Derivatives Level 4 (Investment Advice Diploma) Quiz Exam Quiz 26

The question assesses the understanding of exotic derivatives, specifically barrier options, and their behavior in relation to the underlying asset’s price movement and the knock-in/knock-out barrier. It also tests the understanding of the impact of volatility on option pricing. A down-and-in call option becomes active only when the underlying asset’s price reaches or falls below a specified barrier level. If the barrier is never touched, the option expires worthless. In this scenario, the initial price is 150, and the barrier is 130. The option becomes active when the price hits 130. From that point, the payoff is determined like a regular call option, i.e., the difference between the asset price at expiration and the strike price, if positive. Let’s analyze the scenario. The asset price drops to 130, activating the option. Then, it rises to 145 at expiration. The strike price is 140. Therefore, the payoff is 145 – 140 = 5. The investor paid a premium of £5 for the option. Therefore, the net profit is payoff – premium = 5 – 5 = 0. The example uses a specific numerical scenario to test the application of the concept. The investor’s profit or loss is calculated by considering both the payoff of the option (if any) and the initial premium paid. This requires a deep understanding of how barrier options function and how their value is derived. The distractors focus on common misconceptions about barrier options, such as ignoring the premium or misinterpreting the impact of the barrier being breached.

Let’s break down how to approach valuing an exotic derivative like an Asian option with a twist – a geometrically averaged strike price instead of the more common arithmetically averaged strike. This introduces a layer of complexity requiring careful consideration of the payoff structure. First, understand the standard Asian option. Its payoff at expiry (T) depends on the average price of the underlying asset (S) over a predetermined period. A typical Asian call option with an arithmetic average strike has a payoff of max(0, Average(S) – K), where K is the fixed strike price. An Asian put option has a payoff of max(0, K – Average(S)). Now, consider the geometric average. Instead of summing the prices and dividing by the number of observations, we multiply all the prices together and take the nth root, where n is the number of observations. This introduces a non-linearity that affects the option’s valuation. The key here is the geometrically averaged strike price. The payoff for a call option becomes max(0, S_T – GeometricAverage(S)), and for a put option, max(0, GeometricAverage(S) – S_T), where S_T is the spot price at expiry. The problem introduces a barrier – a knock-out feature. This means the option ceases to exist if the underlying asset’s price touches a certain level (the barrier) during the option’s life. This significantly reduces the option’s value, as there’s a chance it will expire worthless even if the underlying moves favorably. The Monte Carlo simulation is the most appropriate valuation method for this complex derivative. It involves simulating a large number of possible price paths for the underlying asset, taking into account its volatility and drift (risk-free rate). For each path, we calculate the geometric average, check if the barrier has been breached, and calculate the payoff if the option is still alive at expiry. We then average the payoffs across all paths and discount back to the present value to get the option’s price. The risk-neutral drift is calculated as \(r – \frac{\sigma^2}{2}\), where \(r\) is the risk-free rate and \(\sigma\) is the volatility. This drift is used in the simulation to ensure that the expected return on the underlying asset is equal to the risk-free rate, which is a requirement for risk-neutral valuation. The geometric average calculation is performed for each simulated path. If at any point the price touches the barrier, the payoff for that path becomes zero. The final price is the average of all discounted payoffs. The put-call parity relationship doesn’t directly apply here due to the path-dependent nature of the Asian option and the knock-out barrier. Standard put-call parity holds for European options with fixed strikes and no barriers.

The optimal hedge ratio in futures hedging minimizes the variance of the hedged portfolio. It is calculated as the correlation between the change in spot price and the change in futures price, multiplied by the ratio of the standard deviation of the spot price change to the standard deviation of the futures price change. This is often represented as: \[ h = \rho \cdot \frac{\sigma_s}{\sigma_f} \] Where: \( h \) = hedge ratio \( \rho \) = correlation between changes in spot and futures prices \( \sigma_s \) = standard deviation of spot price changes \( \sigma_f \) = standard deviation of futures price changes In this scenario, we are given the correlation (\(\rho = 0.75\)), the standard deviation of spot price changes (\(\sigma_s = 0.03\)), and the standard deviation of futures price changes (\(\sigma_f = 0.04\)). We can directly substitute these values into the formula: \[ h = 0.75 \cdot \frac{0.03}{0.04} = 0.75 \cdot 0.75 = 0.5625 \] Therefore, the optimal hedge ratio is 0.5625. Now, let’s consider why this hedge ratio is optimal. Imagine a farmer who wants to hedge their wheat crop using wheat futures. If the correlation between the spot price of wheat and the futures price is high, it means that they tend to move in the same direction. However, the magnitudes of these movements can differ, captured by the ratio of standard deviations. The hedge ratio adjusts the number of futures contracts needed to offset the price risk in the spot market. If the spot price is more volatile than the futures price (i.e., \(\sigma_s > \sigma_f\)), the hedge ratio will be greater than the correlation, indicating that more futures contracts are needed to hedge each unit of the spot commodity. Conversely, if the futures price is more volatile, the hedge ratio will be less than the correlation. The hedge ratio of 0.5625 means that for every unit of the underlying asset the investor holds, they should short 0.5625 units of the futures contract to minimize the risk. For example, if an investor holds 1000 shares of a stock, they should short 562.5 (which is practically 563) futures contracts to hedge their position effectively. This minimizes the overall variance of the portfolio, providing the best possible risk mitigation strategy given the observed relationship between spot and futures price movements. A higher or lower hedge ratio would lead to either under-hedging (leaving the portfolio exposed to price risk) or over-hedging (introducing unnecessary risk from the futures position).

To determine the breakeven point for the combined strategy, we need to consider the premium paid for the call options and the premium received for the put options. The investor is effectively creating a collar strategy. The breakeven points represent the stock prices at which the strategy neither makes nor loses money. Lower Breakeven Point: This is calculated by subtracting the net premium (premium received from selling puts minus premium paid for buying calls) from the strike price of the short put. \[ \text{Lower Breakeven Point} = \text{Strike Price of Short Put} – (\text{Premium Received for Put} – \text{Premium Paid for Call}) \] \[ \text{Lower Breakeven Point} = 95 – (2.50 – 3.50) = 95 – (-1) = 96 \] Upper Breakeven Point: This is calculated by adding the net premium (premium received from selling puts minus premium paid for buying calls) to the strike price of the long call. \[ \text{Upper Breakeven Point} = \text{Strike Price of Long Call} + (\text{Premium Paid for Call} – \text{Premium Received for Put}) \] \[ \text{Upper Breakeven Point} = 105 + (3.50 – 2.50) = 105 + 1 = 106 \] Therefore, the investor will start making a profit if the stock price goes below 96 or above 106. This strategy profits from stock prices remaining within a specific range, and the breakeven points define the boundaries of acceptable movement. The investor is effectively capping their potential profit and limiting their potential loss. In essence, the investor is betting that the stock price will stay within a certain band. If the price moves significantly outside this band, the investor will either forgo potential gains (above 106) or start incurring losses (below 96). The net premium received or paid adjusts the breakeven points, influencing the range within which the strategy is profitable.

The question revolves around the concept of delta-hedging a short call option position and the impact of a sudden price jump in the underlying asset. Delta-hedging aims to neutralize the risk associated with changes in the underlying asset’s price by dynamically adjusting the position in the underlying asset. A short call option position has a negative delta, meaning the option’s value decreases as the underlying asset’s price increases. To delta-hedge this position, an investor would typically buy shares of the underlying asset. The number of shares to buy is determined by the option’s delta. In this scenario, the investor is perfectly delta-hedged before the price jump. However, the price jump instantaneously changes the option’s delta. Since the price increased, the call option is now deeper in the money, and its delta increases (becomes less negative). This means the investor needs to buy more shares to maintain the delta-neutral position. The investor’s profit or loss depends on the difference between the initial delta hedge and the new delta hedge after the price jump. The initial delta hedge was 5,000 shares. The new delta after the jump is 0.75. This means the investor needs to hold 7,500 shares (10,000 contracts * 0.75 delta). Therefore, the investor needs to buy an additional 2,500 shares (7,500 – 5,000). The cost of buying these additional shares at the new price of £105 is 2,500 * £105 = £262,500. This represents the loss incurred due to the price jump and the need to re-hedge the position. The question tests understanding of delta-hedging dynamics and the impact of sudden price movements on option positions. It requires the candidate to calculate the change in delta exposure and the resulting profit or loss from re-hedging.

To determine the fair price of the Asian option, we need to calculate the average price of the asset over the specified period and then apply the option pricing logic. Since this is a discrete average price option, we will calculate the arithmetic average of the asset prices at the end of each month. 1. **Calculate the Average Price:** Average Price = (105 + 108 + 112 + 109 + 115 + 110) / 6 = 659 / 6 = 109.83 2. **Determine the Payoff:** Since this is a call option, the payoff is max(Average Price – Strike Price, 0). Payoff = max(109.83 – 110, 0) = max(-0.17, 0) = 0 3. **Discount the Payoff:** The risk-free rate is 5% per annum, so the monthly rate is approximately 5%/12 = 0.004167. The option expires in 6 months, so we need to discount the payoff back 6 months. Discount Factor = \(e^{-rT}\) = \(e^{-0.05 \times 0.5}\) = \(e^{-0.025}\) ≈ 0.9753 Present Value of Payoff = 0 * 0.9753 = 0 Therefore, the fair price of the Asian call option is approximately 0. The example illustrates the importance of the averaging period in determining the option’s payoff. Unlike standard European or American options that depend solely on the asset’s price at expiration, Asian options consider the average price over a period. This feature reduces the impact of price volatility and makes Asian options attractive to investors who want to hedge against average price risk, such as commodity producers or consumers. Consider a multinational corporation that imports raw materials monthly. To hedge against fluctuations in the average cost of these materials, the corporation could use an Asian option. The strike price could be set at a level that ensures profitability, and the averaging feature would protect against short-term price spikes. Another application is in the energy sector. An oil refiner might use an Asian option to hedge against the average price of crude oil over a quarter. This would provide more stable hedging compared to using a standard option that only considers the price at the expiration date. The risk-free rate is used to discount the expected payoff back to the present value, reflecting the time value of money. A higher risk-free rate would result in a lower present value of the payoff, and vice versa.

The core of this question revolves around understanding how the margin requirements for futures contracts operate, specifically in the context of intraday price volatility and the potential for margin calls. The initial margin is the amount required to open a futures position, and the maintenance margin is the level below which the account cannot fall. If the account balance drops below the maintenance margin, a margin call is issued, requiring the investor to deposit additional funds to bring the account back to the initial margin level. In this scenario, we must calculate the cumulative effect of the intraday price fluctuations on the investor’s margin account. Each adverse price movement reduces the account balance, and each favorable movement increases it. If the balance falls below the maintenance margin, we need to determine the amount required to restore the account to the initial margin. Here’s the breakdown of the calculations: 1. **Initial Margin:** £8,000 2. **Maintenance Margin:** £6,000 3. **Price Drop 1:** 5 points, each point worth £100. Loss = 5 * £100 = £500. Account balance: £8,000 – £500 = £7,500. 4. **Price Drop 2:** 8 points, each point worth £100. Loss = 8 * £100 = £800. Account balance: £7,500 – £800 = £6,700. 5. **Price Increase:** 3 points, each point worth £100. Gain = 3 * £100 = £300. Account balance: £6,700 + £300 = £7,000. 6. **Price Drop 3:** 12 points, each point worth £100. Loss = 12 * £100 = £1,200. Account balance: £7,000 – £1,200 = £5,800. Since the account balance of £5,800 is below the maintenance margin of £6,000, a margin call is triggered. The investor needs to deposit enough funds to bring the account back to the initial margin of £8,000. Therefore, the margin call amount is £8,000 – £5,800 = £2,200. This scenario highlights the importance of understanding margin requirements and the potential risks associated with leveraged derivatives trading. A seemingly small price fluctuation can trigger a margin call, requiring the investor to deposit additional funds to maintain their position. Failure to meet a margin call can result in the forced liquidation of the position, potentially leading to significant losses. The dynamic nature of intraday price movements and their impact on margin accounts is a critical aspect of derivatives trading that investment advisors must understand and explain to their clients. The initial margin acts as a performance bond, ensuring that the investor can cover potential losses. The maintenance margin serves as an early warning system, alerting the investor to the need for additional funds before losses become unmanageable.

Let’s consider a scenario where a UK-based agricultural cooperative, “Green Harvest,” wants to protect its future revenue from selling wheat. They anticipate harvesting 5,000 metric tons of wheat in six months. The current spot price of wheat is £200 per metric ton. However, Green Harvest is concerned about a potential price drop due to an anticipated bumper crop worldwide. They decide to use wheat futures contracts traded on the London International Financial Futures and Options Exchange (LIFFE) to hedge their price risk. Each LIFFE wheat futures contract is for 100 metric tons. Therefore, Green Harvest needs to sell 50 futures contracts (5,000 metric tons / 100 metric tons per contract = 50 contracts). The current futures price for wheat with a delivery date six months from now is £210 per metric ton. Six months later, the following occurs: * Green Harvest harvests its 5,000 metric tons of wheat. * The spot price of wheat has fallen to £180 per metric ton due to the global oversupply. * The futures price for wheat with immediate delivery is also £180 per metric ton. To calculate Green Harvest’s effective selling price, we need to consider the following: 1. **Sale of Wheat in the Spot Market:** Green Harvest sells its 5,000 metric tons of wheat at the spot price of £180 per metric ton, generating revenue of 5,000 * £180 = £900,000. 2. **Gain or Loss on Futures Contracts:** Green Harvest initially sold 50 futures contracts at £210 per metric ton. They now buy back those contracts (or their equivalent) at £180 per metric ton. The gain per contract is (£210 – £180) * 100 metric tons = £3,000. The total gain on 50 contracts is 50 * £3,000 = £150,000. 3. **Effective Selling Price:** The total revenue from selling the wheat in the spot market is £900,000, and the gain from the futures contracts is £150,000. Therefore, the total effective revenue is £900,000 + £150,000 = £1,050,000. 4. **Effective Price per Metric Ton:** The effective selling price per metric ton is £1,050,000 / 5,000 metric tons = £210 per metric ton. This example illustrates how futures contracts can be used to hedge against price risk. Green Harvest effectively locked in a selling price close to the initial futures price, mitigating the impact of the price decline in the spot market. The key is understanding the relationship between spot and futures prices, and how gains or losses on the futures contracts offset changes in the spot market revenue. The futures market acts as a price discovery mechanism and allows producers to manage their exposure to price volatility. It’s also important to note the potential for margin calls if the futures price moves against Green Harvest before the delivery date, which would require them to deposit additional funds into their margin account.

The question assesses the understanding of exotic derivatives, specifically a cliquet option, and its payoff structure. A cliquet option is a series of forward-starting options, each capping the return for a specific period. The global cap limits the total accumulated return over the entire life of the option. The calculation involves determining the return for each period, applying the local cap, accumulating these capped returns, and finally applying the global cap. Period 1 Return: \( \frac{105 – 100}{100} = 0.05 \) or 5%. Since this is less than the 3% local cap, the accumulated return is 3%. Period 2 Return: \( \frac{108 – 105}{105} = 0.0286 \) or 2.86%. Since this is less than the 3% local cap, the accumulated return becomes 3% + 2.86% = 5.86%. Period 3 Return: \( \frac{100 – 108}{108} = -0.0741 \) or -7.41%. Since this is a negative return, the accumulated return becomes 5.86% – 7.41% = -1.55%. Period 4 Return: \( \frac{110 – 100}{100} = 0.10 \) or 10%. Since this exceeds the 3% local cap, the return for this period is capped at 3%. The accumulated return becomes -1.55% + 3% = 1.45%. Period 5 Return: \( \frac{115 – 110}{110} = 0.0455 \) or 4.55%. Since this exceeds the 3% local cap, the return for this period is capped at 3%. The accumulated return becomes 1.45% + 3% = 4.45%. The global cap is 5%. Since the accumulated return of 4.45% is less than the global cap, the final payoff is based on the 4.45% accumulated return. Therefore, the final payoff is 4.45% of the notional principal. If the notional principal is £1,000,000, the payoff is \( 0.0445 \times 1,000,000 = £44,500 \). A key understanding is that cliquet options offer a path-dependent payoff, meaning the final payoff depends on the sequence of returns over the life of the option. Local caps limit the upside for each period, while the global cap limits the overall accumulated return. This structure provides a balance between participation in market gains and protection against excessive losses. The investor benefits from positive returns in each period, but the caps prevent any single period from disproportionately influencing the final payoff. The global cap ensures that the total return remains within a defined range, providing a degree of certainty for the investor.

To determine the most suitable hedging strategy, we need to evaluate the potential profit or loss under different scenarios and compare them with the outcomes of using various derivative instruments. In this case, the company wants to protect against a decrease in the price of their copper inventory. A forward contract locks in a future selling price, eliminating both upside and downside potential. A put option gives the company the right, but not the obligation, to sell copper at a specified price (the strike price). If the copper price falls below the strike price, the company can exercise the option and sell at the higher strike price, mitigating losses. If the copper price rises, the company can let the option expire and sell the copper at the market price, benefiting from the increase. A call option, on the other hand, would be beneficial if the company wanted to profit from an increase in the copper price, but it doesn’t protect against a price decrease. A short futures position would provide similar downside protection to a forward contract, but with the added complexity of margin calls and daily settlements. Given the company’s primary goal of protecting against a price decrease while retaining the ability to benefit from a price increase, purchasing a put option is the most appropriate strategy. The put option provides a floor for the selling price, limiting potential losses, while still allowing the company to profit if the copper price rises above the strike price.

The correct answer is b) £3.1 million, representing the difference between the realized variance and the implied variance (256 – 225 = 31) multiplied by the notional principal (£0.1 million per variance point). This question tests the understanding of variance swaps and their use in hedging volatility risk. The key is to correctly calculate the payoff based on the difference between realized and implied variance and the notional principal. Here’s a breakdown: 1. **Variance Swap Payoff:** The payoff of a variance swap is determined by the difference between the realized variance and the strike variance

The correct answer is (a). This question assesses the understanding of how exotic derivatives, specifically barrier options, are affected by market volatility and correlation between the underlying assets. In this case, the digital barrier option pays a fixed amount if both assets close above their respective barriers on the expiration date. Here’s a breakdown of why the other options are incorrect and a detailed explanation of why (a) is correct: **Why other options are incorrect:** * **(b)**: While increased volatility in individual assets generally increases option prices, the correlation aspect is crucial here. If the assets are negatively correlated, an increase in individual asset volatility might *decrease* the likelihood of *both* assets simultaneously breaching their barriers, thus decreasing the option’s value. This option oversimplifies the impact of volatility and ignores the crucial role of correlation. * **(c)**: A decrease in correlation would increase the likelihood of at least one asset breaching its barrier, making it less likely that *both* will stay above. This would decrease the value of the option, not increase it. The option also incorrectly states that the volatility would decrease the option’s value, which is the opposite of what would happen. * **(d)**: A decrease in volatility would decrease the option’s value, not increase it. Volatility is a key driver of option prices, and a decrease in volatility generally leads to a decrease in option prices. The effect of correlation is also misunderstood. If the assets are positively correlated, a decrease in correlation would decrease the option’s value, not increase it. **Why (a) is correct:** This option correctly identifies the combined impact of volatility and correlation. Let’s consider a simplified analogy. Imagine two climbers attempting to scale separate icy walls. The digital barrier option pays out only if *both* climbers reach a certain height (the barrier). * **Increased Volatility:** If the weather becomes more volatile (winds gusting, ice shifting more unpredictably), each climber’s chances of both succeeding and failing increase. The increased uncertainty raises the option’s value, as there’s a greater chance of a payout (both succeed) and a greater chance of no payout (at least one fails). * **Increased Correlation:** If the weather conditions on both walls become more correlated (e.g., both walls experience the same wind gusts simultaneously), the climbers’ fates become intertwined. If one climber is likely to succeed, the other is also more likely to succeed, and vice versa. This *increases* the probability of both succeeding together, thus increasing the value of the digital barrier option. Therefore, the combined effect of increased volatility and increased positive correlation between the assets makes it more likely that both assets will close above their respective barriers at expiration, increasing the value of the digital barrier option.

The optimal hedge ratio minimizes the variance of the hedged portfolio. This involves calculating the correlation between the asset being hedged and the hedging instrument (in this case, a futures contract). The formula for the hedge ratio is: Hedge Ratio = Correlation * (Volatility of Asset / Volatility of Futures Contract) First, we need to calculate the volatilities (standard deviations) of both the stock and the futures contract. Given the weekly price changes, we calculate the standard deviation using the formula: Standard Deviation = \(\sqrt{\frac{\sum (x_i – \bar{x})^2}{n-1}}\) Where \(x_i\) is each weekly price change, \(\bar{x}\) is the average weekly price change, and \(n\) is the number of weeks. For the stock: Average weekly price change = (2 + 1 – 3 + 0 + 2) / 5 = 0.4 Variance = \(\frac{(2-0.4)^2 + (1-0.4)^2 + (-3-0.4)^2 + (0-0.4)^2 + (2-0.4)^2}{5-1}\) = \(\frac{2.56 + 0.36 + 11.56 + 0.16 + 2.56}{4}\) = \(\frac{17.2}{4}\) = 4.3 Standard Deviation (Stock) = \(\sqrt{4.3}\) ≈ 2.07 For the futures contract: Average weekly price change = (1 + 0 – 2 + 1 + 1) / 5 = 0.2 Variance = \(\frac{(1-0.2)^2 + (0-0.2)^2 + (-2-0.2)^2 + (1-0.2)^2 + (1-0.2)^2}{5-1}\) = \(\frac{0.64 + 0.04 + 4.84 + 0.64 + 0.64}{4}\) = \(\frac{6.8}{4}\) = 1.7 Standard Deviation (Futures) = \(\sqrt{1.7}\) ≈ 1.30 Now, we calculate the correlation between the stock and the futures contract. The formula for correlation is: Correlation = Covariance / (Standard Deviation of Stock * Standard Deviation of Futures) First, we calculate the covariance: Covariance = \(\frac{\sum (x_i – \bar{x})(y_i – \bar{y})}{n-1}\) Covariance = \(\frac{(2-0.4)(1-0.2) + (1-0.4)(0-0.2) + (-3-0.4)(-2-0.2) + (0-0.4)(1-0.2) + (2-0.4)(1-0.2)}{4}\) Covariance = \(\frac{(1.6)(0.8) + (0.6)(-0.2) + (-3.4)(-2.2) + (-0.4)(0.8) + (1.6)(0.8)}{4}\) Covariance = \(\frac{1.28 – 0.12 + 7.48 – 0.32 + 1.28}{4}\) = \(\frac{9.6}{4}\) = 2.4 Correlation = 2.4 / (2.07 * 1.30) ≈ 2.4 / 2.691 ≈ 0.89 Finally, we calculate the hedge ratio: Hedge Ratio = 0.89 * (2.07 / 1.30) ≈ 0.89 * 1.59 ≈ 1.42 Since the investor wants to hedge a portfolio worth £750,000 and each futures contract covers £500,000, we first determine the equivalent portfolio size in terms of futures contracts: 750,000/500,000 = 1.5. The number of futures contracts to short is then: 1.5 * 1.42 ≈ 2.13. Since you can’t trade fractions of contracts, you would round to the nearest whole number. However, the question specifies to find the *optimal* number of contracts based on the hedge ratio, and rounding introduces error. The closest answer to the calculated value is 2.13, so we consider that to be the most precise answer given the constraints.

Let’s analyze the combined impact of delta hedging a short call option position and the subsequent gamma exposure when the underlying asset’s price experiences a significant move. We will compute the profit or loss arising from adjusting the hedge and the option’s payoff. Assume an investor initially sells a call option on shares of “NovaTech,” a technology company. The option has a strike price of £150, and the investor receives a premium of £8 per share (total £800 for 100 shares). The investor delta hedges this short position by buying shares of NovaTech. Initially, the option’s delta is 0.4. This means the investor buys 40 shares of NovaTech at the current market price of £148 per share, costing £5920 (40 * £148). Now, let’s say the price of NovaTech shares unexpectedly rises to £160 per share. The option is now in the money. The option’s delta increases to 0.8 due to its gamma. The investor needs to adjust their hedge. To maintain a delta-neutral position, the investor must buy an additional 40 shares (80 total shares – initial 40 shares) at the new market price of £160 per share, costing £6400 (40 * £160). The total cost of hedging is £5920 + £6400 = £12320. At £160, the call option is exercised. As the option was sold, the investor is obligated to deliver the shares at the strike price of £150. Since the investor has 80 shares, the investor will need to buy another 20 shares to fulfil the 100 shares, so the cost is £160 * 20 = £3200. The total cost is now £12320 + £3200 = £15520. The investor receives £150 * 100 = £15000 for delivering the shares. The loss from buying and delivering the shares is £15520 – £15000 = £520. However, the investor initially received a premium of £800 for selling the call option. Therefore, the net profit is £800 – £520 = £280. This scenario demonstrates how delta hedging aims to neutralize directional risk, but gamma exposure introduces complexities, especially during significant price movements. The need to rebalance the hedge incurs transaction costs and can impact the overall profitability of the strategy. The analysis illustrates the dynamic nature of derivatives risk management and the importance of considering higher-order Greeks like gamma.

The value of a European call option is influenced by several factors, including the current stock price, the strike price, the time to expiration, the risk-free interest rate, and the volatility of the underlying asset. This question specifically tests the understanding of how changes in the risk-free interest rate affect the option price. According to option pricing theory, a higher risk-free interest rate generally increases the value of a call option. This is because the present value of the strike price decreases, making the option more attractive. Conversely, a lower risk-free interest rate decreases the value of a call option. To calculate the impact, we can use the Black-Scholes model as a conceptual framework, even though the exact formula isn’t necessary for this qualitative assessment. The key is understanding the relationship: call option price is positively correlated with the risk-free rate. In this scenario, the risk-free rate decreases from 5% to 3%. This reduction makes the call option less attractive because the present value of paying the strike price at expiration is now slightly higher than before. To compensate for this decreased attractiveness, the call option price will decrease. The magnitude of the decrease depends on the other parameters (stock price, strike price, time to expiration, volatility), but the direction is clear: a decrease in the risk-free rate leads to a decrease in the call option price. The other options present scenarios where the call option price increases, which contradict the fundamental relationship between the risk-free rate and the call option price. Therefore, the option that reflects a decrease in the call option price is the correct answer.

The correct answer is (a). This question tests understanding of the factors influencing option prices and the application of hedging strategies. Gamma measures the rate of change of an option’s delta with respect to a change in the underlying asset’s price. A high gamma indicates that the delta is very sensitive to price changes, meaning the hedge needs frequent adjustments. Time decay (theta) erodes the value of an option as it approaches expiration, particularly impacting short-dated options. Volatility, often measured by implied volatility, reflects the market’s expectation of future price fluctuations. Higher volatility increases the value of options, as there’s a greater chance of the option moving into the money. The risk-free interest rate also plays a role, affecting the present value of future payoffs. In this scenario, the fund manager is using a delta-neutral strategy, meaning the portfolio’s delta is zero, making it insensitive to small price changes in the underlying asset. However, the gamma is high, implying that the delta changes rapidly. The fund manager must rebalance the hedge frequently to maintain delta neutrality. With short-dated options, time decay accelerates, further complicating the hedging process. Higher volatility increases the value of the options, but also increases the uncertainty in the hedge. The interest rate has a smaller, but still relevant, impact. The most critical factors are the high gamma, which requires frequent rebalancing, and the accelerating time decay, which erodes the option’s value rapidly. The fund manager must carefully consider these factors when choosing the frequency of rebalancing. Failing to do so can lead to significant losses if the hedge becomes ineffective due to rapid changes in the underlying asset’s price or the option’s value.

To answer this question, we need to understand how delta hedging works and how the profit or loss on the hedge is calculated. Delta hedging aims to create a portfolio that is insensitive to small changes in the underlying asset’s price. This is achieved by taking an offsetting position in the underlying asset. The number of units of the underlying asset needed to offset the option’s delta is called the hedge ratio. The profit or loss on the delta hedge is the difference between the change in the option’s price and the cost of maintaining the hedge. In this scenario, the fund initially sells call options and delta hedges by buying shares. As the market moves, the fund needs to rebalance the hedge by buying or selling shares to maintain the desired delta. The profit or loss on the hedge is calculated by considering the cost of buying shares and the proceeds from selling shares. Specifically, we track the changes in the share price and the fund’s corresponding actions: 1. **Initial Hedge:** The fund sells 100 call options with a delta of 0.40. To delta hedge, the fund buys 100 \* 0.40 = 40 shares at £50 each. Cost: 40 \* £50 = £2000. 2. **Price Increase to £52:** The delta increases to 0.50. The fund needs to buy an additional 100 \* (0.50 – 0.40) = 10 shares at £52 each. Cost: 10 \* £52 = £520. 3. **Price Decrease to £48:** The delta decreases to 0.30. The fund needs to sell 100 \* (0.50 – 0.30) = 20 shares at £48 each. Proceeds: 20 \* £48 = £960. 4. **Price Increase to £55:** The delta increases to 0.70. The fund needs to buy an additional 100 \* (0.70 – 0.30) = 40 shares at £55 each. Cost: 40 \* £55 = £2200. 5. **Price Decrease to £45:** The delta decreases to 0.20. The fund needs to sell 100 \* (0.70 – 0.20) = 50 shares at £45 each. Proceeds: 50 \* £45 = £2250. Total cost of buying shares: £2000 + £520 + £2200 = £4720. Total proceeds from selling shares: £960 + £2250 = £3210. Net cost of hedging = Total cost – Total proceeds = £4720 – £3210 = £1510. The call options expire worthless, so the fund keeps the premium received from selling the options. The question states the fund received £800 premium per option, hence total premium is 100 * £800 = £80,000. The overall profit is the premium received minus the cost of hedging: £80,000 – £1510 = £78,490. This example illustrates the dynamic nature of delta hedging and how continuous rebalancing is required to maintain a near-neutral position with respect to price changes in the underlying asset. The profitability of the strategy depends on the magnitude and frequency of these price changes, as well as the initial premium received.

The value of a swap is determined by calculating the present value of the expected future cash flows. In an interest rate swap, these cash flows are the differences between the fixed rate payments and the floating rate payments. To value the swap, we need to discount these future cash flows back to the present using the appropriate discount rates (which are derived from the spot rate curve). First, we calculate the expected floating rate payments for each period. Since the floating rate resets at the beginning of each period, we use the forward rates to forecast these payments. The forward rate is derived from the spot rates using the formula: \[F_{0,1} = \frac{(1 + S_1)}{(1 + S_0)} – 1 \] Where \(S_1\) is the spot rate for the period to the payment date, and \(S_0\) is the spot rate for the period to the previous payment date (or today). Then, we calculate the present value of each cash flow by discounting it back to today using the corresponding spot rate. The present value of a cash flow is given by: \[PV = \frac{CF}{(1 + S)^t}\] Where \(CF\) is the cash flow, \(S\) is the spot rate for the period \(t\). Finally, we sum the present values of all the cash flows to get the value of the swap. A positive value indicates that the swap is an asset for the party receiving the fixed rate (and a liability for the party paying the fixed rate), while a negative value indicates the opposite. In this scenario, the swap has a notional principal of £5,000,000. The fixed rate is 3.5% per annum, paid semi-annually, so each fixed payment is £5,000,000 * 0.035 / 2 = £87,500. The floating rate is reset semi-annually based on LIBOR. The spot rates are given as: 6-month spot rate: 3.0% 12-month spot rate: 3.2% 18-month spot rate: 3.4% 24-month spot rate: 3.6% First, we calculate the forward rates: 6-month forward rate (6 months to 12 months): \(\frac{(1 + 0.032)}{(1 + 0.030)} – 1 = 0.00194 = 0.194\%\) semi-annual rate. Therefore, the floating rate payment at 12 months is £5,000,000 * 0.03194 = £95,820 12-month forward rate (12 months to 18 months): \(\frac{(1 + 0.034)^{1.5}}{(1 + 0.032)} – 1 = 0.00299 = 0.299\%\) semi-annual rate. Therefore, the floating rate payment at 18 months is £5,000,000 * 0.03399 = £101,970 18-month forward rate (18 months to 24 months): \(\frac{(1 + 0.036)^{2}}{(1 + 0.034)^{1.5}} – 1 = 0.00399 = 0.399\%\) semi-annual rate. Therefore, the floating rate payment at 24 months is £5,000,000 * 0.03599 = £107,970 Next, we calculate the net cash flows (Fixed – Floating): At 6 months: £87,500 At 12 months: £87,500 – £95,820 = -£8,320 At 18 months: £87,500 – £101,970 = -£14,470 At 24 months: £87,500 – £107,970 = -£20,470 Finally, we calculate the present values: At 6 months: \(\frac{87500}{(1 + 0.030)} = £84,951.46\) At 12 months: \(\frac{-8320}{(1 + 0.032)^2} = -£7,799.58\) At 18 months: \(\frac{-14470}{(1 + 0.034)^3} = -£13,038.98\) At 24 months: \(\frac{-20470}{(1 + 0.036)^4} = -£17,641.78\) Swap Value = £84,951.46 – £7,799.58 – £13,038.98 – £17,641.78 = £46,471.12 Therefore, the value of the swap to the party receiving fixed is approximately £46,471.12

1. **Understanding the Scenario:** A down-and-out call option becomes worthless if the underlying asset’s price touches or falls below the barrier level. When the asset price is very close to the barrier, the option’s value becomes highly sensitive to even small price changes. This sensitivity is reflected in a large negative gamma near the barrier. 2. **Delta and Barrier Options:** Delta measures the sensitivity of the option price to changes in the underlying asset’s price. For a standard call option, delta is positive (the option price increases as the asset price increases). However, for a down-and-out call near the barrier, the delta can become significantly reduced and even negative as the risk of being knocked out increases. 3. **Approximation:** Since the spot price is very close to the barrier, a small decrease in the spot price will likely cause the option to be knocked out, rendering it worthless. Therefore, the option’s value is highly sensitive to downward price movements. Conversely, a small increase in the spot price won’t significantly increase the option’s value because the barrier effect still looms large. 4. **Calculating Approximate Delta:** Given that the spot price is just above the barrier, the option behaves almost like a binary (digital) option that pays out only if the barrier is not hit. A small decrease in the spot price causes the option to expire worthless. This means the option’s value will drop significantly with even a small decrease in the underlying asset’s price. Therefore, the delta will be highly negative. 5. **The Correct Answer:** The most appropriate answer is -0.95. This reflects the high sensitivity of the option’s price to downward movements in the underlying asset’s price when it’s near the barrier. The option is almost certain to be knocked out with even a slight decrease in the asset’s price. 6. **Why Other Options are Incorrect:** * 0.05: This would be a small positive delta, indicating the option’s price increases slightly with the asset’s price, which is not the case near the barrier for a down-and-out call. * 0.50: This would be a moderate positive delta, which is also not correct near the barrier. * -0.50: This is a negative delta, but not as extreme as -0.95. The option is extremely sensitive to downward price movements near the barrier, so the delta would be closer to -1. Analogy: Imagine you’re holding a fragile glass vase on the edge of a table. The table represents the barrier. If the vase (the underlying asset’s price) is very close to the edge (the barrier), a small nudge (price decrease) will cause it to fall and break (the option is knocked out and becomes worthless). The closer the vase is to the edge, the more sensitive it is to being knocked off. This sensitivity is reflected in the large negative delta.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Green Harvest,” which relies on a specific type of fertilizer imported from a volatile geopolitical region. Green Harvest faces uncertainty about the future price of this fertilizer due to potential supply chain disruptions and currency fluctuations. They decide to use a derivative strategy to hedge their exposure. The cooperative anticipates needing 500 tonnes of fertilizer in six months. The current spot price is £400 per tonne. Futures contracts on a similar fertilizer product are available, each contract representing 10 tonnes, and currently trading at £410 for delivery in six months. Green Harvest decides to hedge 80% of their anticipated need using futures contracts. First, we calculate the total amount to be hedged: 500 tonnes * 80% = 400 tonnes. Next, we determine the number of futures contracts needed: 400 tonnes / 10 tonnes per contract = 40 contracts. If, at the delivery date, the spot price of the fertilizer is £430 per tonne, Green Harvest will buy the fertilizer at this price. Simultaneously, they will close out their futures position, selling the contracts at the prevailing market price. Let’s assume the futures price converges to the spot price at delivery, so they sell at £430. The profit per futures contract is £430 – £410 = £20. The total profit from the futures contracts is 40 contracts * £20 per contract * 10 tonnes/contract = £8,000. Without hedging, Green Harvest would have paid £430 * 400 = £172,000 for the 400 tonnes. With hedging, they effectively paid £172,000 – £8,000 = £164,000. This equates to an effective price of £164,000 / 400 tonnes = £410 per tonne, plus the unhedged amount at the new spot price. However, this profit is not free. Green Harvest had to deposit initial margin and may have faced margin calls if the futures price had moved against them. Furthermore, the basis risk – the difference between the futures price and the spot price – can affect the effectiveness of the hedge. If the futures price hadn’t converged to the spot price, the hedge would have been less effective. The decision to hedge only 80% also introduces some risk, as the remaining 20% is exposed to the spot price fluctuations. This example demonstrates how futures contracts can be used to mitigate price risk, but it also highlights the importance of understanding the nuances of hedging strategies, margin requirements, and basis risk. The FCA would require Green Harvest’s advisor to have assessed their risk appetite and hedging objectives before recommending this strategy.

The question assesses the understanding of exotic derivatives, specifically barrier options, and how their value and delta are affected by the barrier being breached. A down-and-out call option becomes worthless if the underlying asset’s price hits the barrier level before the option’s expiration. Let’s analyze the impact on delta when the barrier is breached. Delta measures the sensitivity of the option’s price to a change in the underlying asset’s price. Before the barrier is hit, the down-and-out call option has a positive delta, similar to a regular call option, albeit smaller due to the barrier risk. If the barrier is breached, the option becomes worthless, and its value becomes independent of the underlying asset’s price. Consequently, the delta drops to zero. Consider a scenario where an investor holds a portfolio hedged with a down-and-out call option. The investor is long the underlying asset and short the down-and-out call option. Initially, the hedge is constructed to offset the price movements of the underlying asset. However, when the barrier is breached, the option’s delta becomes zero. The investor is then effectively unhedged and exposed to the full price risk of the underlying asset. For example, imagine an investor holding 100 shares of a company trading at £50. They purchase 1 down-and-out call option with a strike price of £52 and a barrier at £45. The option’s delta is initially 0.4. The investor is effectively short 40 shares (100 * 0.4 = 40). Now, suppose the company’s share price drops to £45, breaching the barrier. The option becomes worthless, and its delta drops to zero. The investor is now fully exposed to the 100 shares, and the hedge is no longer in place. Therefore, when the barrier is breached, the delta of a down-and-out call option transitions from a positive value to zero. This shift can have significant implications for hedging strategies and portfolio risk management. The question aims to evaluate whether the candidate understands this dynamic and its consequences.

The core of this question revolves around understanding how delta hedging is implemented and its limitations when dealing with options, specifically in the context of gamma. Gamma represents the rate of change of delta with respect to the underlying asset’s price. A high gamma implies that delta changes rapidly, making delta hedging more challenging and requiring more frequent adjustments to maintain a near-neutral position. Transaction costs associated with these adjustments can significantly erode profits, especially in volatile markets or with options that are close to their strike price where gamma is typically highest. The profit calculation needs to account for the initial premium received, the gains or losses from the underlying asset purchases and sales, and the transaction costs incurred during the hedging process. The initial short option position generates a premium income. As the underlying asset price fluctuates, the delta changes, requiring adjustments to the hedge. Buying the underlying asset increases the hedge’s cost, while selling it generates revenue. Transaction costs reduce the overall profitability. The final profit or loss is determined by summing the initial premium, the net gains/losses from hedging, and subtracting the total transaction costs. In this specific scenario, the investor shorts a call option, initially hedging by buying shares. As the price rises, more shares are bought, and when the price falls, shares are sold. The transaction costs are a key factor in determining the final profit. Let’s break down the calculation: 1. **Initial Premium:** £4.50 2. **Initial Hedge:** Delta is 0.4, so 40 shares are bought at £100: Cost = 40 * £100 = £4000 3. **Price Rises to £105:** Delta increases to 0.7. Additional shares to buy = 70 – 40 = 30. Cost = 30 * £105 = £3150 4. **Price Falls to £102:** Delta decreases to 0.5. Shares to sell = 70 – 50 = 20. Revenue = 20 * £102 = £2040 5. **Price Falls to £98:** Delta decreases to 0.2. Shares to sell = 50 – 20 = 30. Revenue = 30 * £98 = £2940 6. **Price Rises to £101:** Delta increases to 0.4. Additional shares to buy = 20. Cost = 20 * £101 = £2020 7. **Option Expires:** The option expires out-of-the-money (strike price is £110, and the final price is £101), so no payout is required. 8. **Total Costs:** (40 * £100 + 30 * £105 + 20 * £101) = £4000 + £3150 + £2020 = £9170 9. **Total Revenue:** (20 * £102 + 30 * £98) = £2040 + £2940 = £4980 10. **Net Hedging Cost:** £9170 – £4980 = £4190 11. **Transaction Costs:** (40 + 30 + 20 + 30 + 20) * £0.10 = 140 * £0.10 = £14 12. **Net Profit:** Initial Premium – Net Hedging Cost – Transaction Costs = £450 – £4190 – £14 = -£3754 The investor experiences a loss due to the dynamic hedging and transaction costs. This loss highlights the challenges of delta hedging, especially when gamma is high and frequent adjustments are needed.

The question assesses the understanding of exotic derivatives, specifically barrier options, and how their payoff structure is affected by market volatility and the correlation between the underlying asset and a related asset. First, let’s break down the scenario. The investor holds a down-and-out call option on a UK energy company (UKEC) stock. The option becomes worthless if the stock price touches the barrier level of £8. The correlation between UKEC and Brent Crude oil is a crucial factor. The current price of UKEC is £12, and the option’s strike price is £12, meaning it’s currently at-the-money. Now, let’s analyze each potential outcome: * **Scenario 1: Increased Volatility in UKEC stock, Negative Correlation with Brent Crude.** Higher volatility in UKEC increases the probability of hitting the barrier, making the option less valuable. The negative correlation with Brent Crude means that when UKEC drops, Brent Crude tends to rise (or vice-versa). This provides a partial hedge. If UKEC falls towards the barrier, the potential rise in Brent Crude might slightly offset the negative impact on the barrier option’s value, but the primary driver is the increased probability of hitting the barrier due to the increased volatility of UKEC. * **Scenario 2: Decreased Volatility in UKEC stock, Positive Correlation with Brent Crude.** Lower volatility reduces the chance of hitting the barrier, making the option more valuable. Positive correlation means that UKEC and Brent Crude move in the same direction. If UKEC rises, Brent Crude also tends to rise, enhancing the potential payoff of the call option (if it stays alive). If UKEC falls, Brent Crude will also fall, increasing the chance of hitting the barrier. * **Scenario 3: Increased Volatility in UKEC stock, Positive Correlation with Brent Crude.** Higher volatility increases the probability of hitting the barrier, decreasing the option’s value. Positive correlation exacerbates the risk. If UKEC falls, Brent Crude also falls, doing nothing to mitigate the risk of hitting the barrier. * **Scenario 4: Decreased Volatility in UKEC stock, Negative Correlation with Brent Crude.** Lower volatility reduces the chance of hitting the barrier, making the option more valuable. Negative correlation provides a hedge. If UKEC falls towards the barrier, Brent Crude might rise, partially offsetting the negative impact. Therefore, decreased volatility in UKEC stock and a negative correlation with Brent Crude would provide the most favorable conditions for the investor, as the option is less likely to be knocked out, and the negative correlation provides a partial hedge.

Let’s analyze the potential profit or loss from exercising or abandoning an American-style put option on a volatile stock, considering transaction costs and early exercise considerations. The breakeven point for the put option holder is the strike price less the premium paid, adjusted for transaction costs. The decision to exercise early depends on the intrinsic value compared to the time value and the impact of dividends. Consider a scenario where an investor holds an American put option on “StellarTech” stock, which is known for its high volatility and unpredictable dividend payouts. The current market price of StellarTech is £42. The put option has a strike price of £50, and the investor paid a premium of £3. Transaction costs for exercising the option are £0.50 per share. StellarTech is expected to announce a significant dividend in the next month, estimated to be £4 per share. If the investor expects the stock price to remain relatively stable until expiration, the investor must decide whether to exercise the put option immediately, hold it until the dividend announcement, or wait until the expiration date. If the investor exercises the put option immediately, the profit would be the difference between the strike price and the current market price, minus the premium paid and transaction costs: (£50 – £42) – £3 – £0.50 = £4.50. However, the investor would forego any potential time value remaining in the option. If the investor holds the option until the dividend announcement, the stock price may decrease by the amount of the dividend. If the stock price decreases to £38 after the dividend announcement, the intrinsic value of the put option would increase to £12 (£50 – £38). However, the time value of the option would decrease as the expiration date approaches. If the investor waits until the expiration date, the decision to exercise depends on the stock price at that time. If the stock price remains at £42, the investor would exercise the option for a profit of £4.50. If the stock price increases above £50, the option would expire worthless, resulting in a loss of the premium paid (£3). The breakeven point for the put option is the strike price less the premium paid: £50 – £3 = £47. This means the stock price must be below £47 at expiration for the investor to make a profit. Considering transaction costs, the breakeven point is adjusted to £47.50. The decision to exercise early depends on whether the intrinsic value of the option exceeds the remaining time value and the potential impact of dividends. If the dividend is expected to significantly decrease the stock price, early exercise may be optimal. However, if the investor believes the stock price will increase, holding the option may be more profitable.

To determine the maximum potential loss for the fund, we need to analyze the worst-case scenario for the short put options and the long call options. The fund has sold 100 put options on stock XYZ with a strike price of 95 at a premium of £5 each and purchased 50 call options on stock XYZ with a strike price of 105 at a premium of £3 each. **Short Put Options:** The maximum loss on a short put option occurs when the stock price falls to zero. In this case, the fund would be obligated to buy the stock at the strike price of 95, but the stock is worthless. The loss per put option is the strike price minus the premium received, which is £95 – £5 = £90. Since the fund sold 100 put options, the total potential loss from the short puts is 100 * £90 = £9,000. **Long Call Options:** The maximum loss on a long call option occurs when the stock price is at or below the strike price at expiration. In this case, the call option expires worthless, and the fund loses the premium paid. The loss per call option is £3. Since the fund purchased 50 call options, the total potential loss from the long calls is 50 * £3 = £150. **Combined Maximum Loss:** The total maximum potential loss for the fund is the sum of the maximum losses from the short puts and the long calls, which is £9,000 + £150 = £9,150. Consider a different scenario: A small investment firm, “Phoenix Investments,” uses similar strategies but incorporates risk management overlays. They sell puts to generate income and buy calls to hedge against significant upside moves in their core portfolio holdings. Imagine Phoenix sells puts on a volatile tech stock, hoping to collect premiums. However, a sudden regulatory change decimates the tech sector, driving the stock price to near zero. The firm faces substantial losses on its put positions. Simultaneously, their long call positions on a different, unrelated stock, intended as a hedge, expire worthless due to a market correction unrelated to the tech sector crash. This highlights how seemingly independent derivative positions can both contribute to losses under adverse market conditions, emphasizing the importance of comprehensive risk management and stress testing. The maximum loss scenario isn’t just theoretical; it represents a real possibility that investment firms must actively manage.

The correct answer is (a). This question tests the understanding of the impact of early exercise on American options, particularly in relation to dividend-paying assets. American options grant the holder the right, but not the obligation, to exercise the option at any time before the expiration date. This early exercise feature introduces complexities, especially when the underlying asset pays dividends. The key principle here is that an American call option on a dividend-paying stock may be exercised early if the present value of the expected dividends exceeds the time value of the option. The time value represents the potential for the option’s price to increase before expiration. If the dividends are large enough, the option holder might prefer to capture the dividends immediately rather than wait for potential future gains. Let’s consider a scenario involving a hypothetical tech company, “Innovate Inc.”, trading at £150 per share. An investor holds an American call option with a strike price of £140, expiring in six months. Innovate Inc. is scheduled to pay a dividend of £15 per share in three months. The risk-free interest rate is 5% per annum. First, calculate the present value of the dividend: \[ PV = \frac{Dividend}{(1 + r)^t} = \frac{15}{(1 + 0.05/4)^1} \approx £14.81 \] Where r is the risk-free rate and t is the time to dividend payment (3 months, or 1/4 of a year). Next, consider the intrinsic value of the option: \[ Intrinsic\,Value = Stock\,Price – Strike\,Price = 150 – 140 = £10 \] Now, compare the present value of the dividend (£14.81) with the potential time value decay if the option is not exercised early. If the time value is less than £4.81, early exercise is rational. The time value of the option reflects the volatility of the underlying asset, the time remaining until expiration, and the risk-free interest rate. If the market anticipates a significant price drop in Innovate Inc. due to broader economic factors or sector-specific headwinds, the time value might be relatively low. In this case, capturing the £14.81 dividend by exercising early becomes more attractive. Options (b), (c), and (d) present incorrect assertions. Option (b) incorrectly states that early exercise is always optimal for dividend-paying stocks. This ignores the trade-off between capturing dividends and retaining the time value of the option. Option (c) incorrectly suggests that American options are never exercised early. This is false, as demonstrated by the dividend scenario. Option (d) incorrectly claims that early exercise only depends on the strike price relative to the stock price, neglecting the critical role of dividends and time value.

To determine the most suitable hedging strategy, we need to consider the investor’s risk aversion and the specific characteristics of the derivative contracts. A risk-averse investor typically prefers strategies that minimize potential losses, even if it means sacrificing some potential gains. In this scenario, the investor wants to hedge against a potential increase in the price of copper. Let’s analyze each option: * **Selling copper futures:** This strategy hedges against a *decrease* in copper prices, not an increase. If copper prices rise, the investor will lose money on the futures contract, offsetting the benefit of holding the physical copper. * **Buying copper call options:** This strategy allows the investor to profit from an increase in copper prices above the strike price, while limiting potential losses to the premium paid for the options. It provides upside potential with limited downside risk. * **Writing copper put options:** This strategy obligates the investor to buy copper at the strike price if the option is exercised. While it generates premium income, it exposes the investor to potential losses if copper prices fall below the strike price. It does not protect against rising copper prices. * **Entering into a copper swap to receive fixed, pay floating:** This strategy would protect against *decreasing* copper prices. The investor receives a fixed price and pays a floating price based on the market rate. If the market rate falls, the investor benefits. Therefore, buying copper call options is the most suitable strategy for a risk-averse investor seeking to hedge against an increase in copper prices. It offers the potential for profit if copper prices rise, while limiting potential losses to the premium paid for the options. The other strategies either expose the investor to losses if copper prices rise or protect against falling prices, which is not the investor’s objective.

Let’s analyze the scenario. The fund manager is considering using options to hedge against a potential market downturn affecting their portfolio, which is heavily weighted in UK equities. The portfolio’s beta is 1.2, indicating it is more volatile than the overall market. The FTSE 100 is currently at 7500, and the manager wants to protect the portfolio’s value over the next six months. To determine the number of FTSE 100 put option contracts needed, we first need to calculate the portfolio’s equivalent index exposure. This is done by multiplying the portfolio’s value by its beta: £150,000,000 * 1.2 = £180,000,000. Next, we need to determine the index points represented by this exposure. Each FTSE 100 index point is worth £10. Therefore, the portfolio’s index point exposure is £180,000,000 / £10 = 18,000,000 index points. Each FTSE 100 option contract covers 1 index point multiplied by a contract multiplier of 1. So, each contract covers 1 index point. Therefore, the number of contracts needed is 18,000,000 index points / 1 index point per contract = 18,000 contracts. However, the fund manager decides to use delta-hedging. The put options have a delta of -0.4. Delta represents the sensitivity of the option price to changes in the underlying asset price. In this case, a delta of -0.4 means that for every 1-point decrease in the FTSE 100, the put option price is expected to increase by 0.4. To account for the delta, we need to adjust the number of contracts. The formula is: Number of contracts = (Index point exposure) / (Contract multiplier * Delta). Therefore, the number of contracts needed is 18,000,000 / (1 * 0.4) = 45,000 contracts. The fund manager should buy 45,000 FTSE 100 put option contracts to hedge their portfolio, considering its beta and the delta of the options. This strategy aims to offset potential losses in the portfolio due to a market downturn by gaining value from the put options. The delta-adjusted hedging approach ensures that the hedge is more precisely calibrated to the portfolio’s risk profile and the option’s sensitivity to market movements. If the fund manager ignored the delta, they would be significantly under-hedged, leaving the portfolio vulnerable to substantial losses if the market declines sharply.

The value of a European call option can be estimated using the Black-Scholes model. The formula is: \[C = S_0N(d_1) – Ke^{-rT}N(d_2)\] where: * \(C\) = Call option price * \(S_0\) = Current stock price * \(K\) = Strike price * \(r\) = Risk-free interest rate * \(T\) = Time to expiration (in years) * \(N(x)\) = Cumulative standard normal distribution function * \(e\) = Euler’s number (approximately 2.71828) and \[d_1 = \frac{ln(\frac{S_0}{K}) + (r + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}}\] \[d_2 = d_1 – \sigma\sqrt{T}\] where: * \(\sigma\) = Volatility of the stock First, we need to calculate \(d_1\) and \(d_2\): \[d_1 = \frac{ln(\frac{105}{100}) + (0.05 + \frac{0.2^2}{2})0.5}{0.2\sqrt{0.5}}\] \[d_1 = \frac{ln(1.05) + (0.05 + 0.02)0.5}{0.2\sqrt{0.5}}\] \[d_1 = \frac{0.04879 + 0.035}{0.14142}\] \[d_1 = \frac{0.08379}{0.14142} \approx 0.5925\] \[d_2 = d_1 – \sigma\sqrt{T}\] \[d_2 = 0.5925 – 0.2\sqrt{0.5}\] \[d_2 = 0.5925 – 0.14142 \approx 0.4511\] Next, we find \(N(d_1)\) and \(N(d_2)\). Using standard normal distribution tables or a calculator, we find: \(N(0.5925) \approx 0.7232\) \(N(0.4511) \approx 0.6739\) Now we can calculate the call option price: \[C = 105 \times 0.7232 – 100 \times e^{-0.05 \times 0.5} \times 0.6739\] \[C = 75.936 – 100 \times e^{-0.025} \times 0.6739\] \[C = 75.936 – 100 \times 0.9753 \times 0.6739\] \[C = 75.936 – 65.643 \approx 10.29\] Therefore, the estimated value of the European call option is approximately £10.29. Consider a portfolio manager at a UK-based investment firm who is evaluating the use of options for hedging and speculative purposes, within the constraints of the FCA regulations. They are considering a European call option on a FTSE 100 stock. The current stock price is £105, the strike price is £100, the risk-free interest rate is 5% per annum, the time to expiration is 6 months, and the volatility of the stock is 20%.