Lecture 7: One factor interest rate models, Arbitrage vs Equilibrium models

Question 1

Explain dynamics of volatility for Vasicek model

Answer 1

For this model we look at the case y=0
we get the following CEV model: sigma(r,t) = sigma

in the vasiceck model, volatility is independent of the level of the short rate, as in the equation:
dr = bdt + sigma*dz
this if referred to as the normal model, it is possible for negative interest rates to be generated.

Question 2

Explain dynamics of volatility for Dothan model

Answer 2

for this model we look at y=1
We get the CEV model: sigma(r,t) = sigma*r
in this model, volatility is proportional to the short rate
known as the proportional model.

Question 3

Explain dynamics of volatility for CIR model

Answer 3

for this model we look at y = 1/2
we get the CEV model sigma(r,t) = sigma*sqrt(r)
known as square root model as volatility is proportional to the square root of the short rate.
negative interest rates are not possible in this model.

Question 4

key arbitrage free models and what differs them

Answer 4

Ho- Lee model - there is no mean reversion and volatility is independent of the short rate, i.e. y=0

Kalotay-williams- fabozzi model - changes in the short rate are modelled by he natural log of r - no allowance for mean reversion is considered

HJM Model is a general time continuous multifactor model.
review handwritten notes for maths equations

Question 5

Types of equlibrium models

Answer 5

review handwritten notes for maths