Stock Prices as Geometric Brownian Motion
Tags: #Financial #EconomicsEquation
$$\frac{\mathrm{d}S(t)}{S(t)} = (a - \delta) \mathrm{d}t + \sigma \mathrm{d}Z(t) \\ S(t) = S(0) e^{(a - \delta - \frac{\sigma^{2}}{2})t + \sigma Z(t)} \\ \mathrm{d}[\ln S(t)] = (a - \delta - \frac{\sigma^{2}}{2}) \mathrm{d}t + \sigma \sigma \mathrm{d} Z(t) \\ S(t) \sim \ln( \ln S(0) + (a - \delta - \frac{\sigma^2}{2})t, \sigma^{2}t)$$Latex Code
\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
\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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