A futures price is currently 40 . It is known that at the...

A futures price is currently 40 . It is known that at the end of three months the price will be either 35 or 45 . What is the value of a three-month European call option on the futures with a strike price of 42 if the risk-free interest rate is $7 \%$ per annum?

Key Concepts

Futures Contracts

A futures contract is a standardized agreement to buy or sell an asset at a predetermined price at a specified time in the future. Unlike a forward contract, futures are traded on exchanges, which helps limit counterparty risk through the use of margin and daily settlement. Understanding futures is crucial in option pricing because options on futures derive their value from the underlying futures prices.

European Call Options

A European call option gives the holder the right, but not the obligation, to buy the underlying asset at a predetermined strike price only at the option’s expiration date. The European style means that the option cannot be exercised before its maturity, which simplifies the pricing model compared to American options that allow early exercise.

Strike Price

The strike price is the predetermined price at which the holder of a call option can purchase the underlying asset at expiration. It is a critical input in the option pricing model because the option's payoff depends on the relationship between the strike price and the underlying asset’s price at expiration.

Binomial Option Pricing Model

The binomial option pricing model is a discrete-time method for valuing options by simulating possible paths that the underlying asset's price might take over the option's life. The model works by dividing the time period into intervals, calculating the value of the option at each possible future state, and then working backwards to obtain the current option value.

Risk-Neutral Valuation

Risk-neutral valuation is a fundamental concept in financial economics that involves adjusting probabilities so that the expected returns of all assets equal the risk-free rate. This allows the option price to be calculated as the discounted expected payoff under these risk-neutral probabilities, simplifying the valuation of derivative securities.

Discounting Cash Flows

Discounting cash flows means converting future amounts into their present value by applying a discount rate, typically the risk-free interest rate. This process is essential in option pricing as it accounts for the time value of money, ensuring that the option’s future expected payoff is appropriately valued in today’s dollars.

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