Question

Day trading. An option to buy a stock is priced at \(\$ 200\). If the stock closes above 30 on May 15 , the option will be worth \(\$ 1000\). If it closes below 20 , the option will be worth nothing, and if it closes between 20 and 30 (inclusively), the option will be worth \(\$ 200\), A trader thinks there is a \(50 \%\) chance that the stock will close in the \(20-30\) range, a \(20 \%\) chance that it will close above 30 , and a \(30 \%\) chance that it will fall below 20 on May 15 .a) Should she buy the stock option? b) How much does she expect to gain? c) What is the standard deviation of her gain?

Short answer

a) Yes, buy it for an expected gain. b) Expected gain is $100. c) Standard deviation is approximately $284.6.

Step-by-step solution

Calculate Expected Gain

To determine whether the trader should buy the option, calculate the expected gain. Use the formula for expected value:\[ E(X) = \sum (X_i \cdot P_i) \]where \(X_i\) is the gain in each outcome and \(P_i\) is the probability of each outcome. The gain for each scenario is:- If the stock closes above 30: Gain is \\(1000 - \\)200 = \\(800.- If the stock closes between 20 and 30: Gain is \\)200 - \\(200 = \\)0.- If the stock closes below 20: Gain is \\(0 - \\)200 = -\$200.Plug in these values:\[ E(X) = (800 \times 0.20) + (0 \times 0.50) + (-200 \times 0.30) \]\[ E(X) = 160 + 0 - 60 = 100 \].

Interpret Expected Gain

The expected gain calculated in Step 1 is \\(100. This means that, on average, the trader gains \\)100 by buying the option. Therefore, it is financially beneficial for her to purchase the option, assuming she repeats situations like this many times.

Calculate Standard Deviation

The standard deviation measures the variation of the outcomes. Use the formula for standard deviation:\[ SD(X) = \sqrt{\sum ((X_i - E(X))^2 \cdot P_i)} \]With \(E(X) = 100\), the calculations for each outcome are:- For closing above 30:\( (800 - 100)^2 \times 0.20 = 49000 \).- For closing between 20 and 30:\( (0 - 100)^2 \times 0.50 = 5000 \).- For closing below 20:\( (-200 - 100)^2 \times 0.30 = 27000 \).Adding these:\[ SD(X) = \sqrt{49000 + 5000 + 27000} = \sqrt{81000} \approx 284.6 \].

Final Step: Conclusion

With an expected gain of \\(100 and a standard deviation of approximately \\)284.6, the trader should recognize both the potential for an average profit and the associated risk. The decision to buy would depend on her risk preference, given that the gain is positive.

Key concepts

Standard Deviation

Standard deviation is a way to measure how much variation or spread there is from the expected value in different scenarios. In the context of stock options, it helps understand the level of risk involved.

When calculating the standard deviation for a stock option, you consider the difference between each possible outcome and the expected gain. You then factor in how likely each outcome is. This can be calculated using the formula:

• \[ SD(X) = \sqrt{\sum ((X_i - E(X))^2 \cdot P_i)} \]

Here, \(X_i\) is each individual outcome, \(E(X)\) is the expected gain, and \(P_i\) is the probability of each outcome.

In our example, the calculated standard deviation was approximately 284.6, indicating a notable variation in possible profits and losses. This high figure suggests that while there is a chance for substantial profit, there's also a risk of significant loss. So, while an option might have a good expected value, it's crucial to weigh that against its standard deviation to understand your risk.

Probability

Probability is a measure of how likely an event is to occur and is expressed as a percentage or a fraction. When calculating expected values in scenarios like stock options, probability helps determine the weight of each possible outcome.

To find the expected gain from buying a stock option, you take each potential gain or loss and multiply it by the probability that particular outcome will occur. Consider this simplified example:

• If stock closes above 30 with 20% probability, the outcome is a gain of \(800.
• If stock closes between 20 and 30 with 50% probability, the gain is \)0.
• If stock closes below 20 with a 30% probability, it results in a loss of \(200.

Combining these probabilities with the corresponding outcomes gives the expected gain:\[ E(X) = (800 \times 0.20) + (0 \times 0.50) + (-200 \times 0.30) = 100 \].

This means that, on average, if these trades were repeated many times, the trader could expect to make \)100 per transaction.

Stock Options

Stock options give investors the right, but not the obligation, to buy or sell a stock at a specified price within a particular time frame. Options can be complex, as their value is influenced by multiple factors including the stock's current price, expected future price movements, and time until expiration.

When deciding whether to purchase a stock option, an investor will consider factors such as expected value and risk indicated by standard deviation. The potential for financial reward is evident when the option's increase in value exceeds its purchase price and associated fees. However, if the stock performs contrary to expectations, the investor may face losses.

• An option to buy (call option) is profitable if the stock's market price exceeds the option's strike price after accounting for the cost of the option itself.
• Conversely, an option becomes worthless if the stock price moves unfavorably past the expiration date.

Thus, a trader must carefully analyze both the risk and potential gain associated with the purchase, examining probabilities and the variation of potential outcomes, as seen from the standard deviation.