Derivatives Level 4 (Investment Advice Diploma) Quiz Exam Quiz 18

The question revolves around the concept of early exercise of American options, specifically in relation to dividend-paying stocks. American options, unlike European options, can be exercised at any time before expiration. However, early exercise is not always optimal. The decision to exercise early depends on several factors, primarily the trade-off between the intrinsic value of the option and the time value remaining. For a call option, early exercise is generally considered when the present value of expected dividends exceeds the time value of the option. This is because by exercising early, the holder captures the dividend, which they would otherwise miss out on. However, they also forgo the potential for further price appreciation of the underlying stock. For a put option, early exercise is more likely when the option is deeply in the money, and interest rates are high. In this scenario, the holder might prefer to receive the strike price immediately and invest it, rather than waiting until expiration. The breakeven point for the swap is where the present value of all future floating rate payments equals the fixed rate. The swap is constructed so that the initial value is zero. The calculation involves discounting future cash flows using the appropriate discount rates (LIBOR in this case). The exotic option example, specifically the barrier option, introduces an additional layer of complexity. A knock-out barrier option ceases to exist if the underlying asset price reaches a certain barrier level. The value of a knock-out option is always less than or equal to the value of a standard option because of the added risk of the barrier being breached. The presence of a dividend further complicates the valuation, as it can influence the likelihood of the barrier being hit. The forward contract scenario examines the relationship between spot prices, future prices, interest rates, and storage costs. The cost of carry model dictates the relationship between these variables. Any deviation from this model presents an arbitrage opportunity.

Let’s break down this complex scenario involving a barrier option and its implications for a portfolio manager under MiFID II regulations. The core concept here revolves around understanding how the “knock-in” feature of a barrier option affects its value and suitability for a client, particularly when considering regulatory requirements like best execution and client appropriateness. The initial calculation focuses on the probability of the underlying asset (the FTSE 100) reaching the barrier level. We’re given a current index value, a barrier level, time to maturity, and volatility. While a precise probability calculation would require complex stochastic modeling (beyond the scope of a single question), we can infer the likelihood based on the proximity of the barrier to the current price and the given volatility. A lower volatility and a barrier further away from the current price suggest a lower probability of hitting the barrier. The key is understanding the impact of this probability on the option’s value and the portfolio manager’s obligations. Under MiFID II, the portfolio manager must act in the client’s best interest. Selling a knock-in barrier option carries a significant risk for the client if the barrier is breached, as the option then comes into existence and potentially exposes the client to further losses. The portfolio manager needs to assess whether the potential premium received adequately compensates the client for this risk, considering their risk profile and investment objectives. Furthermore, the “best execution” requirement under MiFID II extends beyond simply obtaining the highest premium. It also encompasses factors like the probability of the barrier being hit, the potential losses if it is hit, and the overall impact on the client’s portfolio. The portfolio manager must document their rationale for selling the option, demonstrating that they have considered all these factors and acted in the client’s best interest. A failure to adequately assess and document these considerations could lead to regulatory scrutiny and potential penalties. The appropriateness assessment must consider the client’s knowledge and experience with derivatives, their financial situation, and their investment objectives. Selling a complex derivative like a knock-in barrier option to a client who does not fully understand its risks would be a breach of MiFID II regulations. The scenario highlights the importance of not just understanding the mechanics of derivatives, but also the regulatory framework within which they must be used. The question is designed to assess the candidate’s ability to apply these concepts in a practical, real-world situation.

To determine the fair value of the swap, we need to discount each expected cash flow back to its present value and sum them. Since the fixed rate is paid annually, we will have three present value calculations. Year 1: The cash flow is £1,000,000 * (0.055 – 0.045) = £10,000. The discount factor is 1 / (1 + 0.05) = 0.9524. The present value is £10,000 * 0.9524 = £9,524. Year 2: The cash flow is £1,000,000 * (0.055 – 0.045) = £10,000. The discount factor is 1 / (1 + 0.05)^2 = 0.9070. The present value is £10,000 * 0.9070 = £9,070. Year 3: The cash flow is £1,000,000 * (0.055 – 0.045) = £10,000. The discount factor is 1 / (1 + 0.05)^3 = 0.8638. The present value is £10,000 * 0.8638 = £8,638. Summing the present values: £9,524 + £9,070 + £8,638 = £27,232. Therefore, the fair value of the swap is £27,232. In this scenario, a company, “Innovatech,” entered into an interest rate swap to manage its floating-rate debt. Innovatech pays a fixed rate and receives a floating rate. The floating rate has risen above the fixed rate, creating a positive value for Innovatech in the swap. We calculate the present value of the expected future cash flows to determine the fair value of the swap. This involves discounting each future cash flow by the appropriate discount factor, reflecting the time value of money. The sum of these present values represents the fair value of the swap to Innovatech. The calculation assumes that the future floating rates are accurately predicted and that the discount rate reflects the appropriate risk-free rate for the period. This fair value is crucial for Innovatech’s financial reporting and risk management, as it represents the current economic value of the swap agreement. Understanding the fair value allows Innovatech to make informed decisions about hedging strategies and potential termination of the swap. The entire process demonstrates how derivatives, specifically interest rate swaps, are valued and used in corporate finance to manage interest rate risk.

The value of a European call option is determined 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 \(d_1 = \frac{ln(\frac{S_0}{K}) + (r + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}}\) \(d_2 = d_1 – \sigma\sqrt{T}\) \(\sigma\) = Volatility of the stock In this scenario: \(S_0 = 150\) \(K = 155\) \(r = 0.05\) \(T = 0.5\) \(\sigma = 0.25\) First, calculate \(d_1\): \[d_1 = \frac{ln(\frac{150}{155}) + (0.05 + \frac{0.25^2}{2})0.5}{0.25\sqrt{0.5}} = \frac{ln(0.9677) + (0.05 + 0.03125)0.5}{0.25 \times 0.7071} = \frac{-0.0328 + 0.040625}{0.1768} = \frac{0.007825}{0.1768} = 0.04426\] Next, calculate \(d_2\): \[d_2 = d_1 – \sigma\sqrt{T} = 0.04426 – 0.25\sqrt{0.5} = 0.04426 – 0.25 \times 0.7071 = 0.04426 – 0.1768 = -0.13254\] Now, find \(N(d_1)\) and \(N(d_2)\) using standard normal distribution tables or a calculator. \(N(0.04426) \approx 0.5177\) \(N(-0.13254) \approx 0.4473\) Finally, calculate the call option price \(C\): \[C = 150 \times 0.5177 – 155e^{-0.05 \times 0.5} \times 0.4473 = 150 \times 0.5177 – 155e^{-0.025} \times 0.4473 = 77.655 – 155 \times 0.9753 \times 0.4473 = 77.655 – 151.1715 \times 0.4473 = 77.655 – 67.627 = 10.028\] Therefore, the value of the European call option is approximately £10.03. This question tests the application of the Black-Scholes model, a cornerstone of derivatives pricing. The scenario avoids textbook examples by presenting a situation where a fund manager needs to assess the fair value of a call option on a specific stock, forcing candidates to apply the formula rather than simply recognizing a standard problem. The slight variations in the incorrect answers ensure that a superficial understanding of the formula is insufficient.

The question focuses on the impact of correlation on the value of a spread option, specifically a call spread. A call spread involves buying a call option at a lower strike price and selling a call option at a higher strike price on the same underlying asset. The payoff of a call spread is capped, as the profit from the bought call is offset by the sold call if the asset price rises significantly. The value of the spread is influenced by the correlation between the underlying assets, here, specifically, two different grades of crude oil. Lower correlation between the two grades of crude oil would mean that their prices are less likely to move in tandem. If the correlation is low, there is a higher probability that the spread between the two grades will widen. Since the investor benefits from a widening spread (because the spread is structured to profit from the price difference), a lower correlation increases the value of the spread option. Conversely, higher correlation would mean that the prices move more closely together, reducing the likelihood of a significant spread widening and decreasing the option’s value. The initial spread value is calculated using a pricing model (like Black-Scholes) with an assumed correlation. If the actual correlation is lower than assumed, the spread option is undervalued by the model. For example, consider two crude oil grades, West Texas Intermediate (WTI) and Brent. Suppose an investor buys a call option on the spread between WTI and Brent, expecting the spread to widen. The option has a strike price of $5 (Brent – WTI) and expires in 6 months. The initial valuation assumes a correlation of 0.8 between WTI and Brent. If, over the life of the option, the correlation drops to 0.5, the spread is more likely to widen significantly, increasing the value of the option. Consider a hypothetical scenario where, under the initial correlation of 0.8, the model predicts a 20% chance of the spread exceeding $10. With a lower correlation of 0.5, the probability might increase to 35%. This increased probability directly translates to a higher expected payoff and, therefore, a higher option value.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their behavior under different market conditions. The key is to understand how the knock-out barrier affects the option’s payoff. A down-and-out call option becomes worthless if the underlying asset’s price touches or goes below the barrier level at any point during the option’s life. In this scenario, the initial price is £100, and the barrier is set at £80. The option will only pay out if the price remains above £80 throughout the option’s life. The strike price is £105. First, we need to determine if the barrier was breached. The market price fluctuated, and at one point, the price of the underlying asset reached £75, which is below the barrier of £80. This means the option knocked out and became worthless at that point. The final price of the underlying asset at expiration (£110) is irrelevant because the option had already been knocked out. Therefore, the option expires worthless.

The investor’s profit or loss on the short strangle position depends on the price of the underlying asset (FTSE 100 index) at expiration. A short strangle involves selling both a call and a put option with different strike prices. The investor profits if the price at expiration remains between the two strike prices. If the price moves outside this range, the investor incurs a loss. First, determine the total premium received from selling both options: £3.50 (call) + £4.50 (put) = £8.00 per contract. This is the maximum profit the investor can make. Next, calculate the breakeven points. The upper breakeven point is the call strike price plus the total premium received: 7600 + £8.00 = 7608. The lower breakeven point is the put strike price minus the total premium received: 7300 – £8.00 = 7292. Since the FTSE 100 index settled at 7620, it is above the upper breakeven point. The loss on the call option is the difference between the settlement price and the call strike price, minus the premium received: (7620 – 7600) – £3.50 = £20 – £3.50 = £16.50. The put option expires worthless, so the investor keeps the £4.50 premium. The net loss is the loss on the call option minus the premium received from the put option: £16.50 – £4.50 = £12.00 per contract. Since the investor sold 10 contracts, the total loss is £12.00 * 10 * £10 (index multiplier) = £1200. Consider a farmer who sells a short strangle on the price of wheat. They believe the price will remain stable. The call option they sold has a strike price above the current market price, and the put option has a strike price below. If the price of wheat stays within this range, both options expire worthless, and the farmer keeps the premiums, representing their profit. However, if a drought causes the price of wheat to skyrocket, the call option will be in the money, and the farmer will have to deliver wheat at the strike price, incurring a loss. Conversely, if a bumper crop causes the price of wheat to plummet, the put option will be in the money, and the farmer will have to buy wheat at the strike price, also incurring a loss. The short strangle is a strategy that profits from low volatility but carries significant risk if the price moves substantially in either direction.

The question assesses the understanding of how varying strike prices in a butterfly spread strategy affect its maximum profit and risk profile. A butterfly spread, constructed using options with different strike prices, aims to profit from a range-bound market. The key to maximizing profit while minimizing risk lies in the strategic selection of these strike prices. The maximum profit of a long butterfly spread is realized when the underlying asset price at expiration equals the strike price of the short options (the middle strike price). The maximum profit is the difference between the higher strike price and the middle strike price, minus the initial premium paid for the spread. The risk is limited to the premium paid. In this scenario, if an investor narrows the gap between the strike prices (e.g., by using strikes of 95, 100, and 105 instead of 90, 100, and 110), the maximum potential profit decreases. This is because the range within which the strategy can achieve its maximum payout is reduced. However, narrowing the strike prices also reduces the initial cost (premium) of establishing the butterfly spread. This is because the options being bought and sold are closer in value. Consider an analogy: imagine building a bridge across a chasm. A wider chasm (larger difference between strike prices) requires more materials (higher premium) but allows for more traffic (potential profit if the asset price moves within a wider range). A narrower chasm (smaller difference between strike prices) requires less material (lower premium) but can only handle less traffic (smaller profit potential within a narrower range). Therefore, narrowing the strike prices reduces both the maximum potential profit and the initial cost of the butterfly spread. The investor must weigh the trade-off between a potentially larger profit range and a lower initial investment. The optimal strategy depends on the investor’s risk tolerance and their expectation of the underlying asset’s price movement.

The question assesses understanding of how early termination clauses in swaps impact valuation and potential compensation. It requires understanding of replacement cost, break costs, and the specific ISDA Master Agreement framework. The correct approach involves calculating the present value of the remaining cash flows under the original swap, then determining the cost to replace the swap in the market at current rates. The difference, adjusted for any contractual break costs, represents the termination payment. Let’s break down the calculations: 1. **Calculate the present value of remaining cash flows:** This requires discounting each future cash flow (difference between fixed and floating rates) back to the present using appropriate discount factors derived from the yield curve. This step is complex and typically involves iterative calculations or specialized software. We will assume this present value difference (the mark-to-market) is £550,000 in favour of Bank A. This means Bank B would need to pay Bank A £550,000 if the swap was terminated today based solely on market conditions. 2. **Consider the replacement cost:** The present value already represents the replacement cost, as it reflects the market’s current valuation of the swap’s future cash flows. 3. **Account for break costs:** The ISDA Master Agreement allows for break costs, which compensate the non-defaulting party for losses incurred in unwinding hedges related to the terminated swap. In this case, the break cost is 0.1% of the notional principal, which is \(0.001 \times £100,000,000 = £100,000\). 4. **Determine the termination payment:** The termination payment is the replacement cost (present value) plus the break cost. In this case, it’s \(£550,000 + £100,000 = £650,000\). Therefore, Bank B would owe Bank A £650,000. The incorrect options present plausible but flawed calculations, such as only considering the replacement cost without break costs, applying the break cost to the wrong party, or misinterpreting the direction of the present value calculation. This scenario is unique because it combines the theoretical understanding of swap valuation with the practical implications of the ISDA Master Agreement, a crucial aspect of derivatives trading. The break cost component adds a layer of complexity not typically found in basic examples. The specific numerical values and the context of a corporate restructuring make this a novel problem-solving challenge.

The question revolves around the concept of early exercise of American options, specifically focusing on the conditions under which it becomes optimal. The core principle is that an American call option on a non-dividend paying stock should generally not be exercised early. This is because the option’s time value provides more flexibility than the immediate gain from exercising. However, when dividends are involved, the situation changes. If the present value of expected dividends exceeds the time value of the option, early exercise might be optimal to capture those dividends. To determine the optimal strategy, we need to compare the potential gain from exercising early (receiving the stock and subsequently the dividend) with the potential loss of the option’s time value. The time value represents the potential for the stock price to increase further before the expiration date. In this scenario, we need to calculate the present value of the dividend expected in 3 months and compare it with the estimated time value of the option. The present value is calculated using the risk-free rate. 1. **Calculate the present value of the dividend:** The dividend amount is £3.50, and it will be received in 3 months (0.25 years). The risk-free rate is 4% per annum. The present value (PV) is calculated as: \[PV = \frac{Dividend}{(1 + Risk-free\, rate)^{Time}}\] \[PV = \frac{3.50}{(1 + 0.04)^{0.25}} = \frac{3.50}{1.00985} \approx 3.466\] 2. **Calculate the intrinsic value of the option if exercised now:** The intrinsic value is the difference between the stock price and the strike price: \[Intrinsic\, Value = Stock\, Price – Strike\, Price\] \[Intrinsic\, Value = 65 – 60 = 5\] 3. **Estimate the Time Value:** The option premium is £6.50. Time Value is the difference between option premium and intrinsic value. \[Time\, Value = Option\, Premium – Intrinsic\, Value\] \[Time\, Value = 6.50 – 5 = 1.50\] 4. **Compare Present Value of Dividend and Time Value:** The present value of the dividend (£3.466) is greater than the time value of the option (£1.50). This suggests that exercising early to capture the dividend is the optimal strategy. By exercising early, the investor receives the stock, collects the dividend (which has a present value of £3.466), and avoids losing the dividend to someone else. The loss of the option’s time value (£1.50) is less than the benefit of capturing the dividend. Therefore, the investor should exercise the option immediately before the ex-dividend date.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their valuation sensitivity to volatility. The calculation involves understanding how the probability of hitting the barrier changes with volatility and how this affects the option’s price. We’ll use a simplified approach to illustrate the concept. Let’s consider a down-and-out call option with a strike price of £100 and a barrier at £90. The current asset price is £105. The initial volatility is 20%, and we want to assess the impact of an increase in volatility to 25%. We’ll simplify by assuming that a higher volatility increases the probability of hitting the barrier before maturity. Assume, under 20% volatility, the probability of the asset price hitting the barrier before maturity is 10%. This means the option’s value is reduced by 10% of its intrinsic value if the barrier is hit. The intrinsic value is £5 (£105 – £100). Therefore, the barrier reduces the option value by 10% * £5 = £0.50. Now, assume the volatility increases to 25%. This increases the probability of hitting the barrier to 15%. The reduction in option value is now 15% * £5 = £0.75. The difference in the reduction of option value due to the increased probability of hitting the barrier is £0.75 – £0.50 = £0.25. This represents the additional negative impact on the down-and-out call option due to the increased volatility. However, volatility also increases the potential upside of the option. Without considering the barrier, a higher volatility would increase the option’s value. But, for a down-and-out option, the increased probability of hitting the barrier outweighs some of the potential upside gain from the increased volatility. The key takeaway is that barrier options have complex relationships with volatility. While standard options benefit from increased volatility, barrier options can see a decrease in value if the increased volatility significantly raises the probability of the barrier being hit. This is because the option becomes worthless if the barrier is breached. The exact change in price depends on several factors, including the distance to the barrier, the time to maturity, and the risk-free rate. Therefore, calculating the exact price change requires complex models, but this example illustrates the directional impact. The value change will be negative due to the increased probability of hitting the barrier.

To determine the value of the European call option, we use the Black-Scholes model: \[C = S_0N(d_1) – Ke^{-rT}N(d_2)\] Where: * \(C\) = Call option price * \(S_0\) = Current stock price = £95 * \(K\) = Strike price = £100 * \(r\) = Risk-free interest rate = 3% or 0.03 * \(T\) = Time to expiration = 6 months or 0.5 years * \(N(x)\) = Cumulative standard normal distribution function * \(e\) = Euler’s number (approximately 2.71828) First, calculate \(d_1\) and \(d_2\): \[d_1 = \frac{ln(\frac{S_0}{K}) + (r + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}}\] \[d_2 = d_1 – \sigma\sqrt{T}\] Where \(\sigma\) = Volatility = 25% or 0.25 1. Calculate \(d_1\): \[d_1 = \frac{ln(\frac{95}{100}) + (0.03 + \frac{0.25^2}{2})0.5}{0.25\sqrt{0.5}}\] \[d_1 = \frac{ln(0.95) + (0.03 + 0.03125)0.5}{0.25\sqrt{0.5}}\] \[d_1 = \frac{-0.05129 + (0.06125)0.5}{0.25 \times 0.7071}\] \[d_1 = \frac{-0.05129 + 0.030625}{0.176775}\] \[d_1 = \frac{-0.020665}{0.176775} \approx -0.1169\] 2. Calculate \(d_2\): \[d_2 = d_1 – \sigma\sqrt{T}\] \[d_2 = -0.1169 – 0.25\sqrt{0.5}\] \[d_2 = -0.1169 – 0.25 \times 0.7071\] \[d_2 = -0.1169 – 0.176775 \approx -0.2937\] 3. Find \(N(d_1)\) and \(N(d_2)\) using the standard normal distribution table: * \(N(-0.1169) \approx 0.4534\) * \(N(-0.2937) \approx 0.3844\) 4. Calculate the call option price \(C\): \[C = S_0N(d_1) – Ke^{-rT}N(d_2)\] \[C = 95 \times 0.4534 – 100 \times e^{-0.03 \times 0.5} \times 0.3844\] \[C = 95 \times 0.4534 – 100 \times e^{-0.015} \times 0.3844\] \[C = 43.073 – 100 \times 0.9851 \times 0.3844\] \[C = 43.073 – 100 \times 0.3786\] \[C = 43.073 – 37.86\] \[C \approx 5.21\] The estimated value of the European call option is approximately £5.21. The Black-Scholes model provides a theoretical valuation for European-style options. It assumes that the price of the underlying asset follows a log-normal distribution and that the volatility is constant over the option’s life. In real-world scenarios, volatility is rarely constant, and the model’s assumptions may not hold perfectly. However, it remains a widely used tool for option pricing and risk management. The calculation involves several steps, including determining \(d_1\) and \(d_2\), which incorporate the current stock price, strike price, risk-free rate, time to expiration, and volatility. These values are then used to find the cumulative standard normal distribution values, \(N(d_1)\) and \(N(d_2)\), which are crucial for calculating the option price.

The core of this question lies in understanding how delta, gamma, and theta interact to influence the value of a short call option position as the underlying asset’s price and time to expiration change. We need to consider the combined effects of these “Greeks” on the portfolio’s value. Delta (\(\Delta\)) measures the sensitivity of the option price to a change in the underlying asset’s price. A delta of -0.6 for a short call means that for every $1 increase in the underlying asset’s price, the short call option loses $0.60 in value (or the portfolio loses $0.60). Gamma (\(\Gamma\)) measures the rate of change of delta with respect to the underlying asset’s price. A gamma of 0.05 indicates that for every $1 increase in the underlying asset’s price, the delta of the short call will increase by 0.05 (become less negative). Theta (\(\Theta\)) measures the sensitivity of the option price to the passage of time. A theta of -0.02 means that the short call option loses $0.02 in value each day due to time decay. Let’s break down the calculation step-by-step: 1. **Price Change Impact:** The underlying asset increases by $2. The initial impact on the short call option due to delta is: \(\Delta \times \text{Price Change} = -0.6 \times \$2 = -\$1.20\). This means the short call position initially loses $1.20. 2. **Gamma Adjustment:** The delta changes due to gamma. The change in delta is: \(\Gamma \times \text{Price Change} = 0.05 \times \$2 = 0.10\). The new delta is -0.6 + 0.10 = -0.5. The average delta during the $2 price move is approximately the midpoint between the initial and final deltas: (-0.6 + -0.5) / 2 = -0.55. Therefore, the price change impact is refined to: \(\text{Average Delta} \times \text{Price Change} = -0.55 \times \$2 = -\$1.10\). 3. **Theta Impact:** One day passes. The impact on the short call option due to theta is: \(\Theta \times \text{Time Change} = -0.02 \times 1 = -\$0.02\). This means the short call position loses an additional $0.02 due to time decay. 4. **Total Impact:** The total change in the value of the short call position is the sum of the price change impact and the theta impact: \(-\$1.10 + (-\$0.02) = -\$1.12\). Since the option is shorted, a loss in value translates to a negative change. Therefore, the value of the short call position decreases by $1.12. A crucial consideration is the interplay between gamma and delta. Gamma modifies the delta as the underlying price changes, making the initial delta estimate less accurate for larger price movements. Using an average delta refines the calculation. Furthermore, theta acts independently, eroding the option’s value as time passes, irrespective of price movements. Ignoring gamma or theta would lead to a significantly inaccurate valuation of the portfolio change. This scenario emphasizes the importance of understanding and managing these Greeks in a dynamic market environment.

The question assesses understanding of exotic derivatives, specifically barrier options, and their behavior around the barrier level. The key is to recognize that a knock-in call option only becomes active if the underlying asset price touches the barrier level. If the barrier is never reached, the option expires worthless, regardless of the asset’s final price. In this scenario, the investor holds a knock-in call option with a barrier set at 95% of the initial asset price. The initial price is £100, so the barrier is at £95. The asset price fluctuates, but the lowest price it reaches is £96. Since the barrier of £95 was never touched, the option never became active. The investor also holds a standard call option with a strike price of £105. At expiration, the asset price is £110. The payoff for a standard call option is calculated as max(0, Asset Price – Strike Price). In this case, it’s max(0, £110 – £105) = £5. Therefore, the investor’s total profit is £5 (from the standard call option) – £3 (premium paid for the knock-in option) – £7 (premium paid for the standard call option) = -£5. The question requires calculating the payoff of a standard call option and understanding the activation condition of a knock-in barrier option. It tests the ability to apply these concepts in a combined scenario. The incorrect options test common misunderstandings, such as assuming the knock-in option pays off if the asset price is above the strike price at expiration, or incorrectly calculating the payoff of the standard call option.

The question assesses the understanding of delta hedging and its effectiveness in various market scenarios. Delta hedging aims to neutralize the sensitivity of an option portfolio to small changes in the underlying asset’s price. However, its effectiveness is limited when the underlying asset experiences significant price jumps or gaps. Gamma measures the rate of change of delta with respect to changes in the underlying asset’s price. A higher gamma indicates that the delta will change more rapidly, making it more challenging to maintain a perfectly hedged position, especially during large price movements. Vega measures the sensitivity of an option’s price to changes in the volatility of the underlying asset. Changes in volatility can significantly impact the option’s price and the effectiveness of the delta hedge. Theta measures the rate of decline in the value of an option due to the passage of time. While time decay affects the option’s price, it is not the primary factor affecting the effectiveness of delta hedging during significant price jumps. In this scenario, the oil price experiences a sudden and substantial drop due to unexpected geopolitical events. This large price movement renders the initial delta hedge less effective because the delta, which was calculated based on the initial price, is no longer accurate. The larger the price movement, the more inaccurate the delta becomes. The effectiveness of delta hedging is significantly reduced because delta hedging is designed to protect against small price movements, not large gaps. The portfolio manager needs to re-hedge quickly to adjust the hedge ratio to the new price level. The gamma of the portfolio determines how quickly the delta changes, and the vega determines how sensitive the portfolio is to volatility changes, both of which are relevant in this scenario.

Let’s break down this complex scenario involving a quanto swap and its associated risks. A quanto swap is a type of financial derivative where the interest rate of one currency is exchanged for that of another currency, but the notional principal is fixed in one currency. This eliminates exchange rate risk on the notional amount. However, it introduces other complexities, particularly around correlation risk and the potential for unexpected movements in interest rate differentials. The key here is to understand how changes in the interest rate differential between the UK and the US, combined with the correlation between these rates and the performance of the FTSE 100, impact the value of the swap. A positive correlation means that as UK interest rates rise relative to US rates, the FTSE 100 is also likely to rise (or fall less). Conversely, a negative correlation would imply the FTSE 100 is likely to fall (or rise less) as UK rates increase relative to US rates. In this scenario, the fund manager initially hedged against adverse movements in the FTSE 100 related to interest rate differentials. If the correlation between the FTSE 100 and the UK-US interest rate differential weakens significantly (approaches zero), the hedge becomes less effective. The fund is then exposed to potential losses if UK rates rise relative to US rates and the FTSE 100 does not perform as expected (i.e., doesn’t rise or falls). The potential loss can be estimated by considering the sensitivity of the swap to changes in the interest rate differential and the notional principal. The question is designed to assess the understanding of these relationships and the risks inherent in quanto swaps, particularly when correlation assumptions change. The calculation to derive the correct answer involves considering the change in interest rate differential (0.5%), the notional principal (£50 million), and the weakened correlation. The correct answer will reflect the potential loss given the fund’s position and the altered market conditions. The incorrect answers are designed to represent common misunderstandings of quanto swaps and correlation risk, such as overestimating the impact of the change, ignoring the correlation effect entirely, or misinterpreting the direction of the relationship.

The question revolves around the concept of a quanto swap, specifically a currency-protected equity swap. The core principle is that the equity return is converted into a fixed currency (in this case, GBP) at a predetermined exchange rate, shielding the investor from currency fluctuations. The challenge is to determine the appropriate notional principal in USD to achieve a target GBP return, considering the swap’s structure and the initial exchange rate. The calculation involves several steps: 1. **Target GBP Return:** The investor desires a GBP return equivalent to the return on £1,000,000. 2. **Equity Return:** The FTSE 100 return is 8%. This return is applied to the USD notional, but the result needs to be converted to GBP at the fixed exchange rate. 3. **Fixed Exchange Rate:** The swap uses a fixed exchange rate of 1.30 USD/GBP. 4. **Solving for USD Notional:** We need to find the USD notional principal that, when multiplied by the 8% FTSE 100 return and converted to GBP at 1.30 USD/GBP, equals the return on £1,000,000. Let *N* be the USD notional principal. The equation is: \[ \frac{N \times 0.08}{1.30} = 1,000,000 \times 0.08 \] Solving for *N*: \[ N = \frac{1,000,000 \times 0.08 \times 1.30}{0.08} \] \[ N = 1,000,000 \times 1.30 \] \[ N = 1,300,000 \] Therefore, the USD notional principal should be $1,300,000. A crucial understanding is that the fixed exchange rate in a quanto swap eliminates currency risk. If the actual exchange rate at the end of the period is different from 1.30, the investor’s GBP return remains unaffected. This is because the conversion is done at the agreed-upon rate within the swap agreement. Consider a scenario where the actual exchange rate moves to 1.40 USD/GBP. Without the quanto feature, the investor’s GBP return would be lower than expected. The quanto swap provides certainty regarding the GBP return, irrespective of currency movements. This makes it attractive for investors with GBP liabilities or those who prefer to manage currency risk separately. The problem highlights the practical application of quanto swaps in managing currency exposure while participating in foreign equity markets.

To determine the breakeven point for the combined strategy, we need to consider the initial costs and potential profits from both the long call and short put options. The long call option was purchased for £4, and the short put option was sold for £3. This results in a net cost of £1 (4 – 3 = 1). The long call option has a strike price of £100, and the short put option has a strike price of £95. The breakeven point for the long call is the strike price plus the premium paid, which is £100 + £4 = £104. However, the combined strategy also involves a short put, which introduces a different breakeven consideration. If the asset price falls below £95, the short put will be exercised, obligating the investor to buy the asset at £95. This obligation effectively caps the potential profit from the short put. The maximum loss on the short put is the strike price minus the premium received, which is £95 – £3 = £92. However, this loss is offset by the premium received. The combined breakeven point is where the profit from the long call equals the maximum potential loss from the short put, adjusted for the net premium paid. We need to find the asset price at which the profit from the long call covers the initial net cost of £1 and the potential loss from the short put. Let \(S\) be the spot price at expiration. If \(S > 100\), the long call is in the money, and its profit is \(S – 100 – 4 = S – 104\). If \(S < 95\), the short put is in the money for the buyer, and the seller (investor) loses \(95 - S - 3 = 92 - S\). The investor's net profit is \((S - 104) + 3 = S - 101\) if \(S > 100\), and \(0 + (3 – (95 – S)) = S – 92\) if \(S < 95\). We need to find \(S\) such that the combined profit is zero, considering the initial net cost of £1. If the spot price is above £100, the profit from the call option must offset the initial £1 cost. Therefore, \(S – 100 – 4 + 3 = 0\), which simplifies to \(S – 101 = 0\), so \(S = 101\). However, since the short put option introduces a lower strike price of £95, we must consider the possibility of the asset price falling below this level. In this case, the investor would be obligated to buy the asset at £95, and the profit from the long call would be limited. The combined breakeven point is £96. This is because if the asset price is £96, the short put loses £1 (95-96), and the initial £1 net cost is covered. EXPLANATION (Continued): Consider a scenario where an investor uses a combination of a long call and a short put to speculate on a particular stock. The investor purchases a call option with a strike price of £100 for a premium of £4 and simultaneously sells a put option with a strike price of £95, receiving a premium of £3. The net cost of this strategy is £1. If the stock price rises above £100, the call option becomes profitable. For example, if the stock price reaches £110, the call option provides a profit of £6 (£110 – £100 – £4), but we need to subtract the initial cost of £1 to get a net profit of £5. However, if the stock price falls below £95, the investor is obligated to buy the stock at £95. For instance, if the stock price drops to £90, the investor must purchase the stock at £95, resulting in a loss of £5 on the put option. Taking into account the premium received, the net loss is £2 (£5 – £3). The breakeven point is the stock price at which the combined profit or loss from both options equals zero, considering the initial cost. To calculate this, we need to determine the stock price at which the profit from the call option offsets the loss from the put option and the initial net cost. If the stock price stays between £95 and £100, both options expire worthless, and the investor loses the initial cost of £1. If the stock price is above £100, the call option's profit needs to cover the initial cost. If the stock price is below £95, the put option's loss needs to be offset by the initial premium received. The breakeven point is found by considering the net cost and the potential profits and losses from both options. In this case, the breakeven point is £96. At this price, the put option loses £1, offsetting the initial net cost of £1.

The question assesses the understanding of exotic derivatives, specifically a cliquet option, and its payoff structure under different market conditions. A cliquet option is a series of options, typically caps or floors, where the strike price for each period is determined by the performance of the underlying asset in the previous period. The global cap limits the overall return of the cliquet. In this scenario, the investor receives the sum of the returns for each period, subject to both individual period caps and a global cap. To determine the final payoff, we need to calculate the return for each period, apply the period cap, sum the capped returns, and then apply the global cap. Period 1: Return = 8%, Capped Return = min(8%, 5%) = 5% Period 2: Return = -3%, Capped Return = max(-3%, -2%) = -2% Period 3: Return = 6%, Capped Return = min(6%, 5%) = 5% Period 4: Return = 4%, Capped Return = min(4%, 5%) = 4% Sum of Capped Returns = 5% – 2% + 5% + 4% = 12% Global Cap = 10% Final Payoff = min(12%, 10%) = 10% The investor’s payoff is limited by the global cap of 10%, even though the sum of the capped periodic returns is 12%. This example demonstrates the risk management feature of cliquet options, where both individual period performance and overall performance are constrained. The unique aspect of this problem lies in the combination of individual period caps/floors and the global cap, requiring careful consideration of both local and global constraints. Understanding how these caps interact is crucial for determining the final payoff and assessing the suitability of such a derivative for a particular investment strategy. For instance, a portfolio manager using a cliquet option to hedge against downside risk while still participating in potential upside needs to fully grasp these limitations. The scenario also implicitly tests knowledge of path dependency, as the final payoff depends on the sequence of returns.

To determine the value of the American call option, we need to consider the potential outcomes at expiration and discount them back to the present value. The option’s payoff depends on whether the stock price exceeds the strike price at expiration. We’ll use a simplified single-period binomial model. First, calculate the potential stock prices at expiration: Upward movement: \( S_u = S_0 \times u = 100 \times 1.2 = 120 \) Downward movement: \( S_d = S_0 \times d = 100 \times 0.8 = 80 \) Next, calculate the call option payoffs at expiration: If the stock price goes up to 120: \( C_u = \max(S_u – K, 0) = \max(120 – 110, 0) = 10 \) If the stock price goes down to 80: \( C_d = \max(S_d – K, 0) = \max(80 – 110, 0) = 0 \) Now, calculate the risk-neutral probability \( p \): \[ p = \frac{e^{rT} – d}{u – d} = \frac{e^{0.05 \times 1} – 0.8}{1.2 – 0.8} = \frac{1.0513 – 0.8}{0.4} = \frac{0.2513}{0.4} = 0.62825 \] Finally, calculate the present value of the expected payoff: \[ C_0 = e^{-rT} \times [p \times C_u + (1 – p) \times C_d] = e^{-0.05 \times 1} \times [0.62825 \times 10 + (1 – 0.62825) \times 0] \] \[ C_0 = e^{-0.05} \times [6.2825 + 0] = 0.9512 \times 6.2825 = 5.976 \] Therefore, the value of the American call option is approximately £5.98. Consider a similar scenario, but with a twist. Imagine a farmer growing a rare strain of saffron. He anticipates harvesting 10kg of saffron in six months. He wants to lock in a price now to protect against price fluctuations. He enters into a forward contract to sell his saffron at £2,500 per kg. Six months later, the spot price of saffron is £2,200 per kg. The farmer has effectively protected himself from the price decrease. However, if the spot price had risen to £2,800 per kg, he would have missed out on the potential profit. This illustrates the trade-off between hedging and speculation inherent in derivatives. The farmer’s forward contract is similar to the call option in the question, as both instruments are used to manage risk and potentially profit from future price movements. The binomial model helps to quantify these potential outcomes and their present values, providing a framework for making informed decisions.

The question assesses understanding of the impact of early exercise on American options, particularly in the context of dividend-paying assets. The key is recognizing that early exercise is most likely when the dividend payment exceeds the time value of holding the option. The time value represents the potential for the option to increase in value before expiration, discounted by the risk-free rate. To calculate the optimal decision, we need to compare the immediate gain from exercising and capturing the dividend versus the potential future gain from holding the option. 1. **Calculate the intrinsic value:** The intrinsic value of the call option is the current stock price minus the strike price: \(£55 – £50 = £5\). 2. **Consider the dividend:** The dividend is \(£6\). If exercised immediately, the investor receives the intrinsic value plus the dividend, totaling \(£5 + £6 = £11\). 3. **Estimate the time value:** The time value represents the potential for the stock price to increase before expiration. A rough estimate can be made considering the risk-free rate. If the option is held, the potential gain must outweigh the cost of delaying the dividend receipt. The present value of the dividend received in 3 months (0.25 years) at a risk-free rate of 4% is approximately \(£6 / (1 + 0.04 * 0.25) = £5.94\). 4. **Compare exercise vs. hold:** Exercising immediately yields \(£11\). Holding yields a potential gain from the option’s price movement, but this gain must be greater than the foregone dividend of \(£5.94\) (present value). Given the dividend exceeds the intrinsic value and the time until expiry is relatively short, early exercise is likely optimal. 5. **Factor in transaction costs:** Transaction costs of \(£0.25\) would reduce the net benefit of exercising, but the dividend is still significantly higher than the intrinsic value. Therefore, the early exercise is still optimal. The crucial understanding is that the large dividend payment relative to the intrinsic value and the short time to expiration makes early exercise the most profitable strategy. This scenario highlights that option valuation is not just about the underlying asset’s price but also about considering dividends, interest rates, and transaction costs. A similar analysis applies to put options, where early exercise is more likely when interest rates are high, and the option is deep in the money. This example is different from typical textbook examples by including transaction costs and requiring a comparison of present values, making it more realistic.

To determine the most suitable derivative instrument, we need to evaluate the hedge ratio and the specific risk associated with each option. The hedge ratio signifies the quantity of derivative contracts needed to offset the price risk of the underlying asset. In this scenario, the hedge ratio is calculated as the change in the portfolio value for a given change in the index value. First, we calculate the required number of futures contracts. The portfolio’s current value is £5,000,000, and the FTSE 100 index stands at 7,500. Each index point is worth £10. The formula to determine the number of contracts is: \[ \text{Number of Contracts} = \frac{\text{Portfolio Value} \times \beta}{\text{Index Level} \times \text{Multiplier}} \] Where \(\beta\) is the portfolio beta. \[ \text{Number of Contracts} = \frac{5,000,000 \times 1.2}{7,500 \times 10} = \frac{6,000,000}{75,000} = 80 \] Thus, 80 futures contracts are needed to hedge the portfolio. Next, we evaluate the impact of using options. Options provide a more tailored approach to hedging, especially when considering specific risk profiles. A put option gives the holder the right, but not the obligation, to sell the underlying asset at a predetermined price (the strike price). The number of put options needed depends on the desired level of downside protection and the option’s delta. If we use put options with a delta of 0.5, we need to adjust the number of contracts to account for the option’s sensitivity to changes in the underlying index. The adjusted number of contracts is: \[ \text{Number of Option Contracts} = \frac{\text{Portfolio Value} \times \beta}{\text{Index Level} \times \text{Multiplier} \times \text{Delta}} \] \[ \text{Number of Option Contracts} = \frac{5,000,000 \times 1.2}{7,500 \times 10 \times 0.5} = \frac{6,000,000}{37,500} = 160 \] Therefore, 160 put option contracts are needed. Given the need for precise downside protection and the portfolio manager’s view of potential market volatility, put options provide a more flexible and targeted hedging strategy. While futures offer a straightforward hedge, options allow for customization based on risk tolerance and market outlook. The key advantage of options lies in their ability to limit downside risk while still allowing participation in potential upside gains, making them particularly suitable for managing portfolios with specific risk objectives.

The core of this question revolves around understanding the interplay between correlation, volatility, and delta hedging within a derivatives portfolio, specifically concerning options on two distinct but related assets. The challenge lies in recognizing that perfect hedging is impossible when dealing with multiple assets whose price movements are not perfectly correlated. A delta-neutral portfolio aims to neutralize the portfolio’s sensitivity to small changes in the underlying asset’s price. However, when dealing with multiple assets, even if each individual option position is delta-hedged, the overall portfolio is still exposed to risk arising from changes in the correlation between the assets. This is because the delta hedge ratios are calculated based on the current correlation, and any change in this correlation will affect the portfolio’s value. Let’s consider a scenario where an investment firm holds a portfolio consisting of European call options on two stocks, Stock A and Stock B. Each option has a delta of 0.6 and 0.4 respectively. The firm has delta-hedged each position individually by shorting 60 shares of Stock A and 40 shares of Stock B for every 100 call options held. Initially, the correlation between Stock A and Stock B is 0.7. Now, suppose the correlation unexpectedly drops to 0.3 due to unforeseen market events (e.g., a sector-specific regulation impacting one stock more than the other). This change in correlation will affect the overall portfolio value because the hedge ratios were established based on the initial correlation. The drop in correlation implies that the stocks are now moving more independently of each other than previously anticipated. The portfolio’s vulnerability arises from the fact that the individual delta hedges do not account for the joint movement of the assets. The combined effect of the price changes in Stock A and Stock B, given the new correlation, will deviate from the expected outcome based on the initial correlation. The portfolio will experience a loss if Stock A increases in price while Stock B decreases, or vice versa, because the hedges were constructed assuming a stronger positive relationship between the two stocks. This residual risk, stemming from imperfect correlation, is known as correlation risk. Therefore, even with delta hedging, the portfolio remains exposed to losses due to changes in the correlation between the underlying assets. The firm needs to dynamically adjust its hedge positions to account for changes in correlation, using techniques like vega hedging or correlation hedging, to mitigate this risk.

Let’s analyze the scenario step-by-step to determine the most suitable derivative instrument for mitigating the identified risks. First, we need to quantify the potential loss due to currency fluctuations. The initial investment is £5 million, and the expected return is 8% in USD terms. This translates to an expected USD amount of \( £5,000,000 \times (1 + 0.08) \times \text{Spot Rate} \). The risk arises from the potential depreciation of the USD against GBP. If the USD depreciates by 5% against GBP, the realized GBP return will be lower than expected. To hedge against this, the company can use a forward contract to sell the expected USD proceeds at a predetermined exchange rate. This locks in the GBP value of the investment, regardless of future spot rate movements. The forward rate is calculated as: \[ \text{Forward Rate} = \text{Spot Rate} \times \frac{1 + (\text{Interest Rate GBP} \times \text{Time})}{1 + (\text{Interest Rate USD} \times \text{Time})} \] Given the spot rate of 1.25, GBP interest rate of 4%, USD interest rate of 2%, and a time horizon of 1 year, the forward rate is: \[ \text{Forward Rate} = 1.25 \times \frac{1 + (0.04 \times 1)}{1 + (0.02 \times 1)} = 1.25 \times \frac{1.04}{1.02} \approx 1.2745 \] The expected USD amount after 1 year is \( £5,000,000 \times 1.08 \times 1.25 = \$6,750,000 \). Selling this amount forward at 1.2745 guarantees a GBP return of \( \frac{\$6,750,000}{1.2745} \approx £5,296,979 \). The hedge ensures that the company receives approximately £5,296,979 regardless of the spot rate at the end of the year. This mitigates the risk of USD depreciation. A currency option could provide upside potential if the USD appreciates, but it requires an upfront premium. A currency swap is generally used for longer-term hedging and managing interest rate risk in addition to currency risk, which is not the primary concern here. A futures contract, while similar to a forward contract, may not be as customizable to the exact amount and maturity required.

Let’s break down how to determine the profit or loss from a short strangle position, accounting for early exercise of the call option and the subsequent actions taken to mitigate risk. First, understand the components of the strangle: a short call and a short put, both out-of-the-money initially. The investor profits if both options expire worthless, keeping the premiums received. However, if the underlying asset’s price moves significantly, one or both options can be exercised, leading to potential losses. In this scenario, the call option is exercised early. This means the investor is obligated to sell the underlying asset at the call’s strike price (£105). To fulfill this obligation, the investor immediately buys the asset in the spot market at the prevailing price of £112. This purchase results in an immediate loss of £7 per share (£112 – £105). Next, the investor decides to close out the short put position to limit further potential losses. They do this by buying back the put option at the current market price of £1. This purchase incurs an additional cost of £1 per share. The total profit or loss is calculated by considering the initial premiums received from selling both options and subtracting the losses incurred from the call option exercise and the put option buyback. Initial premium received: £4 (call) + £3 (put) = £7 per share Loss from call option exercise: £7 per share Cost to close put option: £1 per share Total profit/loss: £7 (initial premium) – £7 (call loss) – £1 (put cost) = -£1 per share Therefore, the investor experiences a loss of £1 per share. Now, let’s use an analogy to illustrate this. Imagine you’re running a betting shop. You take bets (premiums) that two horses, “Call-Me-Maybe” and “Put-It-Down,” won’t win a race. “Call-Me-Maybe” suddenly surges ahead, and you have to pay out on that bet (loss from call exercise) by buying a winning ticket from someone else at a higher price. Seeing the other horse, “Put-It-Down,” is also gaining momentum, you decide to close that bet (buy back the put) to avoid potentially paying out on that one too, incurring another cost. The total outcome is the initial bets you took minus what you had to pay out and the cost of closing the second bet. This example highlights the core principles: the initial income from selling options, the obligation triggered by exercise, and the cost of managing the position to mitigate further losses. The key takeaway is that while the initial strategy aims to profit from price stability, significant price movements can lead to losses, even with proactive risk management.

Let’s consider a scenario where a UK-based agricultural cooperative, “British Harvest Co-op,” wants to protect its future wheat sales against fluctuating market prices. They decide to use futures contracts. The current spot price of wheat is £200 per tonne. The December wheat futures contract is trading at £210 per tonne. The Co-op expects to harvest 5,000 tonnes of wheat in December. They decide to hedge their risk by selling 50 December wheat futures contracts (each contract representing 100 tonnes). Now, let’s analyze two possible scenarios in December: Scenario 1: The spot price of wheat drops to £190 per tonne. The futures price also drops to £190 per tonne. The Co-op sells their physical wheat in the spot market for £190 per tonne, receiving £190 * 5000 = £950,000. Simultaneously, they close out their futures position by buying back 50 December wheat futures contracts at £190 per tonne. They initially sold these contracts at £210 per tonne, so they make a profit of (£210 – £190) * 50 * 100 = £100,000 on the futures contracts. Their effective selling price is (£950,000 + £100,000) / 5000 = £210 per tonne. Scenario 2: The spot price of wheat rises to £230 per tonne. The futures price also rises to £230 per tonne. The Co-op sells their physical wheat in the spot market for £230 per tonne, receiving £230 * 5000 = £1,150,000. Simultaneously, they close out their futures position by buying back 50 December wheat futures contracts at £230 per tonne. They initially sold these contracts at £210 per tonne, so they make a loss of (£230 – £210) * 50 * 100 = £100,000 on the futures contracts. Their effective selling price is (£1,150,000 – £100,000) / 5000 = £210 per tonne. In both scenarios, the Co-op effectively locked in a selling price close to £210 per tonne, demonstrating the risk-reducing nature of hedging with futures. Basis risk (the difference between spot and futures prices) can cause slight deviations from the target price. For instance, if the spot price is £192 instead of £190 when futures are £190, the effective price would be slightly higher than £210. The key principle here is understanding how hedging allows a business to offset potential losses (or gains) in the spot market with corresponding gains (or losses) in the futures market, thereby stabilizing their income. This is particularly relevant in volatile commodity markets where price fluctuations can significantly impact profitability. The Financial Conduct Authority (FCA) would expect investment advisors to fully understand these mechanisms and their implications for clients.

To determine the most suitable hedging strategy, we must first calculate the potential loss without any hedge. The initial portfolio value is £5,000,000 and the expected volatility is 18%. A 2% decrease in the portfolio value represents a potential loss of £100,000 (2% of £5,000,000). Next, we evaluate each hedging strategy. Strategy A: Selling 50 FTSE 100 futures contracts. Each contract represents £10 per index point. With the FTSE 100 at 7,500, the notional value of each contract is £75,000 (7,500 * £10). 50 contracts have a notional value of £3,750,000 (50 * £75,000). The hedge ratio is calculated as the notional value of the futures contracts divided by the portfolio value: £3,750,000 / £5,000,000 = 0.75. This means the portfolio is 75% hedged. A 2% fall in the FTSE 100 would lead to a gain in the futures position, offsetting 75% of the portfolio loss. The unhedged loss would be 25% of £100,000, or £25,000. Strategy B: Buying 100 put options on the FTSE 100 with a strike price of 7,400. Each option contract covers one index unit and costs £5 per unit. The total cost is £500 per contract (100 index units * £5). For 100 contracts, the total premium is £50,000. If the FTSE 100 falls below 7,400, the put options will be in the money. If the index falls by 2% from 7,500 to 7,350, the intrinsic value of each put option is 7,400 – 7,350 = 50 index points, or £50 per index point. This gives a profit of £5,000 per contract (100 index units * £50 per unit). For 100 contracts, the total profit is £500,000. Subtracting the premium paid (£50,000) gives a net profit of £450,000. This significantly over-hedges the portfolio, resulting in a substantial profit. Strategy C: Entering into a swap to receive a fixed rate and pay a floating rate linked to the FTSE 100. The notional principal is £2,500,000. A 2% decrease in the FTSE 100 would result in a payment from the counterparty, effectively hedging half the portfolio. If the floating rate tracks the FTSE 100, the payment received would offset £50,000 of the portfolio loss (2% of £2,500,000). The unhedged loss would be £50,000. Strategy D: Using exotic derivatives, such as barrier options, to hedge against downside risk. The cost of these options is £10,000, and they are designed to pay out £75,000 if the FTSE 100 falls by 2% or more. The net benefit of this strategy is £75,000 – £10,000 = £65,000. Comparing the results: Strategy A: Unhedged loss of £25,000. Strategy B: Net profit of £450,000. Strategy C: Unhedged loss of £50,000. Strategy D: Net benefit of £65,000. The best strategy is the one that minimises the loss or maximises the profit, while considering the investor’s risk appetite. In this case, Strategy D provides a reasonable hedge at a low cost, offering a net benefit of £65,000. Strategy B provides a high profit but is considered an over-hedge. Strategy A and C both leave the portfolio with unhedged losses.

To determine the most suitable derivative for mitigating the company’s risk, we need to analyze each option’s characteristics and how they align with the specific risk profile. A forward contract locks in a price for future delivery, offering certainty but eliminating potential gains if the market moves favorably. In this case, securing a fixed exchange rate might protect against adverse currency fluctuations, but the company foregoes the opportunity to benefit if the GBP strengthens against the USD. A futures contract, similar to a forward, obligates the company to buy or sell at a predetermined price and date. However, futures are standardized and traded on exchanges, providing liquidity and transparency but also requiring margin calls and daily settlements. This adds complexity and potential cash flow strain. An option provides the right, but not the obligation, to buy or sell at a specific price. A currency option allows the company to hedge against adverse movements while still benefiting from favorable ones. For example, if the company buys a GBP put option, it can sell GBP at a predetermined rate if it weakens against the USD, but if the GBP strengthens, it can let the option expire and exchange the currency at the prevailing market rate. This flexibility comes at the cost of the option premium. A swap involves exchanging cash flows based on different variables. A currency swap could involve exchanging GBP cash flows for USD cash flows at specified intervals, effectively locking in an exchange rate over a longer period. This can be beneficial for long-term projects but may be less flexible than options. Given the company’s desire to protect against downside risk while retaining the potential for upside gains, a currency option is the most suitable derivative. It provides a hedge against adverse currency movements while allowing the company to benefit if the GBP strengthens. The cost of the option premium is a trade-off for this flexibility.

The core of this question lies in understanding how delta hedging works in practice, especially when transaction costs are involved. A perfect delta hedge, in theory, eliminates all directional risk. However, in the real world, every trade incurs costs (brokerage fees, bid-ask spread). These costs erode the profit from the hedge and must be factored into the decision of how frequently to rebalance. The problem involves calculating the profit or loss from a delta-hedged position, considering the cost of rebalancing. The investor initially sells an option and delta hedges by buying the underlying asset. As the asset price changes, the delta changes, requiring rebalancing. The investor rebalances only once. The key is to calculate the cost of the initial hedge, the cost of rebalancing, and the final profit or loss, considering the transaction costs. Here’s the breakdown of the calculation: 1. **Initial Position:** – Sells the option for £5.00. – Delta is 0.4, so buys 0.4 shares at £100 each. – Cost of initial hedge: 0.4 * £100 = £40. – Transaction cost for initial hedge: 0.4 shares * £0.25/share = £0.10. – Net cash flow at time 0: £5.00 (option premium) – £40 (hedge cost) – £0.10 (transaction cost) = -£35.10 2. **Rebalancing:** – Price increases to £105. – Delta increases to 0.7. – Needs to buy an additional 0.7 – 0.4 = 0.3 shares. – Cost of additional shares: 0.3 * £105 = £31.50. – Transaction cost for rebalancing: 0.3 shares * £0.25/share = £0.075. 3. **Closing the Position:** – Price is £105. – Delta is 0.7. – Buys back the option. – Sells the 0.7 shares at £105 each. – Revenue from selling shares: 0.7 * £105 = £73.50 – Total cost of shares bought = £40 + £31.50 = £71.50 – Total transaction costs = £0.10 + £0.075 = £0.175 – Profit from shares = £73.50 – £71.50 – £0.175 = £1.825 4. **Overall Profit/Loss:** – Profit from the shares: £1.825 – Initial cash flow from selling option: £5 – Total cash inflow = £5 + £1.825 = £6.825 – Cost of setting up and rebalancing the hedge = £40 + £0.10 + £31.50 + £0.075 = £71.675 – Net profit/loss = £6.825 – (£71.50 + £0.175) = £6.825 – £71.675 = -£64.85 However, we need to consider the option itself. The investor sold the option for £5. The question doesn’t state the option’s payoff. Since it’s a delta hedge scenario, we assume the option expires. If the price is above the strike, the option is exercised. If the price is below the strike, the option expires worthless. Let’s assume the strike price is £100. The final price is £105, so the option is in the money. The payoff is £105 – £100 = £5. The investor has to pay out £5. Therefore, the net profit/loss = £1.825 – £5 = -£3.175. Now, let’s assume the strike price is £110. The final price is £105, so the option is out of the money. The option expires worthless. Therefore, the net profit/loss = £1.825 – £0 = £1.825 We sold the option for £5 initially, and the cost of setting up the hedge was £40.10. We rebalanced at a cost of £31.575. We then sold the shares for £73.50. So, £5 + £73.50 – £40.10 – £31.575 = £6.825. Therefore, the net profit/loss is £6.825. However, this neglects the crucial fact that the investor SOLD the option. They received £5 for it initially. So the overall profit/loss is affected by whether the option is exercised. If the option is exercised, the investor loses money. If the option expires worthless, the investor keeps the £5 premium. Let’s assume the option expires worthless (strike price above £105). Then, the profit is the £5 premium + profit from the hedge = £5 + £1.825 = £6.825 Let’s assume the option is exercised (strike price below £105, say £100). Then the investor loses £5 (105-100). The profit is the £5 premium + profit from the hedge – £5 = £1.825 Since the question does not specify the strike price, and asks for the profit/loss from the delta hedge, it is asking for the profit/loss from the shares. Therefore, the answer is £1.825.

The question explores the complexities of option pricing and hedging strategies, specifically focusing on the practical implications of gamma and vega exposures in a dynamic market environment. The scenario involves a portfolio manager who has written options and must adjust their hedge due to market movements and time decay. The calculation involves understanding how changes in the underlying asset’s price (affecting delta) and changes in implied volatility (affecting vega) impact the required hedge adjustments. The key is to calculate the new delta exposure after the price change, adjust for gamma, and then account for the change in vega due to the volatility shift. First, we calculate the initial delta exposure. The portfolio manager has written 100 call options, each representing 100 shares, so a total of 10,000 shares are involved. The initial delta is 0.6, meaning the initial delta exposure is 10,000 * 0.6 = 6,000. This represents the number of shares needed to hedge the position. Next, we account for the change in the underlying asset’s price and the impact of gamma. The price increases by £1, and the gamma is 0.05. The change in delta due to gamma is calculated as the gamma multiplied by the price change, multiplied by the number of options: 0.05 * 1 * 10,000 = 500. The new delta is therefore 0.6 + 0.05 = 0.65 per option, so the new delta exposure is 10,000 * 0.65 = 6,500. The manager needs to increase the hedge by 500 shares due to the price movement and gamma. Finally, we consider the impact of the change in implied volatility. The vega is 0.08, meaning that for every 1% change in implied volatility, the option price changes by £0.08. The implied volatility increases by 2%, so the change in the option price due to vega is 0.08 * 2 = £0.16 per option. However, vega does not directly impact the delta. Vega primarily influences the option’s price sensitivity to volatility changes, which is a separate risk factor. The hedge adjustment is based on the delta, which is influenced by the price change and gamma. Therefore, the vega component is not directly used to calculate the number of shares to buy or sell. Therefore, the portfolio manager needs to buy an additional 500 shares to rebalance their delta hedge. The vega component is important for understanding the overall risk profile of the option position but does not directly factor into the delta hedging calculation in this specific scenario. The manager will need to monitor the vega and adjust their strategy accordingly, but the immediate action is to buy 500 shares to maintain delta neutrality.