Derivatives

    Exam weight 5–8%
    96 questions in the bank
    5 subtopics

    Derivatives has a small weight and a fearsome reputation, most of it undeserved at Level I. The exam wants the logic of no-arbitrage pricing and the shape of a payoff, not the heavy machinery that arrives at Level II.

    What you need to be able to do

    • Draw the payoff and profit of a position before reasoning about it
    • Apply put-call parity to find the missing instrument
    • Explain why a forward price is a no-arbitrage result, not a forecast
    • Distinguish the credit and cash-flow consequences of forwards versus futures

    Where candidates lose marks

    Confusing payoff with profit. The premium moves the break-even point and changes the answer to most 'is this position profitable' questions.

    10 free Derivatives practice questions

    Real questions from the CFAQuiz bank, one or two per subtopic, with the full explanation. No account needed.

    1
    Derivative Instrument Features
    medium
    DER-FEAT-01

    A trader goes long 5 crude oil futures contracts at a futures price of 60.00. Each contract is for 1,000 barrels. The initial margin is $6,000 per contract and the maintenance margin is $4,500 per contract. Settlement prices over the next three days are: Day 1 = 58.80, Day 2 = 59.10, Day 3 = 56.90. What variation margin deposit is required at the end of Day 3 to restore the margin account to the initial margin level?

    1. A.$8,000
    2. B.$11,000
    3. C.$15,500
    Show answer and explanation

    Correct answer: C

    Initial margin (total):

    6,000×5=30,0006{,}000 \times 5 = 30{,}000

    Day 1 P&L and balance:

    (58.80−60.00)×1,000×5=(−1.20)×1,000×5=−6,000(58.80 - 60.00) \times 1{,}000 \times 5 = (-1.20) \times 1{,}000 \times 5 = -6{,}000 30,000−6,000=24,00030{,}000 - 6{,}000 = 24{,}000

    Day 2 P&L and balance:

    (59.10−58.80)×1,000×5=0.30×1,000×5=1,500(59.10 - 58.80) \times 1{,}000 \times 5 = 0.30 \times 1{,}000 \times 5 = 1{,}500 24,000+1,500=25,50024{,}000 + 1{,}500 = 25{,}500

    Day 3 P&L and balance:

    (56.90−59.10)×1,000×5=(−2.20)×1,000×5=−11,000(56.90 - 59.10) \times 1{,}000 \times 5 = (-2.20) \times 1{,}000 \times 5 = -11{,}000 25,500−11,000=14,50025{,}500 - 11{,}000 = 14{,}500

    Maintenance margin (total):

    4,500×5=22,5004{,}500 \times 5 = 22{,}500

    Since 14,500 < 22,500, a margin call is triggered to restore the account to the initial margin total of 30,000. Required deposit:

    30,000−14,500=15,50030{,}000 - 14{,}500 = 15{,}500
    • Option A ($8,000): 22,500−14,500=8,00022{,}500 - 14{,}500 = 8{,}000, replenishes only to maintenance, not initial.
    • Option B ($11,000): 2.20×1,000×5=11,0002.20 \times 1{,}000 \times 5 = 11{,}000, uses only Day 3’s loss and ignores prior days’ P&L.
    2
    Forward Commitment Pricing
    medium
    DER-FWD-01

    A 6-month forward contract is written on a stock currently priced at $120. The stock will pay two known cash dividends of $1 each in 2 months and 5 months. The annual risk-free rate is 6% with annual compounding (use the same convention for discounting dividends and compounding to maturity). What is the no-arbitrage forward price?

    1. A.$123.55
    2. B.$121.49
    3. C.$121.52
    Show answer and explanation

    Correct answer: C

    Discount the dividends to the present using 6% annual compounding.

    PV of the dividend at 2 months (0.1667 years):

    PVd1=1(1+0.06)0.1667=0.9903\text{PV}_{d1} = \frac{1}{(1 + 0.06)^{0.1667}} = 0.9903

    PV of the dividend at 5 months (0.4167 years):

    PVd2=1(1+0.06)0.4167=0.9760\text{PV}_{d2} = \frac{1}{(1 + 0.06)^{0.4167}} = 0.9760

    Sum of present values:

    PVdivs=0.9903+0.9760=1.9663\text{PV}_{\text{divs}} = 0.9903 + 0.9760 = 1.9663

    Adjust the spot for dividends:

    S0−PVdivs=120−1.9663=118.0337S_0 - \text{PV}_{\text{divs}} = 120 - 1.9663 = 118.0337

    Grow to maturity (0.5 years at 6% with annual compounding):

    F0=118.0337×(1+0.06)0.5=118.0337×1.029563=121.52F_0 = 118.0337 \times (1 + 0.06)^{0.5} = 118.0337 \times 1.029563 = 121.52

    Therefore, the forward price is $121.52.

    • Option A ($123.55) ignores dividends entirely: 120×1.029563=123.55120 \times 1.029563 = 123.55
    • Option B ($121.49) subtracts the dividends without discounting: (120−2.00)×1.029563=118.00×1.029563=121.49(120 - 2.00) \times 1.029563 = 118.00 \times 1.029563 = 121.49
    3
    Hedging and Risk Management
    medium
    DER-HRM-03

    A dealer is short 50,000 shares of a stock and wants to delta-hedge using call options. Each call has delta 0.60 and each option contract covers 100 shares. How many call option contracts should the dealer buy to establish a delta-neutral position? (Round to the nearest whole number.)

    1. A.Buy 833 call option contracts
    2. B.Buy 834 call option contracts
    3. C.Buy 500 call option contracts
    Show answer and explanation

    Correct answer: A

    The short stock position has total delta:

    Δshares=−1.00×50,000=−50,000\Delta_{\text{shares}} = -1.00 \times 50{,}000 = -50{,}000

    Each call contract contributes delta:

    Δper contract=0.60×100=60\Delta_{\text{per contract}} = 0.60 \times 100 = 60

    Contracts required to offset the delta:

    N=50,00060=833.33…≈833N = \frac{50{,}000}{60} = 833.33\ldots \approx 833

    Rounding to the nearest whole number gives 833 contracts.

    • Option B (834) incorrectly rounds up rather than to the nearest whole number.
    • Option C (500) assumes option delta = 1.00 instead of 0.60: 50,0001.00×100=500\frac{50{,}000}{1.00 \times 100} = 500.
    4
    Options Pricing: Basics
    easy
    DER-OB-1

    A European call option on a non-dividend-paying stock is priced at $4.50. The underlying stock is trading at $52 and the strike price is $50. What is the option's time value?

    1. A.$4.50
    2. B.$6.50
    3. C.$2.50
    Show answer and explanation

    Correct answer: C

    Intrinsic value of a call is max⁡(S0−K,0)\max(S_0 - K, 0).

    Intrinsic (call)=max⁡(52−50,0)=2\text{Intrinsic (call)} = \max(52 - 50, 0) = 2 Time value=Premium−Intrinsic=4.50−2=2.50\text{Time value} = \text{Premium} - \text{Intrinsic} = 4.50 - 2 = 2.50

    Therefore, the time value is 2.50.

    • Option A uses the put's intrinsic (max⁡(K−S0,0)\max(K - S_0, 0)) instead of the call's:
    Time value=4.50−max⁡(50−52,0)=4.50−0=4.50\text{Time value} = 4.50 - \max(50 - 52, 0) = 4.50 - 0 = 4.50
    • Option B subtracts K−S0K - S_0 (a negative number) instead of the call intrinsic:
    Time value=4.50−(50−52)=4.50−(−2)=6.50\text{Time value} = 4.50 - (50 - 52) = 4.50 - (-2) = 6.50
    5
    Options Pricing: Models
    medium
    DER.OPM.04

    In a one-period binomial model for a European call, the current stock price is S0 = 60. Over the next half-year (Δt=0.5)(Δt = 0.5), the stock can go up by u = 1.20 or down by d = 0.90. The strike price is K = 65. The risk-free rate is r = 5% per year, continuously compounded. Price the call today using replication.

    1. A.2.85
    2. B.2.92
    3. C.5.74
    Show answer and explanation

    Correct answer: A

    Up and down prices and payoffs:

    Su=60×1.20=72,Sd=60×0.90=54S_u = 60\times 1.20 = 72,\quad S_d = 60\times 0.90 = 54 Cu=max⁡(72−65,0)=7,Cd=0C_u = \max(72-65, 0) = 7,\quad C_d = 0

    Hedge ratio and bond (replication):

    h=Cu−CdSu−Sd=7−072−54=718=0.3888889h = \frac{C_u - C_d}{S_u - S_d} = \frac{7 - 0}{72 - 54} = \frac{7}{18} = 0.3888889 BerΔt=h Sd−Cd=0.3888889×54−0=21.0B e^{r\Delta t} = h\,S_d - C_d = 0.3888889\times 54 - 0 = 21.0 B=21.0 e−0.05×0.5=21.0 e−0.025=21.0×0.97531=20.4815B = 21.0\,e^{-0.05\times 0.5} = 21.0\,e^{-0.025} = 21.0\times 0.97531 = 20.4815 C0=h S0−B=0.3888889×60−20.4815=23.3333−20.4815=2.85C_0 = h\,S_0 - B = 0.3888889\times 60 - 20.4815 = 23.3333 - 20.4815 = 2.85
    • Option B (2.92): Omits discounting the expected payoff (risk-neutral method), i.e., C0=p Cu=2.9240C_0 = p\,C_u = 2.9240 instead of multiplying by e−rΔte^{-r\Delta t}.
    • Option C (5.74): Uses Sd=S0 (1−d)S_d = S_0\,(1-d) instead of Sd=d S0S_d = d\,S_0, giving Sd=6S_d = 6. Then h=772−6=766=0.10606h = \frac{7}{72-6} = \frac{7}{66} = 0.10606, B=0.63636 e−0.025=0.6203B = 0.63636\,e^{-0.025} = 0.6203, and C0=0.10606×60−0.6203=5.74C_0 = 0.10606\times 60 - 0.6203 = 5.74.
    6
    Derivative Instrument Features
    easy
    DER-FEAT-03

    A firm enters a pay-fixed, receive-floating interest rate swap with a notional principal of $50,000,000. The fixed rate is 2.40% and payments are made quarterly on an ACT/360 basis (assume 90 days in the quarter). The floating rate set at the start of the quarter is 3.10%. At the settlement date for the quarter, what is the net cash flow and its direction for the firm?

    1. A.Receive $387,500
    2. B.Receive $87,500
    3. C.Receive $350,000
    Show answer and explanation

    Correct answer: B

    Day-count fraction:

    90360=0.25\frac{90}{360} = 0.25

    Fixed-leg payment:

    50,000,000×0.024×0.25=300,00050{,}000{,}000 \times 0.024 \times 0.25 = 300{,}000

    Floating-leg payment (rate set at the start of the period):

    50,000,000×0.031×0.25=387,50050{,}000{,}000 \times 0.031 \times 0.25 = 387{,}500

    Net to the pay-fixed, receive-floating party: 387,500−300,000=87,500387{,}500 - 300{,}000 = 87{,}500
    Direction: receive $87,500.

    • Option A: 387,500 is the floating payment alone, ignoring the fixed payment owed.
    • Option C: (0.031−0.024)×50,000,000=350,000(0.031 - 0.024) \times 50{,}000{,}000 = 350{,}000, omits the 0.25 day-count factor.
    7
    Forward Commitment Pricing
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    DER-FWD-02

    The spot exchange rate is 1.1200 USD/EUR. The 6‑month domestic (USD) simple annual interest rate is 4%, and the 6‑month foreign (EUR) simple annual interest rate is 1%. Using interest rate parity with simple interest over the exact 6‑month horizon, compute the 6‑month no‑arbitrage forward rate (USD/EUR).

    1. A.1.1367 USD/EUR
    2. B.1.1200 USD/EUR
    3. C.1.1035 USD/EUR
    Show answer and explanation

    Correct answer: A

    Use interest rate parity with simple interest over T = 0.5 years:

    F0=S0×1+rdT1+rfTF_0 = S_0 \times \frac{1 + r_d T}{1 + r_f T} F0=1.1200×1+0.04×0.51+0.01×0.5=1.1200×1.021.005=1.1200×1.014925=1.1367F_0 = 1.1200 \times \frac{1 + 0.04 \times 0.5}{1 + 0.01 \times 0.5} = 1.1200 \times \frac{1.02}{1.005} = 1.1200 \times 1.014925 = 1.1367
    • Option B (1.1200) ignores the interest differential, using the spot rate only.
    • Option C (1.1035) inverts the ratio: 1.1200×1.0051.02=1.10351.1200 \times \frac{1.005}{1.02} = 1.1035
    8
    Hedging and Risk Management
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    DER-HRM-02

    A manager oversees a $24,000,000 equity portfolio with beta 1.1. She wants to reduce the portfolio beta to 0.8 using index futures. The futures price is 4,000 and the contract multiplier is $50. Assume the futures beta is 1. How many index futures contracts should she short to achieve the target beta?

    1. A.36
    2. B.96
    3. C.132
    Show answer and explanation

    Correct answer: A

    Use the beta-adjustment hedge formula: N=(βp−βtargetβf)×VpF0×QN = \big(\frac{\beta_p - \beta_{target}}{\beta_f}\big) \times \frac{V_p}{F_0\times Q}. With βf=1\beta_f = 1:

    F0×Q=4,000×50=200,000F_0 \times Q = 4,000 \times 50 = 200,000 VpF0×Q=24,000,000200,000=120\frac{V_p}{F_0 \times Q} = \frac{24,000,000}{200,000} = 120 Δβ=1.1−0.8=0.3\Delta\beta = 1.1 - 0.8 = 0.3 N=0.3×120=36N = 0.3 \times 120 = 36

    She should short 36 contracts.

    • Option B uses the target beta instead of the change in beta: N=0.8×120=96N = 0.8 \times 120 = 96.
    • Option C uses the current beta instead of the change in beta: N=1.1×120=132N = 1.1 \times 120 = 132.
    9
    Options Pricing: Basics
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    DER-OB-2

    Using put-call parity for a European option on a non-dividend-paying stock, compute the no-arbitrage price of the put given: call price = 7.00, stock price S0 = 100, strike K = 110, effective annual risk-free rate r = 5%, and time to expiration T = 1 year.

    1. A.$11.76
    2. B.$17.00
    3. C.$2.24
    Show answer and explanation

    Correct answer: A

    Put-call parity (no dividends): p=c+PV(K)−S0p = c + \text{PV}(K) - S_0.

    PV(K)=1101.05=104.7619\text{PV}(K) = \frac{110}{1.05} = 104.7619 p=7.00+104.7619−100=11.7619p = 7.00 + 104.7619 - 100 = 11.7619 p=11.76p = 11.76
    • Option B forgets to discount the strike:
    p=7.00+110−100=17.00p = 7.00 + 110 - 100 = 17.00
    • Option C reverses the signs on S0S_0 and PV(K)\text{PV}(K):
    p=7.00+100−104.7619=2.2381 (= 2.24)p = 7.00 + 100 - 104.7619 = 2.2381 \ (\text{= 2.24})
    10
    Options Pricing: Models
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    DER.OPM.01

    A European call option on a non-dividend-paying stock has the following parameters: current stock price S0 = 50, strike price K = 52, risk-free rate r = 5% (continuously compounded), time to expiration T = 0.5 years, and volatility σ=0.30\sigma = 0.30. Using the Black–Scholes–Merton model, what is the call price?

    1. A.3.34
    2. B.0.95
    3. C.3.90
    Show answer and explanation

    Correct answer: C

    Compute inputs.

    ln⁡ ⁣(S0K)=ln⁡ ⁣(0.961538)=−0.03922\ln\!\left(\frac{S_0}{K}\right) = \ln\!(0.961538) = -0.03922 σT=0.30×0.5=0.21213\sigma\sqrt{T} = 0.30 \times \sqrt{0.5} = 0.21213 d1=−0.03922+(0.05+0.5×0.302)×0.50.21213=−0.03922+0.047500.21213=0.0390d_1 = \frac{-0.03922 + (0.05 + 0.5\times 0.30^2)\times 0.5}{0.21213} = \frac{-0.03922 + 0.04750}{0.21213} = 0.0390 d2=d1−σT=0.0390−0.21213=−0.1731d_2 = d_1 - \sigma\sqrt{T} = 0.0390 - 0.21213 = -0.1731 e−rT=e−0.025=0.97531e^{-rT} = e^{-0.025} = 0.97531 N(d1)≈0.5155,N(d2)≈0.4314N(d_1) \approx 0.5155,\quad N(d_2) \approx 0.4314 c=S0N(d1)−Ke−rTN(d2)=50×0.5155−52×0.97531×0.4314=25.775−21.879=3.90c = S_0 N(d_1) - K e^{-rT} N(d_2) = 50\times 0.5155 - 52\times 0.97531\times 0.4314 = 25.775 - 21.879 = 3.90
    • Option A (3.34): Uses KK without discounting. 50×0.5155−52×0.4314=25.775−22.4328=3.3450\times 0.5155 - 52\times 0.4314 = 25.775 - 22.4328 = 3.34
    • Option B (0.95): Uses variance 0.09 instead of volatility 0.30 in σT\sigma\sqrt{T}, giving d1=−0.1915, d2=−0.2551d_1 = -0.1915,\ d_2 = -0.2551, so c=50×0.4241−52×0.97531×0.3993=21.205−20.251=0.95c = 50\times 0.4241 - 52\times 0.97531\times 0.3993 = 21.205 - 20.251 = 0.95.

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