Derivatives Level 4 (Investment Advice Diploma) Exam Quiz Quiz 38

The question assesses understanding of how a clearing house mitigates counterparty risk in derivatives transactions, specifically focusing on margin requirements and default procedures. The correct answer highlights the role of variation margin in covering daily losses and the use of the default fund in extreme situations. The incorrect answers present plausible but flawed scenarios regarding the operation of the clearing house and the handling of defaults. Here’s a breakdown of why the correct answer is correct and why the incorrect answers are incorrect: * **Correct Answer (a):** Variation margin is collected daily to cover mark-to-market losses, and in the event of a clearing member default, the clearing house first uses the defaulting member’s margin, then its contribution to the default fund, and finally, contributions from non-defaulting members. This reflects the sequential loss allocation mechanism. * **Incorrect Answer (b):** While the clearing house does monitor positions, it does not unilaterally alter contract terms to prevent losses. Altering contract terms would disrupt the market and create uncertainty. The primary mechanism is margin calls and risk management. * **Incorrect Answer (c):** The clearing house does not guarantee profits for all members. Its role is to ensure the integrity of the market and mitigate risk. The clearing house ensures that obligations are met, not that profits are realized. * **Incorrect Answer (d):** While initial margin is important, it’s not the *primary* defense against daily fluctuations. Variation margin serves that purpose. Initial margin is more of a buffer for potential future losses and to cover the period it takes to liquidate a position. Here’s a more detailed explanation of the margining system: * **Initial Margin:** This is the upfront collateral required to open a derivatives position. It’s designed to cover potential losses over a specified time horizon (usually a few days) with a high degree of confidence (e.g., 99%). * **Variation Margin:** This is the daily adjustment to reflect the mark-to-market profit or loss on a derivatives contract. If a trader incurs a loss, they must deposit variation margin to cover the loss. If they make a profit, they receive variation margin. This ensures that positions are always adequately collateralized. * **Default Fund:** This is a pool of funds contributed by all clearing members. It’s used to cover losses in the event that a clearing member defaults and their margin is insufficient to cover their obligations. The clearing house has a waterfall of resources it can use in the event of a default: the defaulting member’s margin, the defaulting member’s contribution to the default fund, and then contributions from non-defaulting members. The entire system is designed to minimize the risk of contagion and ensure that the derivatives market remains stable and efficient. The clearing house acts as a central counterparty, guaranteeing the performance of all trades and mitigating counterparty risk.

The question explores the concept of delta-hedging and how changes in volatility impact the effectiveness of a delta-hedged portfolio. Delta-hedging aims to create a portfolio that is insensitive to small changes in the underlying asset’s price. However, delta is not constant; it changes as the underlying asset’s price changes (gamma) and as volatility changes (vega). The problem focuses on the impact of a sudden, unexpected increase in volatility on a short call option position that is initially delta-hedged. The key is to understand that an increase in volatility increases the value of the call option, requiring the investor to buy more of the underlying asset to maintain the delta-neutral position. The calculation involves determining the change in the call option’s delta due to the volatility change, and then calculating the number of additional shares needed to re-establish the delta hedge. The initial delta is 0.6, meaning for every £1 increase in the underlying asset’s price, the call option’s price increases by £0.6. The portfolio is delta-hedged, so the investor holds short call options and a long position in the underlying asset. When volatility increases from 20% to 25%, the call option’s delta increases to 0.65. This means the call option is now more sensitive to changes in the underlying asset’s price. To re-establish the delta hedge, the investor needs to increase their long position in the underlying asset. The change in delta is 0.65 – 0.6 = 0.05. Since the investor has sold 1000 call options, the total change in delta for the portfolio is 0.05 * 1000 = 50. This means the investor needs to buy 50 additional shares of the underlying asset to maintain a delta-neutral position.

The question revolves around the concept of using options to hedge a portfolio against a potential market downturn, specifically focusing on the impact of implied volatility on the hedge’s effectiveness. The core idea is that as implied volatility rises, the price of options increases, impacting the cost and effectiveness of hedging strategies. We’ll use a protective put strategy as an example. Let’s consider a portfolio worth £1,000,000 tracking the FTSE 100. An investor wants to protect against a potential 10% market decline over the next three months. The FTSE 100 is currently at 7,500. To create a protective put, the investor buys put options with a strike price close to the current index level (at-the-money or slightly out-of-the-money). Let’s assume the investor buys 100 put options contracts (each contract representing 100 shares) with a strike price of 7,400, expiring in three months. Scenario 1: Initial Implied Volatility is 20%. The premium for each put option contract is £500. Total cost of the hedge: 100 contracts * £500 = £50,000, or 5% of the portfolio value. Scenario 2: Implied Volatility Rises to 30%. The premium for each put option contract increases to £750. Total cost of the hedge: 100 contracts * £750 = £75,000, or 7.5% of the portfolio value. If the FTSE 100 declines by 10% (to 6,750), the put options will be in the money. The intrinsic value of each put option will be (7,400 – 6,750) * 100 = £65,000. Total value of the put options: 100 contracts * £65,000 = £6,500,000. However, since the investor only needed to hedge £100,000 (10% of £1,000,000), the excess gain illustrates the leverage inherent in options. The key takeaway is that higher implied volatility increases the cost of the hedge, but also increases the potential payoff if the market declines significantly. The investor must balance the cost of the hedge (option premium) against the desired level of protection and their expectation of market volatility. The effectiveness of the hedge also depends on factors like the strike price of the options, the time to expiration, and the correlation between the portfolio and the underlying asset. The investor must also consider the impact of gamma, which measures the rate of change of delta, and how it affects the hedge’s sensitivity to market movements.

The core concept tested here is the interplay between delta, gamma, and hedging strategies, specifically in the context of a short option position. Delta represents the sensitivity of the option price 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 short option position has a negative gamma, meaning that as the underlying asset price moves, the delta changes in a way that makes the hedge less effective. The initial hedge ratio is calculated using the delta of the option. The change in the underlying asset’s price necessitates rebalancing the hedge. The magnitude of the rebalancing is determined by the gamma of the option and the change in the underlying asset’s price. The cost of rebalancing is the number of shares bought or sold multiplied by the current market price of the underlying asset. The profit or loss on the option position is approximated using the delta and gamma, and the change in the underlying asset’s price. The total profit or loss is the sum of the profit or loss on the option position and the cost of rebalancing. Here’s the breakdown of the calculation: 1. **Initial Hedge:** Short 100 options with a delta of -0.45. This means you need to buy 45 shares to be delta-neutral (100 \* 0.45 = 45). 2. **Price Change:** The underlying asset increases from £50 to £52. 3. **Gamma Effect:** The option’s gamma is 0.05. This means the delta changes by 0.05 for every £1 change in the underlying asset. Since the price increased by £2, the delta changes by 2 \* 0.05 = 0.10. 4. **New Delta:** The new delta is -0.45 + 0.10 = -0.35. The hedge is no longer delta-neutral. 5. **Rebalancing:** To rebalance, you need to sell shares. The number of shares to sell is 100 \* (0.45 – 0.35) = 10 shares. 6. **Rebalancing Cost:** Selling 10 shares at £52 costs 10 \* £52 = £520. 7. **Option Profit/Loss:** Approximate the profit/loss on the option using delta and gamma. Delta effect: -100 \* -0.45 \* £2 = £90. Gamma effect: 0.5 \* 100 \* 0.05 \* (£2)^2 = £10. Total profit on option: £90 + £10 = £100. Since you are short the option, this is a loss of -£100. 8. **Total Profit/Loss:** -£520 (rebalancing cost) – £100 (option loss) = -£620.

The question explores the complexities of hedging a portfolio with options in the presence of non-linear relationships between asset prices and option values, specifically when the portfolio’s value is heavily skewed. A skewed portfolio implies that the portfolio’s potential gains and losses are not symmetrical. The delta of an option measures the sensitivity of the option’s price to changes in the underlying asset’s price. Gamma measures the rate of change of delta with respect to changes in the underlying asset’s price. A high gamma indicates that the delta is very sensitive to changes in the underlying asset’s price, meaning the hedge ratio needs frequent adjustments. Vega measures the sensitivity of the option’s price to changes in the volatility of the underlying asset. Rho measures the sensitivity of the option’s price to changes in the risk-free interest rate. In a skewed portfolio, relying solely on delta-neutral hedging can be insufficient because the delta changes rapidly, especially near the money. A portfolio that is heavily skewed to the upside will benefit greatly from upward price movements, but will suffer disproportionately from downward movements. Therefore, the gamma exposure is significant. Additionally, the volatility of the underlying asset (vega) and interest rate changes (rho) can impact the option’s price and the effectiveness of the hedge. Stress testing and scenario analysis are crucial tools to assess the hedge’s performance under extreme market conditions. In this scenario, the fund manager must consider the combined impact of delta, gamma, vega, and rho, and use stress testing to understand how the hedge will perform under various adverse conditions, especially large downward price movements. The fund manager needs to proactively rebalance the hedge based on these factors to maintain adequate protection.

The question assesses the understanding of put-call parity and how dividends affect the relationship. Put-call parity states: Call Price + Present Value of Strike Price = Put Price + Underlying Asset Price + Present Value of Dividends. Here’s how to solve the problem: 1. **Calculate the present value of the strike price:** The strike price is £105, and the risk-free rate is 4%. The time to expiration is 6 months (0.5 years). The present value is calculated as \(105 / (1 + 0.04)^{0.5} = 105 / 1.0198 = £102.96\). 2. **Calculate the present value of the dividends:** The dividends are £2.50, paid in 3 months (0.25 years). The present value is \(2.50 / (1 + 0.04)^{0.25} = 2.50 / 1.00985 = £2.48\). 3. **Apply the put-call parity formula:** Call Price + Present Value of Strike Price = Put Price + Underlying Asset Price – Present Value of Dividends £8 + £102.96 = Put Price + £100 – £2.48 £110.96 = Put Price + £97.52 4. **Solve for the put price:** Put Price = £110.96 – £97.52 = £13.44 Therefore, the theoretical price of the put option is £13.44. The inclusion of dividends complicates the standard put-call parity relationship. Failing to account for the present value of dividends leads to an incorrect put price. This question tests the candidate’s ability to adjust the put-call parity formula for real-world factors.

The question assesses the understanding of how market makers manage risk associated with options positions, particularly the concept of delta hedging. Delta represents the sensitivity of an option’s price to a change in the underlying asset’s price. A market maker selling options faces the risk that the option’s price will move against them as the underlying asset’s price changes. Delta hedging involves taking offsetting positions in the underlying asset to neutralize this risk. The calculation involves determining the number of shares needed to delta hedge the short call options position. First, we calculate the total delta exposure by multiplying the number of contracts, the shares per contract, and the delta of each call option: 100 contracts * 100 shares/contract * 0.45 delta = 4500 shares. Since the market maker sold the call options, they are short delta, meaning they will lose money if the underlying asset’s price increases. To hedge this short delta position, the market maker needs to buy shares of the underlying asset. Therefore, they need to buy 4500 shares to offset their delta exposure. The example provided is original and reflects a practical scenario faced by market makers. It goes beyond basic definitions and requires the application of the delta hedging concept to a specific situation. It uses original numerical values and parameters to create a unique problem-solving challenge.

The question assesses the understanding of delta hedging, a strategy used to reduce the risk associated with price movements of the underlying asset of an option. Delta represents the sensitivity of an option’s price to a change in the price of the underlying asset. A delta of 0.60 means that for every £1 increase in the price of the underlying asset, the option’s price is expected to increase by £0.60. To delta hedge a short position in 100 call options, an investor needs to buy shares of the underlying asset to offset the potential losses from the options position if the asset price increases. The number of shares to buy is calculated by multiplying the number of options by the delta. In this case, it’s 100 options * 0.60 = 60 shares. The initial investment is 60 shares * £45/share = £2700. If the share price increases to £46, the profit from the shares is 60 shares * (£46 – £45) = £60. The option premium received is £300 (100 options * £3 premium). However, the delta changes as the price of the underlying asset changes. This is known as gamma. If the gamma is 0.04, it means that for every £1 increase in the share price, the delta increases by 0.04. So, when the share price increases from £45 to £46, the delta increases from 0.60 to 0.64. To maintain a delta-neutral position, the investor needs to adjust the hedge by buying additional shares. The change in delta is 0.04, so the investor needs to buy an additional 100 options * 0.04 = 4 shares. The cost of buying these additional shares is 4 shares * £46/share = £184. The profit from the initial 60 shares is £60. The cost of adjusting the hedge is £184. Therefore, the net profit/loss is £60 – £184 = -£124. The total profit/loss, considering the initial option premium received, is £300 – £124 = £176.

A hypothetical portfolio manager at a UK-based wealth management firm is tasked with identifying potential arbitrage opportunities in the derivatives market. The core concept being tested is the understanding and application of put-call parity. This is crucial for ensuring portfolios are accurately priced and for exploiting market inefficiencies. The scenario involves European options on a FTSE 100 stock, which are traded on the London Stock Exchange. The question tests the candidate’s ability to calculate the present value of the strike price using continuous compounding, a standard practice in financial modeling. Furthermore, it requires the candidate to recognize a violation of put-call parity and to construct an arbitrage strategy to profit from this mispricing. The incorrect options are designed to reflect common errors in applying the put-call parity formula or in identifying the correct arbitrage strategy, such as miscalculating the present value or reversing the buy and sell positions. The question also touches on the regulatory landscape, referencing potential scrutiny from the FCA, which adds a layer of realism and emphasizes the importance of ethical considerations in arbitrage activities.

This question tests the understanding of delta hedging and gamma, focusing on the practical implications for managing a portfolio of options. Delta represents the sensitivity of an option’s price to changes in the underlying asset’s price. Gamma, on the other hand, measures the rate of change of delta with respect to the underlying asset’s price. A higher gamma indicates that the delta will change more rapidly, requiring more frequent adjustments to maintain a delta-neutral position. The scenario involves a portfolio manager, Amelia, who is delta-hedging a portfolio of call options on FTSE 100. She aims to maintain a delta-neutral position to protect the portfolio’s value from small price movements in the FTSE 100. However, the portfolio also has a significant gamma. The question explores how Amelia should adjust her hedge as the FTSE 100 experiences a notable price increase. To answer this question, one must understand that when the underlying asset’s price increases, the delta of a call option also increases. Since Amelia is delta-hedging, she needs to buy more of the underlying asset (FTSE 100 futures) to maintain the delta-neutral position. The higher the gamma, the more significant the change in delta for a given change in the underlying asset’s price, and thus the larger the adjustment needed. The calculation of the required adjustment involves understanding the relationship between gamma, delta, and the change in the underlying asset’s price. The change in delta can be approximated as: Change in Delta ≈ Gamma * Change in Underlying Asset Price In this case, Gamma = 0.0005, and the Change in Underlying Asset Price = 100 points. Therefore: Change in Delta ≈ 0.0005 * 100 = 0.05 Since Amelia is short 10,000 call options, the total change in delta for the portfolio is: Total Change in Delta = 0.05 * 10,000 = 500 This means Amelia needs to increase her position in FTSE 100 futures by 500 contracts to maintain a delta-neutral position. Therefore, she needs to buy 500 FTSE 100 futures contracts.

The question assesses understanding of delta hedging, particularly in a scenario where the underlying asset’s volatility deviates from the implied volatility used in the initial hedge calculation. The core principle of delta hedging is to neutralize the portfolio’s sensitivity to small changes in the underlying asset’s price. Delta represents the change in the option’s price for a £1 change in the underlying asset’s price. To maintain a delta-neutral position, the portfolio must be rebalanced continuously as the underlying asset’s price changes or as time passes (delta changes). When actual volatility is lower than implied volatility, the option’s price will change less than predicted by the model used for hedging (often Black-Scholes). This means the hedge will be over-adjusted. Here’s how we approach the calculation: 1. **Initial Hedge:** The portfolio manager sells 100 call options with a delta of 0.6. To hedge this, they buy 60 shares (100 options * 0.6 delta). 2. **Price Drop:** The stock price drops by £1. The option price decreases, but less than expected due to lower realized volatility. 3. **Delta Change:** The option’s delta decreases to 0.5. The manager now needs to reduce their shareholding to maintain delta neutrality. 4. **New Hedge Ratio:** The new hedge requires 50 shares (100 options * 0.5 delta). 5. **Shares to Sell:** The manager needs to sell 10 shares (60 initial shares – 50 new shares). 6. **Profit/Loss Calculation:** Selling 10 shares at the current price (£49) after buying them at £50 results in a loss of £1 per share, totaling £10. The key takeaway is that the effectiveness of delta hedging is highly dependent on the accuracy of the volatility estimate. When realized volatility differs significantly from implied volatility, the hedge will not perform as expected, leading to potential profits or losses. In this case, lower realized volatility led to a small loss due to over-hedging. A crucial element is understanding that delta is not static; it changes with the underlying asset’s price, time to expiration, and volatility. The manager’s task is to dynamically adjust the hedge to maintain a delta-neutral position, a process that incurs transaction costs and is subject to the risk of model misspecification.

The question assesses understanding of delta hedging, specifically how adjustments are made to maintain a delta-neutral position. Delta represents the sensitivity of an option’s price to changes in the underlying asset’s price. A delta of 0.5 means the option price will change by £0.50 for every £1 change in the underlying asset. To hedge this, one would take an offsetting position in the underlying asset. Initially, the portfolio consists of 1000 call options, each with a delta of 0.5. The total portfolio delta is 1000 * 0.5 = 500. To delta-hedge, the portfolio manager sells 500 shares of the underlying asset. The asset price then decreases from £100 to £95. The call option’s delta decreases to 0.3. The new portfolio delta is 1000 * 0.3 = 300. The manager is still short 500 shares, but the desired short position is now 300. The manager needs to reduce the short position by buying back shares. The number of shares to buy back is the difference between the initial hedge and the new hedge requirement: 500 – 300 = 200 shares. This maintains a delta-neutral position, minimizing the portfolio’s sensitivity to small price movements in the underlying asset. The profit or loss from this adjustment depends on the price at which the shares were initially sold and the price at which they were bought back. The shares were initially sold at £100 and bought back at £95, resulting in a profit of £5 per share. The total profit from buying back 200 shares is 200 * £5 = £1000. This profit offsets the losses incurred by the call options due to the decrease in the underlying asset’s price, maintaining the delta-neutral hedge. This example illustrates the dynamic nature of delta hedging and the need for continuous adjustments as the underlying asset’s price and option deltas change. It also demonstrates how a delta-neutral strategy aims to immunize the portfolio against small price fluctuations, focusing on risk management rather than speculation.

This question tests the understanding of put-call parity and its violation in the presence of transaction costs. Put-call parity is a fundamental relationship that links the prices of a European call option, a European put option, a risk-free asset, and the underlying asset. The formula is: \(C + PV(X) = P + S\), where \(C\) is the call option price, \(PV(X)\) is the present value of the strike price, \(P\) is the put option price, and \(S\) is the spot price of the underlying asset. Transaction costs introduce a deviation from the theoretical parity. The presence of transaction costs (brokerage fees) means that arbitrage opportunities are only profitable if the profit exceeds the cost of trading. We must calculate the profit from each potential arbitrage strategy and compare it with the total transaction costs. Strategy 1: Buy Call, Sell Put, Sell Stock, Borrow PV(Strike) * Cost: \(C + PV(X)\) * Benefit: \(P + S\) * Profit: \(P + S – C – PV(X)\) * With transaction costs: \(P + S – C – PV(X) – 4*T\) Strategy 2: Sell Call, Buy Put, Buy Stock, Lend PV(Strike) * Cost: \(P + S\) * Benefit: \(C + PV(X)\) * Profit: \(C + PV(X) – P – S\) * With transaction costs: \(C + PV(X) – P – S – 4*T\) Where T is transaction cost for each trade. Given values: * Call price (C) = £5.50 * Put price (P) = £2.00 * Spot price (S) = £48.00 * Strike price (X) = £50.00 * Risk-free rate (r) = 5% * Time to expiration (t) = 0.25 years * Transaction cost (T) = £0.20 per transaction First, calculate the present value of the strike price: \(PV(X) = \frac{X}{1 + rt} = \frac{50}{1 + 0.05 \times 0.25} = \frac{50}{1.0125} = £49.38\) Now, let’s calculate the profit from Strategy 1, including transaction costs: Profit = \(P + S – C – PV(X) – 4T = 2.00 + 48.00 – 5.50 – 49.38 – 4(0.20) = 50.00 – 54.88 – 0.80 = -5.68\) Next, let’s calculate the profit from Strategy 2, including transaction costs: Profit = \(C + PV(X) – P – S – 4T = 5.50 + 49.38 – 2.00 – 48.00 – 4(0.20) = 54.88 – 50.00 – 0.80 = 4.08\) Since Strategy 2 yields a positive profit after accounting for transaction costs, an arbitrage opportunity exists by selling the call, buying the put, buying the stock, and lending the present value of the strike price.

The question revolves around the concept of delta-hedging a portfolio of options and the impact of gamma on the effectiveness of that hedge. Delta represents the sensitivity of an option’s price to a change in the underlying asset’s price. Gamma, in turn, represents the sensitivity of delta to a change in the underlying asset’s price. A delta-neutral portfolio is constructed to be insensitive to small changes in the underlying asset’s price. However, this neutrality is only valid for small price movements. Gamma introduces convexity to the portfolio’s payoff, meaning that as the underlying asset’s price moves significantly, the delta changes, and the hedge becomes less effective. In this scenario, the fund manager initially delta-hedges the portfolio, aiming for a delta of zero. The positive gamma indicates that as the underlying asset’s price increases, the portfolio’s delta will also increase, and vice versa. The key is to calculate the change in the portfolio’s value due to the change in the underlying asset’s price, considering the impact of gamma. The formula to approximate the change in portfolio value (\(\Delta P\)) is: \[ \Delta P \approx \Delta \times \Delta S + \frac{1}{2} \times \Gamma \times (\Delta S)^2 \] Where: * \(\Delta\) is the portfolio’s delta * \(\Delta S\) is the change in the underlying asset’s price * \(\Gamma\) is the portfolio’s gamma Since the portfolio is initially delta-hedged, \(\Delta = 0\). Therefore, the change in portfolio value is primarily driven by the gamma: \[ \Delta P \approx \frac{1}{2} \times \Gamma \times (\Delta S)^2 \] Given: * \(\Gamma = 5,000\) * \(\Delta S = 2\) (Increase of 2 points) \[ \Delta P \approx \frac{1}{2} \times 5,000 \times (2)^2 = \frac{1}{2} \times 5,000 \times 4 = 10,000 \] The portfolio value increases by approximately £10,000. Because the portfolio was delta neutral, the change in portfolio value comes entirely from the gamma effect.

The question explores the concept of delta hedging and how it’s impacted by discrete hedging intervals, particularly when considering transaction costs. Delta, representing the sensitivity of an option’s price to a change in the underlying asset’s price, is crucial for maintaining a hedged position. The continuous rebalancing assumed in theoretical models is impossible in practice. Discrete hedging introduces errors, especially when transaction costs are considered. Each rebalancing incurs a cost, directly impacting the profitability of the hedge. The optimal hedging frequency balances the cost of frequent rebalancing against the risk of a poorly hedged position due to infrequent adjustments. To calculate the impact, we need to determine the theoretical profit/loss without transaction costs, then subtract the transaction costs incurred during rebalancing. 1. **Initial Hedge:** The portfolio manager sells 100 call options, each with a delta of 0.60. To hedge, they buy \(100 \times 0.60 = 60\) shares of the underlying asset at £50. 2. **Asset Price Increase:** The asset price increases to £52. 3. **New Delta:** The delta increases to 0.70. The manager needs to adjust the hedge by buying an additional \(100 \times (0.70 – 0.60) = 10\) shares. 4. **Asset Price Decrease:** The asset price decreases to £49. 5. **New Delta:** The delta decreases to 0.55. The manager needs to adjust the hedge by selling \(100 \times (0.70 – 0.55) = 15\) shares. 6. **Asset Price Increase:** The asset price increases to £53. 7. **New Delta:** The delta increases to 0.80. The manager needs to adjust the hedge by buying an additional \(100 \times (0.80 – 0.55) = 25\) shares. **Calculations:** * **Cost of Initial Hedge:** \(60 \times £50 = £3000\) * **Cost of Buying 10 Shares (Price Increase to £52):** \(10 \times £52 = £520\) * **Revenue from Selling 15 Shares (Price Decrease to £49):** \(15 \times £49 = £735\) * **Cost of Buying 25 Shares (Price Increase to £53):** \(25 \times £53 = £1325\) **Total Cost of Shares:** \[£3000 + £520 – £735 + £1325 = £4110\] **Value of 60 Shares at £53:** \(60 \times £53 = £3180\) **Profit/Loss on Shares:** \[£3180 – £4110 = -£930\] Now, we need to calculate the option value changes. We’ll assume the option price changes are exactly offset by the delta changes (ideal hedging scenario, without considering gamma). The net effect of the option changes is zero. **Transaction Costs:** The transaction cost is £5 per trade. There are 3 trades: buying 10 shares, selling 15 shares, and buying 25 shares. **Total Transaction Costs:** \[3 \times £5 = £15\] **Net Profit/Loss:** \[ -£930 – £15 = -£945\] Therefore, the portfolio manager’s net profit/loss is -£945.

The core of this question revolves around understanding how implied volatility, derived from options prices, can be used to infer market sentiment and anticipate potential market movements, particularly around significant economic announcements. The VIX, often called the “fear gauge,” is a real-time market index representing the market’s expectation of 30-day volatility. An increase in VIX suggests increased uncertainty and fear, while a decrease suggests stability and complacency. However, the VIX reflects overall market sentiment. To gauge sentiment regarding a specific event, such as the Bank of England’s (BoE) interest rate decision, one must analyze the implied volatility of options contracts with expiration dates closely aligned to the event’s announcement. The scenario presented requires an understanding of volatility smiles (or skews) and how they change in anticipation of significant events. A volatility smile is a graph showing that options further out-of-the-money tend to have higher implied volatilities than at-the-money options. This reflects the market’s higher demand for protection against extreme price movements. Leading up to a major economic announcement, the volatility smile often becomes more pronounced, especially for options expiring shortly after the announcement. This is because market participants anticipate a larger-than-usual price swing in either direction. After the announcement, if the outcome is largely as expected, the uncertainty diminishes, and the volatility smile flattens, leading to a decrease in implied volatility across all strike prices, but particularly for those options most sensitive to the announcement (those with expirations nearest the announcement date). The question also tests the understanding of how gamma, a measure of the rate of change of an option’s delta with respect to changes in the underlying asset’s price, behaves around major events. Gamma is highest for at-the-money options nearing expiration. A high gamma indicates that the option’s delta (and therefore its sensitivity to price changes) is changing rapidly. Before an announcement, the gamma for near-the-money options will be elevated, reflecting the potential for large price swings. After the announcement, if the outcome is as expected, the gamma will decrease as the market stabilizes. Therefore, the correct answer reflects the scenario where implied volatility decreases, the volatility smile flattens, and gamma decreases for near-the-money options after the BoE’s announcement.

The question assesses understanding of hedging strategies using options, specifically a ratio spread. A ratio spread involves buying and selling different numbers of options with the same expiration date but different strike prices. This strategy is typically used when an investor has a specific outlook on the underlying asset’s price movement. The payoff is calculated by considering the premiums paid/received and the intrinsic value of the options at expiration. Here’s how to calculate the payoff and determine the investor’s outlook: 1. **Initial Setup:** * Buy 1 call option with a strike price of £450 for a premium of £20. * Sell 2 call options with a strike price of £480 for a premium of £8 each. 2. **Total Premium Paid/Received:** * Premium paid for the £450 call: £20 * Premium received for the two £480 calls: 2 * £8 = £16 * Net premium paid: £20 – £16 = £4 3. **Payoff Calculation at Expiration:** * **Scenario 1: Price ≤ £450:** All options expire worthless. Payoff = -£4 (the initial premium paid). * **Scenario 2: £450 < Price ≤ £480:** The £450 call is in the money. The payoff is (Price - £450) - £4. The £480 calls expire worthless. * **Scenario 3: Price > £480:** The £450 call and both £480 calls are in the money. The payoff is (Price – £450) – 2 * (Price – £480) – £4 = Price – £450 – 2Price + £960 – £4 = -Price + £506. 4. **Break-even Points and Maximum Profit/Loss:** * The maximum loss is limited to the net premium paid (£4) if the price is below £450. * The maximum profit occurs when the price is at £480. In this case, the profit is £480 – £450 – £4 = £26. * The upper break-even point can be found by setting the payoff equation (Price > £480) to zero: -Price + £506 = 0 => Price = £506. * If the price rises above £506, the strategy results in a loss. 5. **Investor’s Outlook:** * This strategy is best suited for an investor who believes the price will slightly increase but not significantly. They profit if the price rises to £480 but start losing if it rises above £506. The investor is expecting a modest increase in price, capitalizing on the premium received from selling the two higher strike calls while limiting potential gains if the price increases substantially. The strategy benefits from time decay on the short calls, provided the price remains below £480. The investor aims to capture the premium while participating in a limited price increase.

The question assesses understanding of put-call parity and its application in identifying arbitrage opportunities, considering transaction costs. Put-call parity states: \(C + PV(X) = P + S\), where \(C\) is the call option price, \(PV(X)\) is the present value of the strike price, \(P\) is the put option price, and \(S\) is the stock price. An arbitrage opportunity exists when this equation is not balanced. The transaction costs alter the breakeven point, and this needs to be factored in. In this scenario, the initial put-call parity is: £5.50 + £95.12 = £3.00 + £98.00, which gives £100.62 = £101.00. This indicates a slight mispricing. To profit from the mispricing, one should buy the relatively undervalued side (the left-hand side) and sell the relatively overvalued side (the right-hand side). Considering transaction costs, the decision changes. Buying the call and present value of strike price costs £5.50 + £95.12 + £0.10 + £0.10 = £100.82. Selling the put and stock generates £3.00 + £98.00 – £0.10 – £0.10 = £100.80. The cost of buying the call and present value of strike price is higher than the revenue from selling the put and stock, so arbitrage is not profitable. Now consider reversing the trade. Selling the call and present value of strike price generates £5.50 + £95.12 – £0.10 – £0.10 = £100.42. Buying the put and stock costs £3.00 + £98.00 + £0.10 + £0.10 = £101.20. The revenue from selling the call and present value of strike price is lower than the cost of buying the put and stock, so arbitrage is not profitable. Therefore, no arbitrage opportunity exists when accounting for transaction costs. The mispricing is too small to overcome the costs involved in executing the trade.

The question explores the concept of delta hedging and gamma, focusing on how a portfolio manager dynamically adjusts their hedge to maintain a delta-neutral position. The scenario involves a portfolio of options on a volatile stock, requiring the manager to rebalance their hedge as the stock price fluctuates. Gamma measures the rate of change of delta with respect to changes in the underlying asset’s price. A higher gamma implies that the delta is more sensitive to price changes, necessitating more frequent rebalancing. The calculation demonstrates how to determine the number of shares needed to rebalance the hedge after a specific price movement, considering the portfolio’s gamma. The initial delta hedge requires selling 1000 shares to offset the positive delta of the options portfolio. When the stock price increases, the delta of the options portfolio increases due to the positive gamma. This means the portfolio becomes more sensitive to further price increases. To maintain a delta-neutral position, the portfolio manager needs to reduce the short position in the stock. The change in the number of shares required to rebalance the hedge is calculated by multiplying the portfolio’s gamma by the change in the stock price and the number of options. This result indicates how many fewer shares the manager needs to short to keep the portfolio delta-neutral. The formula used is: Change in shares = – (Gamma * Change in Stock Price * Number of Options). In this case, it is: -(0.05 * 2 * 10000) = -1000 shares. The negative sign indicates that the short position needs to be reduced. The manager initially shorted 1000 shares, but now needs to short 1000 shares less. So the number of shares to short now is 1000 – 1000 = 0. Therefore, the portfolio manager needs to close their entire short position to maintain a delta-neutral hedge.

The question revolves around the concept of delta hedging a short call option position and the associated costs when the underlying asset’s price moves significantly. Delta, representing the sensitivity of the option price to changes in the underlying asset’s price, is crucial for hedging. A short call option has a positive delta, meaning as the underlying asset price increases, the option price also increases, causing a loss for the option writer. To hedge, the option writer buys the underlying asset. The number of units to buy is determined by the delta. In this scenario, the delta changes as the underlying asset’s price changes, necessitating adjustments to the hedge. This adjustment is called dynamic hedging. When the asset price increases, the delta increases, requiring the purchase of more of the underlying asset. Conversely, when the asset price decreases, the delta decreases, requiring the sale of some of the underlying asset. These transactions incur costs, which erode the profit from writing the option. The key here is to understand how delta changes (gamma) and how this affects the cost of maintaining a delta-neutral position. Transaction costs are directly proportional to the number of shares bought or sold during the rebalancing. The higher the volatility, the more frequently the hedge needs to be adjusted, and thus, the higher the transaction costs. In this specific case, the initial hedge requires buying 50 shares (delta of 0.5 * 100 options). When the price rises, the delta increases to 0.7, requiring an additional purchase of 20 shares (0.2 * 100). The total cost is then calculated by multiplying the number of shares traded by the transaction cost per share. Therefore, the total transaction cost is (50 + 20) * £0.10 = £7.00.

The core of this question revolves around understanding the practical application of delta hedging in a portfolio exposed to option risk, particularly when dealing with large, discrete trades and transaction costs. Delta hedging aims to neutralize the directional risk of an option position by taking an offsetting position in the underlying asset. The delta of an option represents the sensitivity of the option’s price to a change in the price of the underlying asset. A delta of 0.6 means that for every £1 increase in the underlying asset’s price, the option’s price is expected to increase by £0.60. The key challenge here is to calculate the number of shares needed to offset the delta risk. The fund manager is short options (selling options), which means they are exposed to upside risk if the underlying asset’s price increases. To hedge this risk, the manager needs to buy shares of the underlying asset. The number of shares to buy is determined by the total delta exposure of the options sold. The fund manager sold 5,000 call option contracts, and each contract represents 100 shares. Therefore, the total number of shares represented by these contracts is 5,000 * 100 = 500,000 shares. The combined delta of the options is 0.6. The total delta exposure is then 500,000 * 0.6 = 300,000. This means the manager needs to buy 300,000 shares to offset the delta risk. However, the question introduces transaction costs. The brokerage charges £0.005 per share. The total transaction cost for buying 300,000 shares is 300,000 * £0.005 = £1,500. This cost needs to be considered when evaluating the overall hedging strategy. The fund manager’s initial capital is £10 million. The transaction cost represents a small percentage of the total capital, but it’s crucial for accurate risk management and performance evaluation. The calculation is as follows: 1. Total shares represented by the options: 5,000 contracts * 100 shares/contract = 500,000 shares 2. Total delta exposure: 500,000 shares * 0.6 = 300,000 3. Number of shares to buy: 300,000 shares 4. Transaction cost per share: £0.005 5. Total transaction cost: 300,000 shares * £0.005/share = £1,500 Therefore, the fund manager needs to buy 300,000 shares, incurring a transaction cost of £1,500.

The core concept tested here is the understanding of implied volatility, its relationship to option prices, and how market makers adjust their bid-ask quotes in response to changes in supply and demand. The scenario presents a situation where unusual trading activity suggests a potential information asymmetry. Market makers, being risk-averse, widen their bid-ask spread to compensate for the increased risk of trading against informed participants. The calculation involves understanding how the spread widening affects the potential profit and loss for a market maker executing a trade. The original scenario is designed to assess the candidate’s ability to apply theoretical knowledge to a practical, real-world situation, emphasizing critical thinking and decision-making under uncertainty. The market maker’s initial strategy is to buy at the bid and immediately sell at the ask. The initial profit margin is the difference between the ask and bid prices. When the implied volatility increases, the market maker widens the spread to mitigate risk. This means increasing the ask price and decreasing the bid price. The calculation determines the new bid and ask prices, the new profit margin, and then considers the impact of a large order hitting the market. If the market maker fills the order at the bid price and then has to sell at a lower price due to adverse information, the loss can be substantial. The spread widening is designed to protect against such losses. The original example uses a unique derivative instrument, the “Vol-Indexed Note,” to add complexity and novelty. This type of note is sensitive to changes in volatility, making it a suitable instrument for this scenario. The numerical values are also original, and the scenario does not resemble any standard textbook examples. The question tests not only the calculation of the bid-ask spread but also the underlying reasoning behind market maker behavior. The calculation is as follows: 1. Initial Spread: Ask – Bid = 103.50 – 103.00 = 0.50 2. Spread Widening: 50% increase in the spread means the new spread is 0.50 * 1.50 = 0.75 3. Spread Adjustment: The spread is widened equally on both sides, so the bid decreases by 0.75/2 = 0.375 and the ask increases by 0.75/2 = 0.375 4. New Bid: 103.00 – 0.375 = 102.625 5. New Ask: 103.50 + 0.375 = 103.875 6. Client buys at the new bid of 102.625. 7. Market maker immediately sells at the new ask of 103.875 8. Profit is 103.875 – 102.625 = 1.25

The question revolves around the application of the Black-Scholes model in a scenario involving a company’s strategic decision to hedge its copper exposure using options. The Black-Scholes model is a cornerstone of options pricing theory, and understanding its inputs and sensitivities (the “Greeks”) is crucial for effective risk management. The calculation involves understanding how a change in volatility (Vega) impacts the option’s price and, consequently, the hedging strategy’s effectiveness. The company, ‘CopperCorp,’ needs to determine the number of options required to maintain a delta-neutral position after a significant volatility shift. Delta-neutrality is a hedging strategy that aims to eliminate the portfolio’s sensitivity to small changes in the underlying asset’s price. The formula for calculating the change in option price due to a change in volatility is: Change in Option Price ≈ Vega * Change in Volatility In this case, Vega is given as 0.03 (meaning the option price changes by £0.03 for every 1% change in volatility). The volatility increases by 8% (from 22% to 30%). Therefore, the change in the option price is approximately 0.03 * 8 = £0.24. Since CopperCorp initially bought 20,000 call options at £2.50 each, the total cost was 20,000 * £2.50 = £50,000. The increase in option price due to the volatility change is £0.24 per option, totaling 20,000 * £0.24 = £4,800. To maintain delta neutrality after this volatility shift, CopperCorp needs to account for the increased value of its existing options. If the company were to sell additional options, it would need to sell an amount that offsets the increased value due to volatility. However, the question implies maintaining delta neutrality, not necessarily profiting from the volatility change. The core concept tested here is understanding Vega’s impact and how it necessitates adjustments to a hedging strategy. The example illustrates how a seemingly abstract concept like Vega translates into practical risk management decisions for a corporation.

The core of this question lies in understanding how a delta-neutral portfolio reacts to changes in volatility (vega) and time decay (theta), and how these sensitivities interact with the investor’s view on future market movements. A delta-neutral portfolio is constructed to be insensitive to small changes in the underlying asset’s price. However, it is still subject to other risks, particularly those related to volatility and time. Vega measures the sensitivity of the portfolio’s value to changes in the volatility of the underlying asset. A positive vega indicates that the portfolio’s value increases as volatility increases, while a negative vega indicates the opposite. Theta, on the other hand, measures the sensitivity of the portfolio’s value to the passage of time. Typically, options lose value as they approach their expiration date, resulting in a negative theta. In this scenario, the investor is bearish, expecting the underlying asset’s price to remain relatively stable or decline slightly. Given this view, the investor should ideally benefit from time decay (positive theta) and be negatively correlated to volatility (negative vega). A short straddle position is a classic strategy that aims to profit from low volatility and time decay. It involves selling both a call and a put option with the same strike price and expiration date. The maximum profit is capped at the premium received, while the potential loss is unlimited if the underlying asset’s price moves significantly in either direction. Let’s analyze why a short straddle aligns with the investor’s view. When volatility is high, option prices are inflated, and selling them generates a higher premium. If the investor correctly anticipates low volatility, the value of the options will decrease over time, allowing the investor to buy them back at a lower price and pocket the difference. Furthermore, as the options approach expiration, their time value erodes, benefiting the short straddle position. Now, let’s consider the other options. A long straddle would benefit from a large price movement in either direction, which contradicts the investor’s bearish view. A covered call strategy involves holding the underlying asset and selling call options against it. While it can generate income, it also limits the upside potential if the asset’s price rises. A protective put strategy involves buying put options to protect against a decline in the asset’s price. This strategy is suitable for investors who are bullish but want to hedge against potential losses. Therefore, the most suitable strategy for an investor with a bearish view who expects low volatility is a short straddle, as it benefits from time decay and a decrease in volatility.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Yorkshire Grain,” that wants to protect itself against potential declines in the price of wheat. They decide to use futures contracts traded on the ICE Futures Europe exchange. Understanding basis risk is crucial here. Basis risk arises because the futures contract price is for a standardized grade of wheat delivered at a specific location (say, a port in Rotterdam), while Yorkshire Grain’s wheat might be of a slightly different grade and is located in Yorkshire. To illustrate, suppose the December wheat futures contract is trading at £200 per tonne. Yorkshire Grain expects to harvest and sell their wheat in December. They hedge by selling December wheat futures contracts. However, due to local supply and demand conditions in Yorkshire, their actual selling price might differ from the futures price. This difference is the basis. Now, let’s say at the time of harvest in December, the December wheat futures contract settles at £190 per tonne. Yorkshire Grain closes out their futures position, making a profit of £10 per tonne (£200 – £190). However, they can only sell their actual wheat for £185 per tonne due to a glut of local supply. Their effective selling price is £195 per tonne (£185 actual sale + £10 futures profit). If, instead, local demand for Yorkshire Grain’s wheat was strong, they might have been able to sell their wheat for £205 per tonne. In this case, their effective selling price would still be £195 per tonne (£205 actual sale – £10 futures loss). This illustrates how hedging reduces price volatility but doesn’t eliminate it entirely. The remaining volatility is due to basis risk. Basis risk is calculated as the difference between the spot price of the asset being hedged and the price of the related futures contract at the time the hedge is lifted. In our example, if Yorkshire Grain expected the basis to be £5 (futures price higher than spot price) but it turned out to be £15, the unexpected change in basis would negatively impact the effectiveness of their hedge.

The question revolves around the concept of hedging currency risk using forward contracts, specifically within the context of a UK-based investment firm dealing with Euro-denominated assets. The core challenge is to determine the optimal forward contract strategy to mitigate the risk of adverse exchange rate movements affecting the firm’s profitability when converting Euros back to GBP. We need to calculate the profit or loss arising from a particular hedging strategy, considering both the initial forward rate and the spot rate at the contract’s maturity. The calculation involves the following steps: 1. **Calculate the initial GBP value of the Euro investment:** Multiply the Euro amount (€10,000,000) by the initial spot rate (0.85 GBP/EUR). This gives the baseline GBP value. 2. **Calculate the GBP value at maturity without hedging (using the spot rate):** Multiply the Euro amount (€10,000,000) by the spot rate at maturity (0.82 GBP/EUR). This represents the unhedged outcome. 3. **Calculate the GBP value at maturity with the forward contract:** Multiply the Euro amount (€10,000,000) by the forward rate (0.84 GBP/EUR). This represents the hedged outcome. 4. **Compare the hedged and unhedged outcomes to determine the effectiveness of the hedge:** Subtract the unhedged GBP value from the hedged GBP value to find the profit or loss due to hedging. 5. **Compare the initial GBP value with the hedged outcome to determine the overall profit or loss:** Subtract the initial GBP value from the hedged GBP value to determine the overall impact of the investment and the hedge. In this specific scenario, hedging provided a better outcome than remaining unhedged. However, the initial GBP value was higher than the hedged outcome, resulting in an overall loss. The explanation highlights the importance of understanding the interplay between exchange rate movements, hedging strategies, and their impact on investment returns. It emphasizes that hedging protects against adverse movements but doesn’t guarantee a profit. The forward rate locks in a specific exchange rate, which may be more or less favorable than the future spot rate. This example demonstrates the practical application of forward contracts in managing currency risk within a real-world investment context, requiring a nuanced understanding of derivative valuation and risk management.

Let’s consider a scenario involving a UK-based agricultural cooperative, “GreenHarvest,” which aims to stabilize its wheat prices amidst fluctuating global markets using derivatives. GreenHarvest plans to use wheat futures contracts listed on the ICE Futures Europe exchange to hedge their anticipated wheat harvest. The cooperative expects to harvest 5,000 tonnes of wheat in six months. One ICE wheat futures contract represents 100 tonnes of wheat. To calculate the number of contracts needed for a perfect hedge, GreenHarvest would divide their expected harvest by the contract size: 5,000 tonnes / 100 tonnes/contract = 50 contracts. However, GreenHarvest’s wheat is of a slightly lower grade than the standard wheat specified in the futures contract. This difference in grade introduces basis risk. Let’s assume the current spot price for GreenHarvest’s wheat is £200 per tonne, while the six-month futures price is £210 per tonne. The basis is therefore £10 per tonne (£210 – £200). GreenHarvest’s analysts anticipate that the basis will narrow to £5 per tonne by the delivery date due to improved global wheat supply forecasts. This means that the spot price will increase relatively more than the futures price. If GreenHarvest hedges using 50 futures contracts, they lock in a price close to £210 per tonne. However, if the basis narrows as predicted, GreenHarvest will effectively receive less than expected for their wheat. To account for this basis risk, GreenHarvest could slightly under-hedge, using fewer than 50 contracts. However, under-hedging exposes them to price declines. Now, let’s consider an alternative strategy: using options. GreenHarvest could purchase put options on wheat futures. This would provide downside protection while allowing them to benefit from potential price increases. Suppose a six-month put option with a strike price of £205 per tonne costs £3 per tonne. If the wheat price falls below £205, GreenHarvest can exercise the option and sell their wheat at £205 (minus the option premium). If the price stays above £205, they let the option expire and sell their wheat at the market price. The breakeven price for the put option strategy is the strike price minus the option premium: £205 – £3 = £202 per tonne. This is the minimum price GreenHarvest needs to receive to be better off than not hedging at all. Finally, consider the impact of margin requirements. Futures contracts require initial and maintenance margins. If the price of wheat futures moves against GreenHarvest’s position, they will need to deposit additional margin to maintain their position. This can strain their cash flow. Options, on the other hand, only require the upfront premium payment, providing more predictable cash flow management.

The core of this question lies in understanding how a delta-neutral portfolio, constructed using options, reacts to changes in implied volatility (vega) and the underlying asset’s price (delta). A delta-neutral portfolio is designed to be insensitive to small changes in the underlying asset’s price. However, it is *not* immune to changes in implied volatility. Vega measures the sensitivity of the portfolio’s value to changes in implied volatility. A positive vega means the portfolio’s value increases with increasing volatility, and vice versa. The question also incorporates Gamma, which is the rate of change of delta with respect to changes in the underlying asset’s price. Gamma measures how quickly the delta of a portfolio changes as the price of the underlying asset moves. A portfolio with positive gamma will see its delta increase as the underlying asset price increases, and decrease as the underlying asset price decreases. To solve this, we need to consider the combined effect of the volatility decrease and the price increase. First, the decrease in implied volatility will negatively impact the portfolio’s value because the portfolio has a positive vega. Second, the increase in the underlying asset price will cause the delta of the portfolio to change. Since the portfolio has a positive gamma, the delta will increase as the underlying asset price increases. Since the portfolio was initially delta-neutral, an increase in delta means the portfolio becomes delta-positive, benefiting from the price increase. The net change in portfolio value depends on the magnitude of these two opposing effects. Let’s calculate the approximate impact. A vega of 10 means that for every 1% decrease in implied volatility, the portfolio loses £10. A 3% decrease in implied volatility would therefore result in a loss of approximately 3 * £10 = £30. A gamma of 0.5 means that for every £1 increase in the underlying asset’s price, the delta increases by 0.5. A £2 increase in the underlying asset’s price would increase the delta by 0.5 * 2 = 1. This means the portfolio is now delta-positive by 1. Since the underlying asset price increased by £2, the portfolio’s value would increase by approximately 1 * £2 = £2. Therefore, the net change in portfolio value is approximately -£30 + £2 = -£28. The portfolio value will decrease by approximately £28.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their sensitivity to market volatility and the proximity of the underlying asset price to the barrier. A down-and-out call option becomes worthless if the underlying asset price touches or falls below the barrier level. Therefore, as the asset price approaches the barrier from above, the option’s value decreases due to the increased probability of being knocked out. This is a crucial concept in risk management, especially when advising clients on complex derivative strategies. The option’s sensitivity to volatility also plays a significant role. Higher volatility increases the likelihood of the asset price hitting the barrier, thus reducing the option’s value. The time remaining until expiration is another key factor. As the expiration date approaches, the time window for the asset price to breach the barrier narrows, potentially decreasing the option’s value if the barrier is close to being breached. In this scenario, we need to consider the combined effect of the approaching barrier, increasing volatility, and decreasing time to expiration. The option’s value will be most significantly impacted when all three factors align to increase the probability of the option being knocked out. Therefore, the advisor must understand these dynamics to provide sound advice.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their payoff structures. A down-and-out call option becomes worthless if the underlying asset’s price hits a pre-defined barrier level *before* the option’s expiration. The problem requires calculating the payoff of such an option given a specific price path for the underlying asset. Here’s the breakdown of the solution: 1. **Barrier Breach Check:** Determine if the asset price ever touched or went below the barrier level of £95 at any point during the option’s life. In this case, the price dipped to £92 on day 3, breaching the barrier. 2. **Knock-Out Effect:** Since the barrier was breached, the down-and-out call option is “knocked out” and becomes worthless. This means the option holder receives no payoff, regardless of the asset’s price at expiration. 3. **Payoff Calculation:** Because the option is knocked out, the payoff is £0. Even though the asset price at expiration (£110) is above the strike price (£100), the barrier event renders the option worthless. The key concept here is the *path dependency* of barrier options. Their value depends not only on the final asset price but also on the asset’s price movements throughout the option’s life. A standard call option, in contrast, is *not* path-dependent. Consider an analogy: Imagine you have a ticket to a concert, but the ticket is only valid if you arrive at the venue *before* 8 PM. If you arrive at 8:05 PM, the ticket is worthless, even if the concert is still ongoing and you’d otherwise enjoy it. The barrier option is similar – breaching the barrier is like arriving late, rendering the option worthless, irrespective of the asset’s final value. Another example: A company might use a down-and-out call option to hedge against rising raw material prices, but only if those prices remain within a certain range. If the price drops below a certain level (the barrier), the company’s risk profile changes, and the hedge is no longer needed. Therefore, the option is designed to “knock out” if the price falls below the barrier. The complexity lies in understanding that the final asset price is irrelevant once the barrier has been breached. Many candidates incorrectly calculate the payoff as if it were a standard call option, ignoring the knock-out feature.