A futures price is currently 60 and its volatility is 30 %....

A futures price is currently 60 and its volatility is $30 \%$. The risk-free interest rate is $8 \%$ per annum. Use a two-step binomial tree to calculate the value of a six-month European call option on the futures with a strike price of 60 . If the call were American, would it ever be worth exercising it early?

Key Concepts

Binomial Tree Model

The binomial tree model is a discrete-time approach for option pricing. It breaks down the time to expiration into several intervals and models the possible price movements (up or down) at each step. This method is particularly useful for pricing options when closed-form solutions are unavailable or when accounting for complexities such as early exercise features.

European vs American Options and Early Exercise

This concept distinguishes between option styles based on exercise rights. European options can only be exercised at expiration, while American options allow exercise at any time before expiration. Understanding the early exercise premium, or lack thereof, is vital to determining whether it is ever optimal to exercise an American option before expiration, particularly in cases like options on futures where early exercise decisions depend on factors such as dividends, interest rates, and the cost structure.

Volatility

Volatility measures the degree of variation in the price of the underlying asset and is a critical input in option pricing models. Higher volatility generally increases the value of options because it enhances the possibility of significant favorable movements in the asset’s price, thereby affecting the probability of the option finishing in the money.

Futures Option Pricing

This concept involves valuing options where the underlying asset is a futures contract rather than a spot asset. It requires adjustments in valuation methods because futures prices are already discounted by the cost-of-carry, and the payoff is determined from the futures price movements over time rather than immediate spot prices.

Risk-Neutral Valuation

Risk-neutral valuation is a cornerstone of modern option pricing, where expected future payoffs are discounted at the risk-free rate. Under this approach, probability measures adjusted for risk preferences are used, simplifying the pricing of derivatives by assuming that all investors are indifferent to risk and that the expected return on all assets equals the risk-free rate.

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