Brownian Motion Basics

A stochastic process with value $Z(t)$ is a pure / standard form of Brownian motion

Characteristics:

  1. $Z(0) = 0$
  2. $Z(t+s) – Z(t) \sim N(0,s)$ *s – variance
  3. $Z(t) – Z(t-s)$ is independent of $Z(t+h) – Z(t)$
  4. $Z(t)$ is continuous

Martingale:

  1. $Z(t)$ is a martingale. => E(Change) = 0
  2. $E[Z(t+s) – Z(t)] = 0$
  3. $E[Z(t+s)/Z(t)] = Z(t)$

Properties:

  1. Quadratic variation = T
  2. Cubic or higher order variation = 0
  3. Total variation is infinite

Arithmetic Brownian Motion

Differential Form

$dX(t) = a dt + b dZ(t)$

  1. a -> Drift (constant)
  2. 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]$

$m = ln(S(0))+(\alpha -\delta-0.5\sigma^2)t$ |  $v^2 = \sigma^2 t$

=> $P(S(t) > C) = P(ln(S(t)) > lnC) = P(z > \frac{lnC – m}{v})$

 


 

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