Forwards, futures, swaps, options and credit derivatives — MCQs

Northbridge Options Advisory is preparing recommendations for clients who use equity options for replication, hedging, and return enhancement. Portfolio manager Elena Valdez reviews several positions with an analyst. All options in the review are European, expire in one year unless otherwise stated, and have no interim cash flows. The underlying shares pay no dividends, transaction costs are zero, and investors can borrow and lend at the stated risk-free rates. The first position involves a share trading at USD 52.00. A one-year European call with an exercise price of USD 50.00 trades at USD 6.00. The effective annual risk-free rate is 4.00%. A European put has the same underlying share, exercise price, and expiration date. Valdez asks the analyst to determine the put's no-arbitrage value and to identify the combination of options and risk-free lending that creates the same expiration payoff as one long share. The second position is evaluated using a one-period binomial model. A share currently trades at USD 100.00 and will be worth either USD 120.00 or USD 90.00 in one year. A European call has an exercise price of USD 100.00, and the effective annual risk-free rate is 5.00%. Northbridge will value the call by forming a portfolio of the underlying share and a risk-free borrowing or lending position that exactly reproduces the call payoff in both terminal states. No probability estimate for the two share-price outcomes is required. The analyst is also asked to value a European put with the same exercise price using the same binomial framework. Valdez also owns shares in a mature company and expects the share price to rise only modestly over the next three months. She expects realized volatility to remain low and is willing to sell her shares if their price rises above a specified exercise price. Her primary objective is to generate option-premium income on the existing holding. She does not require a guaranteed minimum sale price and does not expect a large price movement in either direction. Valdez reminds the analyst that a protective put is appropriate when downside protection and retained upside are the principal objectives, whereas a volatility strategy requires an expectation of a sufficiently large price movement to recover the premiums paid. She asks the analyst to use the facts of her mandate rather than select a strategy merely because it provides some form of risk reduction.

Meridian Asset Management is reviewing option positions for valuation, replication, and portfolio insurance. Senior derivatives analyst Priya Shah requires her team to use no-arbitrage relationships rather than subjective expected returns. The options under review are European, may be exercised only at expiration, and are free of counterparty default and transaction costs. Borrowing and lending are available at the stated risk-free rate, and short selling is permitted. For the first analysis, Shah provides prices for a call and its underlying stock and asks the team to infer the value of a put with the same exercise price and expiration. The stock pays no dividends before expiration. The exercise price must therefore be discounted when applying put–call parity. Shah also asks the team to interpret the exposure produced by buying a call and writing a put with identical exercise prices and expiration dates. She emphasizes that the relevant comparison is the combined expiration payoff, including the obligation represented by the exercise price. She also asks the team to consider how the implied synthetic forward price would respond to a change in the risk-free rate. The second analysis uses a one-period binomial model. The stock can move to only the two terminal values specified in Exhibit 1, and the option payoff is based on those terminal stock values. The risk-neutral probability must be inferred from the stock-price tree and the risk-free return. Alternatively, the call may be valued using a replicating portfolio consisting of shares and risk-free borrowing, and the number of shares in that portfolio defines the hedge ratio. Either valid no-arbitrage method produces the same current option value. The final assignment concerns an equity portfolio whose manager is unwilling to sell the shares. The manager expects long-term appreciation but is concerned about a possible six-month market disruption. The manager requires an explicit minimum value for the equity position over that horizon and insists that the hedge be implemented at zero net premium, even if that requires forgoing some upside participation. Shah considers protective puts, covered calls, and collars, but instructs the team to select the strategy whose payoff most closely matches every element of the stated mandate. Exhibit 1 presents the data required for each independent question.

Northbridge Options Desk advises institutional clients on option valuation and replication. A portfolio manager asks analyst David Lin to verify whether observed option prices are consistent with no-arbitrage relationships and to evaluate a proposed covered call position. All options are European and may be exercised only at expiration. The underlying shares pay no dividends, and taxes, transaction costs, bid–ask spreads, and borrowing constraints are ignored. For the first analysis, Lin applies put–call parity to a one-year call and put with identical exercise prices and expiration dates. The effective annual risk-free rate is constant over the period. Cash invested at the risk-free rate grows by a factor of 1.05 over the year. The relevant market inputs appear in Exhibit 1. Lin then uses a one-period binomial model to value a separate call. The share can take only the stated up-state or down-state value at the end of the year. He intends to calculate both the risk-neutral probability and the number of shares in the call's replicating portfolio. The call's expiration payoff is the greater of zero and the share price less the exercise price. The hedge ratio is based on the difference in call payoffs divided by the difference in underlying share values across the two states. The call value is obtained by discounting its risk-neutral expected payoff at the effective annual risk-free rate. The portfolio manager also proposes the covered call described in Exhibit 2. The position combines ownership of one share with the writing of one call on that share. Northbridge evaluates profit at option expiration, including the initial share purchase price and call premium. If the share price finishes above the exercise price, the share is delivered for the exercise price, so further share-price appreciation does not increase profit. If the share price finishes below the exercise price, the call expires unexercised and the position retains the share-price downside, partly offset by the premium received. Each analysis is independent, and no result from one option position is required to value another.

Northbridge Advisory manages option overlays for private and institutional clients. An analyst is reviewing option values and recommended strategies under the standard no-arbitrage assumptions of frictionless markets, no taxes, and unrestricted borrowing and short selling. The first analysis concerns European options on a non-dividend-paying stock. The call and put have the same exercise price and expiration date. The quoted risk-free return is an effective one-year rate, so the exercise price must be discounted for one year when applying put–call parity. The analyst is asked to determine the put's current value from the observed call value and underlying stock price. All option values are stated per share. The analyst separately evaluates a one-period binomial model for another non-dividend-paying stock. At the end of the period, the stock will take one of the two values shown in Exhibit 1. The risk-free return applies over the full period. The call is European, and its payoff is determined solely by its exercise price and the terminal stock price. The model satisfies the no-arbitrage condition because the risk-free gross return lies between the stock's down-state and up-state gross returns. The option value should therefore be calculated from the risk-neutral expected payoff discounted at the one-period risk-free rate. Northbridge is also advising a client who already owns a concentrated equity position. The client is willing to pay an option premium to establish a firm minimum sale value but does not want to surrender gains if the share price rises substantially. The client has no requirement that the strategy be costless. Exhibit 2 provides the objective and two statements considered during the strategy review. For the final statement, the analyst assumes that the call and put in a long straddle are at the money, have the same expiration date, and are held rather than written. Interest rates, the underlying price, dividends, and time to expiration are held constant. The only changing input is expected volatility. Northbridge evaluates the statement based on the standard relationship between option value and volatility rather than on any directional forecast for the stock.

Northbridge Global Fund, a USD-based investor, holds a EUR-denominated equity position that it expects to liquidate in three months. The fund reports performance in USD and therefore faces the risk that the EUR will depreciate against the USD before the equity proceeds are converted. Exchange rates in the case are quoted as USD per EUR; EUR is the base currency and USD is the price currency. The portfolio manager assumes the value of the underlying equity position will remain EUR 12 million during the hedge horizon so that the analysis can focus exclusively on currency risk. The manager is considering three alternatives. The first is a three-month deliverable forward under which Northbridge would exchange the full EUR amount for USD at expiration. This hedge would remove both unfavorable and favorable currency movements. The manager wants the analyst to calculate the USD proceeds using the forward rate rather than the current spot rate. The second alternative is to purchase European EUR put options with a strike of USD 1.0900 per EUR. The options cover the entire EUR 12 million exposure and expire when the equity position is liquidated. The premium is paid in USD at initiation. For the requested expiration analysis, interest on the premium is ignored. The options are cash-settled based on the difference between the strike and the expiration spot rate, but the combined spot conversion and option payoff produce the same gross USD result as converting the EUR amount at the strike whenever the put finishes in the money. Unlike the forward, the put hedge allows Northbridge to benefit if the EUR appreciates. The third alternative is a zero-cost collar. Northbridge would buy the same EUR puts and finance their premium by selling European EUR calls on the same notional amount. The calls have a strike of USD 1.1400 per EUR and the same expiration date. The premiums received and paid are equal. The collar therefore establishes a protected minimum conversion rate but requires Northbridge to surrender gains from EUR appreciation beyond the call strike. The manager's stated preference is to retain favorable EUR participation if the cost is acceptable. She recognizes that the long-put strategy requires an upfront premium, whereas the collar avoids a net premium by capping favorable currency outcomes. Counterparty default, transaction costs, and differences in settlement timing are ignored.

A stock is priced at USD 100 and will pay USD 2 dividends in 3 and 9 months. With a 4% annual effective risk-free rate, the calculated value of a one-year forward price is closest to:

A stock priced at USD 50 will be USD 60 or USD 45 in one period. With a 5% risk-free rate, the calculated value of a European call with a USD 52 exercise price is closest to: