Term Structure of Interest Rates Vasicek CIR Models

Understanding term structure models

Term structure models aim to describe how interest rates evolve across different maturities. The yield curve represents this relationship between interest rates and time to maturity. Both Vasicek and CIR models belong to the class of "short-rate models" that specify the dynamics of the instantaneous interest rate.

The Vasicek model

The Vasicek model describes interest rate movements using the following stochastic differential equation:

$dr_t = \kappa(\theta - r_t)dt + \sigma dW_t$

Where:

• $r_t$ is the instantaneous interest rate
• $\theta$ is the long-term mean level
• $\kappa$ is the speed of mean reversion
• $\sigma$ is the volatility
• $dW_t$ is a Wiener process

The model incorporates mean reversion, reflecting the tendency of interest rates to move toward a long-term average level.

The CIR model

The Cox-Ingersoll-Ross model extends Vasicek by ensuring rates remain positive:

$dr_t = \kappa(\theta - r_t)dt + \sigma\sqrt{r_t}dW_t$

The key difference is the $\sqrt{r_t}$ term in the volatility component, which makes the variance proportional to the level of rates and prevents negative rates.

CIR model properties

1. Mean reversion to $\theta$
2. Positive interest rates when $2\kappa\theta \geq \sigma^2$
3. Non-central chi-square distribution for rates
4. Closed-form solutions for zero-coupon bond pricing

Applications in fixed income markets

Bond pricing

Both models provide analytical solutions for bond prices:

$P(t,T) = A(t,T)e^{-B(t,T)r_t}$

Where $A(t,T)$ and $B(t,T)$ are model-specific functions depending on parameters.

Risk management

The models enable calculation of key risk metrics:

• Duration and convexity
• Interest rate sensitivity
• Value at Risk (VaR)

Model limitations and extensions

Vasicek limitations

1. Possibility of negative rates
2. Constant volatility assumption
3. Single factor dependency

CIR limitations

1. Perfect correlation between volatility and rate level
2. Limited flexibility in fitting yield curves
3. Single factor structure

Modern extensions include:

• Multi-factor versions
• Stochastic volatility
• Jump components

Practical implementation

Calibration process

1. Collect historical rate data
2. Estimate parameters using maximum likelihood
3. Validate model fit
4. Adjust for market prices

Trading applications

• Fixed income analytics
• Interest rate swaps
• Options pricing
• Risk assessment

Model selection considerations

When choosing between Vasicek and CIR:

1. Market environment

   • Low rate environments favor CIR
   • High volatility periods may suit Vasicek

2. Application purpose

   • Derivatives pricing
   • Risk management
   • Portfolio optimization

3. Implementation complexity

   • Computational requirements
   • Data availability
   • Calibration challenges