Chapter 36, Real options Video Solutions, Options, Futures, and Other Derivatives

Problem 1

Explain the difference between the net present value approach and the risk-neutral valuation approach for valuing a new capital investment opportunity. What are the advantages of the risk-neutral valuation approach for valuing real options?

Nick Johnson

Numerade Educator

Problem 2

The market price of risk for copper is 0.5 , the volatility of copper prices is $20 \%$ per annum, the spot price is 80 cents per pound, and the 6-month futures price is 75 cents per pound. What is the expected percentage growth rate in copper prices over the next 6 months?

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Problem 3

Show that if $y$ is a commodity's convenience yield and $u$ is its storage cost, the commodity's growth rate in the traditional risk-neutral world is $r-y+u$, where $r$ is the risk-free rate. Deduce the relationship between the market price of risk of the commodity, its real-world growth rate, its volatility, $y$, and $u$.

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01:03

Problem 4

The correlation between a company's gross revenue and the market index is 0.2 . The excess return of the market over the risk-free rate is $6 \%$ and the volatility of the market index is $18 \%$. What is the market price of risk for the company's revenue?

Breanna Ollech

Numerade Educator

Problem 5

A company can buy an option for the delivery of 1 million units of a commodity in 3 years at $$\$ 25$$ per unit. The 3-year futures price is $$\$ 24$$. The risk-free interest rate is $5 \%$ per annum with continuous compounding and the volatility of the futures price is $20 \%$ per annum. How much is the option worth?

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Problem 6

A driver entering into a car lease agreement can obtain the right to buy the car in 4 years for $$\$ 10,000$$. The current value of the car is $$\$ 30,000$$. The value of the car, $S$, is expected to follow the process $d S=\mu S d t+\sigma S d z$, where $\mu=-0.25, \sigma=0.15$, and $d z$ is a Wiener process. The market price of risk for the car price is estimated to be -0.1 . What is the value of the option? Assume that the risk-free rate for all maturities is $6 \%$.

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Problem 7

Suppose that the spot price, 6-month futures price, and 12 -month futures price for wheat are 250,260 , and 270 cents per bushel, respectively. Suppose that the price of wheat follows the process in equation (36.3) with $a=0.05$ and $\sigma=0.15$. Construct a two-timestep tree for the price of wheat in a risk-neutral world.
A farmer has a project that involves an expenditure of $$\$ 10,000$$ and a further expenditure of $$\$ 90,000$$ in 6 months. It will increase wheat that is harvested and sold by 40,000 bushels in 1 year. What is the value of the project? Suppose that the farmer can abandon the project in 6 months and avoid paying the $$\$ 90,000$$ cost at that time. What is the value of the abandonment option? Assume a risk-free rate of $5 \%$ with continuous compounding.

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02:07

Problem 8

In the example considered in Section 36.5:
(a) What is the value of the abandonment option if it costs $$\$ 3$$ million rather than zero?
(b) What is the value of the expansion option if it costs $$\$ 5$$ million rather than $$\$ 2$$ million?

Adrian Co

Numerade Educator