Systematic calibration of the mean‑reversion parameter a

Develop a systematic methodology to calibrate the (assumed constant) mean‑reversion speed a in the normalized fictitious spot price process s_t, defined by ds_t = a(1 − s_t) dt + sqrt(ξ_t^t) s_t dW_t^1 within the commodity rough‑volatility framework used to price futures options, recognizing that vanilla futures options alone are insufficiently sensitive to identify a and that the current practice is to fix a to a heuristic value.

Background

Within the proposed commodity rough‑volatility model, the normalized fictitious spot price s_t has dynamics ds_t = a(t)(1 − s_t) dt + sqrt(ξ_t^t) s_t dW_t^1, where the function a(t) controls mean reversion and the Samuelson effect. In practice, the authors assume a constant a because vanilla futures options do not provide enough sensitivity to estimate it reliably.

The paper notes that while prior work enriched the calibration set (e.g., including mid‑curve options) to address this, the present study fixes a = 0.5 for calibration to vanilla options. The authors explicitly identify the lack of a systematic calibration method for a as an open question.