Derivatives Level 4 (Investment Advice Diploma) Quiz Exam Quiz 31

The core of this question revolves around understanding the mechanics of variance swaps, particularly how the strike price (K) is determined and how the payoff is calculated. The fair strike price is designed to make the expected payoff of the swap zero at initiation. The payoff is calculated as N * (Realized Variance – Variance Strike), where N is the notional amount. The realized variance is calculated from the daily returns using the formula: Realized Variance = (1/ (n-1)) * Σ(Ri – Average Return)^2 where Ri is the daily return. To solve this problem, we need to: 1. Calculate the average daily return. 2. Calculate the realized variance using the formula. 3. Calculate the payoff of the variance swap. Let’s assume the daily returns are: 0.1%, -0.2%, 0.3%, 0.05%, -0.15%. 1. **Calculate the average daily return:** Average Return = (0.1 – 0.2 + 0.3 + 0.05 – 0.15) / 5 = 0.1 / 5 = 0.02% = 0.0002 2. **Calculate the Realized Variance:** Realized Variance = (1/(5-1)) * [(0.001 – 0.0002)^2 + (-0.002 – 0.0002)^2 + (0.003 – 0.0002)^2 + (0.0005 – 0.0002)^2 + (-0.0015 – 0.0002)^2] Realized Variance = (1/4) * [(0.0008)^2 + (-0.0022)^2 + (0.0028)^2 + (0.0003)^2 + (-0.0017)^2] Realized Variance = (1/4) * [0.00000064 + 0.00000484 + 0.00000784 + 0.00000009 + 0.00000289] Realized Variance = (1/4) * [0.0000163] = 0.000004075 3. **Annualize the Realized Variance:** Since the returns are daily, and assuming 252 trading days in a year, we annualize the variance: Annualized Realized Variance = 0.000004075 * 252 = 0.001027 4. **Calculate the Realized Volatility:** Realized Volatility = √Annualized Realized Variance = √0.001027 ≈ 0.0320 or 3.20% 5. **Variance Strike:** The variance strike is the square of the volatility strike. If the volatility strike is 3%, then the variance strike is 3%^2 = 0.0009 6. **Payoff Calculation:** Payoff = Notional * (Realized Variance – Variance Strike) Payoff = £1,000,000 * (0.001027 – 0.0009) Payoff = £1,000,000 * (0.000127) = £127 A sophisticated investor uses variance swaps not just for directional bets on volatility, but also for hedging complex portfolios. Imagine a fund manager whose portfolio performance is highly sensitive to market volatility. They might use a variance swap to offset potential losses during periods of high volatility. If volatility spikes, the payoff from the variance swap could compensate for the underperformance of their underlying assets. This allows for more stable returns.

Let’s break down this complex scenario. First, we need to calculate the implied volatility using the provided option prices and other parameters. We will use an iterative approach, plugging in different volatility values into an option pricing model (like Black-Scholes) until the calculated option price matches the market price. Since we don’t have the pricing model available for direct calculation, we will focus on understanding the sensitivities (Greeks) and how they influence the decision-making process. The key here is to understand how Gamma and Vega interact. Gamma measures the rate of change of Delta with respect to changes in the underlying asset’s price. Vega measures the sensitivity of the option’s price to changes in volatility. A high Gamma indicates that the Delta will change significantly with even small price movements in the underlying asset. A high Vega indicates that the option price is highly sensitive to changes in implied volatility. In this scenario, the fund manager is using a delta-neutral strategy, meaning the overall portfolio Delta is zero. However, because the options have a non-zero Gamma, the portfolio will not remain delta-neutral as the underlying asset’s price changes. The fund manager is also concerned about changes in implied volatility, as indicated by the Vega. To address these concerns, the fund manager needs to consider the following: 1. **Gamma Risk:** The fund manager could use more options to reduce the overall Gamma of the portfolio. For instance, if the portfolio has positive Gamma, the fund manager could sell options to reduce the Gamma. The number of options required to neutralize the Gamma depends on the Gamma of each option and the desired level of Gamma neutrality. 2. **Vega Risk:** The fund manager could use options with offsetting Vega to reduce the portfolio’s sensitivity to changes in implied volatility. If the portfolio has positive Vega, the fund manager could sell options with negative Vega to reduce the overall Vega. 3. **Cost of Hedging:** The fund manager must consider the cost of implementing these hedges. Buying or selling options will incur transaction costs and may impact the portfolio’s profitability. Given the fund manager’s objective to minimize both Gamma and Vega exposure, the most appropriate action is to use options with offsetting Gamma and Vega characteristics. This will help to maintain delta neutrality while also reducing the portfolio’s sensitivity to changes in implied volatility. The specific strategy will depend on the available options and their respective Gamma and Vega values. For example, the fund manager could implement a variance swap to hedge volatility risk or use a combination of options with different strike prices and expiration dates to manage both Gamma and Vega. The key is to carefully analyze the Greeks of the available options and construct a portfolio that minimizes the desired exposures.

The core concept being tested is the understanding of how different types of derivatives (forwards, futures, options, swaps) respond to changes in underlying asset prices and interest rates, specifically in the context of managing risk and generating returns within a portfolio. The question probes the candidate’s ability to integrate knowledge of derivative characteristics, market dynamics, and regulatory constraints (specifically, the FCA’s conduct rules) to make informed investment decisions. The scenario presents a complex situation where an investment manager must choose the most appropriate derivative strategy to achieve a specific investment objective (generating income while protecting against downside risk) in a volatile market environment. The manager must consider the trade-offs between different derivative types, their costs, and their potential payoffs under various market conditions. The correct answer (a) identifies a strategy that combines selling covered call options to generate income and buying protective put options to limit downside risk. This strategy aligns with the investment objective of generating income while protecting against losses. The incorrect options (b, c, and d) present alternative strategies that are either inconsistent with the investment objective, too risky, or too costly. For example, option (b) involves selling naked put options, which could generate income but exposes the portfolio to unlimited downside risk. Option (c) involves buying call options, which would benefit from rising stock prices but would not generate income or protect against losses. Option (d) involves entering into a short futures contract, which could protect against losses but would also limit potential gains. The question tests the candidate’s ability to apply their knowledge of derivatives to a real-world investment scenario and to make informed decisions based on the client’s objectives and risk tolerance. The question also tests the candidate’s understanding of the FCA’s conduct rules, which require investment managers to act in the best interests of their clients and to manage conflicts of interest.

Let’s analyze the swap scenario. Company Alpha seeks to hedge against rising interest rates on its floating-rate debt, currently pegged to SONIA + 1.5%. They enter a receive-fixed, pay-floating swap with Bank Beta. The notional principal is £5 million, and the swap term is 3 years, with semi-annual payments. The fixed rate is agreed at 4.0% per annum. First, calculate the semi-annual fixed payment: (£5,000,000 * 0.04) / 2 = £100,000. This is what Alpha pays Beta every six months. Now, consider the floating rate. For the first period, SONIA averages 3.0%. Alpha pays SONIA + 1.5% = 4.5% on the notional principal. The semi-annual floating payment is (£5,000,000 * 0.045) / 2 = £112,500. The net payment for the first period is the difference between the floating payment Alpha owes and the fixed payment Alpha receives: £112,500 – £100,000 = £12,500. Alpha pays Bank Beta £12,500. For the second period, SONIA averages 2.0%. Alpha pays SONIA + 1.5% = 3.5%. The semi-annual floating payment is (£5,000,000 * 0.035) / 2 = £87,500. The net payment for the second period is the difference between the floating payment Alpha owes and the fixed payment Alpha receives: £87,500 – £100,000 = -£12,500. Bank Beta pays Alpha £12,500. Now, consider the regulatory implications under EMIR (European Market Infrastructure Regulation). EMIR aims to reduce systemic risk by requiring OTC derivatives to be cleared through a central counterparty (CCP). Given the characteristics of the swap (standard terms, high notional, etc.), it’s highly probable that the swap is subject to mandatory clearing. This means Alpha and Beta must clear the swap through a CCP, which interposes itself between the two parties, guaranteeing performance. If Alpha defaults, the CCP steps in. EMIR also imposes reporting obligations. Both Alpha and Beta must report the swap to a trade repository. The specific reporting fields include the notional amount, maturity date, underlying asset, and counterparty details. Failure to comply with these obligations can result in penalties. The key takeaway is understanding how interest rate swaps work, calculating net payments, and recognizing the crucial regulatory overlay imposed by EMIR, particularly mandatory clearing and reporting obligations.

The core concept tested is the understanding of how gamma impacts a portfolio’s delta as the underlying asset’s price changes. 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 is more sensitive to price movements. The question requires calculating the new delta of the portfolio after a price change in the underlying asset, considering the portfolio’s gamma. Here’s the step-by-step calculation: 1. **Calculate the change in delta:** Change in delta = Gamma * Change in underlying asset price. In this case, the change in delta is 0.04 (Gamma) * £2.50 (Price change) = 0.10 2. **Calculate the new delta:** New delta = Original delta + Change in delta. In this case, the new delta is 0.60 (Original delta) + 0.10 (Change in delta) = 0.70 Therefore, the new delta of the portfolio is 0.70. A key misunderstanding often arises from neglecting the impact of gamma on delta. Many incorrectly assume delta remains constant regardless of price fluctuations. This is only valid when gamma is zero, which is rarely the case, especially with options close to the money. Imagine a portfolio manager, Anya, who is tasked with managing a large equity portfolio. She uses options to hedge the portfolio’s risk. Anya understands that delta measures the portfolio’s sensitivity to changes in the underlying equity’s price. However, she also knows that delta is not static. Gamma, the rate of change of delta, becomes crucial when the equity price experiences significant volatility. If Anya ignores gamma, she might underestimate the portfolio’s risk exposure, especially during periods of market turbulence. For instance, if the portfolio has a positive gamma and the equity price rises sharply, the portfolio’s delta will increase, making it more sensitive to further price increases. Conversely, if the equity price falls, the delta will decrease, reducing the portfolio’s sensitivity to further declines. Anya’s ability to accurately assess and manage gamma is essential for maintaining the portfolio’s desired risk profile. The scenario illustrates the practical implications of gamma in portfolio management and highlights the importance of considering its impact on delta when making hedging decisions. A failure to understand this relationship can lead to significant financial losses.

Let’s consider a scenario involving a power generation company, “Energetica Ltd,” which uses natural gas to produce electricity. Energetica Ltd. wants to hedge against the volatility of natural gas prices over the next year to stabilize their production costs and ensure predictable electricity pricing for their consumers. They enter into a series of monthly natural gas futures contracts. The question explores how Energetica Ltd. might use options on these futures contracts to refine their hedging strategy, specifically to manage potential upside if natural gas prices unexpectedly decrease significantly. Energetica Ltd. initially hedges by buying natural gas futures contracts for delivery each month. This protects them against price increases but also locks them into a price even if the market price drops substantially. To benefit from a potential price decrease, Energetica Ltd. could purchase put options on their existing futures contracts. This strategy allows them to maintain their price protection in case of a price increase while simultaneously enabling them to profit if prices fall below the put option’s strike price. The question asks about the specific action Energetica Ltd. would take if they observe that the market price of natural gas has fallen substantially below the strike price of their put options. The correct action is to exercise their put options. By exercising the put options, Energetica Ltd. can sell the futures contracts at the strike price (which is higher than the current market price) and simultaneously buy them back at the lower market price, effectively realizing a profit. This profit offsets the cost of the futures contracts they initially purchased at a higher price, allowing them to benefit from the market decline. The other options are incorrect because they either represent missed opportunities or actions that would not allow Energetica Ltd. to capitalize on the price decrease. For instance, selling the put options back into the market would only yield the option’s premium, which would likely be small if the market price is far below the strike price. Similarly, holding the options until expiration would only be beneficial if the market price remains below the strike price, but it would not allow them to immediately realize the profit. Ignoring the price decrease altogether would mean Energetica Ltd. would miss out on the opportunity to reduce their costs and improve their profitability.

The key to solving this problem lies in understanding the combined effects of the swap and the FRA. The company effectively transforms a floating-rate liability into a fixed-rate liability using the swap. The FRA then acts as a hedge against fluctuations in the floating rate before the swap becomes effective. First, calculate the effective fixed rate paid due to the swap: 6%. Next, determine the implied forward rate from the FRA. The FRA protects the company against rates exceeding 5.5%. If LIBOR is above 5.5%, the FRA pays the difference. If LIBOR is below 5.5%, the company receives nothing from the FRA, but still benefits from the lower rate. In this scenario, LIBOR is 6% at the FRA settlement date. Therefore, the FRA pays the company (6% – 5.5%) = 0.5%. The company pays 6% due to the swap, but receives 0.5% from the FRA. The net payment is 6% – 0.5% = 5.5%. Before the swap’s effective date, the company pays LIBOR. The FRA hedges this exposure. The overall effective rate is the LIBOR rate minus the FRA payment. If LIBOR is 6%, the effective rate becomes 6% – 0.5% = 5.5%. After the swap becomes effective, the company pays the fixed rate of 6%. The blended rate is calculated as follows: * 3 months (0.25 years) at 5.5% (due to FRA hedge): 0.25 * 0.055 = 0.01375 * 9 months (0.75 years) at 6% (due to swap): 0.75 * 0.06 = 0.045 Total interest paid = 0.01375 + 0.045 = 0.05875 Effective annual rate = 0.05875 / 1 = 5.875% Therefore, the effective annual rate is 5.875%.

Let’s analyze the combined impact of a rolling hedge using Eurodollar futures and a swaption on a corporate treasurer’s interest rate risk. A rolling hedge involves continuously adjusting the hedge position as the hedge horizon approaches, in this case, using Eurodollar futures contracts. Eurodollar futures are cash-settled futures contracts based on the 3-month LIBOR rate. The treasurer is using these to hedge against rising interest rates. A swaption grants the *right*, but not the *obligation*, to enter into an interest rate swap. In this scenario, the treasurer holds a payer swaption, giving the company the right to *pay* a fixed rate and *receive* a floating rate. The initial hedge ratio is calculated by dividing the notional value of the debt by the contract size of the Eurodollar futures contract. The number of contracts is then adjusted each quarter to maintain the hedge. When interest rates rise, the value of the Eurodollar futures contracts will decrease, generating a profit that offsets the increased cost of borrowing. However, the treasurer also has a payer swaption. If rates rise significantly, the swaption becomes valuable, as the treasurer can exercise the option and pay a fixed rate, limiting their exposure to further rate increases. The profit from the swaption then further offsets the rising borrowing costs. The problem assesses the combined impact of these strategies. The treasurer initially hedges with Eurodollar futures, and then uses a payer swaption as an additional protection layer. The combined effect is to limit the company’s exposure to rising interest rates beyond a certain point. The question requires understanding how these derivatives interact and how their combined effect manages interest rate risk. The key is to understand the payoff profiles of both instruments and how they complement each other. The swaption provides a cap on interest rate increases, while the rolling Eurodollar futures hedge provides ongoing protection against smaller rate movements.

Let’s break down this complex scenario involving a quanto swap and its sensitivity to various market factors. A quanto swap involves exchanging cash flows denominated in different currencies, and its valuation and risk management are intricate. The key here is understanding how changes in interest rates and exchange rates impact the swap’s value. First, consider the fixed leg. It pays a fixed rate in USD on a notional principal in EUR. The present value of this leg is calculated by discounting each future payment back to the present using the appropriate USD discount factors. An increase in USD interest rates will decrease the present value of the fixed leg because the discount factors become smaller. Next, consider the floating leg. It pays a floating rate in EUR, converted to USD at a fixed exchange rate. The expected future EURIBOR rates are used to project the EUR cash flows, which are then converted to USD. The present value of this leg is more complex. An increase in EURIBOR will increase the projected EUR cash flows. However, the present value of these increased USD cash flows is still affected by the USD discount factors. The fixed exchange rate in the quanto swap is crucial because it isolates the USD cash flows from direct EUR/USD exchange rate fluctuations. Now, let’s analyze the impact of the correlation between EURIBOR and USD interest rates. A positive correlation means that when EURIBOR increases, USD interest rates also tend to increase. This has a compounding effect on the floating leg’s present value. The increase in EURIBOR increases the projected EUR cash flows, but the increase in USD interest rates decreases the present value of those cash flows when converted to USD and discounted back to the present. Finally, consider the impact of a change in the fixed exchange rate. If the fixed exchange rate increases (e.g., from 1.10 to 1.15), the USD value of the EUR-denominated floating leg payments will increase. This directly increases the value of the floating leg. In our scenario, the positive correlation between EURIBOR and USD interest rates is the most important factor. It reduces the sensitivity of the quanto swap to changes in EURIBOR. If the correlation were zero, the swap would be more sensitive to EURIBOR changes. The fixed exchange rate isolates the swap from direct EUR/USD spot rate movements, making the correlation the primary driver of the swap’s sensitivity. Therefore, the most accurate statement is that the positive correlation between EURIBOR and USD interest rates significantly reduces the sensitivity of the quanto swap to changes in EURIBOR.

Let’s consider a scenario where a fund manager uses options to hedge a portfolio of UK equities against a potential market downturn due to unexpected regulatory changes following Brexit. The fund holds £50 million worth of FTSE 100 stocks. To protect against a decline, the manager purchases put options on the FTSE 100 index. The FTSE 100 index is currently at 7500. The fund manager buys 500 put option contracts with a strike price of 7400, expiring in 3 months. Each contract covers £10 per index point. The premium paid is £2.00 per index point. The total cost of the hedge is: 500 contracts * £10/point * £2.00/point = £10,000. Now, let’s consider two scenarios at expiration: Scenario 1: The FTSE 100 falls to 7000. The put options are in the money. The profit per contract is (7400 – 7000) * £10 = £4,000. Total profit from options: 500 * £4,000 = £2,000,000. Net profit after deducting premium: £2,000,000 – £10,000 = £1,990,000. Scenario 2: The FTSE 100 rises to 7600. The put options expire worthless. The loss is the premium paid: £10,000. Now, let’s consider the impact on the portfolio. If the FTSE 100 falls from 7500 to 7000, the portfolio value will decrease. We need to calculate the beta of the portfolio to estimate the decline. Let’s assume the portfolio has a beta of 1, meaning it moves in line with the FTSE 100. The percentage decline in the index is (7500 – 7000) / 7500 = 6.67%. Therefore, the portfolio value decreases by approximately 6.67%, which is £50,000,000 * 0.0667 = £3,335,000. The net loss is £3,335,000 – £1,990,000 = £1,345,000. If the FTSE 100 rises to 7600, the portfolio value increases by approximately (7600-7500)/7500 = 1.33%, which is £50,000,000 * 0.0133 = £665,000. The net gain is £665,000 – £10,000 = £655,000. This example demonstrates how options can be used to hedge portfolio risk, but it also highlights the costs (premium) and potential limitations (basis risk, imperfect correlation). The effectiveness of the hedge depends on the accuracy of the beta estimate and the correlation between the portfolio and the index. It also shows that while hedging protects against downside risk, it also limits potential upside gains.

The core of this question lies in understanding how margin requirements work in futures contracts, specifically focusing on the impact of intraday price fluctuations and the maintenance margin. The initial margin is the amount required to open a futures position, acting as a performance bond. The maintenance margin is a lower threshold; if the account balance falls below this level due to adverse price movements, a margin call is triggered. The variation margin is the amount needed to bring the account balance back to the initial margin level. In this scenario, the investor starts with an initial margin of £6,000. The price first moves favorably, increasing the account balance, and then moves unfavorably, decreasing the balance. The key is to track the balance after each price movement and compare it to the maintenance margin to determine if a margin call is triggered. 1. **Initial Margin:** £6,000 2. **Price Increase:** The futures price increases by 2 points. Each point is worth £50. Therefore, the account increases by 2 * £50 = £100. New balance: £6,000 + £100 = £6,100. 3. **Price Decrease:** The futures price decreases by 7 points. The account decreases by 7 * £50 = £350. New balance: £6,100 – £350 = £5,750. 4. **Margin Call Check:** The maintenance margin is £5,500. Since £5,750 (current balance) > £5,500 (maintenance margin), no margin call is triggered at this point. 5. **Further Price Decrease:** The futures price decreases by another 5 points. The account decreases by 5 * £50 = £250. New balance: £5,750 – £250 = £5,500. 6. **Margin Call Check:** The maintenance margin is £5,500. Since £5,500 (current balance) = £5,500 (maintenance margin), no margin call is triggered at this point. 7. **Final Price Decrease:** The futures price decreases by 3 points. The account decreases by 3 * £50 = £150. New balance: £5,500 – £150 = £5,350. 8. **Margin Call Trigger:** The maintenance margin is £5,500. Since £5,350 (current balance) < £5,500 (maintenance margin), a margin call is triggered. 9. **Margin Call Amount:** The investor needs to restore the account balance to the initial margin level of £6,000. Therefore, the margin call amount is £6,000 – £5,350 = £650. Therefore, the investor will receive a margin call for £650.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their sensitivity to market volatility and barrier proximity. A down-and-out call option becomes worthless if the underlying asset’s price touches or falls below the barrier level. The closer the current asset price is to the barrier, the higher the probability of the barrier being hit, thus increasing the option’s sensitivity to price movements (delta) and volatility (vega). Furthermore, the question incorporates the effect of interest rates. A rise in interest rates generally increases the value of call options, as the present value of the strike price decreases. However, the dominant factor here is the barrier proximity. To solve this, we need to consider the combined effects. The proximity to the barrier significantly increases the likelihood of the option expiring worthless, overshadowing the impact of the interest rate increase. Therefore, the option’s value will likely decrease, and its sensitivity to volatility will increase due to the heightened risk of hitting the barrier. Delta, measuring the option’s price sensitivity to changes in the underlying asset price, becomes more negative as the barrier gets closer, because a small downward move can trigger the barrier. Vega, measuring the option’s price sensitivity to changes in volatility, also increases, because the probability of hitting the barrier is more sensitive to volatility changes when the asset price is near the barrier. Therefore, the option’s value will likely decrease, its delta will become more negative, and its vega will increase.

To determine the profit or loss from the combined strategy, we need to analyze the outcomes for both the short put and the long call. * **Short Put Analysis:** The investor sells a put option with a strike price of 95. The investor receives a premium of £3. If the stock price stays above 95, the put option expires worthless, and the investor keeps the premium. If the stock price falls below 95, the investor is obligated to buy the stock at 95. * **Long Call Analysis:** The investor buys a call option with a strike price of 105. The investor pays a premium of £2. If the stock price stays below 105, the call option expires worthless, and the investor loses the premium. If the stock price rises above 105, the investor can exercise the call option and buy the stock at 105. * **Combined Strategy:** We need to consider the combined profit/loss at the expiration date, given the stock price of 102. * **Short Put:** Since the stock price (102) is above the strike price (95), the put option expires worthless. The investor keeps the £3 premium. * **Long Call:** Since the stock price (102) is below the strike price (105), the call option expires worthless. The investor loses the £2 premium. * **Overall Profit/Loss:** The investor makes £3 from the short put and loses £2 from the long call. The net profit is £3 – £2 = £1. Therefore, the investor makes a profit of £1 per share. Imagine a small bakery selling “future bread” contracts. They sell a put option on their bread at £1.50 (strike price) for a premium of £0.10, and buy a call option at £2.50 (strike price) for a premium of £0.05. If, at the contract’s expiry, bread is selling for £2.00, the put expires worthless, and the bakery keeps the £0.10. The call also expires worthless, costing them £0.05. Their net profit is £0.10 – £0.05 = £0.05. This illustrates how a combined options strategy can yield profit or loss depending on the asset’s price at expiry relative to the strike prices. This example provides an accessible way to understand the mechanics of options and how their combined effect results in the final profit or loss.

The core of this question lies in understanding how margin requirements work for futures contracts, especially in scenarios involving large price swings. The initial margin is the amount required to open a futures position, and the maintenance margin is the level below which the account balance cannot fall without triggering a margin call. A margin call requires the investor to deposit additional funds to bring the account back up to the initial margin level. The calculation involves tracking the daily price changes, their impact on the account balance, and comparing that balance to the maintenance margin. If the balance falls below the maintenance margin, a margin call is issued to restore the balance to the initial margin. The investor needs to deposit enough funds to cover the deficit and bring the account back to the initial margin level. Let’s break down the scenario: An investor buys 5 gold futures contracts. Each contract represents 100 troy ounces of gold. The initial margin is £5,000 per contract, and the maintenance margin is £4,000 per contract. 1. **Initial Margin:** The total initial margin for 5 contracts is 5 contracts * £5,000/contract = £25,000. 2. **Daily Price Changes:** * Day 1: Price increases by £20/ounce. Profit = 5 contracts * 100 ounces/contract * £20/ounce = £10,000. Account balance = £25,000 + £10,000 = £35,000. * Day 2: Price decreases by £50/ounce. Loss = 5 contracts * 100 ounces/contract * £50/ounce = £25,000. Account balance = £35,000 – £25,000 = £10,000. * Day 3: Price decreases by £30/ounce. Loss = 5 contracts * 100 ounces/contract * £30/ounce = £15,000. Account balance = £10,000 – £15,000 = -£5,000. 3. **Margin Call:** * The maintenance margin per contract is £4,000, so for 5 contracts, it’s £20,000. * On Day 2, the account balance (£10,000) is already below the maintenance margin (£20,000), triggering a margin call. * The investor needs to bring the account back to the initial margin level of £25,000. * The amount needed to meet the margin call on Day 2 is £25,000 (initial margin) – £10,000 (account balance) = £15,000. * On Day 3, the account balance drops to -£5,000. To meet the margin call, the investor needs to deposit £25,000 – (-£5,000) = £30,000. * However, the question asks for the amount needed to meet the margin call on Day 2. Therefore, the investor needs to deposit £15,000 to meet the margin call on Day 2.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their sensitivity to volatility (vega) and the barrier level. A down-and-out barrier option ceases to exist if the underlying asset price touches the barrier level. The vega of a standard option measures the change in the option’s price for a 1% change in the underlying asset’s volatility. For a down-and-out option, a higher volatility increases the probability of the asset price hitting the barrier, thus decreasing the option’s value. This results in a negative vega, unlike standard options where vega is positive. The proximity of the current asset price to the barrier also significantly impacts the option’s value. If the asset price is very close to the barrier, the option is highly sensitive to even small changes in volatility or the asset price itself. This is because a small increase in volatility makes it much more likely that the barrier will be hit, knocking out the option. The vega effect is most pronounced when the underlying asset price is near the barrier because the probability of the option being knocked out is highly sensitive to changes in volatility. If the asset price is far from the barrier, changes in volatility have less impact on the probability of the barrier being hit, and the vega effect is smaller. Consider an analogy: imagine a tightrope walker close to the edge. A slight gust of wind (increased volatility) is much more likely to push them off (knock out the option) than if they were in the middle of the rope. Similarly, if the tightrope walker is already far from the edge, even a strong gust of wind might not be enough to push them off. Therefore, the vega of a down-and-out option is most negative when the underlying asset price is close to the barrier because the option’s value is most vulnerable to being knocked out by even small increases in volatility.

The core concept tested is the understanding of how various derivative instruments (forwards, futures, options, and swaps) can be used to hedge against different types of risk, specifically focusing on a complex, real-world scenario involving a UK-based manufacturing company, currency fluctuations, and interest rate volatility. The question requires the candidate to analyze the company’s exposure and then determine the most appropriate hedging strategy using a combination of derivatives. The correct answer, option a, is derived from the following logic: The company faces two primary risks: a weakening GBP against the EUR and rising interest rates on its GBP borrowing. To hedge against the currency risk, the company should use a forward contract to sell EUR and buy GBP at a predetermined rate, locking in a favorable exchange rate for its EUR revenues. To hedge against the interest rate risk, the company should use an interest rate swap to convert its floating-rate debt into a fixed-rate debt, protecting it from rising interest rates. The incorrect options are designed to be plausible by including derivatives that could be used in related but ultimately incorrect hedging strategies. Option b incorrectly suggests using a currency option, which would provide flexibility but at a premium cost, and a forward rate agreement (FRA), which only hedges a single future interest rate period, not the entire loan. Option c incorrectly suggests using a currency swap, which is more complex and unnecessary for this specific hedging need, and a cap, which only protects against rates exceeding a certain level, not general rate increases. Option d incorrectly suggests using a futures contract, which might be suitable but less directly tailored than a forward contract for a specific revenue stream, and a floor, which is irrelevant to the company’s concern about rising interest rates. The detailed explanation further elaborates on the rationale behind each derivative choice. A forward contract is ideal for hedging a known future cash flow in a foreign currency because it locks in the exchange rate, eliminating uncertainty. An interest rate swap is effective for converting floating-rate debt to fixed-rate debt, providing certainty about future interest payments. The explanation emphasizes that the best hedging strategy depends on the specific risks faced by the company and its risk appetite. The explanation uses the analogy of a “financial shield” to illustrate how derivatives protect against adverse market movements, and a “locking mechanism” to describe the certainty provided by forward contracts and interest rate swaps.

Let’s consider a scenario where a fund manager uses options to hedge a portion of their portfolio against a potential market downturn. The fund manager holds a large position in UK equities and is concerned about an upcoming economic announcement that could negatively impact the market. To protect against this downside risk, the manager decides to purchase put options on the FTSE 100 index. The fund manager buys 100 put option contracts on the FTSE 100 index with a strike price of 7500, expiring in three months. The premium for each put option is £5. Each contract represents 1 index point multiplied by a contract multiplier of £10. The current FTSE 100 index level is 7600. Now, suppose that at expiration, the FTSE 100 index falls to 7300. The put options are in the money, and the fund manager exercises them. The payoff from each put option contract is the difference between the strike price and the index level at expiration, multiplied by the contract multiplier. Payoff per contract = (Strike Price – Index Level at Expiration) * Contract Multiplier Payoff per contract = (7500 – 7300) * £10 = £2000 Since the fund manager holds 100 contracts, the total payoff is: Total Payoff = Payoff per contract * Number of Contracts Total Payoff = £2000 * 100 = £200,000 However, the fund manager paid a premium for these options. The total premium paid is: Premium per contract = £5 * Contract Multiplier = £5 * £10 = £50 Total Premium Paid = Premium per contract * Number of Contracts Total Premium Paid = £50 * 100 = £5,000 The net profit from the options strategy is the total payoff minus the total premium paid: Net Profit = Total Payoff – Total Premium Paid Net Profit = £200,000 – £5,000 = £195,000 This net profit helps offset the losses incurred in the fund manager’s equity portfolio due to the market downturn. This example illustrates how options can be used as an effective hedging tool to protect against downside risk, albeit with the initial cost of the premium. The key here is understanding that the put options provide insurance against a market decline, and the payoff is contingent on the index falling below the strike price at expiration. If the index had stayed above the strike price, the options would have expired worthless, and the fund manager would have only lost the premium paid.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their sensitivity to market volatility (vega) and the barrier level. The scenario presents a situation where a portfolio manager uses a down-and-out call option to hedge a position in a volatile stock. The key is to understand how changes in volatility and the barrier level affect the option’s value. An increase in volatility generally increases the value of standard options. However, for a down-and-out call, increased volatility can increase the probability of the underlying asset hitting the barrier, causing the option to expire worthless. Therefore, the vega of a down-and-out call can be negative under certain conditions. A decrease in the barrier level makes it easier for the underlying asset to hit the barrier, thus decreasing the value of the down-and-out call. The closer the barrier is to the current price, the more sensitive the option is to changes in the barrier level. Considering the combined effects, if the volatility increases significantly while the barrier level decreases, the negative impact of hitting the barrier outweighs the potential positive impact of increased volatility on the call option’s price. This results in a decrease in the option’s value. The initial value of the down-and-out call is £50,000. The increase in volatility has a potential positive effect, but the decrease in the barrier level has a negative effect, and in this case, the negative effect is dominant. Let’s say the increased volatility could have added £10,000 to the option’s value if the barrier hadn’t been breached, but the decreased barrier level causes a loss of £25,000 due to the increased probability of the option expiring worthless. The net change is -£15,000 (£10,000 – £25,000). Therefore, the new value of the option is £35,000 (£50,000 – £15,000).

Let’s analyze how the early termination of a swap impacts its Net Present Value (NPV). The core concept is that the NPV represents the present value of all future cash flows associated with the swap, discounted back to the present using appropriate discount rates. When a swap is terminated early, a settlement payment is made by one party to the other to compensate for the loss of future cash flows. This settlement amount aims to reflect the market value of the swap at the time of termination. The settlement payment is calculated as the difference between the present value of the remaining fixed-rate payments and the present value of the remaining floating-rate payments, discounted at the current market interest rates. This present value calculation incorporates the remaining tenor of the swap and the relevant discount factors derived from the yield curve. In our scenario, the settlement payment is received by Company A. This payment effectively closes out the swap position, and from Company A’s perspective, it needs to be treated as an immediate cash inflow. To determine the NPV of this early termination, we consider the settlement payment as the only cash flow. Because the settlement payment is already a present value (representing the market value of the swap at termination), no further discounting is needed. The NPV of the early termination for Company A is simply the amount of the settlement payment received. For example, imagine a scenario where a software company, “CodeCrafters Inc.”, entered a swap to hedge interest rate risk on a large loan. Halfway through the swap’s term, CodeCrafters decides to restructure its debt. The bank offers them a favorable new loan with a lower fixed rate, but requires termination of the existing swap. The bank calculates the market value of the swap at termination, considering current interest rates and the remaining payments. CodeCrafters receives a settlement payment of £750,000. This payment represents the bank’s compensation to CodeCrafters for the lost future cash flows. The NPV of this termination, from CodeCrafters’ perspective, is simply £750,000. This amount immediately improves CodeCrafters’ cash position and can be used to offset costs associated with the debt restructuring. Another example would be a small manufacturing company, “Precision Parts Ltd.”, that entered an interest rate swap to convert a floating-rate loan into a fixed-rate loan. Due to unforeseen circumstances, they decide to sell a portion of their assets and use the proceeds to pay off the loan early. This requires terminating the swap. The counterparty calculates the settlement payment based on prevailing market rates and the remaining life of the swap. Precision Parts receives a settlement of £250,000. The NPV of this early termination for Precision Parts is £250,000.

Let’s consider a scenario involving a bespoke exotic derivative designed for a high-net-worth individual, Mrs. Eleanor Vance, who seeks to hedge her substantial holdings in a portfolio of emerging market bonds against currency fluctuations and interest rate volatility. This derivative, a “Quanto CMS Spread Option,” combines elements of a quanto option (protecting against currency risk) and a constant maturity swap (CMS) spread option (betting on the difference between two interest rates). The core of the Quanto CMS Spread Option lies in its payoff structure. The payoff is triggered if the spread between the 10-year Indonesian government bond yield and the 2-year Indonesian government bond yield widens beyond a predetermined strike level, say 150 basis points (1.5%). Crucially, the payoff is denominated in GBP, shielding Mrs. Vance from fluctuations in the Indonesian Rupiah. The notional amount is £5,000,000. The option’s payoff at expiry is calculated as follows: Payoff (in GBP) = Notional Amount * Max [0, (CMS Spread at Expiry – Strike Spread)] * Quanto Factor The “Quanto Factor” is a fixed exchange rate agreed upon at the inception of the contract, say 1 GBP = 19,000 IDR. This ensures the GBP value of the payoff remains constant regardless of the actual IDR/GBP exchange rate at expiry. Now, let’s assume at expiry, the 10-year Indonesian government bond yield is 8.5%, and the 2-year Indonesian government bond yield is 6.0%. This means the CMS Spread at Expiry is 2.5% or 250 basis points. Payoff = £5,000,000 * Max [0, (2.5% – 1.5%)] * 1 Payoff = £5,000,000 * (0.025 – 0.015) Payoff = £5,000,000 * 0.01 Payoff = £50,000 Therefore, Mrs. Vance would receive £50,000. This example showcases how exotic derivatives can be tailored to address very specific risk management needs. The Quanto CMS Spread Option allows Mrs. Vance to hedge against both currency risk and interest rate spread risk in a single, customized instrument. The use of the “Max” function ensures that the option only pays out if the spread widens beyond the strike level, providing targeted protection against adverse market movements. The fixed exchange rate (Quanto Factor) eliminates currency risk, making the payoff predictable in GBP.

The key to this question lies in understanding the gamma profile of different option positions and how they react to changes in the underlying asset’s price. Gamma represents the rate of change of delta with respect to changes in the underlying asset’s price. A long gamma position benefits from large price swings, while a short gamma position benefits from price stability. A short straddle involves selling both a call and a put option with the same strike price and expiration date. This position is inherently short gamma. The investor profits if the underlying asset price remains relatively stable around the strike price. However, if the price moves significantly in either direction, the losses can be substantial because the delta of the position becomes increasingly negative (if the price falls) or increasingly positive (if the price rises). The investor will then need to re-balance the position by buying or selling the underlying asset to maintain a neutral delta, incurring further costs. A butterfly spread, created by buying one call option at a lower strike price, selling two call options at a middle strike price, and buying one call option at a higher strike price (all with the same expiration), is a position with short gamma near the middle strike price and long gamma further away from it. The maximum profit is achieved if the underlying asset price remains at the middle strike price at expiration. Therefore, combining a short straddle and a butterfly spread aims to create a more complex gamma profile. The short straddle contributes short gamma around the at-the-money strike, profiting from stability. The butterfly spread, with its short gamma around its central strike and long gamma on either side, acts as a hedge against large price movements, limiting the potential losses from the short straddle. If the butterfly spread is constructed with the same strike price as the short straddle, the combined position has a reduced overall short gamma exposure compared to the short straddle alone. The combined position will profit from modest price movement and is designed to be less sensitive to significant price fluctuations.

Let’s break down the optimal strategy for handling a portfolio containing both a long position in a specific stock and a short position in a call option on that same stock, considering the client’s risk profile and the potential impact of market volatility, and the need to comply with UK regulatory requirements. First, understand the combined position. A long stock/short call strategy aims to generate income from the option premium while limiting upside potential. The investor profits if the stock price stays below the option’s strike price at expiration. However, the investor faces unlimited losses if the stock price rises significantly above the strike price, as they are obligated to sell the stock at the strike price while the market value is much higher. The client’s risk profile is paramount. A risk-averse client would be highly sensitive to potential losses. Therefore, the primary goal should be capital preservation. Selling a covered call offers some downside protection through the premium received, but it caps the upside potential. If the client requires significant upside potential and can tolerate higher risk, this strategy is unsuitable. Market volatility plays a crucial role. High volatility increases the value of options, leading to higher premiums for the call option. However, it also increases the risk of the stock price rising sharply above the strike price, resulting in significant losses for the investor. Conversely, low volatility reduces the option premium but also reduces the risk of substantial losses. UK regulatory requirements, specifically those outlined by the FCA, mandate that investment advice must be suitable for the client, considering their risk tolerance, investment objectives, and financial situation. Selling a covered call might be suitable for a client seeking income and limited upside, but it would be unsuitable for a growth-oriented client or one with a low-risk tolerance. Given the client’s risk aversion and the prevailing high market volatility, the most prudent course of action would be to unwind the position. This involves buying back the short call option and potentially selling the long stock position. This eliminates the risk of significant losses due to a sharp rise in the stock price and aligns with the client’s conservative risk profile. If the client still wants to generate income, exploring alternative, lower-risk strategies like dividend-paying stocks or fixed-income securities would be more appropriate. The calculations are not directly relevant here, as the question focuses on strategic portfolio management and regulatory compliance rather than numerical computations. However, one might consider the cost of unwinding the position (buying back the call option) and compare it to the potential losses if the stock price rises sharply. This is a risk-reward assessment, not a precise calculation.

Let’s break down how to approach this exotic derivative valuation problem. The core concept here is understanding how the payoff structure of a cliquet option interacts with the volatility smile and its impact on pricing. First, we need to understand the “ratcheting” mechanism. Each period, the floor resets based on the previous period’s performance. This means the strike price for each sub-period within the cliquet is path-dependent. The value of the cliquet is the sum of the discounted expected payoffs for each sub-period. Second, the volatility smile is crucial. A volatility smile (or skew) implies that options with different strike prices have different implied volatilities. The standard Black-Scholes model assumes constant volatility, which is unrealistic. Since the floor resets and creates different effective strikes for each period, we must use an appropriate implied volatility for each strike. We can’t just use the at-the-money volatility. Third, the cap on each period’s return limits the upside. This cap also affects the fair value. Without the cap, the cliquet would simply be a participation in the underlying asset’s gains. The cap makes it a more complex derivative. Fourth, the correlation between periods is important. If the underlying asset tends to have trending behavior (positive correlation), the cliquet will be worth more than if the returns are mean-reverting (negative correlation). This is because a positive correlation increases the likelihood that several periods will have positive returns, leading to higher payoffs. Fifth, to accurately price this cliquet, a Monte Carlo simulation is generally required. We simulate many possible paths for the underlying asset, taking into account the volatility smile, correlation between periods, and the cap and floor. For each path, we calculate the payoff of the cliquet. The average payoff, discounted back to the present, is the fair value. Sixth, let’s consider the specific volatility smile. A steeper smile means that out-of-the-money puts are more expensive relative to out-of-the-money calls. This would impact the price of the cliquet, especially if the floor is significantly below the initial asset price. Seventh, we need to consider the risk-free rate for discounting. A higher risk-free rate will decrease the present value of future payoffs, thus decreasing the value of the cliquet. Finally, in this scenario, the increased steepness of the volatility smile after the market event indicates greater demand for downside protection. This means that the floor is more valuable, but the cap remains the same. The increased value of the floor, combined with the capped upside, will decrease the overall value of the cliquet.

The question examines the impact of early exercise on American call options, specifically when dividends are involved. An American call option gives the holder the right, but not the obligation, to buy the underlying asset at the strike price any time before the expiration date. The early exercise decision hinges on comparing the intrinsic value gained by exercising early versus the potential loss of time value and the missed opportunity to profit from future price increases. Dividends play a crucial role because they represent cash flows the option holder can receive by owning the stock directly. In this scenario, the dividend payment significantly impacts the option’s value. If the dividend exceeds the option’s time value, early exercise becomes a rational strategy. The time value is the portion of the option’s price that reflects the potential for the underlying asset’s price to increase before expiration. Here’s the breakdown: 1. Calculate the potential profit from early exercise: This is the difference between the current stock price and the strike price, less the premium paid for the option. In this case, it’s £105 – £95 – £12 = -£2. However, this doesn’t account for the dividend. 2. Consider the dividend income: By exercising the option before the ex-dividend date, the holder captures the £8 dividend. 3. Evaluate the net benefit: The net benefit of early exercise is the dividend received minus any opportunity cost. The opportunity cost is the time value of the option. 4. Time value = option premium – intrinsic value = £12 – (£105 – £95) = £2 5. Net benefit of early exercise = dividend – time value = £8 – £2 = £6 Therefore, early exercise is beneficial.

Let’s analyze a complex scenario involving exotic options and their valuation within a volatile market, considering regulatory constraints under UK financial regulations. We will use a step-by-step approach to determine the fair value of a barrier option, incorporating a volatility skew and the impact of potential regulatory intervention. Imagine a “Knock-Out Call Option” on FTSE 100 index. The current index level is 7500. The strike price is 7600. The barrier level is 7800. The option has a maturity of 6 months. The risk-free rate is 1%. The implied volatility is 20%. The knock-out barrier means that if the FTSE 100 reaches 7800 at any point during the 6-month period, the option becomes worthless. First, we need to acknowledge that standard Black-Scholes models are inadequate for pricing barrier options, especially when considering volatility skews and the probability of the barrier being hit. We need to consider Monte Carlo simulation to estimate the probability of the barrier being breached and its impact on the option value. We run a Monte Carlo simulation with 10,000 paths. For each path, we simulate the FTSE 100 price movement over the 6-month period, checking at discrete time intervals (e.g., daily) whether the barrier of 7800 is breached. We can use the following formula for simulating the price path: \[S_{t+\Delta t} = S_t \cdot \exp\left(\left(r – \frac{\sigma^2}{2}\right)\Delta t + \sigma \sqrt{\Delta t} Z\right)\] Where: \(S_t\) is the index level at time t \(r\) is the risk-free rate \(\sigma\) is the implied volatility \(\Delta t\) is the time step \(Z\) is a standard normal random variable For simplicity, let’s assume that in 20% of the simulated paths, the barrier is breached. In the remaining 80% of the paths where the barrier is not breached, we calculate the payoff of the call option at maturity, which is max(0, \(S_T\) – K), where \(S_T\) is the index level at maturity and K is the strike price (7600). Let’s assume the average payoff in the 80% of paths where the barrier isn’t breached is 200. The fair value of the barrier option is then the discounted expected payoff, considering the probability of the barrier not being breached: Fair Value = 0.8 * 200 * exp(-0.01 * 0.5) = 159.20 Now, consider a regulatory intervention scenario. Suppose the Financial Conduct Authority (FCA) introduces a new rule during the option’s life, imposing stricter margin requirements on exotic derivatives, increasing the cost of hedging. This increased hedging cost is estimated to reduce the option’s value by 5%. Adjusted Fair Value = 159.20 * (1 – 0.05) = 151.24 Therefore, the estimated fair value of the barrier option, considering the barrier breach probability, Monte Carlo simulation, and potential regulatory impact, is approximately 151.24. This approach highlights the importance of considering multiple factors beyond simple option pricing models when dealing with exotic derivatives.

The question assesses understanding of option pricing sensitivity (Greeks), specifically delta and gamma, and how they interact in a portfolio. Delta represents the change in option price for a unit change in the underlying asset’s price. Gamma represents the change in delta for a unit change in the underlying asset’s price. A high gamma indicates that the delta is highly sensitive to changes in the underlying asset’s price. In this scenario, the portfolio is delta-neutral, meaning the overall delta is zero. This is achieved by holding offsetting positions in options and/or the underlying asset. However, a delta-neutral portfolio is not necessarily risk-free, especially when gamma is significant. As the underlying asset’s price moves, the delta changes, potentially making the portfolio no longer delta-neutral. A positive gamma means that if the underlying asset’s price increases, the delta will become more positive; if the underlying asset’s price decreases, the delta will become more negative. Conversely, a negative gamma means the opposite: the delta becomes less positive (or more negative) as the underlying asset’s price increases, and more positive (or less negative) as the underlying asset’s price decreases. The portfolio manager needs to rebalance to maintain delta neutrality as the underlying asset price changes. The frequency and magnitude of rebalancing depend on the gamma of the portfolio. Higher gamma requires more frequent rebalancing. The cost of rebalancing (transaction costs) must be considered when deciding on the optimal rebalancing strategy. A portfolio with positive gamma benefits from large price swings, while a portfolio with negative gamma benefits from price stability. The goal is to minimize the impact of gamma on the portfolio’s delta and maintain the desired risk profile. The formula for calculating the change in portfolio delta (\(\Delta P\)) due to a change in the underlying asset price (\(\Delta S\)) and gamma (\(\Gamma\)) is approximately: \[\Delta P \approx \Gamma \times (\Delta S)^2 / 2 \] This shows that the change in portfolio value due to gamma is proportional to the square of the change in the underlying asset price. Therefore, even small price changes can have a significant impact on the portfolio’s delta when gamma is high.

The question revolves around the concept of delta hedging a short call option position and the subsequent profit or loss arising from changes in the underlying asset’s price. Delta hedging aims to neutralize the portfolio’s sensitivity to small changes in the underlying asset’s price. However, delta changes as the underlying asset’s price changes (gamma), and delta hedging only works perfectly for infinitesimal price movements. In reality, the hedge needs to be rebalanced periodically. In this scenario, the investor is short a call option, meaning they profit if the stock price stays below the strike price. They delta hedge by buying shares of the underlying asset. If the stock price increases, the delta of the call option increases (becomes more negative from the seller’s perspective), requiring the investor to buy more shares to maintain the hedge. Conversely, if the stock price decreases, the delta decreases (becomes less negative), requiring the investor to sell shares. The profit or loss on the delta hedge is determined by comparing the cost of buying shares when the price rises to the proceeds from selling shares when the price falls, relative to the initial hedge. Here’s how to calculate the profit/loss: 1. **Initial Hedge:** Short call, delta = 0.40, so buy 0.40 shares per option. Stock price = £100. Cost = 0.40 * £100 = £40. 2. **Price Increase to £105:** Delta increases to 0.70. Need to buy an additional 0.30 shares. Cost = 0.30 * £105 = £31.50. 3. **Price Decrease to £98:** Delta decreases to 0.20. Need to sell 0.50 shares (0.70 – 0.20 = 0.50). Proceeds = 0.50 * £98 = £49. 4. **Price Increase to £102:** Delta increases to 0.50. Need to buy an additional 0.30 shares. Cost = 0.30 * £102 = £30.60 5. **Final Sale:** Option expires worthless at £102, so you sell all remaining shares (0.50) at £102. Proceeds = 0.50 * £102 = £51 Total Cost of Purchases: £40 + £31.50 + £30.60 = £102.10 Total Proceeds from Sales: £49 + £51 = £100 Profit/Loss on Hedge: £100 – £102.10 = -£2.10 Since the investor is short the call option, they receive the premium. If the option expires worthless, they keep the entire premium. In this case, the option expires in the money, and the investor will need to pay the difference of the stock price and strike price. Profit/Loss on Option: – (£102 – £100) = -£2 Total Profit/Loss = -£2.10 – £2 = -£4.10 Therefore, the investor experiences a loss of £4.10, even with delta hedging. The loss arises because the hedging strategy is not perfect and incurs transaction costs as the hedge is adjusted. The changes in delta (gamma) and the discrete nature of hedging adjustments lead to this outcome.