time step choice impact in Vasicek model simulations

Question

I am trying to make some computations using Vasicek short rate model. Especially I a trying to compare exact expectation(obtained with the formula) and the expectation from Monte Carlo simulation.

• For Exact computation I use:

$E[r_t] = r_0 * exp(-a*t) + (\theta/a)*(1-exp(-a*t))$

 public override double GetExpectation(double r0, double t)
        {
            double expectation = r0 * Math.Exp(-_a * t) + (_theta / _a) * (1 - Math.Exp(-_a * t));
            return expectation;
        }

• For Monte Carlo simulations I use the following method:

  • I compute r from a time t to a time t+dt using: public override double ComputeNextValue(double r0, double dt) { RandomVariableGenerator rvg = RandomVariableGenerator.GetInstance();

    double randomGaussian = rvg.GetNextRandomGaussian();
    double r_t_dt = (_theta - _a*r0)*dt + _sigma * Math.Sqrt(dt) * randomGaussian;
    
    return r_t_dt;
    

    }

  • then a compute a path from 0 to t with dt as time steps using:

public override double ComputeValue(double r0, double t, double dt) { double x = r0;

    for(double slot = dt; slot <= t; slot += dt)
    {
        x = ComputeNextValue(x, dt);
    }
    return x;
}

• Then I compute the Monte Carlo Expectation using:

  public override double ComputeMonteCarloExpectation(double r0, double t, double dt, int nreps) { double sum = 0.0; double value; for (int i = 0; i < nreps; i++) { value = ComputeValue(r0, t, dt); sum += value; } return sum / nreps; }

I use the following parameters:

            double sigma = 0.03;
            double r0 = 0.03;
            double theta = 0.1;
            double a = 0.3;

            int nreps = 1000;
            double t = 1;

For dt = 0.1:
Exact expectation: 0,108618473059879;
Monte Carlo Expectation: 0,0101464832161612

For dt = 1:
Exact expectation: 0,108618473059879;
Monte Carlo Expectation: 0,092058844704742

using dt = 1 leads to a result close to exact value while using dt = 0.1 seems to lead to a result having a 0.1 factor difference with exact one.

I think I am doing something wrong but I can't figure it out. Do you have an idea?

Answer 1

I got help from LutzL and figured out that I forgot to add the r0 term

ComputeNextValue(double r0, double dt) 
{ 
RandomVariableGenerator rvg = RandomVariableGenerator.GetInstance();

double randomGaussian = rvg.GetNextRandomGaussian();
double r_t_dt = r0 + (_theta - _a*r0)*dt + _sigma * Math.Sqrt(dt) * randomGaussian;

return r_t_dt;
}

It solved the problem and the values are now closer