Explain why a sticky cap is more expensive than a similar...

Explain why a sticky cap is more expensive than a similar ratchet cap. In a ratchet cap, the cap rate equals the previous reset rate, $R$, plus a spread. In the notation of the text it is $R_j+s$. In a sticky cap the cap rate equal the previous capped rate plus a spread. In the notation of the text it is $\min \left(R_j, K_j\right)+s$. The cap rate in a ratchet cap is always at least a great as that in a sticky cap. Since the value of a cap is a decreasing function of the cap rate, it follows that a sticky cap is more expensive.

Explain why a sticky cap is more expensive than a similar ratchet cap.
In a ratchet cap, the cap rate equals the previous reset rate, $R$, plus a spread. In the notation of the text it is $R_j+s$. In a sticky cap the cap rate equal the previous capped rate plus a spread. In the notation of the text it is $\min \left(R_j, K_j\right)+s$. The cap rate in a ratchet cap is always at least a great as that in a sticky cap. Since the value of a cap is a decreasing function of the cap rate, it follows that a sticky cap is more expensive.

Key Concepts

Interest Rate Caps

An interest rate cap is a derivative instrument that provides protection to the buyer against rising interest rates. It sets an upper limit (cap rate) on the rate that the buyer will have to pay, and if the benchmark rate exceeds this cap, the seller compensates the buyer for the excess. The value of such a cap depends on various factors, including the cap rate used in its calculations, making it a critical tool in interest rate risk management.

Ratchet Cap

The ratchet cap is a type of interest rate cap where the cap rate is periodically reset to the current reference rate, augmented by a fixed spread. This mechanism means that at each reset, the cap rate is aligned with recent market conditions, ensuring that the rate used for cap payments is at least equal to the current benchmark plus the spread. This structure generally results in a higher effective cap rate, which can reduce the cap's value as a protective option.

Sticky Cap

A sticky cap, in contrast to a ratchet cap, resets its cap rate based on the previous capped rate (after incorporating a spread) rather than the current market reset rate. This mechanism causes the cap to 'stick' to past capped values, possibly resulting in a lower effective cap rate compared to the ratchet structure. Consequently, this lower cap rate increases the cap's payoff potential, thereby making the instrument more valuable and expensive.

Monotonicity of Option Value

The pricing of cap instruments relies on the fact that the value of an option is a decreasing function of the cap rate or strike price – a lower cap rate translates into a higher potential payoff if interest rates exceed the cap. In the comparison between sticky and ratchet caps, the sticky cap's typically lower effective cap rate implies that it yields a higher payoff potential under adverse rate movements, thus making it more expensive than an equivalent ratchet cap.

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