Valuing a Derivative Using a One-Period Binomial Model — MCQs

Rina Shah manages the Meridian Absolute Return Fund, which uses exchange-traded and over-the-counter derivatives to adjust market exposures without immediately trading the underlying assets. She is reviewing three existing positions and preparing a note for the fund's risk committee. All contracts are free of default risk, transaction costs, and taxes, and the applicable risk-free rates are stated in the exhibits. The first position is a long equity forward entered into several months ago. Shah reminds the committee that the delivery price established at initiation does not change, but the contract's value changes as the forward price for a new contract with the same remaining maturity changes. A junior analyst proposes measuring the current value simply as the difference between the current forward price and the contractual delivery price. Shah responds that any cash flow represented by that difference would occur at delivery and must therefore be discounted over the remaining term. The second position is a long commodity futures contract. The fund posts sufficient variation margin and experiences no liquidity constraint from daily settlement. The risk system records the previous and current settlement prices shown in Exhibit 1. Shah asks the analyst to distinguish the daily settlement cash flow from the futures contract's original price and from its cumulative profit since initiation. The third position is an interest rate swap under which Meridian receives a fixed annual rate and pays a floating annual rate. The swap has just passed an annual settlement date, and the floating rate has just reset to the current market rate. Shah uses the bond-replication approach: a receive-fixed swap is valued as a long fixed-rate bond and a short floating-rate bond. Immediately after a reset, the floating-rate bond is worth its notional principal. The remaining fixed payments and valuation assumptions are presented in Exhibit 2. Finally, the committee is considering whether to replace some over-the-counter forward exposure with exchange-traded futures. The relevant underlying's futures price has historically been strongly positively correlated with interest rates. Shah notes that daily settlement changes the timing of gains and losses. Consequently, the ability to reinvest gains or finance losses at contemporaneous interest rates can cause an otherwise comparable futures position to differ in economic value from a forward position. The committee's comparison assumes no meaningful differences in liquidity, counterparty credit exposure, taxes, or transaction costs.

Meridian Capital maintains a derivatives desk that provides valuation and risk reports to several institutional portfolios. At the current valuation date, analyst Priya Shah reviews an existing long forward on an equity index. The contract was entered into previously and has six months remaining until settlement. The index will make a known cash distribution before the contract expires. Shah notes that the current value of the contract must reflect both the distribution and the present value of the contracted delivery price; simply comparing the current index level with the original delivery price would not produce the correct value. Shah next evaluates a commodity exposure for a client considering a six-month forward purchase. The commodity incurs storage costs, but physical ownership also produces a measurable convenience yield because inventory permits the client to satisfy unexpected production requirements. Storage costs and convenience yield are expressed as continuously compounded annual rates. The commodity is an investment asset, markets permit short sales, and transaction costs and taxes may be ignored. Shah will therefore apply the standard cost-of-carry relationship using the data in Exhibit 1. A colleague suggests replacing the commodity forward with an otherwise comparable futures contract. Shah explains that the two prices need not be equal when interest rates are stochastic because futures are marked to market daily. The colleague expects increases in the commodity price to occur primarily when short-term interest rates are also increasing. Margin balances earn the prevailing short-term rate, and there are no material differences in credit quality, liquidity, or contract terms that would otherwise affect the comparison. Finally, Shah values a USD interest rate swap immediately after the floating rate has reset and the current-period floating payment has been made. Meridian pays a fixed rate annually and receives the annual floating rate on the same notional principal. There are two remaining settlement dates. Shah uses the bond-equivalence approach, under which the receive-floating leg is represented by a floating-rate bond and the pay-fixed leg by a fixed-rate bond. All discount factors applicable to the remaining cash flows are supplied in Exhibit 2, and the notional principal itself is not exchanged.

Northbridge Multi-Asset Fund uses derivatives to alter market exposures while minimizing transactions in its physical holdings. The fund's derivative policy permits fully collateralized forwards, exchange-traded futures, and plain-vanilla interest rate swaps. All contracts are assumed free of counterparty default for valuation purposes, and transaction costs and taxes are ignored. Portfolio manager Elena Ortiz first reviews an equity index that pays a continuous dividend yield. She wants to compare the quoted nine-month forward price with its no-arbitrage price. The index level, financing rate, dividend yield, and maturity are shown in Exhibit 1. The financing rate and dividend yield use continuous compounding. Ortiz notes that owning the index requires financing but provides dividends during the forward term. Northbridge also holds an existing long forward contract on gold, unrelated to the equity index position. The contract requires Northbridge to buy one ounce of gold at the delivery price shown in Exhibit 1. The contract was entered into earlier and now has three months remaining. A newly initiated gold forward with the same remaining maturity has the current forward price shown in the exhibit. Ortiz reminds her analyst that the price of a new forward is not the same concept as the value of an existing forward and that the delivery-date difference must be discounted to the valuation date using the risk-free rate. Ortiz next considers reducing the systematic equity exposure of a diversified portfolio without selling its physical securities. The fund will use futures on a broad equity index that is sufficiently correlated with the portfolio for the beta-adjustment method. Futures are quoted in index points, and each contract has the multiplier shown in Exhibit 1. A positive contract quantity represents a purchase and a negative quantity represents a sale. The futures position will be established immediately, and the current futures price is used to measure each contract's exposure. Finally, Northbridge has an existing two-year pay-fixed, receive-floating interest rate swap with annual settlements. Immediately after the most recent settlement, the swap has two remaining payments. The original fixed rate and the current fixed rate on an otherwise comparable par swap are shown in Exhibit 2. The applicable discount factors correspond to the two remaining annual settlement dates. Because the floating-rate side resets to market at each settlement, Ortiz values the existing swap by comparing its contracted fixed rate with the current market fixed rate and discounting the remaining net fixed-rate differences. All rates in Exhibit 2 are annual rates, and the notional principal is not exchanged.

Arden Capital operates a derivatives overlay desk for institutional portfolios. A portfolio manager asks analyst Priya Raman to review several positions after interest rates and equity index levels change. Raman is instructed to distinguish the price of entering a new contract from the value of an existing contract and to report all values from Arden's position perspective. The first position is a long forward on a broad, continuously yielding equity index. The forward has no interim settlement, and the parties are subject to a standard collateral agreement. Raman uses continuous compounding for both the financing rate and dividend yield. She notes that a new forward must have zero value at initiation, but Arden's existing forward may have a positive or negative value because its contractual delivery price was fixed previously. All data required for the forward analysis are provided in Exhibit 1. Raman also reviews an exchange-traded interest rate futures position. The contract is marked to market daily through the clearinghouse, with gains credited and losses debited in cash. The quoted price change must be multiplied by the stated contract multiplier. Arden holds one contract, and no currency conversion is required. Finally, Raman examines a receive-fixed, pay-floating interest rate swap. The swap was worth zero when initiated, but market fixed rates have since changed. For purposes of the review, the swap may be valued as the present value of the difference between its contractual fixed rate and the current market swap rate over the remaining payment dates. The current market rate applies to a replacement swap with the same maturity and payment frequency. The notional principal itself is not exchanged. Exhibit 2 provides the futures settlement information and the swap inputs, including discount factors applicable to the two remaining annual net payments.

Meridian Pension Fund uses derivatives to adjust market exposures without immediately trading the securities held in its cash portfolio. Priya Shah, a derivatives analyst, is reviewing several positions for the fund's investment committee. Unless otherwise stated, contracts have no counterparty default risk, transaction costs, or taxes, and all quoted interest rates apply for the full relevant term. Shah first considers a new forward contract on a dividend-paying equity security. The proposed contract requires no payment at initiation and will be settled by delivery at expiration. The equity's current price, expected dividends, financing rate, and maturity are shown in Exhibit 1. Dividends are known with certainty, and the amount in the exhibit is their present value as of today. The fund also holds an existing long forward on the same equity. Three months have passed since that contract was initiated. Its delivery price was fixed at initiation, but current market conditions differ from the original conditions. Shah will value the contract from the perspective of Meridian, the long party. The remaining dividends shown in Exhibit 1 are expressed as a present value on the valuation date, not as an expiration-date amount. Meridian separately holds long exchange-traded equity index futures. The clearinghouse marks the contracts to market at each day's settlement price. The difference between the current and previous settlement prices is transferred through the margin account using the stated contract multiplier. No deposits or withdrawals unrelated to variation margin occurred during the day. Finally, Meridian is the fixed-rate payer and floating-rate receiver in an interest rate swap. Immediately after the current settlement payment, Shah values the swap using the bond-equivalence approach. The floating-rate and fixed-rate bond values in Exhibit 1 include all remaining swap cash flows and use the same notional principal and valuation date. Shah records several comments about the contracts in Exhibit 2 before preparing her recommendation for the committee.