I am trying to value a real option in the form of a software investment using a simulation. The software investment yields to daily revenues $R_t$ and costs $C_t$. Here are the formulas for these:
$$R_t=a+E_t*b$$
$$C_t=c*O_t$$
$a, b$ and $c$ are constants, $E_t$ is a daily changing exchange rate and $O_t$ is a daily changing commodity price. I have the historical time series of both $E_t$ and $O_t$ available and I assume them to follow a Geometric Brownian Motion over time:
$$dE_t=\mu_EE_tdt+\sigma_EE_tdW_t$$
$$dO_t=\mu_OO_tdt+\sigma_OO_tdW_t$$
In my simulation I want to simulate the daily profits (revenues - costs) for 1 year. So I am using the closed form solution of the GBM to calculate the daily value of $E_t$ and $C_t$:
$$E_t=E_0exp\left(\left(\mu_E-\frac{\sigma_E^2}{2}\right)t+\sigma_E Wt\right)$$
My plan was to calculate the drift $\mu$ and the volatility $\sigma$ of both GBM using the historical data.
My option value $V$ would then be something like:
$$V=max\left(\sum_{t=1}^{365}R_t-C_t;0\right)$$
I know when I purely want to simulate e.g. a stock price it is fine to calculate the drift and volatility from historical data, however I also know that option pricing is done in a risk neutral measure, whereas $\mu$ is substituted by $r$ in the GBM equation. Since I want to price this option. I am really struggling to understand the difference between real world measure and risk neutral measure and especially what I should use in my case. Is the real world measure or the risk neutral measure the right one to use in my case?
