Once the numbers of factors and their identities have been chosen, the rules
regulating their behaviour must be specified. The finance literature has considered
different models for describing risk factor dynamics in a financial market. The most
common assumption is the trajectory of a stochastic process, that is, a probability
distribution for the value of each factor over future time.2
The choice of a particular stochastic process is also the consequence of a long list of questions. Let us quote some of them: Should the specification be made in a
discrete or in a continuous-time framework? Should we allow for jumps in the risk
factors or restrict ourselves to diffusion processes?3 Should we allow for time-varying
parameters in the stochastic process, or remain with constant ones? Should we have
restrictions placed on the drift coefficient, such as linearity or mean-reversion?
Should the stochastic process be Markovian?4 Can we allow for negative values for
the risk factor, if it has a low probability? Carefully answering every question is
essential, as this will determine the quality and future success of a model.
The adoption of a particular stochastic process to model a financial variable is
generally a trade-off between two contradictory goals. On the one hand, the process
should account for various empirical regularities such as normality, fat tails,5 mean
reversion, absorption at zero, etc. On the other, the model should provide a simple
procedure for its application. Accounting for more empirical observations generally
increases the complexity of the model and loses analytical tractability, which is an
undesirable feature.
As an example, let us consider stochastic processes for the stock price. The
arithmetic Bachelier Brownian motion model, despite its simplicity and tractability,
was not really adopted by the financial community as it had the unfortunate
implications of allowing negative stock prices. On the other hand, the Black and
Scholes geometric Brownian motion was widely adopted in the early days of option
pricing; it assumes that log-returns are normally distributed with a constant volatility,
which precludes negative prices and results also in a simple closed-form expression
for the option price. Later, most of the generalizations of the Black–Scholes
option pricing model focused on asset price dynamics, as the geometric Brownian
motion fails to account for excess kurtosis and skewness which are often empirically
observed in most asset return distributions (see Table 14.4). Some authors, including
Merton (1976a,b), Hull and White (1987), Scott (1987), Wiggins (1987), Bates
(1996a,b), have answered by modeling asset prices by a different process, typically
adding a jump component to the original model or considering that the stock price
and the volatility level were following separate correlated stochastic processes. The
resulting models provide a better fit to the empirical data, but lose their analytical
tractability and ease of implementation. This explains why practitioners were very
reluctant to adopt them.
More recently, instead of assuming an exogenous stochastic process for the volatility,
researchers have assumed an empirical distribution of the ARCH/GARCH type
that is much easier to estimate and better fits the data. Then, they have derived consistent option pricing models (see for instance Duan, 1995). Alternatively, the
empirical stock return distribution can be implied from the smile and the term
structure of volatility, and a modified binomial model (with a distorted tree) is derived
to price standard or exotic options. This second approach was explored by Derman
and Kani (1994), Dupire (1994) and Rubinstein (1994). All these new models are
simply pragmatic approaches to cope with model risk. They do not attempt to build
a better economic model; they simply adapt an existing one in order to fit market
data better.
For bond prices, the route was quite different. Bond prices cannot follow geometrical
Brownian motions, since the prices and returns are subject to strong
constraints: typically, zero-coupon bond prices cannot exceed their face value; close
to maturity, the bond price should be close to its face value and the volatility of the
bond returns must be very small. However, a large variety of stochastic processes is
available for choice, and this has also led to a profusion of models.6
We should note here that many authors justified the adoption of a particular
stochastic process for interest rates by its analytical tractability rather than by its
economic significance. For instance, many models assume normally distributed
interest rates, allowing for the possibility of negative values for nominal rates.
Assuming log-normality would solve the problem, but at the cost of losing analytical
tractability. Furthermore, portfolios of assets with log-normal returns are not lognormal.
Therefore, what was an essential problem for stocks is simply ignored for
fixed income assets. 422
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