Consider an economy in discrete time in which interest and
proceeds are paid out at the end of every constant, equally long
time interval. Let
be the price of the stock
at time
and
the corresponding
one period return without dividends. Assume that a price for risk
exists in the form of a risk premium which is added to the risk
free interest rate
to obtain the expected return of the next
period. It seems reasonable to model the risk premium dependent on
the conditional variance. As a basis we assume an ARCH-M-Model
(see Section 12.2.3) with a risk premium, which is a
linear function of the conditional standard deviation:
In (14.3) ,
and
are constant
parameters that satisfy the stationarity and non-negativity
conditions. The constant parameter
can be understood as
the price of one unit of risk.
indicates, as usual,
the set of information available up to and including time
. In
order to simplify the notation our discuss will be limited to the
GARCH(1,1) case.
The above model is estimated under the empirical measure . In
order to deal with a valuation under no arbitrage, similar to
Black-Scholes in continuous time (see Section 6.1),
assumptions on the valuation of risk must be met. Many studies
have researched option pricing with stochastic volatility under
the assumption that the volatility has a systematic risk of zero,
that is, the risk premium for volatility is zero.
Duan (1995) has identified an equivalent martingale
measure
for
under the
assumption that the conditional distribution of the returns are
normal and in addition it holds that
In order to obtain a martingale under the new measure a new error
term, , needs to be introduced that captures the effect of
the time varying risk premium. When we define
, (14.4) leads to the
following model under the new measure
:
The restriction to a quadratic or symmetric news impact curve
is not always optimal, as many empirical
studies of stock returns have indicated. Within the framework of
the above mentioned model these assumptions can lead to a
non-linear news impact function
. The following model is a semi-parametric analogue to
the GARCH model. Under the empirical measure
we obtain
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In this general specification the estimation without additional
information on is a difficult matter, since iterative
estimation procedures would be necessary in order to estimate the
parameters
and the non-parametric function
at
the same time. Therefore we will consider a specific, flexible
parametric model: the Threshold GARCH Model, see Section
12.2. With this model the news impact function can
be written as:
Remember that the innovations are normally distributed. Thus it
follows for the TGARCH model that the unconditional variance,
similar to Theorem 12.10, under the measure is
, where
. The following theorem
gives the unconditional variance for
under
.
Proof:
Let
. Under measure
it holds
that
N
. The
conditional variance
can be written as
By calculating the expected value it can be shown that for the
integral over the negative values it follows that:
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(15.9) |
Because of
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(15.10) |
Thus we obtain
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(15.11) |
Since the unconditional variance is independent of , the
theorem follows.
The function is positive and
for the
realistic case
. We can make the following statement
about the changes in the unconditional variance: for
in (14.8), one obtains the GARCH(1,1) results.
For
(the case of the leverage
effect) the increase in the unconditional
variance is even stronger than the symmetric GARCH case. For
, the unconditional variance is smaller as in
the leverage case, and we can distinguish between two cases: when
the inequality
Naturally the stationary variance has an effect on an option's price: the larger (smaller) the variance is, the higher (lower) is the option price. This holds in particular for options with longer time to maturity where the long-run average of the volatility is the most important determinant of the option's price. Therefore, an option can be undervalued when a GARCH model is used and at the same time a leverage effect is present.
A second feature of the Duan approach is that under and with
positive risk premia the current innovation is negatively
correlated with the next period's conditional variance of the
GARCH risk premium, whereas under
the correlation is zero.
More precisely, we obtain
with the
GARCH parameter
. It is obvious that small forecasts of
the volatility under
(that influences the option's price)
depend not only on the past squared innovations, but also on their
sign. In particular a negative (positive) past innovation for
leads to the fact that the volatility increases
(falls) and with it, the option price. The following theorem
claims that the covariance is dependent on the asymmetry of the
news impact function when a TGARCH instead of a GARCH model is
used.
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(15.13) |
Proof:
First the conditional covariance is deterimined:
In the following we assume that a positive risk premium
exists per unit. Three cases can be identified: for
(in the symmetric case) we obtain
, i.e., the GARCH(1,1) result. For
(the case of the reverse leverage effect) the covariance
increases, and when
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(15.18) |
This also shows that the return of the volatility to a stationary
variance under is different from the symmetric GARCH case. The
negative covariance in the leverage case is actually larger. This
could indicate that options are over (under) valued when for
positive (negative) past innovation a TGARCH process with
is used for the price process and then mistakenly a GARCH model
(
) is used for the volatility forecast.