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Forwards, futures and put-call parity for the CFA® Program Level I

Forward prices reflect spot, financing costs and holding benefits. Learn carry pricing, futures settlement, put-call parity and put-call-forward parity with worked examples.

CFA® Program Level I guideBy High Performer CertsPublished 8 min read

Key facts

  • Under replication assumptions, financing and holding costs raise forward prices; holding benefits reduce them.
  • A fair new forward has zero initial value; its delivery price fixes the payment for the asset at maturity.
  • Futures settle gains and losses through variation margin; settlement timing can affect pricing when rates are uncertain.
  • European put-call parity matches options on the same underlying with the same strike, expiry and contract size.
  • Put-call-forward parity sets call minus put equal to the discounted difference between forward price and strike.

No-arbitrage forward pricing adds financing and holding costs to spot and subtracts holding benefits. Put-call parity prices equivalent European option portfolios consistently; put-call-forward parity expresses the same relationship through a forward price. Futures differ from forwards through the timing of settlement cash flows. (Back, Liu and Loewenstein, carry (secondary source)) (Naranjo, European parity (secondary source)) (Naranjo, forward pricing (secondary source))

This guide supports study for the CFA® Program Level I, run by CFA Institute. (CFA Institute, Level I exam) Our questions complement CFA Institute's own curriculum; they do not replace it. See the course page and the time value of money guide for discounting foundations.

What does no-arbitrage forward pricing assume?

If two tradable strategies deliver identical cash flows, they must have equal prices under no-arbitrage. Otherwise, buying the cheaper strategy and selling the dearer one locks in a surplus. Replication requires access to the underlying and financing, including short selling when needed. (Naranjo, forward pricing (secondary source))

Our examples assume no tax, transaction costs or default, borrowing and lending at the same risk-free rate, and known cash income and carrying costs. Rates are constant annual rates with continuous compounding; time is in years. These are modelling assumptions, not descriptions of every market. All numerical inputs below are illustrative.

Write spot as S0S_0, time to delivery as TT and the financing rate as rr. With no income, other holding benefits or storage costs, financing a purchase today gives: (Naranjo, forward pricing (secondary source))

F0,T=S0erTF_{0,T}=S_0e^{rT}

This is a replication price, not a forecast of the future spot price. Risk premia can separate forward prices from expected spot prices. (Back, Liu and Loewenstein, carry (secondary source))

How do carry costs and benefits change the price?

Storage increases the cost of holding the asset; dividends and other cash income reduce it. (Back, Liu and Loewenstein, carry (secondary source)) Let U0U_0 be the present value of known holding costs and I0I_0 the present value of known cash income before delivery. Under the replication assumptions, with no non-cash holding benefits, financing the purchase and costs, less income, gives:

F0,T=(S0+U0−I0)erTF_{0,T}=(S_0+U_0-I_0)e^{rT}

Discount each cost or benefit from its own payment date before combining it with spot. For an asset without holding costs, this reduces to the cash-dividend forward formula. (Naranjo, forward pricing (secondary source)) Alternatively, a constant continuous income yield qq gives:

F0,T=S0e(r−q)TF_{0,T}=S_0e^{(r-q)T}

For a storable commodity, the carry representation uses constant, continuously compounded rates for proportional storage costs uu and convenience yield yy:

F0,T=S0e(r+u−y)TF_{0,T}=S_0e^{(r+u-y)T}

Here, convenience yield represents the benefit of having physical inventory available. It is not a cash dividend and is often inferred from prices. Storage constraints and difficulty borrowing inventory can prevent full replication, so this representation is not an unrestricted commodity arbitrage rule. (Back, Liu and Loewenstein, carry (secondary source)) (Naranjo, forward pricing (secondary source)) For assumed spot £75, financing 4%, storage 2%, convenience yield 3% and half a year, it gives £75 × exp[(0.04 + 0.02 − 0.03) × 0.5] = £76.133480.

Worked forward pricing and cash-and-carry example

Assume a share costs £92, pays a known £1.20 dividend in three months and has no holding costs. Delivery is in nine months and the financing rate is 4.5%. First discount the dividend, then compound the adjusted spot price:

I0=1.20e−0.045(0.25)≈1.186576I_0=1.20e^{-0.045(0.25)}\approx1.186576F0,T=(92−I0)e0.045(0.75)≈93.930685F_{0,T}=(92-I_0)e^{0.045(0.75)}\approx93.930685

If a forward instead quotes £95, borrow £92, buy the share and sell it forward. Reinvest the dividend from month three to month nine. At delivery, receive £95 plus £1.227306 of accumulated dividend, repay £95.157991 and retain £1.069315 per share. These figures use unrounded calculations. The share's terminal market price cancels because it is delivered for the agreed amount. (Naranjo, forward pricing (secondary source))

Below the replication price, the reverse trade buys the forward, shorts the share and invests the proceeds, allowing for the dividend owed to the share lender. Borrowing restrictions and dealing costs can prevent an apparent discrepancy becoming a tradable profit. (Naranjo, forward pricing (secondary source)) (OCC/OIC, put-call parity (secondary source))

Why is the forward price different from its value?

A fair new forward has zero initial value; its delivery price fixes the payment for the asset at maturity. After inception, the contracted price KK stays fixed. The long contract's value is the discounted difference between today's forward price for the remaining term and that contracted price: (Naranjo, forward pricing (secondary source))

Vtlong=(Ft,T−K)e−r(T−t)V_t^{\mathrm{long}}=(F_{t,T}-K)e^{-r(T-t)}

Separately, suppose the current forward price is £95.10, the contracted price £93.90, six months remain and the rate is 4%. The long's value is (95.10 − 93.90) × exp(−0.04 × 0.5) = £1.176238. The short has the opposite value. (Naranjo, forward pricing (secondary source))

How do futures differ from forwards?

Futures have standardised exchange terms; forwards can have negotiated terms. Our uncollateralised forward settles at maturity, whereas futures gains and losses are settled in cash through variation margin. (Naranjo, forwards and futures (secondary source)) (CME Group, daily cash settlement)

For an illustrative futures multiplier of £20 per price point, a long entered at 250 receives £80 when settlement rises to 254, then pays £60 when it falls to 251. Net cash received is £20, before interest and costs. The interim payment matters even though the final price is above entry. (CME Group, daily cash settlement) (Naranjo, forwards and futures (secondary source))

Under the frictionless model with deterministic rates, comparable forward and futures prices coincide. With uncertain rates, daily gains may earn different reinvestment rates from the financing rates on losses. Positive co-movement between futures prices and rates tends to put futures above forwards; negative co-movement tends to reverse this. (Naranjo, forward pricing (secondary source))

What is European put-call parity?

Match the call and put's underlying, strike, expiry and contract size. For European options on a non-dividend-paying share, call plus discounted strike cash equals put plus share: (Naranjo, European parity (secondary source)) (OCC/OIC, put-call parity (secondary source))

c0+Ke−rT=p0+S0c_0+Ke^{-rT}=p_0+S_0

Both portfolios pay the greater of terminal spot and strike. Our payoff check assumes a £96 strike:

Terminal values per share, excluding initial portfolio costs
Terminal share priceCall + strike cashPut + share
£82£0 + £96 = £96£14 + £82 = £96
£109£13 + £96 = £109£0 + £109 = £109

Assume spot £92, call premium £6.40, strike £96, nine months remaining, a 4.5% rate and no dividends. The implied put premium is:

p0=6.40+96e−0.045(0.75)−92≈7.214065p_0=6.40+96e^{-0.045(0.75)}-92\approx7.214065

Each portfolio costs £99.214065. Parity determines the put from an observed call; it does not separately determine both option premiums. Known dividends change the equation by their present value: (Naranjo, dividend-adjusted parity (secondary source))

c0−p0=S0−I0−Ke−rTc_0-p_0=S_0-I_0-Ke^{-rT}

What is put-call-forward parity?

Substituting the cash-income forward formula into dividend-adjusted put-call parity gives the following relationship. Here the options are on the asset, and the forward delivers that same asset at their expiry. (Naranjo, forward pricing (secondary source)) (Naranjo, dividend-adjusted parity (secondary source))

c0−p0=e−rT(F0,T−K)c_0-p_0=e^{-rT}(F_{0,T}-K)c0+Ke−rT=p0+F0,Te−rTc_0+Ke^{-rT}=p_0+F_{0,T}e^{-rT}

A long call and short put pay ST−KS_T-K, replicating a long forward with delivery price KK. The discounted forward price is today's cost of receiving the asset at expiry; it is not the value of a new forward, which is zero.

Return to the dividend-paying share's forward price, shown rounded to £93.930685, and use its unrounded value. Assume a matching European call costs £6.40 and strike is £96. Then:

c0−p0=(F0,T−96)e−0.045(0.75)≈−2.000641c_0-p_0=(F_{0,T}-96)e^{-0.045(0.75)}\approx-2.000641

The implied put is £8.400641. If strike equals the forward price, matched call and put premiums are equal. Always discount the whole forward-minus-strike difference and adjust dividends consistently. The exact European equality should not be applied unchanged to American options, whose early exercise rights matter. (Naranjo, European parity (secondary source)) (OCC/OIC, put-call parity (secondary source))

Frequently asked questions

How do dividends affect a forward price?

Known cash dividends reduce the price. Subtract their present value from spot before compounding to delivery, assuming no other holding costs.

Is a commodity's convenience yield a cash payment?

No. It represents the benefit of having physical inventory available and is often inferred from prices. Storage and borrowing constraints can prevent full replication.

Why can futures and forward prices differ?

Daily futures settlement creates reinvestment and financing cash flows. When interest rates are uncertain, their interaction with price movements can affect relative prices.

Does exact put-call parity apply to American options?

The European equality should not be used unchanged for American options because early exercise rights matter.

When are matched European calls and puts equally priced?

When their common strike equals the forward price for the same underlying and expiry, under the stated put-call-forward parity assumptions.

Sources

  1. Lorenzo Naranjo, Washington University in St. Louis (lecture notes, secondary source): Forward and Futures Prices — no-arbitrage replication, income, contract value and settlement timing. Accessed 29 October 2026.
  2. Kerry Back, Hong Liu and Mark Loewenstein (academic textbook, secondary source): Pricing and Hedging Derivative Securities — Chapter 3, §§3.2–3.3: expectations, cost of carry and convenience yield. Accessed 29 October 2026.
  3. Lorenzo Naranjo, Washington University in St. Louis (lecture notes, secondary source): Forward and Futures Markets — contract terms, margin, daily settlement and collateral. Accessed 29 October 2026.
  4. CME Group: Money Calculations for Futures and Options — daily cash settlement variation. Accessed 29 October 2026.
  5. Lorenzo Naranjo, Washington University in St. Louis (lecture notes, secondary source): Properties of European Options — put-call parity and matching terminal payoffs. Accessed 29 October 2026.
  6. Lorenzo Naranjo, Washington University in St. Louis (lecture notes, secondary source): The Impact of Dividends — put-call parity adjusted for the present value of cash dividends. Accessed 29 October 2026.
  7. Options Industry Council, Options Clearing Corporation (market education, secondary source): Put/Call Parity — matched contracts, interest, dividends, trading frictions and early exercise. Accessed 29 October 2026.
  8. CFA Institute: CFA Program Level I exam: format and topic weights. Accessed 26 October 2026.

High Performer Certs is an independent study platform. It is not affiliated with, endorsed by, or sponsored by CFA Institute or the Global Association of Risk Professionals (GARP). We are not a CFA Institute Prep Provider and not a GARP Exam Preparation Provider. CFA Institute does not endorse, promote, review or warrant the accuracy or quality of our products and services. GARP does not endorse, promote, review or warrant the accuracy of our products or services. All questions and explanations are our own original material and contain no CFA Institute or GARP exam, curriculum or practice questions. CFA®, Chartered Financial Analyst®, CIPM® and GIPS® are registered trademarks owned by CFA Institute. FRM® and GARP® are trademarks owned by the Global Association of Risk Professionals.

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