# European call down and out option (geometric Brownian motion, Monte Carlo, Euler)

I need to estimate the expected value and the Greeks of an European call down and out option, assuming geometrical Brownian motion of the asset, with Monte Carlo simulation employing Euler discretization scheme. Given are the strike K,S0, Barier(B), volatility (v), risk free rate(r) and time to end (T). I can program the code, but I cant find the mathematics behind the calculations. Can someone show them to me? Thank you in advance.

• While I understand your concern, you're asking essentially lots of topics, such as risk neutral, central-limit-theorem, option sensitivity, discretisation etc. This is a bit too broad. Jul 26 '15 at 13:04
• I recommend Paul Glasserman's "Monte Carlo Methods in Financial Engineering", it's a good read for your question. Jul 26 '15 at 13:05
• If it would be of any help, closed form solution has an R code in RQuantLib package. You can use it to gauge the performance of MC and for any mistake in your coding process. cran.rstudio.com/web/packages/RQuantLib/index.html Oct 26 '15 at 10:38

first, there is a formula for the continuously monitored case.

second, if you use log coordinates the Euler discretization is exact so this should be done.

third, the convergence for discretely monitored to continuously is actually very slow so you will need a lot of steps.

fourth, it's actually better to draw the hitting time to the barrier rather than stepping naively.

fifth, as has already been noted this is a big question so hard to answer well.

sixth, see my paper http://ssrn.com/abstract=1441142 for further discussion, also Glasserman's book is good.

I know this is almost a month old, but...

Unless it is a homework assignment, you could have a look at this paper by Del Moral and Shevchenko, which gives a different estimator than the one you're probably using. It gives both a crude Monte Carlo-estimator and a sequential Monte Carlo-estimator. You probably just want the crude Monte Carlo one. Eq. (27) follows from Eq. (8) and standard MC arguments, as can be found in Glasserman's book. Eq. (8) follows from the tower property, as far as I recall.