Cox Ingersoll and Vasicek Models


I am having a tough time trying to understand these models. Would anyone be able to help me out to understand these two models?

I specifically do not understand the implications of these two models to creating an interest rate term structure.


Let W(t) denote Brownian motion and dX(t) denote the change in process X(t) over small time interval [t, t+dt]. If X(t) is modeled as Vasicek (Hull-White) or Cox-Ingersoll-Ross, the two models state the following:

VASICEK: dX(t) = κ(t) [µ(t) - X(t)] dt + σ(t) dW(t)

COX-INGERSOLL-ROSS (CIR): dX(t) = κ(t) [µ(t) - X(t)] dt + σ(t) X(t)1/2 dW(t)

The most important property of both models is that process X(t) is mean-reverting. It does not sway too far from mean level µ(t) and always comes back. This makes the two models suitable candidates for describing processes that are relatively stable over time and do not exhibit long trends (only short stochastic trends perhaps). Examples: 1) overnight LIBOR rate, 2) CDS spread of a particular AA-rated company over the next 6 months, etc… CIR is always positive while Vasicek is positive in 99.99% scenarios for properly chosen parameters.

Common properties of the two models:

1] If the models are used to describe interest rates and credit spreads then the prices of bonds are available as simple formulas. You may ask: why do we need simple formulas when we can Monte Carlo simulate any process and calculate the price as the discounted average of payoffs in different scenarios? That is true, but when evaluating Greeks (deltas, gamma, …) using a formula makes the calculations quicker and more numerically stable.

2] The equations above allow us to describe changes in several related processes in parallel. In particular, it is easy to ensure that daily changes in several related rates are correlated. Think how important this is in default modeling.

Differences between the two models:

1] Vasicek is normally distributed while CIR has a more complicated distribution.

2] In most settings, CIR is more realistic in terms of how the volatility depends on the current value of the process.

An extensive reference on Vasicek, CIR and interest rate modeling in general is

Brigo, D., & Mercurio, F. (2006). Interest Rate Models - Theory and Practice (2nd ed).

Note: the book is somewhat technical and requires knowledge of probability and stochastic processes… In introductory finance textbooks, oftentimes simpler versions of the models are presented, e. g.

VASICEK: dX(t) = κ [µ - X(t)] dt + σ dW(t)

COX-INGERSOLL-ROSS (CIR): dX(t) = κ [µ - X(t)] dt + σ X(t)1/2 dW(t)

What would we have to know for the exam?

Also, what is the EXACT difference between this model and the arbitrage free valuation model?

Here’s what I have in my notebook:

Both CIR and Vasicek models are single-factor models, that have a mean reverting term. CIR has a volatility drift term that increases as r increases, while Vasicek assumes constant volatility. The Ho-Lee model is calibrated with the current term-structure, is more accurate, and can be used with options.

Also to determine which part of the equation is the short-term interest rate and the mean reverting value for CIR/Vaskicek.

CFA exam will test your breadth in terms of the Learning Outcome Statements, they will never go for depth thank you @Tactics

You will need to know what is mentioned in the LOS. So just focus on the LOS and don’t get intimidated with the heavy-duty formulas. Understand, the concepts.