Definition Of Markovian Rate Processes

In Part Four of this book a number of important interest-rate models have been reviewed. Several 'taxonomic' criteria could have suggested different classifications: one- versus many-factor models; analytical or numerical approaches; market-fitting versus 'market-describing' models; etc. Given that different perspectives could have given rise to equally (un)satisfactory classifications, no special effort has been devoted to the grouping of the models, apart from an understandable effort to introduce more complex models after simpler ones had already been described.

One feature, however, could have provided a particularly meaningful criterion of differentiation: the Markovian or non-Markovian character of the short-rate process. The implications of this important feature of the yield-curve dynamics are explored in this chapter. (See also Carverhill (1992).)

Given a stochastic differential equation of the form dx = p. di + a dz (20.1)

the process for dx will be said to be Markovian if the probability of dx assuming a given value di(i„) at time tn in the future, contingent on dx having assumed at past times r_/v, i-(/v-1), • • •, values di(r_/v), dx(i_(/v-i>), •. up to dx(io) today, i.e. p(dx(i„)|dx(i_w) = dx(r_/v), dx(r_(W-i)) = dx(i_(/v-i)),..., dx(r0) = dx(io)), is simply equal to p(djc(i„)|dx(r_/v) = di(r_/v), dx(r_(/v_i)) = dx(r_(/v_i)),.. ■, dx(r0) = dx(i0))

This 'interpretation' could account convincingly for the observed term structure, but, strictly speaking, one could not rule out the theoretical possibility that market expectations of rates might be very different from what was outlined, and that a very pronounced change in future volatilities might be discounted by the market. For this particular case, especially given the short horizon over which the change in yield-curve shape takes place, any 'interpretation' that did not take into account in a substantial way the role played by expectation of rates would probably be highly implausible. What is intuitive and plausible for a trader might, however, be beyond the reach of the optimisation procedure employed to 'best'-fit the parameter of the model (which for a two-dimensional case might be as many as a dozen). This was the danger highlighted in the 'implied' approach to the calibration of the Longstaff and Schwartz model (which indeed is of the time-homogeneous, two-dimensional type): the fitting (possibly of high numerical 'quality') of a complex yield curve might well be achieved by finding combinations of the parameters which attempt to 'conceal' the fundamental inability of the deterministic model dynamics to account for the 'expectation' of future rates. When this is the case, the ability to reproduce a virtually arbitrary yield curve can be a very undesirable feature. The alternative, i.e. to constrain the volatility inputs to yield 'reasonable' values, would enhance the 'plausibility' of the model, but, at the same time, the price would have to be paid of a possibly serious mispricing of the underlying plain-vanilla instruments.

Moving from time-homogeneous to time-inhomogeneous models, such as the one- or two-factor HW models presented in Chapter 13, the situation becomes fundamentally different: due to the presence of the 'market-fitting' function 8(f), the burden of accounting for however complicated a market yield curve can be easily accommodated by an affine model for any a priori choice of the volatility inputs. The other side of the coin is that the appropriateness of the input volatilities cannot be gauged from the quality of the fit to discount bond prices, which is ensured to be perfect by construction. The correctness of the volatility choice therefore requires the independent pricing of a different set of volatility-dependent instruments. (This state of affairs has already been encountered in the case of the BDT model.) In other words, time-inhomogeneous models do not 'explain', but rather account for the shape of the yield curve.

Exactly the same considerations can be, and have been, repeated for the fitting to cap prices: in the section devoted to the two-factor HW model it was argued that blind 'best-fitting' of the reversion speed and absolute volatility parameters can easily produce implausible curve dynamics. An enlightened combination of fitting, econometrical constraints and intuition can produce more satisfactory, and far more robust, results.

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