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Study guide · Session 2

Derivatives

6-9% of the exam · session 2 · 10 learning modules

This topic covers mortgage-backed securities and prepayment risk, then derivatives: what forwards, futures, swaps and options are, why they are used, how no-arbitrage pricing sets forward prices, how forwards, futures and swaps are valued, what drives option values, and put-call parity. Level I questions test definitions and short calculations: payoffs and profits, forward prices and values, one-period binomial values and parity.

A study feature written by ExamDragon, not part of the real exam. Each heading is one of this topic's 10 learning modules in the 2027 curriculum, named in our own words.

At a glance

Session 2
6-9%

of the exam, as CFA Institute weights it for 2027

Weight
6-9%
Modules
10
Key terms
11

What this guide covers, module by module

  1. DER.1What derivatives are and where they trade
  2. DER.2Forwards, futures, swaps and options compared
  3. DER.3Why issuers and investors use derivatives
  4. DER.4No-arbitrage pricing and the carry of an asset
  5. DER.5Forward contract pricing and value
  6. DER.6Futures versus forwards
  7. DER.7Swaps as bundles of forwards
  8. DER.8What drives an option's value
  9. DER.9Put-call parity and its forward form
  10. DER.10One-step binomial option pricing

DER.1What derivatives are and where they trade

  • A derivative is a contract whose value comes from an underlying: an asset, a rate, an index or a credit event.
  • Derivatives specify the underlying, the size, the expiration and the price or rate at which the parties will transact.
  • Exchange-traded derivatives are standardized, cleared through a central counterparty and marked to market daily.
  • Over-the-counter derivatives are customized and privately negotiated; they carry counterparty risk, though many are now centrally cleared.
  • Clearing houses reduce counterparty risk by becoming the buyer to every seller and the seller to every buyer, backed by margin.
  1. The situationA company needs to hedge a currency payment due on an unusual date in an amount that matches no listed contract.
  2. Step 1Listed futures have set sizes and dates.
  3. Step 2An OTC forward can be written for the exact amount and date.
  4. Where questions turnExchange-traded means standardized and cleared, not customized.
  5. The takeawayCustomization is the main reason to use OTC derivatives; liquidity and transparency are the reasons to use exchange-traded ones.
Practise DER.1

DER.2Forwards, futures, swaps and options compared

  • Forward commitments (forwards, futures, swaps) oblige both parties to perform; contingent claims (options) give the buyer a right, not an obligation.
  • A call gives the right to buy at the exercise price; a put gives the right to sell.
  • Call payoff at expiry = max(0, ST - X); put payoff = max(0, X - ST); profit subtracts the premium for the buyer and adds it for the writer.
  • A credit default swap pays the protection buyer if a credit event hits the reference entity, in exchange for periodic premiums.
  • Forward commitments have linear payoffs; options have kinked, asymmetric payoffs.
  1. The situationAn investor writes a put with exercise price 40 for a premium of 2; at expiry the share is 37.
  2. Step 1The holder exercises: the writer pays 40 - 37 = 3.
  3. Step 2Profit = 2 - 3 = -1 per share.
  4. Where questions turnPayoff and profit differ by the premium; questions ask for one and offer the other.
  5. The takeawayA put writer's maximum gain is the premium, while losses grow as the price falls.
Practise DER.2

DER.3Why issuers and investors use derivatives

  • Benefits: transferring and pricing risk, price discovery, lower transaction costs, high liquidity and the ability to take positions with little capital.
  • Risks: leverage, counterparty and liquidity risk, basis risk and the potential for misuse.
  • Issuers use derivatives mainly to hedge: for example, a swap turns fixed-rate debt into floating-rate debt, and hedge accounting can reduce reported volatility.
  • Investors use derivatives to replicate cash positions, hedge, gain or change exposures and access markets cheaply.
  • Systemic risk can arise when leverage and interconnection spread losses through the financial system.
  1. The situationAn issuer has fixed-rate bonds outstanding and expects rates to fall.
  2. Step 1It enters a swap receiving fixed and paying floating.
  3. Step 2The fixed receipts offset its coupons, leaving a net floating cost.
  4. Where questions turnQuestions confuse the swap leg that converts fixed-rate debt to floating (receive fixed) with its opposite.
  5. The takeawayA receive-fixed swap converts fixed-rate debt into synthetic floating-rate debt.
Practise DER.3

DER.4No-arbitrage pricing and the carry of an asset

  • Arbitrage is a riskless profit from mispriced assets; no-arbitrage pricing assumes it disappears.
  • Replication builds the same payoff from the underlying and borrowing or lending; equal payoffs must have equal prices.
  • Derivative prices come from replication and the risk-free rate, not from investors' expected future spot price.
  • Cost of carry: holding the underlying has costs (financing, storage) and benefits (dividends, coupons, convenience yield).
  • Forward price = (S0 - PV of benefits + PV of costs) x (1 + r)^T.
  1. The situationA share trades at 50 and will pay a dividend with a present value of 1.20 before a 6-month forward expires; r = 4%.
  2. Step 1Remove the dividend: 50 - 1.20 = 48.80.
  3. Step 2Compound at the risk-free rate: 48.80 x 1.04^0.5 = 49.77.
  4. Where questions turnThe forward price is not a forecast of the future spot price; it follows from carry and the risk-free rate.
  5. The takeawayBenefits of holding the underlying lower the forward price; costs raise it.
Practise DER.4

DER.5Forward contract pricing and value

  • At initiation the forward price is set so the contract's value is zero: F0 = S0 x (1 + r)^T for an underlying with no carry.
  • During the life, the long's value = St - F0/(1 + r)^(T - t), adjusted for any remaining carry.
  • At expiration, the long's value = ST - F0; the short's is the opposite.
  • A forward rate agreement (FRA) locks in an interest rate for a future period; its fixed rate is the implied forward rate.
  • A long FRA gains when the reference rate rises, so a future borrower hedges with a long FRA and a future lender with a short FRA.
  1. The situationA long forward was agreed at 105; with six months left, the underlying is 103 and r = 5%.
  2. Step 1PV of the forward price = 105/1.05^0.5 = 102.47.
  3. Step 2Value = 103 - 102.47 = 0.53.
  4. Where questions turnThe forward price is fixed for the contract's life; the forward value changes daily.
  5. The takeawayA forward's value changes with the underlying even though its price is fixed.
Practise DER.5

DER.6Futures versus forwards

  • Futures are standardized forwards traded on exchanges, marked to market daily through a clearing house.
  • Initial margin is posted at the start; if the account falls below maintenance margin, variation margin restores it to the initial level.
  • Daily settlement resets a futures contract's value to zero each day; a forward's value builds up until expiry.
  • With constant or uncorrelated interest rates, futures and forward prices are equal.
  • If futures prices are positively correlated with rates, futures prices are above forward prices; negatively correlated, below.
  1. The situationInitial margin is 5,000 and maintenance 3,500; a long loses 2,000 in one day.
  2. Step 1Balance = 3,000, below maintenance.
  3. Step 2Variation margin = 5,000 - 3,000 = 2,000.
  4. Where questions turnVariation margin brings the account back to initial margin, not to maintenance margin.
  5. The takeawayA margin call restores the initial margin, not just the maintenance level.
Practise DER.6

DER.7Swaps as bundles of forwards

  • A swap is an exchange of a series of cash flows: in a plain vanilla interest rate swap, fixed payments for floating payments on a notional.
  • A swap resembles a series of forwards, but every exchange uses the same fixed rate, so each implied forward is off market.
  • The swap price is the fixed rate that gives the swap zero value at initiation; with discount factors, swap rate = (1 - last DF)/(sum of DFs).
  • After initiation, value changes as rates move: the fixed-rate payer gains when rates rise.
  • Swaps are mostly OTC and often centrally cleared; notional is not exchanged in an interest rate swap.
  1. The situationDiscount factors for years 1 and 2 are 0.98 and 0.95.
  2. Step 1Swap rate = (1 - 0.95)/(0.98 + 0.95).
  3. Step 2= 0.05/1.93 = 2.59%.
  4. Where questions turnThe swap's price (its fixed rate) is set at initiation; its value starts at zero and then moves.
  5. The takeawayThe swap rate is a par rate built from the discount factors of the payment dates.
Practise DER.7

DER.8What drives an option's value

  • Exercise value is what exercise would pay now: max(0, S - X) for a call, max(0, X - S) for a put; option value = exercise value + time value.
  • Moneyness: a call is in the money when S > X, at the money when S = X, out of the money when S < X; a put the reverse.
  • Options cannot be priced by replication without assumptions about price moves; the one-period binomial model uses risk-neutral probability q = (1 + r - d)/(u - d).
  • Binomial value = [q x payoff up + (1 - q) x payoff down]/(1 + r).
  • A higher underlying price or lower exercise price raises calls and lowers puts; higher volatility and longer time usually raise both; a higher risk-free rate raises calls and lowers puts.
  • Benefits of holding the underlying (dividends) lower calls and raise puts; carrying costs do the reverse.
  1. The situationA share at 100 can rise to 120 or fall to 90; r = 5% for the period; a call has exercise price 100.
  2. Step 1q = (1.05 - 0.90)/(1.20 - 0.90) = 0.5.
  3. Step 2Payoffs: 20 up, 0 down.
  4. Step 3Value = (0.5 x 20)/1.05 = 9.52.
  5. Where questions turnThe risk-neutral probability is not the real probability of an up move; using real-world odds gives a wrong value.
  6. The takeawayOption values come from risk-neutral probabilities and the risk-free rate, not from investors' real-world expectations.
Practise DER.8

DER.9Put-call parity and its forward form

  • Put-call parity for European options: c0 + X/(1 + r)^T = p0 + S0 (fiduciary call equals protective put).
  • Rearranged: p0 = c0 - S0 + X/(1 + r)^T, and c0 = p0 + S0 - X/(1 + r)^T.
  • Synthetic positions follow: a long call plus a short put with the same exercise price replicates a long forward-like position.
  • Put-call forward parity replaces S0 with a forward and a bond paying F0: c0 + X/(1 + r)^T = p0 + F0/(1 + r)^T.
  • Parity holds only for European options with the same underlying, exercise price and expiration.
  1. The situationS0 = 52, X = 50, T = 1 year, r = 5%, and the call costs 6.00.
  2. Step 1PV of X = 50/1.05 = 47.62.
  3. Step 2p0 = 6.00 - 52 + 47.62 = 1.62.
  4. Where questions turnThe exercise price enters parity at its present value, not its face amount.
  5. The takeawayParity pins the put price once the call, the share and the risk-free rate are known.
Practise DER.9

DER.10One-step binomial option pricing

  • In one period the underlying moves from S to S x u (up) or S x d (down); the option's payoff is known in each state.
  • Because the option can be replicated with the underlying and borrowing or lending at the risk-free rate, its value does not depend on the actual probability of an up move or on investors' risk preferences.
  • Risk-neutral probability of an up move: q = (1 + rf - d) / (u - d).
  • Option value = [q x payoff up + (1 - q) x payoff down] / (1 + rf).
  • Under risk neutrality the underlying is expected to earn only the risk-free rate, so expected payoffs are discounted at the risk-free rate.
  1. The situationA share trades at USD 40 and will be USD 50 (u = 1.25) or USD 32 (d = 0.80); a call has an exercise price of USD 40; rf = 5% for the period.
  2. Step 1q = (1.05 - 0.80) / (1.25 - 0.80) = 0.25 / 0.45 = 0.5556.
  3. Step 2Payoffs: 10 in the up state, 0 in the down state.
  4. Step 3Call value = 0.5556 x 10 / 1.05 = USD 5.29.
  5. Where questions turnThe risk-neutral probability is a pricing device, not investors' forecast of an up move.
  6. The takeawayThe value is the risk-neutral expected payoff discounted at the risk-free rate.
Practise DER.10

Key terms

Contraction risk
The risk that prepayments speed up when rates fall, shortening a mortgage security's life.
Extension risk
The risk that prepayments slow when rates rise, lengthening a mortgage security's life.
Forward commitment
A contract obliging both parties to transact: forwards, futures and swaps.
Contingent claim
A contract giving its buyer a right without an obligation, such as an option.
Cost of carry
The net cost of holding the underlying: financing and storage less income and convenience.
Forward price
The fixed price agreed today for a trade at expiry, set so the contract starts at zero value.
Variation margin
Funds needed to restore a futures account to its initial margin.
Swap rate
The fixed rate that gives an interest rate swap zero value at initiation.
Time value
The part of an option's price above its exercise value.
Risk-neutral probability
The probability that makes the underlying earn the risk-free rate in a binomial model.
Put-call parity
The no-arbitrage link between European call, put, underlying and risk-free bond prices.

After this guide: You should be able to explain prepayment risk, compare derivative types, price and value forwards, compute option payoffs, a one-period binomial value and parity prices.

Practise Derivatives

Built to the topics and weights CFA Institute publishes for the 2027 Level I curriculum. The module names, explanations, worked examples and traps are ExamDragon's own. Every source this path is built on.