15.2 A Monte Carlo Study

Since the discounted price process is a martingale under the equivalent martingale measure $ Q$, we can utilize the method of risk neutral valuation according to Cox and Ross (1976). The $ Q$ price, $ C_t$, of a call at time $ t$ is given by the discounted conditional expectation of the payments due at maturity, see (6.23)

$\displaystyle C_t = (1+r)^{-\tau} {\mathop{\text{\rm\sf E}}}^Q[\max(S_T-K,0) \mid {\cal F}_t]$ (15.19)

where $ T$ is the maturity date, $ \tau = T - t$ is the time to maturity and $ K$ is the strike price. For European options the arbitrage free price $ P_t$ of a put follows from the Put-Call-Parity (Theorem 2.3), i.e., $ P_t = C_t - S_t +
(1+r)^{-\tau} K$. Since there is no analytical expression in a GARCH or TGARCH model for the expectation in (14.19), we have to calculate the option price numerically. The distribution of the payment function $ \max(S_T-K,0)$ at maturity is simulated in that $ m$ stock processes

$\displaystyle S_{T,i} = S_t \prod_{s=t+1}^T (1+Y_{s,i}), \quad i=1,\ldots,m,$ (15.20)

are generated, where $ Y_{s,i}$ is the return of the $ i-$th replication at time $ s$. Finally the mean of the payment function is discounted by the risk free interest rate

$\displaystyle C_t = (1+r)^{-\tau} \frac{1}{m} \sum_{i=1}^m \max(S_{T,i}-K,0).$ (15.21)

In the simulation study we used the following parameters: $ r = 0,$ $ S_0 = 100,$ $ \tau = 30$ days, $ m = 400\;000,$ $ \lambda = 0.01$. The Moneyness $ S_0/K$ 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, $ T$, 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 $ \alpha + \beta$. For the GARCH(1,1) process it can be shown that the autocorrelation $ \rho_1$ of first order of the squared residuals is given by

$\displaystyle \rho_1 = \alpha(1-\alpha\beta - \beta^2)/(1-2\alpha\beta - \beta^2),$ (15.22)

and $ \rho_j = (\alpha+\beta) \rho_{j-1}$, $ j \ge 2$. These are the autocorrelations of an ARMA(1,1) process, since the quadratic GARCH(1,1) process satisfy a ARMA(1,1) model (see Theorem 12.9). Table 14.1 lists the parameter groups and characteristics of the three types.


Table 14.1: Characterization of the types of GARCH(1,1) models
Type $ \alpha$ $ \beta$ $ \alpha + \beta$ $ \rho_1$
1 0.1 0.85 0.95 0.1791
2 0.5 0.45 0.95 0.8237
3 0.1 0.5 0.6 0.1077


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 $ \omega$ is set so that $ \sigma^2=0.0002$, i.e. the unconditional variance remains constant.

In view of the non-linear news impact function $ g(\cdot)$ we choose the Threshold ARCH model with two asymmetrical cases. In the first case, which we call the leverage case,

$\displaystyle g_1(x) = \omega + 1.2\alpha x^2 \boldsymbol{1}(x<0) + 0.8 \alpha x^2 \boldsymbol{1}(x \ge 0)$

and in the second case, that of the inverse leverage effect,

$\displaystyle g_2(x) = \omega + 0.8\alpha x^2 \boldsymbol{1}(x<0) +1.2 \alpha x^2 \boldsymbol{1}(x \ge 0). $


Table 14.2: Simulation results for selected values of moneyness. Shown are the proportional differences between the GARCH and TGARCH option prices and the Black-Scholes price and the corresponding standard error (SE) of the simulation.
GARCH TGARCH
Leverage Effect Inv. Lev. Eff.
Type Moneyness % diff SE % diff SE % diff SE
0.85 35.947 1.697 0.746 1.359 75.769 2.069
0.90 -0.550 0.563 -12.779 0.498 11.606 0.631
0.95 -6.302 0.261 -9.786 0.245 -3.153 0.278
Type 1 1.00 -3.850 0.132 -4.061 0.125 -3.806 0.139
1.05 -1.138 0.057 -0.651 0.052 -1.692 0.061
1.10 -0.020 0.025 0.347 0.022 -0.400 0.028
1.15 0.162 0.012 0.347 0.010 -0.013 0.014
0.85 199.068 5.847 104.619 4.433 293.704 7.884
0.90 0.489 1.136 -23.964 0.891 22.140 1.469
0.95 -30.759 0.370 -39.316 0.305 -24.518 0.454
Type 2 1.00 -20.975 0.167 -22.362 0.141 -20.804 0.198
1.05 -6.038 0.077 -5.427 0.063 -7.148 0.095
1.10 -0.302 0.042 0.202 0.033 -0.966 0.054
1.15 0.695 0.027 0.991 0.021 0.351 0.037
0.85 -2.899 1.209 -11.898 1.125 6.687 1.297
0.90 -5.439 0.496 -8.886 0.479 -1.982 0.513
0.95 -4.027 0.249 -4.970 0.245 -3.114 0.254
Type 3 1.00 -2.042 0.128 -2.077 0.126 -2.025 0.130
1.05 -0.710 0.055 -0.559 0.053 -0.867 0.056
1.10 -0.157 0.023 -0.047 0.022 -0.267 0.023
1.15 -0.009 0.010 0.042 0.010 -0.059 0.011


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

Fig. 14.3: The difference between the simulated GARCH (solid line) and TGARCH (dotted line) option prices from the BS prices is given as a function of the moneyness for type 1 and the leverage case. The figure above shows the absolute differences, the figure below shows the absolute differences divided by the BS price.

price we observe in principle a similar deviation from Black-Scholes, although with an important difference: in the case of the leverage effect the price of the out of the money options is lower and the price of those in the money is higher than in the GARCH model. This is also plausible: when an option is way out of the money and the maturity date is close, the only way to achieve a positive payment at maturity is when the price of the underlying instrument consecutively increases in value in large jumps. This is, however, less likely in the leverage case, since positive returns have a smaller effect on the volatility than they do in the symmetric case, assuming that the parameter groups named above hold.

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.