Testing for One-Factor Models versus Stochastic Volatility Models

3 A Simulation Experiment

In this section, the small sample performance of the testing procedure proposed in the previous
section will be assessed through a Monte-Carlo experiment. Under the null hypothesis, we consider
a version of the Cox, Ingersoll and Ross (1985) model with a mean reverting component in the
drift,

d Xt = ( κ + μXt )d t + η y∕X_t d W1 ,t.                         (13)

We first simulate a discretized version of the continuous trajectory of Xt under (13). We use
a Milstein scheme in order to approximate the trajectory, following Pardoux and Talay (1985),
who provide conditions for uniform, almost sure convergence of the discrete simulated path to
the continuous path, for given initial conditions and over a finite time span. In order to get a
very precise approximation to the continuous path, we choose a very small time interval between
successive observations (1/5760); moreover, the initial value is drawn from the gamma marginal
distribution of Xt, and the first 1000 observations are then discarded.

We then sample the simulated process at two different frequencies, 1/n and 1/m, and compute
the different test statistics. In particular, the time span has been fixed to five days and five
different values have been chosen for the number of intradaily observations n, ranging from 144
(corresponding to data recorded every ten minutes) to 1440 (corresponding to data recorded every
minute). Therefore, the total number of observations ranges from Tn = 720 to Tn = 7200, where
T denotes the fixed time span expressed in days. Also, the experiment has been conducted for six
different values for m (namely ∣^(Tn) '⁷ /TJ, ∣^(Tn) '75 /TJ, ∣^(Tn) '8 /TJ, ∣^(Tn) '9 /TJ, ∣^(Tn) '95 /TJ
and then the limiting case m = n). The process is repeated for a total of 10000 replications.

Results are reported for two test statistics, namely

L^{( n-1}) rj J

∑ Sn (Xi/n) - RVmrj

Under the conditions stated in Theorem 1, we know that for m/n → 0,

Z_nm -→ ZM = max ∖ZMr∙∖,
,j

and for m = n,

Zn —► Z — max I Zr I ,
ⁿ              j=1,...,J ^rj

13

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