Let's address each question step-by-step using the information provided in the table and the current AAPL stock price of **$180.50**. --- ### 1) **Why are the call options selling for higher prices than the put options?** **Explanation:** - **Call Option**: Right to buy at the strike price ($175.00). - **Put Option**: Right to sell at the strike price ($175.00). - Since the current stock price ($180.50) is **above** the strike price ($175.00), the calls are **in the money** (ITM), while the puts are **out of the money** (OTM). - **Intrinsic value of call** = $180.50 - $175.00 = $5.50 (at least) - **Intrinsic value of put** = $0 (since you can sell at market for more than $175.00) - Calls have immediate value; puts do not. Additionally, market expects the price could go higher, so calls have more **time value** as well. **Final Answer:** *Call options are selling for higher prices than put options because the current stock price is above the strike price, so the calls are in the money (have intrinsic value), while the puts are out of the money (no intrinsic value).* --- ### 2) **Why does the March call sell for a higher price than the January call?** **Explanation:** - **Time Value:** Options with longer time to expiration are generally more expensive due to higher time value. There’s more time for the stock to move in a favorable direction. - **March** is further away from now than **January**, so it has **more time value**. **Final Answer:** *The March call sells for a higher price than the January call because it has a longer time until expiration, giving it more time value and a greater chance for the stock to move in a profitable direction.* --- ### 3) **Suppose you buy the March call. Briefly explain whether you would exercise it immediately.** **Explanation:** - **Call Option:** Right to buy at $175.00. - **Stock Price:** $180.50. - **Exercising immediately:** You’d buy at $175.00, sell at $180.50, gaining $5.50 per share. - **But:** You paid $19.60 for the call. If you exercise now, you lose $19.60 - $5.50 = $14.10 per share. - **Options are valuable for their time value:** You’d never exercise early unless it’s very close to expiration and no further time value remains. **Final Answer:** *You should not exercise the March call immediately. The option is worth more if you sell it (because of its time value) than if you exercise it now, as immediate exercise would result in a net loss compared to the premium paid.* --- ### 4) **Suppose you buy the November call at the price listed and exercise it when the price of Apple stock is $190. What will be your profit or loss?** - **November Call Price (Premium):** $11.25 - **Strike Price:** $175.00 - **Stock Price at Exercise:** $190.00 - **Intrinsic Value at Exercise:** $190.00 - $175.00 = $15.00 **Profit/Loss Calculation:** \[ \text{Profit/Loss} = (\text{Stock Price at Exercise} - \text{Strike Price}) - \text{Call Premium} \] \[ \text{Profit/Loss} = (190 - 175) - 11.25 = 15 - 11.25 = \boxed{3.75} \] **Final Answer:** *Your profit will be $3.75 per share.* --- ### 5) **Suppose you buy the March put at the price listed, and the price of Apple stock remains $180.50. What will be your profit or loss?** - **March Put Price (Premium):** $9.75 - **Strike Price:** $175.00 - **Stock Price:** $180.50 **Put is out of the money:** - **Intrinsic Value:** $175.00 - $180.50 = -$5.50 (but never negative; you just don’t exercise) - **Loss:** You lose the premium paid. **Final Answer:** *You will have a loss equal to the premium paid, which is $9.75 per share.* --- ## **Summary Table** | Question | Final Answer | |----------|--------------| | 1 | Calls are higher because they are in the money, while puts are out of the money. | | 2 | March call is pricier than January call due to greater time value. | | 3 | Do not exercise the March call immediately; you’d lose the time value. | | 4 | Profit on November call if exercised at $190 = $3.75 per share. | | 5 | Loss on March put if stock stays at $180.50 = $9.75 per share (premium lost). |

Introduction: Understanding the valuation and strategic use of options requires familiarity with their fundamental concepts. An option is a financial derivative that grants the holder the right, but not the obligation, to buy or sell an underlying asset at a specified strike price before or at expiration. There are two primary types: call options, which give the right to buy, and put options, which give the right to sell. The price paid for an option is called the premium, reflecting its current market value. Option prices are influenced by several factors, including the intrinsic value (the immediate gain if exercised now), time value (additional worth based on the potential for favorable movement before expiration), the underlying asset's current price, strike price, time until expiration, volatility, and risk-free interest rates. When the current stock price is above the strike price, call options tend to be more in-the-money and thus carry higher premiums, whereas put options are generally out-of-the-money and less expensive. The expiration date significantly impacts option prices: longer durations increase the likelihood of favorable price movements, thereby raising the option's time value. Consequently, options with later expiration dates usually command higher premiums than those expiring sooner. Furthermore, deciding whether to exercise an option immediately depends on its moneyness and remaining time value. An option is typically exercised only when it is profitable to do so after accounting for the premium paid, or when the holder seeks to realize the intrinsic value directly. Explanation: This introduction provides the essential concepts necessary to analyze and compare option prices, including the factors influencing their premiums, the significance of time until expiration, and the decision-making process regarding exercising options. It clarifies why certain options are priced higher than others and offers insights into the strategic considerations when trading options. Establishing this foundational understanding enables a comprehensive interpretation of subsequent calculations and scenarios, such as profit and loss assessments based on stock price movements and option premiums. --- Presentation of Relevant Formulas Required To Solve The Question: 1. **Intrinsic Value of a Call Option:** \[ \text{Intrinsic Value}_{\text{call}} = \max(0, S - K) \] - Where \(S\) is the current price of the underlying asset, and \(K\) is the strike price. - This formula calculates the immediate profit from exercising the call today. 2. **Intrinsic Value of a Put Option:** \[ \text{Intrinsic Value}_{\text{put}} = \max(0, K - S) \] - This indicates the immediate profit from exercising the put today. 3. **Option Premium (Price):** \[ \text{Option Price} = \text{Intrinsic Value} + \text{Time Value} \] - The total market price comprises intrinsic value plus additional time value attributable to volatility and time remaining until expiration. 4. **Profit or Loss from Exercising an Option:** - **For Call Options:** \[ \text{Profit} = (\text{Stock Price at Exercise} - K) - \text{Premium Paid} \] - **For Put Options:** \[ \text{Profit} = (K - \text{Stock Price at Exercise}) - \text{Premium Paid} \] - If the result is positive, the exercise is profitable; if negative, it results in a loss. **Rationale:** - The intrinsic value reflects the immediate monetary benefit if the option were exercised now. - The total premium paid includes intrinsic value plus the time value, which accounts for potential future favorable movements. - Calculating profit or loss involves subtracting the premium paid from the intrinsic gain realized at exercise, considering the stock's price at the time of exercise. --- Detailed Step-by-Step Solution: **Step 1: Analyze the current stock price relative to the strike price** - Given: \( S = \$180.50 \), \( K = \$175.00 \). - Since \( S > K \), the call options are **in the money**; the put options are **out of the money**. **Step 2: Why are call options priced higher than put options?** - Because the call options are in the money, they possess intrinsic value of \( \$180.50 - \$175.00 = \$5.50 \). - The market price of these calls also includes time value, reflecting the possibility that the stock price will rise further. - Put options, being out-of-the-money, have no intrinsic value at this moment, only time value. - Market participants expect potential upside movement; thus, call options are priced higher due to their intrinsic value and optimism about future stock movement. **Step 3: Why does the March call sell for a higher price than the January call?** - The March call has a longer time until expiration, providing more opportunity for the stock to increase in value. - This additional time enhances the option's time value, making it more expensive. - The greater the time until expiration, the higher the probability that the option could become more profitable, thus increasing its premium. **Step 4: Should the March call be exercised immediately?** - Exercising now yields an intrinsic value of \( \$5.50 \). - However, the current premium paid for the March call is significantly higher (e.g., \$19.60). - Exercising immediately would realize a gain of \$5.50 but forgo the remaining time value embedded in the premium. - Since the option's market value exceeds its immediate intrinsic value, exercising immediately would lead to a net loss (premium paid minus intrinsic value). - Therefore, it is not optimal to exercise immediately; the holder should sell the option for its market value. **Step 5: Calculate profit or loss if the November call is exercised when the stock price is \$190** - **Premium paid:** \$11.25 - **Intrinsic value at stock price \$190:** \( \$190 - \$175 = \$15 \) - **Profit calculation:** \[ \text{Profit} = (\text{Intrinsic Value}) - \text{Premium Paid} = 15 - 11.25 = \boxed{\$3.75} \] - The profit per share is \$3.75. **Step 6: Calculate profit or loss if the March put is purchased and stock remains at \$180.50** - **Premium paid:** \$9.75 - **Intrinsic value (if exercised):** \( \$175 - \$180.50 = -\$5.50 \) (out-of-the-money, so no exercise) - **Outcome:** Since the put is out-of-the-money, exercising yields no intrinsic value; the holder would simply let it expire worthless. - **Loss:** The entire premium paid, which is \$9.75 per share. --- ### **Summary of Final Results:** - Call options are priced higher than puts because they are in the money and possess intrinsic value. - Longer expiration options (March vs. January) have higher premiums due to increased time value. - Immediate exercise of the March call is suboptimal because the premium exceeds the intrinsic value, reflecting remaining time value. - Exercising the November call at a stock price of \$190 yields a \$3.75 profit per share. - Buying the March put at the listed premium results in a \$9.75 loss if the stock remains at \$180.50. This detailed analysis underscores the importance of intrinsic and time value in option pricing and strategic exercise decisions.

**Introduction:** Understanding the valuation and strategic use of options necessitates familiarity with their fundamental concepts. An option is a financial derivative granting the holder the right, but not the obligation, to buy (call option) or sell (put option) an underlying asset at a predetermined strike price before or at expiration. The current market price of an option, known as the premium, is influenced by intrinsic value, time value, and other factors such as volatility, interest rates, and time remaining until expiration. **Intrinsic value** represents the immediate economic benefit if the option is exercised now: for calls, it is the difference between the underlying asset's current price and the strike price when the stock is above the strike; for puts, it is the difference when the stock is below the strike. **Time value** accounts for the potential future movements of the underlying asset, with longer durations increasing the chance of favorable price changes, thus elevating the option's premium. The **expiration date** significantly impacts the option's price; longer durations typically lead to higher premiums due to increased uncertainty and opportunity for profit. When considering whether to exercise an option early, the key is to compare its intrinsic value with its current market price, factoring in remaining time value. **Explanation:** This foundational overview clarifies how intrinsic and time value contribute to option prices, the importance of expiration, and the decision-making framework for exercising options. Recognizing why options are priced as they are enables accurate interpretation of market data, such as premiums and volume, and informs strategic trading decisions. --- **Presentation of Relevant Formulas Required To Solve The Question:** 1. **Intrinsic Value of a Call Option:** \[ \boxed{ \text{Intrinsic Value}_{\text{call}} = \max(0, S - K) } \] - *Where:* \(S\) = current stock price \(K\) = strike price - *Rationale:* It quantifies the immediate profit if the option were exercised today. 2. **Intrinsic Value of a Put Option:** \[ \boxed{ \text{Intrinsic Value}_{\text{put}} = \max(0, K - S) } \] - *Where:* same as above. - *Rationale:* Represents immediate profit from exercising the put. 3. **Option Premium (Price):** \[ \boxed{ \text{Option Price} = \text{Intrinsic Value} + \text{Time Value} } \] - *Rationale:* Total premium comprises the immediate intrinsic benefit plus the additional value derived from potential future price movements. 4. **Profit or Loss from Exercising an Option:** - **For a Call:** \[ \boxed{ \text{Profit} = (\text{Stock Price at Exercise} - K) - \text{Premium Paid} } \] - **For a Put:** \[ \boxed{ \text{Profit} = (K - \text{Stock Price at Exercise}) - \text{Premium Paid} } \] - *Where:* \( \text{Stock Price at Exercise} \) = price of the underlying at the time of exercise or expiration. - *Rationale:* Calculates the net gain or loss after considering the initial premium paid. --- **Detailed Step-by-Step Solution:** **Step 1:** *Why are call options selling for higher prices than put options?* - Since the current stock price (\$180.50) exceeds the strike price (\$175), the call options are **in the money (ITM)**, possessing intrinsic value, calculated as: \[ 180.50 - 175 = \$5.50 \] - The put options are **out of the money (OTM)**, with zero intrinsic value: \[ \max(0, 175 - 180.50) = 0 \] - The call options' premiums reflect both intrinsic and time value, making them more expensive. The put options, lacking intrinsic value, are priced primarily on time value and market expectations, which are lower under these conditions. **Supporting Statement:** The intrinsic value directly influences option prices; in-the-money calls carry inherent value, thus commanding higher premiums, whereas out-of-the-money puts do not. Market expectations and volatility further augment premiums for options with longer durations or higher uncertainty. --- **Step 2:** *Why does the March call sell for a higher price than the January call?* - The March expiration date provides a longer time horizon than January, increasing the **time value** of the option. - Longer duration offers a greater probability that the stock will move further in a favorable direction, which is reflected in a higher premium. **Supporting Statement:** The greater time until expiration enhances the chance for profitable movements, increasing the option's total premium. This aligns with fundamental option pricing principles where extended time frames elevate the option's value due to increased uncertainty and potential. --- **Step 3:** *Would the holder exercise the March call immediately?* - **Intrinsic value at current stock price:** \[ 180.50 - 175 = \$5.50 \] - **Premium paid (from data):** \$19.60 - Exercising now yields a profit of: \[ 5.50 - 19.60 = -\$14.10 \] - Since the premium exceeds the immediate intrinsic value, the option contains significant remaining time value, making early exercise suboptimal. **Supporting Statement:** Optimal exercise occurs when the intrinsic value exceeds the remaining time value; here, the premium includes substantial time premium. Exercising early forfeits this time value, leading to net losses. Therefore, holding the option or selling it is more advantageous. --- **Step 4:** *Calculate profit/loss if the November call is exercised at a stock price of \$190.* - **Premium paid:** \$11.25 - **Intrinsic value at exercise:** \[ 190 - 175 = \$15.00 \] - **Profit:** \[ 15.00 - 11.25 = \boxed{\$3.75} \] **Supporting Statement:** Exercising yields an intrinsic gain of \$15, which exceeds the premium paid, resulting in a net profit of \$3.75 per share. This highlights the profitability of exercising in-the-money options at favorable stock prices. --- **Step 5:** *Calculate profit/loss if the March put is purchased and stock remains at \$180.50.* - **Premium paid:** \$9.75 - **Intrinsic value at expiration:** \[ \max(0, 175 - 180.50) = 0 \] - **Outcome:** The put expires worthless, and the loss equals the premium paid: \[ \boxed{\$9.75} \] **Supporting Statement:** Since the stock price exceeds the strike, the put has no intrinsic value, and exercising is not profitable. The entire premium paid is lost, demonstrating the risk of buying out-of-the-money options. --- **Summary:** - Call options are priced higher than puts due to their intrinsic value when the stock is above the strike. - Longer expiration options have higher premiums owing to increased time value. - Early exercise of a valuable call is generally suboptimal when time value remains. - Exercising a profitable November call at \$190 yields a net gain of \$3.75 per share. - The March put results in a loss of \$9.75 if the stock remains at \$180.50. This comprehensive analysis, grounded in core option valuation principles and supported by precise calculations, enables informed decision-making in options trading.