Question

My friend owns a small old house that is worth approximately $1.1 million. Given the improved real estate market, my friend is considering that over the next three years, she would have the option of tearing down this small old house and build a more expensive house. Her research suggests that the current cost of tearing down the old house and building a new more expensive will be approximately $800,000 and that she should assume that the expected cost would increase by 3%/year in the future. The current value of a comparable house to the one she is thinking of building is approximately $1.8 million. Finally, she believes that the standard deviation of the annual changes in homes prices in this price range is approximately 50%/year. Ignoring flexibility, what is the current expected NPV of the investment to tear down the old house and build the more expensive home (assume the cost of construction occur at time 0)? What is the value of the option to tear down the original house and build a new one anytime over the next three years? (Apply a two-period per year binomial model.)

Answer 1

NPV in the table shows the net present value of the investment with respect to year in which house is to be made if not made right now.

0th year	1st year	2nd year	3rd year
-800000	1747572.816	1696672.636	1647254.987
NPV	947572.8155	896672.6364	847254.9868

The Binomial Option Pricing Model (BOPM) with a House Price of 800000, an uptick percentage of 50%, a downtick percentage of 50%, a risk free interest rate of 3%, and time = 3. Since there are 3 periods, our BOPM should contain 2 = 8 items Each subset term can be written using binaryexpansion representation starting at 0 through 8 - 1 = 7 where D = Down and U = Up U = (1 + 0.5) = 1.5 and D = (1 - 0.5) = 0.5

Scenario 1: DDD

Price Adjustment Factor = (0.5)(0.5)(0.5) = 0.125

House Price at time 3 = Initial House Price x Price Adjustment Factor = 800000 x 0.125 = $100,000.00 Call Price at time 3 = max(0, 100000 - 1800000) = $0.00

Put Price at time 3 = max(0, 1800000 - 100000) = $1,700,000.00

Scenario 2: DDU

Price Adjustment Factor = (0.5)(0.5)(1.5) = 0.375

House Price at time 3 = Initial House Price x Price Adjustment Factor = 800000 x 0.375 = $300,000.00 Call Price at time 3 = max(0, 300000 - 1800000) = $0.00

Put Price at time 3 = max(0, 1800000 - 300000) = $1,500,000.00

Scenario 3: DUD

Price Adjustment Factor = (0.5)(1.5)(0.5) = 0.375

House Price at time 3 = Initial House Price x Price Adjustment Factor = 800000 x 0.375 = $300,000.00 Call Price at time 3 = max(0, 300000 - 1800000) = $0.00

Put Price at time 3 = max(0, 1800000 - 300000) = $1,500,000.00

Scenario 4: DUU

Price Adjustment Factor = (0.5)(1.5)(1.5) = 1.125

House Price at time 3 = Initial House Price x Price Adjustment Factor = 800000 x 1.125 = $900,000.00

Call Price at time 3 = max(0, 900000 - 1800000) = $0.00

Put Price at time 3 = max(0, 1800000 - 900000) = $900,000.00

Scenario 5: UDD

Price Adjustment Factor = (1.5)(0.5)(0.5) = 0.375

House Price at time 3 = Initial House Price x Price Adjustment Factor = 800000 x 0.375 = $300,000.00 Call Price at time 3 = max(0, 300000 - 1800000) = $0.00

Put Price at time 3 = max(0, 1800000 - 300000) = $1,500,000.00

Scenario 6: UDU Price Adjustment Factor = (1.5)(0.5)(1.5) = 1.125

House Price at time 3 = Initial House Price x Price Adjustment Factor = 800000 x 1.125 = $900,000.00 Call Price at time 3 = max(0, 900000 - 1800000) = $0.00

Put Price at time 3 = max(0, 1800000 - 900000) = $900,000

Scenario 7: UUD

Price Adjustment Factor = (1.5)(1.5)(0.5) = 1.125

House Price at time 3 = Initial House Price x Price A djustment Factor = 800000 x 1.125= $900,000.00

Call Price at tim e 3 = max(0, 900000 - 1800000) = $0.00

Put Price at time 3 = max(0, 1800000 - 900000) = $900,0 00.00

Scenario 8: UUU

Price Adjustment Factor = (1.5)(1.5)(1.5) = 3.375

House Price at time 3 = Initial House Price x Price A djustment Factor = 800000 x 3.375= $2,700,000.00

Call Price at time 3 = max(0, 2700000 - 1800000) = $900,000.00

Put Price at time 3 = max(0, 1800000 - 2700000) = $0.00