HelloI would like to simulate USDTRL exchange rate with GBM. Can I use observed implied volatilities?lets assume that today USDTRL 1.80 and 1 year implied volatility %10, drift is %8If confidence interval %90, does the expected value of the USDTRL 1.80 * (1+%8) my question is about the value that correspond to %90 confidence interval.Whether it would be close to[1.80 * (1+%8) ] + [ normsinv(0.90) * 0.10 ]or[1.80 * (1+%8) ] * [1+ normsinv(0.90) * 0.10 ]Thanks in advance

Hi,This probably belongs in the student forum. Yes the expected value will be the forward rate, which is what you wrote (with the reservation that the drift in the BGM diffusion is an instantaneous rate so you should really be integrating, which will end up being a continuously compounded rate, not a simple one).There are a few things one has to be careful about. One is that we are operating in the risk-neutral world, not in the actual world, and are dealing with probabilities in the risk-neutral measure (and the forward rate is the expected value in the risk-neutral measure). The other is that the implied volatility is "the wrong number in the wrong formula" and not necessarily a statistical measure as you interpret it. With this in mind, if you only have one ATM volatility, or if it's just for the sake of an exercise, then yes this is the general idea, but you would use the inverse of the ERF function, not normsinv: [1.80 * (1+%8) ] * [1 +/- x * 0.10 ], where x is the argument of erf(x/SQRT(2)) = 0.9, and +/- should be understood to indicate an interval around the forward rate.Hope this helps,Cheers.