Brownian Motion Basics
A stochastic process with value $Z(t)$ is a pure / standard form of Brownian motion
Characteristics:
- $Z(0) = 0$
- $Z(t+s) – Z(t) \sim N(0,s)$ *s – variance
- $Z(t) – Z(t-s)$ is independent of $Z(t+h) – Z(t)$
- $Z(t)$ is continuous
Martingale:
- $Z(t)$ is a martingale. => E(Change) = 0
- $E[Z(t+s) – Z(t)] = 0$
- $E[Z(t+s)/Z(t)] = Z(t)$
Properties:
- Quadratic variation = T
- Cubic or higher order variation = 0
- Total variation is infinite
Arithmetic Brownian Motion
Differential Form
$dX(t) = a dt + b dZ(t)$
- a -> Drift (constant)
- b -> volatility
Non-Differential Form
$X(t) – X(0) = a T + b Z(T)$
Distribution of ABM
$X(T) – X(0) \sim N(at, b^2t)$
Example
$dY(t) = 0.3dt + 0.2 dZ(t)$ | $Y(0) =1$ | Determine distribution of Y(t)
a = 0.3 , b = 0.2 => $Y(t) – Y(0) \sim N(0.3t , 0.2^2t)$
=> $E[Y(t) – Y(0)] = 0.3t => E[Y(t)] = 0.3t + 1$
and $Var[Y(t)-Y(0)] = 0.2^2t$
=> $Var[Y(t)] + Var[Y(0)] = 0.2^2t => Var[Y(t)] = 0.2^2t$
=> $Y(t) \sim N(1+0.3t, 0.2^2t)$
Geometric Brownian Motion
Differential Form
$dX(t) = aX(t) dt + bX(t) dZ(t)$
Differential form of associated ABM
IF X(t) -> GBM
$d[lnX(t)] = (a – 0.5b^2)dt + b dZ(t)$
Non-differential form
$X(t) = X(0)e^{(a-0.5b^2)t +bZ(t)}$
Distribution
$ln[\frac{X(t)}{X(0)}] \sim N[m = (a-0.5b^2)t, v^2 = b^2t]$
Stock price following GBM
$dS(t) = (\alpha – \delta) S(t) dt + \sigma S(t) dZ(t)$
$d[lnS(t)] = (\alpha – \delta – 0.5\sigma ^2) dt + \sigma dZ(t)$
$S(t) = S(0) e^{(\alpha-\delta-0.5\sigma ^2)t + \sigma Z(t)}$
$S(t) = S(0) e^{(\alpha-\delta-0.5\sigma ^2)t + \sigma \sqrt tZ}$
=>$ ln[S(t)] \sim N[m, v^2]$
$m = ln(S(0))+(\alpha -\delta-0.5\sigma^2)t$ | $v^2 = \sigma^2 t$
=> $P(S(t) > C) = P(ln(S(t)) > lnC) = P(z > \frac{lnC – m}{v})$