Derivatives Level 4 (Investment Advice Diploma) Exam Quiz Quiz 09

To determine the breakeven point for the combined option strategy, we need to consider the initial cost of establishing the position and then identify the underlying asset prices at which the strategy becomes profitable. First, calculate the net premium paid: Cost of buying call option = £6 Premium received from writing call option = £2 Net premium paid = £6 – £2 = £4 Now, determine the breakeven points: * **Upper Breakeven Point:** This is the point where the profit from the written call begins to offset the initial cost. Since the strike price of the written call is £110, the upper breakeven point is calculated as: Upper Breakeven Point = Strike Price of Written Call + Net Premium Paid Upper Breakeven Point = £110 + £4 = £114 * **Lower Breakeven Point:** This is the point where the profit from the bought call begins to offset the initial cost. Since the strike price of the bought call is £100, the lower breakeven point is calculated as: Lower Breakeven Point = Strike Price of Bought Call + Net Premium Paid Lower Breakeven Point = £100 + £4 = £104 Therefore, the combined strategy has two breakeven points: £104 and £114. The investor will start making a profit if the underlying asset price is above £114. Consider a scenario where a fund manager, Sarah, believes that a particular stock, “TechFuture,” currently trading at £100, will experience moderate volatility. To capitalize on this view, she implements a bull call spread by buying a call option with a strike price of £100 for £6 and selling a call option with a strike price of £110 for £2. This strategy limits both her potential profit and potential loss. If TechFuture’s price remains below £100, both options expire worthless, and Sarah loses the net premium of £4. If TechFuture’s price rises to £104, Sarah breaks even on the bought call, but the written call is still out-of-the-money. If TechFuture’s price rises to £114, Sarah breaks even on the written call, and her maximum profit is capped. This illustrates how the breakeven points define the range within which the strategy yields either a loss or a profit.

The question assesses the understanding of exotic options, specifically barrier options, and how their value is affected by the presence of a barrier. The scenario involves a ‘knock-out’ barrier option, which ceases to exist if the underlying asset’s price hits the barrier level before expiration. The key concept here is that the barrier significantly alters the option’s payoff profile and, consequently, its price. A standard European option’s value is based on the probability of the underlying asset’s price being above (for a call) or below (for a put) the strike price at expiration. However, a knock-out barrier option introduces an additional condition: the barrier must *not* be breached before expiration. This condition reduces the option’s value compared to an otherwise identical standard European option. The calculation considers the initial option premium (£500), the potential profit if the option were to finish in the money without the barrier, and the probability of the barrier being hit. The reduction in value due to the barrier is estimated by considering the probability of the barrier being hit multiplied by the potential profit forgone. Here’s a step-by-step breakdown of the calculation: 1. **Potential Profit (without barrier):** The underlying asset price is £11,000, and the strike price is £10,500. The potential profit is £11,000 – £10,500 = £500. 2. **Net Potential Profit (without barrier):** Subtract the initial option premium: £500 – £500 = £0. 3. **Probability of Barrier Hit:** The broker estimates a 30% chance (0.3) of the barrier being hit before expiration. 4. **Expected Loss due to Barrier:** Multiply the probability of the barrier hit by the potential profit: 0.3 * £500 = £150. 5. **Adjusted Option Value:** Subtract the expected loss due to the barrier from the initial option value: £500 – £150 = £350. Therefore, the estimated value of the knock-out call option, considering the barrier, is £350. This reflects the reduced risk (and potential reward) for the option writer due to the barrier feature. The option is cheaper because the holder loses all rights if the barrier is touched, making it less valuable than a standard call option.

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 goal is typically to profit from a specific directional movement in the underlying asset while limiting potential losses. The profit/loss profile is calculated by considering the premium paid or received for each option and the payoff at expiration based on the underlying asset’s price. In this scenario, the investor is implementing a 1:2 put ratio spread. This means they buy one put option and sell two put options with a lower strike price. The investor profits if the stock price stays above the higher strike price, has limited profit potential if the stock price falls slightly below the higher strike price, and faces potentially unlimited losses if the stock price falls significantly below the lower strike price. The initial cost of the strategy is the premium paid for the long put minus the premium received from the short puts. The breakeven points are calculated by finding the stock prices at which the strategy results in zero profit or loss. There can be two breakeven points in a ratio spread strategy. To calculate the breakeven points: 1. **Initial Cost:** Premium paid for the £50 put – 2 \* Premium received for the £45 put = £5 – 2 \* £2 = £1. This is the initial cost per share. 2. **Upper Breakeven:** This is simply the strike price of the long put minus the net premium paid: £50 – £1 = £49. 3. **Lower Breakeven:** This is where it gets trickier. Below £45, the investor loses £1 for every £1 drop in the share price from £45, due to selling two puts. We need to find the price where the profit from the long put (with a strike of £50) offsets the losses from the two short puts. The equation is: \[50 – X – 2(45 – X) = \text{Initial Cost}\] \[50 – X – 90 + 2X = 1\] \[X = 41\] Therefore, the lower breakeven point is £41.

The question tests understanding of volatility skew and its implications for option pricing and trading strategies, particularly in the context of earnings announcements. Volatility skew refers to the asymmetry in the implied volatility curve, where out-of-the-money (OTM) puts typically have higher implied volatilities than OTM calls. This skew is often attributed to demand for downside protection. Earnings announcements are significant events that can cause substantial price movements in a stock. Before an earnings announcement, the implied volatility of options on the underlying stock tends to increase, reflecting the heightened uncertainty about the future price. After the announcement, if the actual price movement is less than expected, implied volatility usually decreases sharply, a phenomenon known as volatility crush. The put-call parity theorem states that a portfolio consisting of a long call option and a short put option with the same strike price and expiration date should have the same value as a forward contract on the underlying asset with the same delivery price and date. Any deviation from this parity can present arbitrage opportunities. In this scenario, the fund manager’s observation of a steepening volatility skew before the earnings announcement suggests that the market anticipates a potentially large negative price movement. The increased demand for OTM puts drives up their implied volatility, creating the skew. If the earnings announcement doesn’t result in the expected negative price movement, the implied volatility of the puts will decrease more significantly than that of the calls. The fund manager can exploit this by selling OTM puts before the announcement and buying them back after the volatility crush. The profit arises from the difference between the higher premium received when selling the puts and the lower premium paid when buying them back. The key risk is that the earnings announcement could trigger a substantial negative price movement, causing the puts to become in-the-money and resulting in a loss. The fund manager should consider factors such as the historical volatility of the stock, the market’s expectations for the earnings announcement, and the cost of the options. The strategy is most likely to be profitable if the market’s expectations for a large negative price movement are not met and the volatility crush is significant.

Let’s consider a scenario involving a UK-based agricultural cooperative, “GreenHarvest,” which produces and exports organic wheat. GreenHarvest faces significant price volatility in the global wheat market and seeks to hedge its price risk using wheat futures contracts traded on the London International Financial Futures and Options Exchange (LIFFE), now part of ICE Futures Europe. The cooperative plans to deliver 500 tonnes of wheat in December. To determine the optimal number of contracts, we need to consider the contract size, the spot price, the futures price, and the basis risk. The LIFFE wheat futures contract size is 100 tonnes. The current spot price for organic wheat is £200 per tonne, and the December futures price is £210 per tonne. GreenHarvest expects to harvest 500 tonnes of wheat. The cooperative decides to hedge 80% of its expected harvest. Number of contracts to hedge = (Tonnes to be hedged) / (Contract size) Tonnes to be hedged = 500 tonnes * 80% = 400 tonnes Number of contracts = 400 tonnes / 100 tonnes/contract = 4 contracts Now, let’s analyze the potential outcomes. Suppose the spot price in December turns out to be £190 per tonne, and the futures price converges to £192 per tonne. Revenue without hedging = 500 tonnes * £190/tonne = £95,000 Hedge position: GreenHarvest sold 4 futures contracts at £210 and buys them back at £192. Profit from futures = 4 contracts * 100 tonnes/contract * (£210 – £192)/tonne = 400 * £18 = £7,200 Net revenue with hedging = £95,000 + £7,200 = £102,200 Now, let’s analyze the effect of basis risk. Basis risk is the risk that the spot price and the futures price do not move in perfect correlation. In our case, the spot price decreased by £10 (£200 – £190), while the futures price decreased by £18 (£210 – £192). This difference of £8 represents the basis risk. If the basis risk was zero, the futures price would have decreased by £10 as well, and the hedge would have been perfect. Hedging strategies are never perfect due to basis risk, which arises from factors like transportation costs, storage costs, and differences in the quality of the underlying asset. The effectiveness of the hedge depends on how well the futures price tracks the spot price. Let’s also consider the impact of margin requirements. Initial margin is the amount of money that GreenHarvest must deposit with its broker to open the futures position. Maintenance margin is the level below which the margin account cannot fall. If the margin account falls below the maintenance margin, GreenHarvest will receive a margin call and must deposit additional funds to bring the account back up to the initial margin level. Finally, consider the regulatory aspect. GreenHarvest must comply with the European Market Infrastructure Regulation (EMIR), which requires the clearing of certain OTC derivatives through a central counterparty (CCP). However, since GreenHarvest is using exchange-traded futures, they are already cleared through a CCP, reducing counterparty risk.

The core of this question revolves around understanding how a delta-neutral portfolio reacts to changes in volatility (vega) and time decay (theta), specifically when the underlying asset experiences a significant price movement *after* the portfolio has been initially constructed. A delta-neutral portfolio is designed to be insensitive to small price changes in the underlying asset. However, vega (sensitivity to volatility changes) and theta (sensitivity to time decay) can still impact the portfolio’s value. Furthermore, a large price movement will disrupt the delta neutrality, exposing the portfolio to directional risk. Here’s the breakdown: 1. **Initial Delta Neutrality:** The portfolio is initially constructed to be delta neutral, meaning its value is (ideally) unaffected by small changes in the underlying asset’s price. This is achieved by balancing long and short positions in the underlying asset and/or options. 2. **Impact of Vega:** An increase in implied volatility (vega) will generally increase the value of options, both long and short. However, the *net* impact depends on the portfolio’s overall vega. If the portfolio has a positive vega (more long options than short options), its value will increase with rising volatility. Conversely, a negative vega portfolio will decrease in value. 3. **Impact of Theta:** Theta represents the time decay of an option’s value. As time passes, options lose value, particularly those close to their expiration date. A portfolio with a negative theta (typically the case for option sellers) will lose value due to time decay. A portfolio with a positive theta (typically option buyers) will gain value due to time decay. 4. **Price Movement and Delta Re-establishment:** The crucial element is the *subsequent* price movement of the underlying asset. A large price increase will make the portfolio no longer delta-neutral. If the portfolio was initially delta-neutral and the underlying asset’s price increases significantly, the portfolio will become delta-positive (it will benefit from further price increases). To re-establish delta neutrality, the portfolio manager must *sell* some of the underlying asset or delta-equivalent instruments (e.g., futures or options). 5. **Combining the Effects:** The overall change in portfolio value is a combination of the vega effect, the theta effect, and the cost of re-establishing delta neutrality after the price movement. The re-establishment cost depends on the magnitude of the price movement and the portfolio’s delta before the adjustment. **Example:** Imagine a portfolio consisting of short call options. Initially, the portfolio is delta-neutral by holding the underlying asset. Volatility increases, which hurts the short call positions. Time decay also erodes value. However, the underlying asset price jumps up significantly. The portfolio is now delta-negative, and the manager must sell some of the underlying asset to return to delta neutrality. The sale of the underlying will likely result in a loss, further decreasing the portfolio value.

The question revolves around the concept of hedging currency risk using futures contracts, specifically in the context of a UK-based company importing goods from the US and paying in USD. It tests the understanding of how to calculate the number of futures contracts needed to effectively hedge the currency exposure, considering factors like contract size, spot exchange rates, and the company’s risk tolerance. The calculation involves determining the total USD exposure, converting it to the contract currency (in this case, GBP), and then dividing by the contract size. A key element is understanding that hedging isn’t about eliminating all risk, but rather mitigating the impact of adverse currency movements within a defined risk tolerance. The explanation will also delve into the potential for basis risk (the risk that the spot price and the futures price don’t move perfectly in tandem) and how that impacts the hedging strategy. For instance, imagine a scenario where the spot rate is fluctuating wildly due to unforeseen geopolitical events. The futures market might not fully reflect these fluctuations, creating a discrepancy. A sophisticated understanding of these nuances is crucial for effective risk management. Another crucial concept is the idea of “over-hedging” or “under-hedging”. Over-hedging can protect against currency fluctuations beyond the company’s risk tolerance, but it also limits the potential gains if the exchange rate moves in a favorable direction. Under-hedging, on the other hand, leaves the company exposed to some currency risk, but it also allows them to benefit from favorable movements. The optimal hedging strategy depends on the company’s specific risk appetite and its assessment of the market conditions. The calculation is as follows: 1. **Total USD Exposure:** \$5,000,000 2. **Spot Exchange Rate (GBP/USD):** 1.25 3. **Value of Exposure in GBP:** \(\frac{\$5,000,000}{1.25} = £4,000,000\) 4. **Futures Contract Size:** £50,000 5. **Number of Contracts:** \(\frac{£4,000,000}{£50,000} = 80\) However, the question introduces a risk tolerance level, suggesting the company is willing to accept a certain level of unhedged exposure. If the company is willing to accept a 5% unhedged exposure, then the amount to be hedged is 95% of the total exposure. 6. **Amount to be Hedged:** \(0.95 \times £4,000,000 = £3,800,000\) 7. **Number of Contracts (with risk tolerance):** \(\frac{£3,800,000}{£50,000} = 76\) Therefore, the company should purchase 76 futures contracts to hedge its currency exposure while remaining within its risk tolerance level.

The question assesses the understanding of credit default swaps (CDS) and their pricing mechanisms, specifically how changes in credit spreads and recovery rates impact the CDS spread. The CDS spread is essentially the premium paid to protect against default. A higher credit spread of the underlying asset implies a higher probability of default, thus increasing the CDS spread. Conversely, a higher recovery rate (the amount recovered in the event of default) reduces the loss given default, thereby decreasing the CDS spread. The formula to approximate the CDS spread is: CDS Spread ≈ (Credit Spread) * (1 – Recovery Rate) In this scenario, the initial CDS spread is calculated as: Initial CDS Spread = 4% * (1 – 30%) = 4% * 0.7 = 2.8% Now, the credit spread increases by 1% (from 4% to 5%), and the recovery rate increases by 10% (from 30% to 40%). The new CDS spread is calculated as: New CDS Spread = 5% * (1 – 40%) = 5% * 0.6 = 3.0% The change in the CDS spread is the difference between the new and initial CDS spreads: Change in CDS Spread = 3.0% – 2.8% = 0.2% or 20 basis points. Therefore, the CDS spread increases by 20 basis points. The key here is understanding the inverse relationship between recovery rate and CDS spread, and the direct relationship between credit spread and CDS spread. The example illustrates how these factors interact to determine the price of credit protection.

The question assesses the understanding of hedging strategies using futures contracts, specifically focusing on cross-hedging and basis risk. Cross-hedging involves hedging an asset with a futures contract on a different, but correlated, asset. Basis risk arises because the price movements of the asset being hedged and the futures contract are not perfectly correlated. The optimal number of futures contracts to use in a cross-hedge is determined by the hedge ratio, which minimizes the variance of the hedged portfolio. The hedge ratio is calculated as: Hedge Ratio = Correlation * (Volatility of Asset / Volatility of Futures Contract) In this case, the correlation between the boutique cheese price and the cheddar cheese futures is 0.75. The volatility of the boutique cheese price is 15%, and the volatility of the cheddar cheese futures is 20%. Therefore, the hedge ratio is: Hedge Ratio = 0.75 * (0.15 / 0.20) = 0.5625 Since the cheese maker wants to hedge 100 tonnes of boutique cheese and each cheddar cheese futures contract is for 20 tonnes, the number of futures contracts needed is: Number of Contracts = Hedge Ratio * (Quantity of Asset / Contract Size) Number of Contracts = 0.5625 * (100 tonnes / 20 tonnes per contract) = 2.8125 Since you cannot trade fractions of contracts, the cheese maker should round to the nearest whole number, which is 3 contracts. Basis risk arises because the boutique cheese and cheddar cheese futures prices won’t move perfectly in tandem. This means the hedge will not perfectly eliminate price risk. A positive basis means the spot price of the boutique cheese increases relative to the cheddar futures price, resulting in a loss on the hedge but a gain on the cheese. A negative basis means the spot price of the boutique cheese decreases relative to the cheddar futures price, resulting in a gain on the hedge but a loss on the cheese. The cheese maker must be aware of this residual risk. This scenario tests not just the calculation, but also the understanding of why cross-hedging is used (when a direct hedge isn’t available), how the hedge ratio is derived (minimizing variance), and the implications of basis risk (the hedge is imperfect). The example is unique because it uses boutique cheese, which is not a standard hedging example.

The question assesses understanding of delta hedging, its limitations, and the impact of gamma on hedge effectiveness. Delta hedging aims to neutralize the sensitivity of an option portfolio 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. A delta-neutral portfolio has a delta of zero. However, delta is not constant; it changes as the underlying asset’s price changes. Gamma measures the rate of change of delta with respect to the underlying asset’s price. A high gamma indicates that delta is highly sensitive to changes in the underlying asset’s price, requiring more frequent rebalancing to maintain a delta-neutral position. In this scenario, the portfolio manager initially hedges the short call options by buying shares of the underlying asset. As the underlying asset’s price rises significantly, the delta of the call options increases. Because the portfolio manager did not account for gamma, the delta hedge becomes less effective. The portfolio becomes delta-positive, meaning it will profit if the underlying asset’s price continues to rise and lose if the price falls. The profit from the shares purchased to hedge the short calls is offset by the increasing losses on the short call options, which are now deeply in the money. The net loss reflects the fact that the hedge was not dynamically adjusted to account for the changing delta due to the option’s gamma. To calculate the loss, we need to consider the change in the option’s price and the profit from the hedging asset. The option premium is not relevant to the calculation of the hedge effectiveness after the initial position is established. The initial delta is also not directly used in the final loss calculation as the hedge was not dynamically adjusted. The key is the change in the option’s value due to the price movement of the underlying asset and the profit from the shares held as a hedge. Let’s assume each call option represents one share. The manager sold 100 call options, each representing 100 shares, so a total of 10,000 shares are represented by the options. The share price increased from £100 to £110, a change of £10. The loss on each call option will be close to the intrinsic value change, which is £10, as the option is now deeply in the money. Therefore, the total loss on the call options is approximately 10,000 shares * £10/share = £100,000. The hedge involved buying shares, so the profit on the shares is 10,000 shares * £10/share = £100,000. However, because the hedge was not dynamically adjusted, the portfolio manager will experience a loss, as the delta of the options increased and the hedge was not sufficient to cover the change. The question is designed to test the understanding of dynamic hedging and the impact of gamma. The correct answer is the net loss after accounting for the profit from the shares and the loss from the options.

The question explores the complexities of hedging a portfolio with futures contracts, specifically focusing on the impact of a non-parallel shift in the yield curve on the hedge’s effectiveness. It requires understanding how changes in the yield curve affect the value of both the bond portfolio and the futures contracts used for hedging. The calculation involves determining the initial hedge ratio based on modified duration, then adjusting for the yield curve twist. The key is recognizing that a flattening yield curve will impact short-term and long-term rates differently, affecting the futures contract and the bond portfolio unevenly. First, calculate the initial hedge ratio using the formula: Hedge Ratio = (Portfolio Modified Duration / Futures Modified Duration) * (Portfolio Value / Futures Contract Value) Initial Hedge Ratio = (6.5 / 4.2) * (£50,000,000 / £100,000) = 773.81 Next, consider the impact of the yield curve flattening. A flattening yield curve means short-term rates increase while long-term rates decrease. This affects the value of the bond portfolio and the futures contract. The bond portfolio’s value decreases due to the increase in short-term rates and decreases less from long-term rates, but overall, the effect is negative. The futures contract, linked to a specific maturity, is also affected. Assume the portfolio’s value decreases by 2.8% due to the yield curve shift. The new portfolio value is: New Portfolio Value = £50,000,000 * (1 – 0.028) = £48,600,000 Assume the futures contract price decreases by 1.5% due to the yield curve shift. The new futures contract value is: New Futures Contract Value = £100,000 * (1 – 0.015) = £98,500 Now, calculate the revised hedge ratio: Revised Hedge Ratio = (6.5 / 4.2) * (£48,600,000 / £98,500) = 757.98 The difference between the initial and revised hedge ratios indicates the required adjustment: Adjustment = 773.81 – 757.98 = 15.83 Since the hedge ratio decreased, the fund manager needs to sell 15.83 futures contracts to maintain the hedge. Rounding to the nearest whole number, the manager should sell 16 contracts.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Green Harvest,” which aims to hedge against potential price declines in their upcoming wheat harvest. Green Harvest is exploring the use of futures contracts traded on the ICE Futures Europe exchange. The current futures price for wheat for delivery in six months is £200 per tonne. Green Harvest expects to harvest 5,000 tonnes of wheat in six months. The cooperative’s risk manager anticipates a potential basis risk arising from the difference between the futures price and the local cash price at the time of harvest. To mitigate this basis risk, Green Harvest decides to use a minimum variance hedge ratio. Historical data suggests a correlation of 0.8 between changes in the futures price and changes in the local cash price, with the standard deviation of changes in the cash price being £10 per tonne and the standard deviation of changes in the futures price being £12 per tonne. The minimum variance hedge ratio is calculated as: Hedge Ratio = Correlation * (Standard deviation of cash price changes / Standard deviation of futures price changes). In this case, the hedge ratio = 0.8 * (10/12) = 0.6667. This means Green Harvest should short approximately 66.67% of their expected production in futures contracts. Since each ICE Futures Europe wheat contract is for 100 tonnes, Green Harvest needs to determine the number of contracts to short. Total hedge position = Hedge Ratio * Expected production = 0.6667 * 5,000 tonnes = 3,333.5 tonnes. Number of contracts = Total hedge position / Contract size = 3,333.5 tonnes / 100 tonnes per contract = 33.335 contracts. Since contracts can only be traded in whole numbers, Green Harvest should short 33 contracts.

The question tests understanding of volatility smiles/skews and their implications for option pricing, particularly in the context of exotic options. A volatility smile (or skew) arises when implied volatilities, derived from market prices of options, are plotted against strike prices for options with the same expiration date. In a Black-Scholes world, volatility should be constant across all strikes. However, market realities often show that out-of-the-money puts and calls have higher implied volatilities than at-the-money options, creating a “smile” or “skew” shape. This is often attributed to factors like supply and demand imbalances, fear of market crashes (leading to higher demand for puts), and the inability of the Black-Scholes model to fully capture market dynamics. The presence of a volatility smile directly impacts the pricing of exotic options, particularly those whose payoffs depend on multiple strikes or barrier levels. A standard Black-Scholes model assumes constant volatility, which is incorrect when a smile exists. Using a single implied volatility from an at-the-money option to price an exotic option can lead to significant mispricing. For example, consider a barrier option with a knock-out barrier far below the current asset price. If a volatility skew exists (where out-of-the-money puts have higher implied volatilities), the probability of the barrier being hit is higher than what the Black-Scholes model, using the at-the-money volatility, would predict. Therefore, the barrier option should be priced higher to reflect this increased risk. Conversely, if the barrier is far above the current asset price and a skew exists (where out-of-the-money calls have higher implied volatilities), the barrier option’s price should be adjusted accordingly. In the given scenario, the trader needs to account for the volatility skew when pricing the exotic option. They should not rely solely on the at-the-money implied volatility. Instead, they need to interpolate or extrapolate the implied volatilities from the volatility skew to the relevant strike prices that affect the exotic option’s payoff. This can be done using various techniques, such as fitting a curve to the observed implied volatilities or using a stochastic volatility model. In summary, understanding and correctly accounting for the volatility smile/skew is crucial for accurately pricing and hedging exotic options. Ignoring it can lead to substantial losses.

To determine the fair premium for the catastrophe bond, we need to calculate the expected loss and then add a risk premium to compensate the investor for taking on the risk. First, calculate the expected loss: Expected Loss = (Probability of Hurricane Category 3) * (Loss Given Hurricane Category 3) + (Probability of Hurricane Category 4) * (Loss Given Hurricane Category 4) + (Probability of Hurricane Category 5) * (Loss Given Hurricane Category 5) Expected Loss = (0.10 * £50 million) + (0.05 * £80 million) + (0.02 * £100 million) Expected Loss = £5 million + £4 million + £2 million Expected Loss = £11 million Next, calculate the fair premium by adding the risk premium to the expected loss: Fair Premium = Expected Loss + Risk Premium Fair Premium = £11 million + £4 million Fair Premium = £15 million Therefore, the fair premium for the catastrophe bond is £15 million. The concept being tested here is the valuation of insurance-linked securities, specifically catastrophe bonds. A catastrophe bond transfers specific risks, such as hurricane damage, from an issuer (in this case, a UK-based insurance company) to investors. Investors receive a coupon (premium) in exchange for taking on the risk. If a predefined catastrophic event occurs and causes losses exceeding a specified threshold, the investors may lose part or all of their principal. The expected loss calculation is crucial. It involves weighting the potential loss amounts by their respective probabilities. This is a fundamental risk management technique used not only in insurance but also in broader financial contexts. The risk premium reflects the compensation investors demand for bearing the uncertainty associated with the catastrophe risk. This premium is influenced by factors such as the investor’s risk aversion, the correlation of the catastrophe risk with other investments, and market conditions. A novel aspect here is the application of this valuation in the context of a UK-based insurance company and specific hurricane categories. This differs from standard textbook examples that often use generic probabilities and loss amounts. The question also implicitly touches upon regulatory capital requirements for insurance companies under Solvency II, where such risk transfer mechanisms can play a role in optimizing capital efficiency. The pricing of catastrophe bonds also links to broader concepts of alternative risk transfer and the securitization of insurance risks.

The question assesses understanding of the impact of correlation between assets within a portfolio when using derivatives for hedging. Specifically, it examines how changes in correlation affect the effectiveness of a hedging strategy using options. Here’s the breakdown of the solution: 1. **Understanding the Scenario:** The fund manager is using put options to hedge a portfolio of two assets. Initially, the assets are weakly correlated. The scenario introduces a sudden increase in correlation. We need to determine the impact on the hedging strategy. 2. **Impact of Increased Correlation:** When assets become more correlated, their price movements tend to converge. If one asset declines significantly, the other is more likely to decline as well. This affects the hedging strategy in the following ways: * **Increased Hedging Effectiveness:** With higher correlation, the put options provide a more effective hedge because the portfolio’s overall value is more likely to decline in tandem. The put options will offset a larger portion of the portfolio’s losses. * **Potential for Over-Hedging:** If the correlation becomes very high, the put options might provide more protection than necessary. This is because the portfolio’s decline will be amplified by the higher correlation, and the put options will compensate for this amplified decline. 3. **Gamma and Correlation:** Gamma measures the rate of change of delta with respect to changes in the underlying asset’s price. Higher correlation leads to a more predictable, and therefore, potentially larger movement in the portfolio value for a given move in one of the underlying assets. This translates to a change in the overall portfolio’s gamma. 4. **Calculating the Impact (Illustrative Example):** Let’s assume the portfolio consists of two assets, A and B, each initially valued at £50 million. The fund manager uses put options to hedge against a potential 10% decline. Initially, the correlation between A and B is 0.2. After a market event, the correlation jumps to 0.8. * **Low Correlation Scenario:** With a 10% decline in asset A, asset B might decline by only 2% due to the low correlation. The put options would need to cover a total loss of £5 million (from A) + £1 million (from B) = £6 million. * **High Correlation Scenario:** With a 10% decline in asset A, asset B is likely to decline by 8% due to the high correlation. The put options would now need to cover a total loss of £5 million (from A) + £4 million (from B) = £9 million. This example illustrates that the hedging strategy becomes more effective but also potentially more expensive due to the increased correlation. The portfolio’s gamma also changes, reflecting the increased sensitivity to market movements. Therefore, the most accurate answer is that the hedging strategy becomes more effective, and the portfolio’s gamma increases.

The question concerns the impact of macroeconomic announcements on currency swap valuations. Currency swaps involve exchanging principal and interest payments in different currencies. Macroeconomic announcements, such as inflation reports, GDP figures, and employment data, can significantly impact interest rate expectations and exchange rates, which are crucial components of currency swap valuations. The valuation of a currency swap involves discounting future cash flows (interest payments and principal repayments) using appropriate discount rates derived from the yield curves of the respective currencies. Changes in interest rate expectations directly affect these discount rates. For instance, a higher-than-expected inflation report in the UK might lead to expectations of interest rate hikes by the Bank of England. This would cause the GBP yield curve to shift upwards, increasing the discount rates for GBP cash flows in the swap and potentially decreasing the present value of GBP inflows. Conversely, a weaker-than-expected GDP growth figure in the US might lead to expectations of lower interest rates by the Federal Reserve. This would cause the USD yield curve to shift downwards, decreasing the discount rates for USD cash flows and potentially increasing the present value of USD outflows. Furthermore, macroeconomic announcements can trigger immediate reactions in the spot exchange rate. If the UK inflation report is unexpectedly high, the GBP might appreciate against the USD due to increased investor confidence and expectations of higher returns. This change in the spot exchange rate would affect the present value of the final principal exchange at the swap’s maturity. The net impact on the swap’s value depends on the specific details of the swap (notional amounts, fixed rates, remaining tenor) and the magnitude and direction of the changes in interest rate expectations and exchange rates. To determine the impact, one would need to re-evaluate the swap using updated yield curves and the new spot exchange rate, comparing the new present value to the original present value. The change in value represents the impact of the macroeconomic announcement.

Let’s consider a scenario involving a UK-based agricultural cooperative, “BritCrops,” which wants to protect its future revenue from wheat sales. BritCrops anticipates harvesting 5,000 tonnes of wheat in six months and is concerned about a potential drop in wheat prices due to an expected bumper crop in Eastern Europe. They decide to use futures contracts traded on the London International Financial Futures and Options Exchange (LIFFE) to hedge their price risk. The LIFFE wheat futures contract size is 100 tonnes. To determine the optimal hedging strategy, BritCrops needs to calculate the number of futures contracts required. This involves dividing the total amount of wheat to be hedged by the contract size. In this case, 5,000 tonnes / 100 tonnes per contract = 50 contracts. However, BritCrops also needs to consider basis risk. Basis risk arises because the futures price and the spot price of wheat at the time of harvest may not converge perfectly. This difference, known as the basis, can fluctuate due to factors such as local supply and demand conditions, transportation costs, and storage costs. To estimate the basis risk, BritCrops analyzes historical data and finds that the basis (spot price – futures price) has historically ranged from -£5/tonne to +£3/tonne. This means that the spot price could be £5/tonne lower or £3/tonne higher than the futures price at the time of settlement. BritCrops also considers the impact of storage costs. Storing wheat incurs costs such as warehousing fees, insurance, and potential spoilage. These costs can affect the attractiveness of holding physical wheat versus selling futures contracts. Let’s assume storage costs are estimated at £2/tonne over the six-month period. To refine their hedging strategy, BritCrops considers a “minimum variance hedge ratio.” This ratio aims to minimize the variance of the hedged portfolio by adjusting the number of futures contracts based on the correlation between changes in the spot price and changes in the futures price. If the correlation is high (close to 1), the hedge ratio will be close to 1, meaning a 1:1 hedge is optimal. If the correlation is lower, the hedge ratio will be adjusted accordingly. Suppose BritCrops calculates the minimum variance hedge ratio to be 0.9. This means they should use 0.9 * 50 = 45 futures contracts to minimize risk. Finally, BritCrops must consider margin requirements. LIFFE requires an initial margin deposit for each futures contract to cover potential losses. Suppose the initial margin is £1,000 per contract. BritCrops needs to deposit £1,000 * 45 = £45,000 as initial margin. They also need to monitor their margin account daily and maintain a minimum maintenance margin. If the futures price moves against them, they may receive margin calls, requiring them to deposit additional funds. This example illustrates how a company can use futures contracts to hedge price risk, while also considering basis risk, storage costs, the minimum variance hedge ratio, and margin requirements. It highlights the complexities involved in developing an effective hedging strategy and the importance of understanding the underlying market dynamics.

The question assesses the understanding of hedging strategies using options, specifically a collar strategy, and how changes in volatility affect the profitability of such strategies. A collar involves buying a protective put and selling a covered call, aiming to limit both upside and downside risk. The initial setup cost is crucial; it determines the strategy’s breakeven point. Changes in implied volatility significantly impact option prices (both the put and the call), altering the overall cost and thus the profit potential. Let’s break down the scenario and calculation. Initially, the investor buys a put option at a strike price of 95 for £3 and sells a call option at a strike price of 105 for £2. The net cost of establishing the collar is £3 (put premium paid) – £2 (call premium received) = £1. This £1 represents the initial outlay, which needs to be considered when calculating the final profit or loss. The share price ends at £102. The put option expires worthless because the share price is above the put’s strike price of £95. The call option also expires worthless because the share price is below the call’s strike price of £105. Therefore, there are no gains or losses from the options themselves. However, the initial cost of setting up the collar (£1) must be factored in. Since both options expire worthless, the investor loses the initial £1 cost. Thus, the net profit/loss is -£1. The example highlights how volatility changes can affect the premiums received and paid for options, influencing the overall cost and profitability of the collar strategy. Understanding the interplay between volatility, option premiums, and the underlying asset’s price movement is vital for effective risk management using derivatives. A key takeaway is that a zero net premium collar does not guarantee a zero profit/loss outcome, especially when transaction costs and potential opportunity costs are considered.

The core concept tested here is the understanding of how various Greeks (Delta, Gamma, Vega, Theta) affect a short strangle position, particularly when a significant market event, like an unexpected earnings announcement, occurs. A short strangle involves selling both a call and a put option with different strike prices. The combined effect of these Greeks determines the position’s sensitivity to changes in the underlying asset’s price, volatility, and time decay. Here’s a breakdown of how each Greek impacts the position and how the earnings announcement influences them: * **Delta:** Measures the sensitivity of the option’s price to a change in the underlying asset’s price. A short strangle has a delta that is close to zero when the underlying asset’s price is between the strike prices. However, a large price movement after an earnings announcement can cause the delta to become significantly positive (if the price rises sharply) or negative (if the price falls sharply). * **Gamma:** Measures the rate of change of delta with respect to changes in the underlying asset’s price. A short strangle has a positive gamma. This means that as the price moves further away from the strike prices, the delta will increase (become more positive if the price rises, more negative if the price falls). The earnings announcement, causing a large price swing, will amplify the effect of gamma. * **Vega:** Measures the sensitivity of the option’s price to changes in volatility. A short strangle has a negative vega. This means that if volatility increases, the value of the short strangle will decrease (resulting in a loss). Earnings announcements often lead to increased volatility (implied volatility) as uncertainty about the company’s performance is resolved. * **Theta:** Measures the rate of decline in the value of an option due to the passage of time (time decay). A short strangle has a positive theta. This means that as time passes, the value of the short strangle will decrease (resulting in a profit). However, this effect is generally smaller in the short term compared to the impact of delta, gamma, and vega, especially after a significant event like an earnings announcement. Given the scenario, the company’s stock price *decreases substantially* after the earnings announcement, and implied volatility *increases significantly*. 1. **Delta:** Because the price decreased, the combined delta of the short strangle becomes negative. 2. **Gamma:** The negative delta becomes even *more* negative due to the positive gamma. 3. **Vega:** The increase in implied volatility causes a loss because the strangle has negative vega. 4. **Theta:** Time decay is working in your favor, but the effects of delta, gamma and vega are likely to outweigh the positive theta, at least in the short term. The most significant effects will be from the negative delta (due to the price drop) and the negative vega (due to the volatility increase), amplified by the positive gamma. CALCULATION: This is a conceptual question, so there is no numerical calculation.

The question explores the concept of hedging a portfolio with futures contracts, specifically focusing on the number of contracts needed to achieve a desired beta. Beta measures the volatility of a portfolio relative to the market. To reduce a portfolio’s beta, shorting futures contracts on a market index is a common strategy. The number of contracts required depends on the portfolio’s current beta, the desired beta, the portfolio’s value, the futures contract’s price, and the contract’s beta. First, calculate the required beta reduction: Desired Beta Reduction = Current Beta – Target Beta. Next, determine the total notional value of futures contracts needed to achieve the desired beta reduction: Notional Value = (Beta Reduction * Portfolio Value) / Futures Beta. Finally, calculate the number of futures contracts: Number of Contracts = Notional Value / (Futures Price * Multiplier). In this specific case, the portfolio value is £5,000,000, the current beta is 1.2, and the target beta is 0.6. The FTSE 100 futures contract is priced at £7,500 with a multiplier of 10, and its beta is assumed to be 1. Beta Reduction = 1.2 – 0.6 = 0.6 Notional Value = (0.6 * £5,000,000) / 1 = £3,000,000 Number of Contracts = £3,000,000 / (£7,500 * 10) = 40 Therefore, the portfolio manager should short 40 FTSE 100 futures contracts to reduce the portfolio’s beta to 0.6. This calculation assumes a linear relationship between the number of contracts and the beta reduction, and it doesn’t account for transaction costs or margin requirements. In reality, the manager would need to continuously monitor and adjust the hedge as market conditions change and the portfolio’s composition evolves. Furthermore, the effectiveness of the hedge relies on the accuracy of the beta estimates and the correlation between the portfolio and the FTSE 100.

Let’s analyze the impact of a sudden, unexpected shift in implied volatility on a short strangle option strategy. A short strangle involves selling both an out-of-the-money call option and an out-of-the-money put option on the same underlying asset, with the expectation that the asset price will remain within a defined range. The strategy profits from time decay (theta) and, typically, from a decrease in implied volatility. However, a significant increase in implied volatility can drastically alter the profit and loss profile. Consider an investor, Anya, who has implemented a short strangle on shares of “NovaTech,” a technology company. The stock is currently trading at £150. Anya sells a call option with a strike price of £160 for a premium of £3 and sells a put option with a strike price of £140 for a premium of £2. Her maximum profit is the combined premium received, £5. The breakeven points are £140 – £5 = £135 on the downside and £160 + £5 = £165 on the upside. Now, suppose NovaTech is rumored to be on the verge of announcing a major breakthrough in artificial intelligence, significantly increasing the stock’s potential volatility. The implied volatility on both the call and put options spikes dramatically. This volatility increase directly impacts the value of Anya’s short options positions. The value of both the call and put options increases, creating a potential loss for Anya. The increase in option prices due to the volatility spike erodes the initial premium received. Let’s quantify this. Assume the implied volatility on both options increases by 20%. Using option pricing models (like Black-Scholes), this volatility increase translates to the call option price rising from £3 to £7 and the put option price rising from £2 to £5. Anya now faces a loss of (£7 – £3) + (£5 – £2) = £7 per share. This loss significantly reduces the profit zone of the strangle and increases the risk of substantial losses if the stock price moves significantly in either direction. This example illustrates how a sudden and unexpected volatility spike can adversely affect a short strangle strategy, even if the stock price remains relatively stable. The key takeaway is that short volatility strategies, like strangles, are highly susceptible to volatility risk (vega). Effective risk management requires continuous monitoring of implied volatility and implementing hedging strategies if necessary.

The question assesses the understanding of delta hedging, gamma, and how changes in the underlying asset’s price and time to expiration affect a delta-hedged portfolio. Delta represents the sensitivity of an option’s price to a change in the underlying asset’s price, while gamma represents the rate of change of delta with respect to the underlying asset’s price. A delta-hedged portfolio aims to maintain a delta of zero, neutralizing the portfolio’s sensitivity to small changes in the underlying asset’s price. However, this hedge needs continuous adjustment because delta changes as the underlying asset’s price and time to expiration change. The portfolio’s gamma indicates how much the delta will change for a given change in the underlying asset’s price. Theta represents the sensitivity of the option’s price to the passage of time. A negative theta means the option’s value decreases as time passes, all other factors being equal. In this scenario, the portfolio is initially delta-hedged. A significant increase in the underlying asset’s price will cause the delta of the call options to increase (become more positive) because call options benefit from price increases. Since the portfolio was initially delta-hedged, this increase in delta means the portfolio now has a positive delta. To re-establish the delta hedge, the portfolio manager must sell additional units of the underlying asset to reduce the portfolio’s delta back to zero. The time decay, represented by theta, constantly erodes the value of the call options. The profit/loss calculation can be simplified as follows: The portfolio’s initial delta is zero. The increase in the underlying asset’s price causes the delta to become positive. To re-hedge, the manager sells the underlying asset, which results in a loss because the asset price subsequently declines. This loss is compounded by the negative theta, which erodes the value of the call options. Therefore, the combined effect of the price increase, subsequent decline after re-hedging, and time decay results in a loss for the portfolio.

The question assesses understanding of the interplay between credit default swaps (CDS), bond prices, and recovery rates, and how changes in one affect the others. The core concept is that a CDS provides insurance against the default of a reference entity (here, “AgriCorp”). If the market perceives an increased risk of AgriCorp defaulting, the CDS spread (the premium paid for the insurance) widens. This increased spread reflects the higher cost of insuring against AgriCorp’s potential default. Simultaneously, the price of AgriCorp’s bonds will likely decrease because investors demand a higher yield to compensate for the increased credit risk. The recovery rate is the percentage of the bond’s face value that investors expect to receive in the event of default. The formula to approximate the change in bond price due to changes in the CDS spread and recovery rate is: Change in Bond Price ≈ – (Change in CDS Spread) * (Maturity of CDS) + (Change in Recovery Rate) In this scenario, the CDS spread widens by 50 basis points (0.5%) and the recovery rate decreases by 10% (0.10). Let’s assume a simplified scenario where the maturity of the CDS is 1 year to illustrate the calculation: Change in Bond Price ≈ – (0.005) * (1) + (0.10) Change in Bond Price ≈ -0.005 + 0.10 = 0.095 Change in Bond Price = 9.5% Therefore, the bond price is expected to increase by 9.5% The decrease in bond price is due to the increased CDS spread which indicates a higher risk of default. The increase in bond price is due to the increased recovery rate which indicates a higher amount of money that investors expect to receive in the event of default. The correct answer will reflect the combined effect of these two changes. Incorrect answers might only consider one factor or misinterpret the relationship between the CDS spread, bond price, and recovery rate.

The core of this question lies in understanding how implied volatility affects option prices, and how that translates to the profitability of a specific option strategy, the strangle. A strangle involves buying both an out-of-the-money call and an out-of-the-money put with the same expiration date. The investor profits if the underlying asset price moves significantly in either direction. Implied volatility is the market’s expectation of how much the underlying asset price will fluctuate. Here’s the breakdown of how to approach this problem: 1. **Initial Outlay:** Calculate the total cost of setting up the strangle: Call premium + Put premium = £3.50 + £4.00 = £7.50. This is the maximum loss if the stock price stays within the range defined by the strike prices of the options at expiration. 2. **Breakeven Points:** Determine the breakeven points for the strangle strategy. * *Upper Breakeven Point:* Call Strike Price + Total Premium Paid = £110 + £7.50 = £117.50 * *Lower Breakeven Point:* Put Strike Price – Total Premium Paid = £90 – £7.50 = £82.50 3. **Profit/Loss Calculation:** At expiration, the profit or loss depends on where the stock price lands: * *Scenario 1: Stock Price at £120:* The call option is in the money with an intrinsic value of £120 – £110 = £10. The put option expires worthless. Profit = £10 – £7.50 (initial outlay) = £2.50. * *Scenario 2: Stock Price at £80:* The put option is in the money with an intrinsic value of £90 – £80 = £10. The call option expires worthless. Profit = £10 – £7.50 (initial outlay) = £2.50. * *Scenario 3: Stock Price at £100:* Both options expire worthless. Loss = £7.50 (initial outlay). 4. **Impact of Implied Volatility Change:** An increase in implied volatility *after* the strangle is established generally benefits the strategy. This is because higher volatility suggests a greater likelihood of the stock price moving beyond the breakeven points. However, the question specifies that the volatility increase occurs *after* the strangle is established, but *before* expiration. This detail is crucial. 5. **Final Assessment:** The question asks for the *most likely* outcome. While a volatility spike could lead to greater profit, it’s not guaranteed. The calculated profit/loss at the given stock prices provides a more concrete answer. Therefore, at a stock price of £120 or £80, the profit is £2.50. At a stock price of £100, the loss is £7.50. The volatility increase does not change these calculations, as it occurs *after* the options were purchased and *before* expiration.

The question assesses the understanding of hedging strategies using options, specifically focusing on creating a synthetic put option using a combination of short stock and long call options. The payoff structure of a synthetic put is designed to mimic the payoff of a standard put option, providing downside protection. To determine the profit/loss at expiration, we need to analyze the combined payoffs of the short stock position and the long call option. The investor shorts 100 shares of stock at £50, receiving £5000. They also buy 100 call options with a strike price of £50 at a premium of £5 each, costing £500. If the stock price at expiration is below £50, the call options expire worthless. The investor’s profit from the short stock position is £50 – Stock Price at Expiration (per share), and they lose the £5 premium per share paid for the call options. If the stock price at expiration is above £50, the call options are exercised. The investor’s loss from the short stock position is Stock Price at Expiration – £50 (per share), but they gain Stock Price at Expiration – £50 from exercising the call options. The £5 premium per share paid for the call options is still a loss. In this scenario, the stock price rises to £60. The investor loses £10 per share on the short stock position (Stock Price – Initial Price = £60 – £50 = £10 loss per share). However, the investor gains £10 per share from exercising the call options (Stock Price – Strike Price = £60 – £50 = £10 gain per share). The net effect of the stock and call option positions is zero profit/loss. However, we must deduct the £5 premium paid for each call option. Total loss = Call option cost = £5 * 100 = £500 Therefore, the investor’s overall profit/loss is a loss of £500.

The core of this question lies in understanding how changes in correlation affect portfolio Value at Risk (VaR). VaR estimates the potential loss in value of a portfolio over a specific time period for a given confidence level. When assets within a portfolio are perfectly correlated (correlation = 1), the portfolio’s VaR is simply the sum of the individual asset VaRs. However, when assets are less than perfectly correlated, diversification benefits reduce the overall portfolio VaR. The formula to calculate portfolio VaR with correlation is: \[VaR_{portfolio} = \sqrt{VaR_A^2 + VaR_B^2 + 2 \cdot \rho_{AB} \cdot VaR_A \cdot VaR_B}\] Where: \(VaR_A\) is the VaR of Asset A \(VaR_B\) is the VaR of Asset B \(\rho_{AB}\) is the correlation between Asset A and Asset B In this scenario, we need to calculate the portfolio VaR under the new correlation and then determine the change in VaR compared to the initial perfectly correlated scenario. First, we calculate the portfolio VaR under perfect correlation (\(\rho_{AB}\) = 1): \[VaR_{portfolio, perfect} = VaR_A + VaR_B = £50,000 + £30,000 = £80,000\] Next, we calculate the portfolio VaR with the new correlation (\(\rho_{AB}\) = 0.3): \[VaR_{portfolio, 0.3} = \sqrt{50000^2 + 30000^2 + 2 \cdot 0.3 \cdot 50000 \cdot 30000}\] \[VaR_{portfolio, 0.3} = \sqrt{2500000000 + 900000000 + 900000000}\] \[VaR_{portfolio, 0.3} = \sqrt{4300000000} \approx £65,574.38\] Finally, we calculate the change in VaR: \[Change\ in\ VaR = VaR_{portfolio, perfect} – VaR_{portfolio, 0.3} = £80,000 – £65,574.38 \approx £14,425.62\] Therefore, the portfolio VaR decreases by approximately £14,425.62 due to the change in correlation. This demonstrates the risk reduction benefit of diversification, where lower correlation between assets leads to a lower overall portfolio risk (as measured by VaR). The implications for an investment advisor are significant. Understanding correlation is crucial when constructing portfolios to manage and mitigate risk effectively. Ignoring correlation can lead to underestimation of portfolio risk, potentially exposing clients to unexpected losses.

The question assesses understanding of the impact of implied volatility on option pricing and strategy selection, specifically in the context of earnings announcements. The scenario involves a complex interaction of factors: implied volatility changes, the direction of the underlying asset’s price movement, and the specific characteristics of a short strangle strategy. The key to answering correctly lies in recognizing that a short strangle profits from time decay and stable underlying prices. However, earnings announcements often lead to significant price swings. An increase in implied volatility *before* the announcement inflates option prices, making the strangle more expensive to establish. If the stock price moves *significantly* in either direction *after* the announcement, both the call and put options in the strangle could move into the money, resulting in a substantial loss. The investor benefits only if the price remains within the break-even points. The investor’s decision to close the position early, *after* observing the volatility spike but *before* the earnings announcement, locks in a loss. This loss is primarily due to the increased cost of buying back the options (both call and put) at higher implied volatility levels. The calculation is as follows: 1. **Initial Premium Received:** £3.50 (call) + £2.80 (put) = £6.30 per share. For 10 contracts (1,000 shares): £6.30 * 1000 = £6,300. 2. **Premium Paid to Close:** £5.10 (call) + £4.20 (put) = £9.30 per share. For 10 contracts (1,000 shares): £9.30 * 1000 = £9,300. 3. **Net Loss:** £9,300 (paid) – £6,300 (received) = £3,000. Therefore, the investor incurs a loss of £3,000. The example highlights the risks associated with short volatility strategies around earnings announcements and the importance of understanding implied volatility dynamics. A similar analogy would be betting on a horse race where the odds suddenly shorten (increased implied volatility) just before the race, making it less profitable to bet and more costly to exit your position.

The question explores the complexities of hedging a portfolio with options, specifically focusing on the impact of implied volatility on the hedge ratio and the subsequent adjustments required to maintain a delta-neutral position. The core concept tested is understanding how changes in implied volatility affect option prices and, consequently, the number of options needed to offset the portfolio’s delta. The initial delta of the portfolio is calculated as the sum of the delta of the underlying asset and the delta of the options. A long position in the underlying asset contributes a positive delta, while short call options contribute a negative delta. The hedge ratio, which represents the number of options needed to neutralize the portfolio’s delta, is determined by dividing the negative of the underlying asset’s delta by the delta of a single option. A change in implied volatility directly impacts the delta of the options. Higher implied volatility generally increases the delta of out-of-the-money call options and decreases the delta of in-the-money call options. Since the portfolio is short call options, an increase in implied volatility will further decrease the delta of the short call options, making the overall portfolio more sensitive to changes in the underlying asset’s price. To maintain a delta-neutral position after an increase in implied volatility, the investor must adjust the number of options to compensate for the change in the option’s delta. The adjustment involves calculating the new hedge ratio based on the updated option delta and then either buying or selling additional options to bring the portfolio back to a delta of zero. For example, consider a portfolio consisting of 100 shares of a stock with a delta of 1 per share (total delta of 100) and short 5 call options, each with an initial delta of 0.5 (total delta of -2.5). The initial hedge ratio is calculated as -100 / -2.5 = 40 options per share. If the implied volatility increases, causing the delta of each option to decrease to 0.4, the new hedge ratio becomes -100 / -2.0 = 50 options per share. To maintain a delta-neutral position, the investor would need to sell additional call options. The correct answer reflects the change in the number of options required to maintain delta neutrality after the increase in implied volatility, considering the initial position and the new option delta. The incorrect answers represent plausible but flawed calculations or misunderstandings of the impact of implied volatility on option deltas and hedge ratios.

The core of this question revolves around understanding the impact of implied volatility on option prices, particularly in the context of earnings announcements. Earnings announcements are pivotal events that often lead to significant price movements in a company’s stock. Before an earnings announcement, the market anticipates potential volatility, leading to an increase in implied volatility for options on that stock. This increase in implied volatility directly translates to higher option premiums, as options become more valuable when the underlying asset is expected to fluctuate significantly. After the earnings announcement, the uncertainty surrounding the event typically diminishes. If the announcement is broadly in line with expectations (or if the market has already priced in the potential outcomes), the actual volatility observed in the stock price may be lower than what was implied before the announcement. This leads to a decrease in implied volatility, often referred to as “volatility crush.” As implied volatility decreases, the value of options, especially those close to expiration, decreases rapidly, even if the stock price moves favorably. The question requires candidates to understand not just the theoretical relationship between implied volatility and option prices, but also the practical implications of this relationship in a real-world scenario involving earnings announcements. The scenario involves both a long call option and a short put option, requiring the candidate to consider the combined impact of volatility crush on both positions. The correct strategy involves hedging against the anticipated volatility crush by employing strategies like a short straddle or strangle after the earnings announcement, effectively profiting from the decrease in implied volatility. The calculation is complex because it involves assessing the impact of both positive price movement and volatility crush on the combined option positions. The question tests the candidate’s ability to synthesize knowledge of options pricing, implied volatility, and earnings announcement dynamics to make informed trading decisions.