Forward Commitment and Contingent Claim Features

A forward contract is a type of firm commitment. It's a customized agreement between two parties to buy or sell an underlying asset at a specified price on a future date. Because they are customized, they trade in the over-the-counter (OTC) market.

Key Terminology

• Long Position: The party who agrees to buy the underlying asset. This position profits when the asset price increases.
• Short Position: The party who agrees to sell the underlying asset. This position profits when the asset price decreases.

Payoff at Expiration

The value or payoff of a forward contract at expiration is a zero-sum game; the long's gain is the short's loss, and vice-versa.

The payoff for the long position (the buyer) is:

The payoff for the short position (the seller) is:

Where:

• S_T: The spot price of the underlying asset at the contract's expiration (time T)
• F₀(T): The forward price agreed upon at the start of the contract (time 0) for delivery at time T

Numerical Example

Settlement Scenarios

The outcome for each party depends on how the spot price at expiration compares to the agreed-upon forward price.

If at Expiration...	Buyer's Payoff (Long)	Seller's Payoff (Short)
ST > F0(T) (Spot price is higher)	Positive	Negative
ST < F0(T) (Spot price is lower)	Negative	Positive

Futures are similar to forwards but with key distinctions. They are standardized contracts traded on an exchange, which introduces several important features.

Key Features of Futures

• Daily Settlement of Gains/Losses: Unlike forwards that are settled only at expiration, futures positions are "marked-to-market" (MTM) daily. Gains and losses are calculated and settled every single day.
• Clearinghouse Guarantee: A central clearinghouse guarantees the transaction, effectively eliminating counterparty credit risk for both parties.
• Margin Requirements: To protect the clearinghouse, both parties must deposit an amount of money called a margin.
  • Initial Margin: The amount that must be deposited when opening the position
  • Maintenance Margin: The minimum amount that must be maintained in the trading account. If the balance falls below this level, a margin call is triggered
  • Margin Call: A demand for additional funds to bring the account balance back up to the initial margin level
  • Variation Margin: The additional money deposited to meet a margin call
• Open Interest: This refers to the total number of outstanding (unsettled) futures contracts at any given point in time.

A swap is a firm derivative contract in which two parties agree to exchange a series of cash flows over a period of time.

• Swaps are typically customized contracts that trade in the OTC market
• Conceptually, a swap can be viewed as a portfolio, or a "strip," of forward contracts bundled together
• A swap that involves only a single payment exchange is equivalent to a single forward contract
• A common example is an interest rate swap, where one party agrees to pay a fixed interest rate on a notional amount in exchange for receiving a floating interest rate from the other party

Options are contingent claims, meaning they give the holder a right, but not an obligation. One party has the option to decide whether a trade will be executed based on the underlying asset's value. Options can be traded on exchanges (ETD) or in the OTC market.

Types of Options

• Call Option: Gives the option holder the right to buy an underlying asset at a specified price (the strike price)
• Put Option: Gives the option holder the right to sell an underlying asset at a specified price

Exercise Styles

• European Options: Can be exercised by the holder only on the expiration date
• American Options: Can be exercised by the holder at any time up to and including the expiration date

Option Payoff and Profit Formulas

The payoff is the value of the option at expiration, while the profit also accounts for the initial premium paid or received.

Formula Components:

• S_T: The spot price of the underlying asset at expiration
• X: The exercise or strike price of the option
• c₀: The premium (price) paid by the buyer for a call option
• p₀: The premium (price) paid by the buyer for a put option

Position	Payoff Formula	Profit Formula
Call Buyer (Long Call)	max(0, ST – X)	max(0, ST – X) – c0
Call Seller (Short Call)	–max(0, ST – X)	–max(0, ST – X) + c0
Put Buyer (Long Put)	max(0, X – ST)	max(0, X – ST) – p0
Put Seller (Short Put)	–max(0, X – ST)	–max(0, X – ST) + p0

Numerical Example (Long Call)

Credit derivatives are contracts that transfer credit risk (the risk of default) from one party to another. The two parties are known as the credit protection buyer and the credit protection seller.

• Credit Default Swap (CDS): This is the most common type. The credit protection buyer makes periodic payments to the seller. In return, the seller agrees to pay the buyer a lump sum only if a specified "credit event," such as a default or bankruptcy, occurs.

Directional Views

Different derivative positions can be used to express a bullish view (i.e., you expect the underlying's price to rise). The following positions all benefit from a rise in the price of the underlying asset:

• Long Call Option: The right to buy at a fixed price becomes more valuable as the market price rises
• Short Put Option: As the price rises, it becomes less likely that the put will be exercised, allowing the seller to keep the premium as profit
• Long Forward Position: The obligation to buy at a fixed price becomes profitable as the market price rises above that fixed price

Profit Equivalence

The following formulas illustrate the relationship between the profit on a forward contract and the premium of an option, often used to understand breakeven points or synthetic positions.

• Equivalence of forward profit to a call option premium:
  S_T – F₀(T) = c₀
• Equivalence of forward profit to a put option premium:
  S_T – F₀(T) = p₀