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IFT Notes for Level I CFA® Program
IFT Notes for Level I CFA® Program

R49 Basics of Derivative Pricing and Valuation

Part 3


 

3.2.     Pricing and Valuation of Futures Contracts

Futures contracts have standard terms, are traded on a futures exchange, and are more heavily regulated than forward contracts. Some of its characteristics are listed below:

  • Marking to market: Futures contracts are marked to market on a daily basis. What this means is that if there is a gain or loss on the position relative to the previous day’s closing price, then the gain is credited to the winning account by deducting the amount from the losing account.
  • Credit guarantee: We saw that there was a default risk inherent in every forward contract. In futures contracts, there is a credit guarantee by the futures exchange through the clearinghouse.
  • Clearinghouse: The clearinghouse is the counterparty to every trade on the exchange.
  • Daily cash flow: In a forward contract, the gain or loss is realized at the end of the contract period. But, in a futures contract, there is a cash flow on a daily basis.

Relationship between futures prices and interest rates

  • If futures prices are positively correlated with interest rates, futures contracts are more desirable to holders of long positions than are forwards because there are intermediate cash flows on which interest can be earned. This cash flow can be invested at higher interest rates.
  • If interest rates are constant, or have zero correlation with futures prices, then forwards and futures prices will be the same.
  • If futures prices are negatively correlated with interest rates, then it is more desirable to have forwards than futures to holders of long positions.

Payoffs and Valuation of Futures Contracts

Let us take a simple example to compare the payoffs and valuation of a forward contract with a futures contract over three days. The futures price at the end of every day is given below with the price at initiation being 100.

Gain to Long in a Forward Contract

  Forward contracts Futures contracts
Payoff Ignoring the time value of money, total payoff is the same for forward and futures contract. If the time value of money is considered, then it depends on the correlation of futures and interest rate.
Value Zero at initiation.

Keeps on increasing until expiration of contract.

Zero at initiation. Keeps rising until settlement on the next trading day. Becomes zero once marked to market. For instance, increase from 0 to 1 on day 1.
Credit risk Higher Lower because of daily settlement

3.3.     Pricing and Valuation of Swap Contracts

To understand how swaps work, let us consider a 3-year plain vanilla interest rate swap with annual settlement where the fixed-rate payer pays 10% and the floating rate payer pays an interest based on LIBOR. LIBOR rates at t = 0, 1, and 2 are 9%, 10%, and 11%.

Interpretation:

  • At times 0, 1, and 2, the two parties exchange a series of payments. The fixed rate payer makes a fixed payment of 10% and receives a floating payment based on the value of the underlying at that point in time. Here is it 9%, 10%, and 11% respectively.
  • Like a forward rate agreement, rates are locked in for future. So, a swap is similar to a series of forward contracts, with each contract expiring at specific times, where one party agrees to make a fixed payment and receive a variable payment. But, the prices of the implicit forward contracts embedded in a swap are not equal.
  • It is like getting into multiple forward rate agreements to lock in rates for different periods in the future at a different forward price.

Swap is a series of off-market forward contracts where:

  • Each forward contract is created at a price and maturity equal to the fixed price of the swap with the same maturity and payment dates respectively.
  • This means that the series of FRAs built into a swap are all off-market FRAs – some with positive values and some with negative values.
  • The combined value of the off-market FRAs is zero.

Now, how do we determine the price of a swap that would result in a combined value of its FRAs to be zero? This is discussed in the following section.

Difference between Price and Value of a Swap Contract

Some key points regarding the price and value of a swap or as follows:

  • The value of a swap at initiation is typically equal to zero.
  • The swap price is determined at initiation through a process known as replication. No-arbitrage pricing is the key to pricing a swap. Replication implies that the valuation of a swap price is the present value of all the net cash flow payments from the swap.
  • The value of a swap changes during the life of the contract.

Example of a Swap

Consider a 3-year plain vanilla interest rate swap with annual settlement where the fixed-rate payer pays 10% and the floating-rate payer pays based on LIBOR. LIBOR rates at t = 0, 1, and 2 are 9%, 10%, and 11%.

The fixed rate of the swap is referred to as its price. In this case, it is 10%. A swap can be replicated as follows:

  • Step 1: Buy a floating-rate bond or any asset that pays coupons of unknown value S0, S1, q….SN at times t = 1, 2 ..N.
  • Step 2: Borrow money to purchase this floating rate bond (equivalent to issuing a fixed-rate bond); the payments for the money borrowed are equal fixed-payments of FS0 (t) at t=1, 2, …N. This must have the same cash flow as the swap.
  • The rate at which the money for the floater was borrowed is the price of the swap. Given the no-arbitrage pricing, the fixed rate on the swap must be equal to the fixed rate at which the fixed-rate bond was issued in step 2.
  • The value of the swap during the life of a swap is based on the present value of the expected future cash flows. The cash flows, floating payments in particular, are based on the market price of the underlying.


Derivative Basics of Derivative Pricing and Valuation Part 3