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What is the mathematical basis to say that $u^{2}_{t}/\sigma_{t}^{2}$ will exhibit little auto-correlation in the series? Let's $r_{t}$ be a series of returns and let's assume (Assumption I) it follows a covariance stationary process defined as : $r_{t}=\sigma_{t} z_{t}$ where $z_{t}$ is i.i.d with $E_{t}(z_{t})=0$ and $Var_{t}(z_{t})=1$ ; Then $... 0 By estimating the model parameters using monthly data, you will get monthly estimates. Thus, you will need to multiply them by$12$(or by$\sqrt{12}$for the volatility) in order to get annual estimates. Once you have done this, you can simulate the model using the timestep$\Delta t=\frac{1}{252}\$.

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You need to use more dimensions. If the number of dimensions (i.e. steps) is large, you may also have to use a Brownian bridge as described in the book by Joshi or Jäckel.

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First let me say that in the Black-Scholes model as you have it, there is of course no need for intermediate steps when pricing vanilla calls, since the SDE has the closed-form solution you included. Intermediate steps would be required for complicated payoffs or other SDEs. To answer your question though, you do need to use additional dimensions. Think ...

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