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5.1.1 Term structure models for dummies

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Imagine a portfolio with two identical discount bonds, except that the first $ P(t,T_1)$ expires some time before the second $ P(t,T_2)$ . What is the effect of a market fluctuation, which suddenly rises the spot rate at a time $ t<T_1<T_2$ before the first bond reaches maturity? The bonds are correlated and both will loose some of their original value; since there is more time left for another fluctuation to step back in the opposite direction, it is reasonable to assume that the second bond with a longer time to maturity will be less affected.

Taking advantage of this correlation, Vasicek creates a portfolio with a positive holding in the first bond and a negative holding in the second. By choosing exactly the right balance, this delta-hedging cancels out the uncertain effect from fluctuations and leaves only a deterministic change in the portfolio value. This is then used to calculate the fair price of a bond. The normalized value of the discount function is of course known at the maturity $ P(T,T)=1$ and the calculation is carried out with a forecast of the interest rates backward in time to predict the fair value $ P(T-t,T)$ for an increasing lifetime $ T-t$ .

The VMARKET applet below illustrates the procedure for a bond lifetime with up to RunTime=10 years.

VMARKET applet:  press Start/Stop to simulate the price of a zero-coupon bond backward in time, for a market with a volatile spot rate paying a reward for the associated risk. The plots show the value of the discount function as a function of the spot rate (P[r] in black) for an increasing time to maturity t (Time on the top of the window, in years). Directly derived from that using (2.2.2#eq.1), two plots show the evolution of the yield curve (Y[r] in blue, for a fixed Time) and the term structure of the interest rates (Y[t] in grey, for a fixed SpotRate). The latter acquires a finite value and sweeps across the plot window over the time span of one simulation [0; RunTime] and is best viewed after rescaling with Display.

WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0" SRC="s5img48.gif" ALT="$ r$"> (horizontal axis, chosen to reflect the current market conditions), the discount function $ P(t,T)$ is decreasing backward in time $ t$ . Indeed, investors expect a return from their investment, which shows up as a growth of the discount function when the time runs forward so as to reach exactly one at maturity. The reward can be measured using (2.2.2#eq.1) as a yield $ Y(t,T)=-\log(P)/(T-t)$ and differs from the spot rate $ r$ because of the uncertain evolution of the future rates.

Virtual market experiments: evolving the yield curve
  1. Press Display to study the evolution of the yield curve $ Y(r)$ for a fixed lifetime of the bond (specified under Time) and the term structure of the interest rates $ Y(t)$ that is plotted for the specified SpotRate.
  2. Set Volatility=0 and compare the output obtained for a constant interest rate with the simple discounting previously used in (1.3#eq.6).

Due to the cyclic nature of the economy and the changes in the central bank interest rates, economists generally forecast what may be the future evolution of spot rates $ r(t^\prime)$ with $ t^\prime\in[t,T]$ . This opinion consists of a drift (``the spot rate will fall'') and a volatility (``the spot rate will fluctuate'') that can be estimated from historical values (exercise 1.05).

Masters: one factor models to forecast the term structure of interest rates.
A broad class of models can already be obtained using only one driving term for the uncertainty and assuming a normal distribution of the interest rate increments of the form

$\displaystyle dr = \mu(r,t) dt +\sigma(t) dW(t).$ (5.1.1#eq.1)

Contrary to stock options where the drift scales out of the Black-Scholes equation (3.4#eq.4), the interest rate drifts play a crucial role for the evolution of bond prices. Using the excess return dP/dt-rP=(-m+ls)dP/dr  when the stochastic term is neglected in (3.5#eq.6), different models have been proposed to forecast the evolution of the interest rates.
* The Vasicek model
accounts for a long-term average rate and investors appetite for risk

$\displaystyle \frac{dP}{dt}-rP = \left[a(b-r) +\lambda\sigma\right] \frac{\partial P}{\partial r}$ (5.1.1#eq.2)

The first term is a mean reversion process, where the interest rate is pulled back to the level b at a velocity a.   The second term is proportional to the market price of risk l   and measures the extra return per unit risk expected by the investors (3.5#eq.9).
* The Ho and Lee model
uses the instantaneous forward rate F(0,0,t)   from the market

$\displaystyle \frac{dP}{dt}-rP =\left[\frac{\partial F(0,0,t)}{\partial t} +\sigma^2 t\right] \frac{\partial P}{\partial r}$ (5.1.1#eq.3)

to forecast a drift based on today's expectations without ever saturating.
* The Hull an White model
circumvents this problem with an evolution

$\displaystyle \frac{dP}{dt}-rP =\left[\frac{\partial F(0,0,t)}{\partial t} +a\b...
...+\frac{\sigma^2}{2a}\big(1-\exp(-2at)\big)\right] \frac{\partial P}{\partial r}$ (5.1.1#eq.4)

which reproduces the slope of the initial instantaneous forward rates from Ho and Lee, and later revert back to the long-term average F(0,0,t) with a velocity a.  

* The VMARKET model
(c.f. Vasicek) uses a modulation of the market price of risk

$\displaystyle \frac{dP}{dt}-rP = \big[a(b-r) +\lambda\sigma\cos(2n\pi t/T)\big] \frac{\partial P}{\partial r}$ (5.1.1#eq.5)

to reproduce economic cycles and help you develop and intuition.
Analytical solutions can be found provided that the parameters remain constant [11,19]. A numerical solution is however needed to account for the volatility hump observed in the markets (5.1.1#fig.1): the volatility starts at zero (no uncertainty with bond prices today), reaches a maximum and drops again to zero at maturity (the price equals the face value):

$\displaystyle \sigma(t)\simeq\sigma_\mathrm{max} \left[1.7\left((1-t/T)-(1-t/T)^6\right)\right].$ (5.1.1#eq.6)

Figure 5.1.1#fig.1: Volatility hump during the 10 years lifetime of a bond.

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