Suppose one call option contract with strike = 200 and expiration date 4/21/2017,
and we are given stock price = 180, volatility = 20%, and interest rate = 3%.
Under BS, we can compute call price using closed form solution.
But, here I am talking about simulation (Monte Carlo or other simulation methods).
So, by doing Monte Carlo simulation, we get the call price, say C0.

Based on the above information, how can I get prices for the same call option contract for the next day, the next next day, .....
In other words, I am trying to get time series call prices: C0, C1, C2, C3, ..... Cn.

What happens to the implied volatility during simulation? Will if stay at 20%. If so then you can simply compute the option price during every step of the simulation, adjust the underlying and time to expiration after each step.

What happens to the implied volatility during simulation? Will if stay at 20%. If so then you can simply compute the option price during every step of the simulation,  adjust the underlying and time to expiration after each step.

Right now, volatility is supposed to be constant, i.e., stay at 20% always.
How to adjust the underlying when move to next step? 
If it is simulated using 1000 paths, then at each time step, there are 1000 underlying prices, do we take the average to adjust the underlying?
Would you please explain a little bit more or refer to some papers (books)?
Thanks a lot!
 

Do you know how to simulate the underlying stock?

I am curious how you can do Monte Carlo simulation of an option without simulating the underlying?

I am curious how you can do Monte Carlo simulation of an option without simulating the underlying?

I would never consider that, it would be needless complicated.

I am curious how you can do Monte Carlo simulation of an option without simulating the underlying?

I would never consider that, it would be needless complicated.

Maybe I am not on the same page with you guys.
In my mind, given initial stock  price S0, I simulate 1000 paths for the stock price, time step = 1 day (to make it simple). 
Then, at each time step t (t = 1, 2, 3, ... ), I have 1000 stock prices, one price for one path: S(t, 1), S(t, 2), S(t, 3), S(t, 4), ......, S(t, 1000). 
So, to get the call price for next day, I have 1000 stock prices. They are S(1, 1), S(1, 2), S(1, 3), S(1, 4), ......, S(1, 1000).
From here, I am trying to figure out what underlying price (S1) I should use from the 1000 stock prices to do simulation again to get the call price.
Should I take the average, i.e., S1 = mean{S(1, 1), S(1, 2), S(1, 3), S(1, 4), ......, S(1, 1000)}?
If so, I only need to do the simulation once to get the call price for next day.
Another way in my mind is:
From each point on the next day: S(1, 1), S(1, 2), S(1, 3), S(1, 4), ......, S(1, 1000), do simulation to get call price C(1, 1), C(1, 2), C(1, 3), C(1, 4), ......, C(1, 1000). In other words,
From S(1,1) do simulation to get call price C(1,1)
From S(1,2) do simulation to get call price C(1,2)
From S(1,3) do simulation to get call price C(1,3)
:
:
:
From S(1,1000) do simulation to get call price C(1,1000).
So, in total, I need to do 1000 simulations on the next day.
Then, the call price for next day C1 = mean{ C(1, 1), C(1, 2), C(1, 3), C(1, 4), ......, C(1, 1000)}.
Please let me know what I am mis-understanding here! I am here ask for help!
Please be nice, do not laugh at me!