Since the discounted price process is a martingale under the
equivalent martingale measure , we can utilize the method of
risk neutral valuation according to
Cox and Ross (1976). The
price,
, of a call at time
is given by the discounted conditional expectation of the
payments due at maturity, see (6.23)
![]() |
(15.20) |
![]() |
(15.21) |
In the simulation study we used the following parameters:
days,
.
The Moneyness
varies between 0.85
and 1.15, which corresponds to the usual bandwidth of the traded
option. We are not comparing here the effect of various maturity
dates,
, since many characteristics such as the smile in the
stochastic volatility disappear with increasing time periods. In
general the effects remain qualitatively equal, but become from a
quantitative point of view less important. This has been shown in
numerous experiments; thus we will concentrate on shorter time
periods.
The effect of an asymmetric news impact function on the price of
an option is studied in three different situations which are
characterized by the degree of the short-run autocorrelation of
the squared returns and the persistence, i.e.,
the value from
. For the GARCH(1,1) process it can
be shown that the autocorrelation
of first order of the
squared residuals is given by
![]() |
(15.22) |
|
Type 1 is characterized by a high persistence and a low first
order correlation; type 2 is characterized by a high persistence
and a high first order autocorrelation and type 3 has a low
persistence and a small first order autocorrelation. Type 1 is
typical for financial time series (for daily as well as intra day
data), since one usually observes that the autocorrelation
function of the squared returns diminishes quickly in the first
few lags and then slowly after that. Type 2 describes a situation
with a very strong ARCH effect, and type 3 is similar to the
behavior of heavily aggregated data such as monthly or quarterly.
In every case the parameter is set so that
, i.e. the unconditional variance remains
constant.
In view of the non-linear news impact function we
choose the Threshold ARCH model with two asymmetrical cases. In
the first case, which we call the leverage case,
|
For type 1 and the leverage effect case the simulation results are given in Figure 14.3. We have removed the absolute and the relative difference of the GARCH and the TGARCH prices from the corresponding Black-Scholes price. The relative difference is defined as the absolute difference divided by the Black-Scholes price. Because of the small step length (we assume a step length of 0.01 for moneyness) the functions appear quite smooth. For the GARCH case we obtain the well known result that the price difference to the Black-Scholes displays a U-shape with respect to moneyness. Due to the monotone increase in moneyness of the call price, the relative difference is the largest for options out of the money. The relative difference becomes insignificantly smaller, the more it is in the money. This could explain the frequently observed skewness of the smile effect. For the TGARCH option
Table 14.2 shows the results for the type 2 and 3 and the
case of the inverse leverage effect and for chosen values of
moneyness. For the leverage effect case the described deviation of
the TGARCH price from each GARCH price is visible even for type 2
and 3. In the case of the inverse leverage effect the arguments
are reverse: it is more probable that an out of the money option
can still end up in the money so that TGARCH prices of out of the
money options are higher than the GARCH prices. As one would
expect, the deviations of the simulated GARCH and TGARCH prices
from the Black-Schole prices are the largest for type 2, i.e., for
strong short-run ARCH effects, and are the smallest for the type
with the lowest persistence, type 3. This last statement is to be
expected, since the differences should disappear the closer we get
to the homoscedastic case.