Figure 10-12. Simulated stochastic volatility model paths
Finally, let us take a brief look at the statistics for the last point in time for both data sets,
showing a pretty high maximum value for the index level process. In fact, this is much
higher than a geometric Brownian motion with constant volatility could ever climb, ceteris
paribus:
In [ 40 ]: print_statistics(S[- 1 ], v[- 1 ])
Out[40]: statistic data set 1 data set 2
–––––––––––––––
size 10000.000 10000.000
min 19.814 0.174
max 600.080 0.322
mean 108.818 0.243
std 52.535 0.020
skew 1.702 0.151
kurtosis 5.407 0.071
Jump diffusion
Stochastic volatility and the leverage effect are stylized (empirical) facts found in a
number of markets. Another important stylized empirical fact is the existence of jumps in
asset prices and, for example, volatility. In 1976, Merton published his jump diffusion
model, enhancing the Black-Scholes-Merton setup by a model component generating
jumps with log-normal distribution. The risk-neutral SDE is presented in Equation 10-8.
Equation 10-8. Stochastic differential equation for Merton jump diffusion model
dSt = (r – rJ)Stdt + StdZt + JtStdNt
For completeness, here is an overview of the variables’ and parameters’ meaning:
St
Index level at date t
r
Constant riskless short rate
