The term structure of Interest Rate

Question 1

State the main potential drawback of the Vasicek model and discuss the extent to which this drawback may be a problem

Answer 1

It allows negative interest rates.

The extent of the problem depends on the probability of negative interest
rates…
… within the timescale of the problem in hand (or, for example, less of an issue if the time horizon is short)…
… and their likely magnitude if they can go negative.
It also depends on the economy being modelled, as negative interest rates have been seen in some countries.

Question 2

Explain how the Cox-Ingersoll-Ross model avoids the drawback of negative interest rates

Answer 2

The CIR model does not allow interest rates to go negative.
This is because the volatility under the CIR increases in line with the square root of r(t).
Since this reduces to zero as r(t) approaches zero…
… and provided the volatility parameter is not too large… [½]
… r(t) will never actually reach zero.
… provided σ2 ≤ 2αμ

Question 3

List the desirable characteristics of a term structure model

Answer 3

• The model should be arbitrage-free.
• Interest rates should ideally be positive.
• Interest rates should exhibit some element of mean reversion.
• The model should be computationally tractable / produce simple formulae for
  bond and option prices.
• It should produce realistic dynamics.
• It should give a full range of possible yield curves.
• It should fit historical data.
• Can be calibrated easily to current market data.
• Flexible to cope with a range of derivatives.

Question 4

Properties of the one-factor Vasicek model

Answer 4

• It incorporates mean reversion [½]
• It is time homogenous, i.e. the future dynamics of r(t) only depend upon the current value of r(t) rather than what the present time t actually is.
• It is arbitrage-free.
• It allows negative interest rates.
• It is easy to implement since the characteristic functions of all related quantities are available.
• It has constant volatility

Question 5

Properties of the CIS model

Answer 5

• It incorporates mean reversion…
  … is arbitrage free…
  … and time homogenous.
• Volatility depends on the level of the rates: it is high/low when rates are high/low.
• It does not allow negative interest rates.
• However it is more involving to implement than Vasicek model as it is linked to the chi-squared distribution.
• It is a one factor model