Figure 1

(Color online) Log-log representation of the survival probability $S\left(z,v,\tau \right)$ as a function of $z$ for $\tau =0.5$ (around 11 trading days), $v=\theta$, and $\beta =0.1$. Empty diamonds represent the numerical evaluation of the exact expression given in Eq. (30). Solid circles represent simulation points of the Heston model. We use the realistic parameters values presented in Ref. [27], which are similar to those encountered in actual markets: $\alpha =0.045/\text{day}$, $m^2=8.62\times {10}^{-5}/\text{day}$ and hence $\theta =1.92\times {10}^{-3}$.Reuse & Permissions

Figure 2

(Color online) Log-log representation of the survival probability as a function of the scaled time $\tau$ with $z=0.01$ and $\beta =1$. Circles, squares, and diamonds represent the numerical evaluation of the exact expression (30), for the three values of the initial volatility $v$ shown in the figure. Solid lines correspond to the asymptotic expression given by Eq. (38) in terms of the error function and different curves correspond to different values of $v$. We see that as the volatility becomes smaller the exact expression needs a longer time to converge to the asymptotic expression. When the initial volatility is five times larger than the normal level, the convergence time is about ${10}^5$ trading days. For $v/\theta =100$ this time is much less (about ${10}^3\text{days}$). Let us note that for volatilities thousand times higher than the normal level, the asymptotic curve follows very closely the exact SP for all values of $\tau$, which is in agreement with the results of Sec. 3B. We use the realistic parameter values given in Fig. 1.Reuse & Permissions

Figure 3

(Color online) Log-log representation of the survival probability $S\left(z,v,\tau \right)$ as a function of the initial volatility with $z=0.01$ and $\beta =1$. Diamonds and squares correspond to the numerical evaluation of the exact expression (30) at two instants of time $\tau =0.1$ and $\tau =1000$. Solid lines represent the asymptotic error function given by Eq. (40). Note that the approximation improves as time increases, in agreement with the results of Sec. 3A. We also observe a power-law decay of the form $v^{-1/2}$ with increasing volatility in agreement with the asymptotic estimate given by Eq. (45). We have taken the realistic parameters of Fig. 1.Reuse & Permissions

Figure 4

(Color online) Log-log representation of the hitting probability $W\left(z,v,\tau \right)=1-S\left(z,v,\tau \right)$ as a function of the threshold distance $z$ at time $\tau =0.5$ and $v=\theta$. Diamonds and squares represent the numerical evaluation following the exact expression (30) for two different values of $\beta$. The dotted line corresponds to the asymptotic expression for $\beta$ small given by Eq. (47). The solid line represents the approximate expression obtained through Eq. (51). We have chosen the realistic parameters of the previous figures and solely modify the parameter $k$ to provide different values of $\beta$.Reuse & Permissions

Figure 5

(Color online) Log-log plot of the hitting probability $W\left(z,\tau \right)=1-S\left(z,\tau \right)$ as a function of $z$ for a fixed time ($\tau =1$, approximately 22 trading days) and for different values of $\beta$. The solid line represents the approximation given by Eq. (60)—which, recall, is independent of $\beta$—while dots (i.e., circles, diamonds, and squares) represent the exact result. Equation (60) only works for very mild volatility fluctuations (note the clear divergence between the exact probability and its approximation for $\beta =1$). Parameters are the same as those of Fig. 1.Reuse & Permissions

Figure 6

(Color online) Log-log plot of the hitting probability $W\left(z,\tau \right)=1-S\left(z,\tau \right)$ as a function of $\beta$. Circles and diamonds represent the exact result, obtained through numerical integration of Eq. (56), for $z=0.01$ and two instants of time: $\tau =0.1$ and $\tau =1$. Note that for small values of $\beta$ the hitting probability is insensitive to $\beta$, in agreement with Eq. (60). Moreover, for large values of $\beta$ we observe the power-law decay predicted in the asymptotic estimate given by Eq. (66), i.e., $W\sim {\beta }^{-1}$. Parameters are the same as those of Fig. 1.Reuse & Permissions

Figure 7

(Color online) Log-log plot of the hitting probability $W\left(z,\tau \right)=1-S\left(z,\tau \right)$ as a function of $z$ with $\tau =0.5$ and for three large values of $\beta$. Solid lines display $W\left(z,\tau \right)$ based on the approximation given in Eq. (64), while dots represent the exact result obtained through the numerical integration of Eq. (56). Parameters are the same as those of Fig. 1.Reuse & Permissions

Figure 8

(Color online) Log-log plot of the hitting probability $W\left(z,\tau \right)=1-S\left(z,\tau \right)$ as a function of $z$ with $\tau =3$ and for large and small volatility fluctuations given, respectively, by $\beta =10$ and $\beta =0.1$. Circles and squares represent the exact $W\left(z,\tau \right)$ obtained through Eq. (56). Curves display $W\left(z,\tau \right)$ based on the approximations given in Eq. (60) (dotted line) and Eq. (64) (solid line). Parameters are the same as those of Fig. 1.Reuse & Permissions

Figure 9

(Color online) Numerical solution of Eq. (67) in terms of $\beta$ with a fixed value of the normal level of volatility $\theta =1.92\times {10}^{-3}$ (the same as of the previous figures) and for $\tau =0.7$, $\tau =1.3$, and $\tau =2.0$. Note that for approximately $\beta >10$, $l_c$ slows down its increase. Solid lines represent a logarithmic fit. Parameters are the same as those of Fig. 1.Reuse & Permissions

Figure 10

(Color online) Numerical solution of Eq. (67) in terms of $\theta \tau$ for $\beta =1$, 5, and 10. Solid lines are the power law given in Eq. (69) where the exponent $\gamma$ depends on $\beta$ and takes the values $\gamma =0.3293\pm 0.0051$ for $\beta =1$, $\gamma =0.4212\pm 0.0011$ for $\beta =5$, and $\gamma =0.4358\pm 0.0007$ for $\beta =10$. Parameters are the same as those of Fig. 1.Reuse & Permissions

Figure 11

(Color online) Log-log plot of the ratio $\overline{W}\left(z,\tau \right)$ as a function of $z$ when $\beta =10$ and $\tau =3$. Note the sharp burst as $\left|L\right|$ increases, in agreement with the approximation given by Eq. (71). The inset shows the little decrease in the ratio for small values of $\left|L\right|$ (see also Fig. 8). The turning point between these two behaviors is approximately located at $\left|L\right|=l_c=0.336$.Reuse & Permissions

Figure 12

(Color online) Dow Jones daily data hitting probability as a function of $z=\left|L-x\right|$ with $t=22\text{days}$ for $L-x>0$ (circles) and $L-x<0$ (squares). Heston model solution (solid lines) given by Eq. (56) is also represented with very similar parameters to those obtained in Ref. [27]. For positive critical levels, parameters are $\alpha =0.045$, $m^2=6.2\times {10}^{-5}/\text{day}$, $k=0.0025/\text{day}$, and hence $\beta =0.054$. For negative critical levels, parameters are $\alpha =0.035$, $m^2=8.6\times {10}^{-5}/\text{day}$, $k=0.0034/\text{day}$, and $\beta =0.10$. Dotted line corresponds to the error function of the Wiener return given in Eq. (57) with the volatility estimated from the standard deviation of empirical daily return $\sigma =0.0080{\text{day}}^{-1/2}$.Reuse & Permissions