as a function of x;

for a particle (monochromatic wave) has been shown to be consistent with the group velocity of the wave packet. Normalization, as previously noted herein, is determined by the nonlinearity of the generalized Hamilton-Jacobi equation for the trajectory representation and for the Copenhagen interpretation by the probability of finding the particle in space being unity. Though probability is not needed for tunneling through a barrier, the trajectory interpretation for tunneling is still consistent with the Schrödinger representation without the Copenhagen interpretation. The incident wave with compound spatial modulation of amplitude and phase for the trajectory representation,

has only two spectral components which are the incident and reflected unmodulated waves of the Schrödinger representation.

Trajectories differ with Feynman’s path integrals in three ways. First, trajectories employ a quantum Hamilton’s characteristic function while a path integral is based upon a classical Hamilton’s characteristic function. Second, the quantum Hamilton’s characteristic function is determined uniquely by the initial values of the quantum stationary Hamilton-Jacobi equation, while path integrals are democratic summing over all possible classical paths to determine Feynman’s amplitude. While path integrals need an infinite number of constants of the motion even for a single particle in one dimension, motion in the trajectory representation for a finite number of particles in finite dimensions is always determined by only a finite number of constants of the motion. Third, trajectories are well defined in classically forbidden regions where path integrals are not defined by precept.

Heisenberg’s uncertainty principle shall remain premature as long as Copenhagen uses an insufficient subset of initial conditions (x, p) to describe quantum phenomena. Bohr’s complementarity postulates that the wave-particle duality be resolved consistent with the measuring instrument’s specific properties.

Heisenberg’s uncertainty principle shall remain premature as long as Copenhagen uses an insufficient subset of initial conditions (x, p) to describe quantum phenomena. Bohr’s complementarity postulates that the wave-particle duality be resolved consistent with the measuring instrument’s specific properties. Anonymous referees of the Copenhagen school have had reservations concerning the representation of the incident modulated wave as represented by the equation

before the barrier. They have reported that compoundly modulated wave represented by the above equation is only a clever superposition of the incident and reflected unmodulated plane waves. They have concluded that synthesizing a running wave with compound spatial modulation from its spectral components is nonphysical because it would spontaneously split. By the superposition principle of linear differential equations, the spectral components may be used to synthesize a new pair of independent solutions with compound modulations running in opposite directions. Likewise, an unmodulated plane wave running in one direction can be synthesized from two waves with compound modulation running in the opposite directions for mappings under the superposition principle are reversible.