---
title: "Delta-One and Dividends"
book: "Strategies II: Volatility, Relative Value, Macro and the Bank Desks"
subject: quant
language: en
chapter: 26
exercises: 8
source: https://one-course.com/books/quant/9/en/chapter/26-delta-one-and-dividends
---

# Chapter 26 — Delta-One and Dividends

A bank that sells autocallables is left long dividends it did not want, and it sells them in dividend futures and swaps. Manley and Mueller-Glissmann reported that from 2004 dividend swaps were a major source of profit for multistrategy and macro hedge funds, because market-implied dividends traded at high discounts while dividends grew. In this chapter’s market, issuers’ selling puts dividend futures 4% below expected dividends for each year of maturity. Buying the one-year future and holding it to expiry earns 4.2% a year on average, but loses in one year in five, and 24% in a recession year. The same desk runs [index arbitrage](#def-s2-delta-one-and-dividends-indexarb) and [financing trades](#def-s2-delta-one-and-dividends-financing) on the index future. The build is `firm.deltaone`.

## 26.1 Index arbitrage at a bank

**Definition 26.1 (Index arbitrage).**

*Index arbitrage* is trading an index future against the basket of its constituents when the future’s price departs from its fair value (Book 1, chapter 21) by more than the cost of trading both, and unwinding when the gap closes.

`firm.deltaone` simulates a year of minute-by-minute mispricings of the future around fair value, with a standard deviation of 4 basis points of the index and a half-life of 10 minutes ([Listing 26.1](#lst-s2-delta-one-and-dividends-arb)). Trading the basket costs 2 basis points a side, for a bank that crosses part of it internally (chapter 25), and the future 0.5, so a round trip costs 5 basis points. The desk sells the future and buys the basket when the future is more than 5 basis points rich, the reverse when cheap, and unwinds when the gap closes:

| execution lag | trades a year | bp per trade | bp a year | trades winning |
| --- | --- | --- | --- | --- |
| none | 2 230 | 1.64 | 3 657 | 100% |
| 1 minute | 2 230 | 1.14 | 2 539 | 69.2% |
| 2 minutes | 2 230 | 0.77 | 1 707 | 60.7% |
| 5 minutes | 2 230 | $-0.26$ | $-584$ | 46.9% |

Without a lag the trade cannot lose, since it enters beyond the cost and leaves at zero. Every minute of delay gives back part of a gap that closes in ten, and at five minutes the trade loses money. [Index arbitrage](#def-s2-delta-one-and-dividends-indexarb) is a race whose prize is a basis point, and it belongs to whoever executes a basket fastest and cheapest.

## 26.2 Financing trades

**Definition 26.2 (Financing trade).**

A *financing trade* is holding an asset against a derivative that delivers it later (a basket against a short future, or a total-return swap in which the bank pays a client the asset’s return), earning the financing spread implied by the derivative’s price less the bank’s own cost of funding and of balance sheet.

A future’s price above spot, less dividends, implies a rate at which the market finances the basket. The synthetic implied rate is the overnight rate plus a spread that averages about 20 basis points and reverts with a half-life of 40 days, plus 25 basis points in the last ten days of each quarter, when banks’ balance sheets are scarce ([Figure 26.1](#fig-s2-delta-one-and-dividends-spread)). The bank funds itself at 10 basis points over the overnight rate and charges 5 for the balance sheet, a hurdle of 15. Buying the basket and selling a three-month future locks the implied spread for the quarter:

| entered each quarter | quarters taken | mean spread locked (bp) | bp of notional a year |
| --- | --- | --- | --- |
| at the roll | 10% | 15.4 | 0.4 |
| in the quarter-end window | 67.5% | 40.7 | 8.0 |

At the roll, the spread rarely clears the hurdle and the round-trip cost. In the quarter-end window it usually does: the bank with balance sheet to spare is paid by those who have none. A total-return swap is the same trade in another form, the bank holding the basket and paying the client its return for a spread over funding.

![Three years of the synthetic futures-implied financing spread over the overnight rate, with its quarter-end jumps, against the bank’s own funding and capital hurdle of 15 basis points. Data: s2_deltaone.market.](https://one-course.com/images/onecourse/chapters/quant-9/s2-delta-one-and-dividends/fig-b68781d6e70c.svg)

***Figure 26.1.** Three years of the synthetic futures-implied financing spread over the overnight rate, with its quarter-end jumps, against the bank’s own funding and capital hurdle of 15 basis points. Data: `s2_deltaone.market`.*

## 26.3 Dividend risk from structured issuance

An autocallable’s buyer is in effect short a put on the index; the issuer holds the other side and hedges its delta by owning the index or its future. Higher dividends lower the index’s forward, raise the put’s value and lower the note’s value, a gain to the issuer, which is therefore long dividends. Book 5’s pricer (chapter 18) values a five-year autocallable with annual observations, an 8% Phoenix coupon at a 70% barrier and 60% protection at 101.84 per 100 at a 3% dividend yield. A rise of 0.1 percentage point in the dividend yield lowers it by 0.11 ([Listing 26.2](#lst-s2-delta-one-and-dividends-div)): on a billion of notes, about 1.1 million for 10 basis points of dividend yield, over an expected life of 2.66 years.

**Definition 26.3 (Dividend supply pressure).**

*Dividend supply pressure* is the discount of dividend futures or swaps to expected dividends that arises when hedgers who are naturally long dividends, such as issuers of structured products, sell more than other investors want to buy at fair value.

**As of September 2026 — Dividend futures.**

Eurex’s EURO STOXX 50 Index Dividend Futures (FEXD) pay EUR 100 per point of the index’s dividends over the contract year, with expiries up to ten years, cash settled on the third Friday of the expiry month. Eurex describes dividend derivatives as letting investors take positions in, or hedge, future dividend payments, particularly for structured products and equity options.

## 26.4 Dividend trades

The synthetic dividend market has expected dividends of 100.8 index points next year, growing 3% a year, with 6% of yearly noise and a 12% chance each year of a recession that cuts that year’s dividends by 30% and the index by 25%. Futures trade at expected dividends less 4% for each year of maturity, the planted supply discount, so their term structure slopes down while expected dividends rise ([Figure 26.2](#fig-s2-delta-one-and-dividends-dividends)). A buyer holding each future to expiry, over 20 000 scenarios:

| maturity (years) | 1 | 2 | 3 | 4 | 5 |
| --- | --- | --- | --- | --- | --- |
| expected dividends | 100.8 | 104.7 | 108.7 | 112.9 | 117.2 |
| future | 96.7 | 96.3 | 95.7 | 94.9 | 93.7 |
| mean return a year (%) | 4.2 | 3.9 | 3.8 | 3.9 | 3.9 |
| standard deviation a year (%) | 12.3 | 9.1 | 7.5 | 6.6 | 5.9 |
| 5% quantile a year (%) | $-25.5$ | $-13.2$ | $-9.4$ | $-8.6$ | $-7.1$ |
| chance of a loss (%) | 20.9 | 25.2 | 29.8 | 28.0 | 24.6 |

The one-year trade’s return has a correlation of 0.41 with the index’s return in the same year and loses 24.4% on average in a recession year. The dividend buyer is paid for supplying the insurance that the structured-product issuer does not want, and pays out in the same years as equities. Van Binsbergen, Brandt and Koijen, recovering the prices of dividend strips, found that short-term dividend claims had higher expected returns, Sharpe ratios and volatilities than the index, with CAPM betas below one. Kragt, de Jong and Driessen fitted the EURO STOXX 50 dividend futures curve with two factors, one reverting within a year and one over the business cycle, and found that investors revise the value of dividends beyond the business cycle only a little.

![Expected dividends by year and the synthetic dividend futures curve, 4% below them per year of maturity. Data: s2_deltaone.dividends.](https://one-course.com/images/onecourse/chapters/quant-9/s2-delta-one-and-dividends/fig-e1a8c610655a.svg)

***Figure 26.2.** Expected dividends by year and the synthetic dividend futures curve, 4% below them per year of maturity. Data: `s2_deltaone.dividends`.*

## 26.5 Strategy files

**Strategy file 26.1 — Index arbitrage.**

**Who pays you, and why.** Traders who move the future ahead of the basket or the basket ahead of the future.

**Instruments and venues.** Index futures; the constituent basket; ETFs.

**Signal.** The future against fair value with the market’s financing and dividends.

**Sizing and execution.** Beyond the round-trip cost; the basket executed in one program.

**Costs.** Basket and futures spreads; latency.

**How it dies.** Faster competitors; a wrong dividend or financing input.

**Horizon, capacity, infrastructure.** Minutes; basket execution and low latency.

**Backtest honestly.** Execute after the signal, at the prices then available.

**Sources.** This chapter: 1.14 basis points per trade at a one-minute lag, a loss at five minutes.

**Strategy file 26.2 — Total-return swap financing.**

**Who pays you, and why.** Investors who want the index’s return without funding it; scarce balance sheet at quarter-ends.

**Instruments and venues.** Total-return swaps; basket against futures.

**Signal.** The implied financing spread against the bank’s funding and capital charge.

**Sizing and execution.** Within balance-sheet limits; more in the quarter-end window.

**Costs.** Funding; capital; basket trading.

**How it dies.** Regulation that raises the capital charge; competition for balance sheet.

**Horizon, capacity, infrastructure.** Months; treasury and balance-sheet allocation.

**Backtest honestly.** The bank’s own funding and capital costs at the time.

**Sources.** This chapter: 8.0 basis points of notional a year entered at quarter-end.

**Strategy file 26.3 — Long dividends against structured supply.**

**Who pays you, and why.** Issuers of structured products selling the dividends their hedges leave them with.

**Instruments and venues.** Index dividend futures and swaps.

**Signal.** The discount of the futures to forecast dividends.

**Sizing and execution.** Held to expiry; sized for a recession cut.

**Costs.** Futures spreads; margin.

**How it dies.** Dividend cuts in recessions, including regulatory bans on bank dividends.

**Horizon, capacity, infrastructure.** One to five years.

**Backtest honestly.** Include recession years; forecast dividends as of the trade date.

**Sources.** Manley and Mueller-Glissmann (2008); van Binsbergen, Brandt and Koijen (2012); this chapter: 4.2% a year, $-24.4\%$ in a recession year.

**Strategy file 26.4 — Dividend calendar spread.**

**Who pays you, and why.** Supply concentrated at the maturities structured products need.

**Instruments and venues.** Dividend futures of two maturities.

**Signal.** The curve’s slope against expected dividend growth.

**Sizing and execution.** Weighted so that a parallel change in dividends nets out.

**Costs.** Two legs.

**How it dies.** A recession that cuts near dividends more than far ones.

**Horizon, capacity, infrastructure.** Years.

**Backtest honestly.** A model of the term structure fitted without the future.

**Sources.** Kragt, de Jong and Driessen (2020); no performance figure verified.

## 26.6 Tutorial: dividends nobody wanted

**Goal.** Run [index arbitrage](#def-s2-delta-one-and-dividends-indexarb) with execution lags, [financing trades](#def-s2-delta-one-and-dividends-financing) at the roll and at quarter-end, the issuer’s dividend exposure from an autocallable, and a dividend market with supply pressure. **End state:** the four tables and two figures.

1. **Arbitrage and financing**. `def index_arb (sim: dict , cfg: DeltaOneConfig | None = None , lag: int = 1 ) -> dict : """Sell the future and buy the basket when the future is rich by more than the round-trip cost (the reverse when cheap); unwind when the mispricing crosses zero. Both legs execute `lag` observations after the signal, when part of the mispricing has gone. P&L per trade in bp of notional.""" cfg = cfg or DeltaOneConfig() band = 2 * (cfg.basket_cost + cfg.futures_cost) m, pnl, pos, entry = sim[" mispricing " ], [], 0 , 0.0 for t in range (len (m) - lag): x = m[t] if pos == 0 and abs (x) > band: pos, entry = (-1 if x > 0 else 1 ), m[t + lag] elif pos != 0 and np.sign(x) != -pos: pnl.append(pos * (m[t + lag] - entry) - band) pos = 0 return {" pnl " : np.array(pnl), " band " : band} def financing_trade (sim: dict , cfg: DeltaOneConfig | None = None , offset: int = 0 ) -> dict : """Each quarter, `offset` days into it, buy the basket and sell a future expiring a quarter later, locking the implied spread; the bank pays its own funding and a capital charge, and the round-trip cost. P&L per trade in bp of notional; a trade is taken only when it is expected to pay.""" cfg = cfg or DeltaOneConfig() cost = 2 * (cfg.basket_cost + cfg.futures_cost) locked = sim[" spread " ][offset::QUARTER] net = (locked - cfg.own_funding - cfg.capital_charge) * QUARTER / YEAR - cost take = net > 0 return {" locked " : locked, " net " : net, " taken " : take, " pnl " : np.where(take, net, 0.0 )}` **Listing 26.1.** Fading mispricings after a lag; locking the implied financing spread. code/firm/deltaone/firm_deltaone.py
2. **Dividends**. `def issuer_dividend_exposure (vol: float = 0.2 , r: float = 0.03 , q: float = 0.03 , bump: float = 0.001 , n_paths: int = 40_000 , seed: int = 18 ) -> dict : """A five-year annual-observation autocallable (trigger 100%, 8% Phoenix coupon at a 70% barrier, 60% protection at maturity). Value per 100 at q and q + bump with common random numbers: the issuer, short the note, gains when dividends rise.""" ts = TermSheet(obs_times=(1.0 , 2.0 , 3.0 , 4.0 , 5.0 ), trigger=1.0 , coupon=8.0 , coupon_barrier=0.7 , protection=0.6 ) lo = price(ts, lambda t, s: vol, r, q, n_paths, seed) hi = price(ts, lambda t, s: vol, r, q + bump, n_paths, seed) life = sum ((j + 1 ) * p for j, p in enumerate (lo[" call_probs " ])) + 5.0 * lo[" maturity_prob " ] return {" price " : lo[" price " ], " bumped " : hi[" price " ], " issuer_gain " : lo[" price " ] - hi[" price " ], " expected_life " : life} def dividend_market (cfg: DeltaOneConfig | None = None ) -> dict : """Scenarios of the next `years` of dividends: growth noise and recession years that cut dividends and the index together. Futures for year y trade at expected dividends x (1 - discount x y). Holding returns to expiry.""" cfg = cfg or DeltaOneConfig() rng = np.random.default_rng(cfg.seed + 1 ) n, Y = cfg.scenarios, cfg.years rec = rng.random((n, Y)) < cfg.recession_p cut = math.log(1 - cfg.recession_cut) g = cfg.div_growth - cut * cfg.recession_p + cfg.div_vol * rng.standard_normal((n, Y)) + cut * rec level = cfg.div_level / math.exp(cfg.div_growth) * np.exp(np.cumsum(g, axis=1 )) expected = level.mean(axis=0 ) years = np.arange(1 , Y + 1 ) futures = expected * (1 - cfg.discount * years) ret = level / futures - 1 equity = 0.06 + 0.15 * rng.standard_normal((n, Y)) - 0.25 * rec # the index in the same years return {" expected " : expected, " futures " : futures, " realised " : level, " return " : ret, " annual " : (1 + ret) ** (1 / years) - 1 , " recession " : rec, " equity " : equity}` **Listing 26.2.** The issuer’s dividend exposure; dividend futures with a supply discount. code/firm/deltaone/firm_deltaone.py
3. **Run** `arbitrage()` , `financing()` , `exposure()` , `dividends()` and `fig_deltaone.py` .

**What to change next.** Let the supply discount rise with issuance and fall after a recession; hedge the dividend trade with index puts; add a dividend-ban scenario for banks.

## 26.7 Build: delta-one

**Purpose.** [Index arbitrage](#def-s2-delta-one-and-dividends-indexarb), [financing trades](#def-s2-delta-one-and-dividends-financing), the issuer’s dividend exposure and a dividend market with supply pressure.

**Interface.** `DeltaOneConfig(…)`, `simulate_financing(cfg)`, `index_arb(sim, cfg, lag)`, `financing_trade(sim, cfg, offset)`, `issuer_dividend_exposure(vol, r, q, bump)`, `dividend_market(cfg)`.

**Rules.** Arbitrage beyond the round-trip cost, executed after a lag; [financing trades](#def-s2-delta-one-and-dividends-financing) only when they clear the hurdle; futures at expected dividends less the discount.

**Acceptance tests.** `code/firm/deltaone/tests/`: no lag, no loss; the [financing trade](#def-s2-delta-one-and-dividends-financing) by hand; the issuer is long dividends; no discount, no expected return.

**Stretch.** Dividend swaps with single stocks; supply that responds to issuance; stochastic interest rates.

Sources and further reading

- J. van Binsbergen, M. Brandt and R. Koijen, “On the timing and pricing of dividends”, *American Economic Review* 102(4), 2012.
- R. Manley and C. Mueller-Glissmann, “The market for dividends and related investment strategies”, *Financial Analysts Journal* 64(3), 2008.
- J. Kragt, F. de Jong and J. Driessen, “The dividend term structure”, *Journal of Financial and Quantitative Analysis* 55(3), 2020.
- Eurex, “Dividend derivatives”, factsheet, accessed 25 September 2026.

## 26.8 Exercises

**Exercise 26.1 ★.**

The future is 7 basis points rich and a round trip costs 5. The gap closes completely. What does the arbitrage earn on $50 million?

**Solution of Exercise 26.1.**

$(7 - 5)$ basis points of $50 million: $10 000.

**Exercise 26.2 ★.**

The implied spread is 40 basis points, the bank’s hurdle 15, and a quarter’s round trip costs 5 basis points. What does a quarter’s [financing trade](#def-s2-delta-one-and-dividends-financing) earn on $100 million?

**Solution of Exercise 26.2.**

$(40 - 15) \times 0.25 - 5 = 1.25$ basis points of $100 million: $12 500 for the quarter.

**Exercise 26.3 ★.**

Expected dividends are 108.7 points and the future trades at 95.7. What does holding it to expiry earn if dividends come in as expected, in total and a year over three years?

**Solution of Exercise 26.3.**

$108.7 / 95.7 - 1 = 13.6\%$ in total, $1.136^{1/3} - 1 = 4.3\%$ a year. The simulated mean, 3.8% a year, is lower because recession years cut dividends and the average of annualised returns is below the annualised average.

**Exercise 26.4 ★★.**

Why is the issuer of an autocallable long dividends?

**Solution of Exercise 26.4.**

The note’s buyer is short a put on the index through the protection; the issuer is long it and hedges by holding the index. Higher dividends lower the index’s forward and raise the put’s value to the issuer, lowering the note it owes: a 0.1 point rise in the dividend yield gains it 0.11 per 100.

**Exercise 26.5 ★★.**

Short-term dividend claims earned more than the index in van Binsbergen, Brandt and Koijen. Why might supply explain part of it?

**Solution of Exercise 26.5.**

If issuers of structured products sell short-dated dividends to hedge, the buyers who take them are paid a discount, which shows up as a high expected return on short-term dividend claims; a risk premium and a supply premium are hard to tell apart in prices alone.

**Exercise 26.6 ★★.**

Why does the implied financing spread jump at quarter-ends?

**Solution of Exercise 26.6.**

Banks report and are charged on their balance sheets at quarter-ends, so they shrink them then; the few that can hold the basket against the future demand more for doing so, and the future’s implied financing rate rises.

**Exercise 26.7 ★★★.**

*Coding.* Run `index_arb` with lags of 1, 2 and 5 minutes. At what lag does the trade stop paying, and why?

**Solution of Exercise 26.7.**

Mean P&L per trade is 1.14 basis points at 1 minute, 0.77 at 2, 0.40 at 3, 0.05 at 4 and $-0.26$ at 5: the trade stops paying between four and five minutes. With a half-life of 10 minutes, each minute of delay gives back part of a gap that is only a few basis points beyond the costs.

**Exercise 26.8 ★★★.**

*Find the flaw.* “Dividend futures have earned about 4% a year with low volatility; they are a bond substitute.”

**Solution of Exercise 26.8.**

The return comes with equity risk: the one-year trade correlates 0.41 with the index, loses in about a fifth of years and 24.4% on average in a recession year, when bonds usually do well. It is paid for supplying insurance to structured-product issuers, not for lending.

## 26.9 Problem: Dividends Nobody Wanted

**Problem 26.1.**

Weekend problem — delta-one and dividends

The chapter’s synthetic desk and the public record.

**Part I — Arbitrage.**

1. Define [index arbitrage](#def-s2-delta-one-and-dividends-indexarb) .
2. Describe the mispricing and the costs.
3. Give the results by execution lag.
4. Why can the trade not lose without a lag?

**Part II — Financing.**

5. Define a [financing trade](#def-s2-delta-one-and-dividends-financing) .
6. What does the future’s price imply about financing?
7. Give the results at the roll and at quarter-end.
8. How is a total-return swap the same trade?

**Part III — Dividends.**

9. Why is the autocallable’s issuer long dividends, and by how much?
10. Define [dividend supply pressure](#def-s2-delta-one-and-dividends-supply) .
11. What does the Eurex contract pay?
12. Give the dividend table.

**Part IV — The verdict.**

13. State the *named result* : the dividend trade’s return from the supply discount.
14. What did Manley and Mueller-Glissmann report?
15. What did van Binsbergen, Brandt and Koijen find?
16. What did Kragt, de Jong and Driessen find?
17. When does the dividend trade lose, and why then?
18. Which strategy file needs balance sheet?
19. How does this chapter relate to chapter 12’s swap spreads?
20. In one sentence: what does a delta-one desk sell?

**Solution of Problem 26.1.**

1. Trading an index future against its basket when it departs from fair value by more than the cost of both, unwinding when the gap closes.
2. Minute mispricings with a 4 basis point standard deviation and 10-minute half-life; 2 basis points a side for the basket and 0.5 for the future, 5 for a round trip.
3. 2 230 trades a year: 1.64 basis points each without lag, 1.14 at one minute, 0.77 at two, $-0.26$ at five.
4. It enters beyond the round-trip cost and leaves at zero, so each trade earns at least the excess.
5. Holding an asset against a derivative that delivers it later, earning the implied financing spread less funding and capital.
6. The price above spot less dividends implies the rate at which the market finances the basket.
7. At the roll: 10% of quarters, 15.4 basis points locked, 0.4 a year; in the quarter-end window: 67.5%, 40.7, 8.0 a year.
8. The bank holds the basket and pays the client its return for a spread over funding.
9. The note’s buyer is short a put; the issuer’s hedge makes it long dividends: 0.11 per 100 for 0.1 point of yield, about 1.1 million per billion.
10. The discount of dividend futures to expected dividends when natural holders of dividends sell more than others will buy at fair value.
11. EUR 100 per point of the EURO STOXX 50’s dividends in the contract year, cash settled, up to ten years out.
12. Futures from 96.7 to 93.7 against expected dividends from 100.8 to 117.2; mean returns 3.8–4.2% a year; the one-year trade’s standard deviation 12.3%, 5% quantile $-25.5\%$ , chance of a loss 20.9%.
13. Held to expiry, the one-year dividend future earns 4.2% a year on average from a 4% supply discount, with a loss in about one year in five and $-24.4\%$ in a recession year.
14. From 2004 dividend swaps were a major profit source for hedge funds because implied dividends traded at high discounts and dividends grew.
15. Short-term dividend claims have higher expected returns, Sharpe ratios and volatilities than the index, with betas below one.
16. A two-factor model fits the dividend futures curve, and investors revise dividends beyond the business cycle only a little.
17. In recessions, when dividends are cut and equities fall together.
18. Total-return swap financing.
19. Both earn a spread for providing balance sheet, and both widen when banks’ balance sheets are scarce.
20. Balance sheet and hedging capacity: the index’s return without funding it, and dividends’ risk without the index’s.

## 26.10 Interview questions

**Interview question 26.1 ★ trader.**

How do you compute an index future’s fair value, and what inputs can be wrong?

**Solution of Interview question 26.1.**

Spot less the present value of dividends to expiry, grown at the financing rate to expiry. Dividend forecasts, the financing rate the market uses (not the risk-free rate), withholding tax and the timing of ex-dates can all be wrong.

**Interview question 26.2 ★★ researcher.**

How would you forecast next year’s index dividends?

**Solution of Interview question 26.2.**

Bottom up: each constituent’s announced and forecast dividends, payout policies and earnings, weighted by index weights and ex-dates; checked against dividend futures and option-implied forwards.

**Interview question 26.3 ★★ trader.**

A client wants a three-month total-return swap on an index over quarter-end. How do you price it?

**Solution of Interview question 26.3.**

From the cost of holding the basket over the period: the bank’s funding plus the balance-sheet charge, which is higher over quarter-end, plus dividends passed through and the cost of trading the basket; compare with the futures-implied financing over the same dates.

**Interview question 26.4 ★★ risk.**

What stress would you apply to a book long dividend futures?

**Solution of Interview question 26.4.**

A recession with dividend cuts of 30–50% in the near years, correlated with an equity fall, and a regulatory ban on bank dividends; margin calls on the futures as prices fall.

**Interview question 26.5 ★★ developer.**

Design the system that keeps index fair values up to date intraday.

**Solution of Interview question 26.5.**

A feed of index weights, corporate actions and dividend forecasts; financing curves; real-time spot of the basket; recomputation on each input change; alerts when the future departs from fair value; and versioned inputs for audit.

**Interview question 26.6 ★★★ researcher.**

The future’s price is $F = (S - D) e^{rT}$ with $D$ the present value of dividends to expiry. Show that an error $\delta$ in $D$ shifts fair value by $-\delta e^{rT}$, and say why a dividend forecast error looks like an arbitrage.

**Solution of Interview question 26.6.**

$F(D + \delta) - F(D) = -\delta e^{rT}$. A trader who underestimates dividends by $\delta$ computes a fair value too high by $\delta e^{rT}$, sees the future as cheap by that amount and buys it: the apparent arbitrage is the forecast error, and it loses when the dividends are paid.