Valuing AI Startups with Real Options: Black-Scholes, CRR and Monte Carlo

Introduction: When Uncertainty Is a Source of Value

How do you value a company whose product does not yet exist, whose future revenues are unknown, and whose trajectory depends on strategic decisions that will only be made in the coming months or years? This question, which appears insoluble within traditional business valuation frameworks, is precisely the one faced by investors, venture capital funds and management teams when valuing artificial intelligence startups — whether operating in France, Switzerland, or internationally.

The discounted cash flow (DCF) model, a cornerstone of classical corporate finance, rests on a fundamental assumption: the ability to project future cash flows with a reasonable degree of reliability. For an AI startup at Seed or Series A stage, this assumption is structurally false. Cash flows are non-existent, addressable markets are poorly defined, technology is still under development, and future decisions depend on the evolution of a rapidly changing technological environment. Damodaran himself acknowledges that standard valuation methods simply do not work for young growth companies, or yield figures that are difficult to defend — a powerful analytical tool, but structurally ill-suited to the context of radical uncertainty inherent in AI startups. (Damodaran, Valuing Young, Start-Up and Growth Companies: Estimation Issues and Valuation Challenges, SSRN Working Paper, 12 June 2009, https://ssrn.com/abstract=1418687.)

Real options theory offers a response to this impasse. Born from the application of financial option pricing models to real investment decisions — by Black & Scholes (1973), Cox, Ross & Rubinstein (1979) and Boyle (1977) — it rests on a fundamental conceptual reversal: uncertainty is not an obstacle to valuation, it is a component of value. The higher the volatility, the more the option is worth. And an AI startup is, by definition, one of the most volatile assets in the private investment universe.

This article presents a complete methodology for applying real options to the valuation of AI startups, structured around three major themes. We will first demonstrate why traditional methods are insufficient and why the real options approach is particularly suited to startups — generally and for AI startups specifically. We will then detail the three principal methods — adapted Black-Scholes (BSM), Cox-Ross-Rubinstein binomial tree (CRR) and Monte Carlo simulation — with their complete formulas, parameters and conditions of application. The whole will be illustrated by two concrete numerical case studies: OptiVue (early-stage AI startup) and FlowCast (Series B scale-up), which are two startups for which Hectelion has been mandated, whose names have been modified to preserve confidentiality, applying all three methods to each profile successively.

Definition: What Is a Real Option?

A real option is the right — but not the obligation — to take an investment decision at a future date, under conditions determined today. It is called 'real' as opposed to financial options, which relate to financial assets (equities, currencies, commodities). A real option relates to a real asset: an investment project, an R&D programme, a technology under development, a startup.

The analogy with financial options is direct and powerful. A call option on a share gives the right to purchase that share at a fixed price (the exercise price K), at or before a given date (T), against payment of a premium. If the share price exceeds K before T, the option is exercised and generates a gain. Otherwise, it expires worthless — but the loss is limited to the premium paid.

In the context of an AI startup, the analogy translates directly:

Dixit & Pindyck (Princeton University Press, 1994) formalised this approach in their seminal work Investment Under Uncertainty, demonstrating that the total value of an investment is not reducible to its net present value (NPV), but incorporates an option value linked to managerial flexibility and the partial irreversibility of decisions:

Total Value = NPV (certain cash flows) + Option Value (flexibility + uncertainty)

This equation is the guiding thread of the entire article. It means that a startup may have a negative NPV — and thus appear unviable under DCF — while having a significant positive option value, linked to its expansion, abandonment or pivot options.

Origins and Theoretical Framework

Real options theory emerged at the intersection of quantitative finance and investment strategy. Its foundations rest on four major academic contributions, two of which were awarded the Nobel Prize in Economics.

Black & Scholes — Merton (1973)

In 1973, Fischer Black and Myron Scholes published their option pricing model in the Journal of Political Economy, establishing for the first time an analytical closed-form formula to calculate the theoretical value of a European option. Robert Merton, in the same year, extended the model to the case of assets paying a continuous dividend. Black & Scholes (1973) show that, under certain assumptions — geometric Brownian motion, absence of arbitrage, constant volatility — the value of a European call option has an exact analytical solution. Myron Scholes and Robert Merton received the Nobel Prize in Economics in 1997 for this contribution.

Cox, Ross & Rubinstein — CRR (1979)

Cox, Ross & Rubinstein published in 1979 in the Journal of Financial Economics the binomial model, a discrete approach to option pricing that does not require the continuity assumptions of BSM. By dividing the option's duration into successive periods during which the price of the underlying asset can rise or fall according to determined factors, the CRR model frames sequential decisions in a way particularly suited to real investment projects. For a number of periods tending to infinity, CRR converges to BSM.

Boyle (1977) — Monte Carlo

In 1977, Phelim Boyle published in the Journal of Financial Economics the first application of Monte Carlo simulation to option pricing. By generating a large number of random trajectories of the underlying asset price according to a geometric Brownian motion, then calculating the average discounted payoff across all these trajectories, Monte Carlo allows valuation of options whose complexity precludes an analytical solution — notably options with multiple uncertainty sources, particularly suited to AI startups.

Dixit & Pindyck (1994) and Trigeorgis (1996) — Application to Real Options

The systematic application of these models to real investment decisions was formalised by Dixit & Pindyck in Investment Under Uncertainty (Princeton University Press, 1994) and by Trigeorgis in Real Options: Managerial Flexibility and Strategy in Resource Allocation (MIT Press, 1996). These two founding works establish the theoretical framework and conditions for applying real options to industrial, technological and R&D projects.

The Insufficiency of Traditional Methods for AI Startups

A — The Structural Limitations of DCF for All Startups

DCF is the reference method for business valuation in a context of predictable and stable cash flows. For a startup, its limitations are structural, not circumstantial.

First — the absence of historical cash flows. DCF requires a historical dataset to calibrate growth and profitability assumptions. An AI startup at Seed stage typically has neither revenues nor usable operational history.

Second — excessive sensitivity to terminal value assumptions. In a DCF applied to a startup, the terminal value often represents 80 to 95% of total value. Yet this terminal value depends on long-term growth rate and normalised margin assumptions whose uncertainty for an AI startup is extreme.

Third — penalisation of uncertainty. In DCF, uncertainty is translated into an additional risk premium on the discount rate. The higher the risk, the higher the rate, the lower the value. This treatment is counter-intuitive for a high-potential startup: uncertainty about future cash flows results precisely from the exceptional gain potential — not solely from the risk of loss.

Fourth — ignorance of managerial flexibility. DCF implicitly assumes the project follows a single predefined path. It does not value the manager's ability to accelerate if results are good, reduce investment if signals are negative, or abandon the project if market conditions deteriorate. This flexibility has real value — and DCF ignores it entirely.

B — Why AI Startups Amplify These Limitations

1. Structurally extreme volatility. The implied volatility of an early-stage AI startup typically lies between 70% and 120%, versus 20–30% for a mature listed company and 40–60% for an established SaaS startup. In the BSM formula, option value grows with volatility (σ). Real options therefore capture AI startup value better than DCF, which penalises this same volatility through a high discount rate.

2. AI intellectual property as an option on future markets. A language model, a computer vision algorithm, a recommendation system — these assets do not necessarily generate immediate revenues, but represent real options on potentially very large future markets that are difficult to quantify today.

3. AI development milestones as decision nodes. An AI startup follows specific development stages — proof of concept (POC), minimum viable product (MVP), first pilot clients, industrialisation, scaling. Each milestone is a binary decision point in a CRR tree: continue investment if the milestone is met, abandon or pivot if not.

4. The abandonment option — a real insurance. The ability to stop an AI project if test results are disappointing has a real value, analogous to a put option. This value effectively reduces the investor's risk and justifies a higher valuation than DCF would suggest.

The Five Types of Real Options Applicable to AI Startups

The Five Fundamental Parameters Adapted to AI Startups

Estimating volatility for an AI startup — practical approach

Volatility is the central parameter of all three methods. For an unlisted AI startup, several approaches allow estimation. The first consists of retaining the annualised historical volatility of shares in listed comparable companies in the AI sector — for example, the volatility of NVIDIA (σ ≈ 50–75% over 3 years, depending on market period), C3.ai (σ ≈ 70–110% over 3 years, with peaks in 2022–2023) or a sector AI ETF. The second approach uses the dispersion of AI startup valuations observed in recent transactions (Pitchbook, CB Insights). The third approach, recommended by Damodaran (2005), consists of starting from the closest listed comparable's sector and adjusting upward for the startup's smaller size and earlier development stage.

Presentation of the Two Fictitious Case Studies

Black-Scholes Adapted to AI Startups (BSM)

Complete formula and parameters

C = S · N(d₁) − K · e^(−r·T) · N(d₂)

 d₁ = [ ln(S/K) + (r + σ²/2) · T ] / (σ · √T)

 d₂ = d₁ − σ · √T

Where N(·) denotes the cumulative distribution function of the standard normal distribution.

Advantages of BSM for startups

BSM offers three major advantages in the AI startup valuation context. First, it provides an analytical closed-form solution — a direct formula, without iteration or simulation — that allows instant calculation and complete transparency on the valuation mechanics. Each parameter has a precise, interpretable role, facilitating communication of results to investors and investment committees. Second, it constitutes the convergence reference for the two other methods: CRR converges to BSM for n → ∞, and Monte Carlo converges to BSM for N → ∞. Third, its sensitivity to σ is perfectly suited to high-potential AI startups: high volatility directly generates a higher option value, quantitatively capturing the venture capital logic — asymmetry between limited loss and potentially unlimited gain.

Application — Case 1: OptiVue (BSM)

Parameters: S = €/CHF 4,000k, K = €/CHF 5,000k, T = 3 years, σ = 85%, r = 3.0% (EUR)

1. d₁ = [ ln(4,000/5,000) + (0.030 + 0.85²/2) × 3 ] / (0.85 × √3)
2. d₁ = [ −0.223 + 1.174 ] / 1.472 = 0.645
3. d₂ = 0.645 − 1.472 = −0.827
4. N(d₁) = 0.740   |   N(d₂) = 0.204
5. C = 4,000 × 0.740 − 5,000 × e^(−0.03×3) × 0.204
6. C = 2,963 − 933 = 2,030 k€/kCHF

Limitations of BSM for startups

A key precision on scope: the BSM formulas presented assume no dividend distribution (δ = 0). Where the underlying generates a continuous cash flow, the formula must be adjusted. Despite its elegance, BSM presents three important limitations in the AI startup context: first, it assumes constant volatility — rarely verified for startups whose risk profile changes with each development milestone; second, it models European options (exercisable only at maturity) whereas most startup financing decisions are American (exercisable at any time); third, it does not model sequential decisions — each funding round conditioned on the previous one's success. This is why CRR and Monte Carlo offer indispensable complements.

The Cox-Ross-Rubinstein Binomial Tree (CRR)

Formulas and tree construction

u = e^(σ · √Δt) — up factor

d = e^(−σ · √Δt) = 1/u  — down factor

p = (e^(r·Δt) − d) / (u − d)  — risk-neutral up probability

Advantages of CRR for startups

CRR offers three decisive advantages over BSM in the startup context. First, it natively models sequential decisions: at each tree node, the investor can choose to exercise, abandon or wait — accurately reflecting successive funding rounds (Seed → Series A → Series B). Second, it values American options, exercisable at any time — unlike BSM which only values European options. Third, it is numerically transparent: each node is calculated and interpretable, facilitating dialogue with stakeholders during investment term negotiations.

Application — Case 1: OptiVue (CRR, 3 periods)

Parameters: S = €/CHF 4,000k, K = €/CHF 5,000k, T = 3 years, σ = 85%, r = 3.0%, n = 3, Δt = 1 year

u = e^(0.85 × √1) = 2.340   |   d = 0.427   |   p = 0.315

Backward induction (option values in €/CHF k):

1. C(2,2) = [0.315 × 46,238 + 0.685 × 4,361] × 0.970 = 17,026
2. C(2,1) = [0.315 × 4,361 + 0.685 × 0] × 0.970 = 1,332
3. C(1,1) = [0.315 × 17,026 + 0.685 × 1,332] × 0.970 = 6,091
4. C(1,0) = [0.315 × 1,332 + 0.685 × 0] × 0.970 = 407
5. C(0,0) = [0.315 × 6,091 + 0.685 × 407] × 0.970 = 2,131 k€/kCHF

Limitations of CRR for startups

The CRR model, like BSM, assumes constant volatility across all periods (δ = 0 without carry rate). Its main limitation in the startup context is the exponential growth of the number of nodes for large n — though the recombining property of the tree (u·d = 1) limits this to (n+1)(n+2)/2 nodes, making the calculation tractable for n up to 50. For complex options with multiple simultaneous sources of uncertainty, Monte Carlo becomes indispensable.

Monte Carlo Applied to BSM — Volatility Simulation

The Geometric Brownian Motion (GBM)

S(t + Δt) = S(t) × exp[ (r − σ²/2) × Δt + σ × √Δt × ε ]

Where ε is a random variable drawn from a standard normal distribution N(0,1), simulated at each time step for each trajectory. The term (r − σ²/2) is the risk-neutral adjusted drift (the σ²/2 term is Itô's correction guaranteeing absence of arbitrage), and the term σ × √Δt × ε is the stochastic component generating the diffusion of the price.

Advantages of Monte Carlo for startups

Monte Carlo offers three major advantages for high-uncertainty startups. First, it generates a complete distribution of valuations — not just a central value, but the full spectrum of possible outcomes, explicitly quantifying the asymmetry characteristic of venture capital: loss limited to the premium paid, theoretically unlimited gain potential. Second, it is the only one of the three models able to integrate stochastic volatility — that is, volatility that evolves over time — particularly relevant for AI startups whose risk profile changes radically at each technological milestone. Third, it accommodates non-log-normal return distributions (fat tails, discontinuous jumps), better representing the reality of technology assets in disruption phases.

Application — Case 1: OptiVue (Monte Carlo, 10,000 simulations)

Limitations of Monte Carlo for startups

Monte Carlo's main limitation in the startup context is computational intensity: a meaningful simulation (10,000+ trajectories, monthly time steps over 3–5 years) requires dedicated software — Python, R, or Claude in Excel — rather than native Excel. The analytical BSM estimator used in this article and in the Hectelion Excel model is a reliable proxy (convergence proven for N → ∞), but does not reproduce the full distribution. For precise interval estimation, a true simulation is necessary.

Case 2: FlowCast Scale-up — Application of the Three Methods

FlowCast is a B2B SaaS data analytics scale-up at Series B, posting ARR of €/CHF 3,200k with 85% annual growth. The investing fund evaluates the expansion option: inject an additional €/CHF 15m to accelerate commercial deployment in Europe. The current asset value is estimated at €/CHF 18,000k (based on a sector revenue multiple of 5.6x ARR).

Summary of Valuations — OptiVue and FlowCast Across the Three Methods

Summary Table: Which Method for Which Startup Profile?

Advantages and Limitations of Real Options

Fundamental contributions

The real options approach brings three irreplaceable contributions to AI startup valuation. First, it reconciles uncertainty and value: where DCF penalises uncertainty through the discount rate, real options transform it into a positive component of value. Second, it explicitly values managerial flexibility — the ability to accelerate, reduce, pivot or abandon — which is at the core of venture capital practice. Third, it provides a rigorous and defensible framework for structuring negotiations between founders and investors on valuation clauses, earn-outs and ratchet mechanisms.

Limitations to be aware of

Real options nonetheless present important limitations that Damodaran (2005) describes as the "peril of real options". First, they require estimation of parameters that are difficult to calibrate — notably implied volatility for unlisted assets. Second, they assume financial and real markets are sufficiently integrated for no-arbitrage assumptions to hold. Third, real options do not capture competitive dynamics — the value of an option may evaporate if a competitor exercises the same option first.

A Word from the Managing Director

Valuing an AI startup using traditional methods is often a frustrating exercise for all parties. Founders sense their project is worth far more than DCF suggests. Investors know revenue multiples fail to capture the value of strategic options. Both are right — and this is precisely where real options provide an intellectually coherent solution.

At Hectelion, we use real options not as a valuation oracle — no model is, for an AI startup — but as a tool to structure thinking. What is the true cost of the abandonment option? How much is the flexibility to pivot to an adjacent market worth? What is the value of the expansion option if first pilot clients confirm traction? These questions, framed within the real options framework, allow more balanced negotiations and contractual terms better adapted to the reality of shared risk between founders and investors. Whether we are working on a Swiss mandate in CHF or a cross-border France-Switzerland transaction in EUR, the methodology is the same — only the risk-free rate and currency reference change.

Aristide Ruot, PhD — Founder & Managing Director, Hectelion

Conclusion: Real Options — A Common Language for Uncertainty

The valuation of an AI startup cannot be reduced to a cash flow projection exercise. The very nature of these companies — radical uncertainty about future cash flows, essential managerial flexibility, sequential conditional decisions, intellectual property as an option on future markets — calls for an approach that values uncertainty rather than penalising it.

The three methods presented in this article — adapted Black-Scholes, CRR binomial tree and Monte Carlo simulation — form a complementary triptych. BSM provides the analytical reference and calculation speed. CRR models sequential decisions that are at the heart of the successive round financing process. Monte Carlo generates the complete distribution of outcomes, capturing the fundamental asymmetry of AI startups — loss limited to initial investment, theoretically unlimited gain potential.

These methods do not replace professional judgement. They structure it. And in a context where the valuation of intangible assets and technology startups is becoming a central issue for private equity funds, family offices, and fundraising teams — whether in Paris, Geneva, Zurich or London — mastering real options has become an indispensable competence for any corporate finance practitioner.

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Author

Aristide Ruot, Ph.D.
Founder & Managing Directo

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