Pablo Miranda
Pableins | ME

Pableins | ME

Wiener Process

From June 22, 2020

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Pablo Miranda

Published on Jul 28, 2021

2 min read

In a Wiener process w(t):

  1. If the time intervals [s, t] and [s', t'] do not intersect, then the increments w(t) - w(s) and w(t') - w(s') are independent random variables.
  2. As the increments w(t) - w(s) are normal random variables with E{w(t) - w(s)} = 0 and Var{w(t) - w(s)} = t - s

This way we have that w(ti+1) - w(ti) ~ N(0, ti+1 - ti) for the simulation.

Simulation

Given a normal random variable w(ti+1) = w(ti) + √(ti+1 - ti) due time instants ti is posible to pick them as we see fit, as the moment in which two instants are -sufficiently short- to match to other process simulated on the system.

Brownian Movement

Is given by the following equation:

wiener_0.png

For the simulation we will use

wiener_1.png

Geometric Brownian Movement

Is given by the following stochastic differential equation

wiener_2.png

Its solution can be written in the following way

wiener_3.png

Result

wiener_4.png

wiener_5.png

wiener_6.png

Implementation

Random random = new Random();
const int miu = 1;
const double k = 0.1; //0.02

int days = (int) inputDays.Value;
double price = (double) inputPrice.Value;

chart1.Series[0].Points.AddXY(0, price);

for (int i = 1; i <= days; i++)
{
    //price = price * (1 + k * (random.NextDouble() - 0.5));
    // e^((miu - (price*price/2)) * k + price * random)
    double w = miu + Math.Sqrt(k) * (random.NextDouble() - 0.5);
    price = Math.Exp((miu - (price*price/2)) * k + (price * w));
    chart1.Series[0].Points.AddXY(i, price);
}

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