which, applying Itô's lemma, can be written as:
The empirical facts, however, do not confirm model assumptions. Financial returns exhibit much fatter tails than the Black-Scholes model postulates, see Chapter 1. The common big returns that are larger than six-standard deviations should appear less than once in a million years if the Black-Scholes framework were accurate. Squared returns, as a measure of volatility, display positive autocorrelation over several days, which contradicts the constant volatility assumption. Non-constant volatility can be observed as well in the option markets where ``smiles'' and ``skews'' in implied volatility occur. These properties of financial time series lead to more refined models. We introduce three such models in the following paragraphs.
If an important piece of information about the company becomes public it may cause a sudden change in the company's stock price. The information usually comes at a random time and the size of its impact on the stock price may be treated as a random variable. To cope with these observations Merton (1976) proposed a model that allows discontinuous trajectories of asset prices. The model extends (8.1) by adding jumps to the stock price dynamics:
where is a compound Poisson process with a log-normal
distribution of jump sizes. The jumps follow a (homogeneous) Poisson process
with intensity
(see Chapter 14), which is independent of
. The
log-jump sizes
are i.i.d random
variables with mean
and variance
, which are
independent of both
and
.
The model becomes incomplete which means that there are many possible ways to choose a risk-neutral measure such that the discounted price process is a martingale. Merton proposed to change the drift of the Wiener process and to leave the other ingredients unchanged. The asset price dynamics is then given by:
For the purpose of Section 8.4 it is necessary to
introduce the characteristic function (cf) of
:
These deficiencies were eliminated in a stochastic volatility model introduced by Heston (1993):
for details see Chapter 7. The term
in equation (8.7) simply ensures
positive volatility. When the process touches the zero bound the
stochastic part becomes zero and the non-stochastic part will push
it up.
Parameter measures the speed of mean reversion,
is the average level of volatility, and
is the volatility of volatility.
In (8.8) the correlation
is typically negative, which is consistent with empirical observations
(Cont; 2001). This negative dependence between returns and volatility is
known in the market as the ``leverage effect.''
The risk neutral dynamics is given in a similar way as in the
Black-Scholes model. For the logarithm of the asset price process
one obtains the equation:
The cf is given by:
where
,
and
and
are the initial values for the log-price process and the
volatility process, respectively.
The Merton and Heston approaches were combined by
Bates (1996), who proposed a model with stochastic
volatility and jumps:
Since the jumps are independent of the diffusion part in (8.10), the characteristic function for the log-price process can be obtained as:
where:
is the diffusion part cf and