## Monte-Carlo Simulations

Tags: #Financial #Economics### Equation

$$S(T) = S(0) e^{(a - \delta - \frac{\sigma^2}{2})T + \sigma \sqrt{T} z} \\ S(T) = S(t) e^{(a - \delta - \frac{\sigma^2}{2})(T-t) + \sigma (Z(T) - Z(t))} \\ \text{Variance} \\ e^{-2rT} \times \frac{s^{2}}{n} \\ s^{2} = \frac{1}{n-1} \sum [(g(S_{i}) - \bar{g})]^{2}$$### Latex Code

S(T) = S(0) e^{(a - \delta - \frac{\sigma^2}{2})T + \sigma \sqrt{T} z} \\ S(T) = S(t) e^{(a - \delta - \frac{\sigma^2}{2})(T-t) + \sigma (Z(T) - Z(t))} \\ \text{Variance} \\ e^{-2rT} \times \frac{s^{2}}{n} \\ s^{2} = \frac{1}{n-1} \sum [(g(S_{i}) - \bar{g})]^{2}

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### Introduction

#### Equation

#### Latex Code

#### Explanation

Latex code for the Monte-Carlo Simulations of stock prices. I will briefly introduce the notations in this formulation. Monte-Carlo simulation simulates stock prices, calculate the payoff the option for each of those simulated prices, find the average payoff, and then discount the average payoff. Firstly, we start with iid uniform numbers u_{1} to u_{n}, calculate standard normal variable z_{i} as , convert to normal variable . The variance of the Monte-Carlo estimate is calculated as .

- : The stock price at time T
- : The stock price at time t, which is nearer to final stage stock price S(T)
- : The i-th simulated payoff
- : The variance of stock price

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