Expiry (years)

Figure 19.2 Instantaneous forward rates as a function of maturity for the same values of the reversion speed k, and of the remaining parameters (r(0) = 12.00%, u = 10%, a = 0.06). Notice how more complex shapes can be obtained by changing the volatility parameter. See the text for a discussion of the implications of this observation

Qualitatively similar results hold for the CIR model, although the obtainable shapes are somewhat more complex.

This discussion has provided a more quantitative grounding for the statements, often made throughout the book, about the 'volatility contribution to the shape of the yield curve'. The words of caution against blind cross-sectional fitting of model parameters to market prices should now have a more precise meaning and justification: for time-homogeneous affine models there is an inextricable link between the variance of the short rate and the shape of the yield curve (see, e.g., Equations (19.21), (19.17) and (19.15), and Figures 19.1 and 19.2). From an exogenous set of market prices for discount bonds, which might reflect a deterministic dynamics not compatible with what an affine model can allow, one can easily 'estimate' volatility parameters totally inconsistent with any plausible time-series behaviour of the short rate.

While the results proven above have, strictly speaking, been shown to hold for time-homogeneous models, the general message applies to time-dependent affine models as well, as the following section will show.

19.3 time-inhomogeneous affine models

If the crucial assumption of time-homogeneity is relaxed, the all-important result that allowed one to equate a derivative with respect to maturity with (the opposite of) a derivative with respect to calendar time (see Equation (19.13)) no longer holds true. In order to obtain instantaneous forward rates from discount bond prices one must take the limit as T2 goes to of

„ „ In P(t, T2) - In P(t, Ti) fit, T\, T2) =--' 2) ' (19.23)

Since for general (i.e. not necessarily time-homogeneous) affine models one can write

Equation (19.23) becomes fit T T \ A(f, r2) - A{t, TO - [B(t, T2) - B(t, Ti)]r(f)

The volatility of a forward rate is also immediately obtainable using Ito's lemma from the stochastic part of the process for d/:

df = fj.fdt + -^-crrdz = fi/dt- —;<rrdz (19.26)

or ol where ¡x / depends on the particular choice of numeraire. Notice again that the term structure of volatilities only depends on (the derivative of) B, and not on A.

In the case of a two-factor time-inhomogeneous affine model, characterised by an expression for the discount bond price of the form

Pit, T) = exp[—A(r, T) - Bit, T)r - C(r, T)u] (19.27)

the same type of analysis gives for the instantaneous forward rate

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