CFA Program — Derivatives

36 free practice questions with full explanations.

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Question 1

A trader buys a protective put on a stock she already owns. This combined position is designed primarily to:

A) Eliminate all upside potential in the stock
B) Limit downside risk on the stock position while retaining upside potential, in exchange for paying the put premium
C) Guarantee a fixed profit regardless of the stock's price movement
D) Eliminate the need to ever sell the underlying stock

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Correct answer: B) Limit downside risk on the stock position while retaining upside potential, in exchange for paying the put premium

A protective put (long stock plus a long put option) establishes a floor on the position's value (limiting downside losses to the put's strike price minus the premium paid) while still allowing the investor to benefit from further stock appreciation, at the cost of the put premium.

Question 2

Put-call parity establishes a relationship between the prices of a European call option, a European put option, the underlying asset, and:

A) The risk-free bond (present value of the strike price)
B) The company's dividend payout ratio only
C) The option's implied volatility exclusively
D) The bid-ask spread of the underlying stock

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Correct answer: A) The risk-free bond (present value of the strike price)

Put-call parity states that Call + PV(Strike Price) = Put + Underlying Asset Price (for European options with the same strike and expiration), linking the four instruments through a no-arbitrage relationship.

Question 3

The price of a forward contract at initiation is generally set so that the contract has:

A) Positive value to the long position
B) Zero value to both counterparties at initiation
C) Negative value to the short position only
D) A value equal to the notional amount

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Correct answer: B) Zero value to both counterparties at initiation

A forward contract's price is typically set at initiation such that the contract has zero value to both the long and short counterparties; value subsequently accrues to one side as the underlying asset's price moves relative to the forward price.

Question 4

The binomial option pricing model values an option by:

A) Assuming the underlying asset's price can only move in one direction
B) Modeling the underlying asset's price as moving to one of two possible values (up or down) over each discrete time step, then working backward to value the option
C) Ignoring the risk-free rate entirely
D) Only being applicable to interest rate swaps

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Correct answer: B) Modeling the underlying asset's price as moving to one of two possible values (up or down) over each discrete time step, then working backward to value the option

The binomial model discretizes time into steps where the underlying asset's price can move up or down by specified factors at each step, building a price tree, then values the option by working backward from expiration to the present using risk-neutral probabilities.

Question 5

An interest rate swap in which one party pays a fixed rate and receives a floating rate can be conceptually replicated by:

A) A series of forward rate agreements (FRAs) exchanging fixed for floating cash flows over the life of the swap
B) A single spot currency transaction
C) Purchasing a single zero-coupon bond only
D) A perpetual equity investment

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Correct answer: A) A series of forward rate agreements (FRAs) exchanging fixed for floating cash flows over the life of the swap

An interest rate swap can be conceptually decomposed into a series of forward rate agreements, each exchanging a fixed payment for a floating payment at a specific future settlement date, which is one common approach to valuing swaps.

Question 6

A credit default swap (CDS) buyer is essentially:

A) Selling insurance/protection against the default of a reference entity
B) Buying protection against the default of a reference entity, paying a periodic premium to the seller
C) Guaranteed to receive payment regardless of whether a default occurs
D) Taking on additional credit risk exposure to the reference entity

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Correct answer: B) Buying protection against the default of a reference entity, paying a periodic premium to the seller

The CDS buyer pays a periodic premium to the CDS seller in exchange for protection -- a payment if a defined credit event (e.g., default) occurs with respect to the reference entity, functioning similarly to insurance against credit risk.

Question 7

A portfolio manager uses equity index futures to quickly and efficiently adjust the portfolio's overall market (beta) exposure without transacting in the underlying individual stocks. This is an example of using derivatives for:

A) A strategy that is always illegal for institutional investors.
B) A strategy that guarantees the portfolio will outperform its benchmark.
C) Efficient portfolio management, allowing rapid, lower-cost adjustment of market exposure compared to trading many individual securities.
D) Pure speculation with no relationship to the existing portfolio.

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Correct answer: C) Efficient portfolio management, allowing rapid, lower-cost adjustment of market exposure compared to trading many individual securities.

Using index futures (or other derivatives) to adjust overall portfolio exposure -- such as quickly increasing or decreasing beta exposure -- without transacting in the many underlying individual securities is a classic example of using derivatives for efficient portfolio management, offering speed and lower transaction costs compared to trading individual stocks.

Question 8

Which of the following best describes the relationship between the price of a call option and the price of an otherwise identical put option (same strike, same expiration) on a stock that is expected to pay a dividend before expiration?

A) Dividends have no effect on put-call parity under any circumstances.
B) Put-call parity applies only to non-dividend-paying stocks and cannot be adjusted for dividends.
C) The call price and put price are always exactly equal regardless of dividends.
D) Put-call parity for dividend-paying stocks must incorporate the present value of the expected dividend, since the anticipated dividend affects the relationship between call and put prices, the underlying price, and the present value of the strike.

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Correct answer: D) Put-call parity for dividend-paying stocks must incorporate the present value of the expected dividend, since the anticipated dividend affects the relationship between call and put prices, the underlying price, and the present value of the strike.

When the underlying stock is expected to pay a dividend before option expiration, put-call parity must be adjusted to incorporate the present value of that expected dividend, since anticipated dividends reduce the expected future stock price (all else equal) and thus affect the relative pricing relationship between calls and puts.

Question 9

Which of the following best describes a key risk of writing (selling) an uncovered (naked) call option on a stock the writer does not own?

A) Theoretically unlimited potential loss, since the stock price could rise without limit, requiring the writer to purchase shares at an increasingly high market price to fulfill the obligation if exercised.
B) The writer's maximum loss is always limited to the premium received.
C) This strategy carries no meaningful risk of any kind.
D) The writer is guaranteed to profit regardless of the stock's price movement.

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Correct answer: A) Theoretically unlimited potential loss, since the stock price could rise without limit, requiring the writer to purchase shares at an increasingly high market price to fulfill the obligation if exercised.

Writing an uncovered (naked) call exposes the writer to theoretically unlimited loss, since the underlying stock price has no upper bound, and the writer would need to purchase shares at the prevailing (potentially very high) market price to deliver them if the option is exercised, in stark contrast to a covered call, where the writer already owns the shares.

Question 10

A fixed-income portfolio manager wants to synthetically increase the duration of her bond portfolio without purchasing additional bonds. Which derivative strategy would most directly achieve this?

A) Purchasing a credit default swap unrelated to interest rate exposure.
B) Entering a receive-fixed, pay-floating interest rate swap, or buying interest rate futures/forwards, to add duration exposure synthetically.
C) Entering a pay-fixed, receive-floating interest rate swap, which would decrease duration instead.
D) Selling all bonds and holding cash exclusively.

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Correct answer: B) Entering a receive-fixed, pay-floating interest rate swap, or buying interest rate futures/forwards, to add duration exposure synthetically.

Receiving fixed and paying floating in an interest rate swap effectively adds duration to a portfolio (since the fixed-rate leg behaves like a bond), allowing a manager to synthetically increase overall portfolio duration without transacting in the underlying physical bonds; the opposite (pay-fixed) swap position would reduce duration.

Question 11

Which of the following best describes the "delta" of an at-the-money call option, roughly speaking?

A) Always exactly 0.00 regardless of moneyness.
B) A value that is always negative for any call option.
C) Approximately 0.50, reflecting roughly equal probability-weighted sensitivity to the underlying price moving up or down from the current level.
D) Always exactly 1.00 regardless of moneyness.

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Correct answer: C) Approximately 0.50, reflecting roughly equal probability-weighted sensitivity to the underlying price moving up or down from the current level.

An at-the-money call option typically has a delta near 0.50, reflecting that the option's value is roughly equally sensitive (in a probability-weighted sense) to the underlying price moving up or down from the current level; delta approaches 1.0 as a call moves deeper in-the-money and approaches 0 as it moves deeper out-of-the-money.

Question 12

An investor holds a portfolio of stocks and wants to hedge against a moderate market decline while retaining some upside potential, and is willing to accept a cap on the maximum gain in exchange for reducing the cost of the hedge. Which strategy best fits this objective?

A) Selling all holdings and moving entirely to cash, foregoing all upside.
B) Buying additional shares of the same stocks with no options involved.
C) Selling uncovered (naked) call options against the portfolio.
D) A collar (buying a protective put and selling a covered call, often structured to be low-cost or zero-cost).

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Correct answer: D) A collar (buying a protective put and selling a covered call, often structured to be low-cost or zero-cost).

A collar combines a protective put (providing downside protection below a certain level) with a covered call (capping upside above a certain level, with the premium received offsetting some or all of the put's cost), directly matching an investor's goal of reducing hedging cost while accepting a cap on maximum gains.

Question 13

An analyst estimates the value of a European call option using the Black-Scholes-Merton model and compares it to the option's actual market price, finding the market price is notably higher. This discrepancy, holding the model's other inputs constant, is most likely explained by:

A) The market pricing in a higher implied volatility than the volatility assumption used in the analyst's model input.
B) The option must be mispriced and present a risk-free arbitrage opportunity with certainty.
C) Black-Scholes-Merton always produces prices identical to market prices with no exceptions.
D) The risk-free rate used has no effect on the model's output value.

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Correct answer: A) The market pricing in a higher implied volatility than the volatility assumption used in the analyst's model input.

Since option value is highly sensitive to the volatility assumption, a market price higher than the model-derived value (using the analyst's own volatility estimate) often simply reflects that the market is pricing in a higher implied volatility than the analyst assumed, rather than indicating a clear mispricing or arbitrage opportunity.

Question 14

Which of the following best describes the primary distinction between a "plain vanilla" interest rate swap and a "basis swap"?

A) A basis swap always involves the exchange of physical commodities.
B) There is no meaningful difference between the two types of swaps.
C) A plain vanilla swap can only be used by government entities.
D) A plain vanilla swap typically exchanges a fixed rate for a floating rate, while a basis swap exchanges two different floating rates (based on different reference indices), both applied to the same notional principal.

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Correct answer: D) A plain vanilla swap typically exchanges a fixed rate for a floating rate, while a basis swap exchanges two different floating rates (based on different reference indices), both applied to the same notional principal.

While a plain vanilla interest rate swap typically involves exchanging a fixed rate for a floating rate, a basis swap instead exchanges two different floating rate benchmarks (for example, one reference rate versus another), which can be useful for managing basis risk between different floating rate exposures a party might have.

Question 15

A portfolio manager is considering whether to hedge a long equity position using put options versus using equity index futures. Which of the following is a key advantage of using put options for this purpose, compared to a short futures hedge?

A) Put options allow the portfolio to retain upside participation if the market rises, while a short futures position would offset both downside protection and any potential upside gains.
B) Put options always cost less than an equivalent futures hedge in every situation.
C) Put options eliminate all forms of counterparty risk, unlike futures.
D) Put options provide no downside protection compared to futures.

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Correct answer: A) Put options allow the portfolio to retain upside participation if the market rises, while a short futures position would offset both downside protection and any potential upside gains.

A key structural difference is that a long put position provides asymmetric downside protection while preserving upside participation (since the put is simply not exercised if the market rises), whereas a short futures hedge offsets both downside losses and upside gains symmetrically, meaning the choice between the two involves a tradeoff between the upfront premium cost of options and the retained upside potential.

Question 16

Which of the following best describes a "variance swap"?

A) A type of interest rate swap unrelated to volatility.
B) A derivative contract in which the payoff is based on the difference between realized variance of an underlying asset's returns over the contract period and a variance level agreed upon at inception.
C) A contract that pays out based solely on the direction of the underlying asset's price movement.
D) A contract that can only be settled through physical delivery of the underlying asset.

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Correct answer: B) A derivative contract in which the payoff is based on the difference between realized variance of an underlying asset's returns over the contract period and a variance level agreed upon at inception.

A variance swap allows counterparties to directly trade exposure to realized volatility (specifically variance, the square of volatility) of an underlying asset, with the payoff determined by the difference between the actual realized variance over the contract period and a variance level fixed at inception, providing a relatively pure way to gain or hedge volatility exposure.

Question 17

A structured note embeds a call option on an equity index within a zero-coupon bond structure, designed to provide principal protection with equity-linked upside. If the equity index performs poorly over the note's life, the investor would most likely:

A) Receive a guaranteed fixed coupon regardless of the equity index performance.
B) Be required to make an additional payment to the note issuer.
C) Receive their original principal back at maturity (assuming the issuer does not default), since the zero-coupon bond component is designed to return principal regardless of the embedded option's outcome, though no additional upside would be realized.
D) Lose their entire principal investment, since the embedded call option expired worthless.

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Correct answer: C) Receive their original principal back at maturity (assuming the issuer does not default), since the zero-coupon bond component is designed to return principal regardless of the embedded option's outcome, though no additional upside would be realized.

In a typical principal-protected structured note, the zero-coupon bond component is sized to return the investor's principal at maturity (assuming no issuer default) regardless of how the embedded equity call option performs; if the index underperforms, the option simply expires worthless, and the investor receives no additional upside but does not lose their original principal.

Question 18

Which of the following best describes "gamma" as an option Greek, and its relevance to a delta-hedged position?

A) Gamma measures the sensitivity of an option's price to changes in interest rates.
B) Gamma is always equal to zero for any option position.
C) A delta-hedged position with high gamma never requires any rebalancing.
D) Gamma measures the rate of change of an option's delta with respect to a change in the underlying price, and a delta-hedged position with high gamma will require more frequent rebalancing to maintain the hedge as the underlying price moves.

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Correct answer: D) Gamma measures the rate of change of an option's delta with respect to a change in the underlying price, and a delta-hedged position with high gamma will require more frequent rebalancing to maintain the hedge as the underlying price moves.

Gamma captures how quickly delta itself changes as the underlying price moves; a position with high gamma means delta can shift significantly with even modest underlying price changes, requiring more frequent rebalancing of a delta-hedge to maintain effective neutrality, compared to a low-gamma position where delta remains more stable.

Question 19

The value of a swaption (option to enter a swap) increases with all of the following EXCEPT:

A) Longer time to expiration of the swaption
B) Higher interest rate volatility
C) A decrease in the forward swap rate when holding a payer swaption
D) An increase in the fixed swap rate when holding a payer swaption

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Correct answer: C) A decrease in the forward swap rate when holding a payer swaption

A payer swaption gives the right to pay fixed in a swap (benefiting from rising rates). Its value rises when the forward swap rate rises above the exercise rate. A decrease in the forward rate reduces the payer swaption's value.

Question 20

An investor writes (sells) a put option on Stock A with a strike of $50 and receives a premium of $4. At expiration, the stock is at $44. The investor's profit/loss is:

A) Loss of $6
B) Loss of $2
C) Profit of $4
D) Profit of $2

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Correct answer: B) Loss of $2

The short put is exercised against the investor, who must buy at $50. The investor pays $50 for a stock worth $44, a loss of $6. Offset by the $4 premium received: net loss = $6 − $4 = $2.

Question 21

A firm enters a 3-year pay-fixed swap at 5%. One year later, the market fixed rate for a 2-year swap is 6%. The current value of the swap to the fixed payer is most likely:

A) Positive, because higher rates benefit the fixed payer in the future
B) Negative, because the fixed payer is locked into a below-market rate
C) Zero, because the notional is not exchanged
D) Positive, because the fixed payer would receive a higher fixed rate if entering new

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Correct answer: B) Negative, because the fixed payer is locked into a below-market rate

When market rates rise above the fixed rate, the pay-fixed position has negative value: the party is obligated to pay 5% while new swaps pay 6%, so the existing swap is worth less than a newly negotiated one. The mark-to-market value is negative for the fixed payer.

Question 22

Black's model for pricing interest rate options uses the forward rate as the underlying. Compared to the Black-Scholes model, Black's model:

A) Is identical to Black-Scholes with spot rate substituted for stock price
B) Adjusts for the fact that interest rates, unlike stock prices, are not lognormally distributed
C) Uses the forward price instead of the current spot price, appropriate for options on futures or rates
D) Cannot be applied to caps and floors

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Correct answer: C) Uses the forward price instead of the current spot price, appropriate for options on futures or rates

Black's model modifies Black-Scholes by using the forward price (rather than spot price) as the underlying. It is widely used for pricing interest rate caps, floors, and swaptions where the underlying is a forward rate.

Question 23

An equity swap where one party pays the total return on the S&P 500 and receives a fixed rate has which of the following characteristics for the fixed-rate receiver?

A) Benefits when equity markets rise
B) Benefits when equity markets fall and suffers when they rise
C) Has no exposure to equity market direction
D) Is equivalent to holding a long position in the index

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Correct answer: B) Benefits when equity markets fall and suffers when they rise

The fixed-rate receiver pays the equity return and receives fixed. If the S&P 500 rises, the receiver must pay a larger amount (the equity return) and effectively loses. If equity falls, the counterparty pays the fixed receiver the negative return.

Question 24

A currency swap exchanges fixed USD payments for floating EUR payments. An American firm with EUR revenues and USD debt should enter which position to eliminate currency mismatch?

A) Pay fixed USD, receive fixed EUR
B) Pay fixed EUR, receive fixed USD
C) Pay floating EUR, receive fixed USD
D) Pay fixed USD, receive floating EUR and convert to fixed

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Correct answer: A) Pay fixed USD, receive fixed EUR

The firm has USD debt obligations (fixed payments) and EUR revenue. To match, it should pay fixed USD (to service debt) and receive EUR (matching its revenue). A pay-fixed USD / receive fixed EUR swap eliminates the mismatch.

Question 25

The Black-Scholes-Merton option pricing model assumes, among other things, that:

A) Interest rates are always negative.
B) The underlying asset's returns are lognormally distributed and that volatility and the risk-free rate remain constant over the option's life.
C) The underlying asset price never changes over the life of the option.
D) The option can only be exercised on the first day it is issued.

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Correct answer: B) The underlying asset's returns are lognormally distributed and that volatility and the risk-free rate remain constant over the option's life.

Among its key assumptions, the Black-Scholes-Merton model assumes the underlying asset's returns follow a lognormal distribution (implying continuously compounded returns are normally distributed) and that volatility and the risk-free rate are constant over the life of the option, along with other assumptions like frictionless markets and no dividends (in the basic version).

Question 26

The value of a call option's "delta" measures:

A) The sensitivity of the option's price to a small change in the price of the underlying asset.
B) The option's time to expiration exclusively.
C) The option's strike price exclusively.
D) The risk-free interest rate used in the option pricing model.

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Correct answer: A) The sensitivity of the option's price to a small change in the price of the underlying asset.

Delta measures the rate of change of an option's price with respect to a small change in the underlying asset's price, and is a key sensitivity ("Greek") used in option risk management and hedging (for example, delta-hedging a position).

Question 27

Put-call parity establishes a relationship among the prices of a European call option, a European put option, the underlying asset, and a risk-free bond, all with the same strike price and expiration date. This relationship is primarily used to:

A) Eliminate the need to know the strike price when valuing an option.
B) Apply exclusively to American-style options, never European-style options.
C) Identify arbitrage opportunities and derive the value of one option given the price of the other and the underlying asset.
D) Guarantee that call and put options always have identical prices.

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Correct answer: C) Identify arbitrage opportunities and derive the value of one option given the price of the other and the underlying asset.

Put-call parity defines a no-arbitrage relationship (C + PV(X) = P + S for European options) among a call, a put, the underlying, and the present value of the strike; if this relationship does not hold in the market, an arbitrage opportunity exists, and the formula can also be used to derive the implied value of one option given the others.

Question 28

A European call option with a strike price of $50 and 1 year to expiration is priced at $6.00. The underlying stock trades at $52, and the 1-year risk-free rate is 5%. Using put-call parity, the price of the otherwise identical European put option is closest to:

A) $6.00
B) $1.62
C) $8.00
D) $3.62

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Correct answer: B) $1.62

Put-call parity: P = C - S + PV(X) = C - S + X/(1+r) = 6 - 52 + 50/1.05 = 6 - 52 + 47.62 = $1.62.

Question 29

A trader constructs a "delta-neutral" hedge by combining an option position with an offsetting position in the underlying asset. The primary goal of this strategy is to:

A) Guarantee a large profit regardless of market conditions.
B) Eliminate the passage of time as a factor affecting the position.
C) Maximize the position's sensitivity to underlying price changes.
D) Make the combined position's value insensitive, at least momentarily, to small changes in the price of the underlying asset.

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Correct answer: D) Make the combined position's value insensitive, at least momentarily, to small changes in the price of the underlying asset.

A delta-neutral hedge combines options and the underlying asset in proportions such that the combined position's delta is approximately zero, making its value insensitive to small movements in the underlying asset's price, at least momentarily (since delta itself changes as the underlying price and time change, requiring rebalancing).

Question 30

In the binomial option pricing model, the "risk-neutral probability" used to calculate an option's expected payoff at each node is best understood as:

A) A probability that must always equal exactly 50% at every node.
B) A measure of the option's implied volatility.
C) A probability derived so that the underlying asset's expected return equals the risk-free rate, allowing the option to be valued by discounting expected payoffs at the risk-free rate without needing to know investors' actual risk preferences or the true probability of an up or down move.
D) The actual, real-world probability that investors believe the stock will move up.

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Correct answer: C) A probability derived so that the underlying asset's expected return equals the risk-free rate, allowing the option to be valued by discounting expected payoffs at the risk-free rate without needing to know investors' actual risk preferences or the true probability of an up or down move.

Risk-neutral probabilities are a mathematical construct, not real-world probabilities: they are calibrated so that discounting the underlying asset's expected future value at the risk-free rate reproduces its current price, which conveniently allows the option's expected payoff (computed using these probabilities) to also be discounted at the risk-free rate, bypassing the need to estimate investors' true risk preferences or actual probabilities.

Question 31

An investor constructs a bull call spread by buying a call option with a strike price of $50 for a premium of $4.00, and simultaneously selling a call option on the same underlying stock, with the same expiration date, with a strike price of $60 for a premium of $1.50. The maximum profit, maximum loss, and breakeven stock price at expiration for this strategy are closest to:

A) Maximum profit = $7.50 per share; maximum loss = $2.50 per share; breakeven = $52.50. Net premium paid = $4.00 - $1.50 = $2.50 (maximum loss, occurring if the stock finishes at or below $50). Maximum profit = (Strike spread) - net premium = ($60-$50) - $2.50 = $10.00 - $2.50 = $7.50 (occurring if the stock finishes at or above $60). Breakeven = lower strike + net premium = $50 + $2.50 = $52.50.
B) Maximum profit = $10.00 per share; maximum loss = $4.00 per share; breakeven = $54.00, which incorrectly ignores the premium received from selling the $60 call entirely, treating the position as if only the long $50 call had been purchased.
C) Maximum profit = $2.50 per share; maximum loss = $7.50 per share; breakeven = $52.50, which incorrectly reverses the maximum profit and maximum loss figures relative to their correct values.
D) Maximum profit = unlimited; maximum loss = $2.50 per share; breakeven = $52.50, which incorrectly treats the strategy as if it were simply a long call position with unlimited upside, ignoring the offsetting short call leg that caps the maximum gain at the strike spread minus the net premium.

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Correct answer: A) Maximum profit = $7.50 per share; maximum loss = $2.50 per share; breakeven = $52.50. Net premium paid = $4.00 - $1.50 = $2.50 (maximum loss, occurring if the stock finishes at or below $50). Maximum profit = (Strike spread) - net premium = ($60-$50) - $2.50 = $10.00 - $2.50 = $7.50 (occurring if the stock finishes at or above $60). Breakeven = lower strike + net premium = $50 + $2.50 = $52.50.

A bull call spread's net cost (and maximum loss) equals the premium paid for the long call minus the premium received for the short call: $4.00 - $1.50 = $2.50. Maximum profit is capped at the difference between the two strike prices minus this net premium: ($60-$50) - $2.50 = $7.50, achieved once the stock price reaches or exceeds the higher ($60) strike. Breakeven occurs where the payoff on the long call just offsets the net premium paid: $50 (lower strike) + $2.50 (net premium) = $52.50.

Question 32

An analyst is pricing a 2-year annual-pay interest rate swap using discount factors derived from the benchmark curve: the 1-year discount factor is 0.9615 and the 2-year discount factor is 0.9070. Using the standard swap pricing relationship, the fixed swap rate that equates the present value of the fixed-rate leg to the present value of the floating-rate leg (par value of 1) is closest to:

A) 9.30%, which incorrectly fails to divide the numerator (1 - final discount factor) by the sum of discount factors, reporting only the numerator itself, expressed as a percentage.
B) 4.98%, calculated as Swap fixed rate = (1 - final discount factor) / (sum of all discount factors) = (1 - 0.9070) / (0.9615 + 0.9070) = 0.0930 / 1.8685 = 4.98%.
C) 3.85%, which incorrectly uses only the 1-year discount factor (as 1 - 0.9615) in the numerator, rather than correctly using (1 - the final period's discount factor).
D) 5.00%, which incorrectly rounds the swap rate to a conveniently round number rather than deriving it from the actual discount factor calculation.

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Correct answer: B) 4.98%, calculated as Swap fixed rate = (1 - final discount factor) / (sum of all discount factors) = (1 - 0.9070) / (0.9615 + 0.9070) = 0.0930 / 1.8685 = 4.98%.

The fixed swap rate for a swap with notional/par value of 1 is: Swap rate = (1 - DF_final) / (sum of DF_1 through DF_final) = (1 - 0.9070) / (0.9615 + 0.9070) = 0.0930 / 1.8685 = 0.04977, approximately 4.98%. This rate equates the present value of the fixed leg to the present value of the floating leg (which is always par value at initiation).

Question 33

A stock currently trades at $60. Using a one-period binomial model, the stock price can move up to $72 (an up-factor of 1.20) or down to $54 (a down-factor of 0.90) after one period. The risk-free rate over the period is 5.0%. A European call option on this stock has a strike price of $60. Using risk-neutral valuation, the call option's value today is closest to:

A) $5.71: The risk-neutral probability of the up-move is p = [(1+r) - d] / (u - d) = (1.05 - 0.90) / (1.20 - 0.90) = 0.15/0.30 = 0.50. Payoff if up = max(72-60, 0) = $12; payoff if down = max(54-60, 0) = $0. Call value = [0.50 x $12 + 0.50 x $0] / 1.05 = $6.00 / 1.05 = $5.71.
B) $6.00, which correctly calculates the risk-neutral expected payoff but incorrectly omits discounting that expected payoff back to the present at the risk-free rate.
C) $12.00, which incorrectly reports the full up-state payoff itself as the option's current value, without applying the risk-neutral probability weighting or discounting.
D) $4.29, which incorrectly uses an assumed risk-neutral probability of 0.375 (rather than the correctly calculated 0.50) when computing the expected payoff.

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Correct answer: A) $5.71: The risk-neutral probability of the up-move is p = [(1+r) - d] / (u - d) = (1.05 - 0.90) / (1.20 - 0.90) = 0.15/0.30 = 0.50. Payoff if up = max(72-60, 0) = $12; payoff if down = max(54-60, 0) = $0. Call value = [0.50 x $12 + 0.50 x $0] / 1.05 = $6.00 / 1.05 = $5.71.

Risk-neutral probability p = [(1+r)-d]/(u-d) = (1.05-0.90)/(0.30) = 0.50. Expected payoff = p x C_up + (1-p) x C_down = 0.50 x 12 + 0.50 x 0 = 6.00. Discounting at the risk-free rate: Call value = 6.00/1.05 = $5.71.

Question 34

A European call option on a stock has a premium of $4.50, and the corresponding European put option (same underlying, strike, and expiration) is being valued. The stock currently trades at $52, the strike price is $50, the risk-free rate is 3.0% (annually compounded), and both options expire in 6 months. Using put-call parity, the value of the European put option is closest to:

A) $2.50, which incorrectly computes the put value as simply the difference between the strike price and the stock price ($52 - $50 = $2), adjusted arbitrarily, rather than correctly applying the put-call parity formula.
B) $6.27, which incorrectly adds, rather than subtracts, the current stock price in the put-call parity formula (P = C + S0 + PV(X) - 2 x S0, or an otherwise inconsistent rearrangement).
C) $4.50, which incorrectly assumes the put and call must have identical premiums, ignoring the actual relationship between the stock price, strike price, and time value of money specified by put-call parity.
D) $1.77, calculated using put-call parity: P = C - S0 + X/(1+r)^T = $4.50 - $52.00 + [$50/(1.03)^0.5] = $4.50 - $52.00 + $49.27 = $1.77.

Show answer & explanation

Correct answer: D) $1.77, calculated using put-call parity: P = C - S0 + X/(1+r)^T = $4.50 - $52.00 + [$50/(1.03)^0.5] = $4.50 - $52.00 + $49.27 = $1.77.

Put-call parity: C + PV(X) = P + S0, rearranged as P = C - S0 + PV(X). PV(X) = 50/(1.03)^0.5 = 50/1.014889 = 49.266. P = 4.50 - 52.00 + 49.266 = $1.77.

Question 35

An investor is long a 6x9 forward rate agreement (FRA) with a notional amount of $10,000,000 and a contract (FRA) rate of 5.00%. At settlement (which occurs at the start of the 3-month underlying rate period, in 6 months), the relevant 90-day reference rate is 5.60%, and settlement uses the standard discounted FRA payoff convention. The payment received by the long party at settlement is closest to:

A) $15,000, which correctly computes the numerator (undiscounted payoff) but incorrectly omits the required discounting of the payment back to the start of the underlying rate period, since FRA settlement occurs at the beginning, not the end, of the reference rate period.
B) $60,000, which incorrectly applies the full annualized rate differential of 0.60% directly to the notional without prorating for the 90/360 period fraction or applying the discounting adjustment.
C) $14,793, calculated as Payment = [(Reference rate - FRA rate) x (90/360) x Notional] / [1 + Reference rate x (90/360)] = [(0.056 - 0.050) x 0.25 x $10,000,000] / [1 + (0.056 x 0.25)] = $15,000 / 1.014 = $14,793.
D) -$14,793, which incorrectly reverses the sign of the payoff; since the reference rate (5.60%) exceeds the FRA rate (5.00%), the long party (who benefits from rising rates) receives, rather than pays, this amount.

Show answer & explanation

Correct answer: C) $14,793, calculated as Payment = [(Reference rate - FRA rate) x (90/360) x Notional] / [1 + Reference rate x (90/360)] = [(0.056 - 0.050) x 0.25 x $10,000,000] / [1 + (0.056 x 0.25)] = $15,000 / 1.014 = $14,793.

FRA settlement payments are discounted because they are paid at the start, rather than the end, of the underlying interest rate period. Numerator = (0.056-0.050) x (90/360) x 10,000,000 = 0.006 x 0.25 x 10,000,000 = $15,000. Discount factor = 1 + (0.056 x 90/360) = 1.014. Payment = 15,000/1.014 = $14,793, received by the long party since the reference rate exceeded the FRA rate.

Question 36

A stock currently trades at $50 per share and is expected to pay dividends with a present value of $1.50 over the next year. The 1-year risk-free rate is 4.0% (annually compounded). Using the cost-of-carry model, the no-arbitrage 1-year forward price on the stock is closest to:

A) $52.00, which incorrectly adds the present value of dividends to the spot price rather than subtracting it, before applying the cost-of-carry growth factor.
B) $50.44, calculated as F0 = (S0 - PV(dividends)) x (1+r)^T = ($50.00 - $1.50) x (1.04)^1 = $48.50 x 1.04 = $50.44.
C) $52.06, which incorrectly grows the full, undiscounted spot price of $50.00 at the risk-free rate without first subtracting the present value of expected dividends.
D) $48.50, which correctly subtracts the present value of dividends from the spot price but incorrectly omits the cost-of-carry growth factor (1+r)^T entirely.

Show answer & explanation

Correct answer: B) $50.44, calculated as F0 = (S0 - PV(dividends)) x (1+r)^T = ($50.00 - $1.50) x (1.04)^1 = $48.50 x 1.04 = $50.44.

The cost-of-carry forward pricing formula for a dividend-paying stock: F0 = [S0 - PV(dividends)] x (1+r)^T = [50.00 - 1.50] x (1.04) = 48.50 x 1.04 = $50.44. Dividends reduce the effective cost of carrying the stock (since the holder receives them), lowering the forward price relative to a non-dividend-paying stock with the same spot price.

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