Stock Prices as Geometric Brownian Motion Equation Formula Latex Code and Summary

Stock Prices as Geometric Brownian Motion

Tags: #Financial #Economics

Equation

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

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

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

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)

Explanation

Latex code for Stock Prices as Geometric Brownian Motion.

$S(t)$ : Observed stock price S(t) at time stamp t
$U$ : Any normal random variable
$\mu$ : Drift coefficient
$\sigma$ : Volatility

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Theresa Rivera

Fingers crossed for passing this exam!

2024-03-11 00:00
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Joshua Moore reply to Theresa Rivera

You can make it...

2024-04-09 00:00:00.0
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Barbara Garcia

I'm determined to get a pass on this test.

2023-05-30 00:00
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Doris Gonzalez reply to Barbara Garcia

Nice~

2023-06-03 00:00:00.0
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Kathy Bailey

The only outcome I'll accept is a pass on this test.

2023-04-26 00:00
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Douglas Richardson reply to Kathy Bailey

Best Wishes.

2023-05-10 00:00:00.0
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