Suppose that the spot price, 6-month futures price, and 12...

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.

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.

Key Concepts

Risk-Neutral Pricing

Risk-neutral pricing is a fundamental concept in financial economics where the expected return of an asset is set equal to the risk?free rate. Under this framework, investors are indifferent to risk, and asset prices are determined by discounting expected future payoffs at the risk-free rate. This concept allows for the valuation of derivatives and projects by transforming the probability distribution of future outcomes into a risk-neutral measure, simplifying complex pricing models.

Binomial Tree Construction

The binomial tree (or multi-timestep tree) is a discrete-time model used for valuing options and other derivative securities. It involves constructing a lattice showing the possible future values of the underlying asset over time. Each node represents a possible future price, and the risk-neutral probabilities are used to weight the outcomes. This approach facilitates the numerical valuation of instruments by working backward from replicating portfolio payoffs or project cash flows at maturity.

Commodity Futures Pricing

Commodity futures pricing considers the relationship between current (spot) prices and the prices for future delivery, which reflects the cost of carry including storage, financing, and other costs. In models involving commodities, the dynamics of the production process are often assumed to follow stochastic processes with specified drift and volatility. Understanding these processes is essential for constructing models to forecast future prices and to evaluate forward contracts linked to commodities.

Real Options Analysis

Real options analysis applies option pricing methods to investment decisions, incorporating the value of managerial flexibility in the presence of uncertainty. The abandonment option, for example, allows an investor to exit a project if future conditions are unfavorable, thus limiting losses. This concept is crucial in project valuation as it recognizes that the option to delay, expand, contract, or abandon an investment can add significant value compared to traditional net present value calculations.

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