Derivatives Level 4 (Investment Advice Diploma) Exam Quiz Quiz 30

The core of this question lies in understanding how changes in interest rate volatility impact the price of a swaption, specifically a payer swaption. A payer swaption gives the holder the right, but not the obligation, to *enter* into a swap as the payer of the fixed rate. The swaption’s value is derived from the underlying swap. Here’s the breakdown of why increased interest rate volatility benefits the holder of a payer swaption: 1. **Volatility and Option Value:** Options, including swaptions, increase in value as volatility increases. This is because higher volatility means a wider range of potential future outcomes for the underlying asset (in this case, the swap rate). The option holder benefits from favorable movements while being protected from unfavorable ones. 2. **Payer Swaption and Rising Rates:** A payer swaption is valuable when interest rates rise. If rates rise above the fixed rate specified in the swaption, the holder can exercise the swaption, enter into the swap, pay the fixed rate, and receive the higher floating rate. This generates a profit. 3. **Increased Volatility and Upside Potential:** Higher interest rate volatility increases the likelihood of rates rising significantly above the fixed rate. This increases the potential payoff from exercising the payer swaption. The swaption holder benefits from the possibility of a larger profit if rates rise, while their losses are limited to the premium paid for the swaption if rates fall. 4. **Analogy:** Imagine you have a ticket to buy a house for £300,000 in six months (a swaption). If house prices are stable, the ticket isn’t worth much. But if there’s high volatility in the housing market, with prices potentially soaring or plummeting, the ticket becomes more valuable. If prices soar to £400,000, you can buy the house for £300,000 and make a profit. If prices fall, you simply let the ticket expire, limiting your loss to the ticket price. 5. **Pricing Models:** While the Black-Scholes model isn’t directly applicable to swaptions, the general principle holds: option prices are positively correlated with volatility. More sophisticated models like the Hull-White model, used for pricing interest rate derivatives, also reflect this relationship. Therefore, an increase in interest rate volatility increases the value of a payer swaption because it increases the potential for the underlying swap rate to move favorably (i.e., increase) for the swaption holder.

The question assesses understanding of how implied volatility, time to expiration, and the strike price relative to the current asset price affect the value of a European call option. The core of option pricing lies in understanding the sensitivity of the option price to these parameters. Implied Volatility: Higher implied volatility reflects greater uncertainty about the future price of the underlying asset. This increases the value of a call option because there’s a higher probability the asset price will move significantly above the strike price. A call option benefits from upward price movements. Time to Expiration: A longer time to expiration increases the value of a call option. This is because there is more time for the underlying asset to increase in value beyond the strike price. The option holder has a longer window of opportunity for the option to become profitable. Strike Price: The strike price is the price at which the option holder can buy the underlying asset. A strike price significantly above the current asset price means the option is out-of-the-money. The further out-of-the-money the option is, the lower its value, as a substantial price increase is required for the option to become profitable. Conversely, a strike price below the current asset price means the option is in-the-money, increasing its value. The combined effect of these factors determines the overall value of the call option. The Black-Scholes model, while not explicitly required for calculation here, provides a framework for understanding these relationships. The question tests the conceptual understanding of these sensitivities rather than numerical calculation.

To determine the theoretical price of the Asian option, we need to calculate the average strike price and then apply the Black-Scholes model using this adjusted strike price. The average strike price is calculated as the sum of the strike prices divided by the number of strikes. In this case, the strike prices are £98, £102, and £105. The average strike price is (£98 + £102 + £105) / 3 = £101.67. Next, we use the Black-Scholes model to calculate the option price. The Black-Scholes formula is: \[ C = S_0N(d_1) – Ke^{-rT}N(d_2) \] where: \( C \) = Call option price \( S_0 \) = Current stock price = £100 \( K \) = Strike price = £101.67 (average strike price) \( r \) = Risk-free interest rate = 5% or 0.05 \( T \) = Time to expiration = 6 months or 0.5 years \( N(x) \) = Cumulative standard normal distribution function \( d_1 = \frac{ln(\frac{S_0}{K}) + (r + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}} \) \( d_2 = d_1 – \sigma\sqrt{T} \) \( \sigma \) = Volatility = 20% or 0.20 First, calculate \( d_1 \): \[ d_1 = \frac{ln(\frac{100}{101.67}) + (0.05 + \frac{0.20^2}{2})0.5}{0.20\sqrt{0.5}} \] \[ d_1 = \frac{ln(0.9835) + (0.05 + 0.02)0.5}{0.20\sqrt{0.5}} \] \[ d_1 = \frac{-0.0166 + 0.035}{0.1414} \] \[ d_1 = \frac{0.0184}{0.1414} = 0.1301 \] Next, calculate \( d_2 \): \[ d_2 = d_1 – \sigma\sqrt{T} \] \[ d_2 = 0.1301 – 0.20\sqrt{0.5} \] \[ d_2 = 0.1301 – 0.1414 = -0.0113 \] Now, find \( N(d_1) \) and \( N(d_2) \) using standard normal distribution tables or a calculator. \( N(0.1301) \approx 0.5517 \) \( N(-0.0113) \approx 0.4955 \) Finally, calculate the call option price \( C \): \[ C = 100 \times 0.5517 – 101.67 \times e^{-0.05 \times 0.5} \times 0.4955 \] \[ C = 55.17 – 101.67 \times e^{-0.025} \times 0.4955 \] \[ C = 55.17 – 101.67 \times 0.9753 \times 0.4955 \] \[ C = 55.17 – 49.18 \] \[ C = 5.99 \] Therefore, the theoretical price of the Asian call option is approximately £5.99. This calculation demonstrates the application of the Black-Scholes model to value a derivative with a strike price that is averaged over a period. The averaging feature of Asian options reduces volatility and makes them less sensitive to price fluctuations compared to standard European or American options.

Let’s analyze a scenario involving a UK-based agricultural cooperative (“FarmCo”) that wants to protect itself from fluctuations in wheat prices. FarmCo plans to deliver 5,000 metric tons of wheat in six months. To mitigate price risk, FarmCo decides to use wheat futures contracts traded on the ICE Futures Europe exchange. Each contract represents 100 metric tons of wheat. FarmCo will need 50 contracts (5,000 tons / 100 tons per contract = 50 contracts). The current futures price for wheat with a six-month delivery is £200 per metric ton. FarmCo decides to hedge by selling 50 futures contracts at this price. If, at the delivery date, the spot price of wheat is £180 per metric ton, and the futures price converges to the spot price (also £180), FarmCo will have lost £20 per ton in the physical market but gained £20 per ton in the futures market. Here’s the calculation: * **Loss in physical market:** (£180 – £200) * 5,000 tons = -£100,000 * **Gain in futures market:** (£200 – £180) * 50 contracts * 100 tons/contract = £100,000 * **Net effect:** -£100,000 + £100,000 = £0 However, hedging isn’t perfect due to basis risk. Basis risk is the risk that the futures price and spot price do not converge perfectly at the delivery date. Let’s assume the spot price at delivery is £180, but the futures price is £185. * **Loss in physical market:** (£180 – £200) * 5,000 tons = -£100,000 * **Gain in futures market:** (£200 – £185) * 50 contracts * 100 tons/contract = £75,000 * **Net effect:** -£100,000 + £75,000 = -£25,000 FarmCo experiences a net loss of £25,000 due to basis risk. This example illustrates how hedging with futures can reduce price risk but is not a perfect solution due to the potential for basis risk. The effectiveness of the hedge depends on how closely the futures price tracks the spot price. In this case, the futures price was higher than the spot price at delivery, leading to a loss.

The core of this question revolves around understanding how different option strategies behave around earnings announcements, specifically focusing on the interplay between implied volatility, time decay (theta), and the potential for significant price movements. The investor’s expectation of a large price swing post-earnings is key. A long straddle involves buying both a call and a put option with the same strike price and expiration date. This strategy profits from significant price movement in either direction. The primary risk is time decay (theta), which erodes the value of both options as time passes, and the initial cost of establishing the position. However, leading up to an earnings announcement, implied volatility typically increases, boosting the price of options (a positive vega effect). After the announcement, if the price movement isn’t substantial enough to offset the initial cost and time decay, the straddle loses money. A short strangle involves selling a call and a put option with different strike prices (one out-of-the-money call and one out-of-the-money put) but the same expiration date. This strategy profits if the underlying asset’s price remains within a defined range. The primary risk is a significant price movement outside of this range, which can lead to substantial losses. Short strangles benefit from time decay and a decrease in implied volatility. However, before an earnings announcement, increased implied volatility will negatively impact the position. The calculation to determine the most suitable strategy involves weighing the potential profit from a large price movement against the costs associated with time decay and implied volatility changes. The investor believes the price will move significantly. Therefore, strategies that benefit from large movements are favored. Since the investor expects a large price movement, the long straddle is more appropriate. The short strangle is unsuitable because the investor expects a large price move which will likely result in a loss.

The core concept tested here is the understanding of hedging strategies using options, specifically a ratio spread. A ratio spread involves buying and selling different numbers of options with the same expiration date but different strike prices. The goal is typically to profit from a specific price movement while limiting potential losses. The payoff profile is crucial to understand, as it can be complex. In this scenario, the investor is using a *bearish* ratio spread, anticipating a slight decrease in the underlying asset’s price. They are buying fewer calls at a lower strike and selling more calls at a higher strike. This strategy profits if the price stays below the higher strike but carries unlimited risk if the price rises significantly. The investor aims to benefit from the premium received from selling the two calls at the higher strike price and from the limited upside potential if the stock price declines slightly. However, if the stock price rises substantially, the investor is obligated to sell twice as many shares as they would have if they had just sold one call option. The breakeven point is the price at which the strategy neither makes nor loses money. It is calculated by considering the initial cost (or credit) of setting up the spread and the payoff at different price levels. The maximum profit is achieved when the stock price is at the higher strike price at expiration. At this point, the purchased call expires worthless, and the sold calls expire at the money. The profit is the net premium received. The maximum loss is potentially unlimited if the stock price rises significantly above the higher strike price. The investor is short two calls at the higher strike, meaning they must deliver twice the number of shares. To calculate the breakeven point, we need to consider the cost of the options. The investor buys one call at £95 for £5 and sells two calls at £100 for £2 each, receiving a net credit of £(2*2 – 5) = -£1. This means the investor receives £1 upfront. The breakeven point is calculated as: Higher Strike Price + Net Premium Received / Number of Short Calls. Breakeven Point = £100 + £1/2 = £100.50

Let’s analyze the scenario involving Albion Dynamics and the potential insider trading violation. The key is to understand what constitutes inside information and when its use becomes illegal under UK regulations, specifically the Market Abuse Regulation (MAR). Firstly, inside information is defined as information of a precise nature, which has not been made public, relating, directly or indirectly, to one or more issuers or to one or more financial instruments, and which, if it were made public, would be likely to have a significant effect on the prices of those financial instruments or on the price of related derivative financial instruments. In this case, the CFO’s comment about the “innovative restructuring” leading to a “substantial boost” in Albion Dynamics’ share price qualifies as inside information. It’s precise (restructuring), non-public, directly related to Albion Dynamics, and likely to significantly impact the share price if released. Now, let’s consider John’s actions. He overheard the conversation, which makes him an “accidental insider.” However, the critical factor is whether he *used* this information to trade. Buying shares immediately after overhearing the conversation strongly suggests he did. The FCA (Financial Conduct Authority) would investigate whether John’s trading pattern deviated from his usual investment behavior and whether he could reasonably have come to his investment decision based on publicly available information. The “reasonable investor” test is crucial. Could a reasonable investor, without access to inside information, have concluded that Albion Dynamics was a good investment at that time? If the answer is no, then John’s actions are highly suspect. The FCA would also consider the timing of the trade – its proximity to the inside information – as strong evidence of insider trading. The potential penalties for insider trading are severe, including unlimited fines and imprisonment. Moreover, John’s actions could also lead to civil penalties, such as disgorgement of profits. Let’s examine the options. Option a) is incorrect because it downplays the CFO’s comment, which clearly suggests material non-public information. Option c) is incorrect because, while John’s past performance is irrelevant, his *use* of the inside information is the key violation. Option d) is incorrect because it focuses on the legality of overhearing, which is not the primary issue. The key violation is *acting* on the inside information.

The core of this question lies in understanding how implied volatility derived from option prices reflects market sentiment regarding future price movements of the underlying asset, and how that sentiment is influenced by macroeconomic announcements. Implied volatility is essentially the market’s expectation of how much the price of an asset will fluctuate in the future. A higher implied volatility suggests greater uncertainty and a wider potential range of price outcomes. When a major macroeconomic announcement is anticipated, such as the release of inflation figures by the Office for National Statistics (ONS), investors become more uncertain about the future direction of the market. This uncertainty leads to increased demand for options, particularly those that would profit from large price swings, regardless of direction (i.e., straddles or strangles). This increased demand drives up the prices of these options, which in turn increases their implied volatility. The magnitude of the increase in implied volatility depends on several factors, including the perceived importance of the announcement, the level of uncertainty surrounding the announcement, and the time remaining until the announcement. For example, if the market widely expects the ONS inflation announcement to be benign, the increase in implied volatility will likely be smaller than if there is significant disagreement among economists about the likely outcome. The increase in implied volatility is often temporary, as it tends to decline after the announcement is released and the market has had time to digest the information. This phenomenon is known as volatility crush. The scenario with the FTSE 100 index options is a classic example of this dynamic. The implied volatility of at-the-money options expiring shortly after the ONS announcement will likely be higher than the implied volatility of options expiring well before or well after the announcement. This is because the former options are most sensitive to the uncertainty surrounding the announcement.

The question assesses understanding of delta hedging, gamma, and the profit/loss implications of rebalancing a delta-hedged portfolio. Delta hedging aims to neutralize the portfolio’s sensitivity to small changes in the underlying asset’s price. Gamma measures the rate of change of delta with respect to the underlying asset’s price. A positive gamma means the delta increases as the underlying asset price increases, and decreases as the price decreases. When gamma is positive, the hedge needs to be rebalanced more frequently. When an investor sells options, they typically have negative gamma. To delta hedge, they buy the underlying asset. If the asset price increases, their delta increases (becomes less negative), and they need to buy more of the underlying asset to maintain the delta hedge. Buying high means they lose money on the rebalancing. Conversely, if the asset price decreases, their delta decreases (becomes more negative), and they need to sell the underlying asset to maintain the delta hedge. Selling low means they also lose money on the rebalancing. This “buy high, sell low” dynamic results in a loss due to gamma. The profit/loss from gamma is approximately proportional to -0.5 * Gamma * (Change in Underlying Price)^2. Theta represents the time decay of the option. In this scenario, the investor receives the option premium upfront (positive theta) but incurs losses due to gamma as they rebalance the hedge. The overall profit or loss depends on the magnitude of the gamma losses relative to the theta gain. The problem requires calculating the approximate profit/loss using the given delta, gamma, theta, and price movement information. First, calculate the profit/loss from gamma: Gamma Loss = -0.5 * Gamma * (Change in Underlying Price)^2 * Number of Options Gamma Loss = -0.5 * 0.15 * (£3)^2 * 1000 = -£675 Next, calculate the profit from theta: Theta Profit = Theta * Number of Options Theta Profit = £0.20 * 1000 = £200 Finally, calculate the net profit/loss: Net Profit/Loss = Theta Profit + Gamma Loss Net Profit/Loss = £200 – £675 = -£475 Therefore, the investor experiences an approximate loss of £475.

The question assesses the understanding of option pricing sensitivity to changes in underlying asset volatility, specifically how different option strategies are affected by an increase in implied volatility after the strategy is established. The core concepts are Vega (sensitivity of option price to volatility) and how Vega changes for different option strategies. A short straddle (selling a call and a put with the same strike price and expiration) has negative Vega because you profit from a decrease in volatility and lose from an increase in volatility. A long strangle (buying an out-of-the-money call and put with the same expiration) has positive Vega, benefiting from increased volatility. A covered call (owning the underlying asset and selling a call option) has negative Vega due to the short call position. A protective put (owning the underlying asset and buying a put option) has positive Vega due to the long put position. The calculation is conceptual. No numerical calculation is required, but an understanding of the signs of Vega for each strategy is essential. The correct answer is a covered call strategy because it has a negative Vega, making it vulnerable to losses when volatility increases. A covered call involves selling a call option, which benefits from stable or decreasing volatility. When volatility increases, the price of the call option increases, leading to a loss for the option seller. The investor is long the underlying asset, but the short call position dominates the volatility risk. The other options are incorrect because a short straddle also has negative Vega, but the question specifically asks which of the listed strategies is most vulnerable, and covered call is a more common and simpler strategy for individual investors, making it a more realistic and plausible scenario.

The question assesses the understanding of exotic options, specifically barrier options, and their application in hedging strategies within a portfolio context. A down-and-out put option becomes worthless if the underlying asset’s price falls below a specified barrier level. This feature makes them cheaper than standard put options but also introduces the risk of the option expiring worthless even if the asset price subsequently recovers. The calculation involves determining the potential loss given the option expires worthless and comparing it to the potential loss without any hedging. The difference represents the effectiveness of the hedging strategy, considering the barrier risk. Here’s a step-by-step breakdown: 1. **Calculate the unhedged loss:** If the asset price drops to £80, the unhedged loss is £100 (initial price) – £80 = £20 per share. For 10,000 shares, the total unhedged loss is £20 * 10,000 = £200,000. 2. **Assess the barrier event:** The barrier is triggered because the asset price fell below £85. The down-and-out put option expires worthless. 3. **Calculate the loss with the ineffective hedge:** Since the option expired worthless, the portfolio is effectively unhedged. The loss remains £200,000. 4. **Consider the option premium:** The investor paid a premium of £2 per share for the option, totaling £2 * 10,000 = £20,000. This premium represents an additional cost to the hedging strategy. 5. **Calculate the net loss with the failed hedge:** The total loss is the unhedged loss plus the option premium: £200,000 + £20,000 = £220,000. 6. **Determine the cost of the ineffective hedge:** The question asks for the *cost* of the ineffective hedge, which is the difference between the unhedged loss and the loss with the failed hedge. In this case, the loss increased by the amount of the premium paid, meaning the hedge *cost* the investor an additional £20,000. The analogy here is like buying flood insurance for your house, but the policy has a clause that if the water reaches a certain level *before* flooding your house, the policy is void. If that water level is reached, and then your house floods, you’ve paid for insurance that provides no benefit. The cost of the ineffective hedge is the premium paid for the insurance. A crucial point is understanding that barrier options are cheaper because of the risk that they will expire worthless. This makes them suitable only for investors with specific views on the likely price path of the underlying asset. This question tests the understanding of the downside of barrier options, not just their potential benefits.

The core of this question lies in understanding how basis risk arises in hedging strategies using futures contracts, particularly when the asset being hedged is not perfectly correlated with the underlying asset of the futures contract. Basis risk is the risk that the price of the asset being hedged and the price of the futures contract will not move in a perfectly correlated manner. This can lead to the hedge being less effective than anticipated. Here’s a breakdown of the calculation and the concepts involved: 1. **Initial Position:** The fund manager holds £5,000,000 worth of UK small-cap equities, with a beta of 1.2 relative to the FTSE 250. This means that for every 1% move in the FTSE 250, the small-cap portfolio is expected to move by 1.2%. 2. **Hedging with FTSE 100 Futures:** The fund manager uses FTSE 100 futures to hedge. This introduces basis risk because the FTSE 100 and FTSE 250, while correlated, are not perfectly so. The hedge ratio needs to account for this imperfect correlation. 3. **Hedge Ratio Calculation:** The hedge ratio is calculated as: \[ \text{Hedge Ratio} = \beta \times \frac{\text{Portfolio Value}}{\text{Futures Contract Value}} \] In this case: \[ \text{Hedge Ratio} = 1.2 \times \frac{5,000,000}{75 \times 7,600} = 1.2 \times \frac{5,000,000}{570,000} \approx 10.53 \] This means the fund manager needs to sell approximately 10.53 FTSE 100 futures contracts. Since you can’t trade fractions of contracts, this would typically be rounded to 11 contracts for a more conservative hedge. However, the scenario states 10 contracts are sold. 4. **Market Movement:** The FTSE 100 declines by 5%, and the small-cap portfolio declines by 4%. 5. **Futures Profit/Loss:** The FTSE 100 futures contract price also declines by 5%. The profit from the short futures position is: \[ \text{Profit} = \text{Number of Contracts} \times \text{Contract Size} \times \text{Percentage Change} \times \text{Initial Index Level} \] \[ \text{Profit} = 10 \times 75 \times 0.05 \times 7,600 = £285,000 \] 6. **Portfolio Loss:** The small-cap portfolio declines by 4%: \[ \text{Loss} = \text{Portfolio Value} \times \text{Percentage Change} \] \[ \text{Loss} = 5,000,000 \times 0.04 = £200,000 \] 7. **Net Outcome:** The net outcome is the profit from the futures position minus the loss on the portfolio: \[ \text{Net Outcome} = \text{Futures Profit} – \text{Portfolio Loss} \] \[ \text{Net Outcome} = 285,000 – 200,000 = £85,000 \] This outcome illustrates basis risk. If the hedge were perfect, the profit from the futures would exactly offset the loss in the portfolio. However, because the FTSE 100 and the small-cap portfolio did not move in perfect correlation (and the hedge ratio was not perfectly precise), there is a residual profit. Had the small-cap portfolio declined *more* than expected given the FTSE 100’s movement, the net outcome could have been a loss, even with the hedge in place. The beta of 1.2 is an *expected* relationship, not a guaranteed one. Furthermore, using FTSE 100 futures to hedge a FTSE 250-correlated portfolio introduces another layer of basis risk, as these indices, while related, track different companies and market segments. This question underscores the critical importance of understanding basis risk and the limitations of hedging strategies, especially when using imperfectly correlated hedging instruments.

The question assesses the understanding of delta hedging in a portfolio context, specifically focusing on how changes in the underlying asset’s price and the passage of time (theta decay) impact the hedge. Delta hedging aims to neutralize the portfolio’s sensitivity to small changes in the underlying asset’s price. However, delta is not static; it changes as the underlying asset’s price moves or as time passes. The question requires calculating the new number of shares needed to maintain a delta-neutral position after considering both the price change and the theta decay of the options. 1. **Initial Delta:** The portfolio is initially delta-neutral, meaning the portfolio’s delta is 0. This implies that the initial number of short call options is perfectly offset by the delta of the shares held. 2. **Price Change Impact:** The underlying asset’s price decreases by £2. The options’ delta changes. A call option’s delta typically decreases when the underlying asset’s price decreases. The change in the portfolio delta due to the price change is calculated as: Change in portfolio delta = Number of options \* Change in delta per option \* Price change = 100 \* 0.05 \* (-2) = -10. This means the portfolio’s delta has become -10 due to the price decrease. 3. **Theta Decay Impact:** Theta represents the rate of change of the option’s price with respect to time. In this case, the options’ theta is -0.03, meaning the option’s price decreases by £0.03 per day due to time decay. Since delta is affected by the passage of time, we need to estimate how much the delta changes due to theta. A common approximation is to assume that the delta changes proportionally to theta. A more precise calculation would involve Gamma (the rate of change of delta with respect to the underlying asset’s price), but this approximation is reasonable given the information provided. The change in delta due to theta is: Change in delta per option due to theta = Theta \* Sensitivity Factor = -0.03 \* 0.1 = -0.003. The total change in portfolio delta due to theta is: Change in portfolio delta = Number of options \* Change in delta per option = 100 \* (-0.003) = -0.3. 4. **Total Delta Change:** The total change in the portfolio’s delta is the sum of the changes due to the price change and theta decay: Total change in delta = -10 + (-0.3) = -10.3. 5. **Shares to Rebalance:** To rebalance the portfolio and maintain delta neutrality, the portfolio manager needs to buy shares to offset the negative delta. The number of shares to buy is equal to the absolute value of the total change in delta: Number of shares to buy = 10.3. Since you can only buy whole shares, the portfolio manager needs to buy 10 shares.

To solve this problem, we need to understand how changes in implied volatility affect the price of options, specifically straddles. A straddle consists of buying both a call and a put option with the same strike price and expiration date. The value of a straddle is highly sensitive to changes in implied volatility. The vega of an option measures the sensitivity of the option’s price to changes in implied volatility. Since a straddle consists of both a call and a put, the straddle’s vega is approximately the sum of the vegas of the individual call and put options. An increase in implied volatility will increase the value of both the call and put options, thus increasing the value of the straddle. Conversely, a decrease in implied volatility will decrease the value of both options, decreasing the straddle’s value. In this scenario, the investor is short a straddle. This means they will profit if the straddle’s value decreases and lose if it increases. If implied volatility decreases, the value of the straddle will decrease, and the investor will profit. The profit will be approximately equal to the change in implied volatility multiplied by the negative of the straddle’s vega (since the investor is short). Here’s the calculation: Change in implied volatility = 21% – 18% = 3% = 0.03 Straddle Vega = 14,000 Profit/Loss = – (Change in implied volatility * Straddle Vega) = – (0.03 * 14,000) = -420 Since the investor is short the straddle, a decrease in implied volatility results in a profit. Therefore, the investor makes a profit of £420. Let’s consider an analogy: Imagine you’re running a car insurance company. You sell insurance policies (similar to selling a straddle). If the overall riskiness (volatility) in the driving environment decreases (fewer accidents), your payouts decrease, and you make more profit. Conversely, if the riskiness increases, your payouts increase, and you make less profit or incur a loss.

The question assesses the understanding of delta hedging and its limitations, particularly in the context of gamma risk. Delta hedging aims to neutralize the sensitivity of an option portfolio to small changes in the underlying asset’s price. However, delta is not constant; it changes as the underlying price moves or time passes. Gamma measures the rate of change of delta with respect to the underlying asset’s price. A high gamma implies that the delta hedge needs frequent adjustments to remain effective. The cost of these adjustments (transaction costs) and the discrete nature of trading (one cannot continuously adjust the hedge) can erode the profitability of the delta hedging strategy. In this scenario, the portfolio manager is attempting to delta hedge a short option position. A short option position has negative gamma, meaning that as the underlying asset price increases, the delta becomes more negative (for a short call) or less positive (for a short put). This requires the manager to sell more of the underlying asset as the price rises (or buy back less). Conversely, as the underlying asset price decreases, the delta becomes less negative (for a short call) or more positive (for a short put), requiring the manager to buy back the underlying asset (or sell less). If the underlying asset price is highly volatile and oscillates rapidly, the manager will need to frequently adjust the hedge, incurring significant transaction costs. The profit or loss from delta hedging is determined by the initial premium received for the short option, the cost of adjusting the hedge (transaction costs), and the final payoff of the option. The manager profits if the initial premium received exceeds the cost of hedging and the final payoff to the option buyer. If the underlying asset price moves significantly in one direction, the delta hedge can be adjusted profitably, offsetting the losses from the short option position. However, if the underlying asset price oscillates rapidly, the transaction costs associated with frequent hedge adjustments can outweigh any potential profit. The final profit is calculated by subtracting the hedging costs and option payoff from the initial premium received. In this case, the option premium is £5, the strike price is £100, and the underlying asset price fluctuates between £95 and £105. The manager must frequently adjust the hedge, incurring transaction costs of £6. The option expires in the money, so the payoff to the option buyer is the difference between the final price and the strike price. Since the final price is £105, the payoff is £5. The manager’s profit is therefore £5 (initial premium) – £6 (transaction costs) – £5 (option payoff) = -£6.

The question explores the complexities of hedging a foreign currency liability with futures contracts, specifically considering the impact of basis risk and contract sizing. Basis risk arises because the spot rate and the futures rate do not always move in perfect lockstep. This discrepancy can lead to hedging imperfections. To calculate the optimal number of contracts, we first need to understand the relationship between the liability, the contract size, and the spot/futures rates. The company owes €5,000,000 in 6 months. We want to use GBP/EUR futures contracts to hedge this liability. The contract size is £125,000. The current spot rate is £0.85/€, and the 6-month futures rate is £0.86/€. The target GBP amount to hedge is €5,000,000 * £0.85/€ = £4,250,000 (using the spot rate as a starting point). The number of contracts needed is £4,250,000 / £125,000 = 34 contracts. Now, we need to consider the basis risk. The question states that the spot rate is expected to be £0.87/€ in 6 months. This means the basis (Spot – Futures) has changed. The initial basis was £0.85 – £0.86 = -£0.01. The expected final basis is £0.87 – £0.86 = £0.01. The change in basis is £0.02. Because the spot rate is expected to increase relative to the futures rate, the hedge will be slightly less effective. To account for this, we need to adjust the number of contracts. If we don’t adjust, we risk being under-hedged. The potential unhedged amount due to basis risk is €5,000,000 * (£0.02) = £100,000. This represents the potential additional GBP needed to cover the liability. To hedge this additional amount, we need £100,000 / £125,000 = 0.8 contracts. Since we can’t trade fractions of contracts, we need to consider the impact of rounding. Rounding down to 34 contracts leaves us under-hedged. Rounding up to 35 contracts provides a slight over-hedge, which is generally preferable in liability hedging. Therefore, the optimal number of contracts is 35.

The question revolves around the concept of delta hedging a short call option position and the subsequent adjustments required as the underlying asset’s price changes and time passes. Delta, Gamma, and Theta are crucial Greeks in options trading. Delta represents the sensitivity of the option price to changes in the underlying asset’s price. Gamma represents the rate of change of delta with respect to changes in the underlying asset’s price. Theta represents the rate of change of the option price with respect to time. Here’s how we can solve the problem: 1. **Initial Hedge:** The investor initially sells 100 call options, each representing 100 shares, creating a short position on 10,000 shares (100 contracts * 100 shares/contract). The initial delta of 0.4 indicates that for every £1 increase in the share price, the option price is expected to increase by £0.40 per share. To delta hedge, the investor needs to buy shares equivalent to the delta exposure. Thus, the initial hedge requires buying 4,000 shares (10,000 shares * 0.4). 2. **Price Increase and Delta Change:** The share price increases by £5, and the delta increases to 0.6. This means the investor’s short call option position is now more sensitive to changes in the share price. The new delta exposure is 6,000 shares (10,000 shares * 0.6). To maintain the delta-neutral position, the investor needs to buy an additional 2,000 shares (6,000 – 4,000). 3. **Time Decay and Delta Change:** Over the next week, time passes, and the delta decreases to 0.5. This is due to theta, which erodes the value of the option as it approaches its expiration date. The new delta exposure is 5,000 shares (10,000 shares * 0.5). To rebalance the delta-neutral position, the investor needs to sell 1,000 shares (6,000 – 5,000). Note that we use the previous holding (6,000) as the starting point, not the original 4,000, because the investor already adjusted the position when the price increased. 4. **Total Transactions:** The investor first buys 4,000 shares, then buys an additional 2,000 shares, and finally sells 1,000 shares.

The question assesses the understanding of option pricing sensitivity to changes in underlying asset volatility, specifically focusing on Vega. Vega measures the change in an option’s price for a 1% change in the underlying asset’s volatility. The calculation involves estimating the new option price after a change in volatility and determining the resulting profit or loss. 1. **Initial Option Price:** £5.50 2. **Volatility Increase:** 2% 3. **Vega:** 0.25 (Change in option price per 1% change in volatility) 4. **Change in Option Price:** Vega * Volatility Increase = 0.25 * 2 = £0.50 5. **New Option Price:** Initial Option Price + Change in Option Price = £5.50 + £0.50 = £6.00 6. **Sale Price:** £5.80 7. **Profit/Loss:** Sale Price – New Option Price = £5.80 – £6.00 = -£0.20 Therefore, the investor incurs a loss of £0.20 per option. This calculation demonstrates how changes in implied volatility, a key component of option pricing models like Black-Scholes, directly impact option values. A positive Vega indicates that the option’s price increases with increasing volatility, and vice versa. Understanding Vega is crucial for managing risk in option strategies, especially when dealing with volatile underlying assets or anticipating significant market events that could affect volatility. For instance, consider a portfolio heavily invested in options on a tech stock just before a major product announcement. If the announcement is delayed, implied volatility might decrease, negatively impacting the value of long option positions (positive Vega). Conversely, if the announcement is expected to cause a large price swing, traders might buy options to profit from the anticipated volatility increase. Furthermore, regulatory frameworks like EMIR (European Market Infrastructure Regulation) emphasize the importance of accurate risk assessment, including sensitivity to volatility changes, for OTC derivative transactions. Therefore, understanding and managing Vega is not only essential for individual trading strategies but also for broader regulatory compliance and risk management within financial institutions.

The question revolves around the concept of hedging a portfolio using options, specifically protective puts, and the impact of implied volatility on the effectiveness of this strategy. A protective put involves buying put options on an asset already held in a portfolio. This limits the downside risk of the portfolio, as the put option’s value increases if the asset’s price falls. The cost of this protection is the premium paid for the put option. Implied volatility is a crucial factor in option pricing. It represents the market’s expectation of how much the underlying asset’s price will fluctuate in the future. Higher implied volatility leads to higher option premiums because there’s a greater chance the option will end up in the money. Conversely, lower implied volatility results in lower premiums. In this scenario, the investor is concerned about a potential market downturn and wants to protect their portfolio with protective puts. However, the implied volatility of the put options is unusually high due to recent market turbulence. The investor needs to consider whether the cost of the puts (premium) is justified by the level of protection they offer, given the high implied volatility. The effectiveness of the hedge is determined by comparing the potential loss in the portfolio’s value to the cost of the put options. If the market declines sharply, the put options will offset the loss, but the investor still incurs the cost of the premium. If the market remains stable or rises, the put options expire worthless, and the investor loses the premium. The question requires understanding the trade-off between the cost of the hedge (put option premium) and the level of protection it provides, considering the impact of implied volatility. A high implied volatility increases the cost of the hedge, making it less attractive if the investor believes the market turbulence is temporary and the implied volatility will decrease soon. Conversely, if the investor strongly believes the market will decline, the higher premium might be justified. Consider a portfolio worth £1,000,000. Protective puts with a strike price close to the current market price cost £50,000 due to high implied volatility. If the market declines by 10%, the portfolio loses £100,000, but the put options offset most of this loss. However, if the market remains stable, the investor loses the £50,000 premium. The investor must assess whether the potential loss of £100,000 justifies paying a £50,000 premium for protection. The breakeven point for the protective put strategy is the initial portfolio value minus the premium paid for the put options. In this case, the breakeven point is £1,000,000 – £50,000 = £950,000. If the portfolio’s value falls below £950,000, the protective put strategy is profitable. If the portfolio’s value remains above £950,000, the strategy results in a loss equal to the premium paid.

Let’s analyze a scenario involving a UK-based pension fund aiming to hedge its exposure to potential interest rate increases. The fund holds a substantial portfolio of long-dated UK Gilts. To mitigate the risk of rising interest rates diminishing the value of these Gilts, the fund is considering using a combination of short-dated Sterling Overnight Index Average (SONIA) futures contracts and a receiver swaption. The swaption gives the fund the right, but not the obligation, to enter into a swap where it receives fixed and pays floating based on SONIA. The key is to understand how these instruments work in tandem. SONIA futures provide a relatively short-term hedge, as they are typically based on 3-month periods. However, the pension fund’s liabilities are long-term. The receiver swaption acts as a longer-term hedge, protecting against significant rises in interest rates beyond the timeframe covered by the futures. The effectiveness of this strategy hinges on the correlation between short-term SONIA rates and longer-term swap rates. If short-term rates rise sharply, the SONIA futures will generate profits that offset some of the losses on the Gilt portfolio. If long-term rates also rise significantly, the swaption will become in-the-money, allowing the fund to receive fixed payments at a rate higher than the prevailing market rate, further offsetting the losses. The fund must carefully consider the strike price of the swaption. A lower strike price provides greater protection but comes at a higher premium cost. A higher strike price is cheaper but offers less protection. Furthermore, the fund needs to assess the potential for basis risk, which arises from the imperfect correlation between SONIA futures and the specific yields of the Gilts in its portfolio. Stress testing different interest rate scenarios is crucial to evaluate the overall effectiveness of the hedging strategy. The fund must also consider the regulatory implications under EMIR regarding clearing and reporting obligations for derivative transactions. Finally, the fund must analyze the counterparty risk associated with the swaption, particularly if it’s traded OTC.

The question explores the impact of early exercise on American call options, specifically when dividends are involved. The core principle is that an American call option on a dividend-paying stock might be exercised early if the present value of the expected dividends exceeds the time value of holding the option. This happens because the option holder forgoes receiving the dividends if they wait until expiration. The calculation involves comparing the intrinsic value of the option if exercised now (Stock Price – Strike Price) with the potential gain from holding the option until just before the dividend payment, considering the time value. We must also consider that the dividend payment will reduce the stock price. Let’s break down the logic: 1. **Calculate the potential stock price drop:** The dividend yield is 5%, and the stock price is £100, so the expected dividend is 5% * £100 = £5. This dividend payment will likely decrease the stock price by approximately £5. 2. **Estimate the stock price just before the dividend:** We assume the stock price will remain constant until just before the dividend payment. 3. **Calculate the option’s intrinsic value if exercised now:** This is £100 – £98 = £2. 4. **Consider the time value of the option:** This is the potential gain from holding the option until just before the dividend, less the potential loss from the stock price drop. We’re implicitly assuming that the time value derives primarily from the possibility of the stock price increasing significantly. 5. **Compare the immediate exercise value with the value of holding:** If the dividend is large enough, the investor might prefer to capture the intrinsic value now rather than wait for a potential price increase that might not materialize and will be offset by the dividend-related price drop. 6. **Black-Scholes Consideration (Implicit):** While not directly calculated, the Black-Scholes model informs our understanding of option pricing and the factors influencing early exercise. A high dividend yield, combined with a short time to dividend payment, increases the likelihood of early exercise. The Black-Scholes model would be used to estimate the option’s price at different points in time, considering volatility, time to expiration, and dividend yield. 7. **Risk-Free Rate Consideration:** The risk-free rate is implicitly considered when discounting future cash flows (dividends) back to the present. A higher risk-free rate would decrease the present value of future dividends, making early exercise less attractive. Therefore, the optimal strategy depends on balancing the immediate gain from exercising (£2) against the potential future gain from holding the option, adjusted for the expected dividend impact. In this scenario, the dividend payment is substantial enough relative to the current intrinsic value to warrant considering early exercise.

The question revolves around the concept of hedging a portfolio using options, specifically focusing on the impact of gamma on the hedge’s effectiveness. Gamma measures the rate of change of an option’s delta with respect to changes in the underlying asset’s price. A higher gamma implies that the delta is more sensitive to price fluctuations, requiring more frequent adjustments to maintain a delta-neutral hedge. The scenario involves a portfolio manager using short put options to hedge against downside risk. The key is to understand how changes in the underlying asset’s price affect the portfolio’s delta and how the gamma of the options influences the necessary adjustments. Here’s a breakdown of the calculation: 1. **Initial Portfolio Value:** £5,000,000 2. **Hedge Ratio:** 2% (meaning options cover 2% of the portfolio value) 3. **Notional Value Covered by Options:** £5,000,000 * 0.02 = £100,000 4. **Underlying Asset Price:** £100 5. **Contract Size:** 100 shares per option contract 6. **Number of Contracts:** £100,000 / (£100 * 100) = 10 contracts (short puts) 7. **Initial Delta:** -0.4 (short puts have negative delta) 8. **Portfolio Delta Change:** The portfolio’s delta changes by +50 for every £1 change in the underlying asset price. 9. **Price Increase:** £1 10. **Gamma:** 0.05 per contract. The initial delta of the option position is 10 contracts * -0.4 = -4. The increase in the underlying asset price changes the portfolio delta by +50. The change in delta of the option position due to gamma is: 10 contracts * 0.05 * £1 = 0.5 The new delta of the option position is -4 + 0.5 = -3.5 To maintain a delta-neutral hedge, the portfolio manager needs to offset the portfolio’s delta change (+50) with an equivalent change in the option position’s delta. The portfolio manager must reduce the number of short put options to make the delta change more positive. The number of contracts to unwind is calculated by dividing the delta change by the option’s delta: 50 / 0.4 = 125 contracts. However, we only have 10 contracts. The portfolio manager must buy back (unwind) some of the short put options to decrease the negative delta exposure. The change in the underlying asset price has caused the option’s delta to become less negative, reducing the effectiveness of the hedge. Since the portfolio delta has increased by 50, the manager needs to decrease the magnitude of the negative delta from the options position. Given the gamma of 0.05, each option’s delta will increase by 0.05 for every £1 increase in the underlying asset price. Therefore, the number of options that need to be unwound is the change in portfolio delta divided by the option’s delta, which is 50 / 0.4 = 125. Since the portfolio manager only has 10 contracts, the best action is to unwind all 10 contracts and then buy 115 contracts of the underlying asset to compensate for the remaining delta.

1. **Initial Option Price:** The option is priced at the midpoint of the bid-ask spread: (£5.20 + £5.30) / 2 = £5.25. 2. **Vega Calculation:** The option’s Vega is 0.65. This means that for every 1% change in implied volatility, the option price is expected to change by £0.65. 3. **Volatility Change:** The implied volatility decreases by 2%. 4. **Price Change Due to Volatility:** The option price is expected to decrease by 2 * £0.65 = £1.30. 5. **New Option Price:** The new expected option price is £5.25 – £1.30 = £3.95. 6. **New Bid-Ask Spread Midpoint:** The market maker adjusts the bid-ask spread, and the new midpoint is (£3.80 + £4.00) / 2 = £3.90. 7. **Profit/Loss Calculation:** The trader initially bought the option at £5.25 and can now sell it at the midpoint of the new bid-ask spread, which is £3.90. The loss is £5.25 – £3.90 = £1.35 per option. Therefore, the trader experiences a loss of £1.35 per option. This loss reflects the combined impact of the volatility decrease and the market maker’s adjustment of the bid-ask spread. It’s crucial to consider both Vega and the bid-ask spread when evaluating the impact of volatility changes on option positions. The example illustrates how changes in implied volatility directly affect option prices through Vega. It highlights the importance of understanding option Greeks for effective risk management and trading strategies. The scenario also demonstrates the practical impact of market maker behavior and bid-ask spreads on trading outcomes. A key takeaway is that even with accurate Vega calculations, market dynamics can influence the final profit or loss.

This question tests the understanding of how the shape of the forward curve impacts option strategies, particularly calendar spreads. In contango, there’s a negative carry cost associated with holding the long near-month position.

Let’s consider a scenario involving a UK-based agricultural cooperative, “GreenHarvest,” which exports organic wheat to several European countries. GreenHarvest faces significant price volatility in the wheat market due to unpredictable weather patterns and fluctuating demand. To mitigate this risk, they are considering using wheat futures contracts traded on the London International Financial Futures and Options Exchange (LIFFE). The cooperative needs to determine the optimal number of contracts to hedge their anticipated wheat harvest. First, we need to calculate the hedge ratio. The hedge ratio is the ratio of the size of the position to be hedged to the size of the hedging instrument. In this case, the position to be hedged is GreenHarvest’s expected wheat harvest, and the hedging instrument is the wheat futures contract. The formula for the hedge ratio (HR) is: \[ HR = \frac{\text{Size of the position to be hedged}}{\text{Size of one futures contract}} \] Suppose GreenHarvest expects to harvest 500,000 bushels of wheat. One wheat futures contract on LIFFE represents 5,000 bushels. Therefore, the hedge ratio is: \[ HR = \frac{500,000}{5,000} = 100 \] This means GreenHarvest needs to sell 100 futures contracts to fully hedge their expected harvest. However, this assumes a perfect correlation between the spot price of GreenHarvest’s organic wheat and the futures price of standard wheat. In reality, there will be basis risk, which is the risk that the spot price and the futures price do not move in perfect lockstep. This can be due to differences in quality, location, or timing. To account for basis risk, GreenHarvest could perform a regression analysis of the historical relationship between their organic wheat price and the LIFFE wheat futures price. Suppose this analysis yields a beta (β) of 0.8. This means that for every £1 change in the futures price, GreenHarvest’s organic wheat price tends to change by £0.8. The adjusted hedge ratio, considering basis risk, is: \[ \text{Adjusted HR} = HR \times \beta = 100 \times 0.8 = 80 \] Therefore, GreenHarvest should sell 80 futures contracts to minimize their price risk, taking into account the historical correlation between their specific product and the hedging instrument. This approach helps GreenHarvest to better manage the risk associated with basis, providing a more refined hedge than simply using the naive hedge ratio.

Let’s analyze the scenario involving the exotic derivative, a Cliquet Option, within the context of portfolio hedging and performance measurement for a UK-based investment firm. A Cliquet Option, also known as a ratchet option, offers a series of resets on its strike price, usually annually. This feature makes it attractive for investors seeking participation in market gains while limiting downside risk. The key is to understand how the periodic resets affect the option’s overall payoff and how this payoff profile aligns with specific hedging objectives. The calculation of the Cliquet Option’s payoff involves summing the returns for each period, capped at a maximum level. Let’s assume there are three periods. For each period \(i\), the return \(R_i\) is calculated as: \[R_i = min(max(Return_{underlying}, floor), cap)\] The overall payoff is then: \[Payoff = \sum_{i=1}^{3} R_i\] In our scenario, the fund manager uses the Cliquet Option to hedge against potential downside risk in a portfolio concentrated in FTSE 100 companies. The portfolio’s beta is 1.2. The manager wants to protect the portfolio against significant losses while still participating in market upside. The Cliquet Option provides a mechanism for achieving this. The floor limits the loss in any given period, while the cap limits the gain. The sum of these capped and floored returns determines the final payoff. Consider a scenario where the FTSE 100 returns for the three periods are -10%, 15%, and 5%, respectively. The Cliquet Option has a floor of -5% and a cap of 8%. The returns for each period would be adjusted as follows: Period 1: Return = max(-10%, -5%) = -5% Period 2: Return = min(15%, 8%) = 8% Period 3: Return = min(5%, 8%) = 5% The total payoff would be -5% + 8% + 5% = 8%. This illustrates how the Cliquet Option limits the downside to -5% in the first period and caps the upside in the subsequent periods. Now, consider the alternative strategies and why they might be less suitable. A standard put option provides downside protection but requires a premium payment upfront, which can reduce overall returns if the market does not decline. A covered call strategy generates income but limits upside potential. A collar strategy combines a put option and a call option, offering a defined range of returns but potentially missing out on significant gains. The Cliquet Option offers a dynamic approach, resetting the strike price periodically and allowing participation in market gains while providing a safety net.

The question assesses the understanding of delta hedging, particularly in the context of portfolio management and the impact of market events on the effectiveness of hedging strategies. Delta hedging aims to neutralize the directional risk of an option position by adjusting the underlying asset holdings. However, this strategy is not foolproof and is affected by market volatility, time decay (theta), and discrete hedging intervals. The calculation involves understanding how a sudden price jump affects the delta hedge. Initially, the portfolio is delta-neutral. When the underlying asset’s price jumps, the option’s delta changes. The hedge needs to be rebalanced to account for this new delta. The cost of rebalancing is the number of shares to buy or sell multiplied by the price at which they are bought or sold. Here’s the breakdown of the solution: 1. **Initial Position:** Portfolio is delta-neutral. 2. **Price Jump:** Asset price increases from £100 to £105. 3. **New Delta:** The call option’s delta increases from 0.60 to 0.75 due to the price jump. 4. **Shares to Buy:** To re-establish the delta hedge, the fund manager needs to buy shares equivalent to the change in delta, which is 0.75 – 0 = 0.75 (since the initial position was short one call option, the delta is positive). 5. **Number of Shares:** Since the fund holds 10,000 call options, the number of shares to buy is 0.75 * 10,000 = 7,500 shares. 6. **Cost of Rebalancing:** The cost of buying these shares at £105 is 7,500 * £105 = £787,500. The question specifically tests whether the candidate can apply the concept of delta hedging in a dynamic market environment and calculate the cost associated with rebalancing the hedge after a significant price movement. It also highlights the limitations of delta hedging, as large, unexpected price jumps can lead to substantial rebalancing costs. The question requires a deep understanding of options, hedging, and market dynamics, aligning with the CISI Derivatives Level 4 syllabus.

The question assesses the understanding of put-call parity and its application in identifying arbitrage opportunities, particularly when transaction costs are involved. Put-call parity establishes a relationship between the price of a European call option, a European put option, the underlying asset, and a risk-free bond, all with the same strike price and expiration date. The formula for put-call parity is: \[C + PV(K) = P + S\] Where: * C = Call option price * PV(K) = Present value of the strike price (K) discounted at the risk-free rate * P = Put option price * S = Current price of the underlying asset When transaction costs are present, the arbitrage opportunity is only profitable if the profit from exploiting the mispricing exceeds the transaction costs. In this scenario, we need to determine if the market prices deviate enough from the put-call parity to cover the brokerage fees. 1. **Calculate the Present Value of the Strike Price:** The strike price (K) is £105, and the risk-free rate is 3% per annum. The time to expiration is 6 months (0.5 years). \[PV(K) = \frac{K}{(1 + r)^t} = \frac{105}{(1 + 0.03)^{0.5}} = \frac{105}{1.014889} \approx 103.46\] 2. **Check for Put-Call Parity:** Using the given market prices: * Call option price (C) = £8 * Put option price (P) = £4 * Underlying asset price (S) = £108 Substituting these values into the put-call parity equation: \[8 + 103.46 = 4 + 108\] \[111.46 = 112\] The equation does not hold exactly, indicating a potential arbitrage opportunity. The left side is less than the right side, meaning the call is relatively undervalued and the put is relatively overvalued. 3. **Identify the Arbitrage Strategy:** To exploit this mispricing, we should buy the undervalued assets (the call and the bond representing the present value of the strike price) and sell the overvalued assets (the put and the underlying asset). * Buy the call option for £8 * Buy a risk-free bond that will pay £105 at expiration (costing £103.46 today) * Sell the put option for £4 * Sell the underlying asset for £108 4. **Calculate the Profit Before Transaction Costs:** The profit before transaction costs is the difference between the right side and the left side of the put-call parity equation: \[Profit = 112 – 111.46 = 0.54\] 5. **Calculate Total Transaction Costs:** The brokerage fee is £0.10 per transaction, and we have four transactions (buying the call, buying the bond, selling the put, and selling the asset). \[Total\ Transaction\ Costs = 4 \times 0.10 = 0.40\] 6. **Calculate Net Arbitrage Profit:** Subtract the total transaction costs from the profit before transaction costs: \[Net\ Arbitrage\ Profit = 0.54 – 0.40 = 0.14\] Since the net arbitrage profit (£0.14) is positive, the arbitrage opportunity is profitable even after considering transaction costs.

The question assesses understanding of exotic derivatives, specifically barrier options, and their sensitivity to market volatility. A down-and-out call option becomes worthless if the underlying asset’s price touches or falls below the barrier level. The probability of this occurring is directly related to the asset’s volatility. Higher volatility increases the likelihood of the barrier being breached, reducing the option’s value. The time to maturity also plays a crucial role; a longer time frame provides more opportunities for the barrier to be hit. The interest rate has a less direct impact, primarily affecting the present value of potential future payoffs. The correct approach is to recognize the inverse relationship between volatility and the value of a down-and-out call option. A significant increase in volatility, combined with a relatively long time to maturity, makes it more probable that the barrier will be triggered, rendering the option worthless or significantly reducing its value. The initial price is a red herring, and the precise interest rate change is less relevant than the volatility spike. Let’s consider a hypothetical scenario: Imagine a river with a dam (the barrier). A boat (the asset price) is trying to reach a destination downstream (the option’s potential payoff). High winds (volatility) make the boat’s path erratic and increase the chance it will crash into the riverbank (hit the barrier). The longer the journey (time to maturity), the greater the risk of an accident. A slight change in the river’s current (interest rate) is less likely to cause a crash than the high winds. Therefore, the most likely outcome is a substantial decrease in the option’s value, potentially to zero.

Let’s analyze a scenario involving a UK-based pension fund considering using currency swaps to hedge their exposure to Euro-denominated assets. The pension fund holds €50 million in European equities and wants to protect against a potential depreciation of the Euro against the British Pound. They are evaluating a currency swap with a notional principal of €50 million. The current spot exchange rate is £0.85/€ (meaning £0.85 buys one Euro). The pension fund enters into a currency swap where they agree to pay a fixed interest rate of 1.5% per annum in Euros and receive a fixed interest rate of 3% per annum in British Pounds. The swap has a maturity of three years, with annual payments. Year 1: Euro Payment: €50,000,000 * 0.015 = €750,000 Pound Receipt: £(50,000,000 * 0.85) * 0.03 = £1,275,000 Year 2: Euro Payment: €50,000,000 * 0.015 = €750,000 Pound Receipt: £(50,000,000 * 0.85) * 0.03 = £1,275,000 Year 3: Euro Payment: €50,000,000 * 0.015 = €750,000 Pound Receipt: £(50,000,000 * 0.85) * 0.03 = £1,275,000 At maturity (Year 3), the principal amounts are exchanged back at the initially agreed-upon rate of £0.85/€. This means the pension fund receives €50,000,000 and pays £42,500,000 (£50,000,000 * 0.85). Now, consider a scenario where, at the end of year 1, the exchange rate has moved to £0.80/€. This means the Euro has depreciated against the Pound. The pension fund is considering unwinding the swap. The present value of the remaining payments needs to be calculated. Assume the discount rate for both currencies is 2%. Remaining Euro Payments: €750,000 (Year 2) / 1.02 + €50,750,000 (Year 3) / (1.02)^2 = €735,294.12 + €48,767,058.82 = €49,502,352.94 Remaining Pound Receipts: £1,275,000 (Year 2) / 1.02 + £1,275,000 (Year 3) / (1.02)^2 + £42,500,000 (Year 3) / (1.02)^2 = £1,249,999.99 + £1,225,490.19 + £40,880,597.01= £43,356,087.19 The value of the swap to the pension fund is the difference between the present value of the pound receipts and the present value of the euro payments converted to pounds at the new spot rate: Value = £43,356,087.19 – (€49,502,352.94 * 0.80) = £43,356,087.19 – £39,601,882.35 = £3,754,204.84 This positive value indicates that the swap is an asset to the pension fund because the Euro depreciated. The counterparty would need to pay the pension fund approximately £3,754,204.84 to terminate the swap.