# Real Option Valuation using simulation: real world vs risk neutral measure

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?

• Just like financial options, you often value real options under the risk-neutral measure - to avoid estiming means, real world probabilities and stochastic discount factors (see this answer). One problem with real options: markets may be incomplete and you can't easily find one (unique) risk-neutral measure. That's why some real options are valued using the HJB equation. That avoids the completeness assumption but requires a subjective discount rate (e.g. CAPM) – Kevin Jan 8 at 9:08
• @Kevin Ok there is a lot of new things I'm not familiar with in your answer. I guess in my examle the market is incomplete, since my revenues, costs or the software itself is NOT traded, correct? However, I am not really able to connect your answer to my question (probably due to my lack of understanding). I don't see a way to incorporate the HJB euqation in my simulation so that still leaves me with my initial question if I would be wrong to just compute the drift of my GBMs from historical data keeping in mind in the end I am calculating the option price. – Arely Jan 8 at 10:39