Forwards and futures

~8 min read

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Why it matters

A forward is the simplest derivative: an obligation to transact at a later date at a price agreed today. Futures are the exchange-traded, marked-to-market cousin. Cash-and-carry pricing and the forward–spot relationship show up in every asset class (equity index, FX, rates, commodities) and are a favorite “derive this on the whiteboard” question.

Core concepts

Forward vs. futures. A forward is an OTC contract: customized, usually zero value at inception, settled at \(T\), bilateral counterparty risk. A future is standardized, exchange-traded, cleared, and marked to market daily (variation margin). Payoffs are similar; financing of margin and daily settlement can make futures prices differ slightly from forwards when rates are stochastic.
The forward price \(F(0,T)\) is not a forecast. It is the price that makes the forward contract worth zero today. Under no-arbitrage, it is pinned by cash-and-carry, not by anyone’s expected future spot (unless you add risk premia and the real-world measure).
Cash-and-carry for an investment asset (no income): borrow \(S_0\), buy the asset, hold to \(T\). Cost of carry is financing: \(F(0,T) = S_0 e^{rT}\) (continuous). If the quoted forward is higher, short the forward, buy spot, carry; if lower, reverse. That’s the whole argument.
Income, dividends, storage, convenience.
- Continuous dividend / FX foreign rate \(q\): \(F = S_0 e^{(r-q)T}\).
- Discrete dividends: \(F = (S_0 - PV(\text{divs})) e^{rT}\).
- Commodity with storage cost \(u\) and convenience yield \(y\): \(F = S_0 e^{(r+u-y)T}\).
Contango vs. backwardation. Contango: \(F > S\) (upward-sloping curve). Backwardation: \(F < S\). For commodities this often reflects storage + convenience, not “the market thinks the price will fall” — the curve is a snapshot of today’s contract prices by expiry, not a path of future spots (Khan Academy’s futures-curve videos hammer this).
Value of an existing forward. After inception, a long forward struck at \(K\) is worth \(e^{-rT}(F(t,T) - K)\) (or \(S_t - K e^{-rT}\) for non-dividend assets). Mark-to-market is “replace \(K\) with the new fair forward and pocket the difference, discounted.”
FX: covered interest parity is cash-and-carry in two currencies: \(F = S_0 e^{(r_d - r_f)T}\). A violation is an FX-swap arbitrage, not a view on the currency.

Mental model

  Cash-and-carry (no income):

  t=0                         t=T
  borrow S0 at r              repay S0 e^{rT}
  buy the asset  ------------ deliver asset vs K
  short forward @ K

  If K > S0 e^{rT}: this trade locks K - S0 e^{rT} > 0  (arb)
  If K < S0 e^{rT}: reverse it (short asset, lend S0, long forward)

  Fair K = F(0,T) = S0 e^{rT}

A futures curve is a row of today’s prices for different delivery dates, not a prediction of the spot path.

Interview questions

A stock trades at 100, \(r = 5\%\) continuous, 1-year forward quoted at 108. What do you do? Answer: Fair \(F = 100 e^{0.05} \approx 105.13\). Quoted 108 is rich. Short the forward, buy stock, finance by borrowing 100. At \(T\): deliver stock, receive 108, repay \(\approx 105.13\), pocket \(\approx 2.87\). Classic cash-and-carry.
Why isn’t the forward price the expected future spot? Answer: No-arbitrage pins \(F\) from replicating the delivery with cash-and-carry. \(F = E^Q[S_T]\) under the risk-neutral measure (or \(F = e^{(r-\mu)T} E^P[S_T]\)). Real-world expected spot includes a risk premium; the forward does not have to equal it.
Name three concrete differences between a forward and a futures contract. Answer: (1) OTC vs exchange/cleared. (2) Custom vs standardized. (3) Settlement at \(T\) vs daily mark-to-market + margin. Bonus: counterparty risk vs default-fund through CCP; forwards can have value during life without cash moving until expiry.
An equity index pays a 2% continuous dividend. How does that change \(F\)? Intuition? Answer: \(F = S_0 e^{(r-q)T}\) with \(q=0.02\). Holding the index you receive dividends you would miss if you only held the forward, so the forward must be cheaper than the pure-financing forward. Same pattern as FX with a foreign rate.
Gold is in contango. Does that mean the market expects the gold price to rise? Answer: Not necessarily. Contango can be pure cost of carry (finance + storage). The futures curve is today’s menu of delivery prices, not \(E[S_t]\) plotted against \(t\).

Watch

Forward contract introduction — Khan Academy. A farmer and a baker lock a price; that’s a forward.
Futures introduction — Khan Academy. Standardization, exchange, and why futures exist (counterparty risk).
Arbitraging futures contract — Khan Academy. Cash-and-carry when the futures is mispriced versus spot.