Note on Brownian motion

04 Feb 2019

I just want to summarize a few results that I find useful when dealing with arithmetic and geometric Brownian motions. For simplicity, I looked at unitary time steps. If it’s not the case (e.g., daily rate of change, looked hourly), then some minor adjustment needs to be done.

Arithmetic Brownian motion

The SDE for the ABM is given by \(dI_t = \mu I_0 d_t + \sigma I_0 dW_t\)

The solution is then \(I_d = I_0 (1 + \mu d + \sigma W_d)\) where $W_d$ is a Brownian motion. This means the following moments for the quantity $I_d$,

\[\begin{aligned} \mathbb{E}[I_d] & = I_0 (1 + \mu d) \\ \text{Var}[I_d] & = \sigma^2 I_0^2 d \\ \text{Cov}[I_d, I_{d'}] & = \sigma^2 I_0^2 \min(d,d') \\ \end{aligned}\]

Geometric Brownian motion

You can find some info from Wikipedia, here and here. The SDE for GBM is \(dI_t = \mu I_t d_t + \sigma I_t dW_t\). Then the solution is \(I_d = I_0 \exp \left( (\mu - \sigma^2/2)d + \sigma W_d \right)\). This means the following moments for the quantity $I_d$,

\[\begin{aligned} \mathbb{E}[I_d] & = I_0 e^{\mu d} \\ \text{Var}[I_d] & = I_0^2 e^{2\mu d}(e^{\sigma^2 d} - 1) \\ \text{Cov}[I_d, I_{d'}] & = I_0^2 e^{\mu(d+d')} \left( e^{\sigma^2 \min(d,d')} -1 \right) \end{aligned}\]

For the covariance, you can derive it as

\[\begin{aligned} \text{Cov}[I_d, I_{d'}] & = \mathbb{E}[I_d I_{d'}] - \mathbb{E}[I_d] \mathbb{E}[I_{d'}] \\ & = I_0^2 e^{(\mu-\sigma^2/2)(d+d')} \mathbb{E}[e^{\sigma(W_d + W_{d'})}] - I_0^2 e^{\mu(d+d')} \end{aligned}\]

And the last part is equal to $\mathbb{E}[e^Y]$ where $Y$ is log-normal with parameters $\log Y \sim \mathcal{N}(0, \sigma^2(d+d’+2\min(d,d’)))$, such that $\mathbb{E}[e^Y] = \exp[\sigma^2/2(d+d’+2\min(d,d’)]$ and the result follows.

It is interesting to convert the variables $\mu, \sigma$ into the empirical moments of the quantity $I_d$. This is

\[\begin{aligned} \mu & = \frac1d \log \frac{\mathbb{E}[I_d]}{I_0} \\ \sigma^2 & = \frac1d \log \left( 1 + \frac{\text{Var}[I_d]}{I_0^2 e^{2 \mu d}} \right) \end{aligned}\] [ statistics  brownianmotion  ]