The futures price of a commodity is 90. Use a three-step...

The futures price of a commodity is $$\$ 90$$. Use a three-step tree to value (a) a 9-month American call option with strike price $$\$ 93$$ and (b) a 9-month American put option with strike price $$\$ 93$$. The volatility is $28 \%$ and the risk-free rate (all maturities) is $3 \%$ with continuous compounding.

Key Concepts

American Option

An American option is a type of derivative that can be exercised at any time up to and including its expiration date. This flexibility introduces the possibility of early exercise, which must be considered when valuing the option, as the optimal strategy may involve exercising the option before the final expiration date based on the evolving asset price and remaining time value.

Binomial Tree Model

The binomial tree model is a popular discrete-time method for pricing options by simulating various possible paths that the underlying asset's price can take over its life. In each step of the tree, the asset price either moves up or down by a fixed factor, and the model uses risk-neutral probabilities to compute the expected payoff, which is then discounted back to the present value. This method is particularly useful for American options where early exercise is possible.

Futures Contract

A futures contract is an agreement to buy or sell an asset at a predetermined future date and price. In the context of option pricing, options on futures provide the holder the right, but not the obligation, to enter into such a contract. The underlying asset for the option in this case is the futures price, and pricing these options involves assessing how changes in the futures price over time affect the option's value.

Risk-Neutral Valuation

Risk-neutral valuation is a fundamental concept in option pricing wherein investors are assumed to be indifferent to risk. Under this assumption, the expected payoff from an option is calculated using probabilities that have been adjusted so that the expected return on the underlying asset equals the risk-free rate. This approach allows the discounting of the expected future payoff back to the present value using the risk-free rate.

Volatility

Volatility represents the degree of variation of the underlying asset's price over time, typically measured as an annualized standard deviation. In option pricing, volatility is a key parameter because it impacts the potential price movements in each step of the binomial tree model, thereby affecting the likelihood of the option finishing in or out of the money and ultimately its value.

Continuous Compounding

Continuous compounding is a method used to calculate the growth of an investment assuming that interest is compounded constantly. In option pricing, the risk-free rate is often given with continuous compounding, meaning that expected payoffs are discounted using the exponential function, which provides a more precise method for calculating present values over continuous intervals of time.

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