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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 expires some time before the second . What is the effect of a market fluctuation, which suddenly rises the spot rate at a time 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 and the calculation is carried out with a forecast of the interest rates backward in time to predict the fair value for an increasing lifetime .

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

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 is decreasing backward in time  . 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 and differs from the spot rate because of the uncertain evolution of the future rates.

 Virtual market experiments: evolving the yield curve Press Display to study the evolution of the yield curve for a fixed lifetime of the bond (specified under Time) and the term structure of the interest rates that is plotted for the specified SpotRate. 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 with . 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

 (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

 (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

 (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

 (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

 (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):

 (5.1.1#eq.6)

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