Canonical Research Note: Originally published by the TrendSpike Institutional Systems & Quantitative Finance Desk. Full mathematical proofs, CME SPAN 2 clearing tables, and statutory 26 U.S.C. § 1256 tax accounting breakdowns are available in the primary publication.
1. Continuous-Time Cost-of-Carry Fair Value Pricing
Exchange-traded equity index futures (/ES, /NQ, /RTY) cleared on electronic central limit order books (CLOBs) are tethered to the underlying spot index ($S_t$) via continuous-time financing ($r_{\text{SOFR}}$) and constituent dividend yield ($q$) arbitrage bounds:
$$F_{t,T}^{*} = S_t \cdot e^{(r_{\text{SOFR}} - q)\tau}$$
Whenever pre-market macroeconomic releases drive the live futures price ($F_{\text{mkt}}$) outside the transaction-cost no-arbitrage band ($C_{\text{arb}}$), quantitative index arbitrage desks execute simultaneous Cash-and-Carry baskets at the 9:30 AM ET opening cross.
2. Python Reference Implementation: Implied Repo Rate & Basis Engine
From our open-source repository (abmughl/institutional-systems-and-quant-research):
python
import math
from dataclasses import dataclass
@dataclass
class IndexFuturesContract:
symbol: str
spot_index: float
sofr_rate: float
dividend_yield: float
days_to_expiry: int
market_futures_price: float
def compute_futures_fair_value(contract: IndexFuturesContract) -> dict:
tau = contract.days_to_expiry / 360.0
fair_value = contract.spot_index * math.exp(
(contract.sofr_rate - contract.dividend_yield) * tau
)
basis_spread = contract.market_futures_price - fair_value
forward_dividends = contract.spot_index * contract.dividend_yield * tau
implied_repo = (
(contract.market_futures_price + forward_dividends - contract.spot_index)
/ contract.spot_index
) * (360.0 / contract.days_to_expiry)
return {
"symbol": contract.symbol,
"theoretical_fair_value": round(fair_value, 2),
"basis_spread_pts": round(basis_spread, 2),
"implied_repo_rate_pct": round(implied_repo * 100, 4),
"irc_1256_blended_tax_rate_pct": round((0.60 * 20.0) + (0.40 * 37.0), 2),
}
