Derivatives Level 4 (Investment Advice Diploma) Exam Quiz Quiz 27

The question assesses understanding of exotic derivatives, specifically a cliquet option, and its valuation considering a reset feature. The calculation involves understanding how the reset mechanism impacts the overall payoff. A cliquet option provides a series of capped returns over different periods. The reset feature adjusts the starting point for the subsequent return calculation based on the current level of the underlying asset. Let’s break down the valuation: 1. **Period 1 Return:** The underlying asset increases from 100 to 110. Return = (110-100)/100 = 10%. This is below the cap of 8%, so the period return is 10%. 2. **Reset:** The underlying is now at 110. The next period’s return will be based on this new level. 3. **Period 2 Return:** The underlying asset decreases from 110 to 105. Return = (105-110)/110 = -4.55%. Since the return is negative, it’s compared against the floor of -2%. The actual return is -2%. 4. **Reset:** The underlying is now at 105. The next period’s return will be based on this new level. 5. **Period 3 Return:** The underlying asset increases from 105 to 115. Return = (115-105)/105 = 9.52%. This is above the cap of 8%, so the period return is 8%. 6. **Reset:** The underlying is now at 115. The next period’s return will be based on this new level. 7. **Period 4 Return:** The underlying asset increases from 115 to 120. Return = (120-115)/115 = 4.35%. This is below the cap of 8%, so the period return is 4.35%. 8. **Total Return:** Sum the returns from each period: 10% + (-2%) + 8% + 4.35% = 20.35% 9. **Payoff:** The payoff of the cliquet option is the initial investment multiplied by (1 + total return). Payoff = 1,000 * (1 + 0.2035) = £1,203.50 Therefore, the final payoff of the cliquet option is £1,203.50. The reset feature significantly impacts the returns, as it uses the previous period’s ending level as the new starting point, rather than the initial level. This makes cliquet options path-dependent derivatives. Understanding these path-dependent features is crucial in valuing and managing the risk associated with these exotic derivatives. The capped nature of the returns also limits potential gains, but the floor provides a level of downside protection.

The core concept being tested here is the impact of volatility on option pricing, specifically within the context of exotic options and their sensitivity to changes in the volatility smile. A volatility smile represents the implied volatility of options with the same expiration date but different strike prices. In real-world markets, the volatility smile is rarely flat; it often exhibits a skew or smile shape, reflecting market expectations about future price movements. Exotic options, such as barrier options, are particularly sensitive to the shape of the volatility smile because their payoff depends on the underlying asset reaching a specific barrier level. A knock-out barrier option ceases to exist if the underlying asset price touches the barrier level. Therefore, the probability of hitting the barrier, and thus the option’s value, is directly influenced by the volatility of the underlying asset. If the volatility smile steepens, it indicates that out-of-the-money (OTM) options are becoming more expensive relative to at-the-money (ATM) options. This often suggests increased uncertainty about future price movements, especially in the tails of the distribution. In the scenario presented, the knock-out barrier is set below the current asset price. A steepening volatility smile implies that the probability of the asset price falling to or below the barrier level has increased. This increased probability reduces the value of the knock-out barrier option because there’s a higher chance that the option will be knocked out before it can reach its full potential payoff. Conversely, if the volatility smile flattens, the probability of hitting the barrier decreases, increasing the value of the knock-out barrier option. The question also requires an understanding of the regulatory environment. Under MiFID II, firms must provide best execution, which includes considering the impact of volatility smiles on option pricing. A failure to account for the changing volatility smile when advising a client on a barrier option could be considered a breach of best execution. The FCA would likely scrutinize such a scenario, especially if the client suffered a loss as a direct result of the firm’s negligence in assessing the volatility smile’s impact.

The question assesses understanding of put-call parity and its violation in the presence of transaction costs and early exercise rights on American options. Put-call parity is a no-arbitrage relationship that links the prices of a European call and put option with the same strike price and expiration date. The formula is: \(C + PV(X) = P + S\), where \(C\) is the call option price, \(PV(X)\) is the present value of the strike price, \(P\) is the put option price, and \(S\) is the spot price of the underlying asset. Transaction costs widen the arbitrage band. If the observed prices deviate from the parity relationship by more than the transaction costs, an arbitrage opportunity exists. Early exercise rights on American options also affect the parity relationship, making it an inequality rather than a strict equality. In this scenario, the investor must compare the potential profit from the arbitrage strategy with the transaction costs to determine if the opportunity is worthwhile. The present value of the strike price is calculated using continuous compounding: \(PV(X) = Xe^{-rT}\), where \(X\) is the strike price, \(r\) is the risk-free rate, and \(T\) is the time to expiration. Here’s how to determine if an arbitrage opportunity exists and the potential profit: 1. **Calculate the present value of the strike price:** \(PV(X) = 105 \times e^{-0.05 \times 0.25} = 105 \times e^{-0.0125} \approx 105 \times 0.9876 = 103.70\) 2. **Check for put-call parity violation:** \(C + PV(X) = 8 + 103.70 = 111.70\) \(P + S = 5 + 104 = 109\) 3. **Determine the arbitrage strategy:** Since \(C + PV(X) > P + S\), the investor should buy the put and the underlying asset, and sell the call option. 4. **Calculate the profit before transaction costs:** Profit = \(C + PV(X) – (P + S) = 111.70 – 109 = 2.70\) 5. **Account for transaction costs:** Total transaction costs = \(0.10 + 0.10 + 0.10 = 0.30\) 6. **Calculate the net profit:** Net profit = Profit – Transaction costs = \(2.70 – 0.30 = 2.40\) Therefore, the investor can realize a net profit of £2.40 by executing the arbitrage strategy.

The core of this question revolves around understanding how a delta-neutral portfolio reacts to changes in volatility (vega) and time decay (theta), and how these reactions influence rebalancing decisions, especially when considering transaction costs. 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 represents the portfolio’s sensitivity to changes in the underlying asset’s volatility, while theta represents the portfolio’s sensitivity to the passage of time. When volatility increases, a delta-neutral portfolio with positive vega will become long delta (i.e., its value will increase if the underlying asset’s price increases). To maintain delta neutrality, the portfolio manager must sell some of the underlying asset or buy offsetting options. Conversely, when volatility decreases, the portfolio becomes short delta, and the manager must buy the underlying asset or sell offsetting options. Theta, on the other hand, constantly erodes the value of options as time passes. A delta-neutral portfolio typically has negative theta (especially if it involves short options positions), meaning its value decreases as time goes by. Transaction costs are a critical consideration. Each rebalancing trade incurs costs (brokerage fees, bid-ask spread). Frequent rebalancing to perfectly maintain delta neutrality can erode profits. The manager must weigh the benefits of precise hedging against the costs of trading. In this scenario, the portfolio starts delta-neutral, has positive vega, and negative theta. A volatility spike causes the portfolio to become long delta, necessitating a trade to re-establish delta neutrality. However, the negative theta is also causing a continuous, albeit smaller, loss in value. The optimal strategy balances the need to hedge against volatility changes with the desire to minimize transaction costs, while also considering the portfolio’s theta decay. Here’s the calculation: 1. **Initial State:** Delta-neutral, Vega = 15, Theta = -3 2. **Volatility Spike:** Volatility increases by 2%, causing a delta change of 15 \* 2 = 30 (portfolio becomes long 30 deltas). 3. **Rebalancing Trade:** To restore delta neutrality, the manager needs to sell 30 units of the underlying asset. 4. **Transaction Cost:** Each unit costs £0.50 to trade, so selling 30 units costs 30 \* £0.50 = £15. 5. **Theta Decay:** Over the next week (5 trading days), theta decay is 5 \* -3 = -£15. 6. **Total Cost of Perfect Hedge:** Transaction cost + Theta decay = £15 + £15 = £30. 7. **Partial Hedge (50%):** Hedge only half of the delta exposure (15 deltas). 8. **Transaction Cost (Partial):** 15 \* £0.50 = £7.50. 9. **Remaining Delta Exposure:** 15 deltas. If the underlying moves significantly against the unhedged position, the portfolio will suffer a loss. 10. **Scenario Analysis:** The question requires an assessment of the trade-off between cost and risk. A perfect hedge costs £30. A partial hedge costs £7.50, but leaves the portfolio exposed to delta risk. The best answer balances these considerations.

Let’s analyze a scenario involving a bespoke structured note linked to the performance of a basket of ESG-focused equities and a currency swap. The note promises a return of principal plus a participation rate in the upside of the equity basket, capped at a certain level, but also incorporates a currency swap to hedge against fluctuations between GBP and EUR. The investor needs to understand the combined impact of equity performance, currency movements, and the structured note’s payoff profile. Suppose the structured note has a principal of £1,000,000. The equity basket participation rate is 70%, capped at a 15% total return. A GBP/EUR currency swap is embedded, where the initial exchange rate is 1.15 GBP/EUR. The swap is structured such that any appreciation of the EUR against the GBP beyond 5% negatively impacts the investor’s return, effectively reducing the participation rate by 2% for every 1% increase in EUR strength above the 5% threshold, but no lower than 0%. First, calculate the maximum potential return from the equity basket: 15% of £1,000,000 = £150,000. Now, consider the currency swap. If EUR appreciates by 8% against GBP, it exceeds the 5% threshold by 3%. This triggers a reduction in the participation rate of 3% * 2% = 6%. The effective participation rate becomes 70% – 6% = 64%. Next, let’s assume the equity basket performs exceptionally well, generating a 25% return. However, due to the cap, the return is limited to 15%. The initial return before currency adjustment is £150,000. The EUR appreciation of 8% reduces the participation rate to 64%. However, the cap remains, so the return is still capped at £150,000. Finally, calculate the net return after considering the currency swap’s impact. The effective participation rate is 64%. Since the cap was reached, the investor receives the capped return of £150,000. Therefore, the total return is the principal of £1,000,000 plus the capped return of £150,000, totaling £1,150,000. This example demonstrates the intricate interplay between equity performance, embedded currency swaps, participation rates, and return caps in structured notes. It highlights the importance of understanding how these components interact to determine the final investor payoff.

The question explores the complexities of managing a portfolio using options strategies, specifically focusing on hedging against potential market downturns while simultaneously generating income. It tests the understanding of covered call and protective put strategies, their payoffs, and the implications of combining them. The scenario involves a portfolio manager, Anya, who needs to optimize her strategy based on her market outlook and risk tolerance. The correct answer requires calculating the net profit/loss of the combined strategy under a specific market condition (market decline). This involves understanding the individual payoffs of the covered call and protective put, and then netting them against the initial costs and premiums. The incorrect answers are designed to trap candidates who may miscalculate the option payoffs, misunderstand the combined effect of the strategies, or incorrectly apply the premiums. Here’s a breakdown of the calculation for the correct answer: 1. **Covered Call:** Anya sells a call option with a strike price of 160 for a premium of £8. Since the market declines to £150, the call option expires worthless. Anya keeps the £8 premium. 2. **Protective Put:** Anya buys a put option with a strike price of 155 for a premium of £5. Since the market declines to £150, the put option is exercised. The payoff is Strike Price – Market Price = £155 – £150 = £5. 3. **Initial Portfolio:** The initial value of the portfolio is £158 per share. The market declines to £150, resulting in a loss of £8 per share. 4. **Net Profit/Loss:** Net Loss = Portfolio Loss + Call Premium + Put Payoff – Put Premium = -£8 + £8 + £5 – £5 = £0 Therefore, the combined strategy results in no profit or loss for Anya. The question goes beyond basic definitions by requiring candidates to apply their knowledge to a realistic scenario, calculate payoffs under different conditions, and understand the combined effect of multiple strategies. It also tests their ability to account for premiums and initial portfolio values in determining the overall profit/loss.

The question assesses understanding of exotic derivatives, specifically a barrier option, and its behavior relative to the underlying asset’s price movement and the knock-out barrier. It requires knowledge of how the option’s value changes as the underlying asset approaches and potentially breaches the barrier. The calculation focuses on determining the potential profit/loss given a specific scenario. The key to answering this question is understanding that a knock-out call option becomes worthless if the underlying asset’s price touches or exceeds the barrier *before* the option’s expiration. The holder only benefits if the asset price increases *without* hitting the barrier. In this case, the barrier is at 160. The initial underlying asset price is 140. The call option has a strike price of 150. The option is purchased for £8. Scenario 1: The asset price rises to 155, then falls to 145. The barrier has not been breached. However, the option is now £5 in the money (150 strike vs 155 asset price). If the option is exercised, the profit before considering the premium is £5. Subtracting the £8 premium results in a loss of £3. Scenario 2: The asset price rises to 165. The barrier has been breached. The option is knocked out and becomes worthless. The holder loses the entire premium of £8. Scenario 3: The asset price rises to 158, then falls to 152 at expiration. The barrier has not been breached. The option is £2 in the money (150 strike vs 152 asset price). If the option is exercised, the profit before considering the premium is £2. Subtracting the £8 premium results in a loss of £6. Scenario 4: The asset price rises to 152, then falls to 148 at expiration. The barrier has not been breached. The option is out of the money. The option is worthless and the holder loses the entire premium of £8. Therefore, the most accurate answer reflects the loss of the premium when the barrier is breached.

The question explores the application of put-call parity, a fundamental concept in options pricing, within a context complicated by transaction costs and dividend payments. Put-call parity establishes a relationship between the prices of a European call option, a European put option, a risk-free asset, and the underlying asset. The basic formula is: \(C + PV(K) = P + S\), where C is the call option price, PV(K) is the present value of the strike price, P is the put option price, and S is the spot price of the underlying asset. However, real-world scenarios often deviate from this idealized model. Transaction costs (brokerage fees, bid-ask spreads) and dividends paid during the option’s life introduce complexities. Transaction costs widen the arbitrage-free range, making it less profitable to exploit deviations from parity. Dividends reduce the stock price, impacting the call and put option prices differently. To incorporate these factors, we need to adjust the put-call parity equation. The present value of dividends (PV(Div)) must be subtracted from the stock price side of the equation: \(C + PV(K) = P + S – PV(Div)\). Transaction costs (TC) effectively create a no-arbitrage band. If the cost of establishing the arbitrage trade exceeds the potential profit, the mispricing can persist. Thus, the equation becomes: \(C + PV(K) = P + S – PV(Div) \pm TC\). The plus or minus sign depends on the direction of the arbitrage trade. In this case, we are looking for the maximum price of the call option. The cost of buying/selling the shares, put option, and borrowing or lending cash will increase the upper bound for the call option price. Given the provided data, we first calculate the present value of the strike price: \(PV(K) = \frac{105}{e^{(0.05 \times 0.5)}} = \frac{105}{1.0253} = 102.41\). Next, we calculate the present value of the dividends: \(PV(Div) = \frac{2.00}{e^{(0.05 \times 0.25)}} + \frac{2.00}{e^{(0.05 \times 0.5)}} = \frac{2.00}{1.0126} + \frac{2.00}{1.0253} = 1.975 + 1.951 = 3.926\). The transaction cost is given as £0.75. Now, we rearrange the put-call parity equation to solve for the maximum call price: \(C = P + S – PV(Div) – PV(K) + TC\). Plugging in the values: \(C = 5 + 100 – 3.926 – 102.41 + 0.75 = -0.586\). This negative result indicates that the theoretical call price would be negative to prevent arbitrage. However, option prices cannot be negative. In this scenario, the maximum price of the call option is effectively capped by the cost of arbitrage. Therefore, we must consider the cost of arbitrage in the reverse direction. \(C = P + S – PV(Div) – PV(K) – TC\) implies the *minimum* call price. For the *maximum* call price, the equation becomes \(C + PV(K) + TC = P + S – PV(Div)\), so \(C = P + S – PV(Div) – PV(K) – TC\), meaning we subtract transaction costs. Then \(C = 5 + 100 – 3.926 – 102.41 – 0.75 = -2.086\). Since the call price cannot be negative, the market price of the call option is capped at 0.

The question assesses the understanding of implied volatility, delta, and their combined impact on option pricing, especially in the context of significant market events. Delta represents the sensitivity of an option’s price to changes in the underlying asset’s price. Implied volatility reflects the market’s expectation of future price volatility of the underlying asset; higher implied volatility generally increases option prices, as it suggests a greater chance of the option moving into the money. The scenario involves a short call option, meaning the investor profits if the underlying asset price stays below the strike price. A sudden increase in implied volatility increases the option’s price, causing a loss for the short call position. The delta of a call option is positive and ranges from 0 to 1. If the delta is 0.6, it means that for every £1 increase in the underlying asset’s price, the option price is expected to increase by £0.60. Conversely, if the underlying asset price decreases by £1, the option price is expected to decrease by £0.60. To calculate the approximate loss, we need to consider both the change in implied volatility and the change in the underlying asset’s price. The initial option price is £4. The implied volatility increases, causing the option price to increase to £6. This results in an initial loss of £2 per option. Then, the underlying asset’s price increases by £5. Given a delta of 0.6, this causes the option price to further increase by approximately 0.6 * £5 = £3. Therefore, the total increase in the option price is £2 (due to volatility) + £3 (due to the underlying asset’s price change) = £5. The initial option price was £4, so the new option price is £4 + £5 = £9. The loss is the difference between the new option price and the initial option price, which is £9 – £4 = £5 per option. For 100 options, the total loss is £5 * 100 = £500.

The question assesses understanding of the impact of transaction costs on hedging strategies, specifically when using futures contracts. The calculation involves determining the number of futures contracts needed to hedge a portfolio, then calculating the total transaction costs (brokerage fees) associated with establishing and closing out that hedge. The optimal hedging strategy must consider these costs. First, calculate the number of futures contracts needed: Portfolio Value = £5,000,000 Futures Contract Value = £100,000 Beta of Portfolio = 1.2 Hedge Ratio = Portfolio Beta = 1.2 Number of Contracts = (Portfolio Value / Futures Contract Value) * Hedge Ratio Number of Contracts = (£5,000,000 / £100,000) * 1.2 = 50 * 1.2 = 60 contracts Next, calculate the total transaction costs: Brokerage Fee per contract (round trip) = £25 Total Transaction Costs = Number of Contracts * Brokerage Fee Total Transaction Costs = 60 * £25 = £1500 Now, consider the percentage impact of these costs on the portfolio: Percentage Impact = (Total Transaction Costs / Portfolio Value) * 100 Percentage Impact = (£1500 / £5,000,000) * 100 = 0.03% The key takeaway is understanding that transaction costs, even if seemingly small on a per-contract basis, can accumulate and noticeably affect the overall performance of a hedging strategy, especially for large portfolios. Consider a scenario where the expected return on the hedged portfolio is only 0.05%. In this case, transaction costs of 0.03% would significantly erode the profitability of the hedge. Another important aspect is the frequency of adjustments to the hedge. A dynamic hedging strategy, which involves frequent adjustments to maintain the desired hedge ratio, will incur higher transaction costs than a static hedge. Therefore, portfolio managers must carefully weigh the benefits of dynamic hedging against the associated costs. The breakeven point, where the benefits of hedging outweigh the transaction costs, needs to be determined. A manager might use a lower-cost ETF to hedge if available, even if it is not a perfect hedge. The impact of bid-ask spreads on futures contracts, especially during volatile market conditions, also needs consideration as it adds to the transaction costs.

Let’s consider a scenario involving a UK-based agricultural cooperative, “British Harvest Co-op” (BHC), which produces and exports barley. BHC wants to protect itself from potential declines in barley prices over the next year. They decide to use futures contracts listed on the ICE Futures Europe exchange to hedge their price risk. To determine the number of contracts needed, BHC needs to consider the contract size, the quantity of barley they want to hedge, and the hedge ratio. The hedge ratio isn’t 1:1 because the barley BHC produces isn’t identical to the barley used for futures delivery, introducing basis risk. Let’s assume BHC wants to hedge 50,000 metric tons of barley. Each ICE Futures Europe barley contract represents 100 metric tons. The historical correlation between BHC’s barley price and the ICE Futures Europe barley futures price is 0.8. This correlation is crucial for calculating the optimal hedge ratio. The optimal hedge ratio minimizes the variance of the hedged portfolio and is calculated as the correlation coefficient multiplied by the ratio of the standard deviation of the spot price changes to the standard deviation of the futures price changes. We’ll assume that the standard deviation of BHC’s barley price changes is £5 per ton, and the standard deviation of the ICE Futures Europe barley futures price changes is £6 per ton. The hedge ratio is calculated as: Hedge Ratio = Correlation * (Standard Deviation of Spot Price / Standard Deviation of Futures Price) = 0.8 * (5/6) = 0.667. The number of contracts needed is calculated as: Number of Contracts = (Quantity to Hedge * Hedge Ratio) / Contract Size = (50,000 tons * 0.667) / 100 tons/contract = 333.5. Since you can’t trade fractions of contracts, BHC should round to the nearest whole number, resulting in 334 contracts. This ensures they are as close as possible to the optimal hedge, minimizing basis risk while staying within practical trading constraints.

The question assesses understanding of hedging strategies using options, specifically a ratio spread. A ratio spread involves buying and selling different numbers of options with the same expiration date but different strike prices. The goal is often to profit from a specific directional movement in the underlying asset while limiting potential losses. The payoff calculation involves considering the premium paid or received for each option, and the profit or loss at different price levels of the underlying asset at expiration. We need to calculate the profit or loss at the expiration price of £155. First, calculate the net premium received/paid: Premium received from selling 2 call options: 2 * £2.50 = £5.00 Premium paid for buying 1 call option: 1 * £7.00 = £7.00 Net premium paid: £7.00 – £5.00 = £2.00 Next, calculate the payoff at expiration (S_T = £155): The long call option with a strike price of £150 will be in the money. Payoff from the long call: S_T – K = £155 – £150 = £5.00 The two short call options with a strike price of £160 will be in the money if the price is above £160, but since it’s £155, they expire worthless. Payoff is £0. Total payoff = Payoff from long call – Net premium paid = £5.00 – £2.00 = £3.00 Therefore, the profit from the ratio spread strategy at expiration is £3.00. A crucial aspect of understanding ratio spreads lies in recognizing their risk profile. Unlike simple option strategies, ratio spreads can have unlimited risk on one side. In this case, if the price of the underlying asset rises significantly above £160, the investor would face potentially substantial losses from the two short call options. Conversely, the profit potential is capped if the price remains below £150. This makes ratio spreads suitable for investors who have a specific market outlook and are willing to accept a defined level of risk. Ratio spreads are often used when an investor expects limited price movement in the underlying asset. The strategy aims to generate income from the premiums received from the short options while limiting the potential downside risk with the long option. However, it’s important to carefully consider the potential losses if the market moves significantly in the opposite direction. The investor needs to have a clear understanding of the potential profit and loss at different price levels to effectively manage the risk associated with this strategy.

Let’s analyze a scenario involving a UK-based agricultural cooperative seeking to hedge against price volatility in the wheat market using futures contracts, while also navigating the regulatory landscape under EMIR (European Market Infrastructure Regulation). The cooperative, “Golden Harvest,” needs to understand the implications of clearing obligations and potential margin calls. The core concept here is hedging with futures, combined with the practical implications of EMIR regulations on derivatives trading. We need to consider initial margin, variation margin, and the potential impact of market volatility on these margin requirements. Assume Golden Harvest enters into a wheat futures contract to sell 500 tonnes of wheat at £200 per tonne for delivery in six months. The initial margin requirement is 5% of the contract value, and the maintenance margin is 80% of the initial margin. 1. **Contract Value:** 500 tonnes * £200/tonne = £100,000 2. **Initial Margin:** 5% of £100,000 = £5,000 3. **Maintenance Margin:** 80% of £5,000 = £4,000 4. **Margin Call Trigger:** If the margin account balance falls below £4,000, Golden Harvest will receive a margin call. Now, let’s say the wheat futures price increases to £210 per tonne after one week. This is adverse to Golden Harvest’s short position. The loss is: 500 tonnes * (£210 – £200)/tonne = £5,000 This loss is deducted from the margin account. The new balance is: £5,000 (initial margin) – £5,000 (loss) = £0 Since £0 is below the maintenance margin of £4,000, Golden Harvest will receive a margin call. They need to deposit enough funds to bring the margin account back to the initial margin level of £5,000. Therefore, the margin call amount is £5,000. If, instead, the price had *decreased* to £190, Golden Harvest would have a profit of £5,000, increasing their margin account to £10,000. This would not trigger a margin call. The EMIR regulation adds complexity. Golden Harvest, exceeding certain thresholds, might be obligated to clear the futures contract through a central counterparty (CCP). This involves additional margin requirements and potential CCP default fund contributions. Furthermore, EMIR mandates reporting of derivative transactions to trade repositories, increasing operational burden. The cooperative must also implement robust risk management procedures and comply with regulatory reporting requirements to avoid penalties. The clearing obligation under EMIR is crucial as it reduces counterparty risk but adds to the cost and complexity of hedging.

The question revolves around the concept of hedging a portfolio using options, specifically focusing on achieving a desired portfolio beta. Beta represents the systematic risk of a portfolio relative to the market. Modifying the portfolio beta involves adjusting the portfolio’s sensitivity to market movements. Options, particularly futures options, can be used to achieve this. The key is understanding how the notional value of the futures options contract and the option’s delta affect the overall portfolio beta. The formula to adjust portfolio beta using futures options is: Number of contracts = \[\frac{(Target Beta – Current Beta) \times Portfolio Value}{Option Delta \times Futures Contract Value}\] In this scenario, we aim to *reduce* the portfolio beta. The option delta reflects the change in the option price for a unit change in the underlying asset’s price (the futures contract). A positive delta for a call option means the option price moves in the same direction as the futures price. The number of contracts needs to be calculated carefully, considering the direction of the hedge (buying or selling options). Since we are reducing the beta, we need to consider the impact of the options on the portfolio’s overall sensitivity to market movements. Incorrectly calculating the number of contracts or misunderstanding the impact of option delta will lead to an incorrect hedge. The use of futures options allows for a more precise adjustment of the portfolio’s beta compared to using stocks alone, especially when dealing with large portfolios or specific risk management objectives. The accuracy of the beta adjustment relies heavily on the accurate estimation of the option delta and the stability of the relationship between the futures contract and the underlying portfolio.

The question assesses the understanding of delta hedging and portfolio rebalancing in the context of options trading, specifically considering transaction costs. Delta hedging involves adjusting the portfolio’s position in the underlying asset to offset changes in the option’s value due to small movements in the underlying asset’s price. The delta of an option represents the sensitivity of the option’s price to a change in the underlying asset’s price. A delta-neutral portfolio has a delta of zero, meaning it is theoretically immune to small price changes in the underlying asset. Transaction costs are a crucial consideration in real-world hedging strategies. Frequent rebalancing to maintain delta neutrality can erode profits due to these costs. The optimal rebalancing frequency depends on the trade-off between the cost of rebalancing and the risk of being unhedged. The calculation involves determining the number of shares to trade to adjust the portfolio’s delta. The initial portfolio has a short position in 1,000 call options, each with a delta of 0.60, resulting in a portfolio delta of -600. To achieve delta neutrality, the portfolio needs a long position of 600 shares. The trader initially buys 600 shares. When the underlying asset’s price increases, the call option delta increases to 0.65. The portfolio delta becomes -1,000 * 0.65 = -650. To re-establish delta neutrality, the trader needs to increase the long position in the underlying asset. The change in delta required is 650 – 600 = 50 shares. Therefore, the trader needs to buy an additional 50 shares. The total transaction cost is calculated by multiplying the number of shares traded by the transaction cost per share. The initial trade involves buying 600 shares, and the subsequent rebalancing involves buying 50 shares. The total number of shares traded is 600 + 50 = 650 shares. With a transaction cost of £0.10 per share, the total transaction cost is 650 * £0.10 = £65. This example illustrates the practical challenges of delta hedging, where transaction costs can significantly impact the profitability of the strategy. It highlights the need for careful consideration of rebalancing frequency and transaction costs in managing option portfolios. For instance, a fund manager using options to hedge a large equity portfolio must weigh the benefits of precise delta hedging against the costs of frequent trading. A less frequent rebalancing strategy might be chosen if transaction costs are high, accepting a slightly higher level of risk from imperfect hedging. This decision would be influenced by the fund’s risk tolerance and investment objectives.

The question concerns the impact of behavioral biases, specifically confirmation bias and loss aversion, on derivative pricing and trading strategies. Confirmation bias leads traders to seek out information confirming their existing beliefs, potentially overvaluing or undervaluing derivatives based on selective data. Loss aversion causes traders to feel the pain of a loss more acutely than the pleasure of an equivalent gain, leading to risk-averse behavior that can distort market prices. Consider a scenario where a trader believes a particular technology stock will significantly increase in value due to a new product launch. Affected by confirmation bias, the trader selectively focuses on positive news articles and analyst reports supporting this view, while ignoring negative or contradictory information. This could lead the trader to purchase call options on the stock at a price higher than their fair value, based on an inflated expectation of future price movements. Furthermore, loss aversion might cause the trader to set a very tight stop-loss order on these options. If the stock price experiences a temporary dip, the stop-loss could be triggered, resulting in a loss. The trader’s aversion to this loss might prevent them from holding the position long enough to realize potential gains if the stock price eventually rises as initially anticipated. Now, let’s consider a scenario with Credit Default Swaps (CDS). A portfolio manager, influenced by confirmation bias, believes that a specific corporate bond is highly creditworthy and will not default. The manager might downplay negative credit ratings or financial reports concerning the company, reinforcing their initial belief. Consequently, they may decide to sell protection on a CDS referencing that bond at a premium lower than justified by the actual credit risk. If the company’s financial situation deteriorates unexpectedly, the manager could face significant losses if the bond defaults. Another example is a trader holding a short position in a currency future. Due to loss aversion, the trader might be overly sensitive to small upward price movements, leading them to cover their position prematurely at a small loss. This premature closing prevents them from realizing larger profits if the currency’s value subsequently declines as initially predicted. The correct option will accurately reflect the combined influence of these biases on derivative trading decisions and market prices.

The Black-Scholes model is a cornerstone of options pricing theory. It relies on several key assumptions, including constant volatility, a risk-free interest rate, and the European option style (exercisable only at expiration). When these assumptions are violated, the model’s accuracy diminishes. Volatility smiles and skews are common phenomena in options markets, indicating that implied volatility (the volatility implied by market prices of options) varies depending on the strike price and expiration date. A volatility smile suggests that out-of-the-money options and in-the-money options have higher implied volatilities than at-the-money options. A volatility skew indicates that implied volatilities are systematically higher for either out-of-the-money puts (downward skew) or out-of-the-money calls (upward skew). When using the Black-Scholes model in the presence of a volatility smile, a single volatility input will lead to mispricing of options across different strike prices. Options with strike prices further away from the current asset price (either higher or lower, depending on the smile’s shape) will be systematically undervalued if the at-the-money volatility is used. Conversely, options closer to the money will be relatively overvalued compared to those further out. To account for the volatility smile, traders and analysts often use a volatility surface, which is a three-dimensional plot of implied volatility as a function of strike price and time to expiration. They can then interpolate or extrapolate volatilities for specific strike prices and maturities to improve the accuracy of option pricing and hedging. Another approach is to use models that explicitly incorporate stochastic volatility, such as the Heston model, which allows volatility to vary randomly over time. However, these models are more complex and require more sophisticated calibration techniques. Ignoring the volatility smile can lead to significant errors in risk management, particularly when hedging portfolios of options with different strike prices. For instance, a trader who sells a straddle (selling both a call and a put option with the same strike price and expiration date) based on Black-Scholes pricing using at-the-money volatility might find that the position incurs losses if the underlying asset price moves significantly, as the out-of-the-money options become more valuable than predicted by the model.

Let’s break down how to approach valuing a complex exotic derivative: a Cliquet Option with a participation rate. This option offers a series of periodic resets, where the return for each period is capped and floored, and then summed to determine the overall payoff. The participation rate scales the final accumulated return. The challenge lies in simulating the underlying asset’s price path over multiple periods, applying the cap and floor at each reset date, and then calculating the final payoff based on the accumulated returns and the participation rate. Here’s a step-by-step breakdown of the Monte Carlo simulation: 1. **Simulate Asset Price Paths:** We generate a large number (e.g., 10,000) of possible price paths for the underlying asset over the life of the Cliquet Option. Each path consists of the asset’s price at each reset date. We assume the asset price follows a geometric Brownian motion: \[ S_{t+\Delta t} = S_t \cdot \exp\left(\left(\mu – \frac{\sigma^2}{2}\right)\Delta t + \sigma \sqrt{\Delta t} \cdot Z\right) \] where: * \(S_t\) is the asset price at time \(t\) * \(\mu\) is the expected return of the asset (risk-free rate in a risk-neutral world) * \(\sigma\) is the volatility of the asset * \(\Delta t\) is the time step (time between resets) * \(Z\) is a standard normal random variable 2. **Calculate Periodic Returns:** For each path, we calculate the return for each period (between reset dates): \[ R_i = \frac{S_{t_i} – S_{t_{i-1}}}{S_{t_{i-1}}} \] where: * \(R_i\) is the return for period \(i\) * \(S_{t_i}\) is the asset price at reset date \(t_i\) * \(S_{t_{i-1}}\) is the asset price at the previous reset date \(t_{i-1}\) 3. **Apply Caps and Floors:** We apply the cap and floor to each periodic return: \[ R_i^{CappedFloored} = \max(\text{Floor}, \min(R_i, \text{Cap})) \] where: * `Cap` is the maximum return for each period (e.g., 5%) * `Floor` is the minimum return for each period (e.g., -3%) 4. **Accumulate Returns:** We sum the capped and floored returns for each path: \[ \text{Accumulated Return} = \sum_{i=1}^{n} R_i^{CappedFloored} \] where \(n\) is the number of reset periods. 5. **Apply Participation Rate:** We multiply the accumulated return by the participation rate: \[ \text{Final Payoff} = \text{Participation Rate} \cdot \text{Accumulated Return} \cdot \text{Notional Amount} \] If the result is negative, the payoff is zero. 6. **Discount to Present Value:** We discount the final payoff back to the present value using the risk-free rate: \[ PV = \frac{\text{Final Payoff}}{\exp(rT)} \] where: * \(r\) is the risk-free rate * \(T\) is the time to maturity of the option 7. **Average Across Paths:** We average the present values of the payoffs across all simulated paths to estimate the value of the Cliquet Option. Now, let’s apply this to the specific question.

Let’s break down the concept of calculating the theoretical price of a European call option using a two-step binomial tree. This model helps us approximate how the option’s price changes over time, considering the underlying asset’s potential movements. The core idea is to work backward from the expiration date to the present. First, we need to calculate the up and down factors (u and d) representing the potential price movements. These are derived from the volatility (\(\sigma\)) and the length of each time step (h). We have \(u = e^{\sigma \sqrt{h}}\) and \(d = e^{-\sigma \sqrt{h}}\). Then, we calculate the risk-neutral probability (p) using the formula \(p = \frac{e^{r h} – d}{u – d}\), where r is the risk-free rate. Next, we construct the binomial tree. At each node, we calculate the asset price based on whether it moves up or down. At the expiration date, we determine the option’s payoff at each final node, which is the maximum of (Asset Price – Strike Price, 0) for a call option. Working backward, we calculate the option value at each preceding node using the risk-neutral probabilities. The formula for this is: Option Value = \(e^{-r h} [p \times \text{Option Value (Up)} + (1-p) \times \text{Option Value (Down)}]\). We repeat this process for each time step until we reach the initial node, which represents the theoretical price of the option today. For instance, imagine a stock currently priced at £50. Over two periods, it could go up or down. We use the binomial model to estimate the call option price. If, after the first period, the price goes up, and then up again in the second period, the final payoff will be different than if it goes up then down, or down then up, or down then down. Each path has a probability associated with it, derived from the risk-free rate and volatility. The binomial model cleverly combines all these possibilities, weighted by their probabilities, to arrive at a fair option price. This is not just about predicting the future; it’s about pricing the uncertainty.

The question assesses understanding of hedging strategies using options, specifically the concept of a ratio spread. A ratio spread involves buying and selling different numbers of options with different strike prices but the same expiration date. The key is to understand how changes in the underlying asset’s price affect the profitability of the strategy, considering the different sensitivities of the long and short option positions. Here’s a breakdown of how to calculate the profit/loss: 1. **Initial Setup:** Calculate the net premium paid or received. In this case, the investor buys one call option with a strike price of £150 and sells two call options with a strike price of £160. The net premium is: \[ \text{Net Premium} = \text{Premium Paid for £150 Call} – 2 \times \text{Premium Received for £160 Call} \] \[ \text{Net Premium} = 8 – 2 \times 3 = 8 – 6 = 2 \] The investor pays a net premium of £2. 2. **Scenario Analysis:** Evaluate the profit/loss at the expiration date based on the underlying asset’s price. * **Case 1: Asset Price at £145** Neither call option is in the money. The profit/loss is simply the negative of the net premium paid: \[ \text{Profit/Loss} = -2 \] * **Case 2: Asset Price at £155** The £150 call option is in the money, but the £160 call options are not. Profit from £150 call = £155 – £150 = £5 Net Profit/Loss = £5 – £2 = £3 * **Case 3: Asset Price at £160** The £150 call option is in the money, and the asset price is exactly at the strike price of the short calls. Profit from £150 call = £160 – £150 = £10 Net Profit/Loss = £10 – £2 = £8 * **Case 4: Asset Price at £165** Both the £150 and £160 call options are in the money. Profit from £150 call = £165 – £150 = £15 Loss from two £160 calls = 2 * (£165 – £160) = 2 * £5 = £10 Net Profit/Loss = £15 – £10 – £2 = £3 * **Case 5: Asset Price at £170** Both the £150 and £160 call options are in the money. Profit from £150 call = £170 – £150 = £20 Loss from two £160 calls = 2 * (£170 – £160) = 2 * £10 = £20 Net Profit/Loss = £20 – £20 – £2 = -£2 3. **Maximum Profit:** The maximum profit occurs when the asset price is at the strike price of the short calls (£160). In this case, the long £150 call is in the money, but the short £160 calls are not. The profit is the difference between the asset price and the strike price of the long call, minus the net premium paid: \[ \text{Max Profit} = (£160 – £150) – £2 = £10 – £2 = £8 \] The key to solving this problem is understanding the payoff profile of each option and how they combine to create the overall strategy payoff. The ratio spread strategy is designed to profit from limited upside movement in the underlying asset, with a defined maximum profit and potential for losses if the asset price rises significantly.

Let’s consider a scenario where a UK-based agricultural cooperative, “British Harvest Co-op,” wants to protect itself against potential price declines in its upcoming wheat harvest. They decide to use futures contracts listed on the ICE Futures Europe exchange. The current spot price of wheat is £200 per tonne. The December wheat futures contract is trading at £205 per tonne. British Harvest Co-op expects to harvest 5,000 tonnes of wheat in December. Each futures contract represents 100 tonnes of wheat. To hedge their price risk, British Harvest Co-op sells futures contracts. The number of contracts to sell is calculated as (Total Wheat to Hedge / Contract Size) = 5,000 tonnes / 100 tonnes/contract = 50 contracts. Scenario 1: In December, the spot price of wheat falls to £190 per tonne. The futures price converges to the spot price at £190 per tonne. British Harvest Co-op sells their wheat in the spot market for £190 per tonne, receiving £190/tonne * 5,000 tonnes = £950,000. Simultaneously, they close out their futures position by buying back 50 contracts at £190 per tonne. They initially sold the contracts at £205 per tonne, so their profit on the futures contracts is (£205 – £190) * 100 tonnes/contract * 50 contracts = £75,000. The effective price received is £950,000 (spot market sale) + £75,000 (futures profit) = £1,025,000, or £205 per tonne. Scenario 2: In December, the spot price of wheat rises to £220 per tonne. The futures price converges to the spot price at £220 per tonne. British Harvest Co-op sells their wheat in the spot market for £220 per tonne, receiving £220/tonne * 5,000 tonnes = £1,100,000. Simultaneously, they close out their futures position by buying back 50 contracts at £220 per tonne. They initially sold the contracts at £205 per tonne, so their loss on the futures contracts is (£220 – £205) * 100 tonnes/contract * 50 contracts = -£75,000. The effective price received is £1,100,000 (spot market sale) – £75,000 (futures loss) = £1,025,000, or £205 per tonne. Basis risk arises because the futures price and the spot price may not converge perfectly at the delivery date. The basis is defined as Spot Price – Futures Price. If the basis changes unexpectedly, the hedge may not be perfect. For example, if in Scenario 1, the futures price only converged to £195 instead of £190, the profit on the futures would be smaller, and the effective price received would be lower than the intended £205. This difference is basis risk. Basis risk can arise from transportation costs, storage costs, quality differences, or other market factors.

This question tests understanding of the impact of volatility on option pricing, specifically in the context of exotic options and structured products. The scenario involves a barrier option, where the payoff is contingent on the underlying asset’s price reaching a specific barrier level. The Black-Scholes model, while a cornerstone of option pricing, assumes constant volatility, which is rarely the case in real-world markets. Volatility skew refers to the phenomenon where out-of-the-money puts and in-the-money calls tend to have higher implied volatilities than at-the-money options. This skew is more pronounced in equity markets due to the “leverage effect” (a drop in equity price increases a company’s leverage, making it riskier). The presence of volatility skew significantly affects the pricing and hedging of barrier options. A down-and-out put option, for example, becomes worthless if the underlying asset’s price falls below the barrier. If the volatility skew indicates higher implied volatility for out-of-the-money puts (which are closer to the barrier), the probability of the barrier being breached is higher than what the Black-Scholes model would predict using a single, constant volatility. This leads to a higher price for the down-and-out put option. To accurately price and hedge such options, traders use models that incorporate volatility skew, such as stochastic volatility models or models that use a volatility surface. They also dynamically adjust their hedges based on changes in the volatility skew. Ignoring the volatility skew can lead to significant mispricing and substantial losses, especially for complex derivatives like barrier options embedded in structured products. The correct answer recognizes that the down-and-out put option is likely underpriced by the model because the volatility skew increases the probability of hitting the barrier, thus making the option more valuable.

The question concerns the impact of correlation between assets within a portfolio when using derivatives for hedging. The key concept is that imperfect correlation reduces the effectiveness of a hedge. The hedge ratio is the amount of the derivative needed to offset the risk of the underlying asset. When correlation is perfect (1.0), the hedge ratio is simply the ratio of the standard deviations of the asset and the derivative. However, when correlation is less than perfect, the hedge ratio must be adjusted to account for the diversification benefit obtained by combining assets with less than perfectly correlated returns. The formula for the optimal hedge ratio (h) is: \[h = \rho \cdot \frac{\sigma_A}{\sigma_F}\] Where: * \( \rho \) is the correlation between the asset (A) and the hedging instrument (F) (Futures in this case). * \( \sigma_A \) is the standard deviation of the asset’s returns. * \( \sigma_F \) is the standard deviation of the futures contract’s returns. In this scenario: * \( \rho = 0.75 \) * \( \sigma_A = 0.15 \) (15%) * \( \sigma_F = 0.20 \) (20%) Therefore, the optimal hedge ratio is: \[h = 0.75 \cdot \frac{0.15}{0.20} = 0.75 \cdot 0.75 = 0.5625\] The interpretation is that for every £1 of asset exposure, the investor should short £0.5625 of the futures contract to minimize portfolio variance. This is less than the ratio of standard deviations (0.75) because the correlation is not perfect, implying some diversification benefit. The question then introduces a portfolio of £10,000,000. To calculate the amount of futures contracts needed, we multiply the portfolio value by the hedge ratio: £10,000,000 \* 0.5625 = £5,625,000 Each futures contract has a face value of £1,000,000. Therefore, the number of contracts required is: £5,625,000 / £1,000,000 = 5.625 contracts. Since you cannot trade fractions of contracts, one would typically round to the nearest whole number. However, the question asks for the *theoretically* optimal number, implying we should use the exact calculated value.

To understand this problem, we need to calculate the theoretical price of the exotic derivative, a forward start option, using a modified Black-Scholes model, accounting for the dividend yield and the delayed start. The standard Black-Scholes formula is: \[C = S_0e^{-qT}N(d_1) – Xe^{-rT}N(d_2)\] Where: * \(C\) = Call option price * \(S_0\) = Current stock price * \(X\) = Strike price * \(r\) = Risk-free interest rate * \(q\) = Dividend yield * \(T\) = Time to expiration * \(N(x)\) = Cumulative standard normal distribution function * \(d_1 = \frac{ln(\frac{S_0}{X}) + (r – q + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}}\) * \(d_2 = d_1 – \sigma\sqrt{T}\) * \(\sigma\) = Volatility However, because this is a forward start option, we need to discount the Black-Scholes price back to today’s value using the risk-free rate over the delay period. First, calculate \(d_1\) and \(d_2\) using the given parameters: \(S_0 = 100\), \(X = 100\), \(r = 0.05\), \(q = 0.02\), \(\sigma = 0.20\), and \(T = 1\) (since the option exists for 1 year after the 1-year delay). \[d_1 = \frac{ln(\frac{100}{100}) + (0.05 – 0.02 + \frac{0.20^2}{2})1}{0.20\sqrt{1}} = \frac{0 + (0.03 + 0.02)1}{0.20} = \frac{0.05}{0.20} = 0.25\] \[d_2 = 0.25 – 0.20\sqrt{1} = 0.25 – 0.20 = 0.05\] Next, find \(N(d_1)\) and \(N(d_2)\). Approximating using standard normal distribution tables (or a calculator): \(N(0.25) \approx 0.5987\) \(N(0.05) \approx 0.5199\) Now, calculate the Black-Scholes price: \[C = 100e^{-0.02 \cdot 1}(0.5987) – 100e^{-0.05 \cdot 1}(0.5199)\] \[C = 100(0.9802)(0.5987) – 100(0.9512)(0.5199)\] \[C = 58.68 – 49.45 = 9.23\] Finally, discount this price back one year using the risk-free rate: \[Present\ Value = 9.23 \cdot e^{-0.05 \cdot 1} = 9.23 \cdot 0.9512 \approx 8.78\] Therefore, the theoretical price of the forward start option today is approximately £8.78. The essence of this problem lies in understanding how to adapt the standard Black-Scholes model for exotic derivatives, specifically forward start options. The key adjustment is discounting the future option price back to the present, recognizing the time value of money over the initial delay period. This tests not just the application of the formula but the understanding of its underlying principles and how it can be modified to fit different derivative structures.

The question revolves around the concept of delta hedging a portfolio of options and the subsequent profit or loss arising from the hedge’s performance over a specific period. Delta hedging aims to neutralize the portfolio’s sensitivity to small changes in the underlying asset’s price. However, delta changes as the underlying asset’s price fluctuates (gamma) and as time passes (theta). Therefore, a perfect hedge requires continuous rebalancing. The profit or loss is calculated by considering the cost of rebalancing the hedge (buying or selling the underlying asset) and the payoff from the options. In this scenario, the portfolio manager initially delta hedges by shorting the underlying asset. As the asset price changes, the delta of the options portfolio changes, requiring adjustments to the short position. The profit or loss from the hedging activity is the difference between the gains/losses from the options portfolio and the gains/losses from the hedging transactions. To calculate the profit or loss, we need to: 1. Determine the initial delta hedge: The portfolio is shorted 200 shares initially to offset the positive delta. 2. Calculate the change in delta and the required rebalancing: The delta changes as the price moves. We calculate the number of shares to buy back at each price point. 3. Calculate the cost/benefit of rebalancing: Multiply the number of shares bought/sold by the price at which they were traded. 4. Calculate the profit/loss on the options portfolio: This is the difference between the final value of the options and the initial value. 5. Calculate the overall profit/loss: Sum the profit/loss on the options and the profit/loss from hedging. The example uses specific price points and delta changes to illustrate the rebalancing process and its impact on the overall profit or loss. This tests understanding of dynamic hedging and its implications.

This question tests the understanding of delta hedging, gamma, and the impact of market movements on a hedged portfolio. Delta hedging aims to neutralize the portfolio’s sensitivity to small changes in the underlying asset’s price. Gamma, however, 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 also increases, and vice versa. This requires dynamic adjustments to the hedge. The initial hedge is established to be delta neutral. However, because the option has a gamma of 0.05, the delta of the portfolio changes as the underlying asset’s price changes. 1. **Calculate the change in the option’s delta:** The underlying asset price increases by £2. Therefore, the change in delta is Gamma \* Change in Price = 0.05 \* 2 = 0.1. 2. **Calculate the new delta of the option:** The initial delta of the option was -0.5. The new delta is -0.5 + 0.1 = -0.4. 3. **Calculate the number of shares needed to re-hedge:** To re-hedge, the portfolio needs to be delta neutral again. This means the delta of the shares must offset the delta of the option. Since the option’s delta is now -0.4, the portfolio needs a delta of +0.4 from the shares. This means you need to sell shares to reduce the delta of the shares by 0.1 (0.5-0.4). Since you own 500 options, you need to sell 0.1 * 500 = 50 shares. Therefore, the investor needs to sell 50 shares to re-hedge the portfolio. This dynamic adjustment is crucial because gamma exposes the hedge to risk from larger price movements. In essence, the higher the gamma, the more frequently the hedge needs to be adjusted to maintain delta neutrality. This adjustment process incurs transaction costs, which need to be factored into the overall hedging strategy. Ignoring gamma can lead to significant losses, especially in volatile markets. This question goes beyond basic definitions and requires understanding the practical implications of gamma in a delta-hedged portfolio, a critical aspect of derivatives risk management.

The core of this question revolves around understanding how a delta-neutral portfolio reacts to changes in implied volatility (vega) and the underlying asset price (delta). Maintaining delta neutrality means the portfolio’s value is, theoretically, unaffected by small changes in the underlying asset’s price. However, this neutrality is sensitive to changes in implied volatility. Vega measures this sensitivity. A positive vega indicates that the portfolio’s value increases with rising implied volatility, and vice versa. Gamma measures the rate of change of delta with respect to the underlying asset’s price. The trader’s initial position is delta-neutral, meaning the delta is zero. The portfolio has a positive vega of 1,500, implying that a 1% increase in implied volatility will increase the portfolio’s value by £1,500. The trader wants to hedge against a potential decrease in implied volatility. To achieve this, the trader needs to reduce the portfolio’s vega to zero. The trader can use options to hedge against vega. To reduce vega, the trader needs to sell options. The trader sells 150 call options with a vega of 10 each. The trader’s vega changes by 150 * 10 = 1,500, which offsets the initial positive vega of 1,500. The trader’s new vega is 1,500 – 1,500 = 0. The portfolio’s gamma is -500. Gamma is the rate of change of delta with respect to the underlying asset’s price. A negative gamma means that as the underlying asset’s price increases, the portfolio’s delta decreases, and as the underlying asset’s price decreases, the portfolio’s delta increases. The underlying asset’s price increases by £2. The change in delta is -500 * 2 = -1,000. The trader needs to rebalance the portfolio to maintain delta neutrality. To offset the negative delta, the trader needs to buy shares of the underlying asset. The trader needs to buy 1,000 shares of the underlying asset.

The question focuses on the application of put-call parity in a market anomaly scenario. Put-call parity states that for European-style options with the same strike price and expiration date, the following relationship should hold: \(C + PV(K) = P + S\), where \(C\) is the call option price, \(PV(K)\) is the present value of the strike price, \(P\) is the put option price, and \(S\) is the current stock price. If this relationship is violated, an arbitrage opportunity exists. In this case, the observed market prices deviate from the parity. To identify the arbitrage strategy, we need to determine which side of the equation is undervalued and which is overvalued. Given the prices: Call (\(C = £7\)), Put (\(P = £3\)), Stock (\(S = £98\)), and Strike (\(K = £100\)), with a risk-free rate of 5% and 6 months to expiration, we first calculate the present value of the strike price: \(PV(K) = \frac{K}{1 + r \cdot t} = \frac{100}{1 + 0.05 \cdot 0.5} = \frac{100}{1.025} \approx £97.56\). Now, we check the put-call parity: Left side: \(C + PV(K) = 7 + 97.56 = £104.56\) Right side: \(P + S = 3 + 98 = £101\) Since the left side (\(£104.56\)) is greater than the right side (\(£101\)), the combination of the call option and the present value of the strike is overvalued relative to the combination of the put option and the stock. To exploit this, we should sell the overvalued side (sell the call and lend to replicate the present value of the strike) and buy the undervalued side (buy the put and buy the stock). Therefore, the arbitrage strategy involves: 1. Selling the call option for \(£7\). 2. Lending \(£97.56\) (the present value of the strike price). 3. Buying the put option for \(£3\). 4. Buying the stock for \(£98\). The initial cash flow is: \(7 + (100/1.025) – 3 – 98 = 7 + 97.56 – 3 – 98 = £3.56\). At expiration, if the stock price is above £100, the call option will be exercised against us, and we will deliver the stock we bought. The loan will be repaid with £100, perfectly offsetting the strike price. If the stock price is below £100, we will exercise our put option, selling the stock for £100, which we use to repay the loan. Thus, regardless of the stock price at expiration, the strategy yields a risk-free profit of £3.56 initially.

The question assesses understanding of put-call parity, a fundamental concept in options pricing. Put-call parity states the relationship between the prices of a European call and a European put option with the same strike price and expiration date. It assumes no arbitrage opportunities exist. The formula is: C + PV(X) = P + S Where: C = Call option price P = Put option price S = Current stock price X = Strike price PV(X) = Present value of the strike price, discounted at the risk-free rate to the expiration date. This is calculated as \(X e^{-rT}\), where r is the risk-free rate and T is the time to expiration in years. In this scenario, we need to determine if an arbitrage opportunity exists and, if so, how to exploit it. First, calculate the present value of the strike price: PV(X) = \(520 * e^{-0.04 * 0.5}\) = \(520 * e^{-0.02}\) ≈ \(520 * 0.9802\) ≈ 509.70 Now, check if the put-call parity holds: Call + PV(Strike) = Put + Stock 65 + 509.70 = 12 + 550 574.70 ≠ 562 Since the left side (574.70) is greater than the right side (562), the call option is relatively overpriced compared to the put option and the underlying asset. To exploit this arbitrage opportunity, we should sell the overpriced side (the call option) and buy the underpriced side (the put option and the stock). Arbitrage strategy: 1. Sell the call option for 65. 2. Buy the put option for 12. 3. Buy the stock for 550. 4. Borrow the present value of the strike price, 509.70, which will be repaid at expiration. Initial cash flow: 65 (sell call) – 12 (buy put) – 550 (buy stock) + 509.70 (borrow) = 12.70 At expiration (6 months): * If the stock price is above 520, the call option will be exercised against you. You will have to deliver the stock, which you bought for 550. The put option will expire worthless. Your net cash flow will be 520 (from the strike) – 550 (cost of stock) + 509.70 (repay borrowed amount) = 0, plus the initial profit. * If the stock price is below 520, the call option will expire worthless. You will exercise the put option, selling the stock for 520. Your net cash flow will be 520 (from put) – 550 (cost of stock) + 509.70 (repay borrowed amount) = 0, plus the initial profit. The arbitrage profit is the initial cash flow, which is 12.70.

The core of this question lies in understanding how implied volatility impacts option pricing, specifically in the context of earnings announcements. Implied volatility is the market’s expectation of future volatility, derived from option prices. Earnings announcements are typically periods of heightened uncertainty, leading to a surge in implied volatility. This increase directly affects option premiums. The Black-Scholes model, while having limitations, is a fundamental tool for understanding option pricing. The model incorporates factors like the underlying asset’s price, strike price, time to expiration, risk-free interest rate, and implied volatility. A higher implied volatility, all else being equal, results in a higher option price. This is because a greater expected volatility increases the probability of the underlying asset’s price moving significantly in either direction, benefiting both call and put option holders. After the earnings announcement, if the actual volatility is lower than the implied volatility priced into the options beforehand, the implied volatility will decrease, causing the option premiums to decline. This phenomenon is known as volatility crush. The extent of the decline depends on how much the market overestimated the actual volatility during the earnings period. In this scenario, we need to calculate the initial call option price using the Black-Scholes model with the pre-earnings implied volatility (40%) and then calculate the call option price again using the post-earnings implied volatility (25%). The difference between these two prices represents the change in the option premium due to the volatility crush. Given the provided data: * Underlying asset price (S) = £100 * Strike price (K) = £105 * Time to expiration (T) = 0.25 years (3 months) * Risk-free interest rate (r) = 5% * Pre-earnings implied volatility (\(\sigma_1\)) = 40% * Post-earnings implied volatility (\(\sigma_2\)) = 25% We’ll calculate the call option prices using the Black-Scholes formula: \[C = S \cdot N(d_1) – K \cdot e^{-rT} \cdot N(d_2)\] where: \[d_1 = \frac{ln(\frac{S}{K}) + (r + \frac{\sigma^2}{2})T}{\sigma \sqrt{T}}\] \[d_2 = d_1 – \sigma \sqrt{T}\] and N(x) is the cumulative standard normal distribution function. First, calculate \(d_1\) and \(d_2\) for both pre-earnings and post-earnings volatility: Pre-earnings: \[d_1 = \frac{ln(\frac{100}{105}) + (0.05 + \frac{0.40^2}{2})0.25}{0.40 \sqrt{0.25}} = \frac{-0.0488 + 0.0325}{0.20} = -0.0815\] \[d_2 = -0.0815 – 0.40 \sqrt{0.25} = -0.0815 – 0.20 = -0.2815\] Post-earnings: \[d_1 = \frac{ln(\frac{100}{105}) + (0.05 + \frac{0.25^2}{2})0.25}{0.25 \sqrt{0.25}} = \frac{-0.0488 + 0.0156}{0.125} = -0.2656\] \[d_2 = -0.2656 – 0.25 \sqrt{0.25} = -0.2656 – 0.125 = -0.3906\] Using a standard normal distribution table or calculator: N(-0.0815) ≈ 0.4676 N(-0.2815) ≈ 0.3893 N(-0.2656) ≈ 0.3955 N(-0.3906) ≈ 0.3481 Now, calculate the call option prices: Pre-earnings Call Option Price: \[C_1 = 100 \cdot 0.4676 – 105 \cdot e^{-0.05 \cdot 0.25} \cdot 0.3893 = 46.76 – 105 \cdot 0.9876 \cdot 0.3893 = 46.76 – 40.28 = 6.48\] Post-earnings Call Option Price: \[C_2 = 100 \cdot 0.3955 – 105 \cdot e^{-0.05 \cdot 0.25} \cdot 0.3481 = 39.55 – 105 \cdot 0.9876 \cdot 0.3481 = 39.55 – 36.06 = 3.49\] The change in the call option premium is: \[\Delta C = C_2 – C_1 = 3.49 – 6.48 = -2.99\] Therefore, the call option premium decreases by approximately £2.99 due to the volatility crush.