Reading 67

Derivative instruments fall into two broad families. Forward commitments bind both parties to act at a future date — these include forwards, futures, and swaps. Contingent claims create rights that pay only when a specified event occurs — these include options and credit default swaps.

Forward Contracts

A forward contract is a private agreement between two parties: the buyer (long) commits to purchase, and the seller (short) commits to deliver, a specified physical or financial asset at an agreed price on a future settlement date.

• Long position: profits when the underlying's price at settlement exceeds the forward price; loses when it is lower.
• Short position: profits and loses are exactly the mirror image — gains if the price falls below the forward price.

Forwards are customised, traded over-the-counter, and have no daily settlement, which means counterparty credit risk accumulates over the life of the contract.

Futures Contracts

A futures contract is the exchange-traded, standardised cousin of the forward. Versus an otherwise identical forward, futures offer:

• A liquid secondary market
• Greater regulation and price transparency (more disclosure)
• A central clearinghouse that becomes counterparty to both sides — minimising counterparty credit risk
• Daily mark-to-market settlement of gains and losses

Margin Mechanics

Both buyer and seller post margin — cash or eligible collateral — with the exchange. Unlike securities margin, no loan is involved and there is no interest charge; the margin is collateral protecting the clearinghouse.

• Initial margin — required deposit before trading; roughly one day's maximum expected price move on the contract's notional.
• Maintenance margin — minimum balance the account must maintain. If daily mark-to-market drops the balance below this floor, the trader must top up back to the initial margin level (not just to the maintenance level — this differs from equity margin calls). Failure to meet the call results in the position being closed by the exchange.
• Settlement price — typically the average price of trades over a short window at the end of the session; used to mark accounts.

At day's end, the gain on one side is credited and the matching loss debited, and both sides are repositioned at the new settlement price.

Swaps

A swap is an agreement to exchange a series of payments on multiple settlement dates over a stated period (e.g., quarterly for two years). On each date, the two cash flows are netted; only the difference is paid by the party with the larger obligation.

Swaps are dealer-market instruments. Without a central counterparty, both sides face counterparty credit risk. When clearing exists, margin and mark-to-market may further reduce that exposure.

Fixed-for-Floating Interest Rate Swap — Mechanics

Take a 2-year quarterly-pay swap on a $10 million notional with a fixed rate of 2% and a floating rate of 90-day SOFR.

• Fixed-rate payment each quarter \( = \$10\text{M} \times 0.02 / 4 = \$50{,}000 \).
• The floating-rate payment for quarter 1 is set at swap initiation using the then-current SOFR (advance set, settle in arrears for the floating leg). So both flows are known at the start of period 1.
• If 90-day SOFR is 1.6% at the end of quarter 1, the quarter-2 floating payment \( = \$10\text{M} \times 0.016 / 4 = \$40{,}000 \). The fixed payer therefore net pays \( \$50{,}000 - \$40{,}000 = \$10{,}000 \) to the floating payer.

The swap rate is set at inception so that the swap is worth zero to both sides. As expectations of future short rates change, the swap acquires positive value to one side and negative value to the other.

Hedging use. A firm with a 2-year quarterly-pay floating-rate note can enter the swap as the fixed-rate payer. The floating payments it receives from the swap counterparty fund the floating coupon on its debt; it is left paying a fixed rate — converting floating-rate liability into fixed-rate liability and removing the rate uncertainty.

Swap as a series of forwards. A swap can be decomposed into a series of forward contracts in which the underlying is the floating rate and the forward price is the fixed rate. Each "forward" settles on one swap settlement date; the net of fixed and floating produces the same cash flow as the swap. (Interest-rate forwards often settle at the start of a period; the cash flows are the present-value equivalent of the end-of-period swap payment.)

Credit Default Swaps (CDS)

A credit default swap is structured differently from a vanilla swap. The protection buyer makes fixed payments on each settlement date. The protection seller pays only if the underlying reference security suffers a defined credit event — for example, a bond default, corporate bankruptcy, or involuntary restructuring.

When a credit event happens, the protection seller delivers an amount that compensates for the loss in value of the reference security. The fixed premiums represent the yield premium on the reference bond — i.e., the compensation bondholders receive for the expected loss (probability of default × expected loss given default).

Use cases:

• A risky-bond holder can hedge default risk by buying CDS protection.
• The protection seller earns the credit-spread premium and takes on default risk — a risk profile economically similar to owning the reference bond.

Options — Calls and Puts

An option contract on 100 shares of a stock is a useful concrete case.

• A put option gives the buyer the right but not the obligation to sell 100 shares at the exercise price (strike) during the option's life. The put writer (seller) is obligated to buy the shares at that strike if the holder exercises.
• A call option gives the buyer the right (not the obligation) to buy 100 shares at the strike. The call writer is obligated to deliver the shares if exercised.

One-way payoff. If a put has strike $25 and the stock trades at $25 or higher at expiration, the holder will let it expire — there is no reason to sell at $25 when the market price is at least $25. Whether the stock is at $25 or $1,000, the put expires worthless and the writer keeps the premium. Below $25, the holder exercises and the writer must buy at $25; the holder's net payoff is essentially \( (25 - S) \times 100 \).

Unlike forwards, futures, and swaps (which start at zero value), options are sold for a price — the option premium.

Value at expiration (payoff to holder):

• Call: \( \text{Payoff}_{\text{call}} = \max(0,\; S - X) \)
• Put: \( \text{Payoff}_{\text{put}} = \max(0,\; X - S) \)

Where \( S \) is the underlying price at expiration and \( X \) is the strike. If \( S < X \) for a call (or \( S > X \) for a put), the option expires worthless.

Profit at expiration:

• Long: payoff − premium paid.
• Short (writer): premium received − payoff at expiration.

Risk profile:

• Buyer of a put or call: maximum loss = premium paid (no further obligation).
• Writer of a call: theoretically unlimited loss because \( S \) has no upper bound.
• Writer of a put: maximum loss = strike \( X \) (since \( S \) cannot fall below 0), so the maximum payoff to a put holder is \( X \).

Call Option Profits and Losses

Consider a call with premium = $5, strike \( X = \$50 \). The buyer pays $5 to the writer.

• If \( S \le \$50 \) at expiration, the call expires worthless. Buyer is out $5; writer keeps $5.
• For \( \$50 < S < \$55 \), the call has intrinsic value but the buyer has not yet recovered the premium.
• Breakeven for both sides occurs at \( S = X + \text{premium} = \$55 \).
• For \( S > \$55 \), the long call gains $1 per $1 above $55; the short call loses $1 per $1.

Key takeaways:

• Maximum loss to the call buyer = $5 (the premium), occurring at any \( S \le \$50 \).
• Profit potential to the call buyer is unlimited; loss to the writer is unlimited.
• The holder exercises whenever \( S > X \) at expiration.
• Maximum profit to the writer = $5 (the premium), at any \( S \le \$50 \).
• Long-and-short profits sum to zero — options are a zero-sum game.

Put Option Profits and Losses

Consider a put with premium = $5, strike \( X = \$50 \). The buyer pays $5 to the writer.

• If \( S \ge \$50 \) at expiration, the put expires worthless. Buyer loses $5; writer keeps $5.
• Breakeven = \( X - \text{premium} = \$45 \).
• If \( S < \$45 \), the long put earns $1 per $1 of decline below $45; the short put loses $1 per $1.

Key takeaways:

• Maximum loss to the put buyer = $5 (premium) at any \( S \ge \$50 \).
• Maximum gain to the put buyer = strike − premium = $50 − $5 = $45 (achieved if \( S = 0 \)). The writer's potential loss equals this same amount.
• Breakeven = \( X - \text{premium} = \$45 \).
• Maximum profit to the put writer = $5 (the premium).
• Buyer's gain (loss) always equals writer's loss (gain) — zero-sum.

A forward commitment is a legally binding promise to perform some action in the future — both parties are obligated. Forward commitments include forwards, futures, and most swaps.

A contingent claim is a claim to a payoff that depends on a particular event. Options are contingent claims because the payoff depends on whether \( S \) is above or below \( X \). Credit default swaps are also contingent claims because the protection seller pays only if a credit event occurs.