Financial Mathematics for Cryptocurrencies

Chapter 16Overview of Interest Rate Models

Interest rate models provide essential frameworks for pricing rate-dependent products and modeling staking rewards in DeFi. This chapter surveys key models and their applications to digital asset valuation.

• Present the Cox-Ingersoll-Ross (CIR) model and its square-root process for mean-reverting rates.
• Examine the SABR model and its applications to cryptocurrency markets.
• Provide an overview of jump-diffusion models for capturing rate discontinuities.

We begin with foundational frameworks, starting with the CIR model. Table 16.1 provides a comparative overview of the models discussed.

TABLE 16.1 Comparison of Interest Rate Models Presented in This Chapter for Rate Modeling and Pricing

CIR	SABR	SVJC	HJM
Mean reversion	Yes	No	Yes	Flexible
Stochastic volatility	No	Yes	Yes	Yes
Jump processes	No	No	Yes	No
Closed-form solution	Yes	Approximate	No	No
Complexity	Low	Medium	High	Very high

16.1 CIR MODEL

The Cox-Ingersoll-Ross (CIR) model [300] was introduced in 1985 to model interest rates. It is a one-factor model, which means that there is only one stochastic process. It is commonly used for short-term rates, alongside the Hull-White model, as discussed in Remark 11.6.

Definition 16.1 (Cox-Ingersoll-Ross [CIR] Model):

The Cox-Ingersoll-Ross (CIR) model is a stochastic process defined as

with the rate of mean-reversion, the long-term mean, the volatility, ...