The Dallas/Fort Worth International Airport would like to buy the option to purc

Question

The Dallas/Fort Worth International Airport would like to buy the option to purchase a large parcel of land on the edge of the city of Grapevine from a real estate investor. The option would give the airport the right to buy the land in one year for $10 million. Real estate experts estimate that the land will be worth either $12 million or $8 million in one year. The current value of the land is $9.5 million. If the risk-free rate of interest is 5% per year, what is the fair market value of the option? [Hint: You can use the replicating portfolio approach or binomial pricing.]

Explanation / Answer

Solution:

Strike price (K) = $10 m, Underlying asset in this case is land with current value (S) = $9.5 m

Upper value in future (Su) = $12 m Lower value in future (Sd) = $8 m , Risk free rate ( R) = 5%

Using replicating portfolio,

At time t =0, S= $9.5 m ,
At time t = 1, Su = $12 m , Sd = $8 m

Number of units of underlying asset = 1, Let B be the borrowing required

Starting at time t=1, and moving backwards,

Su * 1 - (1.05 * B) = 10
12 - 1.05 B = 10
2 = 1.05 * B
B = 2/1.05 = 1.90

Now, value of option = Value of replicating portfolio = Current value of land * Number of units - Borrowing needed
                                                                                  = 9.5 * 1 - 1.9
                                                                                 = 7.595
Hence, fair value of option = $7.6

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