Geometric Brownian Motion
Tags: #Financial #EconomicsEquation
$$Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\mu t + \sigma Z(t)]} \\ E(e^{kU}) = e^{kE(U) + \frac{1}{2}k^{2}\text{Var}(U)} \\ E[Y^{k}(t)] = Y^{k}(0) e^{(k\mu + \frac{1}{2}k^{2}\sigma^{2})t} \\ \ln Y(t) \sim N(\ln Y(0) + \mu t, \sigma^{2} t)$$Latex Code
Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\mu t + \sigma Z(t)]} \\
E(e^{kU}) = e^{kE(U) + \frac{1}{2}k^{2}\text{Var}(U)} \\
E[Y^{k}(t)] = Y^{k}(0) e^{(k\mu + \frac{1}{2}k^{2}\sigma^{2})t} \\
\ln Y(t) \sim N(\ln Y(0) + \mu t, \sigma^{2} t)
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Introduction
Equation
Latex Code
Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\mu t + \sigma Z(t)]} \\
E(e^{kU}) = e^{kE(U) + \frac{1}{2}k^{2}\text{Var}(U)} \\
E[Y^{k}(t)] = Y^{k}(0) e^{(k\mu + \frac{1}{2}k^{2}\sigma^{2})t} \\
\ln Y(t) \sim N(\ln Y(0) + \mu t, \sigma^{2} t)
Explanation
Latex code for the Geometric Brownian Motion.
: Observed value Y(t) at time stamp t
: Any normal random variable
: Drift coefficient
: Volatility
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