The Signature Cheyette Model

We introduce the Signature Cheyette model, a class of Heath--Jarrow--Morton term-structure models whose forward-rate volatility is linear in a truncated Stratonovich signature. Shuffle multiplication closes the HJM drift. For finite exponential maturity bases, every cross term remains signature linear and the curve has a finite-dimensional realization; the one-exponential caserecovers the two-state Cheyette model. The signature lift is a polynomial diffusion with finite moment systems. These results make the shuffle algebra operational rather than decorative and separate the maturity-basis dimension from the sparse signature closure required by a chosen loading. Thus higher-level path dependence can be implemented without retaining the full truncated tensor state space. In an orthogonal subclass, the volatility excludes the Brownian rate driver. The second Cheyette state is the conditional variance of the rate state, and a filtering reduction with Malliavin nondegeneracy proves unspanned stochastic volatility. Discounted bonds are true martingales, forward measures preserve the volatility law, bond options are random-variance mixtures, and swaptions admit conditional Jamshidian representations. Single-forward implied variance is symmetric in log-moneyness, so leverage skew lies outside this exact subclass. For leveraged loadings, a tilted-diffusion explosion criterion exactly characterizes martingality. Riccati transforms quantify skew for certified level-one models.Deterministic transforms, a level-two sparse implementation, a machine-checked sum-of-squares certificate, and a mean-reverting CIR comparison verify tractability. A historical ECB exercise is reported only as a negative boundary diagnostic; it is not option-market validation. Code, frozen data, tests, and vector figures reproduce the reported numerical claims.