# Symplectic Manifolds

The canonical example of the n-symplectic manifold is that of the frame bundle, so the question is whether this formalism can be generalized to other principal bundles, and distinguished from the quantization arising from symplectic geometry on the prototype manifold, the bundle of linear frames, a good place to motivate the formalism.

Let us start with an n-dimensional manifold M, and let π : LM → M be the space of linear frames over a base manifold M, the set of pairs (m,ek), where m ∈ M and {ek},k = 1,···,n is a linear frame at m. This gives LM dimension n(n + 1), with GL(n,R) as the structure group acting freely on the right. We define local coordinates on LM in terms of those on the manifold M – for a chart on M with coordinates {xi}, let

qi(m,ek) = xi ◦ π(m,ek) = xi(m)

πji(m,ek) = ej ∂/∂xj

where {ej} denotes the coframe dual to {ej}. These coordinates are analogous to those on the cotangent bundle, except, instead of a single momentum coordinate, we now have a momentum frame. We want to place some kind of structure on LM, which is the prototype of n-symplectic geometry that is similar to symplectic geometry of the cotangent bundle T∗M. The structure equation for symplectic geometry

df= _| X dθ

gives Hamilton’s equations for the phase space of a particle, where θ is the canonical symplectic 2-form. There is a naturally defined Rn-valued 1-form on LM, the soldering form, given by

θ(X) ≡ u−1[π∗(X)] ∀X ∈ TuLM

where the point u = (m,ek) ∈ LM gives the isomorphism u : Rn → Tπ(u)M by ξiri → ξiei, where {ri} is the standard basis of Rn. The Rn-valued 2-form dθ can be shown to be non-degenerate, that is,

X _| dθ = 0 ⇔ X = 0

where we mean that each component of X dθ is identically zero. Finally, since there is also a structure group on LM, there are also group transformation properties. Let ρ be the standard representation of GL(n, R) on Rn. Then it can be shown that the pullback of dθ under right translation by g ∈ GL (n,R) is Rg dθ = ρ(g−1) · dθ.

Thus, we have an Rn-valued generalization of symplectic geometry, which motivates the following definition.

Let P be a principal fiber bundle with structure group G over an m-dimensional manifold M . Let ρ : G → GL(n, R) be a linear representation of G. An n-symplectic structure on P is a Rn-valued 2-form ω on P that is (i) closed and non-degenerate, in the sense that

X _| ω = 0 ⇔ X = 0

for a vector field X on P, and (ii) ω is equivariant, such that under the right action of G, Rg ω = ρ(g−1) · ω. The pair (P, ω) is called an n-symplectic manifold.

Here, we have modeled n-symplectic geometry after the frame bundle by defining the general n-symplectic manifold as a principal bundle. There is no reason, however, to limit ourselves to this, since we can let P be any manifold with a group action defined on it. One example of this would be to look at the action of the conformal group on R4. Since this group is locally isomorphic to O(2, 4), which is not a subgroup of GL(4, R), then forming a O(2,4) bundle over R4 cannot be thought of as simply a reduction of the frame bundle.

# High Frequency Markets and Leverage

Leverage effect is a well-known stylized fact of financial data. It refers to the negative correlation between price returns and volatility increments: when the price of an asset is increasing, its volatility drops, while when it decreases, the volatility tends to become larger. The name “leverage” comes from the following interpretation of this phenomenon: When an asset price declines, the associated company becomes automatically more leveraged since the ratio of its debt with respect to the equity value becomes larger. Hence the risk of the asset, namely its volatility, should become more important. Another economic interpretation of the leverage effect, inverting causality, is that the forecast of an increase of the volatility should be compensated by a higher rate of return, which can only be obtained through a decrease in the asset value.

Some statistical methods enabling us to use high frequency data have been built to measure volatility. In financial engineering, it has become clear in the late eighties that it is necessary to introduce leverage effect in derivatives pricing frameworks in order to accurately reproduce the behavior of the implied volatility surface. This led to the rise of famous stochastic volatility models, where the Brownian motion driving the volatility is (negatively) correlated with that driving the price for stochastic volatility models.

Traditional explanations for leverage effect are based on “macroscopic” arguments from financial economics. Could microscopic interactions between agents naturally lead to leverage effect at larger time scales? We would like to know whether part of the foundations for leverage effect could be microstructural. To do so, our idea is to consider a very simple agent-based model, encoding well-documented and understood behaviors of market participants at the microscopic scale. Then we aim at showing that in the long run, this model leads to a price dynamic exhibiting leverage effect. This would demonstrate that typical strategies of market participants at the high frequency level naturally induce leverage effect.

One could argue that transactions take place at the finest frequencies and prices are revealed through order book type mechanisms. Therefore, it is an obvious fact that leverage effect arises from high frequency properties. However, under certain market conditions, typical high frequency behaviors, having probably no connection with the financial economics concepts, may give rise to some leverage effect at the low frequency scales. It is important to emphasize that leverage effect should be fully explained by high frequency features.

Another important stylized fact of financial data is the rough nature of the volatility process. Indeed, for a very wide range of assets, historical volatility time-series exhibit a behavior which is much rougher than that of a Brownian motion. More precisely, the dynamics of the log-volatility are typically very well modeled by a fractional Brownian motion with Hurst parameter around 0.1, that is a process with Hölder regularity of order 0.1. Furthermore, using a fractional Brownian motion with small Hurst index also enables to reproduce very accurately the features of the volatility surface.

The fact that for basically all reasonably liquid assets, volatility is rough, with the same order of magnitude for the roughness parameter, is of course very intriguing. Tick-by-tick price model is based on a bi-dimensional Hawkes process, which is a bivariate point process (Nt+, Nt)t≥0 taking values in (R+)2 and with intensity (λ+t, λt) of the form

Here μ+ and μ are positive constants and the functions (φi)i=1,…4 are non-negative with associated matrix called kernel matrix. Hawkes processes are said to be self-exciting, in the sense that the instantaneous jump probability depends on the location of the past events. Hawkes processes are nowadays of standard use in finance, not only in the field of microstructure but also in risk management or contagion modeling. The Hawkes process generates behavior that mimics financial data in a pretty impressive way. And back-fitting, yields coorespndingly good results.  Some key problems remain the same whether you use a simple Brownian motion model or this marvelous technical apparatus.

In short, back-fitting only goes so far.

• The essentially random nature of living systems can lead to entirely different outcomes if said randomness had occurred at some other point in time or magnitude. Due to randomness, entirely different groups would likely succeed and fail every time the “clock” was turned back to time zero, and the system allowed to unfold all over again. Goldman Sachs would not be the “vampire squid”. The London whale would never have been. This will boggle the mind if you let it.

• Extraction of unvarying physical laws governing a living system from data is in many cases is NP-hard. There are far many varieties of actors and variety of interactions for the exercise to be tractable.

• Given the possibility of their extraction, the nature of the components of a living system are not fixed and subject to unvarying physical laws – not even probability laws.

• The conscious behavior of some actors in a financial market can change the rules of the game, some of those rules some of the time, or complete rewire the system form the bottom-up. This is really just an extension of the former point.

• Natural mutations over time lead to markets reworking their laws over time through an evolutionary process, with never a thought of doing so.

Thus, in this approach, Nt+ corresponds to the number of upward jumps of the asset in the time interval [0,t] and Nt to the number of downward jumps. Hence, the instantaneous probability to get an upward (downward) jump depends on the arrival times of the past upward and downward jumps. Furthermore, by construction, the price process lives on a discrete grid, which is obviously a crucial feature of high frequency prices in practice.

This simple tick-by-tick price model enables to encode very easily the following important stylized facts of modern electronic markets in the context of high frequency trading:

1. Markets are highly endogenous, meaning that most of the orders have no real economic motivation but are rather sent by algorithms in reaction to other orders.
2. Mechanisms preventing statistical arbitrages take place on high frequency markets. Indeed, at the high frequency scale, building strategies which are on average profitable is hardly possible.
3. There is some asymmetry in the liquidity on the bid and ask sides of the order book. This simply means that buying and selling are not symmetric actions. Indeed, consider for example a market maker, with an inventory which is typically positive. She is likely to raise the price by less following a buy order than to lower the price following the same size sell order. This is because its inventory becomes smaller after a buy order, which is a good thing for her, whereas it increases after a sell order.
4. A significant proportion of transactions is due to large orders, called metaorders, which are not executed at once but split in time by trading algorithms.

In a Hawkes process framework, the first of these properties corresponds to the case of so-called nearly unstable Hawkes processes, that is Hawkes processes for which the stability condition is almost saturated. This means the spectral radius of the kernel matrix integral is smaller than but close to unity. The second and third ones impose a specific structure on the kernel matrix and the fourth one leads to functions φi with heavy tails.

# Dance of the Shiva, q’i (chee) and Tibetan Sunyata. Manifestation of Mysticism.

अनेजदेकं मनसो जवीयो नैनद्देवाप्नुवन्पूर्वमर्षत् ।
तद्धावतोऽन्यान्नत्येति तिष्ठत् तस्मिन्नापो मातरिश्वा दधाति ॥

The self is one. It is unmoving: yet faster than the mind. Thus moving faster, It is beyond the reach of the senses. Ever steady, It outstrips all that run. By its mere presence, the cosmic energy is enabled to sustain the activities of living beings.

तस्मिन् मनसि ब्रह्मलोकादीन्द्रुतं गच्छति सति प्रथमप्राप्त इवात्मचैतन्याभासो गृह्यते अतः मनसो जवीयः इत्याह ।

tasmin manasi brahmalokādīndrutaṃ gacchati sati prathamaprāpta ivātmacaitanyābhāso gṛhyate ataḥ manaso javīyaḥ ityāha |

When the mind moves fast towards the farthest worlds such as the brahmaloka, it finds the Atman, of the nature of pure awareness, already there; hence the statement that It is faster than the mind.

नित्योऽनित्यानां चेतनश्चेतनानाम्
एको बहूनां यो विदधाति कामान् ।
तमात्मस्थं योऽनुपश्यन्ति धीराः
तेषां शान्तिः शाश्वतं नेतरेषाम् ॥

nityo’nityānāṃ cetanaścetanānām
eko bahūnāṃ yo vidadhāti kāmān |
tamātmasthaṃ yo’nupaśyanti dhīrāḥ
teṣāṃ śāntiḥ śāśvataṃ netareṣām ||

He is the eternal in the midst of non-eternals, the principle of intelligence in all that are intelligent. He is One, yet fulfils the desires of many. Those wise men who perceive Him as existing within their own self, to them eternal peace, and non else.

Eastern mysticism approaches the manifestation of life in the cosmos and all that compose it from a position diametrically opposed to the view that prevailed until recently among the majority of Western scientists, philosophers, and religionists. Orientals see the universe as a whole, as an organism. For them all things are interconnected, links in a chain of beings permeated by consciousness which threads them together. This consciousness is the one life-force, originator of all the phenomena we know under the heading of nature, and it dwells within its emanations, urging them as a powerful inner drive to grow and evolve into ever more refined expressions of divinity. The One manifests, not only in all its emanations, but also through those emanations as channels: it is within them and yet remains transcendent as well.

The emphasis is on the Real as subject whereas in the West it is seen as object. If consciousness is the noumenal or subjective aspect of life in contrast to the phenomenal or objective — everything seen as separate objects — then only this consciousness can be experienced, and no amount of analysis can reveal the soul of Reality. To illustrate: for the ancient Egyptians, their numerous “gods” were aspects of the primal energy of the Divine Mind (Thoth) which, before the creation of our universe, rested, a potential in a subjective state within the “waters of Space.” It was through these gods that the qualities of divinity manifested.

A question still being debated runs: “How does the One become the many?” meaning: if there is a “God,” how do the universe and the many entities composing it come into being? This question does not arise among those who perceive the One to dwell in the many, and the many to live in the One from whom life and sustenance derive. Despite our Western separation of Creator and creation, and the corresponding distancing of “God” from human beings, Western mystics have held similar views to those of the East, e.g.: Meister Eckhart, the Dominican theologian and preacher, who was accused of blasphemy for daring to say that he had once experienced nearness to the “Godhead.” His friends and followers were living testimony to the charisma (using the word in its original connotation of spiritual magnetism) of those who live the life of love for fellow beings men like Johannes Tauler, Heinrich Suso, the “admirable Ruysbroeck,” who expressed views similar to those of Eastern exponents of the spiritual way or path.

In old China, the universe was described as appearing first as q’i (chee), an emanation of Light, not the physical light that we know, but its divine essence sometimes called Tien, Heaven, in contrast to Earth. The q’i energy polarized as Yang and Yin, positive and negative electromagnetism. From the action and interaction of these two sprang the “10,000 things”: the universe, our world, the myriads of beings and things as we perceive them to be. In other words, the ancient Chinese viewed our universe as one of process, the One energy, q’i, proliferating into the many.

In their paintings Chinese artists depict man as a small but necessary element in gigantic natural scenes. And since we are parts of the cosmos, we are embodiments of all its potentials and our relationship depends upon how we focus ourselves: (1) harmoniously, i.e., in accord with nature; or (2) disharmoniously, interfering with the course of nature. We therefore affect the rest: our environment, all other lives, and bear full responsibility for the outcome of our thoughts and acts, our motivations, our impacts. Their art students were taught to identify with what they were painting, because there is life in every thing, and it is this life with which they must identify, with boulders and rocks no less than with birds flying overhead. Matter, energy, space, are all manifestations of q’i and we, as parts thereof, are intimately connected with all the universe.

In India, the oneness of life was seen through the prism of successive manifestations of Brahman, a neuter or impersonal term in Sanskrit for divinity, the equivalent of what Eckhart called the Godhead. Brahman is the source of the creative power, Brahma, Eckhart’s Creator; and also the origin of the sustaining and supporting energy or Vishnu, and of the destructive/regenerative force or Siva. As these three operate through the cosmos, the “world” as we know it, so do they also through ourselves on a smaller scale according to our capacity. Matter is perceived to be condensed energy, Chit or consciousness itself. To quote from the Mundaka Upanishad:

By the energism of Consciousness Brahman is massed; from that: Matter is born and from Matter Life and Mind and the worlds . . .

In another Hindu scripture, it is stated that when Brahma awakened from his period of rest between manifestations, he desired to contemplate himself as he is. By gazing into the awakening matter particles as into a mirror, he stirred them to exhibit their latent divine qualities. Since this process involves a continuous unfoldment from the center within, an ever-becoming, there can never be an end to the creativity — universal “days” comprising trillions of our human years, followed by a like number of resting “nights.”

We feel within ourselves the same driving urge to grow that runs through the entire, widespread universe, to express more and more of what is locked up in the formless or subjective realm of Be-ness, awaiting the magic moment to come awake in our phase of life.

Tibetan metaphysics embraces all of this in discussing Sunyata, which can be viewed as Emptiness if we use only our outer senses, or as Fullness if we inwardly perceive it to be full of energies of limitless ranges of wave-lengths/frequencies. This latter aspect of Space is the great mother of all, ever fecund, from whose “heart” emerge endless varieties of beings, endless forces, ever-changing variations — like the pulsing energies the new physicists perceive nuclear subparticles to be.

In the Preface to his Tao of physics Fritjof Capra tells how one summer afternoon he had a transforming experience by the seashore as he watched the waves rolling in and felt the rhythm of his own breathing. He saw dancing motes revealed in a beam of sunlight; particles of energy vibrating as molecules and atoms; cascades of energy pouring down upon us from outer space. All of this coming and going, appearing and disappearing, he equated with the Indian concept of the dance of Siva . . . he felt its rhythm, “heard” its sound, and knew himself to be a part of it. Through this highly personal, indeed mystical, experience Capra became aware of his “whole environment as being engaged in a gigantic cosmic dance.”

This is the gist of the old Chinese approach to physics: students were taught gravitation by observing the petals of a flower as they fall gracefully to the ground. As Gary Zukav expresses it in his Dancing Wu Li Masters: An Overview of the New Physics:

The world of particle physics is a world of sparkling energy forever dancing with itself in the form of its particles as they twinkle in and out of existence, collide, transmute, and disappear again.

That is: the dance of Siva is the dance of attraction and repulsion between charged particles of the electromagnetic force. This is a kind of “transcendental” physics, going beyond the “world of opposites” and approaching a mystical view of the larger Reality that is to our perceptions an invisible foundation of what we call “physical reality.” It is so far beyond the capacity or vocabulary of the mechanically rational part of our mind to define, that the profound Hindu scripture Isa Upanishad prefers to suggest the thought by a paradox:

तदेजति तन्नैजति तद्दूरे तद्वन्तिके ।
तदन्तरस्य सर्वस्य तदु सर्वस्यास्य बाह्यतः ॥

It moves. It moves not.  It is far, and it is near. It is within all this, And It is verily outside of all this.

Indeed, there is a growing recognition mostly by younger physicists that consciousness is more than another word for awareness, more than a by-product of cellular activity (or of atomic or subatomic vibrations). For instance, Jack Sarfatti, a quantum physicist, says that signals pulsating through space provide instant communication between all parts of the cosmos. “These signals can be likened to pulses of nerve cells of a great cosmic brain that permeates all parts of space (Michael Talbot, Mysticism and the New Physics).” Michael Talbot quotes Sir James Jeans’ remark, “the universe is more like a giant thought than a giant machine,” commenting that the “substance of the great thought is consciousness” which pervades all space. Or as Schrödinger would have it:

Consciouness is never experienced in the plural, only in the singular….Consciouness is a singular of which the plural is unknown; that; there is only one thing and that, what seems to be a plurality is merely a series of different aspects of this one thing, produced by a deception (the Indian Maya).

Other phenomena reported as occurring in the cosmos at great distances from each other, yet simultaneously, appear to be connected in some way so far unexplained, but to which the term consciousness has been applied.

In short, the mystic deals with direct experience; the intuitive scientist is open-minded, and indeed the great discoveries such as Einstein’s were made by amateurs in their field untrammeled by prior definitions and the limitations inherited from past speculations. This freedom enabled them to strike out on new paths that they cleared and paved. The rationalist tries to grapple with the problems of a living universe using only analysis and whatever the computer functions of the mind can put together.

The theosophic perspective upon universal phenomena is based on the concept of the ensoulment of the cosmos. That is: from the smallest subparticle we know anything about to the largest star-system that has been observed, each and all possess at their core vitality, energy, an active something propelling towards growth, evolution of faculties from within.

The only “permanent” in the whole universe is motion: unceasing movement, and the ideal perception is a blend of the mystical with the scientific, the intuitive with the rational.