Methodology
The Fundamental Equation
C(T, M, t) = S(T, M) * D(M, t)
Total expected cost equals spot cost times the decay factor. At spot (t = 0), D = 1 and you get the exact, observable cost.
Ticker Price beta
The anchor of the model: beta is the published output token price at the origin provider, in USD per million tokens ($/M).
MODELSync output reference price
MODEL.BBatch output price
Structural Greeks
| Greek | Definition | Range |
|---|---|---|
| r_in | Input/output price ratio | 0.20 - 0.50 |
| r_cache | Cache price as fraction of output | 0.01 - 0.10 |
| r_think | Thinking token price ratio | 0.50 - 1.00+ |
| r_batch | Batch discount ratio | 0.40 - 0.60 |
Effective Input Rate
r_in_eff = r_in_depth * (1 - eta) + r_cache * eta
Combines context-depth pricing (for tiered pricing models) with cache discounts. eta is your cache hit ratio (0 to 1).
kappa - The Task's Delta
kappa = 1 + (n_in / n_out) * r_in_eff
kappa is both the context cost multiplier and your delta to beta movements. If beta moves by $1/M, your task cost moves by kappa * n_out * 10^-6.
Spot Cost
S = beta * [n_out + n_in * r_in_eff + n_think * r_think] * 10^-6
Decay Rate theta
theta is the continuous monthly decay rate, estimated from historical price data. It absorbs all sources of price decline into a single continuous rate.
theta > 0Price declining (typical)
theta = 0Stable pricing
theta < 0Price increasing (rare)
Forward Price
beta_fwd(M, t) = beta(M) * e^(-theta(M) * t)
Published at standard tenors: 1M, 3M, 6M.