Stochastic calculus

Geometric Brownian motion

The solution behind Black–Scholes, and the reason returns are modelled as lognormal.

St=S0exp⁡ ⁣[(μ−σ22)t+σWt]S_t = S_0\exp\!\left[\left(\mu - \tfrac{\sigma^2}{2}\right)t + \sigma W_t\right]

Where

−σ2/2-\sigma^2/2
The Itô correction, which is why the median lags the mean.

Assumptions

  • Constant μ\mu and σ\sigma, continuous paths, no jumps.

Sanity check. E[St]=S0eμt\mathbb{E}[S_t] = S_0e^{\mu t} exactly, while the median is S0e(μ−σ2/2)tS_0e^{(\mu - \sigma^2/2)t}.

Where this is taught

Geometric Brownian motion · QuantMax