3.15.1 Path Signatures from Zero

Basic Prerequisites: none beyond the idea of a curve and an integral (this page stands alone). The formal machinery begins at 02 · The Signature Algebra.

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1. Intuition & Practical Objective

A time series is a path - a curve through a state space, drawn between a start time and an end time. The most naive summary of that path is its endpoint: where it ended. For a stock, that is the close price; for vol, the current level. This page is built on one idea, stated as a challenge:

The endpoint throws away almost everything. The signature is the systematic upgrade - an ordered, infinitely-deep hierarchy of summaries in which level 1 is the endpoint, level 2 is the area swept between coordinates (the shape of the loop), level 3 is the signed volume, and so on, such that the full collection recovers the path up to a harmless reparametrisation.

Why finance should care: nearly every derivative contract is a path-dependent functional - an Asian depends on the whole average, a barrier on whether the path crossed a level, a variance swap on the integral of squared increments, a cliquet on a sequence of forward returns. The signature is the natural coordinate system for all of them: by a theorem (Lyons; §03) continuous path-functionals can be approximated as linear combinations of signature terms. Before any algebra, you need to feel the hierarchy, and that is the whole job of this page.

Three intuitions to carry out:

1. Level 1 is the increment - it cannot see shape. The integral $\int dX$ is just $X_T-X_0$. Two paths that begin and end identically are indistinguishable at level 1 no matter how different their shapes.
2. Level 2 is the area - it sees the loop. The double integral $\int {\int }_{s<t}d{X}_s^{i}d{X}_t^j$ encodes, through its antisymmetric part, the oriented area swept between coordinates $i$ and $j$. This is where “the path went clockwise around the loop” first registers. For two assets, the level-2 cross terms are literally the covariance structure of the increments - this is why the signature, not the endpoint, is the object an ML model should consume (§05–§06).
3. Levels 1, 2, 3, … are a real hierarchy. Information that is invisible at level $k$ routinely becomes visible at level $k+1$. The signature is only “complete” in the limit; the depth you need is set by the question you ask.

The practical objective: be able to look at a path and know which level of its signature answers your question, and to appreciate that the signature is a lossless-in-the-limit summary rather than an arbitrary set of hand-picked features.

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2. Mathematical Ground Truth & Derivations

2.1 The hierarchy of iterated integrals

Let $X:[0,T]\to {\mathbb{R}}^d$ be a path of bounded variation (for now - roughness is handled in §04). Write its coordinates $X^1,\dots ,X^d$. The signature of $X$ is the collection of all iterated integrals, indexed by words $w=i_1\cdots i_k$ over the alphabet $\{1,\dots ,d\}$:

$$S_{0,T}^{i_1\cdots i_k}(X)={\int }_{0<t_1<\cdots <t_k<T}d{X}_{t_1}^{i_1}\cdots d{X}_{t_k}^{i_k},S_{0,T}^{\varnothing }=1.$$

Level $k$ (the word has $k$ letters) lives in the $k$-fold tensor power $({\mathbb{R}}^d{)}^{\otimes k}$; collecting all levels gives an element of the tensor algebra. The ordering $0<t_1<\cdots <t_k$ is what makes the hierarchy hierarchical: you integrate level-$k$ information against more of the path to get level $k+1$.

2.2 Why level 1 is blind and level 2 is not

Level 1 is immediate: $S_{0,T}^i={\int }_0^{T}d{X}_t^i=X_T^i-X_0^i$ - a single number per coordinate, the endpoint (for a path starting at the origin). Level 2 splits into symmetric and antisymmetric parts:

$$S_{0,T}^{ij}=\underset{\text{symmetric, from level 1}}{\underbrace{\frac{1}{2}{S}^i{S}^j}}+\underset{\text{antisymmetric, the area}}{\underbrace{\frac{1}{2}(S^{ij}-S^{ji})}}.$$

The Lévy area $\frac{1}{2}(S^{ij}-S^{ji})$ is exactly the oriented area swept by the path between coordinates $i$ and $j$ (Green’s theorem). It is not recoverable from the endpoint - the loop that returns to its start has $S^i=0$ yet carries arbitrary area. This is the cleanest demonstration that the signature is a genuinely higher-order object.

2.3 Same endpoints, different path - where each level fails

A one-parameter family of shapes all share their endpoints; each successive level of the signature is the “next” invariant that tells them apart. §3 makes this numerical: two paths with identical level-1 and identical level-2 signatures are still separated at level 3. The moral: truncating the signature at level $N$ is a real modelling choice - it commits you to seeing only the shape features of depth $\le N$.

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3. Computational Implementation - the endpoint, the area, and the depth of the hierarchy

We compute iterated integrals of piecewise-linear paths by explicit sums (exact, no simulation): level 1 is a sum of increments; level 2 is ${\sum }_{k<l}{\Delta }_k^i{\Delta }_l^j+\frac{1}{2}{\sum }_k{\Delta }_k^i{\Delta }_k^j$ (the half-diagonal term is the within-segment contribution); level 3 is the analogous triple sum. Stdlib only, deterministic.

Reading the output.

• The endpoint is blind. P and Q have identical level-1 signatures $S^1=[3,3]$ (and therefore identical $S^{11}=4.5$) yet are completely different paths - one goes right then up, the other up then right.
• The area is the first shape detector. Their Lévy areas are $+4.5$ and $-4.5$: the orientation of the loop is reversed. Green’s theorem turns this into the signed area between the curve and its chord, and it is antisymmetric (swapping coordinates flips the sign) - exactly the kind of invariant a covariance-blind endpoint summary can never produce.
• A closed loop has zero endpoint but full area. The loop R starts and ends at the origin, so $S^1=[0,0]$; yet its area is $+9$ and its level-2 matrix is $S^{12}=9$, $S^{21}=-9$. If all you look at is “where did it end,” you have missed the entire loop - which is the entire payoff of a barrier or a variance swap.
• The hierarchy is real: level 3 sees what level 2 cannot. U and V share the entire level-1 and level-2 signature ($S^1=[0,3]$, area $+6$, all of $S^2$) - you literally cannot tell them apart at depth 2 - yet their level-3 integrals $S^{112}$ differ ($9.000$ vs $10.500$). The shape difference between “bulge on the right” and “bulge on the left” lives one level higher. This is why the signature is an infinite object: there is always one more level of shape.

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4. Failure Modes & First-Principles Breakdowns

1. “The close price is the signal.” For path-dependent payoffs (Exotic & Path-Dependent Options) the endpoint is the least informative scalar in the whole hierarchy - a barrier cares about the maximum, an Asian about the average, a variance swap about the increments. Level 1 is the right answer only when the payoff is genuinely $F(S_T)$.
2. Using a 1D signature on univariate data. A single coordinate has no area: $S^{11}=\frac{1}{2}(S^1{)}^2$ is a function of the endpoint, so the raw 1D signature is the endpoint in disguise. The fix (time augmentation, lead-lag) is the subject of 05 · Failure Modes & Practice.
3. Stopping at level 2 “because the area is the interesting bit.” As the U/V example shows, depth-2-identical paths separate at depth 3. The level you need is set by the question; a truncated signature is a deliberate approximation with a measurable information price.
4. Reading the signed area as an unsigned shape measure. The Lévy area is oriented: $+4.5$ and $-4.5$ are opposite orientations, and for assets this orientation is exactly the sign of the cross-covariance (lead vs lag matters, §05). Symmetrising it away discards the signal.

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5. References

• Chevyrev & Kormilitzin (2016), A Primer on the Signature Method in Machine Learning, arXiv:1603.03788
• Lyons (1998), Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14(2)
• Hambly & Lyons (2010), Uniqueness for the signature of a path of bounded variation and the reduced path group, Annals of Mathematics 171
• Friz & Victoir (2010), Multidimensional Stochastic Processes as Rough Paths

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6. Connected Graph Bridges

• Base: Stochastic Calculus & Itô’s Lemma · Linear Algebra & Matrices · Calculus & Optimization
• Continue: 02 · The Signature Algebra · Index Hub
• Sibling: Lookbacks & Asians (payoffs that are path functionals) · Econometrics & Time Series