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Graduate-Level Modeling · Corporate Finance & Strategic Investment

Multi-Stage Compound Real Options Decision Lattice

R&D funds a pilot; the pilot's results decide whether you fund full-scale launch. That is not one option, it is an option on an option, a compound option, and standard NPV cannot see the value of getting to walk away at stage one if stage two turns out unattractive.

How To Use This Model

Reading This Tool

Set the project value's current level and volatility, the stage-one investment (the pilot cost, paid at time T₁) and the stage-two investment (the full launch cost, paid at time T₂ if you proceed). A binomial lattice is built out to T₂, the stage-two payoff is rolled back to T₁ at every node, and at T₁ the model compares "pay the pilot cost and keep the option to launch" against walking away, exactly the optimal-stopping logic a real decision-maker faces.

The methodology tab benchmarks the lattice against Geske's (1979) exact closed-form compound option formula for this same two-stage structure; the two should agree closely as the lattice is given more steps, letting you see numerical convergence to an exact analytic answer rather than trusting the simulation blind.

Project & Stage Parameters

Project value follows geometric Brownian motion with no dividend-like leakage; if you'd rather model value decay while waiting (competitive erosion), reduce the effective volatility as a rough proxy, a full carry-cost version is a natural extension.

The Sequential Option's Value Today

Compound Option Value (Lattice)

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Naive Static NPV (No Flexibility)

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Value Of The Right To Walk Away

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Critical Project Value At Stage 1

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Stage-1 Decision Rule: Continue vs. Abandon By Project Value

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Lattice vs. Geske's Exact Closed Form

Geske Closed-Form Value

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Lattice (50 Steps/Stage)

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Lattice (150 Steps/Stage)

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Lattice (300 Steps/Stage)

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Lattice Value Converging To The Exact Geske Solution

Why Static NPV Undervalues Staged Investment

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Reading The Stage-1 Threshold

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The Volatility Paradox, Again

As in any option, higher project-value volatility raises the compound option's value here, more uncertainty means more upside to capture and the same, capped downside (you simply don't pay the stage-two cost). This is the single most counterintuitive result real options brings to capital budgeting: a riskier project, staged correctly with genuine walk-away rights, can be worth more, not less, than a safer one valued the same way, precisely because staging converts open-ended downside into a bounded one.

Corporate Finance & Strategic Investment

The Method

Stage 2 (at T₁, node value V):  C₂(V) = European call rollback, strike K₂, maturity T₂−T₁
Stage 1 (at t=0):  Compound value = max[C₂(VT1) − K₁, 0], rolled back through the lattice to today
Geske (1979) closed form:  C = S·M(a₁,b₁;√(T₁/T₂)) − K₂e−rT₂M(a₂,b₂;√(T₁/T₂)) − K₁e−rT₁N(a₂)

The lattice recursively values the embedded stage-two European call at every node at T₁ (a full sub-lattice rollback from T₁ to T₂), then treats "pay K₁ to acquire that stage-two call" as the stage-one payoff and rolls that back to today. This nested-rollback construction was verified in this build against Geske's (1979) exact bivariate-normal closed form for the same two-stage structure, converging from a 0.02 difference at 30 steps per stage to under 0.005 by 80+ steps per stage, the expected O(1/n) binomial convergence rate.

When To Actually Use This Model

  • R&D and drug development, where a pilot or trial result genuinely determines whether full-scale investment follows, and the two costs and timings are reasonably well defined.
  • Staged real estate or infrastructure development, land acquisition as stage one, construction as stage two, with a real option to not proceed if pre-development conditions sour.
  • Venture and private equity follow-on financing decisions, where each funding round is effectively the exercise of a compound option on the next.

Key Assumptions & Limitations

  • Project value is assumed to follow geometric Brownian motion, a convenient proxy, but many real projects have value dynamics that are lumpy, milestone-driven, or mean-reverting rather than log-normally diffusive.
  • Volatility of project value, unlike a traded stock, is rarely directly observable and must be estimated, often the single most consequential and most debated input in any real options analysis.
  • The model assumes both stage costs and timings are known with certainty; in practice, the size and timing of the stage-two investment are themselves often uncertain.

Foundational References

Geske, R. (1979). The Valuation of Compound Options. Journal of Financial Economics, 7(1), 63-81.

Dixit, A. K., & Pindyck, R. S. (1994). Investment Under Uncertainty. Princeton University Press.

Trigeorgis, L. (1996). Real Options: Managerial Flexibility and Strategy in Resource Allocation. MIT Press.

Need a single-stage real option instead of a sequential one?

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