Forward and Futures Prices

Options, Futures and Derivative Securities

Lorenzo Naranjo

Spring 2026

No-Arbitrage

No-Arbitrage Pricing

  • We determine the forward price by assuming it is hard to make risk-free profits in financial markets.
  • An arbitrage costs nothing today and pays something in some states without ever losing money, or generates income today with no future obligation.
  • Traders would exploit an arbitrage at a massive scale until prices moved to eliminate it.
  • We treat futures and forward prices with the same maturity as equal; we come back to when they differ.

What the Arbitrage Requires

  • We must be able to buy the asset, or borrow it and sell it short, carry it to delivery at a known cost and without risk, and deliver.
  • Stocks, stock indices, and convertible currencies qualify: no storage, no deterioration, and a known dividend or interest rate.
    • The cash price is the spot price, and the forward price follows by arbitrage.
  • When the asset cannot be carried, the carry in the forward price is an implied rate read from market prices.
    • Currencies under capital controls and physical commodities.

Assets That Can Be Carried

The Forward Price

  • The forward price of a non-dividend paying asset with maturity T years is F = S e^{r T}, where S is the spot price and r is the risk-free rate with continuous compounding.
  • Buy the asset for S and sell it forward for F: at T you deliver and receive F for sure.
  • A strategy that costs S and pays F for sure must satisfy S = F e^{-r T}.

Example: Forward Price of a Non-Dividend Paying Stock

  • Consider a non-dividend paying stock trading at $40.
  • The risk-free rate is 5% per year with continuous compounding.
  • What is the 3-month forward price? F = 40 e^{0.05 \times 3/12} = \$40.50
  • What if the forward price was higher or lower than $40.50?

Example: Forward Price Arbitrage (1)

  • Suppose that the spot price of a non-dividend-paying stock is $40, the 3-month forward price is $43 and the 3-month interest rate is 5% per year with continuous compounding.
  • Is there an arbitrage opportunity?
t = 0 t = 3/12
Short forward 0.00 43 - S_{T}
Borrow 42.47 -43
Long stock -40.00 S_{T}
Total 2.47 0
  • Yes, the forward price is too high!

Example: Forward Price Arbitrage (2)

  • Suppose that the spot price of a non-dividend paying stock is $40, the 3-month forward price is $39 and the 3-month interest rate is 5% per year with continuous compounding.
  • Is there an arbitrage opportunity?
t = 0 t = 3/12
Long forward 0.00 S_{T} - 39
Invest -38.52 39
Short stock 40.00 -S_{T}
Total 1.48 0
  • Yes, the forward price is too low!

Assets Paying Cash Dividends

  • The holder of a forward does not receive the dividends paid before delivery.
  • The forward price of an asset paying cash dividends is F = (S - D) e^{r T}, where D is the present value of the dividends paid during the life of the forward.

Example: Forward Price of a Dividend Paying Stock

  • Consider a stock that currently trades at $50.
  • The stock is expected to pay dividends of $1.15 and $1.20 in two and five months, respectively.
  • The risk-free rate is 5% per year with continuous compounding.
  • The present value of the dividends paid during the life of the forward contract is: D = 1.15 e^{-0.05 \times 2/12} + 1.20 e^{-0.05 \times 5/12} = 2.32
  • The 6-month forward price of the stock is: F = (50 - 2.32) e^{0.05 \times 6/12} = 48.89

Example: Forward Price Arbitrage (3)

  • Suppose that in the previous example the observed forward price is $50.20.
  • Is there an arbitrage opportunity?
t = 0 t = 2/12 t = 5/12 t = 6/12
Short forward 0.00 50.20 - S_{T}
Loan 1 1.14 -1.15
Loan 2 1.18 -1.20
Loan 3 48.96 -50.20
Long stock -50.00 1.15 1.20 S_{T}
Total 1.28 0 0 0
  • Yes, the forward price is too high!

Assets Paying a Dividend Yield

  • A stock index behaves as if it paid dividends continuously at a yield \delta.
  • Reinvesting the dividends, e^{-\delta T} units today grow to one unit at T.
  • Buy e^{-\delta T} units for S e^{-\delta T} and sell forward: the payoff is F for sure, so F = S e^{(r - \delta) T}.

Example: Index Futures

  • An index pays a dividend yield of 3% per year with continuous compounding and is currently at 4,300.
  • The risk-free rate for all maturities is 1% per year continuously-compounded.
  • The 6-month futures price is F = 4300 e^{(0.01-0.03) \times 6/12} = 4257.21
  • Because the dividend yield is higher than the risk-free rate, the futures price is less than the spot price.

Index Arbitrage

  • If F < S e^{(r - \delta) T}: buy futures, short e^{-\delta T} units of the index, invest F e^{-r T}.
  • If F > S e^{(r - \delta) T}: sell futures, buy e^{-\delta T} units of the index, borrow F e^{-r T}.
  • In the example, if the futures trades at 4,300:
t = 0 t = 6/12
Short futures 0 4300 - S_{T}
Buy e^{-\delta T} units of the index -4235.98 S_{T}
Borrow F e^{-r T} 4278.55 -4300
Total 42.57 0
  • With the E-mini’s $50 multiplier, that is $2,128.50 per contract.

Exchange Rates

  • The EUR/USD rate is the number of US dollars per euro: a quote of 1.08 means \$1.08 \Leftrightarrow \text{€}1.
    • The euro is the base currency, the asset being priced.
    • The dollar is the quote currency, the units of the price.
  • For a European investor, $1 is worth 1/1.08 = \text{€}0.93.
  • EUR/USD and GBP/USD use the USD as quote currency, but most pairs use the dollar as base currency: USD/JPY, USD/CNY, USD/CLP.

Covered Interest Parity

  • A unit of the base currency deposited in a bank earns the base currency’s interest rate, which plays the role of \delta: F = S e^{(r - r_{f}) T}, where r is the quote currency’s rate and r_{f} the base currency’s rate.
  • For USD/JPY, r_{f} is the dollar rate and r the yen rate.
  • Forward points are 10,000 \times (F - S) for most pairs (100 for yen pairs).
  • Daily turnover in outright FX forwards: about $1.8 trillion (BIS, April 2025).

Example: Currency Forward

  • The current GBP/USD exchange rate is 1.30.
  • The interest rates in USD and GBP are 1% and 3% per year with continuous compounding, respectively.
  • The 9-month GBP/USD forward price is then F = 1.30 e^{(0.01 - 0.03) \times 9/12} = 1.28065, or 10{,}000 \times (1.28065 - 1.30000) = -193.5 \text{ forward points.}

When the Underlying
Cannot Be Carried

Non-Deliverable Forwards

  • Capital controls stop foreign investors from holding local deposits: Indian rupee, Korean won, Taiwan dollar, Brazilian real.
  • Offshore investors trade non-deliverable forwards: settled in dollars against an official fixing, never delivered.
  • Offshore investors cannot carry the currency, so no arbitrage ties the NDF to the onshore rate.
  • The implied foreign rate is an output of the market: r_{f}^{*} = r - \frac{1}{T} \ln\left(\frac{F}{S}\right)

Example: NDF Implied Rate

  • Spot is $0.2000 per unit, the one-year NDF is $0.1900, and the dollar rate is 4%. r_{f}^{*} = 0.04 - \ln\left(\frac{0.1900}{0.2000}\right) = 9.13\%
  • With an onshore deposit rate of 12%, covered interest parity would give 0.2000 e^{0.04 - 0.12} = \$0.1846, below the NDF price.
  • Capital controls prevent offshore investors from exploiting the gap, and it persists.

Commodity Futures

  • The spot price is the nearby futures S_{t} = F(t, T_{1}), and the futures curve is the primitive.
  • Storing the physical good and delivering it is not risk free: the basis moves, and managing it is the business of merchants (Working 1953).
  • No arbitrage ties the futures curve to the cash price.
  • Among futures, no-arbitrage applies: holding futures needs no storage, so spreads and options on futures are priced against the curve.

The Price of Storage

  • The spread between a deferred and the nearby contract is the price of storage (Telser 1958):
    • storage cost, including interest, minus the convenience yield of stocks on hand.
    • Large when stocks are plentiful, negative when they are scarce, across 31 commodities from 1971 to 2010 (Gorton et al. 2013).
  • Summarize it with an implied yield \delta: F(t, T) = S_{t} e^{(r - \delta)(T - T_{1})}
  • Like the NDF rate, \delta is read from prices, not plugged into a formula. Gold: cheap storage, large stocks, \delta close to zero.

Example: Oil Futures

  • The nearby crude oil futures, expiring in one month, trades at $95 per barrel.
  • The contract expiring in one year trades at $93, and the interest rate is 5%.
  • The implied yield is \delta = 0.05 - \frac{1}{11/12} \ln\left(\frac{93}{95}\right) = 7.32\%
  • \delta > r, so the curve slopes down: the market pays for oil now rather than later, a sign that stocks are scarce.

Negative Oil Prices

  • April 2020: inventories rose and storage at Cushing, Oklahoma, the NYMEX delivery point, was filling up.
  • On April 20, the day before expiration, traders long the May contract could not find buyers willing to take oil with no place to store it.
  • The May contract, then the spot price, settled at about -\$38; the June contract settled at about $20.
  • The spread of about $58 was the price of one month of storage at Cushing.

WTI Spot Price

Line chart of the WTI crude oil spot price, the nearest futures contract, since 2018, including the brief drop below zero in April 2020.

Forwards vs. Futures

Valuing an Existing Forward Contract

  • A forward contract is worth zero when it is first negotiated.
  • Afterwards it may have a positive or negative value.
  • Suppose that K is the delivery price and F is the forward price for a contract that would be negotiated today.
  • Offsetting the old contract with a new one at F locks in a sure cash flow at T:
    • The value of a long forward contract is (F - K) e^{-r T}.
    • The value of a short forward contract is (K - F) e^{-r T}.

Example: Valuing an Existing Forward Position

  • You entered into a short forward contract some time ago when the forward price was $200.
  • Today the contract has 6 months until maturity and the current forward price is $190.
  • The current risk-free rate is 5% per year with continuous compounding.
  • If we buy a forward today, that would lock in a certain cash flow in six months of 200 - 190 = \$10.
  • The present value today of this cash flow is 10 e^{-0.05 \times 6/12} = \$9.75 which is the value of the short forward contract.

Futures vs. Forward Prices

  • Futures settle daily, so a futures position is worth zero after each settlement; forwards settle at maturity.
  • If interest rates are deterministic, futures and forward prices are equal.
  • Positive correlation between rates and the futures price: futures price above forward price, since gains arrive when rates are high.
  • Negative correlation: the reverse, as for SOFR futures, whose price (100 - rate) falls when rates rise.
  • For interest rate futures, traders adjust the observed futures rate to obtain the forward rate: a convexity adjustment.

Futures Prices vs. 
Expected Spot Prices

Predicting Future Prices

  • Futures prices are unbiased predictors of future spot prices only if investors require no risk premium.
  • Let \mu be the rate at which investors discount a claim to one unit of the asset at T.
    • \mu - r is the risk premium, from systematic risk or from hedgers paying speculators.
  • Valuing one unit of the asset at T two ways, discounting \operatorname{E}(S_{T}) at \mu or a sure F at r: F e^{-r T} = \operatorname{E}(S_{T}) e^{-\mu T} \quad \Rightarrow \quad F = \operatorname{E}(S_{T}) e^{(r - \mu) T}

Futures Prices and the Risk Premium

Risk Premium Expected Return Futures vs. Expected Price
Zero \mu = r F = \operatorname{E}(S_{T})
Positive \mu > r F < \operatorname{E}(S_{T})
Negative \mu < r F > \operatorname{E}(S_{T})
  • Determining the risk premium of a commodity is not an easy task in practice.

The Evidence on Commodity Premiums

  • Keynes (1930) and Hicks (1939): net short hedgers pay speculators, so \mu > r. Working (1953): hedgers are merchants trading the basis.
  • Telser (1958): no upward drift in cotton and wheat futures.
  • Cootner (1960): hedging is net short after the harvest and can be net long before, so the premium changes sign over the season.
  • Dusak (1973): systematic risk and average returns near zero for wheat, corn, and soybean futures.
  • Bessembinder (1992): hedgers’ net positions still predict returns. Gorton et al. (2013): inventories do, positions do not.

References

Bessembinder, Hendrik. 1992. “Systematic Risk, Hedging Pressure, and Risk Premiums in Futures Markets.” Review of Financial Studies 5 (4): 637–67.
Cootner, Paul H. 1960. “Returns to Speculators: Telser Versus Keynes.” Journal of Political Economy 68 (4): 396–404.
Dusak, Katherine. 1973. “Futures Trading and Investor Returns: An Investigation of Commodity Market Risk Premiums.” Journal of Political Economy 81 (6): 1387–406.
Gorton, Gary B., Fumio Hayashi, and K. Geert Rouwenhorst. 2013. “The Fundamentals of Commodity Futures Returns.” Review of Finance 17 (1): 35–105.
Hicks, John R. 1939. Value and Capital. Clarendon Press.
Keynes, John Maynard. 1930. A Treatise on Money. Macmillan.
Telser, Lester G. 1958. “Futures Trading and the Storage of Cotton and Wheat.” Journal of Political Economy 66 (3): 233–55.
Working, Holbrook. 1953. “Futures Trading and Hedging.” American Economic Review 43 (3): 314–43.