MonteCarlo Simulations
Tags: #Financial #EconomicsEquation
$$S(T) = S(0) e^{(a  \delta  \frac{\sigma^2}{2})T + \sigma \sqrt{T} z} \\ S(T) = S(t) e^{(a  \delta  \frac{\sigma^2}{2})(Tt) + \sigma (Z(T)  Z(t))} \\ \text{Variance} \\ e^{2rT} \times \frac{s^{2}}{n} \\ s^{2} = \frac{1}{n1} \sum [(g(S_{i})  \bar{g})]^{2}$$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})(Tt) + \sigma (Z(T)  Z(t))} \\ \text{Variance} \\ e^{2rT} \times \frac{s^{2}}{n} \\ s^{2} = \frac{1}{n1} \sum [(g(S_{i})  \bar{g})]^{2}
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Introduction
Equation
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})(Tt) + \sigma (Z(T)  Z(t))} \\ \text{Variance} \\ e^{2rT} \times \frac{s^{2}}{n} \\ s^{2} = \frac{1}{n1} \sum [(g(S_{i})  \bar{g})]^{2}
Explanation
Latex code for the MonteCarlo Simulations of stock prices. I will briefly introduce the notations in this formulation. MonteCarlo 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 MonteCarlo 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 ith simulated payoff
 : The variance of stock price
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