Chapter 17: The term structure of interest rates Flashcards
8 Desirable characteristics of a term structure model
- The model should be arbitrage free
- Interest rates should ideally be positive.
- Interest rates should be mean-reverting
- Bonds and derivative contracts should be easy to price.
- It should produce realistic interest rate dynamics.
- It should fit historical interest rate data adequately.
- It should be easy to calibrate to current market data.
- It should be flexible enough to cope with a range of derivatives
General SDE for r(t)
dr(t) = a(t, r(t))dt + b(t, r(t)) dW(t)
- a(t, r) is the drift
- b(t, r) is the volatility
- W(t) is a standard Brownian motion under the real-world measure P.
Market price of risk (expression)
γ(t, T) = {m(t, T) - r(t)} / S(t, T)
γ(t, T) represents the excess expected return over the risk-free rate per unit of volatility in return for an investor taking on this volatility.
Risk premium on a bond
γ(t, T) S(t, T) = m(t, T) - r(t)
5 Desirable Properties of:
the one-factor Vasicek model
- arbitrage free
- does not prescribe the short rate process
- is tractable (allows closed form analytical solutions to a wide range of derivatives)
- encompasses mean reversion
- allows for a wide range of yield curves
6 Properties of:
Cox-Ingersoll-Ross model
Same as Vasicek:
- Incorporates mean reversion
- Arbitrage free
- Time homogeneous
Different from Vasicek:
- Does not allow negative interest rates
- More involving to implement than Vasicek model (linked to the chi-squared distribution)
- Volatility depends on the level of the rates: it is high/low when rates are high/low
- It is a one factor model
Time homogeneous
The future dynamics of r(t) only depend upon the current value of r(t) rather than what the present time t actually is.
Vasicek Model: L
L = μ - σ²/α²
Vasicek Model: β
β = σ² / 2α
Auxiliary process for solving the Vasicek differential equation
Xt = r exp{at}
Value at redemption of a company according to the merton model
value at redemption = min(F(t), L)
Where F(t) is the gross value of the company at time t and L is the outstanding debt/bonds
3 limitations of 1-factor term structure models
- Not perfect correlation across maturities
- Different volatility phases
- Pricing complex derivatives
Gain on a risk-free portfolio
dV(t) = r(t)V(t) dt
where r(t) is the return on the risk-free asset.
6 Deficiencies of the Vasicek Term Structure Model
- Does not prevent negative interest rates
- Is difficult to use to obtain humped yield curves
- Has a lack of time dependence of parameters which is not compatible with empirical evidence
- Gives, in the long run, spot rates normally distributed - which is not compatible with empirical evidence
- Implies perfect instantaneous correlation of bond prices, which is not compatible with empirical evidence
- Will need to be re-parameterised as the yield curve evolves.
Compare the properties of the Vasicek and the Cox-Ingersoll-Ross models for interest rates
- Vasicek and CIR both encompass mean reverting short rates
- Both generate arbitrage-free yield curves
- In both models, the parameters are time invariant
- Vasicek permits negative rates, CIR does not
- Vasicek is more mathematically tractable
- CIR enforces a non-negative lower bound on yields.
