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You have the right idea, but it seems you don't know $\mu$, so using it in your error check doesn't seem correct. Also, checking the result every 10,000 iterations may not be optimal for deciding when to stop. To be clear, let $E(X) = \mu$ and $Var(X) = \sigma$. We're invoking the CLT when we write $$ P\left( \left|\frac{\bar{X}_n - ...


Yes, that's an excellent approach. The only time it might go wrong is if, say, you are integrating on some extreme tail event without using importance sampling. For example, let's say you were simulating expected loss on a portfolio of five bonds issued by the USA, Germany, Norway, Sweden and the Netherlands. After 10,000 simulations, there's a chance you ...


I wouldn't repeat the same algorithm on Excel, because if you make a mistake in your Python code, it's likely that you'll also make the same mistake in your Excel code. Quants usually test an implementation with an analytical formula (not always possible). You should start off with something easy by pricing an European option with your MC algorithm. You ...


there has been a huge amount of work on this. In terms of numerical studies, see my paper Beveridge, Christopher and Joshi, Mark S. and Tang, Robert, Practical Policy Iteration: Generic Methods for Obtaining Rapid and Tight Bounds for Bermudan Exotic Derivatives Using Monte Carlo Simulation (January 23, 2009). Available at SSRN: ...


for the L2 norm you have to take a square root and then you get $1/\sqrt{n}$ convergence as before.


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 J├Ąckel.


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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