# Terminal Variance in the Heston Model

I am trying to understand the basics of financial models.

Random Walk as a model for asset prices.

We use gaussian random numbers to generate a Gaussian Random walk. The variance of the terminal value (cumulative sum of random steps) divided by the square root of the number of steps is $$1$$ due to central limit theorem.

$$S_0=0$$ and $$\displaystyle S_{n}=\sum _{j=1}^{n}Z_{j}$$

$$\displaystyle Var\left(\dfrac{S_n}{\sqrt{n}}\right)=1$$

Black Scholes Model

Asset price follows a geometric brownian motion.

The variance of the terminal value ($$\ln {\frac {S_{t}}{S_{0}}}$$) is:

$$\displaystyle \ln {\frac {S_{t}}{S_{0}}}=\left(\mu -{\frac {\sigma ^{2}}{2}}\,\right)t+\sigma W_{t}\,$$

$$\displaystyle Var\left(\ln {\frac {S_{t}}{S_{0}}}\right)=\left(\mu -{\frac {\sigma ^{2}}{2}}\,\right)^2t+\sigma^2t,$$

The simulations also return values as expected from these closed form equations.

Heston Model

$$\displaystyle dS_t=\mu S_t dt + \sqrt{v_t}S_t dW_{1,t}$$

$$\displaystyle dv_t=\kappa (\theta-v_t)dt + \sigma\sqrt{v_t} dW_{2,t}$$

where $$\mathbb{E}[dW_{1,t}dW_{2,t}]=\rho dt$$

From what I understand, there is no closed form solution for $$Var(S_t)$$. Now, when I simulate

  clear;clc;

s0=100;        % Initial stock price
mu=0;          % expected return
v0=0.05;       % Initialising value for variance
kappa=1.6;     % mean reversion rate AND 2*kappa*theta geq sigma^2
theta=0.1;     % long run variance
sigma=0.2;     % volatility of variance
rho = 0.0;     % Correlation between Stock and its Volatility Process

n_sim=10^4;    % count of simulations
n=1000;        % number of steps

dt=1/n;
ss=ones(n_sim,n)*s0;
vv=ones(n_sim,n)*v0;
for i=1:n_sim    % multiple simulated paths
z=randn(n,2);    % To generate two independent N(0,1) variables
for t=2:n        % single path generation
Z_v = z(:,1);
Z_s = rho*z(:,1)+sqrt(1-rho^2)*z(:,2);
% Discretize the log variance
vv(i,t)=vv(i,t-1)*exp((1/vv(i,t-1))*(kappa*(theta-vv(i,t-1))-(sigma^2/2))*dt+(sigma*sqrt(dt/vv(i,t-1))*Z_v(t)));
% Discretize the log stock price
ss(i,t)=ss(i,t-1)*exp((mu-(vv(i,t-1)/2))*dt+sqrt(vv(i,t-1)*dt)*Z_s(t));
end
end
x=log(ss(:,n)/s0);
vv1=vv(:,n);
mean_vv1=mean(vv1);  % comes out as 0.09

var_sn=var(x);       % comes out as 0.075


I get a value of $$0.075$$. What "meaning" do I draw from it ?

Its not anywhere near the long run variance ($$\theta$$) or $$v_t|t=1$$. When should the long run variance be achieved ?

Any pointers and suggestions would be helpful. Please Advise.

## 1 Answer

From the equations of the model it is clear that $v_t$ is the instantaneous variance of the log-returns, not the terminal annualised variance of the log-asset price.

Put differently, you are you confusing $$v_t \approx \text{var}(\ln(S_{t+\delta t}/S_t))/\delta t$$ with $$\text{var}(\ln(S_t))/t$$ presumably because in the Black-Scholes framework these are both equal to $\sigma^2$.

If you want to look at something near $v_0$ or $\theta$ you should take the mean of the process $(v_t)_{t \geq 0}$ itself (vv in your above code) for different times $t$ and see how it starts from $v_0$ to gradually reach $\theta$.