The spot price of copper is 0.60 per pound. Suppose that...

The spot price of copper is $$\$ 0.60$$ per pound. Suppose that the futures prices (dollars per pound) are as follows: $$\begin{array}{ll}\hline 3 \text { months } & 0.59 \\6 \text { months } & 0.57 \\9 \text { months } & 0.54 \\12 \text { months } & 0.50 \\\hline\end{array}$$ The volatility of the price of copper is $40 \%$ per annum and the risk-free rate is $6 \%$ per annum. Use a binomial tree to value an American, call option on copper with an exercise price of $\$ 0.60$ and a time to maturity of 1 year. Divide the life of the option into four 3 -month periods for the purposes of constructing the tree. (Hint: As explained in Section 14.7, the futures price of a variable is its expected future price in a riskneutral world.)

The spot price of copper is $$\$ 0.60$$ per pound. Suppose that the futures prices (dollars per pound) are as follows:
$$\begin{array}{ll}\hline 3 \text { months } & 0.59 \\6 \text { months } & 0.57 \\9 \text { months } & 0.54 \\12 \text { months } & 0.50 \\\hline\end{array}$$
The volatility of the price of copper is $40 \%$ per annum and the risk-free rate is $6 \%$ per
annum. Use a binomial tree to value an American, call option on copper with an exercise price of $\$ 0.60$ and a time to maturity of 1 year. Divide the life of the option into four 3 -month periods for the purposes of constructing the tree. (Hint: As explained in Section 14.7, the futures price of a variable is its expected future price in a riskneutral world.)

Key Concepts

Binomial Tree Model

The binomial tree model is a discrete-time framework used for pricing options by modeling the potential future movements of the underlying asset over several time steps. At each node, the asset price can move up or down by certain factors derived from the asset’s volatility, and option values are computed backwards from expiration to the present using risk-neutral probabilities. This approach is especially useful for American options which allow early exercise.

American Option Pricing

American options differ from European options by allowing early exercise at any point up to expiry. This feature requires a robust pricing model, such as the binomial tree, to account for the possibility of optimal early exercise at each node, balancing the immediate exercise value with the expected continuation value. This early exercise feature is particularly important for options on commodities where storage, convenience yield, or other factors might make early exercise optimal.

Futures Price

Futures prices represent the market’s expectation of the future spot price of a commodity in a risk-neutral world. They are critical in commodity option pricing because, under the risk-neutral measure, the current futures price is equal to the expected future spot price, which simplifies the valuation of options that derive their value from the underlying commodity.

Risk-Neutral Valuation

Risk-neutral valuation is a fundamental concept in financial engineering where all assets are assumed to grow at the risk-free rate. This allows for the pricing of derivatives by discounting the expected future payoffs under the risk-neutral probability measure, rather than dealing with investor-specific risk preferences. It plays a crucial role in the pricing of options using binomial trees or other models.

Discrete Time Valuation

Discrete time valuation involves breaking the option’s life into a set number of periods and evaluating the underlying asset’s possible price movements step by step. This method simplifies the analysis and computation of option prices, particularly when using models like the binomial tree, by approximating continuous price changes through a series of discrete, easily managed steps.

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