Trading and Financial Markets

application

The domain where the residual is the product. Everywhere else a nonzero energy is a diagnostic; here it is the trade signal.

Sources: original to this vault (design and analysis; no single paper).

The relations

No-arbitrage conditions are exact relations among observable prices, with no input and no output:

relation	form
put–call parity	$C - P = S - K e^{- r T}$
triangular FX	$r_{A \to B} r_{B \to C} r_{C \to A} = 1$ (log-linear)
index vs constituents	$I = \sum_{i} w_{i} S_{i}$
cash-and-carry	$F = S e^{(r - q) T}$
cross-listing / ADR	$P_{local} = P_{foreign} \times FX$

Several are linear — or linear in logs — which puts them squarely in the fragment where LinearConstraintFactor is exact. None designates an output: parity does not say which of the four legs is “the answer”, and in practice you infer whichever one you cannot observe cleanly.

What is observed

Quotes from multiple venues, at different times, with bid–ask spreads, varying liquidity and missing instruments. Asynchrony is not a nuisance here, it is the microstructure — two venues quoting the same relation at slightly different timestamps is where a large part of the signal lives.

That is a strong case for Time as a Base: a price is a trajectory queried at a time, not a value attached to a tick.

What would be learned

A volatility surface — fitted, not derived, and feeding the option legs.
An illiquid or exotic instrument’s pricing model, where no closed form applies.
A microstructure noise model — how much of an observed deviation is spread versus signal.

Mixed with exact parity relations in one graph, which is the grey-box case: the relations you trust constrain the components you fitted.

What a residual means

A mispricing. The graded energy of Scalar and Multivariate Energy gives per-relation attribution: not “something is off by 3 basis points” but which parity is violated and by how much, with the uncertainty of the estimate attached.

And the uncertainty is the operationally important half. A deviation of two basis points means nothing if the posterior width is five; the same deviation is a trade if the width is a quarter. Point-estimate arbitrage screens cannot make that distinction and are famous for firing on noise.

One model, several questions

Which leg is stale? Clamp the liquid legs, infer the illiquid one, compare with its quote.
What is the implied state? Clamp all quotes, read the latent (implied vol, implied dividend, implied borrow).
What if this leg moved? Clamp a hypothetical and propagate.

Same relations, different clamps.

What would be hard

Returns are heavy-tailed. The Gaussian fragment is where every exactness result in this project lives, and financial innovations are decidedly not Gaussian. This is messages §1’s gap in its most commercially painful form: you want particle or heavy-tailed beliefs and combine does not support them.
The relations hold only approximately. Transaction costs, borrow costs, settlement and short constraints all widen parity into a band. That is expressible — a band is a soft constraint with a suitable noise model — but it means the “exact relation” framing is already an idealisation.
Regime change. The learned components are non-stationary, and nothing in the framework tracks that a fitted factor has gone stale.
Latency. If the residual is the signal, the inference has to run at market speed. See Parallelism and Compilation — the current implementation is measured at quadratic in the number of factors.

Lenticulum — theory vault

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