Derivatives Level 4 (Investment Advice Diploma) Exam Quiz Quiz 32

The core of this question revolves around understanding how changes in correlation impact the effectiveness of a hedging strategy, specifically a short hedge using futures contracts. The key is to recognize that a perfect hedge (eliminating all risk) is only achievable when the correlation between the asset being hedged and the hedging instrument (the futures contract) is perfect (correlation = 1). Any deviation from perfect correlation introduces basis risk, which reduces the effectiveness of the hedge. The hedge ratio is calculated to minimize variance, but its success hinges on the degree of correlation. In this scenario, the initial hedge ratio is calculated based on the initial correlation. When the correlation decreases, the initial hedge ratio becomes suboptimal. The portfolio is now less effectively hedged. To determine the impact, we must consider that a lower correlation implies a weaker relationship between the price movements of the asset and the futures contract. This means that the futures contract will offset less of the asset’s price fluctuations. Specifically, the percentage of the portfolio’s value protected by the hedge will decrease. The hedge ratio is designed to minimize the variance of the combined portfolio (asset + hedge). When the correlation falls, the initial hedge ratio no longer minimizes variance; the variance increases, implying less effective risk reduction. The hedge ratio is calculated as: \[ \text{Hedge Ratio} = \rho \frac{\sigma_s}{\sigma_f} \] Where: * \( \rho \) is the correlation between the spot price and the futures price. * \( \sigma_s \) is the standard deviation of the spot price changes. * \( \sigma_f \) is the standard deviation of the futures price changes. Initially, the hedge ratio is 0.8 * (0.10 / 0.12) = 0.6667. This means 0.6667 futures contracts are sold for each unit of the asset being hedged. If the correlation drops to 0.6, the *optimal* hedge ratio changes to 0.6 * (0.10 / 0.12) = 0.5. However, the *actual* hedge ratio remains at 0.6667. The effectiveness of the hedge has decreased because the hedge is now “over-hedged” relative to the new correlation. The original hedge protected a certain percentage of the portfolio’s value. The new, lower correlation means the hedge now protects a smaller percentage. To quantify this, we recognize that the initial hedge was designed to offset a certain amount of price movement based on the higher correlation. With a lower correlation, the same hedge now offsets a smaller proportion of the potential price movement.

The question revolves around the concept of hedging currency risk using futures contracts, specifically in the context of a UK-based company importing goods priced in USD. The key is to understand how futures contracts can lock in an exchange rate and mitigate the risk of adverse currency movements. The company needs to buy USD, so it should buy USD futures. The calculation involves determining the number of contracts needed to hedge the exposure. Each contract covers a specific amount of USD (e.g., USD 125,000). The company’s total USD exposure is USD 7,500,000. Therefore, the number of contracts needed is the total exposure divided by the contract size. Number of contracts = Total USD exposure / Contract size = 7,500,000 / 125,000 = 60 contracts. The company will buy 60 USD futures contracts to hedge their USD exposure. A crucial element is understanding basis risk. Basis risk arises because the spot rate and the futures rate may not converge perfectly at the delivery date. This difference can result in a hedging outcome that is slightly different from the intended result. For instance, if the spot rate strengthens against the futures rate, the company may pay slightly more for the USD than initially anticipated when the hedge was put in place. Conversely, if the spot rate weakens more than the futures rate, the company may end up paying less. Consider a scenario where the spot rate is £0.80/USD when the company needs to make the payment, but the futures contract settles at £0.79/USD. The company effectively bought USD at £0.79/USD through the futures contract, which is better than the spot rate. However, this also illustrates the potential for basis risk to result in a less-than-perfect hedge. The question also touches on the regulatory aspects of using derivatives for hedging. Under UK regulations (e.g., FCA rules), companies must ensure that their use of derivatives is appropriate for their risk profile and that they have adequate risk management controls in place. This includes understanding the potential impact of margin calls and the need to monitor the hedge’s effectiveness regularly.

The core of this question lies in understanding how hedging with futures contracts works in the context of basis risk. Basis risk arises because the price of the asset being hedged (in this case, crude oil) may not move perfectly in tandem with the price of the futures contract used for hedging. This difference can stem from various factors like location differences (the oil refinery is in Rotterdam, but the futures contract might be based on Brent crude), quality differences in the oil, or the time to delivery of the futures contract versus the actual sale date of the refined product. To solve this, we need to calculate the expected revenue considering the potential price movements of both the refined product and the futures contract. First, calculate the initial hedge position by selling futures contracts to cover the expected production. Then, determine the impact of the price changes on both the physical asset (refined oil) and the futures contracts. The profit or loss on the futures contracts will offset some of the change in the value of the refined oil. Finally, consider the basis risk, which is the difference between the spot price of the refined product and the futures price at the time of settlement. Here’s the calculation: 1. **Initial Hedge:** The refinery expects to produce 100,000 barrels of refined oil. Each ICE Brent Crude Oil futures contract covers 1,000 barrels. Therefore, the refinery sells 100 futures contracts (100,000 / 1,000). 2. **Refined Oil Revenue:** The price of refined oil decreases from $85 to $80 per barrel, resulting in a revenue of 100,000 * $80 = $8,000,000. 3. **Futures Contract Profit/Loss:** The futures price decreases from $83 to $79 per barrel, a decrease of $4 per barrel. The refinery sold 100 contracts, each covering 1,000 barrels, resulting in a profit of 100 * 1,000 * $4 = $400,000. 4. **Total Revenue with Hedge:** The total revenue is the refined oil revenue plus the profit from the futures contracts: $8,000,000 + $400,000 = $8,400,000. 5. **Unhedged Revenue:** If the refinery hadn’t hedged, the revenue would simply be 100,000 * $80 = $8,000,000. 6. **Hedge Effectiveness:** The hedge increased revenue by $400,000. Now, let’s consider a slightly different scenario to illustrate basis risk further. Suppose the futures price only decreased to $81 instead of $79. Then, the profit on the futures contracts would be 100 * 1,000 * ($83 – $81) = $200,000. The total revenue would be $8,000,000 + $200,000 = $8,200,000. This illustrates that the hedge is not perfect, and the refinery still experiences some impact from the price decrease, albeit less than if they hadn’t hedged at all. Another point to consider is the concept of “rolling” the hedge. If the refinery needs to hedge production beyond the expiry of the initial futures contracts, they would need to close out their existing positions and open new positions in contracts with a later expiry date. This process introduces additional basis risk and transaction costs. Finally, the refinery could also consider using options instead of futures. For example, they could buy put options on crude oil. This would provide downside protection while allowing them to benefit if the price of oil actually increased. However, options have an upfront cost (the premium), which needs to be factored into the hedging strategy.

The question explores the application of put-call parity to identify arbitrage opportunities when transaction costs are involved. Put-call parity is a fundamental concept in derivatives pricing, stating that a portfolio consisting of a European call option and a present value of the strike price should have the same value as a portfolio consisting of a European put option and the underlying asset. The formula for put-call parity 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 current stock price. In this scenario, transaction costs are introduced, which can create deviations from the theoretical put-call parity relationship, potentially leading to arbitrage opportunities. The key is to identify situations where the cost of creating a synthetic asset is less than the cost of the actual asset, or vice versa, after accounting for transaction costs. Here’s how to determine the arbitrage strategy: 1. **Calculate the theoretical put-call parity value:** Determine the expected price based on the put-call parity formula. In this case, the theoretical relationship is: Call + PV(Strike) = Put + Stock. 2. **Incorporate transaction costs:** Add transaction costs to the appropriate sides of the equation based on whether you are buying or selling the assets. Buying increases costs, while selling generates revenue (net of costs). 3. **Identify mispricing:** Compare the cost of the synthetic asset (e.g., synthetic stock created using call and put options) with the actual asset. If the synthetic asset is cheaper, buy the synthetic asset and sell the actual asset. If the synthetic asset is more expensive, sell the synthetic asset and buy the actual asset. 4. **Execute the arbitrage:** Implement the identified strategy by simultaneously buying the undervalued portfolio and selling the overvalued portfolio to lock in a risk-free profit. In this specific case, the present value of the strike price is calculated as \(PV(K) = \frac{K}{(1 + r)^t} = \frac{105}{(1 + 0.05)^{0.5}} = 102.47\). The strategy involves comparing the cost of buying the stock directly versus creating a synthetic stock using the call and put options, considering the transaction costs. By selling the call and buying the put, one creates a short position in the synthetic stock. The overall cost of each strategy is then compared, and the arbitrage involves choosing the cheaper strategy to replicate the stock position. The profit is derived from the difference in costs, adjusted for transaction costs.

The question focuses on calculating the theoretical price of a European call option using the Black-Scholes model and then assessing the impact of transaction costs on a hedging strategy using that option. The Black-Scholes model is a cornerstone of options pricing, and understanding its application is crucial. Transaction costs, however, often overlooked in theoretical models, can significantly erode the profitability of hedging strategies. First, we calculate the d1 and d2 values required for the Black-Scholes formula: \[d_1 = \frac{ln(\frac{S}{K}) + (r – q + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}}\] \[d_2 = d_1 – \sigma\sqrt{T}\] Where: S = Current stock price = 450 K = Strike price = 460 r = Risk-free interest rate = 0.05 q = Dividend yield = 0.02 σ = Volatility = 0.25 T = Time to expiration = 0.5 \[d_1 = \frac{ln(\frac{450}{460}) + (0.05 – 0.02 + \frac{0.25^2}{2})0.5}{0.25\sqrt{0.5}} = \frac{-0.022 + (0.03 + 0.03125)0.5}{0.25 \times 0.707} = \frac{-0.022 + 0.030625}{0.17675} = 0.0482\] \[d_2 = 0.0482 – 0.25\sqrt{0.5} = 0.0482 – 0.17675 = -0.12855\] Next, we use the cumulative standard normal distribution function, N(x), to find N(d1) and N(d2). Approximating: N(0.0482) ≈ 0.5192 N(-0.12855) ≈ 0.4489 Now, we apply the Black-Scholes formula: \[C = SN(d_1)e^{-qT} – Ke^{-rT}N(d_2)\] \[C = 450 \times 0.5192 \times e^{-0.02 \times 0.5} – 460 \times e^{-0.05 \times 0.5} \times 0.4489\] \[C = 450 \times 0.5192 \times 0.99005 – 460 \times 0.9753 \times 0.4489\] \[C = 231.834 \times 0.99005 – 448.638 \times 0.4489\] \[C = 229.527 – 201.492 = 28.035\] Therefore, the theoretical price of the call option is approximately 28.04. Now consider the transaction costs: Shares purchased for hedging: Delta = N(d1) = 0.5192. For 100 options, you need 52 shares (rounding up to ensure adequate hedge). Cost of buying 52 shares: 52 * 450 * 1.001 = 23423.4 Initial option premium received: 100 * 28.04 = 2804 Net initial cost: 23423.4 – 2804 = 20619.4 At expiration, the stock price is 470. Value of shares: 52 * 470 = 24440 Payoff of 100 call options: (470-460) * 100 = 1000 Net profit: 24440 + 1000 – 20619.4 = 3820.6 Now consider transaction costs at expiration: Cost of selling 52 shares: 52 * 470 * 0.999 = 24440 * 0.999 = 24171.56 Net profit: 24171.56 + 1000 – 20619.4 = 4552.16 – 1000 = 4552.16

To determine the most suitable hedging strategy, we need to calculate the potential profit/loss from each strategy under the given scenarios. This involves understanding how options pricing is affected by changes in the underlying asset’s price and volatility. Scenario 1: Share price increases to 160p, Volatility increases to 35% * **Covered Call:** The call option will be exercised. Profit from the share = 160p – 120p = 40p. Loss from the call option = 20p premium received – (160p – 140p) = -20p. Net profit = 40p – 20p = 20p. * **Protective Put:** The put option expires worthless. Profit from the share = 160p – 120p = 40p. Cost of the put option = 10p. Net profit = 40p – 10p = 30p. * **Straddle:** Both options will be in the money. Profit from the call option = (160p – 140p) – 15p = 5p. Profit from the put option = 0 – 15p = -15p. Profit from the share = 160p – 120p = 40p. Net profit = 40p + 5p – 15p = 30p. * **Bull Spread:** The 150p call will be exercised, while the 170p call expires worthless. Profit from 140p call = (160p – 140p) – 10p = 10p. Loss from 170p call = 5p. Net profit = 10p – 5p = 5p. Profit from the share = 160p – 120p = 40p. Net profit = 40p + 5p = 45p. Scenario 2: Share price decreases to 90p, Volatility decreases to 15% * **Covered Call:** The call option expires worthless. Profit from the share = 90p – 120p = -30p. Profit from the call option premium = 20p. Net loss = -30p + 20p = -10p. * **Protective Put:** The put option will be exercised. Loss from the share = 120p – 90p = -30p. Profit from the put option = (130p – 90p) – 10p = 30p. Net profit = -30p + 30p = 0p. * **Straddle:** The put option will be in the money. Loss from the call option premium = 15p. Profit from the put option = (130p – 90p) – 15p = 25p. Loss from the share = 90p – 120p = -30p. Net profit = -30p – 15p + 25p = -20p. * **Bull Spread:** Both options expire worthless. Profit from the share = 90p – 120p = -30p. Loss from the bull spread premium = 10p + 5p = 15p. Net loss = -30p – 15p = -45p. Comparing the net profits/losses across both scenarios: * Covered Call: Scenario 1 (20p), Scenario 2 (-10p) * Protective Put: Scenario 1 (30p), Scenario 2 (0p) * Straddle: Scenario 1 (30p), Scenario 2 (-20p) * Bull Spread: Scenario 1 (45p), Scenario 2 (-45p) The protective put provides the most consistent outcome, limiting losses in a down market while still allowing participation in an up market.

The question assesses the understanding of how implied volatility impacts option pricing, specifically focusing on the vega risk metric and its relationship to time decay (theta). A key concept is that vega is highest for at-the-money (ATM) options with longer times to expiration. An increase in implied volatility will increase the price of all options, but the increase is more pronounced for ATM options. Theta, on the other hand, is the rate at which an option’s value decays with the passage of time. For ATM options, theta is generally negative, meaning the option loses value as time passes. However, deep in-the-money (ITM) or out-of-the-money (OTM) options can have positive theta near expiration. The scenario presented combines these concepts to test the candidate’s ability to apply them in a practical trading situation. To calculate the impact, we need to consider the vega of the option and the change in implied volatility. Vega represents the change in option price for a 1% change in implied volatility. In this case, the option has a vega of 0.05, and implied volatility increases by 3%. Therefore, the price increase due to volatility is 0.05 * 3 = 0.15. The theta of -0.02 means the option loses 0.02 in value per day. Over 5 days, the loss is 0.02 * 5 = 0.10. The net change in the option price is the increase due to volatility minus the decrease due to time decay: 0.15 – 0.10 = 0.05. Therefore, the option price is expected to increase by £0.05.

This question tests the candidate’s understanding of exotic derivatives, specifically barrier options, and their sensitivity to market movements near the barrier. The scenario involves a ‘knock-in’ call option, where the option only becomes active if the underlying asset’s price touches the barrier. The key here is to understand how the probability of the barrier being hit changes as the asset price approaches the barrier, and how this affects the option’s value and delta. The incorrect options present plausible but flawed reasoning regarding the impact of the barrier proximity on the option’s value and delta. The correct answer considers that as the asset price nears the barrier, the probability of the option being ‘knocked in’ increases. This increased probability dramatically changes the option’s delta, as it transitions from zero (before being knocked in) to a delta similar to a regular call option. Here’s a breakdown: 1. **Initial State:** Before the barrier is hit, the knock-in option has a delta close to zero because it’s essentially worthless unless the barrier is breached. 2. **Approaching the Barrier:** As the asset price approaches the barrier, the probability of the barrier being hit increases significantly. This makes the option more valuable because it’s more likely to become a standard call option. 3. **Delta Change:** The delta of the option will increase dramatically as the underlying price gets closer to the barrier. The closer the price is to the barrier, the higher the chance of the option being activated, so the delta will change from close to zero to a value close to that of a vanilla call option (assuming the barrier is hit). 4. **Gamma Considerations:** Gamma, which measures the rate of change of delta, will also be high near the barrier. This is because a small change in the underlying asset’s price can dramatically change the probability of the barrier being hit, and thus significantly change the delta of the option. The other options are incorrect because they misunderstand the dynamics of a knock-in option near the barrier. Option b) incorrectly states that the delta decreases, which is the opposite of what happens. Option c) incorrectly focuses on the time to maturity without considering the barrier effect. Option d) incorrectly states that the delta remains constant, which is not true as the probability of the barrier being breached changes.

The question assesses the understanding of delta hedging and its limitations when dealing with significant price jumps in the underlying asset, a common scenario in volatile markets. Delta hedging aims to neutralize the risk associated with small price movements by adjusting the portfolio’s position in the underlying asset. However, it is most effective for small, incremental changes. When a large, unexpected price jump occurs, the delta hedge becomes less effective because the delta itself changes rapidly (a concept known as gamma). The hedge is calibrated to the old price and delta, not the new reality. Furthermore, transaction costs associated with rebalancing the hedge after a large price movement can erode profits and make continuous hedging impractical. The payoff from the option is path-dependent to some extent, as early large moves can influence the overall profitability of the strategy. In the scenario, the initial delta hedge is designed to protect against small price fluctuations in the stock. However, the sudden 15% drop significantly alters the option’s delta. The hedge, initially set to offset the risk of small movements, is now insufficient to cover the losses incurred due to the large price change. Rebalancing the hedge immediately after the drop would incur transaction costs. The overall profit will be less than expected because the delta hedge only works perfectly for infinitesimal movements, and the discrete jump renders the initial hedge inadequate. The initial hedge ratio was calculated based on the assumption of gradual price changes. The sudden jump invalidates this assumption, leading to a hedging error. Let’s consider an example: Suppose an investor delta hedges a short call option position. The underlying stock is trading at £100, and the option has a delta of 0.5. The investor sells the call option and buys 50 shares to delta hedge. If the stock price unexpectedly drops to £85, the option’s delta might decrease to 0.2. The initial hedge of 50 shares is now an over-hedge, and the investor would need to sell shares to rebalance. However, the losses incurred from the stock price drop before rebalancing will reduce the overall profit. Transaction costs associated with selling shares further decrease the profit. If the option expires in the money, the investor’s profit will be significantly less than expected due to the initial inadequate hedge and subsequent rebalancing costs.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their payoff structures under different market conditions. Barrier options are path-dependent options whose payoff depends on whether the underlying asset’s price reaches or breaches a pre-defined barrier level during the option’s life. In this scenario, we have a knock-out put option. This means the option ceases to exist if the underlying asset’s price touches or exceeds the barrier level. The investor is essentially betting that the asset price will decrease below the strike price without ever hitting the barrier. The calculation involves determining if the barrier was breached and, if not, calculating the payoff of a standard put option. The barrier was set at 105. The highest price reached during the option’s life was 103, so the barrier was not breached. Therefore, the option did not knock out. The payoff of a put option is given by: Payoff = max(Strike Price – Final Asset Price, 0). In this case, the strike price is 100 and the final asset price is 90. Therefore, the payoff is max(100 – 90, 0) = 10. Since the investor purchased 1000 options, the total payoff is 10 * 1000 = £10,000. This question requires a deep understanding of barrier options, payoff calculations, and the ability to apply these concepts in a specific market scenario. It also emphasizes the path-dependent nature of barrier options, differentiating them from standard European or American options. Understanding regulatory implications is crucial, as the sale of complex products like barrier options is subject to suitability assessments under FCA guidelines to protect retail investors. Furthermore, the scenario highlights the importance of monitoring the underlying asset’s price throughout the option’s life, as the payoff is contingent on the barrier not being breached.

The question assesses understanding of the impact of market microstructure on derivative pricing, particularly the effect of bid-ask spreads on option prices. The correct answer recognizes that the bid-ask spread represents a transaction cost that must be factored into arbitrage-free pricing models. Wider spreads increase the cost of hedging and arbitrage, leading to a wider range of acceptable option prices. Consider a scenario where a market maker is quoting options on a highly volatile stock. The theoretical price of a call option, derived from a model like Black-Scholes, might be £5.00. However, due to the bid-ask spread, the market maker is willing to buy the option for £4.90 and sell it for £5.10. An arbitrageur attempting to exploit a mispricing relative to a synthetic option position (created using the underlying stock and a risk-free bond) must account for this £0.20 spread. If the spread is wider, say £0.50 (buy at £4.75, sell at £5.25), the arbitrage opportunity needs to be larger to overcome the transaction cost. Furthermore, the existence of a bid-ask spread affects hedging strategies. A delta-neutral hedge involves continuously adjusting the position in the underlying asset to offset changes in the option’s price. Each adjustment incurs transaction costs due to the bid-ask spread. A wider spread necessitates less frequent adjustments, as smaller price discrepancies become unprofitable to exploit. This, in turn, increases the risk of the hedge, as the portfolio is less perfectly matched to the option’s exposure. The market maker will therefore widen the bid-ask spread to compensate for this increased risk. Finally, regulatory factors, such as MiFID II requirements for best execution, influence how firms manage bid-ask spreads. Firms must demonstrate that they are obtaining the best possible price for their clients, which may involve executing trades across multiple venues to minimize the impact of spreads. This added complexity and cost can further widen spreads, particularly for less liquid derivatives.

The question revolves around the impact of a sudden, unexpected regulatory change on the valuation of a complex structured note incorporating a credit default swap (CDS) referencing a basket of UK corporate bonds. The key is understanding how increased capital reserve requirements imposed by the Prudential Regulation Authority (PRA) on banks holding these notes would affect the CDS spread, the overall note valuation, and the actions a portfolio manager might take. The increase in capital reserve requirements directly impacts the cost of holding the structured note for banks. Banks, facing higher capital costs, will demand a higher return on the note to compensate. This increased return requirement translates to a widening of the CDS spread, as the perceived risk associated with the underlying corporate bonds increases due to the increased cost of hedging that risk. The valuation of the structured note is inversely related to the CDS spread. A wider CDS spread means the market perceives a higher probability of default or credit deterioration in the underlying bonds. This higher perceived risk reduces the present value of the future cash flows from the note, leading to a lower overall valuation. Given this scenario, a portfolio manager holding the structured note has several options. They could hold the note, hoping the regulatory impact is temporary. They could hedge the increased credit risk by purchasing additional protection on the underlying bonds (though this will be more expensive now). They could reduce their exposure by selling a portion of the note. Or, they could attempt to restructure the note to reduce its capital weighting under the new regulations. The calculation of the impact is as follows: 1. **Initial CDS Spread:** 150 basis points (bps) or 1.5% 2. **Increase in Capital Reserve Requirement:** 2% 3. **Estimated Increase in CDS Spread:** We assume the CDS spread widens to reflect the increased cost of capital for banks holding the structured note. A reasonable estimate is that the CDS spread widens by half the increase in the capital reserve requirement, reflecting the banks’ increased cost of doing business. So, the increase is 1% (half of 2%). The new CDS spread is therefore 1.5% + 1% = 2.5% or 250 bps. 4. **Impact on Note Valuation:** The exact impact on the note’s valuation requires a complex pricing model, but we can approximate it. Assuming the note has a maturity of 5 years and the discount rate is primarily driven by the CDS spread, we can estimate the present value impact. A simplified approximation would be to consider the present value of the increased cost of protection. The present value factor for a 5-year annuity at a rate of 2.5% is approximately 4.5. The increase in cost of protection is 1% per year. Therefore, the approximate decrease in the note’s value is 4.5 * 1% = 4.5%. If the note was initially valued at par (100), the new approximate value is 100 – 4.5 = 95.5. Therefore, the portfolio manager should expect the note’s value to decrease by approximately 4.5% and should consider reducing their exposure.

Let’s break down how to determine the maximum profit potential of a butterfly spread, and why the other options are incorrect. A butterfly spread is a neutral options strategy designed to profit from a stock trading in a narrow range. It involves buying one call option with a lower strike price, selling two call options with a middle strike price, and buying one call option with a higher strike price. All options have the same expiration date. The maximum profit occurs when the stock price at expiration equals the strike price of the short calls (the middle strike). The formula for maximum profit is: Maximum Profit = (Middle Strike – Lower Strike) – Net Premium Paid Let’s assume the following strikes and premiums: * Buy 1 Call @ Lower Strike (K1) = 95, Premium = £7 * Sell 2 Calls @ Middle Strike (K2) = 100, Premium = £4 each (Total £8) * Buy 1 Call @ Higher Strike (K3) = 105, Premium = £2 Net Premium Paid = £7 + £2 – £8 = £1 Maximum Profit = (£100 – £95) – £1 = £5 – £1 = £4 Now, consider why the other options are wrong: * **Option B (Incorrect):** Miscalculates the net premium or incorrectly applies the strike price differences. It might assume profit is simply the difference between the highest and lowest strike, ignoring the premium paid. * **Option C (Incorrect):** May stem from confusing the butterfly spread with another strategy, like a condor spread, or by adding the premium instead of subtracting. * **Option D (Incorrect):** Could arise from incorrectly calculating the net debit and then subtracting the difference between the high and middle strike prices, misunderstanding how profit is realized at the middle strike. The key to a butterfly spread is that the maximum profit is capped and occurs only at the middle strike price. Understanding the interplay of the long and short calls and the initial premium is crucial.

The question assesses understanding of how implied volatility, time to expiration, and the risk-free rate influence option prices, specifically focusing on the *Rho* of an option. Rho measures the sensitivity of an option’s price to changes in the risk-free interest rate. A higher risk-free rate generally increases the price of call options and decreases the price of put options, all else being equal. The magnitude of this effect is greater for options with longer times to expiration. Here’s a breakdown of why the correct answer is what it is, and why the others are not: * **Impact of Risk-Free Rate:** An increase in the risk-free rate makes it more attractive to hold the underlying asset, increasing the call option price. For put options, the effect is reversed. * **Time to Expiration:** The longer the time to expiration, the greater the impact of changes in the risk-free rate on the option’s price. This is because the interest rate effect has more time to accumulate. * **Implied Volatility:** Implied volatility reflects the market’s expectation of future price fluctuations. While implied volatility affects the overall option price, it does not directly dictate the *direction* of the price change due to changes in the risk-free rate (that is Rho). It mainly affects the *magnitude* of the change in the option price, which is captured by Vega. * **Moneyness:** The “moneyness” of an option (whether it’s in-the-money, at-the-money, or out-of-the-money) affects the magnitude of Rho. At-the-money options typically have the highest Rho. The correct answer accurately reflects these relationships. The incorrect answers misrepresent the direction or magnitude of the impact of the risk-free rate, time to expiration, and implied volatility on option prices. For example, suggesting that higher implied volatility *decreases* the impact of risk-free rate changes is incorrect, as implied volatility primarily affects Vega.

Let’s consider a scenario where a fund manager uses options to manage portfolio risk. The fund holds a significant position in UK equities and is concerned about a potential market downturn triggered by unexpected changes in the Bank of England’s monetary policy. To protect the portfolio, the manager implements a collar strategy using FTSE 100 index options. The fund manager sells call options with a strike price significantly above the current market price (out-of-the-money) and simultaneously buys put options with a strike price somewhat below the current market price (out-of-the-money). The premiums received from selling the calls partially offset the cost of buying the puts. This strategy provides downside protection while limiting potential upside gains. Now, consider how changes in implied volatility affect the value of the collar. If implied volatility increases, the value of both the call and put options will increase. However, the puts, which provide downside protection, will generally increase in value more than the calls, as they are further in the money relative to the expected market move in a downturn scenario. This is because the delta of the put options will become more negative, indicating a greater sensitivity to downward price movements. The net effect of increased volatility on the collar will be positive, as the increased value of the puts outweighs the increased value of the calls. Conversely, if implied volatility decreases, the value of both the call and put options will decrease. The puts will generally decrease in value more than the calls, leading to a negative effect on the collar’s overall value. This is because the delta of the put options will become less negative, indicating a reduced sensitivity to downward price movements. To calculate the approximate change in the collar’s value, we can use the concept of vega. Vega measures the sensitivity of an option’s price to changes in implied volatility. The vega of the put options will be positive, indicating that their value increases with increasing volatility. The vega of the call options will also be positive, but typically smaller. The net vega of the collar will be the difference between the vega of the puts and the vega of the calls. The change in the collar’s value can be approximated by multiplying the net vega by the change in implied volatility. For example, if the net vega of the collar is 0.05 (meaning the collar’s value changes by £0.05 for every 1% change in implied volatility) and implied volatility increases by 2%, the collar’s value would increase by approximately £0.10.

The question assesses understanding of option pricing models, specifically the Black-Scholes model, and the impact of implied volatility on option prices. It tests the ability to calculate the theoretical price of a call option using the Black-Scholes formula and to interpret the relationship between implied volatility and option premiums. The scenario involves a complex financial instrument (convertible bond) with an embedded call option, requiring the candidate to isolate and value the option component. The Black-Scholes model is used to determine the theoretical price of a European-style call option. The formula is: \[C = S_0N(d_1) – Ke^{-rT}N(d_2)\] Where: * \(C\) = Call option price * \(S_0\) = Current stock price * \(K\) = Strike price * \(r\) = Risk-free interest rate * \(T\) = Time to expiration (in years) * \(N(x)\) = Cumulative standard normal distribution function * \(e\) = the exponential constant (approximately 2.71828) And \(d_1\) and \(d_2\) are calculated as follows: \[d_1 = \frac{ln(\frac{S_0}{K}) + (r + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}}\] \[d_2 = d_1 – \sigma\sqrt{T}\] Where: * \(\sigma\) = Volatility of the stock In this scenario: * \(S_0 = £100\) * \(K = £110\) * \(r = 5\%\) or 0.05 * \(T = 1\) year * \(\sigma = 20\%\) or 0.20 First, calculate \(d_1\): \[d_1 = \frac{ln(\frac{100}{110}) + (0.05 + \frac{0.20^2}{2})1}{0.20\sqrt{1}}\] \[d_1 = \frac{ln(0.9091) + (0.05 + 0.02)1}{0.20}\] \[d_1 = \frac{-0.0953 + 0.07}{0.20}\] \[d_1 = \frac{-0.0253}{0.20} = -0.1265\] Next, calculate \(d_2\): \[d_2 = d_1 – \sigma\sqrt{T}\] \[d_2 = -0.1265 – 0.20\sqrt{1}\] \[d_2 = -0.1265 – 0.20 = -0.3265\] Now, find the values of \(N(d_1)\) and \(N(d_2)\). We can approximate these values, assuming \(N(-0.1265) \approx 0.45\) and \(N(-0.3265) \approx 0.37\). Finally, calculate the call option price \(C\): \[C = 100 \times 0.45 – 110 \times e^{-0.05 \times 1} \times 0.37\] \[C = 45 – 110 \times e^{-0.05} \times 0.37\] \[C = 45 – 110 \times 0.9512 \times 0.37\] \[C = 45 – 38.02\] \[C = 6.98\] Therefore, the theoretical price of the embedded call option is approximately £6.98. Now, let’s consider the impact of implied volatility. Implied volatility is the market’s expectation of future volatility, derived from option prices. A higher implied volatility means the market expects larger price swings in the underlying asset. This increased uncertainty raises the value of both call and put options because there’s a greater chance the option will end up in the money. In the context of a convertible bond, the embedded call option benefits from higher implied volatility. If the implied volatility used in pricing the convertible bond was significantly lower than the market’s current implied volatility for similar options, the bond might be undervalued. Conversely, if the implied volatility used was higher, the bond might be overvalued. Therefore, accurately assessing and incorporating implied volatility is crucial for pricing and trading derivatives, especially those embedded within more complex financial instruments.

To determine the fair price of the Asian option, we must calculate the arithmetic average of the asset’s price over the specified period. The payoff of an Asian call option is the maximum of zero and the difference between the average price and the strike price, which is expressed as max(0, Average Price – Strike Price). 1. **Calculate the Arithmetic Average:** \[ \text{Average Price} = \frac{S_1 + S_2 + S_3 + S_4 + S_5}{5} = \frac{105 + 108 + 112 + 109 + 115}{5} = \frac{549}{5} = 109.8 \] 2. **Determine the Payoff:** \[ \text{Payoff} = \max(0, \text{Average Price} – \text{Strike Price}) = \max(0, 109.8 – 110) = \max(0, -0.2) = 0 \] 3. **Discount the Payoff:** \[ \text{Present Value} = \frac{\text{Payoff}}{(1 + r)^n} = \frac{0}{(1 + 0.05)^5} = 0 \] Therefore, the fair price of the Asian call option is £0. In this context, imagine a UK-based exporter, “Thames Exports,” using an Asian option to hedge against currency fluctuations. Thames Exports will receive payments in USD over the next five months. They fear the GBP/USD exchange rate might become unfavorable, reducing their GBP revenue. An Asian option, in this case, provides a payoff based on the *average* exchange rate over the period, not just the final rate. This makes it less sensitive to extreme daily fluctuations than a standard European or American option, offering a smoother hedge. If the average exchange rate is better than their strike price, they don’t exercise the option. If it’s worse, the option provides a payout that offsets some of their losses. The exporter’s risk management team must understand how the averaging mechanism affects the option’s price and its effectiveness as a hedging tool under various market conditions. Regulatory compliance, specifically under EMIR, requires Thames Exports to accurately value and report these derivative positions, adding another layer of complexity.

This question explores the application of put-call parity in a complex scenario involving transaction costs and dividends. Put-call parity is a fundamental relationship that defines the theoretical price relationship between European put and call options, a corresponding stock, and a risk-free bond. The formula is: Call Price + Present Value of Strike Price = Put Price + Stock Price + Present Value of Dividends. Transaction costs introduce a deviation from the theoretical parity. The investor faces costs when buying or selling the underlying asset or the options. This cost impacts the arbitrage profit and thus the bounds of the parity. Similarly, dividends paid during the option’s life reduce the stock price, impacting the parity relationship. The present value of these dividends needs to be factored in. To find the range within which no-arbitrage exists, we need to consider the transaction costs when buying or selling the assets to exploit the mispricing. We calculate the upper and lower bounds by considering the costs incurred when establishing a specific arbitrage position. Upper Bound: This represents the highest call price where arbitrage is still possible. We calculate this by assuming we buy the stock, buy a put, and sell a call. The cost includes the stock price, the put price, and the transaction costs for buying the stock and the put. The revenue is from selling the call and the risk-free return on the strike price. Lower Bound: This represents the lowest call price where arbitrage is still possible. We calculate this by assuming we sell the stock, sell a put, and buy a call. The cost includes the call price and the transaction costs for buying the call. The revenue is from selling the stock, the put price, and the risk-free return on the strike price. In this case, the upper bound is calculated as: Stock Price + Put Price + PV of Dividends – PV of Strike Price + Transaction Costs on Stock and Put = 52 + 4 + 1.5 – (50 / (1.05)) + (0.01 * 52) + (0.01 * 4) = 52 + 4 + 1.5 – 47.62 + 0.52 + 0.04 = 10.44 The lower bound is calculated as: Stock Price + Put Price + PV of Dividends – PV of Strike Price – Transaction Costs on Stock and Put = 52 + 4 + 1.5 – (50 / (1.05)) – (0.01 * 52) – (0.01 * 4) = 52 + 4 + 1.5 – 47.62 – 0.52 – 0.04 = 9.32 Therefore, the call price must be between £9.32 and £10.44 for no arbitrage to exist.

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 underlying asset and expiration date but different strike prices. This strategy is often used when an investor has a specific outlook on the underlying asset’s price movement and wants to generate income or reduce the cost of a hedge. The calculation involves determining the net premium received or paid, and then analyzing the potential profit or loss at different price levels of the underlying asset at expiration. The breakeven point is where the profit or loss is zero. The maximum profit is capped due to the short calls, while the potential loss is theoretically unlimited if the price rises significantly. Here’s a breakdown of the calculation: 1. **Initial Setup:** * Buy 100 put options with a strike price of 95 at a premium of £3.50 each. Total cost: 100 \* £3.50 = £350. * Sell 200 call options with a strike price of 105 at a premium of £1.50 each. Total received: 200 \* £1.50 = £300. * Net Premium: £300 (received) – £350 (paid) = -£50 (Net debit of £50). 2. **Breakeven Point:** * The breakeven point is where the profit/loss is zero. We need to consider the net premium paid and the potential profit from the puts and losses from the calls. * If the stock price is below 95, the puts will be in the money. If the stock price is above 105, the calls will be in the money. * Let’s denote the breakeven point as *B*. * Profit from puts = 95 – *B* (per share). * Loss from calls = (*B* – 105) \* 2 (per share, since 200 calls). * Net Profit/Loss = 100 \* (95 – *B*) – 200 \* (*B* – 105) – 50 = 0 * 9500 – 100*B* – 200*B* + 21000 – 50 = 0 * 30500 – 300*B* – 50 = 0 * 30450 = 300*B* * *B* = 30450 / 300 = 101.5 3. **Maximum Profit:** * The maximum profit occurs when the stock price is at the strike price of the short calls (105) or higher, as the calls will offset the puts. * If the stock price is 95 or lower, the maximum profit is limited to the strike price of the short calls (105). * If the stock price is between 95 and 105, the puts will be in the money, but the calls will not. * Maximum Profit = (105 – 95) \* 100 (from the puts) – £50 (net premium) = £1000 – £50 = £950. 4. **Maximum Loss:** * The maximum loss occurs when the stock price rises significantly above the strike price of the short calls (105). * Since there are 200 calls short, the loss can be significant. * Loss = 200 \* (Stock Price – 105) – 100 \* (95 – Stock Price) + 50 * This loss is theoretically unlimited as the stock price rises. However, the put options provide some downside protection. Therefore, the breakeven point is £101.5.

Let’s break down the calculation and the rationale behind using options to manage currency risk in a global supply chain. First, we need to calculate the potential loss from the unfavourable currency movement. The initial exchange rate is £1 = $1.25, and the adverse rate is £1 = $1.20. This means for every pound, the company receives $0.05 less. The total exposure is £5 million. Therefore, the potential loss is £5,000,000 * $0.05/£ = $250,000. Now, let’s analyze the cost of the put options. The company buys 50 put options contracts, each covering £100,000. The premium per contract is $600. The total premium paid is 50 * $600 = $30,000. If the exchange rate falls to £1 = $1.20, the put options will be exercised. The strike price is £1 = $1.24. The profit per pound from the put options is $1.24 – $1.20 = $0.04. The total profit from the put options is £5,000,000 * $0.04/£ = $200,000. To determine the net outcome, we subtract the cost of the options from the profit generated by exercising them: $200,000 – $30,000 = $170,000. Finally, we subtract this net profit from the potential loss to find the overall loss: $250,000 (potential loss) – $170,000 (net profit from options) = $80,000. The key here is understanding how put options act as insurance against adverse currency movements. The company pays a premium (the cost of the options) to secure the right to sell pounds at a predetermined exchange rate (the strike price). If the exchange rate falls below the strike price, the company exercises the options, mitigating some of the losses. If the exchange rate remains above the strike price, the company lets the options expire, and the only cost is the premium paid. The scenario illustrates a common hedging strategy used by multinational corporations to protect their profit margins from currency fluctuations. By using derivatives like put options, companies can create a more predictable financial outcome, even in volatile markets. This is particularly important for companies with significant international transactions, as currency risk can significantly impact their profitability. This example showcases the practical application of derivatives in managing financial risk and highlights the importance of understanding option pricing and hedging strategies.

The question assesses the understanding of delta-hedging, specifically when the underlying asset experiences a sudden price jump. Delta-hedging aims to neutralize the directional risk of an option position by holding an offsetting position in the underlying asset. The delta of an option represents the sensitivity of the option’s price to a change in the underlying asset’s price. In this scenario, the delta is 0.6, meaning the option’s price is expected to change by £0.60 for every £1 change in the underlying asset’s price. Initially, the fund manager holds 60,000 shares to delta-hedge the short put options position. When the share price jumps to £11, the delta of the put options changes. Since put options become less sensitive to price changes as the underlying asset’s price increases (moving further out-of-the-money), the delta decreases to 0.2. This means the fund manager now needs to hold fewer shares to maintain the delta-neutral position. The new number of shares required is calculated as the new delta (0.2) multiplied by the number of options written (300,000): 0.2 * 300,000 = 60,000 shares. The fund manager initially held 60,000 shares. Now the manager needs to hold 60,000 shares. The change in shares required is 60,000 – 60,000 = 0. Therefore, the fund manager should not trade any shares to rebalance the portfolio.

The question assesses the understanding of delta hedging in a portfolio context, specifically focusing on rebalancing frequency and transaction costs. The core concept revolves around maintaining a delta-neutral portfolio to mitigate directional risk. However, frequent rebalancing, while theoretically keeping the delta close to zero, incurs transaction costs that can erode profits. Conversely, infrequent rebalancing exposes the portfolio to greater directional risk as the underlying asset’s price moves. The optimal rebalancing frequency strikes a balance between these two opposing forces. To illustrate, consider a portfolio manager, Anya, holding a long position in 1,000 shares of a stock currently priced at £50 and simultaneously shorting call options on the same stock to implement a covered call strategy. Initially, the portfolio is delta-neutral. If Anya rebalances daily, the transaction costs (brokerage fees, bid-ask spreads) might amount to £5 per rebalance. If she rebalances weekly, the portfolio’s delta might deviate significantly from zero during volatile periods, potentially leading to losses if the stock price moves adversely. The breakeven point for rebalancing frequency can be conceptualized as the point where the cost of rebalancing equals the expected loss from delta drift. Delta drift refers to the change in the portfolio’s delta due to changes in the underlying asset’s price and the passage of time. If the expected loss from delta drift exceeds the cost of rebalancing, more frequent rebalancing is justified, and vice versa. Mathematically, one could approximate the optimal rebalancing frequency by analyzing historical volatility, transaction costs, and the portfolio’s delta sensitivity. A more sophisticated approach involves simulating various rebalancing frequencies and evaluating the risk-adjusted return of each strategy. For example, one could use Monte Carlo simulation to model stock price movements and calculate the expected profit or loss for different rebalancing intervals. The question requires an understanding of the trade-offs involved and the ability to qualitatively assess the impact of different factors on the optimal rebalancing frequency. It is crucial to recognize that the optimal frequency is not static but rather depends on market conditions, the portfolio’s characteristics, and the investor’s risk tolerance.

The core of this question revolves around understanding how volatility skew affects option pricing, particularly in the context of exotic options like barrier options. Volatility skew refers to the phenomenon where implied volatility differs across different strike prices for options with the same expiration date. Typically, equity options exhibit a “volatility smile” or “volatility smirk,” where out-of-the-money puts (lower strikes) have higher implied volatilities than at-the-money options, reflecting investor demand for downside protection. This skew is critical for pricing barrier options because the probability of hitting the barrier is directly influenced by the volatility at and around the barrier level. The Black-Scholes model assumes constant volatility, a significant limitation when dealing with options where the volatility skew is pronounced. Using a single implied volatility number from an at-the-money option to price a barrier option can lead to significant mispricing, especially if the barrier is far from the current asset price. To accurately price barrier options, practitioners often use techniques that incorporate the volatility skew, such as using a volatility surface (a three-dimensional plot of implied volatility against strike price and time to expiration) or employing more sophisticated models like stochastic volatility models. In this scenario, the down-and-out put option becomes worthless if the underlying asset price hits the barrier. If the barrier is below the current asset price and implied volatility is higher at lower strikes (due to the volatility skew), the probability of the barrier being hit is underestimated if we use the at-the-money volatility. This underestimation of the barrier-hitting probability leads to an underestimation of the option’s value reduction due to the “out” feature. Therefore, the option is overpriced when using the at-the-money volatility. Conversely, if the barrier were above the current asset price and the option was an up-and-out call, the option would be underpriced using the at-the-money volatility. The adjustment needed accounts for the higher probability of the barrier being triggered, reflecting the actual market perception of risk as captured by the volatility skew.

The question explores the application of put-call parity to identify arbitrage opportunities in the context of equity options and forward contracts, considering transaction costs. Put-call parity is a fundamental concept in options pricing theory, stating a relationship between the prices of a European call option, a European put option, the underlying asset, and a risk-free bond. The 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. In this scenario, transaction costs are introduced, making the arbitrage condition slightly more complex. The investor needs to consider these costs when determining if an arbitrage opportunity exists. The present value of the strike price is calculated using the risk-free rate: \(PV(K) = \frac{K}{(1+r)^T}\), where \(K\) is the strike price, \(r\) is the risk-free rate, and \(T\) is the time to expiration. To identify an arbitrage opportunity, the investor compares the cost of creating a synthetic asset with the market price of the actual asset. If the synthetic asset is cheaper, the investor can buy the synthetic asset and sell the actual asset, profiting from the price difference less transaction costs. Conversely, if the synthetic asset is more expensive, the investor can sell the synthetic asset and buy the actual asset. In this case, the investor compares the cost of buying the put option and the underlying asset (and selling the call option) to the cost of a forward contract. The investor needs to buy the cheaper asset and sell the expensive asset, profiting from the price difference. The transaction costs reduce the potential arbitrage profit, and the investor needs to ensure that the profit after transaction costs is positive. To determine the optimal strategy, the investor calculates the potential profit from each possible arbitrage trade, taking into account the transaction costs for each component of the trade. The investor then chooses the trade that maximizes the profit after transaction costs. If the profit is negative for all possible trades, then no arbitrage opportunity exists. The forward price is calculated as \(F = S(1+r)^T\).

The question revolves around the concept of delta hedging a short call option position and the subsequent profit or loss arising from changes in the underlying asset’s price and implied volatility. The delta of a call option represents the sensitivity of the option’s price to a change in the underlying asset’s price. Delta hedging involves taking an offsetting position in the underlying asset to neutralize the risk associated with changes in the asset’s price. The initial delta hedge is constructed by buying shares equal to the call option’s delta. As the underlying asset’s price changes, the option’s delta also changes, requiring adjustments to the hedge. This process is known as dynamic hedging. The profit or loss from delta hedging depends on the accuracy of the delta estimates and the frequency of hedge adjustments. Implied volatility is a crucial factor in option pricing. It reflects the market’s expectation of the underlying asset’s future price volatility. Changes in implied volatility affect the option’s price and delta, thus impacting the effectiveness of the delta hedge. An increase in implied volatility generally increases the value of both call and put options. The calculation involves determining the number of shares to buy or sell at each rebalancing point based on the new delta, and then calculating the profit or loss from these transactions. The profit or loss on the option position itself also needs to be considered. The total profit or loss is the sum of the profit or loss from the hedge and the profit or loss from the option. Here’s how to calculate the profit/loss: 1. **Initial Hedge:** Sells 100 call options, each with a delta of 0.4, and buys 40 shares (100 options \* 0.4 delta). 2. **Price Increase to £105:** Delta increases to 0.6. Buys an additional 20 shares (100 options \* (0.6 – 0.4)). Cost = 20 shares \* £105 = £2100. 3. **Price Decrease to £102:** Delta decreases to 0.5. Sells 10 shares (100 options \* (0.5 – 0.6)). Revenue = 10 shares \* £102 = £1020. 4. **Price Decrease to £98:** Delta decreases to 0.3. Sells 20 shares (100 options \* (0.3 – 0.5)). Revenue = 20 shares \* £98 = £1960. 5. **Price Increase to £100:** Delta increases to 0.4. Buys 10 shares (100 options \* (0.4 – 0.3)). Cost = 10 shares \* £100 = £1000. 6. **Total Cost of Buying Shares:** 40 shares \* £100 + £2100 + £1000 = £4000 + £2100 + £1000 = £7100. 7. **Total Revenue from Selling Shares:** £1020 + £1960 = £2980. 8. **Net Cost of Hedging:** £7100 – £2980 = £4120. 9. **Option Value Change:** The call option expires worthless, so the profit from the short call position is the premium received, which is £3 per option \* 100 options = £300. 10. **Overall Profit/Loss:** £300 (option profit) – £4120 (hedging cost) = -£3820. Therefore, the overall outcome is a loss of £3820.

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 absence of arbitrage opportunities. While the model provides a theoretical price, real-world market prices often deviate due to factors like supply and demand, transaction costs, and investor sentiment. Implied volatility, derived by inverting the Black-Scholes formula using observed market prices, reflects the market’s expectation of future volatility. A higher implied volatility suggests greater uncertainty and, consequently, a higher option price. Gamma, a key “Greek” letter, measures the rate of change of an option’s delta with respect to changes in the underlying asset’s price. A high gamma indicates that the option’s delta is highly sensitive to price movements, making it crucial for hedging strategies, especially when the underlying asset price is near the option’s strike price. The question explores the relationship between implied volatility, gamma, and the potential for arbitrage when market prices deviate from theoretical values. The key is to understand how a mispriced option, influenced by high implied volatility and gamma, can create an arbitrage opportunity if exploited correctly. Let’s break down the arbitrage strategy: 1. **Identify the Mispricing:** The market price of the call option is higher than the Black-Scholes price, indicating overvaluation. This overvaluation is driven by high implied volatility. 2. **Hedge the Position:** Since the option has a gamma of 0.1, we need to dynamically hedge our short position. Gamma represents the change in Delta for a unit change in the underlying asset. 3. **Calculate the Profit:** The profit is the difference between the price received for selling the option and the cost of hedging, considering the change in the underlying asset’s price and the option’s gamma. The initial Black-Scholes price is calculated using the standard formula. However, the market price is higher due to high implied volatility. The profit is calculated by shorting the option at the market price and hedging using the delta. As the stock price increases, the delta changes, and the hedge needs to be adjusted. The profit is the difference between the initial premium received and the cost of the hedging adjustments. Profit = Premium Received – (Change in Stock Price * Delta) – Cost of Hedge Adjustment Profit = £7.50 – (£1 * 0.5) – (£1^2 * 0.1/2) = £7.50 – £0.5 – £0.05 = £6.95

The question assesses the understanding of option pricing models, specifically the Black-Scholes model, and how dividends impact option prices. The Black-Scholes model is a cornerstone in derivatives valuation, and its application in dividend-paying scenarios is a crucial skill for investment advisors. The core principle here is that dividends reduce the stock price on the ex-dividend date. This reduction in stock price impacts the call option price negatively, as the call option holder benefits from an increase in the stock price. The put option price, conversely, is positively impacted by the dividend, as the put option holder benefits from a decrease in the stock price. The adjusted Black-Scholes formula for European call options on dividend-paying stocks is: \(C = S_0e^{-qT}N(d_1) – Xe^{-rT}N(d_2)\) Where: \(S_0\) = Current stock price \(q\) = Continuous dividend yield \(T\) = Time to expiration \(X\) = Strike price \(r\) = Risk-free interest rate \(N(x)\) = 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 of the stock In this scenario, a single discrete dividend is paid. To adjust for this, we subtract the present value of the dividend from the current stock price before applying the standard Black-Scholes formula. The present value of the dividend is calculated as \(D * e^{-r*t}\), where D is the dividend amount, r is the risk-free rate, and t is the time until the dividend payment. 1. **Calculate the present value of the dividend:** Dividend = £2.00 Time to dividend payment = 0.25 years Risk-free rate = 5% Present Value of Dividend = \(2.00 * e^{-0.05 * 0.25} = 2.00 * e^{-0.0125} \approx 2.00 * 0.9876 = £1.9752\) 2. **Adjust the stock price:** Adjusted Stock Price = \(50.00 – 1.9752 = £48.0248\) 3. **Apply the Black-Scholes Model (using adjusted stock price):** \(d_1 = \frac{ln(\frac{48.0248}{50}) + (0.05 + \frac{0.2^2}{2})0.5}{0.2\sqrt{0.5}} = \frac{ln(0.9605) + (0.05 + 0.02)0.5}{0.2 * 0.707} = \frac{-0.0403 + 0.035}{0.1414} = -0.0375\) \(d_2 = -0.0375 – 0.2\sqrt{0.5} = -0.0375 – 0.1414 = -0.1789\) 4. **Find N(d1) and N(d2):** \(N(d_1) = N(-0.0375) \approx 0.4851\) \(N(d_2) = N(-0.1789) \approx 0.4289\) 5. **Calculate the call option price:** \(C = 48.0248 * 0.4851 – 50 * e^{-0.05 * 0.5} * 0.4289 = 23.30 – 50 * 0.9753 * 0.4289 = 23.30 – 20.95 = £2.35\) Therefore, the estimated price of the European call option is approximately £2.35.

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 profit/loss profile of a ratio spread depends on the relative movements of the underlying asset’s price and the chosen strike prices. In this case, a 1:2 put ratio spread is created, which means buying one put option and selling two put options with a lower strike price. To calculate the maximum profit, we need to consider the premium paid/received, the strike prices, and the potential outcomes at expiration. The investor buys one put option with a strike price of 500 (premium paid: £15) and sells two put options with a strike price of 480 (premium received: 2 * £6 = £12). The net premium paid is £15 – £12 = £3. The maximum profit occurs when the asset price is equal to the lower strike price. In this scenario, the bought put option (strike 500) expires in the money, and the sold put options (strike 480) also expire in the money. The profit from the bought put option is (500 – 480) – £15 = £20 – £15 = £5. The loss from each sold put option is 480 – 480 – £6 = -£6. The total loss from the two sold put options is 2 * (0 – £6) = -£12. The maximum profit is calculated as the difference between the strike prices minus the net premium paid: (500 – 480) – £3 = £20 – £3 = £17. The maximum loss occurs when the asset price is above the higher strike price. In this case, all options expire worthless, and the investor loses the net premium paid, which is £3. If the asset price falls below the lower strike price, the profit is limited. The profit is calculated as the difference between the strike prices minus the net premium paid. The breakeven point can be calculated by considering the payoff of the bought put and the two sold puts. Therefore, the maximum profit for this strategy is £17.

The core of this question revolves around understanding how implied volatility derived from options prices can be used to infer market sentiment, specifically regarding future dividend payouts. When a company announces a special dividend, or when there is uncertainty about the size or timing of future dividends, it affects the stock price and, consequently, option prices. The put-call parity theorem, which states that \(C – P = S – PV(K) – PV(Div)\), where C is the call option price, P is the put option price, S is the spot price, K is the strike price, PV(K) is the present value of the strike price, and PV(Div) is the present value of expected dividends, becomes crucial. Any significant deviation from this parity suggests an arbitrage opportunity or a mispricing due to dividend expectations. In this scenario, the implied volatility of put options being significantly higher than call options, particularly for near-the-money options, indicates that the market anticipates a significant drop in the stock price. This drop is likely attributed to an expected dividend payout. If the market consensus were that no special dividend was coming, the implied volatilities of puts and calls would be more closely aligned. The higher put volatility signals increased demand for downside protection (buying puts), driven by the fear of the stock price decreasing due to the dividend. The calculation focuses on estimating the expected dividend based on the observed implied volatility skew. We can use a modified version of the put-call parity, acknowledging the implied volatility difference, to back out the market’s expectation of the dividend. For simplicity, we can assume the difference in implied volatility directly translates to a price difference reflecting the present value of the expected dividend. Given the spot price of £150, strike price of £150, risk-free rate of 5%, and time to expiration of 0.5 years, we can calculate the present value of the strike price as \( PV(K) = \frac{K}{e^{rT}} = \frac{150}{e^{0.05 \times 0.5}} \approx 146.32 \). The difference in option prices, reflecting the implied volatility skew, is £7 (£10 for the put – £3 for the call). Therefore, using the put-call parity, we can estimate the present value of the expected dividend: \[ PV(Div) = S – PV(K) – (C – P) = 150 – 146.32 – (3 – 10) = 150 – 146.32 + 7 = 10.68 \] To find the expected dividend amount, we need to discount this present value back to the dividend payment date (0.5 years): \[ Expected\ Dividend = PV(Div) \times e^{rT} = 10.68 \times e^{0.05 \times 0.5} \approx 10.95 \] Therefore, the market is implicitly pricing in an expected dividend of approximately £10.95.

The core of this question revolves around understanding how various factors influence option prices, specifically focusing on the “Greeks.” Theta measures the rate of decline in the value of an option due to the passage of time. Vega measures the sensitivity of an option’s price to changes in the volatility of the underlying asset. Rho measures the sensitivity of an option’s price to changes in interest rates. Gamma measures the rate of change of an option’s delta with respect to changes in the price of the underlying asset. The Black-Scholes model is the cornerstone for option pricing, factoring in the underlying asset’s price, strike price, time to expiration, volatility, and risk-free interest rate. Let’s consider a European call option on a stock. The current stock price is £100, the strike price is £105, the time to expiration is 6 months (0.5 years), the risk-free interest rate is 5%, and the volatility is 20%. To determine the most significant factor affecting the option’s price over the next week, we need to consider the relative magnitudes of the “Greeks” and the likely changes in their corresponding factors. * **Theta:** Let’s assume the option has a theta of -0.05. This means the option loses £0.05 in value each day due to time decay. Over a week (7 days), this would be a loss of 7 * 0.05 = £0.35. * **Vega:** Let’s assume the option has a vega of 0.20. If volatility increases by 1% (0.01), the option price increases by 0.20. * **Rho:** Let’s assume the option has a rho of 0.03. If the interest rate increases by 1% (0.01), the option price increases by 0.03. * **Gamma:** Let’s assume the option has a gamma of 0.02. If the stock price increases by £1, the delta of the option increases by 0.02. This means the option price will increase, but the impact is secondary to the initial price change. Now, consider plausible changes in the underlying factors: * Volatility might increase by 2% due to an upcoming earnings announcement. This would result in a price change of 2 * 0.20 = £0.40 due to vega. * Interest rates are unlikely to change significantly in a week. A change of 0.1% would result in a price change of only 0.1 * 0.03 = £0.003 due to rho. * The stock price might fluctuate by £2. This would have a direct impact on the option price and also affect its delta, but the initial impact due to the price change itself would likely be more significant than the gamma effect. Comparing these potential impacts, the change in volatility (vega) and the passage of time (theta) are the most significant factors. The question requires determining which is *most* significant. If volatility is expected to increase by 2%, the impact (£0.40) would be greater than the time decay (£0.35).