Which of the following derivatives provide payoff s that are non-linearly related to the payoff s of the underlying? A . Options B . Forwards C . Interest rate swaps
The correct answer and explanation is:
The correct answer is: A. Options
Explanation:
Derivatives are financial instruments whose value depends on the value of an underlying asset, such as stocks, bonds, interest rates, or currencies. The relationship between the payoff of a derivative and the payoff of the underlying can be either linear or non-linear.
1. Options (Non-linear payoff)
Options give the holder the right, but not the obligation, to buy or sell an asset at a predetermined price (strike price) before or at expiration. The payoff profile of options is inherently non-linear because:
- For a call option, the payoff is zero if the underlying asset price is below the strike price (option is out of the money), but increases linearly beyond the strike price (option is in the money).
- For a put option, the payoff is zero if the underlying asset price is above the strike price, but increases linearly as the asset price falls below the strike price.
This creates a payoff curve that is convex and asymmetric, with limited downside risk (premium paid) but potentially unlimited upside for calls. The option payoff doesn’t move in a simple straight line with the underlying price—it has a “kink” at the strike price, which is a hallmark of non-linear payoffs.
2. Forwards (Linear payoff)
A forward contract is an agreement to buy or sell an asset at a specified price on a future date. The payoff at maturity is: Payoff=ST−K\text{Payoff} = S_T – K
where STS_T is the underlying asset price at maturity and KK is the agreed forward price.
This payoff is directly proportional to the change in the underlying price — it increases or decreases linearly, without any curvature or thresholds. Hence, forwards have linear payoffs.
3. Interest Rate Swaps (Linear payoff)
An interest rate swap involves exchanging fixed interest rate payments for floating rate payments. The cash flows are linearly related to the underlying interest rates because the payments depend on the difference between fixed and floating rates, which move proportionally.
Therefore, interest rate swaps also have linear payoffs.
Summary:
- Options have non-linear payoffs due to the asymmetric and convex nature of their payoff profiles around the strike price.
- Forwards and interest rate swaps have linear payoffs, moving in direct proportion to the underlying asset’s price or interest rates.
This non-linearity is what makes options particularly useful for hedging, speculation, and risk management strategies, offering asymmetric risk/reward profiles that cannot be replicated by linear instruments.