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Classic asset price model in the continuous-time limit using a Wiener process notation can be written as $$ dS_t=\mu S_tdt+\sigma S_t dX $$ where $S_t$ is the stock price (not the stock return) and $dX$ is an independent random variable with normal distribution. If we eliminate the drift ($\mu = 0$) and only focus on randomness as asked in your question we ...


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For the first question, it is the standard assumption to make for stock returns if no other information is given. That's not to say it's a great assumption, but there it is clearly the only one that can be justified in this context. For the second part, independence of returns tells you that investment for T years has cumulative variance $T \sigma^2$ (when ...


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