Tradable Itô Signatures: A Model-Free, Interpretable Framework for Dynamic Hedging
Xin Guo, Binnan Wang, Ruixun Zhang
q-fin.CP, q-fin.MF, stat.ML
2026-07-15
Discretized Itô signature terms are tradable self-financing bases. Option hedges become linear combinations; a signature kernel helps most on path-dependent SPX contracts.
Dynamic hedging has to approximate a nonlinear, often path-dependent, payoff on a discrete trading grid using only the underlyings and cash. Black-Scholes delta is exact for European options under geometric Brownian motion and brittle once the contract or the market leaves that box. Deep hedging fits positions with a neural net. That is flexible, hard to audit, sample-hungry, and usually tied to estimating future conditional expectations or to a heavy nonlinear training loop.
Path signatures turn a price path into iterated-integral features whose linear span can approximate continuous functionals of the path. The approximation theorems sit most naturally on Stratonovich or rough-path signatures. Trading needs the non-anticipative Itô integral. Existing signature hedges typically lift the path with a Hoff lead-lag transform and then estimate expected future signatures, which explodes the state dimension and adds an extra estimation layer.
The main theorem is that every non-constant discretized Itô signature coordinate is the terminal gain of an adapted, self-financing strategy in the underlyings and cash. Coordinates that use only the time component are cash. Higher-order positions are built from the already realized PnL of lower-order strategies, with no auxiliary options.
Hedging then has three steps. Fit the payoff as a linear combination of truncated Itô coordinates with OLS or Lasso. Map each coordinate to its self-financing basis strategy. Add those strategies with the same coefficients. The fitted weights are the trades. There is no second translation from feature space into an adapted rule.
Theory fills the Itô approximation gap. Discrete coordinates converge in Lp to their continuous Itô counterparts. For diffusions that admit a deterministic map sending the diffusion matrix to the identity, finite-order Itô expansions uniformly approximate continuous path functionals. A high-dimensional regression bound then splits out-of-sample mean-square hedging error into discretization, approximation, and estimation. Empirically, a signature kernel reweights historical paths toward those close to the current trajectory, a response to non-stationary market data.
Simulations use risk-neutral GBM with four contracts: European call and put, geometric Asian call, floating-strike lookback put. Daily rebalancing, n=250, training sizes from 2^6 to 2^15, 10,000 test paths, 10 seeds. In the small-sample regime the Itô hedge is stable and beats the Stratonovich-signature baseline across all four payoffs. The neural net is unreliable until the sample is large, at which point it can catch up or slightly win. Runtime is the larger gap. For a European call at training size 2^8, mean times are 0.12s for Itô, 69.43s for Stratonovich, and 1,286.73s for the net; at 2^15 the same numbers are 19.18s, 6,998.05s, and 39,007.69s.
The market study uses S&P 500 weekly options from 2011 to 2025, 436,135 contracts in the test set. Mean error at expiry, times 10^{-3}: Black-Scholes 6.007 overall; unweighted Itô 5.744 with a 73.3% win rate against BS; signature-kernel Itô 5.445 and 83.9%; the neural net 8.836 and 26.1%. Kernel weighting helps most on short-dated and out-of-the-money buckets.
Path-dependent contracts have no closed-form delta, so the baseline is a Monte Carlo hedge. Kernel-weighted Itô has the lowest mean error on all four exotic payoffs. Lookback calls: Monte Carlo 26.66, unweighted Itô 19.55, kernel 12.90, neural net 38.25. Unweighted Itô loses to Monte Carlo on Asians; the kernel flips that ranking. Path-similarity weighting is doing real work in rolling, non-stationary estimation.
This is interpretability that reaches the trade blotter. The features are tradable basis strategies, and the coefficients are positions. No conditional-expectation forecast, no black-box hedge net. For desks that re-estimate on short windows and have to explain positions, a linear Itô basis is closer to a usable tool than deep hedging. It changes the representation, it does not drive hedging error to zero.
The approximation theorem needs invertibility and conservative row fields of the diffusion matrix; not every local-vol model qualifies. Unweighted Itô is worse than Monte Carlo on real-path Asians, so the kernel and the rolling window are load-bearing knobs without a selection theory. Exotic contracts are synthesized on the cash index, not OTC prints. High-frequency data, multi-asset books, and adaptive signature depth are left open. The simulation baseline is one deep-hedging net and one Stratonovich pipeline.