a mixed approach, using Euler for the index and the noncentral chi square-based exact
approach for the volatility process.
Figure 10-11 shows the simulation results as a histogram for both the index level process
and the volatility process:
In [ 38 ]: fig, (ax1, ax2) = plt.subplots( 1 , 2 , figsize=( 9 , 5 ))
ax1.hist(S[- 1 ], bins= 50 )
ax1.set_xlabel(‘index level’)
ax1.set_ylabel(‘frequency’)
ax1.grid(True)
ax2.hist(v[- 1 ], bins= 50 )
ax2.set_xlabel(‘volatility’)
ax2.grid(True)
An inspection of the first 10 simulated paths of each process (cf. Figure 10-12) shows that
the volatility process is drifting positively on average and that it, as expected, converges to
v = 0.25:
In [ 39 ]: fig, (ax1, ax2) = plt.subplots( 2 , 1 , sharex=True, figsize=( 7 , 6 ))
ax1.plot(S[:, : 10 ], lw=1.5)
ax1.set_ylabel(‘index level’)
ax1.grid(True)
ax2.plot(v[:, : 10 ], lw=1.5)
ax2.set_xlabel(‘time’)
ax2.set_ylabel(‘volatility’)
ax2.grid(True)
Figure 10-11. Simulated stochastic volatility model at maturity
