Book 1. Market Risk

FRM Part 2

MR 16. The Vasicek and Gauss+ Models

Presented by: Sudhanshu

Module 1. Vasicek Model and Gauss+ Models

Module 1. Vasicek Model and Gauss+ Models

Topic 1. Modeling Interest Rates

Topic 2. Vasicek Model

Topic 3. Gauss+ Model: Overview

Topic 4. Gauss+ Model: Interpretation

Topic 5. Vasicek vs. Gauss+ Model

Topic 6. Applying Gauss+ Model

Topic 7. Estimating Gauss+ Model Parameters

Topic 1. Modeling Interest Rates

• Modeling interest rates is important for many finance applications, including risk management and the pricing of interest rate securities and derivatives. The most useful interest rate models typically:

  • Provide a process for how the short-term rate evolves.

  • Explain the current term structure of interest rates within an acceptable margin of error.

  • Maintain consistency with implied volatilities.

• An interest rate model generally shows how the short-term rate changes as a function of other variables. For instance, a basic model might be defined as       drt​=σdwt​, where $\sigma$ is the volatility of interest rate changes and                            is a random factor.

• In practice, a term structure model should meet the following objectives:

  • Be consistent with market pricing of basic fixed-income instruments, for  example - allowing the model to be calibrated to observed bond prices.

  • Be consistent with market pricing of derivative instruments, allowing the model to be calibrated to observed prices of interest rate derivatives.

  • Mirror tendencies market participants expect from interest rate movements, such as reflecting mean reversion and the market's volatility structure of rates.

Topic 2. Vasicek Model

• The Vasicek model uses a single factor to model the change in short-term interest rates over time.

  • ​Main Characteristic: Mean reversion, where the short-term rate (rt​) tends to revert to its long-term average ($\theta$). This forecasts a rise for below-average rates and a decline for above-average rates.

  • Equation: The behavior is modeled by the following equation:

    $$dr=k(\theta -r)dt+\sigma dw$$

    where $k$ is the speed of mean reversion, $\theta$ is the long-run value of the short-term rate and $r$ is the current interest rate level.

  • Applications: It is useful for simple applications like pricing zero-coupon bonds and applying basic hedging techniques. Its simplicity helps analysts understand the basic effects of mean reversion and stochastic shocks on rates.

  • Drawbacks:

    • It fails to replicate complex term structure shapes and real-world volatility patterns.

    • It fails to account for observed changes in volatility across maturities.

    • It fails to adequately incorporate macroeconomic trends and monetary policy shifts.

• Definition: The Gauss+ model is a three-factor, cascade-style term structure model designed to address the limitations of simpler models like Vasicek.
• Core Factors
  • Short-term factor (   )
  • Medium-term factor (     )
  • Long-term factor (  )
• Each factor mean reverts to the next longer-term level, creating a chain reaction in rate evolution:

•             : Positive mean-reversion parameters
•  : Constant term that includes rate expectations and risk premiums
•      

• The Gauss+ model is constructed in response to certain interest rate dynamics, including: (1) mean reversion, (2) correlation between factors, (3) consistency with the volatility term structure, and (4) pricing consistency.

mtm_

Topic 3. Gauss+ Model: Overview

• Mean reversion: Consider the first equation:
  • When               , rate increases toward medium-term factor; when               , rate decreases; equilibrium at               . Results in convergence cascade where                                  , with only μ fixed and other factors dynamically moving.
• Economic Interpretation of Factors
  •    (short-term): Monetary policy rates (e.g., fed funds rate)
  •      (medium-term): Business cycle, monetary policy effects
  •    (long-term): Inflation expectations, productivity growth

• Factor dynamics: The short-term rate reacts quickly to the medium-term factor, the medium-term factor adjusts more gradually to the long-term factor, and the long-term factor shifts the slowest over time.

• Regarding the correlation between factors, the changes in                are related through the parameter

• Consistency with volatility: The model can be calibrated to replicate historically observed hump-shaped volatility, with low volatility on the short end, higher volatility in the middle, and declining volatility on the long end of the interest rate curve.

• Pricing Consistency: The model offers flexibility to be consistent with the pricing of simple instruments (like bonds and swaps) and interest rate derivatives (like caps and floors).

mtm_    

Topic 4. Gauss+ Model: Interpretation

Practice Questions: Q1

Q1. Which of the following statements best describes the structure of the Gauss+ model?
A. A single-factor model with constant volatility.
B. A model without any mean reversion or volatility assumptions.
C. A two-factor model with mean reversion to a fixed long-term rate.
D. A cascade model where each factor mean reverts to the next level.

Practice Questions: Q1 Answer

Explanation: D is correct.

The Gauss+ model uses a three-factor cascade structure where the short-term rate (    ) mean reverts to the medium-term factor (    ), which mean reverts to the longterm factor (   ).

Practice Questions: Q2

Q2. The role of the medium-term factor in the Gauss+ model is that the factor:
A. remains constant over time.
B. directly determines the pricing of long-term bonds.
C. reflects economic influences, such as business cycles.
D. represents the risk premium in the interest rate structure.

Practice Questions: Q2 Answer

Explanation: C is correct.

The medium-term factor represents broader economic influences like business
cycles and monetary policy effects. It acts as an intermediate driver between the short-term and long-term interest rate factors, reflecting the model’s cascading mean-reversion structure.

Topic 5. Vasicek Vs. Gauss+ Model

• Vasicek Model Limitations
  • Single volatility parameter: Calibrates to only one point on interest rate curve, limiting flexibility in curve shape and ability to accurately match rates for effective hedging
  • Derivative pricing weakness: Cannot capture observable interest rate derivative prices due to inability to model complex volatility structures; deficiencies addressed by Gauss+ model
• Gauss+ Model Advantages
  • ​Extended Vasicek framework: Designed to better align with actual market behaviors and pricing dynamics
  • Enhanced capability: Prices wider range of instruments and maintains consistency with market participants' interest rate expectations, making it preferable to simpler Vasicek model

ρ\rho

mtm_    

Practice Questions: Q3

Q3. What is a main difference or similarity in the volatility assumption between the Vasicek model and Gauss+ model?
A. Both models assume constant volatility.
B. Both models have stochastic volatility components.
C. The Vasicek model incorporates stochastic volatility, while the Gauss+ model assumes deterministic volatility.

D. The Vasicek model assumes constant volatility, while the Gauss+ model can replicate hump-shaped volatility term structures.

Practice Questions: Q3 Answer

Explanation:  D is correct.

The Vasicek model assumes constant volatility, which limits its ability to reflect the hump-shaped term structure of volatilities observed in practice. In contrast, the Gauss+ model incorporates variable volatilities, capturing higher volatility for intermediate maturities and declining volatility for longer terms.

Topic 6. Applying Gauss+ Model

• Factors m and l do not proxy specific forward rates (unlike r, which represents short-term rate like fed funds); instead, they influence short-term rate evolution, allowing forward rates to be inferred from resulting interest rate path distributions.

• To demonstrate how the Gauss+ model is applied—specifically, how changes in rt, mt, and lt are generated—let’s simulate one interest rate path using U.S. Treasury zero- coupon data from January 2014 to January 2022. The estimated parameters for the Gauss+ model are as follows:

• The change in the short-term rate, drt​, is calculated using the first of the cascade equations.

• Example Calculation of drt​    : Let below parameters are given       :      

  •  rt​ = 4.5%, mt​ = 5.25%, lt​=4%, αr​=1.0547, αm​=0.6358, αl​=0.0165 and $dt$=1 month=0.083 years (=1/12)

  • ​Let

  • drt​=−αr​(rt​−mt​)dt=−1.0547×(0.045−0.0525)×0.83=0.0007

  •  

  •  

  • ​Result: The change in the short-term rate is $+0.325\%$.

• Therefore, rt+1​ = rt​ +drt​   = 0.045+0.0007=4.57%; mt+1​ = mt​ + dmt​ = 0.0525-0.0052=4.73% and lt+1​ = lt​ + dlt​  = 0.04+0.0016=4.16%

Practice Questions: Q4

Q4. Assume the short-term rate is 5.0% and the medium-term rate is 6.0%. By using the Gauss+ model, if the corresponding mean reversion speed is 1.3 and the time step is three months, what is the
change in the short-term rate?
A. –0.325%.
B. +0.325%.
C. –3.250%.
D. +3.250%.

Practice Questions: Q4 Answer

Explanation: B is correct.

The change in     is calculated as follows:

Thus, the short-term rate should increase by 0.325%.

Topic 7. Estimating Gauss+ Model Parameters

• The parameters of the Gauss+ model can be estimated using maximum likelihood techniques or a simplified approach when random components are not used to determine mean reversion speeds.
• Simplified Estimation Procedure:

1. Select Proxy: Start by selecting a proxy for the short-term rate (e.g., fed funds rate).

2. Regression: Regress the changes in the short-term rate against changes in a medium-term rate (e.g., 2-year rate) and a longer-term rate (e.g., 10-year rate).

3. Find Mean Reversion Speeds: Find the mean reversion speeds αr​, αm​, and αl​ that replicate the sensitivities seen in Step 2. This is possible because the regression coefficients depend only on the mean reversion parameters, not the volatility parameters.

4. Determine Volatility Term Structure: Determine the volatility term structure from market data, either through the observed volatility of rates or the volatilities used by the market to price interest rate derivatives.

5. Find Volatility and Correlation Parameters: Find the volatility and correlation parameters (σm​, σl​, and $\rho$) that can replicate the volatility term structure found in Step 4.

6. Find Long-Term Reversion Level ($\mu$): Find the long-term reversion level ($\mu$) by minimizing the squared errors of observed yields versus estimated yields.

7. Once estimated, the Gauss+ framework can: (1) precisely model rate term structures and their volatilities; (2) account for rate expectations and risk premiums reflected in the rate term structure; and (3) provide a robust tool for pricing, hedging, and analyzing interest rate products.

Practice Questions: Q5

Q5. When using empirical data to estimate parameters for the Gauss+ model, which of the following parameters is calibrated by minimizing the squared errors of observed yields relative to modelpredicted yields?
A. Long-term target.
B. Volatility parameter.
C. Correlation parameter.
D. Mean reversion speeds.

Practice Questions: Q5 Answer

Explanation: A is correct.

The long-term reversion level (μ) is calibrated to align the model’s predicted yields with observed yields, minimizing squared errors across all maturities.