FM_int_vol

FM_int_vol computes the integrated variance of a diffusion process via the Fourier-Malliavin estimator

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Syntax

  • V_int=FM_int_vol(x,t,T)example
  • V_int=FM_int_vol(x,t,T,Name,Value)example

Description

example

V_int =FM_int_vol(x, t, T) Example of call of FM_int_vol with default value of N.

example

V_int =FM_int_vol(x, t, T, Name, Value) Example of call of FM_int_vol with custom choice of N.

Examples

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  • Example of call of FM_int_vol with default value of N.

    • The following example estimates the integrated volatility of a Heston model from a discrete sample. The Heston model assumes that the spot variance follows a Cox-Ingersoll-Ross model. 

    % Heston model simulation
T=1;  
n=23400;  
parameters=[0,0.4,2,1];
Rho=-0.5;
x0=log(100); 
V0=0.4;
[x,V,t]=Heston1D(T,n,parameters,Rho,x0,V0); 
% Integrated volatility estimation
V_int=FM_int_vol(x,t,T);

  • Example of call of FM_int_vol with custom choice of N.

    • The following example estimates the integrated volatility of a Heston model from a discrete sample. The Heston model assumes that the spot variance follows a Cox-Ingersoll-Ross model. 

    % Heston model simulation
T=1; 
n=23400;  
parameters=[0,0.4,2,1];
Rho=-0.5;
x0=log(100); 
V0=0.4;
[x,V,t]=Heston1D(T,n,parameters,Rho,x0,V0); 
% Integrated volatility estimation
V_int= FM_int_vol(x,t,T,'N',5000);

    Input Arguments

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    x — Observed process 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

    T — Estimation horizon. Scalar.

    Data Types: single| double

    Name-Value Pair Arguments

    Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

    Example: 'N',400

    N —Cutting frequency.scalar.

    If N is not specified, it is set equal to floor(length(x)-/2).

    Example: 'N',400

    Data Types: single | double

    Output Arguments

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    V_int —Integrated variance. Scalar

    Values of the integrated variance.

    More About

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    Additional Details

    We assume that the vector x contains discrete observations from a diffusion process following the Ito stochastic differential equation $$dx(t)= \sigma(t) \ dW(t) + b(t) \ dt,$$ where $W$ is a Brownian motions defined on the filtered probability space $(\Omega, (\mathcal{F}_t)_{t \in [0,T]}, P)$, while $\sigma$ and $b$ are random processes, adapted to $\mathcal{F}_t$.

    See the References for further mathematical details.

    The integrated variance of the process $x$ on $[0,T]$ is defined as $$IV_{[0,T]}:=\int_0^T \sigma^2(t)\, dt.$$ For any positive integer $n$, let $\mathcal{S}_{n}:=\{ 0=t_{0}\leq \cdots \leq t_{n}=T \}$ be the observation times. Moreover, let $\delta_l(x):= x(t_{l+1})-x(t_l)$ be the increments of $x$.

    The Fourier estimator of the integrated volatility on $[0,T]$ is given by $$T c_0(\sigma_{{n},N})={T^2\over {2N+1}} \sum_{|k|\leq N} c_{k}(dx_{n})c_{-k}(dx_{n}),$$ where $$c_k(dx_{n})= {1\over {T}} \sum_{l=0}^{n-1} e^{-{\rm i}\frac{2\pi}{T}kt_l}\delta_{l}(x).$$

    References

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

    Sanfelici, S., Toscano, G. (2024), The Fourier-Malliavin Volatility (FMVol) MATLAB toolbox, available on ArXiv.

    See Also

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