Hi,I am a happy owner of PWOQF.I am checking the spreadsheets and I see that when the brownian motion is explained (chapter 7), Paul uses the following formula (which I understand):P(t)=P(t-1)*(1+drift*time+vol*sqrt(time)*inv.random)But on the spreadsheet on montecarlo simulation, he uses:P(t)=P(t-1)*e^((riskfreerate - o.5*vol^2)+vol*sqrt(time)*inv.random)Well....I understand why the drift is changed by the risk free rate, I also see this is continuous interest rate and the first is compounding....but I dont see why o.5vol^2 is substracted from the riskfree interest rate. This is driving me crazy....!!! What would happen if I just use:P(t)=P(t-1)*e^((riskfreerate+vol*swrt(time)*invrandom)....THANKS A LOT!!!!!

The substraction comes from the fact that this dynamics is under the risk-neutral probability. We must have E(dS/S)=r*dt, and therefore, the drift term is modified by GirsanovHope it helps

The subtraction of o.5vol^2 comes from applying Ito's lemma to ln S. It has nothing to do with the dynamics being under the risk neutral measure. You get this term also in the real world measure. Start by assuming that the share price follows:dS = drift.S dt + vol.S dW(t)Now use Ito's lemma to find the process for d[ln S(t)] :d[ln S(t)] = (drift - 0.5.vol^2) dt + vol dW(t)Integrate to give:ln S(t) = ln(S(0)) + (drift - 0.5.vol^2) t + vol W(t)Or:S(t) = S(0) exp [(drift - 0.5.vol^2).t + vol. W(t)]

I have a side (stupid) question. In the Black-Scholes framework log returns follow a normal distribution and stock prices a lognormal one. dS/S the compounded returns are normally or lognormally distributed?

From looking at the equation dS/S = drift. dt + vol. dW(t) you'd expect dS/S to be normally distributed coz dW(t) is.Another side question: whats the difference of log returns (ln(S(t+1)/S(t)) and dS/S, in terms of use and interpretation?I'm thinking that dS/S is only relevant over an infinitesimally small time period whilst log returns can be used over larger time periods, and is a sort of average of the continuously compounded dS/S's over this period, but I don't really know.Hope this makes sense!