Chapter 36: Real Options

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Capital Investment Appraisal

Traditional capital budgeting uses Net Present Value (NPV):

$NPV=∑t=0nCFt(1+r)t\text{NPV} = \sum_{t=0}^{n} \frac{CF_t}{(1 + r)^t}$

If NPV > 0, accept the project. However, NPV assumes a passive management approach — the project is undertaken now and continues to completion as planned.

In reality, managers have flexibility:

Delay the project
Expand if demand is strong
Contract if demand is weak
Abandon if conditions deteriorate
Switch inputs/outputs

These flexibilities are real options — their value can make a negative-NPV project worthwhile.

Extension of the Risk-Neutral Valuation Framework

Financial Options vs. Real Options

Feature	Financial Option	Real Option
Underlying	Traded asset (stock)	Non-traded (project value, commodity)
Traded market	Yes, exists	Typically no
Observable price	Yes, continuous	No, must be estimated
Risk-neutral pricing	Straightforward	Requires market price of risk

Pricing Real Options

For real options where the underlying is not traded, we need to estimate the market price of risk $λ\lambda$ to convert from the real-world growth rate $μ\mu$ to the risk-neutral growth rate $r$ :

$μrisk-neutral=μ−λσ\mu_{\text{risk-neutral}} = \mu - \lambda \sigma$

The approach:

If a traded asset exists that perfectly correlates with the project value, use its risk-neutral drift ( $r$ , or $r - q$ for dividend-paying)
If no such asset exists, estimate the market price of risk (hardest step)
Apply standard option pricing techniques (binomial trees, Black–Scholes–Merton, Monte Carlo)

Estimating the Market Price of Risk

For a project, the market price of risk relates the project's risk premium to its systematic risk (CAPM beta):

$λ=ρ⋅μm−rσm\lambda = \rho \cdot \frac{\mu_m - r}{\sigma_m}$

where $ρ\rho$ = correlation between project returns and market returns, $μm\mu_m$ = expected market return, $σm\sigma_m$ = market volatility.

Equivalently, the risk-neutral drift rate for the project value:

$μrisk-neutral=μ−β⋅(μm−r)\mu_{\text{risk-neutral}} = \mu - \beta \cdot (\mu_m - r)$

where $β=ρσp/σm\beta = \rho \sigma_p / \sigma_m$ .

Evaluating Options in an Investment Opportunity

Option to Delay

A project can be delayed to wait for more favorable conditions. The value is similar to an American call option on the project. The decision rule: invest when the project value exceeds a critical threshold. The threshold is typically higher than the investment cost to account for the value of waiting.

Example: Oil company considering drilling. $I =$ 100M investment. Current value of developed reserves $V_0 =$ 90M. NPV = − $10 M (n e g a t i v e) . B u t i f o i l p r i ces a r e v o l a t i l e ($ \sigma = 30% $), t h eo pt i o n t o w ai t ha ss i g ni f i c an t v a l u e . U s in g a r e a l o pt i o n s f r am e w or k, t h eo pt ima l t h r es h o l d mi g h t b e$ V^* = $150 M, an d t h eo pt i o n v a l u e mi g h t b e$ 15M — making the overall project value positive.

Option to Expand

If the initial project succeeds, the company can expand capacity. This is a call option on additional capacity. The initial investment buys a growth option.

Option to Abandon

If conditions deteriorate, the company can sell or shut down the project. This is a put option — the project value has a floor equal to the abandonment/salvage value.

Option to Switch

Flexibility to change inputs (e.g., fuel type for a power plant) or outputs (e.g., product mix for a refinery). This is an exchange option (Margrabe formula).

Interactions Between Options

Multiple real options interact — the value of an expansion option may depend on whether the abandonment option exists. Option values are not additive; the portfolio of options must be valued together.

Valuing Amazon.com (Business Snapshot Example)

In 1999, Amazon's stock price implied an enormous market capitalization relative to its current cash flows. A real options analysis showed that the value came from:

Call option on e-commerce growth: If e-commerce became mainstream, Amazon's infrastructure gave it a valuable position
Option to expand into new product categories (books → electronics → everything)
Repeated options: Each category expansion created more growth options

Traditional DCF analysis showed Amazon was overvalued; real options analysis could justify the valuation — if growth options were correctly priced. The subsequent history validated the real options perspective for Amazon.