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

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

Let Z1,Z2,Z_{1},Z_{2},\ldots be a sequence of independent random variables with the standard normal distribution. And let Δt\Delta t be a "small" increment of time. Then define

Wn=i=1nZiΔtW_{n} = \sum_{i=1}^n Z_{i} \sqrt{\Delta t}

amd W0=0W_0 =0. Then WnW_{n} would be the simulated value of the Wiener process W(t)W(t) at time t=nΔtt=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:

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

dX(t)=X(t)(μdt+σdW(t))d X(t) = X(t) \left( \mu dt + \sigma dW(t) \right)

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

Xn+1=Xn+Xn(μΔt+σ(Wn+1Wn)) X_{n+1} = X_{n} + X_{n} \left( \mu\Delta t + \sigma (W_{n+1}-W_n) \right)

Simulation

Mean-Reversion

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

Correlated Geometric Brownian Motions

dX(t)=X(t)(μXdt+σXdWX(t))dY(t)=Y(t)(μYdt+σYdWY(t))WX(t)=W1(t)WY(t)=ρW1(t)+1ρ2W2(t)W1W2\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

dX(t)=X(t)(μdt+σ(t)dWX(t))dσ(t)=Θ(Ωσ(t))dt+Ξσ(t)dWσ(t)WXWσ\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.