Derivatives Level 4 (Investment Advice Diploma) Exam Quiz Quiz 10

The core of this question revolves around understanding how a delta-neutral portfolio reacts to changes in implied volatility (vega) and the time remaining until expiration (theta). A delta-neutral portfolio is constructed to be insensitive to small changes in the underlying asset’s price. However, it remains sensitive to other factors, primarily vega and theta. Vega measures the portfolio’s sensitivity to changes in implied volatility. A positive vega means the portfolio’s value increases with increasing volatility, and vice versa. Theta measures the portfolio’s sensitivity to the passage of time. Typically, options lose value as they approach expiration (time decay), resulting in a negative theta. In this scenario, the fund manager needs to rebalance the portfolio to maintain delta neutrality after the implied volatility decreases and time passes. Decreasing volatility will negatively impact a portfolio with positive vega, and the passage of time will negatively impact a portfolio with negative theta. To offset these changes and maintain delta neutrality, the manager needs to adjust the positions in the underlying asset. The calculation involves understanding the relationship between delta, vega, theta, and the required adjustment in the underlying asset. We can use the following logic: 1. **Impact of Volatility Change:** A decrease in implied volatility of 1% will decrease the portfolio’s value by vega * change in volatility = 1,000 * -0.01 = -£10. This means the portfolio loses £10 due to the volatility decrease. 2. **Impact of Time Decay:** The passage of one day will decrease the portfolio’s value by theta = -£50. This means the portfolio loses £50 due to time decay. 3. **Total Impact:** The total loss in portfolio value is -£10 – £50 = -£60. 4. **Delta Adjustment:** To maintain delta neutrality, the manager needs to offset this loss by adjusting the position in the underlying asset. Since the portfolio is delta-neutral, any change in its value due to volatility and time decay needs to be compensated by an equal and opposite change through trading the underlying asset. To compensate for a loss of £60, the manager needs to make a profit of £60 from the underlying asset. Given the delta of the underlying asset is 0.5, each unit change in the underlying asset’s price results in a £0.5 change in the portfolio’s value. 5. **Required Change in Underlying Asset:** To achieve a £60 profit, the manager needs to buy or sell a number of units of the underlying asset. The number of units can be calculated as: Required Profit / Delta of Underlying Asset = £60 / 0.5 = 120 units. Since the portfolio lost value, the manager needs to *buy* 120 units of the underlying asset to increase the portfolio’s value and offset the losses from volatility and time decay, thus maintaining delta neutrality.

The question revolves around the impact of a sudden, unexpected regulatory change on the valuation of a complex derivative product – a callable interest rate swap. The core concept being tested is the ability to assess how regulatory shifts can alter market perceptions of risk and, consequently, the pricing of derivatives. The callable feature introduces an additional layer of complexity, requiring an understanding of its impact on the swap’s value and how regulatory changes might affect the likelihood of the call option being exercised. Here’s a step-by-step approach to understanding the correct answer: 1. **Baseline Valuation:** Before the regulatory change, the swap’s value is determined by discounting expected future cash flows based on prevailing interest rate curves and the creditworthiness of the counterparties. The call option’s value is priced using models like Black-Scholes or similar interest rate option pricing models, considering factors like volatility and time to maturity. 2. **Regulatory Impact:** The new regulation increases capital requirements for banks holding complex derivatives, making it more expensive for them to maintain such positions. This increases the cost of doing business for the bank. 3. **Swap Valuation Adjustment:** The increased capital requirement translates to a higher cost of carry for the bank. This higher cost reduces the attractiveness of the swap to the bank, making it more expensive for the bank to maintain the swap on its books. The bank will seek to reduce the swap’s value to offset the increased capital cost. 4. **Call Option Impact:** The call option gives the bank the right, but not the obligation, to terminate the swap. The bank will only exercise the call option if it is economically advantageous to do so. 5. **Combined Effect:** The increased capital requirement makes it more likely that the bank will exercise the call option. The bank will exercise the call option if the value of the swap is less than the cost of maintaining the swap on its books. 6. **Fair Value Adjustment:** The bank will seek to reduce the swap’s value to offset the increased capital cost. This will result in a reduction in the fair value of the swap. The fair value of the swap will be reduced by the amount of the increased capital cost. Therefore, the most likely outcome is a decrease in the fair value of the callable interest rate swap, driven by the increased capital requirements and the higher probability of the bank exercising the call option.

The core of this question revolves around understanding how delta changes with respect to changes in the underlying asset’s price (gamma) and how time decay affects option value (theta). We must consider the combined impact of these Greeks on a short call option position. First, we need to understand how gamma affects delta. Gamma represents the rate of change of delta with respect to the underlying asset’s price. A positive gamma means that as the underlying asset’s price increases, the delta of the option also increases (becomes less negative for a short call). Conversely, as the underlying asset’s price decreases, the delta decreases (becomes more negative for a short call). Next, we need to consider theta. Theta represents the rate of change of the option’s price with respect to time. For a short call option, theta is generally negative, meaning the option loses value as time passes, all other factors being equal. This decay accelerates as the option approaches its expiration date. Now, let’s analyze the scenario. The portfolio manager is short call options. A short call option has a negative delta (the portfolio profits if the underlying asset price decreases) and a positive gamma (the delta becomes less negative as the underlying asset price increases). The theta is negative (the option loses value as time passes). The underlying asset’s price increases significantly. This increase in price will cause the delta of the short call option to increase (become less negative) due to the positive gamma. This means the portfolio manager’s short call position becomes more sensitive to further upward price movements. Simultaneously, time is passing, causing the option to lose value due to theta. However, the significant price increase will likely outweigh the impact of theta, at least initially. The portfolio manager needs to decide whether to close out the position (potentially at a loss) or adjust the hedge to account for the increased delta. The most prudent course of action depends on the portfolio manager’s risk tolerance and outlook for the underlying asset. However, understanding the combined impact of gamma and theta is crucial for making an informed decision. The increase in the underlying asset price necessitates a review of the hedge ratio, potentially requiring the purchase of more of the underlying asset to maintain a delta-neutral position.

1. **Barrier Option Characteristics:** A knock-out barrier option ceases to exist if the underlying asset’s price touches the barrier. This introduces path dependency, making it more complex to hedge than standard options. 2. **Delta Hedging:** Delta represents the sensitivity of the option price to a change in the underlying asset’s price. A delta-neutral portfolio is constructed to offset this risk. 3. **Gamma Hedging:** Gamma represents the rate of change of delta with respect to the underlying asset’s price. It measures the instability of the delta hedge. High gamma means the delta hedge needs frequent adjustments. 4. **Volatility Impact:** Near the barrier, the option’s delta and gamma become highly sensitive to volatility. Increased volatility increases the probability of the asset price hitting the barrier, causing the option to knock out. This necessitates more frequent and larger adjustments to the hedge. 5. **The “Sticky Delta” Misconception:** The concept of “sticky delta” refers to the observed phenomenon where an option’s implied volatility moves inversely with the underlying asset’s price. If the underlying asset price rises, implied volatility tends to decrease, and vice-versa. This is more pronounced for options closer to the money or near barriers. However, in the context of a knock-out barrier option, this effect is less reliable due to the discrete nature of the knock-out event. The hedge must still account for the high gamma and potential for the option to cease existing. 6. **Hedging Frequency and Cost:** Due to the heightened sensitivity near the barrier, the portfolio manager must increase the frequency of hedging. Each adjustment incurs transaction costs. The increased volatility amplifies the need for these adjustments, driving up the overall hedging cost. 7. **Calculation:** Although a precise numerical calculation isn’t required in this specific question, the underlying principle involves understanding that hedging costs are directly related to the frequency of adjustments, which is driven by gamma and volatility. The higher the gamma and volatility, the more frequent the adjustments, and the higher the cost. 8. **Original Example:** Imagine a company using a down-and-out put option to hedge against a potential decline in the value of its inventory of a specialized metal. The barrier is set close to the current market price to minimize the option premium. However, the metal market experiences a sudden surge in volatility due to geopolitical instability. The company’s portfolio manager must now actively monitor and adjust the hedge more frequently than initially planned, incurring higher transaction costs. The risk of the option knocking out becomes a significant concern, requiring further adjustments to the hedging strategy, potentially involving more expensive options or alternative hedging instruments. 9. **Key Takeaway:** Managing exotic options, especially barrier options, requires a deep understanding of their sensitivity to volatility and the need for dynamic hedging strategies. Ignoring the impact of increased volatility near the barrier can lead to significant hedging losses.

Let’s analyze a scenario involving a UK-based agricultural cooperative, “Golden Harvest,” which aims to protect its future wheat sales from price volatility using futures contracts. Golden Harvest anticipates selling 5,000 tonnes of wheat in six months. They decide to hedge using wheat futures contracts traded on the ICE Futures Europe exchange. Each contract represents 100 tonnes of wheat. The current futures price for wheat deliverable in six months is £200 per tonne. Golden Harvest decides to short (sell) 50 futures contracts (5,000 tonnes / 100 tonnes per contract). Six months later, the spot price of wheat is £180 per tonne. Golden Harvest sells its wheat in the spot market at this price. Simultaneously, they close out their futures position by buying back 50 futures contracts at £180 per tonne. Here’s the calculation: 1. **Initial Futures Position:** Short 50 contracts at £200/tonne. 2. **Final Futures Position:** Buy 50 contracts at £180/tonne. 3. **Profit from Futures:** (£200 – £180) * 100 tonnes/contract * 50 contracts = £100,000. 4. **Spot Market Sale:** 5,000 tonnes * £180/tonne = £900,000. 5. **Effective Price Received:** (£900,000 + £100,000) / 5,000 tonnes = £200/tonne. Now, let’s consider the impact of basis risk. Basis risk arises because the spot price and the futures price may not converge perfectly at the delivery date. Suppose, instead of converging at £180, the futures price closed at £185. The profit on the futures position would be (£200 – £185) * 100 * 50 = £75,000. The effective price received would be (£900,000 + £75,000) / 5,000 = £195/tonne. This illustrates how basis risk can reduce the effectiveness of a hedge. Furthermore, margin requirements play a crucial role. Initial margin is the amount Golden Harvest must deposit with its broker to open the futures position. Maintenance margin is the level below which the margin account cannot fall. If the futures price moves against Golden Harvest, and their margin account falls below the maintenance margin, they will receive a margin call, requiring them to deposit additional funds. The regulatory framework, such as EMIR, mandates clearing of standardized OTC derivatives through central counterparties (CCPs) to mitigate counterparty risk. This ensures that even if one party defaults, the CCP guarantees the performance of the contract.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their sensitivity to volatility changes, coupled with the impact of correlation between the underlying asset and the barrier. The scenario presented is designed to test not only the knowledge of how barrier options function but also the ability to synthesize this knowledge with real-world market dynamics. The core concept revolves around the fact that an up-and-out call option becomes worthless if the underlying asset’s price touches or exceeds the barrier level before the option’s expiration. The initial calculation of the option’s price, considering volatility and the correlation between the asset and the barrier, is crucial. The calculation involves understanding the inverse relationship between volatility and the price of an up-and-out call option when the asset price is near the barrier. An increase in volatility near the barrier increases the probability of the barrier being hit, thus decreasing the option’s value. The correlation aspect introduces another layer of complexity. A negative correlation implies that when the asset price increases, the probability of the barrier being breached decreases, thereby increasing the option’s value. Let’s assume the initial price of the up-and-out call option is \(P_0\). The implied volatility of the underlying asset is \(\sigma_0 = 20\%\), and the correlation between the asset and the barrier is \(\rho_0 = 0.3\). The barrier level is \(B\), and the current asset price is close to \(B\). Now, the implied volatility increases to \(\sigma_1 = 25\%\), and the correlation changes to \(\rho_1 = -0.2\). The change in option price \(\Delta P\) can be approximated by considering the combined effect of volatility and correlation. The volatility effect (\(\Delta P_\sigma\)) is negative since the increased volatility raises the probability of hitting the barrier. The correlation effect (\(\Delta P_\rho\)) is positive because the negative correlation reduces the probability of hitting the barrier. The combined effect is: \[\Delta P = \Delta P_\sigma + \Delta P_\rho\] Given the initial price of the option is £5.00, a reasonable estimate for the new price, considering the opposing effects of increased volatility and decreased correlation, would be slightly lower than the initial price but not drastically so, as the negative correlation partially offsets the volatility increase.

The question explores the complexities of delta hedging a short call option position, specifically when the underlying asset’s price experiences a significant gap after an earnings announcement. The key is understanding how gamma, which measures the rate of change of delta, affects the hedge’s effectiveness when price movements are large and sudden. Here’s a breakdown of the calculation and concepts: 1. **Initial Hedge:** The portfolio manager sells 100 call options, each representing 100 shares, so a total of 10,000 shares equivalent. The initial delta of 0.6 means for every £1 increase in the share price, the option price increases by £0.60. To delta-hedge, the manager buys 6,000 shares (10,000 * 0.6). 2. **Earnings Announcement and Price Gap:** The share price jumps from £50 to £60 *before* the manager can rebalance. This significant price change invalidates the initial delta hedge because delta itself changes. 3. **New Delta:** The delta increases to 0.8 after the price jump. This means the manager now *needs* to hold 8,000 shares to be delta-neutral (10,000 * 0.8). 4. **Rebalancing Trade:** The manager needs to buy an additional 2,000 shares (8,000 – 6,000) at the new price of £60. 5. **Cost of Rebalancing:** The cost is 2,000 shares * £60/share = £120,000. 6. **Option Payoff:** Since the options are short calls, the manager loses money if the share price exceeds the strike price. Assuming the strike price is £55 (this is an implicit assumption required to make the options in-the-money and the delta to shift as described; without this assumption, the problem is unsolvable), each option is now £5 in the money (£60 – £55). The total payoff for the option holder is £5 per share * 10,000 shares = £50,000. The *loss* for the option writer (the portfolio manager) is £50,000. 7. **Total Loss:** The total loss is the cost of rebalancing *plus* the option payoff: £120,000 + £50,000 = £170,000. This example demonstrates the limitations of delta hedging when gamma is significant and price jumps occur. Delta hedging works best for small price movements. A large price jump exposes the portfolio to gamma risk, requiring costly rebalancing. Furthermore, the example implicitly assumes a strike price to make the options in-the-money and the delta shift as described. This is a critical element for the scenario to work as intended. Ignoring the impact of gamma can lead to substantial losses, especially around significant events like earnings announcements. The scenario highlights the importance of considering higher-order Greeks and stress testing portfolios under various market conditions. It also shows how a seemingly well-hedged position can quickly become unhedged due to market volatility.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their sensitivity to market movements and barrier proximity. A down-and-out call option becomes worthless if the underlying asset’s price touches or falls below the barrier level before the option’s expiration. The sensitivity to the underlying asset’s price changes increases as the asset price approaches the barrier. This is because a small adverse movement can trigger the “out” feature, leading to a total loss of the option’s value. The time decay (theta) also accelerates as the barrier is approached, as the probability of the barrier being hit increases, reducing the option’s value. Vega, representing sensitivity to volatility, generally decreases as the barrier is approached, because the option’s lifespan is potentially shortened, and the impact of volatility is capped by the barrier. Rho, the sensitivity to interest rate changes, has a complex relationship and can increase or decrease depending on the specific parameters and proximity to the barrier. The key concept is that barrier options behave differently from standard options, particularly near the barrier. The combined effect of these sensitivities needs to be understood for effective risk management and trading strategies. Here’s the breakdown of why option a) is the correct response: * **Delta:** As the underlying asset price nears the barrier, the delta of a down-and-out call option increases in magnitude (becomes more negative if the price is above the barrier) because a small price decrease will cause the option to expire worthless. * **Theta:** Theta, the time decay, increases (becomes more negative) as the barrier is approached because there is less time for the asset to move favorably before the barrier is hit. * **Vega:** Vega, the sensitivity to volatility, generally decreases as the barrier is approached because the potential upside of volatility is limited by the barrier. If the barrier is breached, the option becomes worthless, regardless of future volatility. * **Rho:** Rho’s behavior is complex and depends on the specific parameters. In some cases, it might increase slightly, but generally, its impact is less pronounced than the other Greeks near the barrier. The other options present incorrect relationships between the Greeks and the proximity to the barrier.

The question focuses on the practical application of delta-hedging in a portfolio containing options, considering transaction costs. Delta-hedging aims to neutralize the portfolio’s sensitivity to small changes in the underlying asset’s price. The delta of an option indicates how much the option’s price is expected to change for every £1 change in the underlying asset’s price. To maintain a delta-neutral position, one needs to offset the option’s delta by taking an opposite position in the underlying asset. Transaction costs, however, introduce a layer of complexity. Each adjustment to the hedge incurs a cost, which must be factored into the decision of whether to rebalance. A common approach is to establish a “delta-neutral range.” Only when the portfolio’s delta moves outside this range is it deemed cost-effective to rebalance. In this scenario, calculating the number of shares to buy or sell involves several steps: 1. **Calculate the initial portfolio delta:** This is the sum of the deltas of all options held in the portfolio. 2. **Determine the target delta:** The target delta is typically zero for a delta-neutral portfolio. 3. **Calculate the delta change needed:** This is the difference between the current portfolio delta and the target delta. 4. **Calculate the number of shares to trade:** Divide the delta change needed by the delta of a single share (which is 1). 5. **Consider transaction costs:** Determine if the benefit of rebalancing (reducing delta exposure) outweighs the cost of the transaction. Rebalancing is only justified if the delta moves outside the pre-defined delta-neutral range. For instance, if a portfolio has a delta of 0.60 (meaning the portfolio’s value is expected to increase by £0.60 for every £1 increase in the underlying asset’s price) and the investor wants to delta-hedge, they need to sell 60 shares of the underlying asset to offset the positive delta. This would bring the portfolio’s overall delta closer to zero, reducing its sensitivity to small price movements in the underlying asset. However, if the transaction cost to buy or sell shares is £5 per trade, and the portfolio’s delta fluctuates within a range of 0.55 to 0.65, the investor might choose not to rebalance unless the delta moves outside this range. This is because the cost of constantly rebalancing the portfolio to maintain perfect delta neutrality would outweigh the benefits. In this specific question, the initial portfolio delta is 750. The investor wants to reduce it to be within the range of -200 to 200. To achieve this, the investor needs to sell shares to reduce the positive delta. Since the delta is outside the range, the investor must rebalance. They must sell enough shares to bring the delta within the acceptable range. This involves considering the transaction costs and ensuring that the rebalancing strategy is cost-effective in the long run.

Let’s analyze a scenario involving a UK-based agricultural cooperative (“Co-op Harvest”) that seeks to hedge against fluctuating wheat prices using futures contracts traded on the ICE Futures Europe exchange. The Co-op anticipates harvesting 500,000 bushels of wheat in three months. They are concerned about a potential price drop and want to lock in a minimum selling price. The current futures price for wheat with a three-month delivery is £5.00 per bushel. The Co-op decides to hedge 75% of their expected harvest, which is 375,000 bushels. Each futures contract covers 5,000 bushels, so they need to purchase 375,000 / 5,000 = 75 contracts. Now, consider the basis risk. Basis risk arises because the futures price and the spot price at the time of delivery may not converge perfectly. Assume that the Co-op expects the basis to be £0.10 per bushel (spot price lower than futures price) at the time of harvest. This means they expect to sell their wheat for £0.10 less than the futures price. If, at harvest time, the spot price is £4.80 per bushel, and the futures price is £4.90 per bushel, the actual basis is £0.10 (as expected). The Co-op sells their wheat in the spot market for £4.80 per bushel. Simultaneously, they close out their futures position by selling the contracts at £4.90 per bushel (having initially bought them at £5.00). The gain on the futures contracts is (£5.00 – £4.90) * 375,000 = £37,500. The revenue from selling wheat in the spot market is £4.80 * 375,000 = £1,800,000. The effective selling price is (£1,800,000 + £37,500) / 375,000 = £4.80 + £0.10 = £4.90 per bushel. If the actual basis turns out to be different from the expected basis, it will impact the effective selling price. For example, if the spot price is £4.70 and the futures price is £4.90, the basis is £0.20. In this case, the effective selling price would be lower than expected. The basis risk is the risk that this difference between the expected and actual basis will negatively impact the hedging strategy. The impact of regulatory requirements such as EMIR (European Market Infrastructure Regulation) also needs to be considered. EMIR mandates clearing of certain OTC derivatives through central counterparties (CCPs) to reduce counterparty risk. If the Co-op were using OTC wheat forwards instead of exchange-traded futures, they might be subject to EMIR’s clearing obligations, which involve margin requirements and reporting obligations, increasing the cost and complexity of their hedging strategy.

The question assesses the understanding of hedging strategies using options, specifically focusing on a ratio spread. A ratio spread involves buying and selling different numbers of options with the same expiration date but different strike prices. The investor aims to profit from a specific market movement (or lack thereof) while limiting potential losses. To solve this, we need to calculate the net premium paid or received, and then analyze the profit/loss at the expiration date based on different stock prices. The investor buys 100 call options with a strike price of £45 (costing £3 each) and sells 200 call options with a strike price of £50 (receiving £1 each). 1. **Calculate the initial premium:** * Cost of buying 100 calls at £45 strike: 100 contracts * 100 shares/contract * £3/share = £30,000 * Revenue from selling 200 calls at £50 strike: 200 contracts * 100 shares/contract * £1/share = £20,000 * Net premium paid: £30,000 – £20,000 = £10,000 2. **Analyze profit/loss at expiration based on different stock prices:** * **Stock price below £45:** All options expire worthless. The investor loses the net premium paid, which is £10,000. * **Stock price at £47:** Only the bought calls are in the money. * Profit from bought calls: 100 contracts * 100 shares/contract * (£47 – £45) = £20,000 * Net Profit: £20,000 – £10,000 (initial premium) = £10,000 * **Stock price at £50:** The bought calls are in the money, and the sold calls are at the money. * Profit from bought calls: 100 contracts * 100 shares/contract * (£50 – £45) = £50,000 * Net Profit: £50,000 – £10,000 (initial premium) = £40,000 * **Stock price at £52:** Both the bought and sold calls are in the money. * Profit from bought calls: 100 contracts * 100 shares/contract * (£52 – £45) = £70,000 * Loss from sold calls: 200 contracts * 100 shares/contract * (£52 – £50) = £40,000 * Net Profit: £70,000 – £40,000 – £10,000 (initial premium) = £20,000 * **Stock price at £55:** Both the bought and sold calls are in the money. * Profit from bought calls: 100 contracts * 100 shares/contract * (£55 – £45) = £100,000 * Loss from sold calls: 200 contracts * 100 shares/contract * (£55 – £50) = £100,000 * Net Profit: £100,000 – £100,000 – £10,000 (initial premium) = -£10,000 Therefore, the maximum profit occurs when the stock price is at £50 at expiration.

The question assesses understanding of the impact of macroeconomic indicators, specifically inflation expectations, on derivative pricing, particularly interest rate swaps. The correct answer requires recognizing that increased inflation expectations generally lead to higher interest rates across the yield curve, which, in turn, impacts swap rates. The fixed rate payer in an interest rate swap benefits when rates rise, as the fixed rate they are paying becomes more attractive relative to the floating rate. The calculation involves understanding the relationship between inflation expectations, interest rates, and the net present value (NPV) of the swap. Let’s assume the initial swap terms: Notional Principal = £10,000,000; Fixed Rate = 2.0% (paid annually); Floating Rate = SONIA (Sterling Overnight Index Average) + 0.5%. The swap has 3 years remaining. Scenario 1 (Before Inflation Expectations Increase): We’ll assume SONIA averages 1.5% over the next 3 years, making the floating rate 2.0%. The NPV is close to zero because the fixed and floating rates are nearly equal. Scenario 2 (After Inflation Expectations Increase): Inflation expectations rise, causing the yield curve to shift upwards. Now, the market expects SONIA to average 3.5% over the next 3 years, making the floating rate 4.0%. The fixed rate payer now receives significantly more than they pay. Calculating the approximate NPV change: Year 1: Fixed payer receives 4.0% – 2.0% = 2.0% more on £10,000,000 = £200,000 Year 2: Fixed payer receives 4.0% – 2.0% = 2.0% more on £10,000,000 = £200,000 Year 3: Fixed payer receives 4.0% – 2.0% = 2.0% more on £10,000,000 = £200,000 Discounting these cash flows back to the present using a discount rate (let’s assume 4% for simplicity): Year 1: £200,000 / (1.04)^1 = £192,307.69 Year 2: £200,000 / (1.04)^2 = £184,911.24 Year 3: £200,000 / (1.04)^3 = £177,799.27 Total NPV change = £192,307.69 + £184,911.24 + £177,799.27 = £555,018.20 Therefore, the interest rate swap’s value would increase by approximately £555,018.20 for the fixed-rate payer due to the rise in inflation expectations. This illustrates how macroeconomic factors directly influence derivative valuations.

The core concept being tested is the calculation of the theoretical futures price, incorporating cost of carry and the impact of dividends. The cost of carry model considers storage costs, interest forgone on the investment, and any income received (like dividends). In this case, we need to adjust the future price downwards to account for the dividends received during the contract’s life. First, we need to calculate the present value of the dividends. Dividend 1 of £0.50 is paid in 3 months (0.25 years), so its present value is \(0.50 \times e^{-0.05 \times 0.25} = 0.50 \times e^{-0.0125} \approx 0.50 \times 0.9876 \approx 0.4938\). Dividend 2 of £0.75 is paid in 9 months (0.75 years), so its present value is \(0.75 \times e^{-0.05 \times 0.75} = 0.75 \times e^{-0.0375} \approx 0.75 \times 0.9632 \approx 0.7224\). The total present value of dividends is \(0.4938 + 0.7224 = 1.2162\). Next, we calculate the future price without considering dividends: \(F_0 = S_0 \times e^{rT} = 50 \times e^{0.05 \times 1} = 50 \times e^{0.05} \approx 50 \times 1.0513 \approx 52.565\). Finally, we subtract the present value of the dividends from the future price to get the adjusted future price: \(F_0^{adj} = 52.565 – 1.2162 \approx 51.35\). This entire process showcases a common, yet nuanced, derivatives pricing problem. A common pitfall is simply subtracting the dividend amounts directly, rather than discounting them to present value. Another is forgetting to compound the spot price forward at the risk-free rate. The question tests understanding of both the cost of carry model and present value calculations in a derivative pricing context. The distractors are deliberately chosen to reflect these common errors.

The question assesses the understanding of the impact of early exercise on American options, specifically focusing on dividend-paying stocks. American options can be exercised at any time before expiration, which introduces complexities not present in European options. When a stock pays a dividend, the stock price is expected to drop by approximately the dividend amount on the ex-dividend date. This price drop can make early exercise optimal for American call options, especially if the dividend is large and the option is deep in the money. The holder might prefer to capture the dividend income and reinvest it, rather than waiting until expiration and potentially losing the dividend benefit due to the price decline. The calculation considers the intrinsic value of the option (stock price minus strike price), the present value of the dividend, and the time value of the option. We need to determine if the benefit of receiving the dividend now outweighs the potential loss of time value by exercising early. Here’s the breakdown of the decision-making process: 1. **Calculate the Intrinsic Value:** This is the immediate profit if the option is exercised now. In this case, it’s the stock price (\$115) minus the strike price (\$100), resulting in \$15. 2. **Calculate the Present Value of the Dividend:** Since the dividend is paid in one month, we discount it back to today using the risk-free rate. The formula is: \[PV = \frac{Dividend}{(1 + RiskFreeRate)^{Time}}\] In this case, PV = \[\frac{6}{(1 + 0.05)^{1/12}}\] = \$5.975 3. **Estimate the Time Value:** Time value represents the potential for the option’s value to increase before expiration. A rough estimate can be obtained by comparing the option price with its intrinsic value. Here the option price is \$17 and the intrinsic value is \$15, so the time value is approximately \$2. 4. **Compare Early Exercise Benefit with Time Value Loss:** If the present value of the dividend exceeds the time value, early exercise might be optimal. The net benefit of early exercise is the present value of the dividend minus the time value: \$5.975 – \$2 = \$3.975. This suggests that early exercise is beneficial. 5. **Consider Transaction Costs:** Transaction costs reduce the net benefit of early exercise. If the transaction costs are high enough, they could outweigh the benefit of capturing the dividend. In this case, transaction costs of \$0.25 per share reduce the net benefit to \$3.725. 6. **Final Decision:** Because the net benefit of early exercise (\$3.725) is positive, and because the option is deep in the money, it is likely optimal to exercise the option early. This strategy captures the dividend and avoids the expected price drop.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Green Harvest,” which exports organic barley to several European countries. Green Harvest faces price volatility in the barley market and seeks to hedge its price risk using futures contracts traded on the ICE Futures Europe exchange. The cooperative’s treasurer, Emily, needs to determine the optimal number of contracts to hedge their anticipated barley harvest. First, Emily needs to quantify the price risk. Suppose Green Harvest expects to harvest 500,000 bushels of barley in three months. The current spot price of barley is £4.00 per bushel, but Emily is concerned about a potential price decline. She decides to use barley futures contracts to hedge against this risk. Each ICE Futures Europe barley futures contract represents 100 metric tons of barley. Converting bushels to metric tons, assuming 1 metric ton is approximately 45.93 bushels, Green Harvest’s harvest is equivalent to approximately 10,886 metric tons (500,000 bushels / 45.93 bushels/metric ton). The number of futures contracts required is calculated by dividing the total metric tons to be hedged by the contract size: 10,886 metric tons / 100 metric tons/contract = 108.86 contracts. Since futures contracts are traded in whole numbers, Emily should round up to 109 contracts to ensure adequate coverage. However, Emily also needs to consider the hedge ratio. The hedge ratio is the ratio of the size of the position to be hedged to the size of the hedging instrument. In this case, the hedge ratio is close to 1:1. Now, consider basis risk, which arises from the imperfect correlation between the spot price of Green Harvest’s barley and the futures price of the ICE Futures Europe barley contract. If the correlation is not perfect, the hedge may not completely eliminate price risk. Suppose Emily estimates the correlation coefficient between the spot price and the futures price to be 0.9. The optimal hedge ratio would be the quantity to be hedged multiplied by the correlation coefficient, divided by the contract size. Optimal number of contracts = (10,886 metric tons * 0.9) / 100 metric tons/contract = 98 contracts (rounded to the nearest whole number). Finally, Emily needs to consider the regulatory implications. Under the European Market Infrastructure Regulation (EMIR), Green Harvest may be subject to mandatory clearing requirements if its derivatives positions exceed certain thresholds. Emily must ensure that Green Harvest complies with EMIR’s reporting and clearing obligations, potentially requiring them to clear their futures contracts through a central counterparty (CCP).

The core of this question lies in understanding how changes in the underlying asset’s price and time to expiration impact the value of a European put option, specifically within the context of the Black-Scholes model. We need to consider both the directional impact (positive or negative) and the relative magnitude of these impacts. A higher asset price generally *decreases* the value of a put option, as the option becomes less likely to be “in the money” at expiration. Conversely, as time to expiration decreases, the value of a put option generally *decreases*, as there is less time for the underlying asset’s price to move favorably. However, the *magnitude* of these changes is crucial. The Black-Scholes model uses the concept of “Greeks” to quantify these sensitivities. While we aren’t explicitly calculating the Greeks here, the question assesses understanding of their underlying principles. In this scenario, the asset price *increases significantly* (5%), while the time to expiration *decreases only slightly* (1 week out of 6 months). The large increase in the asset price will have a much greater negative impact on the put option’s value than the small decrease in time to expiration. Therefore, the put option’s value will decrease overall. To illustrate, imagine a put option on a stock with a strike price of £100. If the stock price is currently £95, the put option has some intrinsic value. If the stock price suddenly jumps to £100, the intrinsic value disappears. This significant price movement overshadows the effect of a week less time to expiration. Now, consider two identical put options, one expiring in 6 months and the other in 5 months and 3 weeks. If the stock price remains constant, the option with slightly less time to expiration will be marginally less valuable. However, a 5% increase in the underlying asset price is a far more significant event than a week less time to expiration. This is because the put option’s value is much more sensitive to changes in the underlying asset’s price (Delta) than to changes in time to expiration (Theta), especially when the price change is substantial.

The question assesses understanding of delta hedging, specifically when a portfolio’s delta is not perfectly offset and the underlying asset experiences a price jump. Delta hedging aims to neutralize a portfolio’s sensitivity to small changes in the underlying asset’s price. However, delta is not constant; it changes as the underlying asset’s price changes (this is gamma). A significant price jump exposes the limitations of static delta hedging. The initial portfolio value is £5,000,000, and the initial delta is 0.35, meaning the portfolio’s value will change by approximately £0.35 for every £1 change in the underlying asset. To delta hedge, the fund manager shorts shares worth 0.35 * £5,000,000 = £1,750,000. When the underlying asset’s price jumps by 5%, the portfolio’s value increases by (1 – delta) * (percentage change) * (portfolio value) = (1 – 0.35) * 0.05 * £5,000,000 = £162,500. The short position loses 5% of its value: 0.05 * £1,750,000 = £87,500. The net change in the portfolio value is £162,500 – £87,500 = £75,000. The new portfolio value is £5,000,000 + £75,000 = £5,075,000. Now, the fund manager needs to rebalance the hedge. The new delta is 0.45, meaning the target short position is now 0.45 * £5,075,000 = £2,283,750. The fund manager needs to short an additional £2,283,750 – £1,750,000 = £533,750 worth of shares. The percentage change in the portfolio due to the rebalancing is (amount of new shares shorted / original portfolio value) = (£533,750 / £5,000,000) * 100% = 10.675%. This calculation demonstrates the dynamic nature of delta hedging and the need to rebalance the hedge when significant price movements occur. The rebalancing cost is proportional to the change in delta and the portfolio’s value. Failing to rebalance exposes the portfolio to increased risk.

The core of this question revolves around understanding how implied volatility, as derived from option prices, reflects the market’s expectation of future volatility, and how this expectation can be exploited using variance swaps. A variance swap is a forward contract on future realized variance. Its payoff is directly related to the difference between the realized variance of an asset’s returns over a specified period and a pre-agreed variance strike (K_var). The fair variance strike is essentially the market’s expectation of the average variance over the life of the swap. The VIX index is a measure of the implied volatility of S&P 500 index options. Squaring the VIX gives us an implied variance. When the VIX is used to price a variance swap, it is used to derive the fair variance strike. If a portfolio manager believes that the market is underestimating future volatility (i.e., the realized variance will be higher than the implied variance), they would enter into a long variance swap position. Conversely, if they believe the market is overestimating future volatility, they would enter into a short variance swap position. The payoff of a variance swap is calculated as \(N \times (Realized Variance – K_{var})\), where N is the notional amount of the swap. Realized variance is calculated as the sum of the squared returns over the period, annualized. In this case, we’re given the annualized realized volatility of 22%, so the realized variance is \(0.22^2 = 0.0484\). The variance strike is derived from the VIX, which is 20%, so the variance strike is \(0.20^2 = 0.04\). The payoff is therefore \(£1,000,000 \times (0.0484 – 0.04) = £1,000,000 \times 0.0084 = £8,400\). Since the portfolio manager took a long position, they receive this amount. Now, let’s consider a more nuanced example. Imagine a hedge fund manager, Amelia, who specializes in volatility arbitrage. She notices that the implied volatility of FTSE 100 options is unusually low compared to historical realized volatility during periods of economic uncertainty related to Brexit negotiations. Amelia believes that the market is underpricing the potential for increased volatility as the negotiations progress. She enters into a long variance swap with a notional of £5 million, betting that the realized variance will exceed the variance strike derived from the implied volatility. If Brexit negotiations become particularly contentious, leading to significant market swings, Amelia’s long variance swap position will profit handsomely. This profit can offset potential losses in her other portfolio holdings due to the increased market volatility, acting as a hedge while simultaneously capitalizing on her insight into market mispricing of volatility. This illustrates how variance swaps can be used not only for hedging but also for expressing specific views on future market volatility.

The core of this question revolves around understanding how implied volatility affects option pricing, particularly in the context of exotic options where standard models like Black-Scholes might not directly apply. The scenario introduces a barrier option, which adds complexity to the pricing. Implied volatility is the market’s expectation of future volatility and is a crucial input in option pricing models. An increase in implied volatility generally increases the value of an option because it suggests a higher probability of the underlying asset reaching a profitable price level for the option holder. However, the impact on barrier options is nuanced. For a down-and-out call option, if implied volatility increases, the probability of the underlying asset price fluctuating and hitting the barrier before expiry also increases. This reduces the option’s value, as hitting the barrier renders it worthless. The specific sensitivities (Greeks) are not explicitly required to solve this, but understanding their underlying concepts is crucial. Delta represents the sensitivity of the option price to changes in the underlying asset price. Gamma represents the rate of change of delta with respect to changes in the underlying asset price. Vega represents the sensitivity of the option price to changes in implied volatility. Theta represents the sensitivity of the option price to the passage of time. Rho represents the sensitivity of the option price to changes in interest rates. In this case, the dominant effect of increased implied volatility is the increased probability of breaching the barrier. The correct answer will reflect this inverse relationship between implied volatility and the price of a down-and-out call option.

The core of this question revolves around understanding the impact of correlation on portfolio risk when derivatives, specifically options, are used for hedging. A perfect hedge is rarely achievable in reality due to various factors, including imperfect correlation between the asset being hedged and the hedging instrument. The effectiveness of a hedge is directly related to the correlation coefficient; a correlation of +1 indicates a perfect positive correlation, meaning the assets move in tandem, while -1 indicates a perfect negative correlation, ideal for hedging. A correlation of 0 suggests no linear relationship. In this scenario, we are dealing with an imperfect negative correlation. The formula to calculate the variance of a portfolio comprising two assets (the original portfolio and the hedging instrument) is: \[\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho_{1,2}\sigma_1\sigma_2\] Where: * \(\sigma_p^2\) is the portfolio variance * \(w_1\) and \(w_2\) are the weights of asset 1 and asset 2 in the portfolio, respectively * \(\sigma_1\) and \(\sigma_2\) are the standard deviations of asset 1 and asset 2, respectively * \(\rho_{1,2}\) is the correlation between asset 1 and asset 2 In our case: * \(w_1 = 1\) (the entire portfolio) * \(w_2 = -0.5\) (short position in options representing 50% of the portfolio value) * \(\sigma_1 = 0.20\) (standard deviation of the original portfolio) * \(\sigma_2 = 0.40\) (standard deviation of the options) * \(\rho_{1,2} = -0.8\) (correlation between the portfolio and the options) Plugging these values into the formula: \[\sigma_p^2 = (1)^2(0.20)^2 + (-0.5)^2(0.40)^2 + 2(1)(-0.5)(-0.8)(0.20)(0.40)\] \[\sigma_p^2 = 0.04 + 0.04 + 0.064\] \[\sigma_p^2 = 0.144\] The portfolio standard deviation (\(\sigma_p\)) is the square root of the portfolio variance: \[\sigma_p = \sqrt{0.144} = 0.12\] Therefore, the portfolio standard deviation is 12%. The key takeaway is that even with a negative correlation, the hedge is not perfect, and the resulting portfolio risk depends on the magnitude of the correlation and the volatility of the hedging instrument.

The question tests the understanding of delta hedging and its effectiveness in different market scenarios. Delta hedging aims to create a portfolio that is insensitive to small changes in the underlying asset’s price. However, its effectiveness diminishes when the underlying asset price experiences large, sudden jumps or gaps. This is because delta, which measures the sensitivity of the option price to changes in the underlying asset price, changes as the underlying asset price changes. When the underlying asset price jumps significantly, the delta hedge needs to be rebalanced immediately to maintain its effectiveness. If the rebalancing cannot be done instantaneously, the hedge will be imperfect, and the portfolio will be exposed to risk. Gamma measures the rate of change of delta. A higher gamma means that delta changes more rapidly as the underlying asset price changes. Therefore, a portfolio with a high gamma requires more frequent rebalancing to maintain its delta neutrality. In a scenario where a stock price gaps down significantly overnight due to unexpected negative news, the delta hedge will not be effective in protecting the portfolio from losses. This is because the delta hedge is based on the assumption that the underlying asset price will move gradually. When the underlying asset price gaps down, the delta hedge will be significantly off, and the portfolio will experience a loss. The loss will be greater for a portfolio with a higher gamma, as the delta will change more rapidly as the underlying asset price changes. To mitigate this risk, traders often use other hedging strategies in conjunction with delta hedging, such as gamma hedging or vega hedging. Gamma hedging involves buying or selling options to reduce the portfolio’s gamma. Vega hedging involves buying or selling options to reduce the portfolio’s sensitivity to changes in volatility. The calculation to determine the profit or loss involves understanding how the delta hedge is constructed and how it responds to changes in the underlying asset price. Initially, the portfolio is delta neutral. However, when the stock price gaps down, the delta hedge becomes ineffective, and the portfolio experiences a loss. The loss is equal to the change in the stock price multiplied by the number of shares that are shorted to create the delta hedge. In this case, the initial delta is 0.5, and the number of shares shorted is 500 (0.5 * 1000). When the stock price gaps down from £100 to £80, the loss is £20 per share, and the total loss is £20 * 500 = £10,000.

The question assesses the understanding of delta hedging and its limitations, particularly when dealing with options on assets that experience significant price jumps. Delta hedging aims to create a portfolio that is insensitive to small changes in the underlying asset’s price. However, it relies on continuous adjustments and assumes that price movements are relatively smooth. When an asset experiences a sudden, large price jump (a “gap”), the delta of the option can change dramatically, rendering the existing hedge ineffective. The Black-Scholes model, while widely used, assumes continuous price movements and constant volatility. It doesn’t account for the possibility of price jumps. In a jump scenario, the actual change in the option’s price will likely deviate significantly from what the Black-Scholes model predicts. The delta hedge, based on the Black-Scholes delta, will therefore be insufficient to offset the loss (or gain) on the option. Let’s illustrate with a novel example: Imagine a small biotech company, “GeneLeap,” developing a revolutionary cancer treatment. Its stock price is currently £50. You hold a short call option on GeneLeap with a strike price of £55. You delta hedge this position. The company announces unexpectedly positive clinical trial results. The stock price instantly jumps to £80. The call option’s value increases far more than the delta hedge anticipated because the jump violates the continuous price movement assumption of Black-Scholes. Your hedge fails to fully protect you against this sudden price change. The loss on the short call option will be substantially larger than the gains from the delta hedge. The magnitude of the loss is influenced by the size of the price jump, the time to expiration, and the option’s vega (sensitivity to volatility). Since the jump significantly increases the probability of the option expiring in the money, its value increases dramatically. The calculation will look like this: 1. **Option Value Increase:** The call option, initially out-of-the-money, becomes deeply in-the-money. Its value increases significantly, let’s say from £1 to £25 due to the jump. 2. **Delta Hedge Gain:** The delta hedge, which involved shorting a portion of GeneLeap stock, generates a profit as the stock price increased. However, this profit is limited because the hedge was calculated based on the initial stock price and a smaller expected price movement. 3. **Net Loss:** The net loss is the difference between the option value increase and the delta hedge gain. If the delta hedge gained £10, the net loss would be £25 – £10 = £15.

The question focuses on the impact of correlation between assets within a hedging strategy, specifically using options. A correlation of 1 indicates a perfect positive relationship, meaning the assets move in tandem. This significantly impacts the effectiveness of a hedging strategy. In this scenario, the hedge is designed to protect against losses in the primary asset (the UK bank shares). The core concept is understanding how correlation affects the hedge ratio and the overall cost and effectiveness of the hedge. With perfect positive correlation, the hedge becomes almost redundant in its traditional sense because any loss in the underlying asset is perfectly mirrored by the hedging instrument (the index options). This impacts the optimal number of options needed, potentially reducing it significantly. To calculate the profit or loss, we need to consider: 1. The initial cost of the options. 2. The change in value of the bank shares. 3. The change in value of the options, considering the perfect correlation. In this scenario, the bank shares decline by 15%. The options, due to the perfect correlation, also decline by 15% of their initial value. The profit or loss is calculated as the change in value of the shares plus the change in value of the options (which will be negative since the options lost value). The calculation is as follows: * Initial value of shares: £1,000,000 * Decline in share value: £1,000,000 * 0.15 = £150,000 * Initial cost of options: £50,000 * Decline in option value: £50,000 * 0.15 = £7,500 * Net loss: £150,000 (loss on shares) + £7,500 (loss on options) = £157,500 Therefore, the investor experiences a net loss of £157,500. The perfect correlation diminishes the hedging benefit, resulting in a loss close to the unhedged loss on the shares, plus the loss on the options.

Let’s consider a scenario involving a portfolio manager, Anya, at a UK-based investment firm, “Global Investments Ltd.” Anya is tasked with hedging the firm’s significant holding of FTSE 100 stocks against potential downside risk stemming from upcoming Brexit negotiations. Anya decides to use FTSE 100 futures contracts for hedging. The current FTSE 100 index level is 7,500. Each futures contract represents £10 per index point. Anya’s portfolio has a market value of £37,500,000. First, we calculate the number of futures contracts needed: Number of contracts = Portfolio Value / (Index Level * Contract Multiplier) Number of contracts = £37,500,000 / (7,500 * £10) = 500 contracts Now, let’s analyze the margin requirements. 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. Suppose the exchange stipulates an initial margin of £7,500 per contract and a maintenance margin of £6,000 per contract. Anya must deposit £7,500 * 500 = £3,750,000 as initial margin. Consider a scenario where the FTSE 100 index falls to 7,300. The loss per contract is (7,500 – 7,300) * £10 = £2,000. For 500 contracts, the total loss is £2,000 * 500 = £1,000,000. The margin account balance falls to £3,750,000 – £1,000,000 = £2,750,000. Now, calculate the number of contracts that have fallen below the maintenance margin: Margin shortfall per contract = Initial Margin – (Initial Index – New Index) * Contract Multiplier Margin shortfall per contract = £7,500 – (7,500 – 7,300) * £10 = £7,500 – £2,000 = £5,500 Since the maintenance margin is £6,000, there is no margin call yet, as each contract still has £5,500 of value. Now, suppose the FTSE 100 falls to 7,000. The loss per contract is (7,500 – 7,000) * £10 = £5,000. Total loss = £5,000 * 500 = £2,500,000. The margin account balance is £3,750,000 – £2,500,000 = £1,250,000. Margin per contract = £1,250,000 / 500 = £2,500. Since this is below the maintenance margin of £6,000, a margin call is triggered. The amount of the margin call is the difference between the initial margin and the current margin level, which is £7,500 – £2,500 = £5,000 per contract. The total margin call is £5,000 * 500 = £2,500,000. The key is to understand how margin calls are triggered and calculated based on the index movements and the initial and maintenance margin levels. This requires a thorough grasp of futures contracts mechanics and risk management principles.

The question assesses the 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. The payoff is calculated by considering the premiums paid or received and the intrinsic value of the options at expiration. The maximum profit is achieved when the underlying asset price is at the short call’s strike price. The maximum loss occurs if the asset price rises significantly above the short call’s strike price. In this specific case, the investor buys 100 ABC call options with a strike price of £95 for a premium of £5 each and sells 200 ABC call options with a strike price of £100 for a premium of £2 each. The initial cost is (100 * £5) – (200 * £2) = £500 – £400 = £100. If the price of ABC shares rises to £105 at expiration: – The 100 long calls with a £95 strike price will have an intrinsic value of £10 (£105 – £95). – The 200 short calls with a £100 strike price will have an intrinsic value of £5 (£105 – £100). The profit/loss from the long calls is 100 * (£10 – £5) = £500 (intrinsic value minus initial premium). The profit/loss from the short calls is 200 * (£2 – £5) = -£600 (initial premium minus intrinsic value). The net profit/loss is £500 – £600 = -£100. Therefore, the overall loss is £100.

The question assesses the understanding of delta-hedging a portfolio of options and the impact of market events on the hedge. Specifically, it tests the ability to calculate the required adjustment to a delta-neutral portfolio after a significant price movement in the underlying asset. The initial portfolio is delta-neutral, meaning its delta is zero. This is achieved by holding a certain number of shares of the underlying asset to offset the delta of the options. The delta of the options is given as 0.6 per option, and the portfolio contains 10,000 options, resulting in a total option delta of 6,000. To be delta-neutral, the portfolio must hold -6,000 shares (short 6,000 shares if each option has delta of 0.6). When the underlying asset price increases by £2, the option delta changes. The question provides the gamma of the options, which measures the rate of change of delta with respect to the underlying asset price. The gamma is 0.0001 per option. Therefore, for 10,000 options, the portfolio gamma is 1. The change in the portfolio delta due to the price movement is calculated as: Change in Delta = Portfolio Gamma * Change in Price = 1 * £2 = 2000. This means the portfolio delta has increased by 2000. To re-establish delta neutrality, the portfolio must be adjusted to offset this change. Since the delta has increased, the portfolio needs to sell 2000 shares of the underlying asset. The calculation is as follows: 1. Initial option delta: 10,000 options * 0.6 = 6,000 2. Delta-neutral position: Hold -6,000 shares 3. Portfolio gamma: 10,000 options * 0.0001 = 1 4. Change in delta: 1 * £2 = 2000 5. Adjustment needed: Sell 2000 shares to reduce the delta back to zero. Therefore, the fund manager needs to sell 2000 shares to re-establish delta neutrality.

The question tests understanding of delta hedging, gamma, and the impact of market volatility on a derivatives portfolio. Specifically, it assesses the ability to calculate the required adjustment to a delta-hedged position given a change in the underlying asset’s price and the portfolio’s gamma. The calculation involves using the gamma to estimate the change in delta, and then determining the number of futures contracts needed to re-hedge the portfolio. First, we need to calculate the change in the delta of the portfolio. The formula to approximate the change in delta using gamma is: Change in Delta ≈ Gamma * Change in Underlying Price In this case, Gamma = 250 and Change in Underlying Price = £2. Change in Delta ≈ 250 * £2 = 500 This means the delta of the portfolio has increased by 500. Since the initial delta hedge was neutral (delta = 0), this increase in delta means we now have a delta of +500. To re-hedge, we need to reduce the delta back to zero. Since each futures contract has a delta of 1 (representing one unit of the underlying asset), we need to sell 500 futures contracts to offset the increased delta. Therefore, the portfolio manager should sell 500 futures contracts. Now, let’s consider the implications in a real-world context. Imagine a fund manager using options to hedge a large equity portfolio against market downturns. The portfolio is delta-hedged, meaning its sensitivity to small price changes is neutralized. However, the portfolio also has gamma, representing the rate of change of the delta. If the market experiences a sudden price movement, the delta hedge becomes imperfect. The gamma tells us how much the delta will change for each unit change in the underlying asset’s price. In this scenario, the fund manager needs to actively manage the delta hedge by adjusting the position in futures contracts (or other hedging instruments) as the underlying asset’s price changes. Failure to do so exposes the portfolio to increased risk, as the hedge becomes less effective. The manager must consider transaction costs and market liquidity when making these adjustments. Frequent re-hedging can reduce risk but also increase costs, while infrequent re-hedging can leave the portfolio vulnerable to larger price swings. Another important consideration is the impact of volatility. Higher volatility leads to larger price swings, which in turn cause larger changes in the delta of the portfolio. This means the manager needs to re-hedge more frequently and potentially use more sophisticated hedging strategies to manage the increased risk. Understanding the relationship between delta, gamma, and volatility is crucial for effective risk management in a derivatives portfolio.

This question assesses the understanding of hedging strategies using options, specifically focusing on delta-neutral hedging and the impact of gamma on portfolio rebalancing. The scenario involves a portfolio manager using call options to hedge a short stock position. The key is to understand how delta changes as the stock price fluctuates and how gamma necessitates rebalancing to maintain a delta-neutral position. The calculation involves determining the initial hedge ratio (number of options needed), tracking the change in delta due to a price movement, and calculating the number of additional options required to re-establish delta neutrality. Initial Delta of the Portfolio: The portfolio is short 10,000 shares, so the initial delta from the stock position is -10,000. Initial Hedge Ratio: The call option has a delta of 0.6. To achieve delta neutrality, the number of call options needed is: \[\frac{10,000}{0.6} = 16,666.67\] Since options are traded in contracts of 100 shares, the portfolio manager needs to buy 167 contracts (rounding up to ensure sufficient coverage), or 16,700 options. Change in Stock Price: The stock price increases by £1. Change in Option Delta: The option’s gamma is 0.04. The change in delta due to the £1 increase in stock price is: \[0.04 \times 1 = 0.04\] New Option Delta: The new delta of the call option is: \[0.6 + 0.04 = 0.64\] New Portfolio Delta from Options: The total delta from the options is: \[16,700 \times 0.64 = 10,688\] New Overall Portfolio Delta: The overall portfolio delta is: \[-10,000 + 10,688 = 688\] Options Needed to Rebalance: To re-establish delta neutrality, the portfolio manager needs to reduce the overall delta by 688. Since each option has a delta of 0.64, the number of options to sell is: \[\frac{688}{0.64} = 1075\] Therefore, the portfolio manager needs to sell approximately 1075 call options to rebalance the portfolio and maintain delta neutrality. This calculation highlights the dynamic nature of delta hedging. Gamma represents the rate of change of delta. As the underlying asset’s price moves, the option’s delta changes, requiring adjustments to the hedge. Ignoring gamma can lead to a poorly hedged portfolio, especially when the price of the underlying asset experiences significant fluctuations. The frequency of rebalancing depends on the portfolio’s gamma and the desired level of delta neutrality. Higher gamma portfolios require more frequent rebalancing. This example demonstrates the practical application of delta-gamma hedging in managing market risk.

The question involves understanding the interplay between implied volatility, time decay (theta), and the potential impact of earnings announcements on short straddle positions. A short straddle involves selling both a call and a put option with the same strike price and expiration date. The strategy profits if the underlying asset price remains relatively stable. However, it is highly vulnerable to significant price movements, especially around events like earnings announcements. Here’s how to break down the calculation and reasoning: 1. **Initial Position:** Selling a straddle generates an initial premium income of £750 (call) + £600 (put) = £1350. This is the maximum profit if the stock price remains at £100 at expiration. 2. **Impact of Time Decay (Theta):** Theta represents the rate at which an option’s value decreases as time passes. Since the investor holds a short option position, theta is beneficial. Over the two weeks (14 days), the call option decays by 14 * £15 = £210, and the put option decays by 14 * £12 = £168. Total decay benefit: £210 + £168 = £378. 3. **Impact of Implied Volatility Increase:** Implied volatility (IV) reflects the market’s expectation of future price fluctuations. A short straddle is negatively affected by increasing IV because it increases the value of both the call and put options, making it more expensive to buy them back to close the position. The call option increases by £300, and the put option increases by £250. Total volatility increase impact: £300 + £250 = £550. 4. **Impact of Stock Price Movement:** The stock price increases to £106. * Call Option: The call option is now in the money. Its value is approximately the intrinsic value plus any remaining time value. Assuming minimal remaining time value (since it’s close to expiry), the call option’s value is £6 (intrinsic value). However, we must consider the volatility impact. The net effect is approximately £600 (new value) – £750 (original premium) – £210 (theta) + £300 (volatility) = -£60. * Put Option: The put option is out of the money and likely has minimal value due to time decay and being far from the strike. The value is approximately £0. The net effect is £0 – £600 (original premium) – £168 (theta) + £250 (volatility) = -£518. 5. **Net Profit/Loss:** * From Premiums: £1350 * From Time Decay: £378 * From Volatility Increase: -£550 * From Stock Movement: Call Option Loss: -£60 and Put Option Loss: -£518 Total Profit/Loss = £1350 + £378 – £550 – £60 – £518 = £600 Therefore, the net profit/loss is £600. This scenario highlights the complex interplay of factors affecting derivatives positions, especially the combined impact of time decay, volatility changes, and underlying asset price movements. A key takeaway is the vulnerability of short volatility strategies (like short straddles) to unexpected volatility spikes, particularly around significant events.

The question assesses the understanding of the impact of macroeconomic announcements on derivative pricing, specifically focusing on interest rate futures. The scenario involves a surprise announcement of higher-than-expected inflation figures, which leads to an anticipated increase in interest rates by the Bank of England (BoE). The key concept here is the inverse relationship between interest rates and the price of interest rate futures. A higher-than-expected inflation announcement signals that the BoE is likely to increase interest rates to combat inflation. Interest rate futures contracts are agreements to buy or sell a debt instrument at a predetermined future date and price. When interest rates are expected to rise, the value of these contracts decreases because the fixed interest rate they offer becomes less attractive compared to the higher prevailing market rates. The calculation involves determining the expected price change of a short-dated (3-month) interest rate future contract following the announcement. We assume a direct correlation between the expected interest rate hike and the futures price movement. If the market anticipates a 0.25% (25 basis points) increase in interest rates, the futures price will decrease by a corresponding amount. The initial futures price is given as 97.50. Since futures prices are quoted as 100 minus the interest rate, a price of 97.50 implies an interest rate of 2.50%. A 0.25% increase in the expected interest rate translates to a decrease of 0.25 in the futures price. Therefore, the expected futures price after the announcement is calculated as: \[ \text{New Futures Price} = \text{Initial Futures Price} – \text{Expected Interest Rate Increase} \] \[ \text{New Futures Price} = 97.50 – 0.25 = 97.25 \] The plausible incorrect answers are designed to test common misunderstandings. One incorrect answer assumes a positive correlation between interest rates and futures prices. Another considers only half of the interest rate increase. The final incorrect answer applies the change to the implied interest rate rather than the futures price directly. This question tests the candidate’s ability to apply theoretical knowledge to a practical scenario, understand the market dynamics, and perform the necessary calculations accurately.