CFA Derivatives Pricing Notes: Forwards, Futures and Options Valuation [Level 1]

CFA derivatives pricing answers one question: what is a forward, future, or option worth today, and what price makes the contract fair? The answer comes from no arbitrage, not from a forecast. A forward price equals the cost of buying the asset now and carrying it to delivery. A futures price follows the same logic, with daily settlement. An option value comes from a risk neutral binomial tree or from put call parity. This note covers the pricing and valuation outcomes in the CFA Level 1 Derivatives and Risk Management topic, weighted 6 to 9 percent in the 2027 exam, about 11 to 16 of the 180 questions, according to CFA Institute. Find the full topic set in our CFA notes hub. For the instrument basics, start with our CFA derivatives notes.

CFA derivatives pricing in one view

CFA derivatives pricing rests on four tools: the forward price from cost of carry, the futures price from daily settlement, the one period binomial model with a risk neutral probability, and put call parity for European options. Learn each formula, check the conditions, and practice the worked examples below.

Price and value differ: the price is the delivery price in the contract, while the value is what the contract is worth, and it is zero at initiation. Every formula here is a no arbitrage result, so use the compounding convention the exam states.

CFA derivatives pricing: forward pricing and the cost of carry

The forward price is the delivery price agreed today, set so the contract has zero value at initiation. If the forward price were higher than the cost of buying and carrying the asset, an arbitrageur could sell the forward, buy the asset with borrowed funds, and profit risk free. The forward price is therefore the cost of carry, not a forecast.

With no income and no storage cost, discrete compounding gives F0(T) = S0 x (1 + r)^T, where S0 is the spot price, r is the annualized risk free rate, and T is the time to maturity. Continuous compounding gives F0 = S0 x e^(rT).

Income and storage costs adjust the starting point: income lowers the forward price, while storage and insurance costs raise it. F0(T) = [S0 – PV0(I) + PV0(C)] x (1 + r)^T, where PV0(I) is the present value of income, and PV0(C) is the present value of carrying costs.

The value of a forward during its life

The forward price stays fixed, but the contract value changes as the spot price moves. For a long position at time t: Vt(T) = St – F0(T) / (1 + r)^(T – t), where St is the spot price at time t. At expiry: VT(T) = ST – F0(T) for the long position; the short position is the negative of this.

Take a forward price of $105.00, one year to delivery, at a 5 percent risk free rate. After six months, the spot price is $110.00. The long forward is worth $110.00 minus $105.00 discounted for six months: $110.00 minus $102.47, giving $7.53.

CFA derivatives pricing: futures versus forwards

Futures and forwards are the same promise, but they settle differently. A forward settles only at expiry, so its price stays fixed. A future is marked to market daily: gains are credited and losses debited to the margin account each day, the value resets to zero, and at maturity the futures price converges to the spot price.

In most Level 1 questions, the futures price equals the forward price: when rates are constant, or when futures prices and rates are uncorrelated. Unless a question states a correlation assumption, treat the two prices as equal.

When rates move with futures prices, the prices differ. If futures prices are positively correlated with interest rates, the futures price is higher: a long futures position receives cash gains when prices rise, which happens when rates are also high, so those gains can be reinvested at attractive rates. If the correlation is negative, the futures price is lower.

CFA derivatives pricing: binomial option valuation

The one period binomial model values an option from its payoffs at expiry. The stock moves up by a factor u or down by a factor d, giving payoffs Cu and Cd. A risk neutral probability, p, weights those payoffs, which are then discounted at the risk free rate r: p = [(1 + r) – d] / (u – d), and c0 = [p x Cu + (1 – p) x Cd] / (1 + r).

The risk neutral probability is not the real world probability of a rise. It is the probability under which the expected stock return equals the risk free rate, so the actual probability of an up move is irrelevant. A perfectly hedged portfolio of the option, the stock, and borrowing must earn the risk free rate, so risk neutral valuation prices the hedge, not a forecast.

Validity condition: d < (1 + r) < u, so p stays between zero and one. If p falls outside that range, there is an arbitrage opportunity or an input error.

Put call parity for European options

Put call parity is a no arbitrage relationship between a call, a put, the stock, and a risk free bond, all with the same expiry. It is not a pricing model and does not depend on volatility. For European options on a non dividend paying stock, with the same underlying, strike X, and expiry T, c0 + X / (1 + r)^T = p0 + S0.

The two sides are portfolios with the same payoff at expiry: a protective put, long stock plus long put, costs S0 + p0, while a fiduciary call, long call plus a bond paying X at expiry, costs c0 + PV(X).

Rearranged, the formula prices one option from the other: c0 = S0 + p0 – X / (1 + r)^T, and p0 = c0 + X / (1 + r)^T – S0. Parity applies to European options only, for the same underlying, strike, expiry, and quantity.

Worked examples: CFA derivatives pricing in numbers

Example 1: forward price, discrete versus continuous

A stock trades at $100.00, the risk free rate is 5 percent, and a forward matures in one year, with no income and no storage cost. Discrete compounding: F0 = $100.00 x (1.05)^1 = $105.00. Continuous compounding: F0 = $100.00 x e^(0.05) = $105.13. Use the convention the exam states. Carry variants: with income of present value $5.00, the discrete forward price becomes ($100.00 – $5.00) x 1.05 = $99.75, and with storage costs of present value $3.00, it becomes ($100.00 + $3.00) x 1.05 = $108.15.

Example 2: one period binomial call valuation

A stock trades at $100.00 and can rise by a factor of 1.20 or fall by 0.90 over one period. The risk free rate is 5 percent. Value the one period European call, strike $100.00, no dividends. At expiry, the stock is $120.00 up or $90.00 down, so the call pays $20.00 up or $0.00 down. p = (1.05 – 0.90) / (1.20 – 0.90) = 0.50, and 0.90 < 1.05 < 1.20, so the condition holds. Call value today: c0 = (0.50 x $20.00) / 1.05 = $9.52.

Cross check: the matching put pays $10.00 down and $0.00 up, so it is worth (0.50 x $10.00) / 1.05 = $4.76. Then $9.52 + $95.24 = $104.76 = $4.76 + $100.00. The two sides agree. The real world probability was never needed.

Exam traps

• A forward price is not a prediction. A high forward price means high cost of carry, not a high expected spot price.
• Price versus value. The price is fixed at initiation, when value is zero; during the contract, the value changes.
• Compounding convention. Discrete and continuous formulas give different numbers, so use the convention the question states.
• Income lowers, storage raises. The signs in [S0 – PV0(I) + PV0(C)] are easy to flip.
• Risk neutral is not real world. p is a pricing tool, not a prediction; a real world probability in the question is a distractor.
• Put call parity needs European options. The standard form fails for American options, different strikes, or different expiries.
• Futures versus forwards price. Unless the question states a correlation, treat the two prices as equal.
• Margin does not change the price. Futures margin is a performance deposit, not a pricing input.
• Time pressure. With about 90 seconds per question across 180 questions, practice until the formula setup is automatic.

One page revision box: CFA derivatives pricing

Topic	Formula	Key point
Forward price	F0(T) = S0 x (1 + r)^T	No arbitrage; zero value at initiation
Forward price, income and costs	F0(T) = [S0 – PV0(I) + PV0(C)] x (1 + r)^T	Income lowers; storage raises
Forward value during life	Vt(T) = St – F0(T) / (1 + r)^(T – t)	At expiry: VT(T) = ST – F0(T)
Futures versus forwards	Equal if rates constant or uncorrelated	Positive correlation: futures higher
Binomial model	p = [(1 + r) – d] / (u – d); c0 = [p x Cu + (1 – p) x Cd] / (1 + r)	Same structure for a put
Put call parity	c0 + X / (1 + r)^T = p0 + S0	European only

Derivatives and Risk Management is 6 to 9 percent of the Level 1 exam from February 2027. Successful candidates report over 300 hours on average per level, according to CFA Institute. Keep this note in our CFA notes hub and redo the examples from memory before your next mock exam.