# Vanilla European options: Monte carlo vs BS formula

I have implemented a monte carlo simulation for a plain vanilla European Option and I am trying to compare it to the analytical result obtained from the BS formula. Assuming my monte carlo pricer is correctly implemented, am I supposed to get the very same result with both methods (Monte Carlo and BS analytical formula)?

• with large amount of simulated paths (>200 000) - yes, you should have very close match. – Vytautas Jun 18 '11 at 17:19
• @Vytautas. Thanks. What do you mean by close match? Up to how many decimals for instance? – balteo Jun 18 '11 at 17:25
• You shouldn't be able to reject that the two solutions are different. – Richard Herron Jun 20 '11 at 3:07
• The number of iterations is just a function of what size standard errors with which you are comfortable. – Richard Herron Jun 20 '11 at 3:08
• Lets than 1 cent is considered good enough – Vytautas Jun 21 '11 at 15:34

Fundamentally this is no different from other simulation-based estimation---see this little experiment in R:

R> set.seed(42)
R> rowMeans(replicate(200,sapply(1:6,
+>          FUN=function(x) mean(rnorm(10^x)), simplify=TRUE)))
 -2.47827e-02 -9.46800e-03  2.38226e-03 -1.08650e-03  9.41395e-05  1.06759e-05
R>


We are calculating the mean of a $N(0,1)$ vector for sample sizes from $10^1$ to $10^6$. That is then repeated 200 times, and we are calculating the mean of the 200 draws at the different sample sizes.

We find that by and large, the mean gets closer to zero. But even at $10^6$, repeated 200 times, we are still pretty far from 'zero'. That is the way it goes with simulation, and it pays to get a feel for this.

So while you have perfect benchmark with your analytical Black-Scholes result, you will be hard-pressed to get the difference to vanish completely.

• Thanks! Just one last doubt remains: up to how many decimals do I need to get the difference to vanish? – balteo Jun 19 '11 at 12:35

As long as your simulations are independents, you can calculate some confidence interval thank's to the Central Limit Theorem and see if this interval is encompassing the true BS price.

Regards