# OptimalCuttingFrequency

OptimalCuttingFrequency computes the optimal cutting frequency for the Fourier estimator of integrated variance

## Syntax

• Nopt=OptimalCuttingFrequency(x,t)example

## Description

OptimalCuttingFrequency computes the optimal cutting frequency for running the Fourier estimator of the integrated variance on noisy timeseries data. Note that this function calls function autocorr which needs the Econometric toolbox.

 Nopt =OptimalCuttingFrequency(x, t) Computation of the optimal cutting frequency.

## Examples

expand all

### Computation of the optimal cutting frequency.

Optimal cutting frequency for estimating the integrated variance from a vector x of noisy observations of a univariate diffusion process.

% Generate data.
n=1000;
dt=1/n;
t=0:dt:1;
x=randn(n,1)*sqrt(dt);
% generate the diffusion process
x=[0;cumsum(x)];
noise_to_signal =0.5;
sigma_eps = noise_to_signal*std(diff(x));
noise=sigma_eps*randn(size(x));
% add noise, which is i.i.d. N(0,sigma_eps^2)
x=x+noise;
Nopt = OptimalCuttingFrequency(x,t); % optimal cutting frequency
ivar=FE_int_vol(x,t,'N',Nopt);
disp(['The optimal cutting frequency is: ' num2str(Nopt)])
disp(['The value of the integrated variance is: ' num2str(ivar)])
The optimal cutting frequency is: 168
The value of the integrated variance is: 1.1166


## Input Arguments

### x — Observation values. Vector.

Row or column vector containing the observed values.

Data Types: single| double

### t — Observation times. Vector.

Row or column vector with the same length of x containing the observation times

Data Types: single| double

## Output Arguments

### Nopt —Optimal cutting frequency. Scalar

Integer representing the optimal cutting frequency.

We assume our timeseries data are noisy observations $\tilde x$ from a diffusion process following the Ito stochastic differential equation $$dx(t)= \sigma(t) \ dW(t) + b(t) \ dt,$$ where $W$ is a Brownian motion on a filtered probability space. Let $\sigma$ and $b$ be random processes, adapted to the Brownian filtration.

The integrated variance of the process over the time interval $[0,T]$ is defined as $$\int_0^T \sigma^2(t) dt.$$ For any positive integer $n$, let ${\cal S}_{n}:=\{ 0=t_{0}\leq \cdots \leq t_{n}=T \}$ be the observation times.

The observations are affected by i.i.d. noise terms $\eta(t_i)$ with mean zero and finite variance $$\tilde x(t_i)=x(t_i)+\eta(t_i).$$ See the Reference for further mathematical details.

Moreover, let $\delta_i(\tilde x):= \tilde x(t_{i+1})-\tilde x(t_i)$ be the increments of $\tilde x$.

The optimal cutting frequency $N$ for computing the Fourier estimator of the integrated variance from noisy timeseries data is obtained by minimization of the estimated MSE.

The Fourier estimator of the integrated variance over $[0,T]$, is then defined as $$\widehat\sigma^{2}_{n,N}:= {T^2 \over {2N+1}}\sum_{|s|\leq N} c_s(d\tilde x_n) c_{-s}(d\tilde x_n),$$ where for any integer $k$, $|k|\leq N$, the discretized Fourier coefficients of the increments are $$c_k(d\tilde x_{n}):= {1\over {T}} \sum_{i=0}^{n-1} e^{-{\rm i} {{2\pi}\over {T}} kt_i}\delta_i(\tilde x).$$

## References

Mancino, M.E., Recchioni, M.C., Sanfelici, S. (2017), Fourier-Malliavin Volatility Estimation. Theory and Practice, "Springer Briefs in Quantitative Finance", Springer.