Conjuncted: Gauge Theory

Weyl introduced as a phase factor an exponential in which the phase α is preceded by the imaginary unit i, e.g., e^+iqα(x), in the wave function for the wave equations (for instance, the Dirac equation is (iγ^μ∂_μ − m)ψ = 0). It is here that Weyl correctly formulated gauge theory as a symmetry principle from which electromagnetism could be derived. It had been shown that for a quantum theory of charged particles interacting with the electromagnetic field, invariance under a gauge transformation of the potentials required multiplication of the wave function by the now well-know phase factor. Yang cited Weyl’s gauge theory results as reported by Pauli as a source for Yang-Mills gauge theory; although Yang didn’t find out until much later that these were Weyl’s results. Moreover, Pauli did not explicitly mention Weyl’s geometric interpretation. It was only much after Yang and Mills published their article that Yang realized the connection between their work and geometry. Yang says

What Mills and I were doing in 1954 was generalizing Maxwell’s theory. We knew of no geometrical meaning of Maxwell’s theory, and we were not looking in that direction.

For the wave equations to be gauge invariant, i.e., have the same form after the gauge transformation as before, the local phase transformation ψ(x) → ψ(x)e^+iα(x) has to be accompanied by the local gauge transformation

A_μ → A_μ − q⁻¹∂_μα(x) —– (1)

This dictates that the ∂_μ in the wave equations be replaced by the covariant derivative ∂_μ + iqA_μ in order for the ∂_μα(x) terms to cancel each other. This pair of phase factor-gauge transformation is not unique. Another pair that retains gauge symmetry and results in the same covariant derivative has the q included in the phase factor, i.e., ψ(x) → ψ(x)e^+iqα(x) paired with

A_μ → A_μ − ∂_μα(x) —– (2)

The fact that this pairing is not unique is not surprising since a change in the phase factor and gauge transformation have no physical significance.

Advertisement

Speculatively Accelerated Capital

Is high frequency trading good or bad? A reasonable answer must differentiate. Various strategies can be classified as high frequency; each needs to be considered separately before issuing a general verdict.

First, one should distinguish passive and active high frequency strategies. Passive strategies engage in non-designated market making by submitting resting orders. Profits come from earning the bid-ask spread and liquidity rebates offered by exchanges. Active strategies involve the submission of marketable orders. Their profit often directly translates into somebody else’s loss. Consequently, they have raised more (and eloquent) suspicion (including FLASH BOYS by Michael Lewis). Active strategies typically exploit short-term predictability of asset prices. This is particularly evident in order anticipation strategies, which

ascertain the existence of large buyers or sellers in the marketplace and then trade ahead of these buyers or sellers in anticipation that their large orders will move market prices (Securities and Exchange Commission, 2014, p. 8).

Hirschey demonstrates that high frequency traders indeed anticipate large orders with the help of complex algorithms. Large orders are submitted by institutional investors for various reasons. New information (or misinformation) on the fundamental asset value is one of them. Others include inventory management, margin calls, or the activation of stop-loss limits.

Even in the absence of order anticipation strategies, large orders are subject to execution shortfall, i.e. the liquidation value falls short of the mark-to-market value. Execution shortfall is explained in the literature as a consequence of information asymmetry (Glosten and Milgrom) and risk aversion among market makers (Ho and Stoll).

Institutional investors seek to achieve optimal execution (i.e. minimize execution shortfall and trading costs) with the help of execution algorithms. These algorithms, e.g. the popular VWAP (volume weighted average price), are typically based on the observation that price impact depends on the relative volume of an order: Price impact is lower when markets are busy. When high frequency traders detect such an execution algorithm, they obtain information on future trades and can earn significant profits with an order anticipation strategy.

That such order anticipation strategies have been described as aggressive, predatory  and “algo-sniffing” (MacKenzie) suggests that the Securities and Exchange Commission is not alone in suspecting that they “may present serious problems in today’s market structure”. But which problems exactly? There is little doubt that order anticipation strategies increase the execution shortfall of large orders. This is bad news for institutional investors. But, to put it bluntly, “the money isn’t gone, it’s just somewhere else”. The important question is whether order anticipation strategies decrease market quality.

Papers on the relationship between high frequency trading and market quality have identified two issues where the influence of high frequency trading remains inconclusive:

• How do high frequency traders influence market efficiency under normal market conditions?

An important determinant of market efficiency is volatility. Zhang and Riordan finds that high frequency traders increase volatility, Hasbrouck and Saar finds the opposite. Benos and Sagade point out that intraday volatility is “good” when it is the result of price discovery, but “excessive” noise otherwise. They study high frequency trading in four British stocks, finding that high frequency traders participate in 27% of all trading volume and that active high frequency traders in particular “can significantly amplify both price discovery and noise”, but “have higher ratios of information-to-noise contribution than all other traders”.

• Do high frequency traders increase the risk of financial breakdowns? Bershova and Rakhlin echo concerns that liquidity provided by (passive) high frequency traders could be

fictitious; although such liquidity is plentiful during ‘normal’ market conditions, it disappears at the first sign of trouble

and that high frequency trading

has increasingly shifted market liquidity toward a smaller subset of the investable universe […]. Ultimately, this […] contributes to higher short-term correlations across the entire market.

Thus, high frequency trading may be beneficial most of the time, but dangerous when markets are under pressure. The sociologist Donald MacKenzie agrees, arguing that high frequency trading leaves no time to react appropriately when something goes wrong. This became apparent during the 2010 Flash Crash. When high frequency traders trade ahead of large orders in their model of price impact, they cause price overshooting. This can lead to a domino effect by activating stop-loss limits of other traders, resulting in new large orders that cause even greater price overshooting, etc. Empirically, however, the frequency of market breakdowns was significantly lower during 2007-2013 than during 1993-2006, when high frequency trading was less prevalent.

Even with high-quality data, empirical studies cannot fully entangle different strategies employed by high frequency traders, but what is required instead is an integration of high frequency trading into a mathematical model of optimal execution. It features transient price impact, heterogeneous transaction costs and strategic interaction between an arbitrary number of traders. High frequency traders may decrease the price deviation caused by a large order, and thus reduce the risk of domino effects in the wake of large institutional trades….

Some content on this page was disabled on May 30, 2018 as a result of a DMCA takedown notice from W.W. Norton. You can learn more about the DMCA here:

https://en.support.wordpress.com/copyright-and-the-dmca/

Deleuzian Speculative Philosophy. Thought of the Day 44.0

Deleuze’s version of speculative philosophy is the procedure of counter-effectuation or counter-actualization. In defiance of the causal laws of an actual situation, speculation experiments with the quasi-causal intensities capable of bringing about effects that have their own retro-active power. This is its political import. Leibniz already argued that all things are effects or consequences, even though they do not necessarily have a cause, since the sufficient reason of what exists always lies outside of any actual series and remains virtual. (Leibniz) With the Principle of Sufficient Reason, he thus reinvented the Stoic disjunction between the series of corporeal causes and the series of incorporeal effects. Not because he anticipated the modern bifurcation of given necessary causes (How?) and metaphysically constructed reasons (Why?), but because for him the virtuality of effects is no less real than the interaction of causes. The effect always includes its own cause, since divergent series of events (incompossible worlds) enter into relation with any particular event (in this world), while these interpenetrating series are prior to, and not limited by, actual relations of causality per se. In terms of Deleuze, cause and effect do not share the same temporality. Whereas causes relate to one another in an eternal present (Chronos), effects relate to one another in a past-future purified of the present (Aion). Taken together, these temporalities form the double structure of every event (Logic of Sense). When the night is lit up by a sudden flash of lightning, this is the effect of an intensive, metaphysical becoming that contains its own destiny, integrating a differential potentiality that is irreducible to the physical series of necessary efficient causes that nonetheless participate in it. In order for such a contingent conjugation of events to be actualized (i.e. for effects to influence causes and become individuated in a materially extended state of affairs), however, its impersonal and pre-individual presence must be trusted upon. This takes a speculative investment or amor fati that forms its precursive reason/ground. In Difference and Repetition, Deleuze refers to this will to speculate as the dark precursor which determines the path of a thunderbolt in advance but in reverse, as though intagliated by setting up a communication of difference with difference. It is therefore the differenciator of these differences or in- itself of difference. We find a paradigmatic example of this will to make a difference in William James’ The Will to Believe when he writes: We can and we may, as it were, jump with both feet off the ground into a world of which we trust the other parts to meet our jump and only so can the making of a perfected world of the pluralistic pattern ever take place. Only through our precursive trust in it can it come into being. (James) As Stengers explains, we can and do speculate each time we precursively trust in the possibility of connecting, of entering into a (partial) rapport that cannot be derived from the ground of our current, dominant premises. Or as Deleuze writes: the dark precursor is not the friend (Difference and Repetition) but rather the bad will of a traitor or enemy, since the will does not precede the presubjective cruelty of the event in its involuntariness. At the same time, however, we never jump into a vacuum. We always speculate by the milieu, since a jump in general could never be trusted: If a jump is always situated, it is because its aim is not to escape the ground in order to get access to a higher realm. The jump, connecting this ground, always this ground, with what it was alien to, has the necessity of a response. In other words, the ground must have been given the power to make itself felt as calling for new dimensions. (Stengers) Indeed, if speculative thought cannot be detached from a practical concern, Deleuze at the same time states that [t]here is no other ethic than the amor fati of philosophy. (What is Philosophy?) Speculative reasoning is thus an art of pure expression or efficacy, an art of precipitating events: an art that detects and affirms the possibility of other reasons insisting as so many virtual forces that have not yet had the chance to emerge but whose presence can be trusted upon to make a difference.

In Deleuze’s own terms, there is no such thing as pure reason, only heterogeneous processes of rationalization, of actualizing an irrational potential: There is no metaphysics, but rather a politics of being. (Deleuze) For this reason, the method of speculative philosophy is the method of dramatization. It is a method that distributes events according to a logic that conditions the order of their intelligibility. As such it belongs to what in Difference and Repetition is referred to as the proper order of reasons: differentiation-individuation-dramatisation-differenciation. A book of philosophy, Deleuze famously writes in the preface, should be in part a very particular species of detective novel, in part a kind of science fiction. On the one hand, the creation of concepts cannot be separated from a problematic milieu or stage that matters practically; on the other hand, it seeks to deterritorialize this milieu by speculating on the quasi-causal intensity of its becoming-other.

Some content on this page was disabled on May 4, 2020 as a result of a DMCA takedown notice from Columbia University Press. You can learn more about the DMCA here:

https://en.support.wordpress.com/copyright-and-the-dmca/