Options and Real Options Concept Card Template

Options and Real Options Concept Card: [Course, case or project]

A concept card for options as they appear in the MBA core corporate finance course and its follow-on electives: payoffs and profit diagrams, put-call parity, the binomial model (Cox, Ross and Rubinstein 1979), the Black-Scholes-Merton formula (Black and Scholes 1973; Merton 1973) and real options, the options to expand, abandon and defer that managers hold inside projects (Myers 1977 coined the term; Dixit and Pindyck 1994). Every value is computed and shown with its Excel formula. It is for MBA students, candidates preparing for finance and strategy interviews, and managers who must judge whether "strategic value" in an investment case is real. Worked example A is fully solved, B is half solved and Exercise C is left to you; self-test answers sit in a separate key. Source basis: the original papers cited in each section and standard textbooks named as further reading; all wording, notation and examples are original. Northfield Outdoor Co. and Corvane Robotics are fictional, and Northfield is treated as listed only for the purpose of the example. Replace every [bracketed] value and every row marked Example. This is a learning aid, not investment advice.

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Contents

1. When to use it
2. The idea in plain language
3. Formulas and notation
4. Assumptions and limits
5. Real options: expand, abandon, defer
6. How managers use it in practice
7. Worked example A (fully solved)
8. Worked example B (half solved)
9. Exercise C (unsolved)
10. Connections to other cards
11. Common mistakes and misconceptions
12. Self-test questions
13. My notes and next steps

When to use it

One-sentence rule: use option pricing whenever a payoff is asymmetric, meaning someone has the right but not the obligation to act after uncertainty is resolved; a static NPV that assumes a fixed plan undervalues such rights.

The idea in plain language

A call gives the right to buy an asset at a fixed price (the strike) on or before a date; a put gives the right to sell. European options can be exercised only at expiry, American options at any time up to it. The holder pays a premium today and exercises only if it pays to do so, so the downside is limited to the premium while the upside is open. That asymmetry is why volatility raises option value: more spread means bigger gains in the good states, while the bad states are cut off at zero.

Options are priced by no-arbitrage: a combination of the underlying asset and borrowing or lending can be built to copy the option's payoff over a short period. If two portfolios have the same payoff in every state, they must have the same price today. This removes investors' risk preferences from the problem, so you can value the option as if everyone were risk-neutral: weight the payoffs with "risk-neutral probabilities" and discount at the risk-free rate. The binomial model shows this step by step; Black-Scholes-Merton is its limit when the steps become very small.

Real options apply the same logic to managers' flexibility. A project that can be expanded if demand is strong is a call on the expansion; one that can be shut down and sold is a project plus a put. Myers (1977) introduced the term "real options" when he described a firm's growth opportunities as options on future investment.

Formulas and notation

Payoffs at expiry: call $\max(S_T - K, 0)$; put $\max(K - S_T, 0)$. Profit = payoff minus the premium (ignoring the interest on the premium for a short-dated option).

Put-call parity (Stoll 1969; European options on a non-dividend-paying stock):

One-step binomial (stock moves to $uS_0$ or $dS_0$ over a period of length $\Delta t$):

Cox, Ross and Rubinstein (1979) parameters for $n$ steps: $\Delta t = T/n$, $u = e^{\sigma\sqrt{\Delta t}}$, $d = 1/u$.

Black-Scholes-Merton (European, no dividends):

Direction of effects (European options, no dividends):

Assumptions and limits

• Lognormal prices and constant volatility (BSM). Real markets show fat tails and volatility that changes with the strike and over time (the volatility smile). Traders quote implied volatility instead of trusting one number.
• Continuous trading, no transaction costs, borrowing and lending at $r$. Replication is only approximate in practice.
• No dividends in the basic formulas. Dividends lower call values and raise put values; adjust $S_0$ by the present value of dividends, or use a dividend yield. For real options, cash flows lost while waiting act like dividends.
• European exercise in BSM. American puts, and American calls on dividend-paying stocks, can be worth more because early exercise can be optimal; use a binomial tree.
• A traded underlying. For real options the "stock" is the present value of project cash flows, which does not trade. Valuing it as if it did assumes a traded asset or portfolio that tracks it (the spanning assumption). The numbers are then a disciplined estimate, not a market price.
• Volatility of a project is hard to estimate. Use the volatility of comparable firms' asset returns, a simulation of the project's value, or a range, and show the sensitivity.

Real options: expand, abandon, defer

McDonald and Siegel (1986) and Dixit and Pindyck (1994) showed that when investment is irreversible and the future is uncertain, a firm should invest only when the project's value exceeds its cost by a premium, because investing kills the option to wait. The simple rule "invest if NPV is above zero" is then too aggressive. Waiting has a cost too: cash flows forgone, and the risk that a competitor takes the opportunity.

How managers use it in practice

• Turn "strategic value" into numbers. A negative-NPV pilot may be worth doing because it buys an option on a large follow-on; the option value must exceed the pilot's negative NPV (Worked example A, part A4).
• Design projects for flexibility. Modular plants, short leases, staged contracts and pilot markets create options that are cheap to buy and valuable when the future is uncertain.
• Know when to wait. For irreversible investments in very uncertain markets, waiting for information can be worth more than starting now (part A6).
• Read securities as bundles of options. Convertible bonds, warrants, employee stock options and callable debt all contain options; their cost to shareholders is the option value, not the cash paid today.
• Hedge exposures. Treasurers buy puts or caps to set a floor or ceiling on currency, commodity and interest costs; see finance/fx-interest-rate-hedging-policy.
• Diagnose agency problems. Equity of a highly levered firm behaves like a call, which explains why shareholders of a distressed firm may prefer risky projects (mba/mba-capital-structure-card).
• Practical shortcut. Luehrman (1998) proposed reading a real option from two numbers, $S/PV(K)$ and $\sigma\sqrt{T}$, before building a full model; part A4 computes both.

Worked example A (fully solved)

A1. Black-Scholes-Merton (Example). Northfield Outdoor Co. shares (fictional, treated as listed) trade at USD 50.00. Price six-month European options with strike USD 52.00; risk-free rate 4 percent (continuous); volatility 30 percent; no dividends.

The put is worth more than the call because the strike is above the spot price: the put is in the money and the call is out of the money. Parity holds to four decimals, so the two prices are consistent. A delta of 0.5062 means a USD 1 rise in the share price raises the call by about USD 0.51.

A2. Payoff and profit diagram. Using the premiums above, the long call breaks even at $52.00 + 3.80 = 55.80$ and its maximum loss is 3.80; the long put breaks even at $52.00 - 4.77 = 47.23$ and its maximum profit, if the share price fell to zero, is 47.23.

A3. Binomial trees.

One step, by replication. Same share at 50.00 and strike 52.00, one year, rate 4 percent continuous; the share can rise by a factor of 1.20 to 60.00 or fall by a factor of 0.85 to 42.50.

Replication and risk-neutral pricing give the same value, as they must. Note that $q$ is not the real probability of an up move; it is the probability that makes the share earn the risk-free rate.

Two steps, CRR, with the A1 inputs ($T$ = 0.5, $\Delta t$ = 0.25).

Early exercise of the put is optimal at the down node because receiving the strike now and earning interest on it is worth more than keeping the option alive. That is why an American put is worth more than a European one, here by 0.2582.

Convergence. As steps are added, the European CRR value approaches the BSM value of 3.7967, oscillating on the way.

A4. Option to expand (Example, Corvane Robotics). Corvane can run a pilot of an automated cold-storage line with a static NPV of USD -15m. If the pilot works, a full roll-out in three years would cost USD 150m; the present value today of the roll-out's cash flows is USD 120m, with a volatility of 40 percent (from listed automation peers); the risk-free rate is 4 percent. The roll-out is a European call with $S$ = 120, $K$ = 150, $T$ = 3.

Committing to both stages today would destroy value; buying the right to decide later is worth it. The result depends heavily on the 40 percent volatility: check it in a sensitivity table before presenting it.

A5. Option to abandon (Example). A Corvane service contract business is worth 100 today. In a year it will be worth 140 or 70. At that point the assets can be sold for 85. Risk-free rate 5 percent (annual compounding in this part).

A6. Option to defer (Example, in the spirit of Dixit and Pindyck 1994). Corvane holds a one-year exclusive right to build a robotics plant for 1,000. Built today, the plant is worth 1,100 (NPV 100). In a year its value will be 1,500 or 700, and the investment cost stays 1,000. Risk-free rate 5 percent (annual).

A positive NPV today is not enough: waiting is worth more here because it avoids building in the bad state. Lost cash flows (here 80) act like dividends and reduce the value of waiting, but not enough to change the decision. A competitor that could take the opportunity would reduce it further.

Worked example B (half solved)

A share trades at USD 40.00. Price one-year European options with strike 40.00; risk-free rate 3 percent (continuous); volatility 25 percent; no dividends.

Exercise C (unsolved)

1. Abandonment. A Corvane regional unit is worth 200 today. In a year it will be worth 260 or 160. At that point the unit can be sold to a competitor for 170. Risk-free rate 4 percent (annual). Compute the risk-neutral probability, the value of the unit with the option to sell, and the value of the option. In one sentence, say what the option is worth to a board deciding whether to sign a contract that removes the right to sell.
2. Parity arbitrage. A non-dividend share trades at 50.00. One-year European options with strike 50.00 trade at 6.20 (call) and 5.00 (put); risk-free rate 4 percent (continuous). What put price does parity imply? Describe the trades that lock in a riskless profit, and the cash you receive today.

Connections to other cards

Common mistakes and misconceptions

• Using real probabilities and the risk-free rate together. Either risk-neutral probabilities with the risk-free rate, or real probabilities with a risk-adjusted rate that changes with the option's risk; the second is rarely practical for options.
• "Higher volatility makes the option riskier, so it is worth less." Volatility raises option value because the holder keeps the upside and avoids the downside.
• Mixing rate conventions. BSM uses a continuously compounded rate; converting an annual rate of 4 percent gives $\ln(1.04)$ = 3.92 percent. Use one convention throughout a tree.
• Forgetting dividends or lost cash flows. A call on a dividend-paying share, or an option to defer a project that produces cash meanwhile, is worth less than the no-dividend formula says.
• Using BSM for American puts. Early exercise can be optimal; use a binomial tree.
• Treating any uncertainty as an option. An option needs a right to act after news arrives, at a known cost. If the firm cannot or will not act, there is no option value.
• Double counting flexibility. If the DCF already assumes the firm expands only in good states, adding an expansion option counts it twice.
• Using the volatility of the firm's equity for a project. Equity volatility includes leverage; use asset or project volatility.
• Ignoring competition. A shared option (any rival can enter) is worth less than a proprietary one; waiting can hand the opportunity to a competitor.
• Presenting one real option value without a range. Inputs are estimates; show the sensitivity to $\sigma$ and $T$.

Self-test questions

1. Draw the payoff at expiry of a long call, a short call, a long put and a short put, each with strike 100.
2. State put-call parity and use it to explain why a call and a put with the same strike and expiry have the same value when $S_0 = K e^{-rT}$.
3. In A3, why does the option value not depend on the real probability of an up move?
4. Which inputs raise a call's value, and which raise a put's?
5. Why can early exercise of an American put be optimal, while early exercise of an American call on a non-dividend-paying share never is?
6. In A4, which number would you test first before presenting the result, and why?
7. Map the right to sell a factory for its scrap value into an option: type, $S$, $K$ and $T$.
8. Why does the option to defer fall when waiting means losing cash flows or facing a competitor?
9. Explain in two sentences why the equity of a firm close to default behaves like an out-of-the-money call.
10. A manager says "the project has negative NPV but strong strategic value". What three questions do you ask?

My notes and next steps

Sources and further reading

• Stoll, H. R. (1969). The relationship between put and call option prices. Journal of Finance, 24(5), 801–824.
• Black, F., and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654.
• Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1), 141–183.
• Merton, R. C. (1974). On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance, 29(2), 449–470.
• Myers, S. C. (1977). Determinants of corporate borrowing. Journal of Financial Economics, 5(2), 147–175.
• Cox, J. C., Ross, S. A., and Rubinstein, M. (1979). Option pricing: a simplified approach. Journal of Financial Economics, 7(3), 229–263.
• McDonald, R., and Siegel, D. (1986). The value of waiting to invest. Quarterly Journal of Economics, 101(4), 707–727.
• Dixit, A. K., and Pindyck, R. S. (1994). Investment under Uncertainty. Princeton University Press.
• Trigeorgis, L. (1996). Real Options: Managerial Flexibility and Strategy in Resource Allocation. MIT Press.
• Luehrman, T. A. (1998). Investment opportunities as real options: getting started on the numbers. Harvard Business Review, 76(4).
• Microsoft. NORM.S.DIST function, Microsoft Support documentation (checked 2026-10-28).
• Textbooks (further reading, by name only): Hull, Options, Futures, and Other Derivatives; Berk and DeMarzo, Corporate Finance; Brealey, Myers and Allen, Principles of Corporate Finance; McDonald, Derivatives Markets.
• Open resource: OpenStax, Principles of Finance (2022), free online at https://openstax.org (check the licence on the book page).

Answer key

Worked example B. (2) $d_2$ = -0.0050. (3) $N(d_2)$ = 0.4980. (4) $PV(K)$ = 38.8178. (5) Call = 4.5394. (6) Put = 4.5394 - 40 + 38.8178 = 3.3572. (7) $d$ = 0.8380, $q$ = 0.4984. (8) Two-step call = 4.0903. (9) With only two steps the share can end at just three prices, a coarse approximation of the lognormal distribution; the value converges to BSM (4.5394) as the number of steps grows.

Exercise C. (1) $q$ = (1.04 × 200 - 160)/(260 - 160) = 0.48. Value with the option = [0.48 × 260 + 0.52 × 170]/1.04 = 205.00, so the option is worth 5.00. Signing a contract that removes the right to sell gives up 5.00 of value today, and the board should require at least that much in return. (2) Parity put = 6.20 - 50.00 + 50 $e^{-0.04}$ = 6.20 - 50.00 + 48.0395 = 4.24. The quoted put (5.00) is too expensive: sell the put, buy the call, sell the share short and lend 48.04. Cash today = 5.00 - 6.20 + 50.00 - 48.04 = 0.76; at expiry the positions net to zero whatever the share price.

Self-test.

1. Long call: zero below 100, rising one for one above. Short call: the mirror image (zero, then falling). Long put: falling one for one to zero at 100, then zero. Short put: the mirror image.
2. $C - P = S_0 - Ke^{-rT}$. If $S_0 = Ke^{-rT}$, the right side is zero, so $C = P$.
3. Because the option can be replicated with the share and borrowing; the share's price already reflects investors' views about the real probability and risk, so the option price follows from the share price without it.
4. Call: higher $S_0$, $\sigma$, $T$, $r$; lower $K$. Put: higher $K$ and $\sigma$, usually higher $T$; lower $S_0$ and $r$.
5. Exercising a put early gives the strike now, which earns interest; when the share is far below the strike, that gain exceeds the remaining option value. Exercising a call early pays the strike early and gives up the insurance, so with no dividends it is always better to keep or sell the call.
6. The 40 percent volatility, because the option value and the decision rest on it and it is estimated from peers, not observed for this project.
7. An American put: $S$ = PV of the factory's continuing cash flows, $K$ = scrap value, $T$ = the period during which the sale is possible.
8. Lost cash flows act like dividends that the holder of the waiting option does not receive, and a competitor can take the opportunity, cutting the option's life; both reduce the gain from waiting.
9. Shareholders receive the firm's value above the face value of debt at maturity and nothing below it, which is the payoff of a call on the firm's assets with the debt as strike. Near default, the call is out of the money, so shareholders gain from more risk, even at the lenders' expense.
10. Which specific decision can we take later, and at what cost? What is the underlying value and how volatile is it, from which evidence? Is the option ours alone, or can competitors take it, and is it already counted in the DCF?

Revision history

Standard and licence note

Based on option pricing theory as published in the original works cited above (Black and Scholes 1973, Merton 1973, Cox, Ross and Rubinstein 1979) and on real options analysis (Myers 1977; McDonald and Siegel 1986; Dixit and Pindyck 1994), as taught across MBA corporate finance courses; no single standard governs these methods. The explanations, notation, worked examples and layout are original; no textbook text, published case or exam question is reproduced. Every number was computed in Python and is shown with the matching Excel formula. The template is not affiliated with or endorsed by any business school, publisher or exchange, and it is not investment advice. Template licence: MIT.