Stock Prices as Geometric Brownian Motion

Tags: #Financial #Economics

Equation

dS(t)S(t)=(aδ)dt+σdZ(t)S(t)=S(0)e(aδσ22)t+σZ(t)d[lnS(t)]=(aδσ22)dt+σσdZ(t)S(t)ln(lnS(0)+(aδσ22)t,σ2t)

Latex Code

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                     \frac{\mathrm{d}S(t)}{S(t)} = (a - \delta) \mathrm{d}t + \sigma \mathrm{d}Z(t) \\
S(t) = S(0) e^{(a - \delta - \frac{\sigma^{2}}{2})t + \sigma Z(t)} \\
\mathrm{d}[\ln S(t)] = (a - \delta - \frac{\sigma^{2}}{2}) \mathrm{d}t + \sigma \sigma \mathrm{d} Z(t) \\
S(t) \sim \ln( \ln S(0) + (a - \delta - \frac{\sigma^2}{2})t, \sigma^{2}t)

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Introduction

Equation



Latex Code

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\frac{\mathrm{d}S(t)}{S(t)} = (a - \delta) \mathrm{d}t + \sigma \mathrm{d}Z(t) \\
S(t) = S(0) e^{(a - \delta - \frac{\sigma^{2}}{2})t + \sigma Z(t)} \\
\mathrm{d}[\ln S(t)] = (a - \delta - \frac{\sigma^{2}}{2}) \mathrm{d}t + \sigma \sigma \mathrm{d} Z(t) \\
S(t) \sim \ln( \ln S(0) + (a - \delta - \frac{\sigma^2}{2})t, \sigma^{2}t)

Explanation

Latex code for Stock Prices as Geometric Brownian Motion.

  • : Observed stock price S(t) at time stamp t
  • : Any normal random variable
  • : Drift coefficient
  • : Volatility

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