Choosing Real-World or Risk-Neutral Measures for Real Option Valuation

Summary

The document poses a valuation problem for a software investment whose daily revenues depend on an exchange rate and whose costs depend on a commodity price. The proposed simulation models both underlying prices as geometric Brownian motions estimated from historical data, sums profits over a year, and treats the positive payoff as an option value. Its central question is whether historical real-world drifts or risk-neutral drifts should be used when valuing the investment.

The material frames a key distinction between forecasting expected outcomes under the real-world measure and pricing contingent claims under a risk-neutral measure. It contains the question but no answer, so it does not specify how to calibrate either process, discount the payoff, or model dependence between exchange rates and commodity prices. Those omissions matter: the stated payoff alone is not a complete valuation method, and historical drift estimates do not automatically determine a market-consistent option price.

Key ideas

• The example links software revenues to an exchange rate and operating costs to a commodity price.
• The proposed simulation models the underlying prices with geometric Brownian motions fitted to historical data.
• A real-world measure is typically used to describe expected real outcomes, while risk-neutral valuation is used for market pricing.
• The document raises the choice of measure but does not resolve it or provide a complete pricing procedure.
• Dependence between the simulated drivers and the treatment of discounting are important unresolved modeling details.

Tags

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


# 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?

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