Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives — MCQs

The Northbridge Derivatives Desk advises an institutional client that uses equity forwards and futures to obtain market exposure without immediately reallocating its cash securities. A derivatives analyst is reviewing Position Alpha, a proposed one-year forward on a non-defaultable equity security. The security currently trades at USD 100.00 and will pay a known USD 3.00 dividend six months from today. The dividend is paid to the registered owner of the security and is not received by the long forward holder. The one-year forward has no interim settlement, and the continuously compounded risk-free rate is 4.00% for all relevant maturities. Transaction costs, taxes, bid–ask spreads, and short-sale restrictions can be ignored. The analyst also reviews Position Beta, an existing long equity forward with six months remaining. Its contractual delivery price is USD 98.00, whereas a newly initiated six-month forward on the same underlying would have a delivery price of USD 103.00. The forward requires no collateral payments before maturity, and counterparty credit risk is assumed to be negligible. The analyst notes that the contract's current value must not be confused with either its original value at initiation or its expiration payoff. A portfolio manager then asks whether an exchange-traded futures contract can be valued as though it were an otherwise identical forward. The futures contract is marked to market daily. The manager expects changes in the underlying asset's price to be strongly positively correlated with changes in short-term interest rates. Margin balances earn the prevailing short-term rate, and there are no material differences in contract specifications, liquidity, or counterparty risk that would otherwise affect the comparison. Finally, the desk identifies a one-year forward on a non-income-producing asset quoted at USD 106. The asset's spot price and financing cost imply a no-arbitrage forward price of USD 104. The asset can be bought, sold short, and stored without cost, and all transactions can be executed simultaneously. The borrowing and lending rates are identical. The desk must identify the trades that lock in the pricing discrepancy without exposure to the asset's terminal market price. The desk also considers the opposite case, in which the same asset's market forward price falls below its no-arbitrage value, requiring the mirror-image arbitrage strategy.

Harlow Asset Management uses derivatives in several client mandates. A derivatives analyst is reviewing the pricing of a proposed equity forward, the value of an existing forward, and the value of an interest rate swap. All contracts are free of default risk, transaction costs, and taxes. Short selling is permitted, and borrowing and lending are available at the stated risk-free rates. For the proposed equity forward, the underlying share will pay a single known cash dividend before the forward matures. The analyst notes that the dividend must be removed from the spot price at its present value before the net asset value is carried forward at the risk-free rate. Another employee instead proposes carrying the full spot price to maturity and ignoring the dividend because the forward holder does not receive it. The existing equity forward was entered into at a delivery price of USD 79.00 and now has nine months remaining. Harlow holds the long side. The current forward price shown in Exhibit 1 is the delivery price that would make a newly initiated nine-month forward have zero value. The contract is not collateralized, and its current value is required today rather than at expiration. The appropriate discount rate is the continuously compounded nine-month risk-free rate in Exhibit 1. Harlow also holds a receive-fixed, pay-floating interest rate swap. The swap has exactly one year remaining and payments occur semiannually. The fixed rate is quoted annually, so each fixed coupon equals one-half of the annual fixed-rate payment. The swap is valued immediately after a floating-rate reset and payment, making the floating-rate bond worth par. The next two discount factors are provided in Exhibit 2. No principal is exchanged, but the bond-equivalence valuation includes notional principal in both legs because it cancels when the legs are netted. Finally, the analyst compares exchange-traded futures with otherwise similar over-the-counter forwards. Futures are marked to market daily through a clearinghouse, while the forward's gain or loss generally accumulates until settlement. The analyst recognizes that clearing and margining reduce, but do not necessarily eliminate, counterparty and liquidity risks. She also notes that neither daily settlement nor standardization ensures that a futures hedge will perfectly match the underlying exposure.

Meridian Asset Management uses derivatives to manage an equity portfolio and a liability-sensitive fixed-income allocation. Priya Shah, the firm's derivatives analyst, is reviewing several transactions under the assumption that markets are frictionless, borrowing and lending occur at the stated risk-free rate, and all contracts are free of default risk unless otherwise noted. For an equity index that pays dividends continuously, Shah is asked to calculate a six-month forward price. She notes that the underlying index cannot be valued as if it paid no income because an investor holding the index receives dividends during the forward term. The relevant market inputs are presented in Exhibit 1. Meridian also entered a long equity index forward three months ago. The contract requires settlement based on a delivery price of 4,180, has three months remaining, and has a multiplier of USD 50 per index point. A newly initiated three-month forward currently has a price of 4,230. The existing contract requires no interim cash flows, and its current value must reflect the present value of the difference between the current forward price and the contracted delivery price. Separately, Shah is adjusting the systematic exposure of a diversified equity portfolio. The chief investment officer expects broad equity volatility to increase and directs Shah to reduce, but not eliminate, the portfolio's market exposure. The portfolio and futures data are provided in Exhibit 2. The futures contract is written on an index considered an appropriate proxy for the portfolio's systematic exposure. Shah may transact only in whole contracts and should use the number closest to the calculated hedge quantity. Finally, Meridian has entered a plain-vanilla interest rate swap as the fixed-rate payer and floating-rate receiver. The swap had zero value at initiation. Immediately afterward, the fixed rate on a newly initiated swap with otherwise identical terms rises, with no change in Meridian's credit quality or the expected floating-rate cash flows other than those implied by the higher market rate. Shah must determine the directional effect on Meridian's existing swap position.

Meridian Asset Management operates a derivatives desk that supports equity and fixed-income portfolios. Elena Ruiz, a derivatives analyst, is reviewing several existing and proposed positions. Unless otherwise stated, all contracts are entered into at market terms, interest rates and growth factors are consistent with the relevant contract maturities, and taxes and transaction costs are ignored. The equity portfolio is considering a six-month forward on one share of Lydian Corporation. Lydian is expected to pay dividends before the contract expires. Ruiz intends to use a cost-of-carry framework in which the underlying's continuous dividend yield reduces the effective cost of carry relative to the risk-free rate. The relevant share price, continuously compounded dividend yield, and continuously compounded risk-free rate are presented in Exhibit 1. The desk also owns an existing long equity forward with three months remaining until expiration. The contract has a delivery price of USD 100.00. A newly initiated forward with the same underlying and remaining maturity would have a delivery price equal to the current forward price shown in Exhibit 1. The existing contract is not collateralized, but the parties are assumed to be default-free for valuation purposes. Ruiz notes that the contract's current value can differ from both its initiation value and its expiration payoff. The discount factor for the remaining term is also provided in Exhibit 1. For the fixed-income portfolio, Meridian recently entered a plain-vanilla interest rate swap under which it pays a fixed rate and receives a floating reference rate. The swap had zero value at initiation. Immediately afterward, the market fixed rate for an otherwise identical swap increased, while the credit quality of both parties and the expected floating-rate payment dates remained unchanged. The analyst's conclusion concerning the effect on Meridian's position appears in Exhibit 2. Finally, the desk is comparing exchange-traded futures with customized OTC forwards. The futures positions are centrally cleared, subject to initial and maintenance margin, and settled through daily marking to market. The OTC forwards are bilateral contracts without daily cash settlement or third-party guarantees. Meridian would face the dealer directly throughout the life of each forward. Ruiz's comments concerning the forward, swap, and counterparty exposures appear in Exhibit 2.

Northstar Asset Management uses derivatives to modify market exposures without selling underlying securities. Analyst Elena Ruiz is reviewing two equity forward contracts and an interest rate swap. All contracts are free of default risk, transaction costs, and taxes, and short selling is permitted with full use of the proceeds. Contract Alpha is a new one-year forward on a non-index-dividend-adjusted equity security. Because the security will make two known cash distributions before the contract expires, Ruiz plans to use the income-adjusted cost-of-carry relationship. She notes that the distributions must be removed from the spot price at their present values before the net carrying amount is accumulated to expiration. Another analyst instead proposes compounding the entire spot price and ignoring the distributions. Contract Beta is an existing long forward with six months remaining. Its original delivery price differs from the forward price currently available for a new contract with the same underlying and expiration date. Ruiz states that Contract Beta's current value is not simply its expected expiration payoff. Rather, the difference between the current forward price and the existing delivery price must be discounted over the remaining term. Ruiz also compares otherwise identical forward and futures contracts. Daily settlement allows interim futures gains and losses to be reinvested or financed at prevailing rates. Consequently, when changes in the underlying price are correlated with interest rates, the relative value of the daily settlement feature can cause futures and forward prices to differ. The comparison assumes that both contracts have the same underlying asset and expiration date. Finally, Northstar holds a receive-fixed, pay-floating interest rate swap. The valuation occurs immediately after the floating payment has been settled and the floating rate has reset. Ruiz applies the bond-equivalent approach, treating the position as long a fixed-rate bond and short a floating-rate bond. The market yield curve is flat, and the same annual rate applies to each remaining cash flow. No principal is exchanged under the actual swap; the notional-equivalent principal payment is used only to implement the bond valuation approach. The relevant terms and market inputs are presented in Exhibits 1 and 2.

An airline must lock in the purchase price of a nonstandard quantity of jet fuel on a customized delivery date. The instrument most appropriate for this objective is:

An airline must buy fuel on an irregular date and wants a binding contract customized to its required quantity. The instrument most appropriate for the airline is:

An investment manager requires an exchange-traded derivative that is settled daily and creates a symmetric obligation for both counterparties. The manager should most likely use:

An exchange requires standardized derivatives to clear through a central counterparty. The resulting risk transformation is best described as:

A risk manager replaces an OTC forward with an otherwise comparable exchange-traded futures contract. The change most likely: