Library/CFA (Chartered Financial Analyst)/Derivatives (CFA Program Curriculum 2026 • Level I • Volume 7)/Learning Module 6 Pricing and Valuation of Futures Contracts

Learning Module 6 Pricing and Valuation of Futures Contracts

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Futures basics and pricing at initiation5 min

Futures are standardized, exchange-traded derivatives with a futures price f0(T) set at initiation to satisfy no-arbitrage; for an asset with no income or storage costs, f0(T)=S0(1+r)^T (or continuous compounding f0(T)=S0 erT). When ownership involves income (I) or costs (C), f0(T) = [S0 - PV0(I) + PV0(C)](1+r)^T. The principal distinction between forwards and futures is daily mark-to-market and variation margin for futures, which reset the contract to zero each day and produce realized cash flows during the life of the contract; forwards typically settle only at maturity. The cumulative realized gain or loss of a futures contract approximates a comparable forward at maturity, but daily settlement changes cash-flow timing and creates pricing differences when interest rates vary or when futures prices correlate with interest rates. If interest rates are constant or futures prices and rates are uncorrelated, forward and futures prices match.

Key Points

Futures price at initiation equals spot compounded at risk-free rate (adjust for income/costs).
Daily mark-to-market and margins distinguish futures from forwards.
Cumulative realized P/L by maturity approximates forward profit/loss.
Forward and futures prices equal if rates constant or uncorrelated to futures.

Futures versus forward cash flows and MTM5 min

Futures daily settlement and margining reset the futures contract MTM to zero at each close and produce realized cash flows (variation margin). For a comparable forward, MTM accumulates but is only settled at maturity. Practical impacts: margin calls, maintenance margin, initial margin, and financing/reinvestment effects change effective returns when rates or correlation with prices vary. Example: gold futures with specified initial/maintenance margins — daily price changes reduce margin balances and trigger calls; forward MTM would not require intermediate cash. Another implication: investors who need to finance margin calls effectively experience exposure to short-term borrowing and reinvestment rates.

Key Points

Variation margin creates daily realized P/L for futures holders.
Forward holders have credit exposure until maturity due to no daily settlement.
Reinvestment/financing of variation margin creates returns linked to interest rates.
Margining mechanics (initial, maintenance) determine liquidity exposure.

Interest-rate futures, BPV, and convexity bias5 min

Interest-rate futures are quoted as price = 100 - yield (yield expressed in percent). For short-term MRR underlying, futures contract BPV = Notional × 0.0001 × Period (e.g., 90/360). Futures yield linear price-yield relationship; FRA/fwd contracts settle as discounted cash flows and show non-linear (convex) price-yield changes, creating a convexity bias: FRA PV differs slightly from futures BPV-derived payoff because FRA settlement is PV of final cash flow discounted at realized MRR. Bias increases with maturity/discounting period. Example: a three-month FRA vs futures for 1 bp move differs by small PV discount factor.

Key Points

Interest-rate futures price convention is 100 - yield.
BPV (basis point value) quantifies dollar impact per bp move.
FRAs use discounted settlement, causing a convexity bias vs futures.
Bias typically small for short maturities, more for longer.

Central clearing and narrowing of futures-forward differences4 min

Central clearing of OTC derivatives has created futures-like margining for many forwards because dealers must post variation and initial margins to CCPs, and dealers often require similar collateral from end users. This reduces the cash-flow differences between cleared OTC forwards and exchange-traded futures, narrowing price differences that previously arose from distinct margin and settlement timing.

Key Points

CCP margining imposes daily or frequent collateraling for cleared OTC derivatives.
Dealer requirements pass margin costs to end users, making OTC cash flows similar to futures.
Central clearing reduces pricing differences between futures and forwards.

Practical pricing examples and rate curves6 min

Pricing examples include: commodity futures with storage costs—add PV(storage) to spot before compounding; FX forwards: F0,f/d(T)=S0,f/d e^{(r_f - r_d)T}; implied forward rates from zero rates via (1+zA)^A(1+IFR)^{B-A}=(1+zB)^B and bootstrapping zero curve from coupon bonds; FRA cash settlement equals discounted difference (MRR - IFR) × Notional × Period; swap par rate solved by equating PV(floating)=PV(fixed) using zero rates as discount factors. Use BPV and contract specs for interest-rate futures hedging (hedge ratio, contract size).

Key Points

Include PV of income (dividends) or costs in forward/futures pricing.
FX forwards reflect interest-rate differential between currencies.
Bootstrapping derives zero rates and discount factors; implied forwards derived from zeros.
Swap par rate found by equating PV of floating and fixed payments.

Questions

Question 1

A non-dividend-paying stock currently trades at EUR125 and the one-year risk-free rate is 1% (annual discrete). What is the one-year no-arbitrage futures price f0(T) for a contract on one share?

Learning Module 6 Pricing and Valuation of Futures Contracts

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