Figure 10-7. Simulated square-root diffusion at maturity (Euler scheme)
Figure 10-8 then shows the first 10 simulated paths, illustrating the resulting negative,
averagef drift (due to x 0 > ) and the convergence to = 0.02:
In [ 24 ]: plt.plot(x1[:, : 10 ], lw=1.5)
plt.xlabel(‘time’)
plt.ylabel(‘index level’)
plt.grid(True)
Figure 10-8. Simulated square-root diffusion paths (Euler scheme)
Let us now get more exact. Equation 10-6 presents the exact discretization scheme for the
square-root diffusion based on the noncentral chi-square distribution with
degrees of freedom and noncentrality parameter .
Equation 10-6. Exact discretization for square-root diffusion
The Python implementation of this discretization scheme is a bit more involved but still
quite concise:
