Real Options Valuation

Dixit & Pindyck (1994) · Managerial Flexibility under Uncertainty

Family 08Corporate financeDP / binomial

Classical NPV assumes the firm commits irrevocably at time zero. In reality, managers can defer investment to learn more, expand if things go well, abandon if they go badly, or switch inputs/outputs. Myers (1977), Dixit & Pindyck (1994), and Trigeorgis (1996) showed that these real options can be valued using the same no-arbitrage machinery as financial options: the project value $V_t$ follows a stochastic process and the decision rule is the solution to a stopping or dynamic-programming problem. The expanded valuation is $\text{ROV} \ge \text{NPV}$; the gap is the value of flexibility.

Educational Purpose

Educational demonstration of real-options valuation. Not investment, corporate-finance, or M&A advice. Real-options identification, parameter estimation, and market-comparable traded-twin assumptions are contested in practice.

The model

Option to defer on a binomial tree

OR family: Real options Solver class: DP on lattice Measure: Risk-neutral $\mathbb{Q}$ Realism: ★★★ Exact

Setup: option to defer a one-shot investment

A project will generate stochastic present-value $V_T$ if undertaken; the investment cost $I$ is known and constant. Without the option to defer, the NPV rule says invest iff $V_0 \ge I$. With the option to defer, the value of waiting is an American call on $V$ with strike $I$.

Underlying dynamics under $\mathbb{Q}$ $$ dV_t = (r - \delta) V_t \, dt + \sigma V_t \, dW^{\mathbb{Q}}_t $$

$V_t$ = project present value; $\delta$ = “convenience yield” / cash-flow payout rate; $\sigma$ = project value volatility; $r$ = risk-free rate.

Binomial valuation

Cox-Ross-Rubinstein tree on $V$: at each node, the decision rule is $\max\{\, V - I, \; e^{-r\Delta t}[\,p V_u + (1-p) V_d\,] \,\}$.

Backwards induction for the option to defer $$ F_{n,j} \;=\; \max\!\Bigl\{\, V_{n,j} - I, \;\; e^{-r\Delta t}\bigl[ p\, F_{n+1,j+1} + (1-p)\, F_{n+1,j} \bigr] \,\Bigr\} $$

$F_{n,j}$ = value of the investment opportunity at node $(n,j)$. Exercise region is where $V - I$ dominates continuation.

Dixit-Pindyck hurdle

In continuous time with geometric Brownian motion, the optimal exercise threshold is not $V^* = I$ (as NPV suggests) but higher:

Optimal investment threshold $$ V^* \;=\; \frac{\beta_1}{\beta_1 - 1}\, I, \qquad \beta_1 \;=\; \frac{1}{2} - \frac{r-\delta}{\sigma^2} + \sqrt{\!\left(\frac{r-\delta}{\sigma^2} - \frac{1}{2}\right)^2 + \frac{2r}{\sigma^2}} $$

$\beta_1 > 1$ is the positive root of the characteristic equation. $\beta_1/(\beta_1 - 1) > 1$ always — so you wait longer than NPV would suggest, by a factor that grows with $\sigma$ and $\delta$.

Taxonomy of real options

Option to defer (American call on $V$ with strike $I$; Dixit-Pindyck canonical case).
Option to expand (compound option; pay an expansion cost $I_2$ later for upside $\alpha V$).
Option to abandon (American put with strike = salvage value $A$).
Option to switch (swap one output/input for another; two correlated underlyings).
Staged investment (R&D pipelines, pharma trials; sequential compound options).

Interactive solver

Option to defer

Project value $V_0$

$100

Investment cost $I$

$100

Horizon $T$ (years)

3.0y

Project vol $\sigma$

25%

Risk-free $r$

4.0%

Payout $\delta$

4.0%

Steps $N$

NPV (invest now)

—

ROV (with defer)

—

Flexibility value

—

Hurdle $V^*$

—

Decision

—

Gold: ROV (with option to defer). Blue dashed: static NPV $= V_0 - I$. Vertical red line: current $V_0$. Vertical gold: Dixit-Pindyck hurdle $V^*$. Gap = flexibility value.

Key references

Dixit, A. K. & Pindyck, R. S. (1994).

Investment under Uncertainty.

Princeton University Press. ISBN 978-0-691-03410-2.

Myers, S. C. (1977).

Determinants of Corporate Borrowing.

Journal of Financial Economics, 5(2), 147–175. doi:10.1016/0304-405X(77)90015-0

Brennan, M. J. & Schwartz, E. S. (1985).

Evaluating Natural Resource Investments.

The Journal of Business, 58(2), 135–157. doi:10.1086/296288

McDonald, R. & Siegel, D. (1986).

The Value of Waiting to Invest.

The Quarterly Journal of Economics, 101(4), 707–727. doi:10.2307/1884175

Trigeorgis, L. (1996).

Real Options: Managerial Flexibility and Strategy in Resource Allocation.

MIT Press. ISBN 978-0-262-20102-5.

Cox, J. C., Ross, S. A. & Rubinstein, M. (1979).

Option Pricing: A Simplified Approach.

Journal of Financial Economics, 7(3), 229–263. doi:10.1016/0304-405X(79)90015-1

Copeland, T. & Antikarov, V. (2001).

Real Options: A Practitioner's Guide.

Texere. ISBN 978-1-58799-028-4.