Option Value: Samuelson-McKean approach

Vacant land is a call option: the owner has the right, but not the obligation, to pay the construction cost K and receive a built property worth V. Because the owner can build at any time, this is an American option without expiry, and it can be priced with the classic formula for perpetual American warrants by Paul Samuelson and Henry McKean (Samuelson, 1965, with McKean’s mathematical appendix).

Move the sliders to see how the land value (blue) changes, and when it becomes optimal to develop (the dashed line). Tick “Compare σ_V” to add a second land value curve with a different volatility (green).

The formula

With y_V the cash yield of the built property (its cap rate), σ_V the volatility of property values, r_f the risk-free rate and g_K the real growth rate of construction costs:

• “Construction yield”: y_K = r_f − g_K
• Option elasticity: η = [ y_V − y_K + σ_V²/2 + √( (y_K − y_V − σ_V²/2)² + 2 y_K σ_V² ) ] / σ_V²
• Optimal development trigger (hurdle value): V* = K · η / (η − 1)
• Land value: C₀ = (V* − K) (V₀ / V*)^η while V₀ < V*, and V₀ − K (build now!) above the trigger.

Things to try

• Volatility. More uncertainty makes the option to wait more valuable: land value rises, and the development trigger V* moves to the right. Developers wait for a larger cushion before they build.
• Asset yield. A higher yield on the built property means you forgo more income by waiting, so it pays to build earlier.
• Construction cost growth. If costs are expected to rise quickly, waiting becomes expensive too.
• Elasticity. Below the trigger, a 1% change in property values moves land values by η%. Land is a leveraged bet on the property market, and its values are much more volatile than those of built property.

The assumptions behind the formula (frictionless, efficient markets; property values following a random walk; known and constant volatility) are strong for real estate. Treat the numbers as a benchmark, not a price.

References

• Samuelson, P. A. (1965). Rational Theory of Warrant Pricing. Industrial Management Review, 6(2), 13–31. With McKean, H. P. Jr. (1965). Appendix: A Free Boundary Problem for the Heat Equation Arising from a Problem in Mathematical Economics. Industrial Management Review, 6(2), 32–39. Reprinted in Henry P. McKean Jr. Selecta, Springer 2015: doi:10.1007/978-3-319-22237-0_11.
• Geltner, D., Miller, N., Van de Minne, A., Eichholtz, P., Lindenthal, T. & Shen, J. Commercial Real Estate Analysis for Investment, Finance, and Development (4th ed., 2026), on real options and land valuation.