Pricing Financial Forwards and Futures: formula sheet

Key formulas

Long forward payoff at maturity

Payoff = (S_T − K) × units

S_T is the spot price at maturity; K is the delivery price. Can be negative.

Short forward payoff at maturity

Payoff = (K − S_T) × units

Mirror image of the long. Long payoff + short payoff = 0.

Futures daily gain or loss

Daily P&L = (F_t − F_(t−1)) × contract size × number of contracts (long); reverse sign for short

F is the futures settlement price. The result goes into the margin account.

Margin account balance

New balance = old balance + daily P&L − withdrawals + deposits

Check it against maintenance margin each day.

Margin call amount

Deposit = initial margin − current balance

Applies when the balance falls below maintenance margin. Top up to initial margin, not to maintenance.

Total futures gain if held to maturity

Total P&L = (F_final − F_initial) × size (long)

The sum of daily settlements. At maturity the futures price converges to the spot price.

Forward price, continuous compounding

F₀ = S₀ × e^(rT)

r is the continuously compounded risk-free rate, T in years. No income, no storage cost.

Forward price, discrete compounding

F₀ = S₀ × (1 + r)^T

Use when r is quoted as an annual effective rate. With m compounding periods per year: S₀ × (1 + r/m)^(mT).

Value of an existing long forward

f = S₀ − K × e^(−rT)

K is the delivery price fixed in the contract and T is the remaining life. At initiation K = F₀ so f = 0.

Cost of carry

F₀ − S₀ = S₀ × (e^(rT) − 1)

With no income, carry is pure financing cost.

Arbitrage profit at T

If F > S₀e^(rT): profit = F − S₀e^(rT). If F < S₀e^(rT): profit = S₀e^(rT) − F

Profit at delivery date, risk-free, no initial outlay.

Forward price, no income

F0 = S0 × e^(rT)

Continuous compounding. Use as the base case.

Forward price, known cash income

F0 = (S0 − I) × e^(rT)

I = present value of all income paid before maturity, discounted at r for the time to each payment.

Present value of income

I = Σ Di × e^(−r × ti)

Di is the cash amount paid at time ti.

Forward price, continuous yield

F0 = S0 × e^((r − q)T)

q is the dividend yield, annualised, continuously compounded.

Currency forward (covered interest parity)

F0 = S0 × e^((r − rf)T)

S0 and F0 are domestic currency per one unit of foreign currency. r is domestic, rf is foreign.

Discrete compounding version of parity

F0 = S0 × (1 + r × T) ÷ (1 + rf × T)

Use for simple interest periods, such as money market rates under one year.

Forward with proportional storage cost and convenience yield (continuous)

F = S × e^((r + u − y)T)

r = risk-free rate, u = storage cost as a yearly rate, y = convenience yield, all continuously compounded.

Forward with known dollar storage costs

F = (S + U) × e^(rT)

U = present value of all storage costs over the life, discounted at r. Storage paid at the end of the period needs discounting first.

Cost-of-carry upper bound

F ≤ S × e^((r + u)T)

If F exceeds this, buy the commodity, sell the forward and earn arbitrage. The reverse arbitrage (sell the commodity, buy the forward) fails because holders of the physical commodity value the convenience of holding it and will not sell it to capture the arbitrage.

Implied convenience yield

y = r + u − ln(F ÷ S) ÷ T

Use continuous compounding. Rearranged from the pricing formula.

Curve shape rule

Contango: F > S. Backwardation: F < S.

Contango roughly when r + u > y. Backwardation roughly when y > r + u.

Value of long forward, no income

f = S₀ − K·e^(−rT)

Continuous compounding. For a short forward, the value is the negative of this.

Value of long forward using the forward price

f = (F₀ − K)·e^(−rT)

Works for any underlying. F₀ is today's forward price for the same maturity.

Forward price, no income

F₀ = S₀·e^(rT)

Cost-of-carry result, so f = 0 when K = F₀.

Value with known income (PV of income I)

f = S₀ − I − K·e^(−rT)

I is the present value of income during the life of the contract.

Value with continuous yield q

f = S₀·e^(−qT) − K·e^(−rT)

Use for stock indices and foreign currencies, where q is the dividend yield or the foreign rate.

Futures vs expected spot (systematic risk)

F₀ = E(S_T)·e^((r − k)T). If k > r: F₀ < E(S_T). If k = r: F₀ = E(S_T). If k < r: F₀ > E(S_T).

k is the required return on the asset, set by its systematic risk as in CAPM. Applies to an asset with no income or storage costs. Positive systematic risk means k > r.

Cash received by the short (Treasury futures)

Cash = (Futures quote × CF) + Accrued interest

Per $100 face value. Multiply by 1,000 for one $100,000 contract.

Conversion factor

CF = Price per $1 par of the bond at a 6% yield, semiannual compounding

Maturity rounded down to the nearest 3 months. Coupon above 6% gives CF above 1.

Cheapest-to-deliver

Delivery cost = Quoted bond price − (Futures quote × CF)

The bond with the lowest delivery cost is the CTD. Use the quoted price, not the cash price.

Theoretical Treasury futures quote

F quote = [(S − I) × e^(rT) − Accrued at delivery] ÷ CF

S is the CTD cash price. I is the present value of coupons paid before delivery. r is the risk-free rate and T is the time to delivery.

Eurodollar (and 3-month SOFR) futures quote

Quote = 100 − Rate (in %)

Rate is the 3-month annualised rate. A quote of 96.00 means 4.00%.

Eurodollar contract value

Value = $10,000 × [100 − 0.25 × (100 − Quote)]

1 bp = $25 on $1 million notional. A 0.01 change in the quote is worth $25.

Convexity adjustment

Forward rate = Futures rate − ½ × σ² × T1 × T2

σ is the annual standard deviation of the short rate change. T1 is the futures maturity and T2 is the maturity of the underlying rate, both in years. Use decimals, e.g. 0.012 for 1.2%.

Minimum variance hedge ratio

h* = ρ × σS ÷ σF

σS and σF are standard deviations of spot and futures price changes over the hedge period. ρ is their correlation. Also equals the regression slope.

Number of contracts

N* = h* × QA ÷ QF

QA is the size of the position hedged (units of the asset). QF is the size of one futures contract (units).

Hedge effectiveness

Effectiveness = ρ²

The proportion of variance of the spot position eliminated by the optimal hedge.

Variance of hedged position

σ²hedged = σS² × (1 − ρ²) per unit

Applies when the optimal h* is used. Multiply by the squared size of the exposure for total variance.

Basis

Basis = S − F

Spot minus futures. Some texts use futures minus spot, so check the definition given in the question.

Tailed hedge (value form)

N* = h* × VA ÷ VF

VA is the dollar value of the exposure. VF is the dollar value of one futures contract (futures price × contract size).

Contracts for changing portfolio beta

N = (β* − β) × P ÷ A

P is portfolio value, A is the value of one futures contract (index level × multiplier), β is current beta, β* is target beta. Positive means buy, negative means sell. To fully hedge, β* = 0 and N = −β × P ÷ A.

Quick revision

• No-income forward: F = S × e^(rT), with r the continuously compounded rate.
• Known yield q: F = S × e^((r − q)T).
• Known cash income with present value I: F = (S − I) × e^(rT).
• Storage cost rate u adds to carry: F = S × e^((r + u − y)T), where y is convenience yield.
• Convenience yield lowers the forward price and reflects the benefit of holding the physical asset.
• Value of a long forward with delivery price K and no income: f = S − K × e^(−rT).
• A new forward has value zero at inception, since K is set equal to F.
• Futures are marked to market daily; forwards usually settle at maturity.
• Futures and forward prices can differ when interest rates are correlated with the asset price.
• Eurodollar futures quotes are 100 minus the rate, so a price rise means a fall in the implied rate.
• Bond futures: the short chooses the cheapest-to-deliver bond, and conversion factors adjust for coupon and maturity.
• Minimum-variance hedge ratio: h* = ρ × σS ÷ σF; contracts = h* × QA ÷ QF.

Common mistakes

• Calling the margin call amount maintenance margin minus balance. Fix: The call restores the account to the initial margin. Use initial margin − balance.
• Getting the sign wrong for a short position. Fix: For a short, a rise in price is a loss. Write the position sign before you calculate.
• Using months directly as T, for example T = 6 instead of 0.5. Fix: Always convert to years first. Write T = months ÷ 12 before touching the formula.
• Applying e^(rT) when the rate is quoted as an annual effective rate, or the reverse. Fix: Read the compounding wording. Use (1 + r)^T for annual rates and e^(rT) for continuous rates. If they differ, the answers differ slightly and options are built to catch this.
• Subtracting the undiscounted income instead of its present value. Fix: Always discount each payment to today at r before subtracting from S0. Then compound the remainder to maturity.
• Discounting income for the full time T instead of the time to each payment. Fix: Use each payment's own time ti in e^(−r × ti).
• Adding the convenience yield instead of subtracting it. Fix: Remember it is a benefit of holding the physical good, so it works like a dividend yield and reduces the forward price.
• Forgetting to discount a dollar storage cost paid later. Fix: Add the present value of storage costs to S, then compound the sum at r.
• Valuing the forward as S₀ − K with no discounting. Fix: Always discount K, or discount F₀ − K, back from T to today.
• Saying a new forward has value equal to the forward price. Fix: The forward price is the delivery price K that makes the contract's value zero at inception. A new forward has value zero; its price is a delivery price, not a value.

Exam tips

• Read the position carefully. Long or short decides every sign, and exam options often include the reversed sign.
• On margin questions, check whether the call is triggered by 'below' maintenance and whether the top-up is to initial margin.
• For conceptual items, remember the standard contrasts: OTC vs exchange, customised vs standardised, settlement at maturity vs daily, credit risk vs clearinghouse.
• Check the units: contract size times number of contracts is a frequent trap.
• Keep each margin table to a few lines. A basic calculator is enough; the arithmetic is simple, so accuracy matters more than speed.
• Read the compounding basis in the question before calculating. It is the most common trap in the options.
• Use the sanity check that F₀ > S₀ when r > 0 and there is no income to rule out wrong options quickly.
• Arbitrage questions usually ask for the strategy or the profit. Practise stating the three legs: borrow or invest, spot trade, forward trade.