The implementation of the HW-extended Vasicek model, as described above, represents a stochastic process of the short- term interest rate of the form [2] :
The three time-dependent parameters allow three characteristics of the initial term structure to be fitted (see [ 29 ]). The time-dependent drift parameter ( t ) allows the model to exactly match the initial interest rate term structure; the time-dependent short-term interest rate volatility ƒ ( t ) defines the volatility of the short-term interest rate at times in the future; time-dependent reversion speed a ( t ) specifies the relative volatilities of long and short-term interest rates, hence replicating the initial volatility term structure. However, fitting all three time-dependent parameters to current market data results in an over-parameterisation of the model, which may cause undesirable side effects [ 29 ]. Resulting future term structures of volatilities may take on implausible shapes , leading to mispricing of exotic options. For this reason implementations with constant reversion speed and short-term interest rate volatility a and ƒ respectively, are recommended. The volatility term structure will no longer be fitted exactly but only approximated. However, the model will display a stationary volatility term structure which allows more control over future values of model parameters and hence more accurate, robust pricing of exotic derivatives.
The resulting process of the short-term interest rate is:
with associated functional form of B ( t, T ) reducing to [3] :
Hence the initial time t = 0 value is:
Since the time t = 0 term structure of interest rates is known we find the initial term structure of discount factors (zero coupon bond prices) P ( r, 0, ·), as shown in (13.3). Consequently the initial values of the A ( t, T ) coefficient may be found as:
Since P ( r, 0, ·) and r (0) are known, the exact values of A (0, ·) are determined by the chosen value of parameter a . The initial discount bond prices can be exactly reproduced for any arbitrarily specified value of a . This does not say anything about the correctness of this parameter value. Incorporation of additional market data dependent on volatility parameters, such as interest rate options, is required to allow a correct choice of a .
As before, once the initial values A (0, ·) and B (0, ·) are known, any future values A ( t, T ) and B ( t, T ) may be calculated using (13.9) and (13.10). For constant a and ƒ these reduce to [4] :
Similarly, the forward bond price volatility (13.7), required to value each of the m zero coupon bond options making up the coupon bond option, reduces to:
Calibration of the model to observable market prices involves retrieving values of ƒ and a such that these market prices may be recovered from the model.
[2] See equation (7.4).
[3] See equation (7.25).
[4] These can easily be shown to be equivalent to the functional forms specified in (7.25), (7.26) Chapter 7 for constant a and ƒ .