Question

Find the expected gain (or loss) for a holder of a European call option with strike price $$\$ 90$$ to be exercised in 6 months if the stock price on the exercise date may turn out to be $$\$ 87, \$ 92$$ or $$\$ 97$$ with probability \(\frac{1}{3}\) each, given that the option is bought for $$\$ 8,$$ financed by a loan at $$9 \%$$ compounded continuously.

Short answer

The expected loss is \$ 5.36.

Step-by-step solution

- Identify the payoff of the call option

A call option gives the holder the right to buy the stock at the strike price. If the stock price exceeds the strike price, the payoff is the stock price minus the strike price; otherwise, the payoff is zero. Let the stock price be denoted by \( S \), and the strike price \( K = 90 \). The payoff is: Payoff = \( \text{max}(S - K, 0) \).

- Calculate the payoff for each stock price scenario

For different stock prices, calculate the payoff as follows: - If the stock price is \( S = 87 \): Payoff = \( \text{max}(87 - 90, 0) = 0 \) - If the stock price is \( S = 92 \): Payoff = \( \text{max}(92 - 90, 0) = 2 \) - If the stock price is \( S = 97 \): Payoff = \( \text{max}(97 - 90, 0) = 7 \)

- Calculate the expected payoff

The expected payoff is the probability-weighted average of all possible payoffs. Given the probabilities are equal i.e., \( \frac{1}{3} \) for each scenario: Expected Payoff = \( \frac{1}{3} \times 0 + \frac{1}{3} \times 2 + \frac{1}{3} \times 7 \) Expected Payoff = \( \frac{1}{3} \times 9 = 3 \).

- Calculate the cost of the option including the loan

The option is bought for \( \$ 8 \), and financed by a loan with an interest rate of 9% compounded continuously. The cost of the loan after 6 months is: Cost = \( 8 \times e^{(0.09 \times 0.5)} \) Using the approximation \( e^x \approx 1 + x \) for small x, Cost ≈ \( 8 \times (1 + 0.045) = 8 \times 1.045 = 8.36 \).

- Calculate the expected gain (or loss)

The expected gain (or loss) is the expected payoff minus the cost of the option. Expected Gain (Loss) = Expected Payoff - Cost = 3 - 8.36 = -5.36.

Key concepts

European call option

A European call option gives the holder the right, but not the obligation, to purchase a stock at a predetermined price, known as the strike price, on a specific future date. This differs from American options, which can be exercised at any time before the expiration date.
A key element to remember is that with a European call option, if the stock price at the time of exercise is higher than the strike price, the holder can buy the stock at the lower strike price and potentially sell it at the current market price for a profit. However, if the stock price is below the strike price, the option expires worthless, making the payoff zero.
For instance, in the exercise provided, the strike price is $$\$90$$. If the stock price at maturity is above $$\$90$$, the option has value. If it is below $$\$90$$, it does not.

Expected payoff

The expected payoff of a European call option is the average outcome based on different possible future stock prices, weighted by the probability of each outcome. It essentially tells us what we can expect to gain on average from holding the option.
In our example, the stock can end up at three different prices: $$\$87$$, $$\$92$$, or $$\$97$$, each with a probability of \(\frac{1}{3}\). The payoffs for these prices are calculated as follows:

• For $$\$87$$: Payoff = max($$87 - 90$$, 0) = 0
• For $$\$92$$: Payoff = max($$92 - 90$$, 0) = 2
• For $$\$97$$: Payoff = max($$97 - 90$$, 0) = 7

To find the expected payoff, we take the probability-weighted average of these payoffs:
\[ \text{Expected Payoff} = \frac{1}{3} \times 0 + \frac{1}{3} \times 2 + \frac{1}{3} \times 7 = 3 \]

Financial mathematics

Financial mathematics involves using mathematical techniques to solve problems in finance, such as calculating the value of financial instruments like options. In this exercise, we used several mathematical concepts:

• Compounding interest: The loan was compounded continuously, a common method in financial mathematics. The formula to calculate the cost after 6 months is: \[ \text{Cost} = 8 \times e^{(0.09 \times 0.5)} \]
• Probability and expected value: The expected payoff was found using the formula: \[ \text{Expected Payoff} = \sum (\text{probability} \times \text{payoff}) \]
• Approximation techniques: Using the approximation \( e^x \approx 1 + x \) for small values of \(x\), to simplify calculations.

These tools are fundamental in finance to assess the value and risk of different investments.

Option pricing

Option pricing is the process of determining the cost of an option. Several factors affect an option's price, including the underlying stock's price, the strike price, time to expiration, interest rates, and volatility.
To determine if an investment in a call option is profitable, one must compare the option's cost with its expected payoff. In this exercise, the cost included the price of the option and the interest on the loan used to purchase it. The formula used was:
Cost = 8 \( \times e^{(0.09 \times 0.5)} \)
This simplifies to approximately $$\$8.36$$.
The expected gain or loss is then calculated as:
Expected Gain (Loss) = Expected Payoff - Cost
In this case: Expected Gain (Loss) = 3 - 8.36 = -5.36
This means the holder expects to lose $$\$5.36$$ on average. Understanding these principles helps investors make informed decisions about buying and selling options.