I'm pricing some derivative securities with Monte Carlo Simulation.
I don't know population follows which distribution. But When I some calculation with my samples(all the each path) ,for example $X = ${$X_i | 1≤i≤10,000$ and $i ⊂ N$}, it seems follows normal distribution. I mean $X$~$N(μ,σ^2)$
I know that we can apply Law of Large Numbers to my sample path. So if the population have mean $a$ and variance $b$ then $X$ have mean $a$ and variance $b/n^{\frac{1}{2}}$ (Assume that number of the sample is $n$)
However it doesn't guarantee that the sample paths follows 'normal 'distribution.
But just seeing the result of my sample path it seems to follow normal distribution. What makes me confusing is that in my monte carlo simulation the number of sample is just 1 by each sample path!
I mean I don't calculate every new 10,000 paths with 20,000 times. I just get 10,000 paths then averaging once. Then the average value is the price of my derivative security.
I know that CLT gets the power when the $n$ is goes to ∞. Considering just weak version, at least $n > 30$
But I have 10,000 sample paths with each of it constructed with just one! I mean n is 1!
In Short my question is : Does every result sample path from monte carlo simulation follow normal distribution regardless of the number of the sample $n$?