LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
A listed European option depends on where the underlying finishes, not on the route it took. The volatility surface, the grid of implied volatilities across strikes and expiries, encodes those ending distributions and stays silent about the path. Exotic contracts such as barriers, Asians, and lookbacks pay differently on different routes to the same ending price. A path signature is a stack of iterated integrals that compresses a path, so the expected exotic payoff can be approximated by a weighted sum of expected signature terms, in the same spirit as a Taylor series of moments.
Why it matters
Market makers can often quote vanilla options from supply and demand. Black-Scholes, the 1973 formula that links price to a few inputs, is then used in reverse: four inputs plus the market price imply the fifth, an implied volatility. Doing that for many strikes and maturities builds the surface. Discounted risk-neutral expectation, the average payoff after removing a risk premium and discounting, ties that formula to a probabilistic calculation through the Feynman-Kac link.
That surface is not a path model. Two parameter sets can be calibrated, fitted to the same surface, and still generate different families of paths. Local volatility, stochastic volatility, and mixtures such as local-stochastic volatility or the Heston-Bergomi family can share terminal distributions and disagree about the journey. The disagreement is model risk, the chance that the paths you simulate are the wrong paths. For a European payoff the disagreement may not change the expiry check. For a barrier it can decide whether the contract pays at all.
The idea in plain words
Three paths can start together and finish together. A European call pays the same on all three, because only the final price enters the payoff. A lookback, an Asian average, or a knockout barrier does not. Two models can be fitted to the same volatility surface and still disagree about the journey. Local volatility, stochastic volatility, and mixtures in the Heston-Bergomi family can share terminal distributions and differ on paths. That disagreement is model risk. For a barrier, it can decide whether the contract pays.
A payoff that depends only on the final price can be expanded as a polynomial. A risk-neutral expectation turns the polynomial into moments of the terminal distribution. Inside Black-Scholes, adding moments drives the approximation toward the closed form. In the lecture’s check, the match is already tight around the seventh moment.
A path-dependent payoff needs the whole trajectory. The signature of a continuous path of bounded variation is a stack of iterated integrals: a trivial entry, then a vector, then a matrix, then higher-rank tensors. Expected signature terms play the role moments played. In a Black-Scholes barrier example with a known analytic price, a signature stopped at level four produced 1.7376 against a reference of 1.7378. Higher levels capture more interaction along the path. Averaging over paths is why the weights sit on expected signatures, not on one coordinate. Packages such as Signatory compute the truncated object in Python.
How to use it
Use signatures when the contract’s payoff literally depends on the path and you already trust a model enough to simulate paths that match the liquid surface. Calibrate that model to today’s vanilla market, simulate, compute truncated signatures, and fit the combination that reproduces known prices in a setting where you can check them. Packages such as Signatory compute the truncated signature in Python.
What usually goes wrong
Calibrating only the surface and forgetting that many path models share it. Quoting an exotic from that calibration without a spread for model risk, inventory, liquidity, and the counterparty. Treating a European-style moment expansion as if it saw the path. And using signatures as features because the algebra looks modern, without a payoff that depends on the path.
Questions people ask
Does the volatility surface tell you the path?
No. It encodes terminal risk-neutral distributions for the listed contracts. Different dynamics can match those distributions and still disagree on barriers and averages.
Why is a signature like a Taylor series?
A Taylor series rebuilds a terminal payoff from powers of the final price. A signature expansion rebuilds a path payoff from iterated integrals of the whole path. Expectation turns both into a finite weighted sum.
Are signatures a trading signal?
Not in this argument. They summarize paths so an exotic price can be approximated. A bet on returns needs a separate economic case.