Dividend Futures And Forward Modeling

When to Use

Use this skill in equity index desk trading, forward curve construction, and index arbitrage strategies. Equities and stock indices (S&P 500, EURO STOXX 50) distribute discrete cash dividends, so the theoretical forward price $F(0, T)$ must account for the present value of expected discrete dividends $\text{PV}(D)$. Dividend futures — Eurex EURO STOXX 50 Index Dividend Futures (product ID FEXD) and CME S&P 500 Annual Dividend Index Futures (SDA) — isolate dividend risk so quants can trade dividend expectations independently of spot.

The engine computes two different dividend measures, and confusing them is the most common way to misprice this product:

When NOT to Use

• Single-stock American options with discrete dividends — the escrowed-dividend forward here does not handle early-exercise boundaries. Use a dividend-aware binomial/PDE model.
• Continuous-yield index approximations — if you are pricing a broad index over a long horizon with a continuous $q$, this engine's discrete-schedule machinery adds precision you are not using. Its value is near-dated, where the exact ex-date placement matters.
• Stochastic-dividend or stochastic-rate valuation — the fair value here is the deterministic expected accrual. It carries no convexity adjustment and no dividend-volatility term, so it will not price options on dividend futures (Eurex OEXD).
• Sizing a trade directly from the output — all figures are per unit (per share, or per index point). They are not scaled by a contract multiplier.

Prerequisites

• Spot price $S_0 > 0$, risk-free rate $r$ (decimal, e.g. 0.05 for 5%; negative rates are supported), time to maturity $T > 0$ in years.
• Expected discrete dividend events: gross amount $D_i$, payment time $t_i^{pay}$, and where they differ, the ex-date $t_i^{ex}$.
• Withholding tax rate per event in $[0, 1)$ where cross-border tax applies.
• Market forward price or dividend futures quote, in the same units and currency as the spot.
• Round-trip transaction cost estimates — separately for the forward and reverse legs.

Workflow

1. Ingest the dividend schedule with both dates. A dividend has an ex-date (when eligibility is fixed) and a payment date (when cash arrives); they are different, and this engine keeps them apart. Supply ex_time_years whenever it differs from payment_time_years — omitting it makes the engine assume they coincide, which misplaces a dividend that goes ex just before expiry and pays after it. Flag special/extraordinary dividends with is_special=True.
2. Bound the accrual window at both ends. Filtering only on $t_i \le T$ is not enough. A dividend that has already gone ex must be excluded: including it overstates $\text{PV}(D)$, understates the theoretical forward, and manufactures a false ARBITRAGE_SHORT_FORWARD_LONG_SPOT against an honest quote. For a dividend future, set accrual_start_years to the contract's accrual start — the index resets to zero after the leading contract expires, so pre-window dividends belong to the previous contract. Check excluded_dividend_ids on the report to confirm the window did what you intended.
3. Compute PV/FV on net dividends, accrual on gross.
   • $\text{PV}(D) = \sum_i D_i (1 - \tau_i) e^{-r t_i^{pay}}$ over $t_i^{ex}$ in the window.
   • $\text{FV}(D) = \sum_i D_i (1 - \tau_i) e^{r (T - t_i^{pay})}$.
   • Dividend-future fair value $= \sum_i D_i$ over ordinary (non-special) events in the window, gross.
4. Price the forward. $F_{\text{theo}}(0, T) = (S_0 - \text{PV}(D)) e^{r T} = S_0 e^{r T} - \text{FV}(D)$. Both identities are computed and agree; if they diverge, the schedule has been mutated between calls.
5. Audit the spread, then decide by direction — not by magnitude. $\Delta = F_{\text{market}} - F_{\text{theo}}$.
   • $\Delta > \text{cost}_{\text{fwd}} \implies$ ARBITRAGE_SHORT_FORWARD_LONG_SPOT (sell the rich forward, buy and carry the shares).
   • $\Delta < -\text{cost}_{\text{rev}} \implies$ ARBITRAGE_LONG_FORWARD_SHORT_SPOT. This leg is not the mirror image. It requires borrowing the stock, so it carries a borrow fee, recall risk, and may be impossible in hard-to-borrow names; the short also pays gross manufactured dividends while a long holder receives them net of withholding. Set reverse_arbitrage_cost_threshold_usd explicitly — the symmetric default systematically overstates reverse-leg opportunities.
   • Otherwise NO_ARBITRAGE. The spread must strictly exceed the threshold.
6. Read the report before acting. estimated_gross_profit_usd is $|\Delta|$ before costs; estimated_net_profit_usd subtracts the threshold that actually applied, reported in applied_cost_threshold_usd. Treat a non-empty warnings list — especially a non-positive theoretical forward — as a data-quality stop, not a trade.

Full procedure: see references/workflows.md. Standards reference: see references/standards.md. Printable pre-flight checklist: see assets/checklist.md.

Common Pitfalls

• Netting withholding tax out of the dividend-futures fair value. The dividend-point indices behind FEXD and SDA accumulate gross dividends. Applying a 15% withholding to the accrual understates the fair value of a $4.00 accrual by $0.60 — larger than most arbitrage thresholds, so the error alone can flip the signal.
• Summing special dividends into the index accrual. Special/extraordinary dividends are excluded from the dividend indices, but they do depress the forward, because the shareholder receives the cash. One schedule, two different filters.
• Treating the payment date as the ex-date. Eligibility is set on the ex-date; cash timing is the payment date. Collapsing them misprices any dividend that goes ex near expiry, and the error is largest exactly where near-dated forwards are most traded.
• Leaving already-ex dividends in the feed. A dividend paid last quarter is not a claim on a forward buyer. Including one at $2.00 cuts a $105.13 theoretical forward to $103.03 and reports a $2.10 "arbitrage" that does not exist.
• Using a continuous dividend yield for individual stocks. Applying a continuous $q$ to single-stock forwards instead of the discrete schedule misprices near-term contracts, where a single ex-date dominates.
• Assuming symmetric arbitrage costs. The reverse cash-and-carry needs stock borrow; pricing both legs off one threshold generates reverse-leg signals that cannot be executed.
• Trading estimated_gross_profit_usd as if it were a P&L. It is per unit and before costs. Multiply by the venue multiplier (Eurex FEXD: EUR 100/point; CME SDA: USD 250/point) and subtract the threshold before sizing.
• Letting a NaN through. A NaN spot or dividend used to yield a confident NO_ARBITRAGE, because every NaN comparison is False. Non-finite inputs are now rejected — do not catch and ignore that error.

Verification

• Instantiate DividendForwardModelingEngine. Input $S_0 = 100.0$, $r = 5%$, $T = 1.0$. Add two dividends of $2.00 at $t_1 = 0.25$ and $t_2 = 0.75$. Expect $\text{PV}(D) \approx 3.9015$, $\text{FV}(D) \approx 4.1016$, and $F(0,T) \approx 101.0256$ — and confirm $(S_0 - \text{PV}(D))e^{rT}$ equals $S_0 e^{rT} - \text{FV}(D)$. Submit a market forward of $104.00 and verify ARBITRAGE_SHORT_FORWARD_LONG_SPOT with gross profit $\approx 2.97$ and net $\approx 2.47$ at a $0.50 threshold.
• Re-run the same schedule with withholding_tax_pct=0.15 on both events: $\text{PV}(D)$ must fall to $\approx 3.3163$ while fair_value_dividend_future_points stays at 4.00 gross, and the theoretical forward must rise (a taxed holder loses less to dividends).
• Add a $5.00 is_special=True dividend: it must enter $\text{PV}(D)$ but leave the accrual at 4.00.
• Confirm the window guards: a dividend at $t = 1.5$ and one at $t = -0.25$ must both land in excluded_dividend_ids, and the stale one must leave $\text{PV}(D) = 0$ rather than flagging arbitrage.
• Confirm invalid inputs raise DividendModelError: NaN/Inf spot or rate, $S_0 \le 0$, $T \le 0$, withholding_tax_pct outside $[0,1)$, negative dividend amounts, and an ex-date after its payment date.
• Run python -m unittest discover -s skills/dividend-futures-and-forward-modeling/scripts.

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