Finally, once the model is theoretically fully specified, we reach the estimation
procedure. Model risk is often confused with estimation risk. This confusion is easily
explainable: if poor estimates of the parameters are used, the model’s results will be
misleading. In addition, the problem of reconciling imperfect theory with reality is
often transformed into that of determining the most probable parameter values for a
specified stochastic model so that the output of the model is as close as possible to
the observations.
Several econometric techniques can be found in the literature to estimate the
observable parameters of a model from empirical data. The most popular are maximum
likelihood estimation (MLE), the generalized method of moments (GMM), and the estimation of a state space model with a Kalman filter.7 These techniques proceed
under the possibly false presumption of a correct model specification. Furthermore,
they are not always well suited for the estimation of some particular stochastic
processes parameters. For instance, using maximum likelihood estimation with
jump-diffusion models–such as Merton’s (1976a,b) option pricing model–is inconsistent.
8 Despite this, it is often encountered in the financial literature.
For unobservable parameters, matters are slightly more complicated. For instance,
in applying option pricing models a systematic difficulty is the estimation of the
volatility of the underlying variable, which is not directly observable. A first guess
may be to assume that the past is a good indicator of volatility in the future, and to
use historical volatility, which is defined as the annualized standard deviation of the
natural logarithms of the price relatives. But there are strong critiques against the
use of the simple historical volatility. First, recent data should be given more weight
than more distant data, as it conveys more information. A solution may be exponential
smoothing, that is, to give progressively less weight to the more distant prices.
Second, the more information, the better the model. Therefore, why don’t we use the
information available in the intradaily high and low values in addition to closing
prices? Finally, in the case of time-varying volatility, the considerable literature on
ARCH and GARCH models explicitly addresses the issue of optimally estimating
conditional volatility.
An additional problem is the choice of the data period to estimate model parameters.
Statistically, using more data will provide better estimates. Financially, using more
data will often include outdated situations.
To avoid using an inappropriate time frame, good market practice is to use current
market data to compute an implied volatility. The procedure requires the use of an
explicit analytical pricing formula, that is, of a model and its hypothesis. And it is
not unusual to see the implied volatility calculated using the Black and Scholes
(1973) model plugged into an alternative model! We are back to the problem of model
risk . . .
The estimation problem is not restricted to the volatility. On the fixed-income side,
a very similar observation applies to the ‘short-term rate’, which plays an essential
part in most interest rate models. The ‘short-term rate’ is in fact an instantaneous
rate of return, which is, in general, unobservable. Therefore, if the estimation of its
parameters is based on a pure time-series approach, it is necessary to approximate
the spot rate with the yield on a bond with non-zero maturity. The proxy is typically
an overnight, a weekly or a monthly rate. But authors do not really agree on
a common proxy for their tests. For instance, Stanton (1997) uses the yield on a
3-month Treasury-bill; Chan et al. (1992) use the one-month Treasury-bill yield; Ait-
Sahalia (1996a,b) uses the one-week Eurodollar rate; Conley et al. (1997) use the
Federal Funds rate. All these rates are rather different and subject to important
microstructure effects. This can bias the estimated parameters for the spot rate,
leading to important model risk effects. Let us recall that in the mid-1970s, Merrill
Lynch lost more than US$70 million in a stripping operation of 30-year government
bonds just because they used a yield curve calculated from inadequate instruments
to price their product.
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