Chapter 17 - The Term Structure Of Interest Rates

Question 1

Main uses of the term structure interest rate models (4)

Answer 1

They are:

• by bond traders looking to identify and exploit price inconsistencies
• for calculating the price of interest rate derivatives
• by investors with a portfolio involving bonds or loans who want to set up a hedged position
• for asset-liability modelling

Question 2

Desirable characteristics of a term structure model (8)

Answer 2

1. The model should be arbitrage-free
2. Interest rates should be positive
3. Interest rates should be mean-reverting
4. Bonds and derivative contracts should be easy to price
5. It should produce realistic interest rate dynamics
6. It should fit historical interest rate data adequately
7. It should be easy to calibrate to current market data
8. It should be flexible enough to cope with a range of derivatives

Question 3

Explain what is meant by term structure models

Answer 3

Models that describe the dynamics of B(t,T), r(t), f(t,T) and R(t,T) over time are called, amongst other things, term structure models or models for the term structure of interest rates or interest rate models.

Question 4

Market price of risk

Answer 4

It represents the excess return over the risk free rate per unit of volatility in return for an investor taking on this additional volatility.

Question 5

How do modellers use the risk-neutral approach to pricing in practice?

Answer 5

When modellers use this approach to pricing, from the practical point of view they normally start by specifying the dynamics of r(t) under Q in order to calculate bond prices. Second, they specify the market price of risk as a component of the model, and this allows us to determine the dynamics of r(t) under P.

Question 6

Discuss the drawbacks of the Vasicek model

Answer 6

The Vasicek model suffers from a number of drawbacks. The most obvious one of these is that interest rates can go negative. Sometimes this is not a particular problem: for example, if the probability of negative rates is small, possibly because the time horizon is short. In other cases the probability and severity of negative interest rates can be significant.

Question 7

Main advantage of the Hull-White model

Answer 7

An advantage of the Hull-White model is that it allows us to price interest rate linked contracts more accurately.
This is important for a variety of reasons.
1. In insurance, the fair value of fixed liabilities (with no option characteristics) must accurately reflect the current observed term structure of interest rates. The use of the Vasicek or CIR model, even after fitting the model to the current term structure, might introduce some bias into the fair value.
2. Bond and interest rate derivatives traders want to be able to quote prices that are in line with prices being quoted by other traders. This is facilitaded by the use of models like the HW model and other, more sophosticated, no-arbitrage models.

Question 8

Main drawback of the Hull-White model

Answer 8

The HW model suffers from the same drawback as the Vasicek model, namely that the interest rates might become negative.

Question 9

3 main limitations of one-factor term-structure models

Answer 9

1. If we look at historical interest rate data we can see that changes in the prices of bonds with different terms to maturity are not perfectly correlated, as one would expect to see if a one-factor model was correct.
   [Sometimes we even see, for example, that the short-dated bonds fall in price while long-dated bonds go up. Recent research has suggested that around three factors, rather than one, are required to capture most of the randomness in bonds of different durations.]
2. If we look at the long run of historical data we find that there have been sustained periods of both high and low interest rates with periods of both high and low volatility.
   [Again, these are features that are difficult to capture without introducing more random factors into a model. This issue is especially important for two types of problems in insurance:
   - the pricing and hedging of long-dated insurance contracts with interest rate guarantees.
   - asset-liability modelling and long-term risk-management.]
3. We need more complex models to deal effectively with derivative contracts that are more complex than, say standard European call options.
   [For example, any contract that makes reference to more that one interest rate should allow these rates to be less than perfectly correlated.]

Question 10

Discuss in general terms the use of multifactor models

Answer 10

There are a great many multifactor models. Often these have been designed in order to tackle specific problems. Students should always remember that models that were designed for and are good for one specific task may not be suitable for quite a different task. For example, models that are widely used by investment banks for pricing and hedging short-term derivatives may not be suitable for use in long-term asset liability modelling studies.

Question 11

One factor models can be used as tools for the valuation of what two things?

Answer 11

One factor models have their place as tools for the valuation of simple liabilities with no option characteristics, or short-term, straightforward derivative contracts.