FE_spot_vol

FE_spot_vol estimates the instantaneous variance of a diffusion process by the Fourier estimator with Dirichlet kernel

Syntax

  • spotvar=FE_spot_vol(x,t)example
  • spotvar=FE_spot_vol(x,t,Name,Value)example
  • [spotvar,tau]=FE_spot_vol(___)example

Description

example

spotvar =FE_spot_vol(x, t) Example of call of FE_spot_vol with just two input arguments.

example

spotvar =FE_spot_vol(x, t, Name, Value) FE_spot_vol called with optional input argument N and M.

example

[spotvar, tau] =FE_spot_vol(___) Example of call of FE_spot_vol providing both estimation values and times.

Examples

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  • Example of call of FE_spot_vol with just two input arguments.
  • Generates price and instantaneous variance from the Constant Elasticity of Variance model [S. Beckers, The Journal of Finance, Vol. 35, No. 3, 1980] and estimates the instantaneous variance via the Fourier method.

    n=1000;  T=1; t=0:T/n:T; S0=100;
    [S,sigma]=CEVmodel(t,S0); % data generation
    x=log(S); % log-price
    spotvar=FE_spot_vol(x,t);
    plot(spotvar(2:end-1))
    ylabel('Spot variance')
    Click here for the graphical output of this example (link to Ro.S.A. website). Graphical output could not be included in the installation file because toolboxes cannot be greater than 20MB. To load locally the image files, download zip file http://rosa.unipr.it/fsda/images.zip and unzip it to <tt>(docroot)/FSDA/images</tt> or simply run routine <tt>downloadGraphicalOutput.m</tt>

  • FE_spot_vol called with optional input argument N and M.
  • Generates price and instantaneous variance from the Constant Elasticity of Variance model [S. Beckers, The Journal of Finance, Vol. 35, No. 3, 1980] and estimates the instantaneous variance via the Fourier method.

    n=21600;  T=1; t=0:T/n:T; S0=100;
    [S,sigma]=CEVmodel(t,S0); % data generation
    x=log(S); % log-price
    N=floor(n/2); M=floor(n^0.5); % cutting frequencies
    spotvar=FE_spot_vol(x,t,'N',N,'M',M);
    plot(spotvar(2:end-1))
    ylabel('Spot variance')
    Click here for the graphical output of this example (link to Ro.S.A. website). Graphical output could not be included in the installation file because toolboxes cannot be greater than 20MB. To load locally the image files, download zip file http://rosa.unipr.it/fsda/images.zip and unzip it to <tt>(docroot)/FSDA/images</tt> or simply run routine <tt>downloadGraphicalOutput.m</tt>

  • Example of call of FE_spot_vol providing both estimation values and times.
  • Generates price and instantaneous variance from the Constant Elasticity of Variance model [S. Beckers, The Journal of Finance, Vol. 35, No. 3, 1980] and estimates the instantaneous variance via the Fourier method.

    n=21600;  T=1; t=0:T/n:T; S0=100;
    [S,sigma]=CEVmodel(t,S0); % data generation
    x=log(S); % log-price
    N=floor(n/2); M=150; % cutting frequencies
    [spotvar,tau]=FE_spot_vol(x,t,'N',N,'M',M);
    plot(tau,spotvar)
    xlabel('time')
    hold on; plot(tau,sigma(1:n/(2*M):end));
    ylabel('Spot variance')
    title('Estimated value (blue), True value (red)')
    Click here for the graphical output of this example (link to Ro.S.A. website). Graphical output could not be included in the installation file because toolboxes cannot be greater than 20MB. To load locally the image files, download zip file http://rosa.unipr.it/fsda/images.zip and unzip it to <tt>(docroot)/FSDA/images</tt> or simply run routine <tt>downloadGraphicalOutput.m</tt>

    Input Arguments

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    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

    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',500 , 'M',218

    N —cutting frequency.scalar.

    If N is not specified, it is set equal to (length(x)-1)/2.

    Example: 'N',500

    Data Types: single | double

    M —maximum Fourier frequency for spot variance reconstruction.scalar.

    If M is not specified, it is set equal to sqrt(length(x))*log(length(x)).

    Example: 'M',218

    Data Types: single | double

    Output Arguments

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    spotvar —Spot variance. Vector

    Row vector with length 2M+1 of spot variance values at the time grid tau.

    tau —Estimation times. Vector

    Row vector with length 2M+1 where the spot variance is estimated.

    More About

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    Additional Details

    We assume our timeseries data are discrete observations from a diffusion process $x$ following the It\^o 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.

    See the Reference for further mathematical details.

    The squared diffusion coefficient $\sigma^2(t)$ is called instantaneous variance of the process $x$.

    For any positive integer $n$, let ${\cal S}_{n}:=\{ 0=t_{0}\leq \cdots \leq t_{n}=T \}$ be the observation times. Moreover, let $\delta_i(x):= x(t_{i+1})-x(t_i)$ be the increments of $x$.

    The Fourier estimator of the instantaneous variance for $t \in (0,T)$, is defined as $$\widehat \sigma^2_{n,N,M}(t):= \sum_{|k|\leq M} \left(1- {|k|\over M}\right) e^{{\rm i} {{2\pi}\over {T}} tk} c_k(\sigma^2_{n,N}),$$ where $$c_k(\sigma^2_{n,N}):= {T \over {2N+1}}\sum_{|s|\leq N} c_s(dx_n) c_{k-s}(dx_n),$$ where for any integer $s$, $|s|\leq 2N$, the discretized Fourier coefficients of the increments are $$c_s(dx_{n}):= {1\over {T}} \sum_{i=0}^{n-1} e^{-{\rm i} {{2\pi}\over {T}} st_i}\delta_i(x).$$

    References

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

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