Heston1D

Heston1D simulates observations and instantaneous variances from the Heston model

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Syntax

Description

Heston1D simulates, using the Euler–Maruyama method, observations and instantaneous variances from the model by [S. Heston, The Review of Financial Studies, Vol. 6, No. 2, 1993].

example

x =Heston1D(T, n, parameters, rho, x0, V0) Example of call of Heston1D for obtaining process observations only.

example

[x, V] =Heston1D(___) Example of call of Heston1D for obtaning process observations and volatility values.

example

[x, V, t] =Heston1D(___) Example of call of Heston1D for obtaning process observations, volatility values and sampling times.

Examples

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  • Example of call of Heston1D for obtaining process observations only.

    • Generates observations from the Heston model. 

    T=1;  
n=23400;   
parameters=[0,0.4,2,1];
rho=-0.5;
x0=log(100); 
V0=0.4;
x = Heston1D(T,n,parameters,rho,x0,V0);  
figure
plot(x)
ylabel('Observations')
title('Heston model')
    Click here for the graphical output of this example (link to Ro.S.A. website).

  • Example of call of Heston1D for obtaning process observations and volatility values.

    • Generates observations and volatilities from the Heston model. 

    T=1; 
n=23400; 
parameters=[0,0.4,2,1];
rho=-0.5;
x0=log(100); 
V0=0.4;
[x,V] = Heston1D(T,n,parameters,rho,x0,V0);  
figure
subplot(2,1,1)
plot(x)
ylabel('Observations')
title('Heston model')
subplot(2,1,2)
plot(V)
ylabel('Spot Variances')
title('Heston model')
    Click here for the graphical output of this example (link to Ro.S.A. website).

  • Example of call of Heston1D for obtaning process observations, volatility values and sampling times.

    • Generates observations, volatilities and sampling times from the Heston model. 

    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);  
figure
subplot(2,1,1)
plot(t,x)
xlabel('Time')
ylabel('Observations')
title('Heston model')
subplot(2,1,2)
plot(t,V)
xlabel('Time')
ylabel('Spot Variances')
title('Heston model')
    Click here for the graphical output of this example (link to Ro.S.A. website).

    Input Arguments

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    T — Time horizon. Scalar.

    Data Types: single| double

    n — Number of simulated observation. Scalar.

    Data Types: single| double

    parameters — Model parameters. Vector.

    Vector of dimension 4.

    Data Types: single| double

    rho — Leverage parameter. Scalar.

    Data Types: single| double

    x0 — Initial observation value. Scalar.

    Data Types: single| double

    V0 — Initial variance value. Scalar.

    Data Types: single| double

    Output Arguments

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    x —Process observations. Vector

    Vector of dimension n+1.

    V —Spot variance values. Vector

    Vector of dimension n+1.

    t —Observation times. Vector

    Vector of dimension n+1.

    More About

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    Additional Details

    The Heston model [S. Heston, The Review of Financial Studies, Vol. 6, No. 2, 1993] is given by the following stochastic differential equations: $$\left\{\begin{array}{l} dx_t=\left( \mu-\frac{1}{2} \sigma^2_t \right) \, dt + \sigma_t \, dW_t \\ d\sigma^2_t=\theta \, \left( \alpha-\sigma^2_t \right) \, dt + \gamma \, \sigma_t \, dZ_t \end{array}\right. ,$$ where $\mu$ is a real-valued constant, $\theta, \, \alpha$ and $\gamma$ are positive constants, and $W$ and $Z$ are Brownian motions with correlation $\rho$.

    References

    Heston, S. (1993), A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies, Vol. 6, No. 2.

    See Also

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