The evolution of the art and science underpinning financial asset pricing has produced a plethora of equity pricing models designed to capture the “dynamics” of the underlying equity security.
In this brief introductory lecture I have elected to focus on an equity pricing model, developed and justified by wave theory technology. Heeding the warnings of Hendry et al (1984) that “measurement without theory is valueless” and reinforcing the findings of Marek Musiela and Marek Rutkowski that pricing and hedging models are expected “to reproduce the observed typical behaviours of all pertinent risk factors”, the exploitation of the sinusoidal wave within wave theories, and the modelling of the transformed sinusoidal wave within mathematical finance techniques does successfully reproduce the typical behaviour of stochastic equity security behaviours three months into the future. (source: “An analysis of a select number of equity securities for the purposes of description and modelling”, MSc Financial Mathematics, the University of Exeter, 2005 – 2006)
From this brief introductory lecture into the wave theory underpinning the equity pricing model, readers are invited to consider their interest in viewing “An analysis of a select number of equity securities for the purposes of description and modelling.”
To begin with, please consider the shape of the graph charting stochastic equity security behaviours
Please note the absence in any pattern in the upward and downward movements rendering such equity securities stochastic in their behaviour.
For the purposes of demonstrating the efficacy of wave theories to chart stochastic equity security behaviours, please consider the series of graphs comprising Graph 2.
Please note the distinguishing regularity of the shape of the curve (repetitive after 360°, and reaching the maximum and minimum turning point frequencies of ± 1).
Please note the absence in any symmetry in the sinusoidal wave exploited. In particular, there is no repetition in the shape of the graph after every 360°, and the maximum and minimum turning points have frequencies that lie outside of ± 1.
Please note that the shape of the graph of the sinusoidal wave exploited, joined by maximum and minimum turning points does reinforce the abrupt changes in direction and quantum reflective of Weiner financial assets.
The uneven directionless paths of Weiner assets may be expressed mathematically by the following formula:
S (t + Δ t) – S (t) = Δ Ŝ t = (S) S Δ t + θ (S) S e √ Δ t + o (Δ t) |
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where | e √ Δ t + o (Δ t) is the increment to the Weiner process | ||
is the current mean rate of return | |||
θ² is the variance of returns |
From this formula, please note the capability of the Weiner increment (containing √ Δ t in its formula) to express financial asset returns that lie outside the mean-variance constraints of several alternative/competing valuation methodologies. It is worthy of note that the Weiner increment is known to produce the shape of Graph 1, and by analogy it gives credence to the exploitation of the sinusoidal wave, depicted at Graphs 2(b) – 2(c), to chart and forecast stochastic equity security behaviours.
The progression from the sinusoidal wave to the sinusoidal wave exploited, as demonstrated by Graphs 2(a) – (c), is justified within wave theory by the formulation of a process called “over-modulation.” Modulation represents the transformation of carrier signals to other parts of the frequency spectrum in the interests of ensuring communication carrier efficiency. Over-modulation or a distortion of the modulation envelop produces “splatter” (where the depth of the modulation may exceed 100% because the modulating signal exceeds the capacity of the carrier wave). In wave theory, amplitude, angle and phase, each combine to produce a modulation envelop to fit a distorting modulating signal. Within wave theories the non-sinusoidal or distorted waveforms have ample expression if the modulating signal is sinusoidal.
Mathematically, the sinusoidal modulating signal may be given by:
Vmt = Vm sin wmt |
and the modulating signal’s phase consisting of a carrier and modulated frequency is given by:
vc = vc sin (wct + modulating index * peak phase deviation * sin wmt) |
The above depicted modulation of the carrier wave is describable in words as follows:
…as the modulating signal voltage increases in the positive direction the frequency of the carrier wave increases also and it reaches its maximum value when the modulating signal voltage is at its peak value…as the modulating signal voltage falls to zero the carrier frequency returns to its unmodulated value…and when the modulating signal becomes negative the instantaneous carrier frequency falls below its unmodulated value and reaches its lowest value when the modulating signal voltage is at its negative peak value…after this point the modulating signal voltage increases towards zero and the instantaneous carrier frequency increases modulated…towards its unmodulated value
From Graph 4 please note that the process of over-modulation has transformed the sinusoidal wave to create a distorted waveform. Pictorially the sinusoidal wave modulated takes the shape of two separate waveforms. Please note that the first part of the carrier wave modulated is relatively steeper than the second part of the carrier waveform modulated. This sequence of long-steep and short-flat waveforms does amply demonstrate the propensity of wave theory to chart stochastic equity security behaviours.
Within wave theory, the options available to mathematicians charting the behaviour of stochastic equity securities include amplitude modulation, angular modulation, phase modulation and variations on each of the aforementioned themes. The equities research “An analysis of a select number of equity securities for the purposes of description and modelling” demonstrates the efficacy of amplitude modulation amongst competing wave theory formulae. Freed from the rigidities inherent in formulae incorporating some form of phase modulation, amplitude modulation provides the flexibility which captures the dynamics of equity security behaviours.
Amplitude modulation is given mathematically by the following formula:
a cos x |
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where | a is the amplitude in wave theory (/transformed a variable representing the volatility of an equity security) | ||
x is the angle in wave theory (/transformed a variable representing the appreciation/depreciation of an equity security) | |||
cos is a functionality in wave theory (/transformed it represents a function that can be used to describe equity security values) |
The equity pricing model, consisting of “a cos x” transformed is flexible from the absence in any requirement for regularity in the values assumed by parameters ‘a’ and ‘x’. Additionally, please note that parameters ‘a’ and ‘x’ comprise the complete set of factors affecting equity securities pricing from microeconomic and macroeconomic sensitivities to sectoral/market volatility and trading activity. I have contended that the wave theory decomposition into ‘a’ and ‘x’, transformed and modelled, is capable of accurately forecasting the price of equity securities three months into the future, provided that such future values are entirely determinable from the current parameters inherent within ‘a’ and ‘x’. (source: “An analysis of a select number of equity securities for the purposes of description and modelling”)
The wave theory decomposition, transformed and modelled, has application to equity pricing, derivative finance, corporate finance, structured finance, risk management and asset management.
Should you have any queries in respect of the above lecture or if you are considering your interest in viewing/purchasing “An analysis of a select number of equity securities for the purposes of description and modelling”, please do not hesitate to contact Caroline Aneja at carolineaneja@googlemail.com