Stochastic calculus and brownian motion

Black Scholes Stock follows Geometric Brownian motion

Geometric Brownian motion -

\[ S_t = S_0 e^{\alpha t} \] Share price S grows exponentially over time, is just a constant

Adding “Noise” to this exponential curve. This Noise is brownian motion \[ S_t = S_0 e^{\alpha t + \beta B_{t}} \] \[\beta \] is a constant that is difficult to find.

\[ \frac{S_t}{S_0} = e^{\alpha t + \beta B_{T}} \]

Calculus requires smooth

Stochastic calculus has jagged randomness

S_t = S_0 e^{rt}

= rS_0 e^{rt} = rS_{t}

Share price \[ \frac{S_t}{S_0} = e^{\alpha t + \beta B_{T}} \] that \(\beta B_{T}\) adds the “noise” to the exponential curve. \[\frac{dS}{dt} = [r + \]