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The futures price is the agreed-upon price for a commodity or financial instrument that is to be delivered at a future date. It fluctuates because of supply and demand factors in the market. In this exercise, the futures price starts at 40 and represents the market consensus for the future price at the time of the transaction.
Within three months, this price is expected to either decrease to 35 or increase to 45. This range shows the potential outcomes influenced by different market scenarios. A trader uses this information to gauge the potential payoff of a futures contract, which in turn influences decisions about options like the European call option.

• If the futures price increases to 45, it provides an opportunity for profit.
• If it decreases to 35, the option may have no intrinsic value.

Understanding these dynamics helps traders manage risks and plan their strategies around expected price changes.

Risk-neutral probabilities are a critical concept for pricing options, as they allow us to assess the market's expectations of future outcomes without considering investors' risk preferences. Here’s how it works:
In the example, we need to calculate the probability that aligns the expected future price with the current futures price of 40. By setting up a basic equation with the potential future prices, we solve for the risk-neutral probability \( p \), leading to the solution \( p = 0.5 \).
In this case, this means there's an equal likelihood (50%) of the price being 35 or 45 in three months, under the risk-neutral assumption.

• Risk-neutral probability allows for the simplification necessary for fair pricing by eliminating risk preferences.
• It helps focus on the intrinsic value of the options given future expectations.

By utilizing these probabilities, the expected payoff can be accurately estimated and is crucial in determining the option's worth.

The strike price is a fixed price at which the holder of an option can buy (in the case of call options) the underlying futures contract. In this exercise, the strike price is set at 42.
The strike price is important because it serves as a threshold for determining whether exercising the option is profitable. When the futures price surpasses the strike price, there’s a positive payoff dependency.

• If the futures price is 45, then exercising the call option is advantageous, yielding a payoff of \(45 - 42 = 3\).
• If the futures price is below the strike price, like 35 in this example, exercising the call option incurs a loss, so the logical move is not to exercise it.

This understanding is foundational in options trading as it guides decision-making regarding exercising options.

The risk-free interest rate represents the return on an investment with zero risk, typically aligned with government bonds of short maturity. Here, it’s given as 7% per annum.
In calculating option values, the risk-free rate is used to discount future payoffs back to their present value. For this European call option, you're required to find the present value of the expected payoff calculated using risk-neutral probabilities.
The calculation involves creating a discount factor using this risk-free rate. Since the time frame is three months, this is accounted for in the discount factor calculation:\[ e^{-0.07 \times \frac{3}{12}} = 0.9826 \]
Multiplying this factor by the expected payoff (1.5) gives the present value of around 1.47.

• The risk-free rate is crucial in making comparisons between potential investments, allowing for a fair measure of opportunities differing in risk.
• It’s important to understand how it plays into present valuing expected future cash flows, making it a key factor in options pricing.

Understanding how to use this rate ensures accurate valuation of financial instruments like call options.