# Financial Simulations

Some notes on time series simulations.

# Continuous Time Series

## Wiener Process

A wiener process or a Brownian random walk is a continuous-ish time series with the following properties

• Gaussian increments: for $t>s$, $W(t)-W(s)$ is normally distributed with zero mean and a variance of $t-s$. In notation, $W(t)-W(s)\sim\mathcal N (0,t-s)$. Essentially, the increments of a Wiener process is a "white noise" process.
• Memoryless increments: for $t>s$, $W(t)-W(s)$ is independent of $W(s)$.
• $W(0) = 0$

Simulation of the Wiener process is straightforward. Just simulate the small increments and accumulate.

Let $Z_{1},Z_{2},\ldots$ be a sequence of independent random variables with the standard normal distribution. And let $\Delta t$ be a "small" increment of time. Then define

$W_{n} = \sum_{i=1}^n Z_{i} \sqrt{\Delta t}$

amd $W_0 =0$. Then $W_{n}$ would be the simulated value of the Wiener process $W(t)$ at time $t=n \Delta t$.

Simulation

## Geometric Brownian Motion

Geometric brownian motion, also known as the lognormal random walk, is a simple, but relatively realistic, model of prices. It consists of two parameters:

• Drift ($\mu$). The tendency for value to go up ($\mu > 0$) or down ($\mu < 0$). If the drift is zero, then the time series is a martingle.
• Volatility ($\sigma$): The degree to which changes in prices are due to random fluctuations.

Geometric brownian motion can be put in terms of a stochastic differential equation:

$d X(t) = X(t) \left( \mu dt + \sigma dW(t) \right)$

One method of simulating this SDE is to use the simulated values $W_n$ of the Wiener process $W(t)$ to obtain $X_n$, the simulated values of $X(t)$:

$X_{n+1} = X_{n} + X_{n} \left( \mu\Delta t + \sigma (W_{n+1}-W_n) \right)$

Simulation

## Mean-Reversion

\begin{aligned} dX(t) = \upsilon (\bar X - X(t)) dt + \sigma X(t) dW(t) \end{aligned}

## Correlated Geometric Brownian Motions

\begin{aligned} dX(t) &= X(t) (\mu_X dt + \sigma_X dW_{X} (t)) \\ dY(t) &= Y(t) (\mu_{Y} dt + \sigma_{Y} dW_{Y} (t)) \\ W_{X}(t) &= W_{1}(t) \\ W_{Y}(t) &= \rho W_{1}(t) + \sqrt{1-\rho^2} W_{2}(t) \\ W_1 &\perp W_2 \end{aligned}

## Stochastic Volatility

\begin{aligned} dX(t) &= X(t) (\mu dt + \sigma (t) dW_{X} (t)) \\ d\sigma (t) &= \Theta (\Omega - \sigma(t)) dt + \Xi \sigma(t) dW_{\sigma}(t) \\ W_{X} &\perp W_{\sigma} \end{aligned}

Created 2020-04-08. Last updated 2020-06-22. View source.