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I simulated sample paths to approximate the price of a vanilla European call and then plotted a graph comparing this to the value achieved from the Black Scholes. Why do these values diverge as the option volatility increases?

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  • $\begingroup$ It could be the fact you're doing relatively fewer number of simulations, than what you should. Maybe trying increasing the number of simulations by a factor of 100 or maybe 1000. If the issue still persist, I would be interested to seeing some code to figure out what maybe the issue $\endgroup$ – user23564 Mar 18 at 19:53
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    $\begingroup$ How does MC work: it takes the average of a large number N of $(S_T-K)^{+}$ values, the more these values are spread out the more difficult it is to estimate their mean from N observations. $\endgroup$ – noob2 Mar 18 at 20:16
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    $\begingroup$ In addition to what noob2 says, depending on the implementation of the montecarlo, you can also start to introduce biases when vol gets too high - for example if you're operating in spot space rather than log space, then numerically paths can round to zero, which is an absorbing barrier - this will drag down the forward and introduce a bias. $\endgroup$ – will Mar 19 at 21:34
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Each path is evolved based on the vol and a random number. The higher the vol the more the paths will diverge.

Paths will diverge if you increase time as well.

The solution is to increase the number of paths as vol or time increases to get a standard deviation of terminal values that you are comfortable with.

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