abarletta / viximpv

VIX Implied Volatility Toolbox (MATLAB)

Latest downloads

MATLAB Toolbox installer (recommended)
Zip archive containing all codes

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Getting Started

This toolbox computes approximate values of the Black & Scholes implied volatility of VIX options using a perturbative technique applied to the infinitesimal generator of the underlying process. The modelling setup requires that the VIX index dynamics is explicitly computable as a smooth transformation φ of a purely diffusive, one-dimensional Markov process Y. Specifically:

where W denots a Brownian motion. More details can be found in this paper. Please note that this tool is not a standalone software, but it fully relies on the MATLAB suite.

Prerequisites

This code has been tested on MATLAB R2017a but it should work on any version of MATLAB that supports symbolic calculus. The following MATLAB Toolboxes are required to ensure full compatibility of the code:

• Symbolic Math Toolbox

Installing rndfittool

There are two options to install the VIX Implied Volatility Toolbox on your machine.

Installing viximpv as MATLAB App (recommended)

• Download the MATLAB Toolbox installer.
• Double-click on the file to start the installation process.
• If the double-click does not work you may alternatively open the file by dragging it into the MATLAB command window.
• Call the main function by viximpv.
• If the installation does not work switch to the next method.

Installing viximpv through zip archive

• Download the zip archive containing all the necessary resources.
• Extract the archive contents into a local folder.
• Set the folder containing the extracted file as MATLAB current folder or, alternatively, add it to the MATLAB path list.
• Call the main function by viximpv.

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Usage

Syntax

[Sigma,Futures]=VIXIMPV(Mu,Eta,Phi,Y0,K,T,Order)

Input

• Mu: Drift coefficient of Y (function handle)
• Eta: Diffusion coefficient of Y (function handle)
• Phi: Function mapping Y to VIX (function handle or symbolic function)
• Y0: Initial value of Y (scalar)
• K: Strike values (vector)
• T: Maturity (scalar)
• Order: Expansion order (integer between 0 and 4)

Output

• Sigma: Approximate VIX implied volatility
• Futures: Approximate VIX futures price

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Examples

All the examples provided below are contained in this MATLAB script.

Heston model

Definition

MATLAB code

% Setting model parameters
Lambda=3;
Theta=0.04;
Epsilon=0.5;
Y0=0.035;
% Drift
Mu=@(y) Lambda*(Theta-y);
% Diffusion
Eta=@(y) Epsilon*sqrt(y);
% Function Phi
Tau=30/365;
a=(1-exp(-Lambda*Tau))/(Lambda*Tau);
b=Theta*(1-a);
Phi=@(y) sqrt(a*y+b);
% Setting maturity and strikes
T=1/48;
K=.13:.01:.25;
% Computing approximate implied volatilities
Sigma2=viximpv(Mu,Eta,Phi,Y0,K,T,2);
Sigma3=viximpv(Mu,Eta,Phi,Y0,K,T,3);
Sigma4=viximpv(Mu,Eta,Phi,Y0,K,T,4);
% Plotting
plot(K,[Sigma2; Sigma3; Sigma4],'LineWidth',2);
axis tight;
grid on;

Figure: Implied volatility as function of the log-moneyness.

Mean-reverting CEV model

Definition

MATLAB code

% Setting model parameters
Lambda=3;
Theta=0.04;
Epsilon=1.5;
Delta=1;
Y0=0.035;
% Drift
Mu=@(y) Lambda*(Theta-y);
% Diffusion
Eta=@(y) Epsilon*(y.^Delta);
% Phi
Tau=30/365;
a=(1-exp(-Lambda*Tau))/(Lambda*Tau);
b=Theta*(1-a);
Phi=@(y) sqrt(a*y+b);
% Setting maturity and strikes
T=1/48;
K=.13:.01:.25;
% Computing approximate implied volatilities
Sigma2=viximpv(Mu,Eta,Phi,Y0,K,T,2);
Sigma3=viximpv(Mu,Eta,Phi,Y0,K,T,3);
Sigma4=viximpv(Mu,Eta,Phi,Y0,K,T,4);
% Plotting
plot(K,[Sigma2; Sigma3; Sigma4],'LineWidth',2);
axis tight;
grid on;

Figure: Implied volatility as function of the log-moneyness.

Exp-OU model

Definition

MATLAB code

% Setting model parameters
Lambda=10;
Theta=0.02;
Epsilon=3.5;
Y0=-3.3;
% Drift
Mu=@(y) Lambda*(log(Theta)-y);
% Diffusion
Eta=@(y) Epsilon;
% Phi
syms x y;
Tau=30/365;
g=exp(exp(-Lambda*x).*y+log(Theta)*(1-exp(-Lambda*x))+Epsilon^2/(4*Lambda)*(1-exp(-2*Lambda*x)));
Phi=(1/sqrt(Tau))*sqrt(int(g,x,0,Tau));
% Setting maturity and strikes
T=1/48;
K=.13:.01:.25;
% Computing approximate implied volatilities
Sigma2=viximpv(Mu,Eta,Phi,Y0,K,T,2);
Sigma3=viximpv(Mu,Eta,Phi,Y0,K,T,3);
Sigma4=viximpv(Mu,Eta,Phi,Y0,K,T,4);
% Plotting
plot(K,[Sigma2; Sigma3; Sigma4],'LineWidth',2);
axis tight;
grid on;

Figure: Implied volatility as function of the log-moneyness.