What are the best practices for estimating volatility in real options valuation?

Under the ("big") assumption of the Black-Scholes framework, the underlying follows a GBM (Geometric Brownian Motion), dS(t) = mu S(t) + sigma S(t) dW(t). A property of W(t) is that W(t) - W(s) = W(t-s) ~ N(0, t-s) for t > s. Roughly speaking, we care about where we are and want to go (in probability - distribution law), so what happen before (historical) does not matter. Nonetheless, historical analysis might lead to helpful insights about the volatility behavior.

When estimating volatility, I always start by analyzing historical data. For instance, when I managed a recent investment project, I looked at the past performance of similar ventures over the last five years. I collected data and computed standard deviations to understand potential variability in returns. This gave me a solid foundation, but I didn't stop there. I also considered current market trends and potential future disruptions. Combining historical analysis with a forward-looking perspective helped me make more informed decisions. Remember, while past behavior offers insights, always factor in future uncertainties for a comprehensive volatility assessment.

While I have solid quantitative data, I’ve found that expert opinions often reveal critical insights that data alone can't capture. In previous projects, leveraging your insights on market trends and technological developments has significantly enhanced our models. Could we discuss potential future shifts and how they might influence our volatility estimates? Maintaining objectivity in your input will be key. Thank you for your time and guidance!

Under the assumption that the market-quoted options' prices were obtained using some model, e.g., the Black-Scholes formula V(S,t) = S(t) N(d1) - e^(-r(T-t)) K N(d2) the unknown parameter in this formula is the volatility (sigma). Hence, we can try to find the volatility number (using some root-finding method) such that the price calculated using the formula and the market-quoted are as close as possible. This volatility is called the implied (model) volatility and described by Rebonato as [1, p. 169]: "(...) the wrong number to put in the wrong formula to get the right price (...)." [1] R. Rebonato. Volatility and Correlation: The Perfect Hedger and the Fox. Wiley, 2004.

Implied volatility has significantly enhanced my approach to investment decisions. By utilizing it, I've been able to gauge market expectations more accurately, leading to more informed and strategic choices. For instance, using the Black-Scholes formula, I've been able to identify undervalued options, resulting in substantial returns. This forward-looking measure, unlike historical data, offers a real-time snapshot of market sentiment, which has been instrumental in my investment success. Embracing implied volatility has not only increased my confidence but also my profitability in trading, demonstrating its invaluable role in modern finance.

Monte Carlo simulation has been transformative in my work on real options valuation, allowing me to better estimate volatility and manage risk. By running numerous simulations, I've been able to create comprehensive probability distributions of potential returns. This technique's ability to incorporate various scenarios and assumptions has provided me with a more nuanced understanding of uncertainty. Personally, it's improved my decision-making process and increased my confidence in presenting data-driven insights to stakeholders. The flexibility and depth of Monte Carlo simulation have been invaluable in navigating complex financial landscapes and ensuring robust strategic planning.

Scenario analysis has been incredibly beneficial in my professional journey. By creating various future scenarios—best-case, worst-case, and most likely case—I have gained a comprehensive understanding of potential volatilities. This approach was particularly useful when I managed complex real options affected by multiple factors. For instance, during a project evaluation, scenario analysis helped me anticipate challenges and prepare strategic responses, ensuring successful outcomes. This method not only enhances decision-making but also