Black-Scholes for the land: modelling savannah response to rain as a dynamic world model
Abstract
Carbon credits, offsets and index insurance rest on counterfactuals: what the land would have done under the same weather without management, or what a bad season cost. Current baselines fail in two ways: some ignore rain, others condition on rain linearly and annually. But the true response lags the rain, depends on hidden state such as moisture from other sources, and responds differently from pixel to pixel. Focussing on grassland pasture as a proof of concept, we fit a temporal world model of MODIS greenness conditioned on CHIRPS rain over two 45 km^2 sites in Laikipia, Kenya: a transition density map for greenness given recent and current rainfall. We use it, conceptually, as Black-Scholes is used. Inverting the model on observed greenness it yields an implied residual, everything the land did that the rain does not explain. Two capabilities drop out of this solution: 1) attribution, what the land did beyond the rain; 2) sensitivities, derivatives of the rollout with respect to rain and state, close analogues of the Greeks. On Laikipia test sites, our learned model recovers three-month greenness with 23% less error than a history-only baseline and 13% less than the best hand-written model on unseen years. In the 2022 drought, the land ended 0.17 NDVI below normal; the rain that failed explains roughly half of that, and under normal rain with the same residual it would still have ended 0.10 below. Delta, the derivative of greenness with respect to rain, is computed at every pixel and averages 0.075 NDVI per extra 100 mm of rain. A Delta at every pixel is an exposure to rain that can be hedged, allowing efficient pricing of contracts and accurate attribution.
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BibTeX
@misc{pitts2026blackscholes,
title = {{Black-Scholes for the land: modelling savannah response to rain as a dynamic world model}},
author = {Thomas Pitts},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03744},
eprint = {2610.03744},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03744},
}
RIS
TY - UNPB
TI - Black-Scholes for the land: modelling savannah response to rain as a dynamic world model
AU - Thomas Pitts
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.03744
UR - https://arxiv.org/abs/2610.03744
AB - Carbon credits, offsets and index insurance rest on counterfactuals: what the land would have done under the same weather without management, or what a bad season cost. Current baselines fail in two ways: some ignore rain, others condition on rain linearly and annually. But the true response lags the rain, depends on hidden state such as moisture from other sources, and responds differently from pixel to pixel. Focussing on grassland pasture as a proof of concept, we fit a temporal world model of MODIS greenness conditioned on CHIRPS rain over two 45 km^2 sites in Laikipia, Kenya: a transition density map for greenness given recent and current rainfall. We use it, conceptually, as Black-Scholes is used. Inverting the model on observed greenness it yields an implied residual, everything the land did that the rain does not explain. Two capabilities drop out of this solution: 1) attribution, what the land did beyond the rain; 2) sensitivities, derivatives of the rollout with respect to rain and state, close analogues of the Greeks. On Laikipia test sites, our learned model recovers three-month greenness with 23% less error than a history-only baseline and 13% less than the best hand-written model on unseen years. In the 2022 drought, the land ended 0.17 NDVI below normal; the rain that failed explains roughly half of that, and under normal rain with the same residual it would still have ended 0.10 below. Delta, the derivative of greenness with respect to rain, is computed at every pixel and averages 0.075 NDVI per extra 100 mm of rain. A Delta at every pixel is an exposure to rain that can be hedged, allowing efficient pricing of contracts and accurate attribution.
ER -