Pricing an option on a house's future value

Question

In $6$ months a house will be worth $X$ times its value today, where $X\sim\text{Unif}(0.6,1.6)$ . The owner offers a buyer a contract granting the right, but not the obligation, to buy the house in $6$ months at today's value. Ignoring interest, what is the fair price of this contract as a fraction of the house's current value?

Short answer

$0.18$ (18% of today's value). This is a call struck at today's value, paying $(X-1)^+$ per unit of value. Its expectation is $\int_1^{1.6}(x-1)\,dx=\tfrac{0.6^2}{2}=0.18$ .

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Lessons behind this question

Conditional Expectation and Variance Revisited
Conditional Probability, Total Probability, and Bayes' Rule
Worked Examples — Probability Trees, Monty Hall, and Bayesian Updating

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