20.1 Using Black Scholes

Problem. You have the opportunity to purchase a piece of land for $0.4 million which has known

reserves of 200,000 barrels of oil. The reserves are worth $5.3 million based on the current crude oil price

of $26.50 per barrel. The cost of building the plant and equipment to develop the oil is $5.7 million, so it

is not profitable to develop these reserves right now. However, development may become profitable in the

future if the price of crude oil goes up. For simplicity, assume there is a single date in 1.0 years when you

can decide whether to develop the oil or not. Further assume that all of the oil can be produced

immediately. Using historical data on crude oil prices, you determine that the mean value of the reserves

is $6.0 million based on a mean value of the one-year ahead oil price of $30.00 per barrel and the standard

deviation 30.0%. The riskfree rate is 6.0% and cost of capital for a project of this type is 13.80%. What is

the project’s NPV using the Black-Scholes call formula? What would the NPV be if you committed to

develop it today no matter what and thus, (incorrectly) ignored the option to develop the oil only if it is

profitable?

Solution Strategy. One year from now, you will develop the oil if it is profitable and won't develop it if it

is not. Thus, the payoff is Max (Value of the Reserves - Cost of Development, 0). This is identical to the

payoff of a call option, where the Cost of Development is the Exercise Price and the Value of the

Reserves Now is the Asset Price Now. Calculate the NPV using the Black-Scholes call formula. Calculate

the NPV Ignoring Option projecting expected cash flows from developing the oil no matter whether it is

profitable or not and discounting these expected cash flows back to the present.

FIGURE 20.1 Spreadsheet for Real Options Using Black-Scholes.

How To Build Your Own Spreadsheet Model.

NPV Using Black-Scholes

1. Start with the Black Scholes Option Pricing - Basics Spreadsheet and Change the Inputs.

Open the spreadsheet that you created for Black Scholes Option Pricing – Basics and immediately

save the spreadsheet under a new name using the File | Save As command. Add three rows by

selecting the range A9:A11 and clicking on Insert | Rows. Then delete five rows by selecting

A20:A24, clicking on Edit | Delete..., selecting Entire Row, and click on OK. Relabel the input

labels in the range A4:A11 and enter the new inputs values into the range B4:B11. Lock in the

first eleven rows as titles by selecting cell A12 and clicking on Window | Freeze Panes.

2. NPV Using Black-Scholes. The NPV of the proposal to purchase the land is the difference

between the Value of the Real Option (as calculated by the Black-Scholes call formula) and the

Cost of the Real Option (equal to the cost of the land). Enter =B18-B9 in cell B19.

NPV Ignoring Option

3. Expected Cash Flows. The Date 0 Expected Cash Flow is a (negative) payment for the cost of

the land. Enter =-B9 in cell B23. The Date 1 Expected Cash Flow is Date 1 Expected Asset

Value minus Exercise Price (Cost of Development). Enter =B10-B7 in cell C23.

4. Present Value of Expected Cash Flows. Calculate the Present Value of the Expected Cash Flow

= (Expected Cash Flow) / ((1 + Discount Rate)^Date Number). Enter =B23/((1+$B$11)^B22) in

cell B24 and copy it to cell C24. The Discount Rate $B$11 uses an absolute reference because it

stays constant from date to date.

5. NPV Ignoring Option. Calculate the NPV Ignoring Option by summing all of the present value

of cash flow terms. Enter =SUM(B24:C24) in cell B25.

We obtain opposite results from the two approaches. The NPV Using Black-Scholes is positive $0.20

million, whereas the Ignore Option NPV is negative ($0.14) million. NPV Ignoring Option incorrectly

concludes that the project should be rejected. This mistake happened precisely because it ignores the

option to develop oil only when profitable and avoid the cost of development when it is not. NPV Using

Black-Scholes correctly demonstrates that the project should be accepted. This is because the value of the

option to develop the oil when profitable is greater than the cost of the option (i.e., the cost of the land).

Problem. You have the opportunity to purchase a piece of land for $0.4 million which has known

reserves of 200,000 barrels of oil. The reserves are worth $5.3 million based on the current crude oil price

of $26.50 per barrel. The cost of building the plant and equipment to develop the oil is $5.7 million, so it

is not profitable to develop these reserves right now. However, development may become profitable in the

future if the price of crude oil goes up. For simplicity, assume there is a single date in 1.0 years when you

can decide whether to develop the oil or not. Further assume that all of the oil can be produced

immediately. Using historical data on crude oil prices, you determine that the mean value of the reserves

is $6.0 million based on a mean value of the one-year ahead oil price of $30.00 per barrel and the standard

deviation 30.0%. The riskfree rate is 6.0% and cost of capital for a project of this type is 13.80%. What is

the project’s NPV using the Black-Scholes call formula? What would the NPV be if you committed to

develop it today no matter what and thus, (incorrectly) ignored the option to develop the oil only if it is

profitable?

Solution Strategy. One year from now, you will develop the oil if it is profitable and won't develop it if it

is not. Thus, the payoff is Max (Value of the Reserves - Cost of Development, 0). This is identical to the

payoff of a call option, where the Cost of Development is the Exercise Price and the Value of the

Reserves Now is the Asset Price Now. Calculate the NPV using the Black-Scholes call formula. Calculate

the NPV Ignoring Option projecting expected cash flows from developing the oil no matter whether it is

profitable or not and discounting these expected cash flows back to the present.

FIGURE 20.1 Spreadsheet for Real Options Using Black-Scholes.

How To Build Your Own Spreadsheet Model.

NPV Using Black-Scholes

1. Start with the Black Scholes Option Pricing - Basics Spreadsheet and Change the Inputs.

Open the spreadsheet that you created for Black Scholes Option Pricing – Basics and immediately

save the spreadsheet under a new name using the File | Save As command. Add three rows by

selecting the range A9:A11 and clicking on Insert | Rows. Then delete five rows by selecting

A20:A24, clicking on Edit | Delete..., selecting Entire Row, and click on OK. Relabel the input

labels in the range A4:A11 and enter the new inputs values into the range B4:B11. Lock in the

first eleven rows as titles by selecting cell A12 and clicking on Window | Freeze Panes.

2. NPV Using Black-Scholes. The NPV of the proposal to purchase the land is the difference

between the Value of the Real Option (as calculated by the Black-Scholes call formula) and the

Cost of the Real Option (equal to the cost of the land). Enter =B18-B9 in cell B19.

NPV Ignoring Option

3. Expected Cash Flows. The Date 0 Expected Cash Flow is a (negative) payment for the cost of

the land. Enter =-B9 in cell B23. The Date 1 Expected Cash Flow is Date 1 Expected Asset

Value minus Exercise Price (Cost of Development). Enter =B10-B7 in cell C23.

4. Present Value of Expected Cash Flows. Calculate the Present Value of the Expected Cash Flow

= (Expected Cash Flow) / ((1 + Discount Rate)^Date Number). Enter =B23/((1+$B$11)^B22) in

cell B24 and copy it to cell C24. The Discount Rate $B$11 uses an absolute reference because it

stays constant from date to date.

5. NPV Ignoring Option. Calculate the NPV Ignoring Option by summing all of the present value

of cash flow terms. Enter =SUM(B24:C24) in cell B25.

We obtain opposite results from the two approaches. The NPV Using Black-Scholes is positive $0.20

million, whereas the Ignore Option NPV is negative ($0.14) million. NPV Ignoring Option incorrectly

concludes that the project should be rejected. This mistake happened precisely because it ignores the

option to develop oil only when profitable and avoid the cost of development when it is not. NPV Using

Black-Scholes correctly demonstrates that the project should be accepted. This is because the value of the

option to develop the oil when profitable is greater than the cost of the option (i.e., the cost of the land).