Derivatives Level 4 (Investment Advice Diploma) Quiz Exam Quiz 03

The question assesses the understanding of exotic derivatives, specifically barrier options, and their payoff structures. The scenario involves a “knock-out” call option, where the option becomes worthless if the underlying asset price touches a pre-defined barrier level before the expiration date. The calculation involves determining whether the barrier was breached and then calculating the payoff if the barrier was not breached and the option is in the money at expiration. First, we need to determine if the barrier of £155 was breached at any point during the option’s life. The question states that the price dipped to £152, which is below the barrier level, thus triggering the knock-out feature. Since the barrier was breached, the option is knocked out, and its value at expiration is £0, regardless of the underlying asset’s price at expiration. The explanation should highlight the key differences between standard options and barrier options. A standard call option would have a payoff of max(S – K, 0), where S is the spot price at expiration and K is the strike price. However, the barrier option’s payoff is conditional on the barrier not being breached. This condition significantly alters the risk profile of the option. The explanation should also emphasize the impact of barrier levels on option pricing. Knock-out options are generally cheaper than standard options because the barrier feature reduces the probability of a payoff. Conversely, knock-in options, which only become active when the barrier is breached, are generally more expensive. The explanation should also consider the implications of different barrier types (up-and-out, down-and-out, up-and-in, down-and-in) on option strategies and risk management.

To determine the fair price of the exotic Asian option, we need to understand how the averaging period affects the option’s value. Since the averaging period only covers the first 6 months of the year-long contract, the holder is exposed to the volatility of the underlying asset for the remaining 6 months without any averaging benefit. This makes the option less sensitive to price fluctuations at the beginning but still highly sensitive towards the end of the period. Let’s consider a simplified scenario. Imagine two investors: Alice and Bob. Alice holds a standard European call option on the same asset, while Bob holds the exotic Asian option described in the question. If the asset price skyrockets in the last month, Alice will benefit significantly. Bob, however, will see his gains tempered by the earlier, potentially lower, prices that have already been averaged in. The effect of interest rates must also be considered. The higher the interest rates, the less the present value of the expected payoff, and hence the lower the option value. Given the specific parameters: initial asset price (£100), strike price (£100), volatility (20%), risk-free rate (5%), and a one-year term with a 6-month averaging period, the fair price will be less than a standard Asian option with a full-year averaging period. A standard Asian option would smooth out the price fluctuations over the entire year, providing more protection against late-term volatility. Based on these considerations, the fair price for the exotic Asian option will be £6.25. This price reflects the reduced averaging period and the overall impact of the given parameters on the option’s value.

The core of this question revolves around understanding how margin requirements work in futures contracts, particularly when dealing with adverse price movements and the concept of “marking to market.” It’s crucial to grasp that futures contracts are marked to market daily, meaning profits or losses are credited or debited to the account each day based on the settlement price. When a trader incurs losses, the margin account balance decreases. If this balance falls below the maintenance margin, a margin call is triggered, requiring the trader to deposit additional funds to bring the account back up to the initial margin level. The calculation involves determining the total loss incurred, subtracting it from the initial margin, and then calculating the amount needed to restore the account to the initial margin. In this scenario, the initial margin is £8,000, and the maintenance margin is £6,000. The trader experiences losses over three days. We need to calculate the cumulative loss and then determine if a margin call is triggered and, if so, its size. Day 1: Loss of £1,000. Margin balance: £8,000 – £1,000 = £7,000 Day 2: Loss of £1,500. Margin balance: £7,000 – £1,500 = £5,500 Day 3: Loss of £500. Margin balance: £5,500 – £500 = £5,000 Since the margin balance of £5,000 is below the maintenance margin of £6,000, a margin call is triggered. To calculate the margin call amount, we need to determine how much money is required to bring the margin balance back to the initial margin level of £8,000. Margin call amount = Initial margin – Current margin balance Margin call amount = £8,000 – £5,000 = £3,000 Therefore, the trader needs to deposit £3,000 to meet the margin call. Now, let’s consider a slightly different scenario to further illustrate the concept. Imagine a gold futures trader who initially deposited £10,000 as initial margin, with a maintenance margin of £7,500. If, after a week of trading, the trader’s account balance falls to £7,000 due to unfavorable price movements, a margin call would be issued. The trader would then need to deposit £3,000 to bring the account balance back to the initial margin of £10,000. This demonstrates the continuous monitoring and adjustment of margin accounts in futures trading to mitigate counterparty risk. Another example: A trader initiates a short position in FTSE 100 futures with an initial margin of £5,000 and a maintenance margin of £4,000. Over two days, the market rallies unexpectedly, resulting in losses of £800 on day one and £700 on day two. The margin account balance decreases to £3,500 (£5,000 – £800 – £700). Because this is below the maintenance margin, the trader receives a margin call for £1,500 (£5,000 – £3,500) to restore the account to the initial margin level.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their valuation sensitivity to volatility changes. A down-and-out barrier option ceases to exist if the underlying asset’s price hits a predetermined barrier level. The sensitivity to volatility, often referred to as Vega, is not uniform across all price levels relative to the barrier. Near the barrier, the option’s value becomes highly sensitive to volatility. An increase in volatility near the barrier increases the probability of the barrier being hit, thus extinguishing the option and decreasing its value. Conversely, far from the barrier, the option behaves more like a standard option, and increased volatility typically increases its value. However, the effect near the barrier dominates when considering the overall portfolio impact, especially when the portfolio is constructed to profit from stable prices. The calculation is conceptual rather than numerical, focusing on understanding the directional impact. The portfolio is designed to profit from stable prices. The down-and-out put option near its barrier will lose value if volatility increases because there’s a higher chance of the barrier being breached, and the option expiring worthless. This loss offsets the gains from other portfolio components that benefit from stable prices. The key is recognizing the non-linear relationship between volatility and barrier option value, especially close to the barrier. A portfolio designed for stable prices, containing a down-and-out put option near its barrier, will likely suffer a loss if volatility increases because the increased probability of the barrier being hit outweighs any potential gains from the underlying asset’s price movements. The other portfolio components designed to profit from stable prices will not be able to compensate for the loss in the barrier option’s value.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their sensitivity to volatility, time decay, and the barrier level relative to the asset price. The investor’s belief that volatility will decrease and the asset price will remain stable near the barrier level are crucial to evaluating the potential profitability of the chosen barrier option. A knock-out call option becomes worthless if the underlying asset’s price touches the barrier level before expiration. The investor anticipates decreased volatility, which reduces the probability of the asset price reaching the barrier. If the asset price stays near the barrier but doesn’t breach it, the option retains some value due to the potential for upward movement before expiry. As time passes (time decay), the value erodes if the asset doesn’t move significantly above the strike price. A knock-in call option only becomes active if the barrier is breached. Decreased volatility reduces the chance of the barrier being hit, making the option less likely to become active and therefore less valuable. If the asset price remains near the barrier, the option will remain inactive and lose value due to time decay. A down-and-out put option becomes worthless if the asset price falls below the barrier. If the investor believes the asset price will remain stable near the barrier (but not fall below it) and volatility will decrease (reducing the likelihood of breaching the barrier), this option could be profitable. The investor profits if the asset price stays above the barrier and declines, allowing the put option to increase in value. A down-and-in put option only becomes active if the asset price falls below the barrier. The investor believes the asset price will remain stable near the barrier and volatility will decrease, making it less likely that the barrier will be breached. Therefore, this option is less likely to become active and provide a profit.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their valuation implications when a barrier is breached before maturity. A down-and-out barrier option becomes worthless if the underlying asset’s price hits the barrier level. Therefore, if the barrier is breached early, the option ceases to exist, and its value drops to zero. In this scenario, calculating the probability of breaching the barrier before the option’s expiry is irrelevant because the question focuses on the option’s value *immediately after* the barrier is breached. At that precise moment, the option is worthless. The initial price of the underlying asset and the barrier level are provided to create a realistic context. The details about the volatility and risk-free rate, while important for pricing options generally, are distractors in this specific question, as the option’s value is definitively zero once the barrier is hit. The explanation emphasizes the key characteristic of a down-and-out barrier option: its extinction upon hitting the barrier. It highlights that the option’s value is path-dependent, meaning its value depends on the path the underlying asset takes. Once the barrier is breached, the path becomes irrelevant; the option is immediately worthless. A common misconception is that the option still retains some value after the barrier is breached, perhaps based on the potential for the underlying asset to rebound. However, the “out” feature is absolute; once triggered, the option is terminated, regardless of subsequent price movements. Another misconception is trying to apply standard option pricing models (like Black-Scholes) after the barrier is hit, which is incorrect because the option no longer exists. The question tests the understanding of this fundamental aspect of barrier options.

The core of this question revolves around understanding the mechanics of a swaption, specifically a payer swaption, and how its value is derived from the underlying swap rate and the strike rate. A payer swaption gives the holder the right, but not the obligation, to enter into a swap where they pay the fixed rate and receive the floating rate. The value of the swaption at expiration depends on the difference between the market swap rate at that time and the strike rate specified in the swaption agreement. If the market swap rate is higher than the strike rate, the swaption is in the money, and the holder would exercise it. The payoff is essentially the present value of the difference between these rates, applied to the notional principal, over the life of the swap. To calculate the payoff, we first determine the difference between the market swap rate and the strike rate: 6.5% – 5.75% = 0.75% or 0.0075. This difference represents the advantage gained by exercising the swaption. Next, we need to calculate the present value of this advantage over the remaining life of the swap. Since payments are semi-annual, there are 6 periods (3 years * 2). The present value factor is calculated using the formula for the present value of an annuity: \[PV = \frac{1 – (1 + r)^{-n}}{r}\] where \(r\) is the semi-annual discount rate (market swap rate / 2 = 6.5% / 2 = 3.25% or 0.0325) and \(n\) is the number of periods (6). Plugging in the values, we get: \[PV = \frac{1 – (1 + 0.0325)^{-6}}{0.0325} \approx 5.4172\]. This factor is then multiplied by the rate difference (0.0075) and the notional principal (£10,000,000) to find the payoff: Payoff = 5.4172 * 0.0075 * £10,000,000 = £406,290. This example showcases the practical application of swaption valuation, moving beyond theoretical definitions to a real-world scenario. It also highlights the importance of understanding present value calculations in derivative pricing. A common error is failing to correctly determine the number of periods or using the annual rate instead of the semi-annual rate in the present value calculation. Another mistake is not understanding the payoff structure of a payer swaption, which only has value if the market swap rate exceeds the strike rate. The question challenges the candidate to apply their knowledge in a non-textbook setting, requiring a nuanced understanding of the interplay between interest rates, present value, and derivative contracts.

The core of this question revolves around understanding how changes in implied volatility affect the price of a European-style call option, particularly in the context of hedging a short option position. Vega, representing the option’s sensitivity to volatility changes, is crucial here. A short call option position means the investor profits if the underlying asset price remains below the strike price or rises slowly, and loses if the price rises sharply. An increase in implied volatility increases the option’s price, creating a loss for the short option holder. To hedge against this risk, one needs to buy options whose value increases with volatility. In this scenario, the investor has a short position in call options, meaning they will lose money if the price of the call option increases. The primary driver of call option price increases (other factors being constant) is an increase in implied volatility. Vega measures the sensitivity of an option’s price to changes in implied volatility. A positive Vega means the option price increases when volatility increases, and vice versa. The investor needs to offset the negative impact of increasing volatility on their short call position. Therefore, they need to buy options with a positive Vega. Here’s how we calculate the necessary hedge: 1. **Calculate the total Vega exposure of the short call options:** The investor has sold 100 call option contracts, and each contract covers 100 shares. Therefore, the total number of shares represented by the short call options is 100 contracts * 100 shares/contract = 10,000 shares. The Vega of each short call option is -0.04 (negative because it’s a short position). The total Vega exposure is -0.04 * 10,000 = -400. 2. **Determine the number of long call options needed to offset the Vega exposure:** The hedging instrument is a call option with a Vega of 0.08. To offset the -400 Vega exposure, the investor needs to buy a number of call options such that their total Vega equals +400. Let ‘N’ be the number of call options needed. Then, N * 0.08 * 100 = 400 (since each option contract covers 100 shares). 3. **Solve for N:** N = 400 / (0.08 * 100) = 50. Therefore, the investor needs to buy 50 call option contracts to hedge their Vega exposure. The key is that the investor is *short* options. When volatility rises, the value of options increases, hurting the short position. They must therefore *buy* options to offset this risk. The vega tells us how much the option price will change for each 1% change in volatility. The number of contracts needed is determined by the ratio of the total vega exposure of the short position to the vega of the hedging instrument, adjusted for the contract size.

The core of this question lies in understanding how various factors influence the price of an American-style put option and, crucially, how those factors interact. A key concept is the time value of an option. As time to expiration increases, the value of an American put option generally increases because there’s more opportunity for the underlying asset’s price to fall below the strike price. However, this effect is not linear and is heavily influenced by other factors, particularly volatility and interest rates. Increased volatility almost always increases the value of an option (both puts and calls). Higher volatility means a greater probability of large price swings, which benefits the option holder. For a put option, increased volatility increases the likelihood of the underlying asset’s price falling significantly below the strike price, leading to a higher payoff. Interest rates have a more nuanced effect. Higher interest rates generally decrease the present value of future cash flows. For a put option, the payoff is received when the option is exercised (or at expiration). Higher interest rates reduce the present value of that potential payoff, thus decreasing the option’s price. However, this effect is typically less pronounced than the effects of volatility and time to expiration. In this scenario, we have two opposing forces: the increase in time to expiration (which tends to increase the put option’s value) and the increase in interest rates (which tends to decrease the put option’s value). The magnitude of the volatility increase is crucial. A small increase in volatility might not be enough to offset the negative impact of higher interest rates, especially if the time to expiration is relatively short. A substantial increase in volatility, however, can easily outweigh the interest rate effect, leading to an overall increase in the put option’s price. The early exercise feature of American options adds another layer of complexity, as it gives the holder the right to exercise at any time, making them generally more valuable than their European counterparts. Therefore, the correct answer is the one that reflects a large enough increase in volatility to overcome the negative impact of increased interest rates and the relatively short time to expiration.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their sensitivity to market movements, particularly around the barrier level. A down-and-out put option becomes worthless if the underlying asset’s price touches or goes below the barrier level before the option’s expiration. The closer the underlying asset’s price is to the barrier, the more sensitive the option’s value becomes to small price changes. This sensitivity is heightened by the volatility of the underlying asset. To calculate the probability of breaching the barrier, we consider the asset’s volatility and the time remaining until expiration. A higher volatility increases the likelihood of the asset price hitting the barrier. The shorter the time to expiration, the less time the asset has to breach the barrier, reducing the probability. The current price relative to the barrier is also critical; the closer the price is to the barrier, the higher the probability of it being breached. In this scenario, the trader believes the option will expire worthless. This belief is based on the probability of the underlying asset’s price breaching the barrier. To calculate the probability, we can use a simplified approach considering the asset’s volatility and the proximity to the barrier. The probability of the barrier being breached can be approximated using a normal distribution, where the standard deviation is a function of the volatility and time to expiration. Given a volatility of 20% and 3 months (0.25 years) until expiration, the standard deviation of the asset’s price movement is \( \sigma = \text{volatility} \times \sqrt{\text{time}} = 0.20 \times \sqrt{0.25} = 0.10 \). The distance to the barrier is \( \text{distance} = \text{current price} – \text{barrier} = 95 – 90 = 5 \). The probability of breaching the barrier can be approximated by calculating how many standard deviations the barrier is away from the current price: \( z = \frac{\text{distance}}{\sigma} = \frac{5}{10} = 0.5 \). Using a standard normal distribution table, a z-score of 0.5 corresponds to a probability of approximately 69.15% that the price will be above the current price. Therefore, the probability of the price being below the barrier (i.e., the barrier being breached) is \( 1 – 0.6915 = 0.3085 \) or 30.85%. However, since it is a down-and-out option, we are interested in the probability of the barrier being hit at least once during the option’s life. This requires a more complex calculation using reflection principles or Monte Carlo simulations, which are beyond the scope of a quick estimation. A reasonable estimate, considering the proximity to the barrier and the volatility, would be around 40%. The trader’s action of selling the option at a significantly reduced price reflects their assessment that the probability of the barrier being breached is high enough to render the option worthless.

The question assesses the understanding of delta hedging a short call option position, focusing on the practical application of delta, transaction costs, and the impact of discrete hedging adjustments. 1. **Calculate Initial Delta:** The delta of a call option represents the sensitivity of the option’s price to a change in the underlying asset’s price. A delta of 0.6 means that for every £1 increase in the asset’s price, the option price is expected to increase by £0.6. Since the portfolio is short 100 call options, the initial portfolio delta is -100 * 0.6 = -60. This means the portfolio is short 60 units of the underlying asset. 2. **Hedge Initial Delta:** To delta hedge, the portfolio manager needs to buy 60 units of the underlying asset to offset the short delta. The cost of buying these shares is 60 shares * £50/share = £3000. Transaction costs are 0.1% of the transaction value, so the transaction cost is 0.001 * £3000 = £3. The total cost of the initial hedge is £3000 + £3 = £3003. 3. **Recalculate Delta After Price Change:** The underlying asset’s price increases to £52. The call option’s delta increases to 0.7. The new portfolio delta is -100 * 0.7 = -70. 4. **Adjust the Hedge:** The portfolio manager needs to adjust the hedge to maintain delta neutrality. The portfolio is now short 70 units of the underlying asset, but the manager already owns 60 units. Therefore, the manager needs to buy an additional 10 units of the underlying asset. The cost of buying these shares is 10 shares * £52/share = £520. Transaction costs are 0.1% of the transaction value, so the transaction cost is 0.001 * £520 = £0.52. The total cost of the hedge adjustment is £520 + £0.52 = £520.52. 5. **Total Cost:** The total cost of delta hedging is the sum of the initial hedge cost and the cost of the hedge adjustment: £3003 + £520.52 = £3523.52. This scenario highlights the dynamic nature of delta hedging. Unlike a static hedge, delta hedging requires continuous adjustments as the underlying asset’s price and the option’s delta change. The transaction costs associated with each adjustment also need to be considered when evaluating the profitability and effectiveness of delta hedging. The frequency of adjustments is a trade-off between precision and cost. More frequent adjustments reduce delta exposure but increase transaction costs.

Let’s consider a scenario where a portfolio manager, tasked with managing a UK-based pension fund, uses a combination of futures and options to hedge against interest rate volatility. The fund holds a significant amount of UK Gilts. The manager is concerned that an unexpected rise in UK interest rates, driven by inflationary pressures, could negatively impact the value of the Gilt portfolio. To hedge, the manager could sell short-dated UK Gilt futures contracts. A rise in interest rates would likely cause the price of these futures to fall, generating a profit that offsets the losses in the Gilt portfolio. However, this approach would limit the fund’s upside if interest rates unexpectedly decline. To overcome this limitation, the manager could instead use options. Specifically, they could buy put options on UK Gilt futures. This strategy provides downside protection while allowing the fund to benefit from a potential fall in interest rates. If interest rates rise, the put options will increase in value, offsetting losses in the Gilt portfolio. If interest rates fall, the options will expire worthless, but the fund will benefit from the increased value of its Gilt holdings. The manager could also employ a more complex strategy using a combination of futures and options. For instance, they could use a collar strategy, buying put options for downside protection and simultaneously selling call options to offset the cost of the put options. This strategy limits both the upside and downside potential but provides a cost-effective hedge. The specific choice of hedging strategy depends on the manager’s risk tolerance, market outlook, and cost considerations. Factors like the implied volatility of options, the correlation between the Gilt portfolio and the futures contracts, and the liquidity of the options market must be carefully considered. The manager must also comply with all relevant regulations, including those outlined by the FCA regarding derivatives trading and suitability for pension funds.

Let’s consider a scenario involving a UK-based agricultural cooperative, “GreenHarvest Co-op,” which exports organic wheat to several European countries. GreenHarvest faces significant price volatility due to fluctuations in both the GBP/EUR exchange rate and the price of wheat on the global market. To mitigate these risks, they enter into a series of derivative contracts. The cooperative uses forward contracts to lock in a GBP/EUR exchange rate for future wheat sales denominated in Euros. They also use wheat futures contracts traded on the London International Financial Futures and Options Exchange (LIFFE) to hedge against declines in the price of wheat. Now, imagine that GreenHarvest’s treasurer, Sarah, is evaluating the effectiveness of their hedging strategy. She notes that while the forward contracts have performed as expected, the futures contracts have not perfectly offset the losses from declining wheat prices. This is due to basis risk, which arises because the wheat futures contract on LIFFE is for a standardized grade of wheat, whereas GreenHarvest’s organic wheat has a slightly different quality and price. Also, the delivery location for the LIFFE futures contract is a specific port in the UK, while GreenHarvest’s actual delivery points are spread across several European locations. To analyze the situation, Sarah calculates the correlation between the price changes in GreenHarvest’s organic wheat and the price changes in the LIFFE wheat futures contract. She finds a correlation coefficient of 0.85. This indicates a strong, but not perfect, positive correlation. Sarah also examines the historical price data and observes that the spot price of GreenHarvest’s wheat tends to trade at a premium of £5 per tonne above the LIFFE futures price. This premium represents the basis. To improve their hedging strategy, GreenHarvest could explore other hedging instruments, such as over-the-counter (OTC) options on wheat, which can be customized to match the specific characteristics of their organic wheat. They could also consider adjusting their hedging ratio to account for the basis risk. For example, if they want to hedge 1,000 tonnes of organic wheat, they might use slightly fewer than 1,000 LIFFE wheat futures contracts to compensate for the imperfect correlation. Sarah also needs to consider the regulatory implications of using derivatives. As a UK-based entity, GreenHarvest is subject to the European Market Infrastructure Regulation (EMIR), which requires them to report their derivative transactions to a trade repository. They must also comply with the requirements for clearing certain types of OTC derivatives through a central counterparty (CCP). Failure to comply with EMIR could result in significant fines and other penalties.

Let’s analyze the potential outcomes for both Party A and Party B under different interest rate scenarios. We need to consider the fixed rate paid by Party A, the floating rate received by Party A (based on LIBOR), and the notional principal to determine the net cash flows. The key is to understand how changes in LIBOR affect the overall profitability of the swap for each party. First, let’s calculate the fixed payment made by Party A: \(0.03 \times \$5,000,000 = \$150,000\). Next, we need to calculate the floating rate payments received by Party A under each LIBOR scenario: Scenario 1: LIBOR at 2% Floating payment: \(0.02 \times \$5,000,000 = \$100,000\) Net cash flow for Party A: \(\$100,000 – \$150,000 = -\$50,000\) (Party A pays \$50,000) Net cash flow for Party B: \(\$150,000 – \$100,000 = \$50,000\) (Party B receives \$50,000) Scenario 2: LIBOR at 4% Floating payment: \(0.04 \times \$5,000,000 = \$200,000\) Net cash flow for Party A: \(\$200,000 – \$150,000 = \$50,000\) (Party A receives \$50,000) Net cash flow for Party B: \(\$150,000 – \$200,000 = -\$50,000\) (Party B pays \$50,000) Scenario 3: LIBOR at 5% Floating payment: \(0.05 \times \$5,000,000 = \$250,000\) Net cash flow for Party A: \(\$250,000 – \$150,000 = \$100,000\) (Party A receives \$100,000) Net cash flow for Party B: \(\$150,000 – \$250,000 = -\$100,000\) (Party B pays \$100,000) Now, let’s consider the risk implications. Party A benefits when LIBOR rises above 3% (the fixed rate they are paying). This suggests that Party A believed LIBOR would increase. Conversely, Party B benefits when LIBOR stays below 3%. Therefore, Party B likely believed LIBOR would remain stable or decrease. A key consideration is basis risk, which arises when the floating rate received does not perfectly correlate with the floating rate paid. In this case, since Party A receives LIBOR and Party B effectively receives a fixed rate (by paying LIBOR), there is no explicit basis risk *within the swap itself*. However, both parties are exposed to market risk. Party A is betting on rising interest rates, while Party B is betting on stable or falling interest rates. Finally, counterparty risk is always present in swaps. This is the risk that the other party will default on their obligations. Both Party A and Party B face counterparty risk.

Let’s consider a scenario involving a UK-based agricultural cooperative, “GreenHarvest,” which produces organic wheat. GreenHarvest wants to protect itself against a potential drop in wheat prices before their harvest in six months. They are considering using futures contracts traded on the ICE Futures Europe exchange. The current spot price of organic wheat is £200 per tonne. The six-month futures price is £210 per tonne. GreenHarvest expects to harvest 1000 tonnes of wheat. To hedge their price risk, they decide to sell 10 wheat futures contracts (each contract representing 100 tonnes). After three months, an unexpected drought hits Europe, causing wheat prices to surge. The spot price increases to £240 per tonne, and the three-month futures price rises to £250 per tonne. GreenHarvest decides to close out their hedge by buying back 10 wheat futures contracts. Here’s the calculation: Initial futures price: £210 per tonne Final futures price: £250 per tonne Profit per tonne from futures: £250 – £210 = £40 per tonne Total profit from futures contracts: £40/tonne * 1000 tonnes = £40,000 Spot price increase: £240 – £200 = £40 per tonne Revenue from selling wheat in the spot market: £240/tonne * 1000 tonnes = £240,000 Effective selling price with hedge: (Revenue from spot market + Profit from futures) / Total tonnes Effective selling price: (£240,000 + £40,000) / 1000 = £280 per tonne Now, consider if GreenHarvest did *not* hedge. They would have simply sold their wheat at the spot price of £240 per tonne, resulting in £240,000 revenue. The hedge increased their effective selling price. This example illustrates how futures contracts can be used to hedge price risk. GreenHarvest locked in a minimum selling price by selling futures contracts. While they missed out on some potential upside due to the price increase, they protected themselves against a price decrease. The scenario also highlights the importance of understanding basis risk (the difference between the spot price and the futures price), which can impact the effectiveness of the hedge. It also showcases how regulatory frameworks like those established by the FCA influence the transparency and risk management practices surrounding derivatives trading in the UK agricultural sector.

The correct answer involves calculating the expected payoff of the Asian option, considering the discrete averaging period and the probabilities associated with each price outcome. We need to find the average strike price over the period and then calculate the payoff based on whether the final asset price is above or below this average. First, calculate the possible average strike prices. The asset price can either increase by £5 or decrease by £3 each day. Day 1: Initial price is £100. Day 2: Price can be £105 or £97. Day 3: Four possible prices: £110, £102, £100, £94. The possible average prices are calculated as follows: Scenario 1: £100, £105, £110. Average = (£100 + £105 + £110) / 3 = £105 Scenario 2: £100, £105, £102. Average = (£100 + £105 + £102) / 3 = £102.33 Scenario 3: £100, £97, £100. Average = (£100 + £97 + £100) / 3 = £99 Scenario 4: £100, £97, £94. Average = (£100 + £97 + £94) / 3 = £97 The probabilities are: Scenario 1: Probability = 0.6 * 0.6 = 0.36 Scenario 2: Probability = 0.6 * 0.4 = 0.24 Scenario 3: Probability = 0.4 * 0.6 = 0.24 Scenario 4: Probability = 0.4 * 0.4 = 0.16 The final asset prices are: £110, £102, £100, £94. The payoff for each scenario is calculated as max(Average Price – Final Asset Price, 0): Scenario 1: max(£105 – £110, 0) = 0 Scenario 2: max(£102.33 – £102, 0) = £0.33 Scenario 3: max(£99 – £100, 0) = 0 Scenario 4: max(£97 – £94, 0) = £3 The expected payoff is: (0 * 0.36) + (£0.33 * 0.24) + (0 * 0.24) + (£3 * 0.16) = 0 + £0.0792 + 0 + £0.48 = £0.5592 Therefore, the expected payoff of the Asian put option is approximately £0.56. This example demonstrates how an Asian option’s payoff depends on the average price of the underlying asset over a specified period, making it less sensitive to price fluctuations at maturity compared to standard European or American options. The averaging mechanism reduces the impact of extreme price movements, which can be advantageous in volatile markets. The probabilities associated with each price path are crucial in determining the expected payoff.

Let’s break down this exotic derivative pricing scenario. The derivative in question is a Cliquet option, specifically an Asian Cliquet. This means the return is capped and floored *periodically*, and the underlying is an average of prices over a period. We need to consider several factors: the initial index level, the volatility, the risk-free rate, the dividend yield, the cap and floor levels, and the number of resets. The key here is understanding how the periodic resets affect the overall valuation. We can model this using a Monte Carlo simulation. We simulate multiple price paths for the index, calculate the return for each period, apply the cap and floor, and then average the returns across all paths. This gives us the expected payoff of the Cliquet option. To discount this back to the present value, we use the risk-free rate. Assume the initial index level is \(S_0 = 5000\). The annual volatility is \( \sigma = 20\%\). The risk-free rate is \(r = 5\%\). The dividend yield is \(q = 1\%\). The cap is \(+5\%\) and the floor is \(-3\%\) per period. There are 4 quarterly resets (i.e., \(n = 4\) periods per year). 1. **Simulate Price Paths:** Generate a large number (e.g., 10,000) of possible price paths for the index over the year, using a geometric Brownian motion model: \[ dS = (r – q)S dt + \sigma S dW \] Where \(dW\) is a Wiener process. Discretize this into quarterly steps. 2. **Calculate Periodic Returns:** For each path, calculate the return for each quarter: \[ R_i = \frac{S_i – S_{i-1}}{S_{i-1}} \] Where \(S_i\) is the index level at the end of quarter \(i\). 3. **Apply Cap and Floor:** Apply the cap and floor to each quarterly return: \[ R_{i, \text{capped}} = \text{min}(\text{max}(R_i, -0.03), 0.05) \] 4. **Calculate Total Return:** Sum the capped quarterly returns to get the total capped return for each simulated path: \[ R_{\text{total}} = \sum_{i=1}^{4} R_{i, \text{capped}} \] 5. **Average Total Returns:** Average the total capped returns across all simulated paths: \[ \bar{R}_{\text{total}} = \frac{1}{N} \sum_{j=1}^{N} R_{\text{total}, j} \] Where \(N\) is the number of simulated paths. 6. **Calculate Option Value:** Discount the average total return back to the present value: \[ \text{Option Value} = S_0 \times e^{-rT} \times (1 + \bar{R}_{\text{total}}) \] Where \(T = 1\) year. Assume the Monte Carlo simulation yields an average total capped return of 8%. \[ \text{Option Value} = 5000 \times e^{-0.05 \times 1} \times (1 + 0.08) = 5000 \times 0.9512 \times 1.08 \approx 5136.48 \] The value is then expressed as a percentage of the initial index level, so \[ \frac{5136.48 – 5000}{5000} = 0.0273 \approx 2.73\% \]

To determine the most suitable derivative strategy for mitigating risk in this unique scenario, we must analyze the potential outcomes of each strategy under the given market conditions. Let’s consider a modified butterfly spread using call options. The investor buys one call option with a strike price of £95 (call option A), sells two call options with a strike price of £100 (call option B), and buys one call option with a strike price of £105 (call option C). The premiums paid are £6 for call option A, £3 for each call option B sold, and £1 for call option C. The net cost of this strategy is £6 – (2 * £3) + £1 = £1. If the share price remains at £100 at expiration, call option A is worth £5, call option B is worth £0, and call option C is worth £0. The profit is £5 – £1 = £4. If the share price goes to £95, all options expire worthless, resulting in a loss of £1. If the share price goes to £105, call option A is worth £10, call option B is worth £5 each, and call option C is worth £0. The profit is £10 – (2 * £5) + £0 – £1 = -£1. A short strangle involves selling both a call option and a put option with different strike prices. In this case, selling a call option with a strike price of £105 and a put option with a strike price of £95. This strategy profits if the share price remains within the range of £95 and £105. However, it exposes the investor to unlimited losses if the share price moves significantly in either direction. A long straddle involves buying both a call option and a put option with the same strike price. In this case, buying a call option and a put option with a strike price of £100. This strategy profits if the share price moves significantly in either direction. However, it requires a significant price movement to cover the premiums paid for both options. A protective put strategy involves buying a put option to protect against a decline in the share price. In this case, buying a put option with a strike price of £100. This strategy provides downside protection but limits the upside potential. Given the investor’s specific risk aversion and the desire to profit from low volatility while mitigating potential losses, the modified butterfly spread using call options offers the most suitable balance. It allows for profit if the share price remains near the current level, while limiting potential losses if the share price moves significantly in either direction. The short strangle is too risky due to unlimited losses. The long straddle requires a significant price movement to be profitable. The protective put limits upside potential.

The value of a European call option can be determined using the Black-Scholes model: \[C = S_0N(d_1) – Ke^{-rT}N(d_2)\] where: \(S_0\) = Current stock price \(K\) = Strike price \(r\) = Risk-free interest rate \(T\) = Time to expiration (in years) \(N(x)\) = Cumulative standard normal distribution function \[d_1 = \frac{ln(\frac{S_0}{K}) + (r + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}}\] \[d_2 = d_1 – \sigma\sqrt{T}\] where: \(\sigma\) = Volatility of the stock First, calculate \(d_1\): \[d_1 = \frac{ln(\frac{55}{60}) + (0.05 + \frac{0.25^2}{2})0.5}{0.25\sqrt{0.5}}\] \[d_1 = \frac{ln(0.9167) + (0.05 + 0.03125)0.5}{0.25 \times 0.7071}\] \[d_1 = \frac{-0.0870 + (0.08125)0.5}{0.1768}\] \[d_1 = \frac{-0.0870 + 0.040625}{0.1768}\] \[d_1 = \frac{-0.046375}{0.1768} = -0.2623\] Next, calculate \(d_2\): \[d_2 = d_1 – \sigma\sqrt{T}\] \[d_2 = -0.2623 – 0.25\sqrt{0.5}\] \[d_2 = -0.2623 – 0.25 \times 0.7071\] \[d_2 = -0.2623 – 0.1768 = -0.4391\] Now, find \(N(d_1)\) and \(N(d_2)\) using standard normal distribution tables or a calculator. \(N(-0.2623) \approx 0.3965\) \(N(-0.4391) \approx 0.3304\) Finally, calculate the call option price \(C\): \[C = S_0N(d_1) – Ke^{-rT}N(d_2)\] \[C = 55 \times 0.3965 – 60e^{-0.05 \times 0.5} \times 0.3304\] \[C = 21.8075 – 60e^{-0.025} \times 0.3304\] \[C = 21.8075 – 60 \times 0.9753 \times 0.3304\] \[C = 21.8075 – 58.518 \times 0.3304\] \[C = 21.8075 – 19.334\] \[C = 2.4735\] Therefore, the value of the European call option is approximately 2.47. This calculation demonstrates how the Black-Scholes model integrates various factors such as the current stock price, strike price, time to expiration, risk-free interest rate, and volatility to derive a theoretical value for the option. A nuanced understanding of each variable’s impact is crucial. For example, higher volatility generally increases the option’s value, while a longer time to expiration provides more opportunity for the underlying asset to move favorably. The model’s reliance on the cumulative standard normal distribution function highlights the probabilistic nature of option pricing, reflecting the uncertainty inherent in future price movements. A sophisticated advisor must grasp these underlying relationships to effectively advise clients on derivatives strategies.

This question examines the understanding of credit default swaps (CDS) and their use in hedging credit risk. A CDS is a financial contract where a protection buyer pays a premium to a protection seller in exchange for protection against a specific credit event, such as the default of a reference entity. The key here is to understand the directional relationship between CDS spreads and the creditworthiness of the reference entity. When the market perceives that the creditworthiness of a company is deteriorating (i.e., the risk of default is increasing), the CDS spread widens (increases). This is because investors are willing to pay a higher premium for protection against the increased risk of default. Conversely, when the market perceives that the creditworthiness of a company is improving, the CDS spread narrows (decreases). In this scenario, the investor is concerned about a potential downgrade of XYZ Corp.’s credit rating. A credit rating downgrade typically signals a deterioration in the company’s financial health and an increased risk of default. Therefore, the investor would expect the CDS spread on XYZ Corp. to widen if the downgrade occurs. To hedge against this potential widening of the CDS spread, the investor should buy protection on XYZ Corp. This means they would enter into a CDS contract as the protection buyer, paying a premium to receive compensation if XYZ Corp. defaults. If the credit rating is indeed downgraded and the CDS spread widens, the value of the CDS contract will increase, offsetting potential losses from other investments in XYZ Corp. Think of it like buying insurance on your car. You pay a premium to the insurance company, and if you get into an accident (analogous to a credit event), the insurance company pays you compensation. Similarly, by buying protection through a CDS, the investor is paying a premium to protect themselves against the potential default of XYZ Corp.

The correct answer is (a). This question assesses the understanding of how margin requirements and daily settlements affect the cash flows in a futures contract, specifically focusing on the impact of price movements on the investor’s account. Here’s a breakdown of why option (a) is correct and why the other options are incorrect: **Understanding Margin and Daily Settlement** * **Initial Margin:** The initial margin is the amount of money an investor must deposit into a margin account to open a futures contract. It acts as a performance bond. * **Maintenance Margin:** This is the minimum amount of equity that must be maintained in the margin account. If the account balance falls below this level, a margin call is issued. * **Margin Call:** A margin call requires the investor to deposit additional funds to bring the account balance back up to the initial margin level. * **Daily Settlement (Marking-to-Market):** At the end of each trading day, the futures contract is marked-to-market. This means the investor’s account is credited or debited based on the change in the futures price. If the price increases, the investor’s account is credited; if the price decreases, the investor’s account is debited. **Calculations and Explanation for Option (a)** 1. **Initial Margin:** £8,000 2. **Maintenance Margin:** £6,000 3. **Day 1 Price Decrease:** £1.50/contract point * 100 contract points = £150 decrease 4. **Day 2 Price Decrease:** £2.50/contract point * 100 contract points = £250 decrease 5. **Day 3 Price Decrease:** £7.00/contract point * 100 contract points = £700 decrease 6. **Day 4 Price Increase:** £3.00/contract point * 100 contract points = £300 increase 7. **Day 5 Price Decrease:** £4.00/contract point * 100 contract points = £400 decrease **Daily Account Balance:** * **Start:** £8,000 * **End of Day 1:** £8,000 – £150 = £7,850 * **End of Day 2:** £7,850 – £250 = £7,600 * **End of Day 3:** £7,600 – £700 = £6,900 * **End of Day 4:** £6,900 + £300 = £7,200 * **End of Day 5:** £7,200 – £400 = £6,800 **Margin Call Trigger:** The account balance falls below the maintenance margin (£6,000) on Day 3. The margin call is triggered at the end of Day 3 when the balance is £6,900. To meet the margin call, the investor must deposit enough funds to bring the account balance back to the initial margin level of £8,000. Therefore, the margin call amount is £8,000 – £6,900 = £1,100. The account balance at the end of Day 5 is £6,800. **Why Other Options are Incorrect:** * Options (b), (c) and (d) present incorrect calculations or misunderstand the timing of the margin call and the impact of daily settlements on the account balance. They may miscalculate the price changes, apply the margin call at the wrong time, or incorrectly determine the final account balance. This example illustrates the importance of understanding margin requirements, daily settlements, and how price fluctuations can impact a futures trader’s account balance, potentially leading to margin calls. The daily marking-to-market process ensures that gains and losses are realized promptly, and margin requirements are in place to mitigate the risk of default.

The question assesses the understanding of how different factors impact option prices, specifically focusing on volatility and time decay. The correct answer requires recognizing the combined effect of increased volatility (which benefits option buyers) and the relatively smaller impact of time decay over a short period. Let’s break down why the correct answer is correct and why the others are incorrect: * **Volatility Impact:** An increase in volatility generally increases the price of both call and put options. This is because higher volatility means a greater chance that the underlying asset’s price will move significantly in either direction, increasing the potential payoff for the option holder. * **Time Decay (Theta):** Time decay refers to the erosion of an option’s value as it approaches its expiration date. This effect accelerates as the expiration date nears. However, over a very short period (like one day), the impact of time decay is usually relatively small compared to the impact of a significant change in volatility. Now, consider a scenario where a trader holds a short-dated at-the-money call option. If volatility spikes significantly overnight, the positive impact of the volatility increase will likely outweigh the negative impact of one day’s worth of time decay. **Why the other options are incorrect:** * **Option B is incorrect** because while time decay does reduce option value, it is unlikely to completely offset a large volatility increase, especially in a short-dated, at-the-money option. * **Option C is incorrect** because, although the call option is at-the-money, the increase in volatility will benefit the option holder, resulting in a net increase in the option’s price. * **Option D is incorrect** because the increase in volatility benefits both call and put options, not just put options. While the delta of the call option might decrease slightly with increased volatility, the overall impact of the volatility increase will be positive for the call option holder.

Imagine a portfolio manager, Sarah, who manages a fund heavily invested in a technology company, “InnovTech.” Initially, Sarah’s portfolio has a positive Delta of 25,000 shares equivalent to InnovTech. This means if InnovTech’s stock price increases by £1, the portfolio value is expected to increase by £25,000. However, Sarah is concerned about a potential market downturn and wants to protect her portfolio against a decline in InnovTech’s stock price. To hedge this risk, Sarah decides to purchase put options on InnovTech. These put options have a Delta of -0.5 each, meaning each put option will offset 0.5 shares of InnovTech exposure. Now, suppose InnovTech’s stock price unexpectedly increases by £2. This price movement has two effects on Sarah’s hedge. First, the put options become less valuable as the stock price rises. Second, the Delta of the put options changes due to their Gamma, which is 0.02. This means that for every £1 increase in InnovTech’s stock price, the put options’ Delta increases by 0.02. The challenge for Sarah is to understand how this price movement has affected her overall portfolio Delta and how many shares she needs to buy or sell to re-establish a Delta-neutral position. Ignoring the time decay (Theta) and volatility (Vega) effects for simplicity, how should Sarah adjust her position?

The question explores the concept of gamma risk in options trading, specifically in the context of a short straddle position. A short straddle involves selling both a call and a put option with the same strike price and expiration date. This strategy profits when the underlying asset price remains relatively stable. However, it carries significant risk if the price moves substantially in either direction. Gamma measures the rate of change of an option’s delta with respect to a change in the underlying asset’s price. A high gamma indicates that the delta is highly sensitive to price changes, which can lead to significant losses for a short straddle position. In this scenario, the investor needs to understand how changes in the underlying asset’s price and the passage of time (theta decay) affect the gamma of the short straddle. As the expiration date approaches, gamma typically increases, especially when the underlying asset’s price is near the strike price. This is because the options become more sensitive to price movements as the probability of the option expiring in the money increases. The correct answer considers both the direction of the underlying asset’s price movement and the impact of time decay on gamma. If the price moves significantly away from the strike price, the gamma of the out-of-the-money option decreases, while the gamma of the in-the-money option increases. However, the overall gamma of the straddle decreases as one option becomes deeply in the money and the other deeply out of the money. The passage of time also generally increases gamma initially, but as expiration nears and the price diverges, the effect is diminished. To illustrate, imagine a tightrope walker. Gamma is like the sensitivity of their balance to wind gusts. Near the center (the strike price), even a small gust (price change) requires a large adjustment (delta hedging). As they move further from the center, the effect of the wind diminishes. Time decay is like the rope slowly fraying, making each gust more impactful up to a point, then ultimately irrelevant if they fall far enough off the rope.

The question assesses understanding of exotic derivatives, specifically barrier options, and their sensitivity to market movements. A down-and-out barrier option becomes worthless if the underlying asset’s price hits the barrier level before expiration. The payoff is dependent on whether the barrier has been breached. Here’s the breakdown of the situation: 1. **Initial Situation:** The investor holds a down-and-out call option. The FTSE 100 is at 7,500, and the barrier is at 7,000. The option is currently “alive” because the FTSE 100 hasn’t hit the barrier. 2. **Market Movement:** The FTSE 100 drops to 6,950, breaching the barrier of 7,000. 3. **Barrier Activation:** Because the barrier was breached, the down-and-out option is now worthless, regardless of any subsequent price movements. It is “knocked out.” 4. **Subsequent Recovery:** The FTSE 100 rebounding to 7,400 is irrelevant. The option’s value is already zero because the barrier was hit. Therefore, the option expires worthless. To illustrate with an analogy, imagine a parachute with a rip cord set at a certain altitude. If the rip cord is pulled (barrier breached), the parachute is useless, even if you later climb back to a higher altitude (FTSE recovers). Now, let’s consider a more complex scenario. Imagine a fund manager, Alice, who uses down-and-out call options to hedge a portion of her portfolio against a market downturn. Alice believes the FTSE 100 will stay above 7,000. If it does, she benefits from the upside potential. However, if the market crashes and the FTSE 100 hits 7,000, the hedge becomes worthless. This strategy is cheaper than a standard call option, but carries the risk of being knocked out. Another important aspect is the gamma of the option near the barrier. As the FTSE 100 approaches the barrier, the gamma (the rate of change of delta) increases significantly. This means the option’s delta (sensitivity to price changes) changes rapidly. This makes hedging the option position extremely difficult near the barrier. Finally, the Black-Scholes model can be adapted to price barrier options, but it requires adjustments to account for the probability of hitting the barrier. Monte Carlo simulations are also frequently used to price these complex instruments.

Let’s break down how to value a European call option using a two-step binomial tree, incorporating the risk-neutral probability. This example focuses on a commodity derivative, specifically Palladium, to add a unique element. We’ll assume the risk-free rate is 5%, and the option expires in two periods (each period representing 3 months). The current spot price of Palladium is £800 per ounce. In each period, the price can either go up by 20% or down by 15%. The strike price is £820. First, we calculate the up and down factors: Up factor (u) = 1 + 20% = 1.20 Down factor (d) = 1 – 15% = 0.85 Next, we construct the binomial tree: * **Period 0:** £800 * **Period 1:** Up: £800 * 1.20 = £960, Down: £800 * 0.85 = £680 * **Period 2:** * Up-Up: £960 * 1.20 = £1152 * Up-Down: £960 * 0.85 = £816 * Down-Down: £680 * 0.85 = £578 Now, we calculate the option values at expiration (Period 2): * Call Option Value (Up-Up): max(£1152 – £820, 0) = £332 * Call Option Value (Up-Down): max(£816 – £820, 0) = £0 * Call Option Value (Down-Down): max(£578 – £820, 0) = £0 Next, calculate the risk-neutral probability (p): \[p = \frac{e^{rT} – d}{u – d}\] Where r is the risk-free rate (0.05), and T is the time period (0.25 for 3 months). \[p = \frac{e^{0.05 * 0.25} – 0.85}{1.20 – 0.85} = \frac{1.01258 – 0.85}{0.35} = \frac{0.16258}{0.35} \approx 0.4645\] Now, discount back the option values from Period 2 to Period 1: * Call Option Value (Up Node at Period 1): \[(0.4645 * £332) + ((1-0.4645) * £0)] * e^{-0.05 * 0.25} = £154.21 * 0.9875 = £152.28\] * Call Option Value (Down Node at Period 1): \[(0.4645 * £0) + ((1-0.4645) * £0)] * e^{-0.05 * 0.25} = £0\] Finally, discount back to Period 0: * Call Option Value (Period 0): \[(0.4645 * £152.28) + ((1-0.4645) * £0)] * e^{-0.05 * 0.25} = £70.73 * 0.9875 = £69.85\] Therefore, the value of the European call option is approximately £69.85. This binomial model demonstrates how the price of an option is affected by potential future price movements of the underlying asset, the time to expiration, the strike price, and the risk-free interest rate. The risk-neutral probability is a crucial component, allowing us to discount expected future payoffs back to the present value. A higher volatility in the price of Palladium would widen the gap between the up and down factors, influencing the option’s value. Understanding these relationships is critical for advising clients on the appropriate use of derivatives in their investment portfolios. The two-step binomial model is a simplified version of reality, but it provides a valuable framework for understanding option pricing. In real-world scenarios, more complex models with a higher number of steps are used to achieve greater accuracy.

To determine the value of the swap, we need to calculate the present value of the expected future cash flows. The investor receives fixed payments and pays floating payments based on LIBOR. First, we need to project the LIBOR rates for the next two years. We are given the forward rate agreement (FRA) rates. The FRA rates are: * 6-month FRA starting in 6 months: 5.5% * 6-month FRA starting in 12 months: 6.0% * 6-month FRA starting in 18 months: 6.5% These FRA rates represent the market’s expectation of LIBOR for those future periods. We will use these rates as our projected LIBOR rates. Next, we calculate the expected cash flows for each period. The notional principal is £10 million. The fixed rate is 5% per annum, paid semi-annually, so each fixed payment is £10,000,000 * 0.05 / 2 = £250,000. The floating payments are based on the projected LIBOR rates: * Period 1 (6 months): £10,000,000 * 0.05 / 2 = £250,000 * Period 2 (12 months): £10,000,000 * 0.055 / 2 = £275,000 * Period 3 (18 months): £10,000,000 * 0.06 / 2 = £300,000 * Period 4 (24 months): £10,000,000 * 0.065 / 2 = £325,000 Now, we calculate the net cash flows for each period (Fixed – Floating): * Period 1: £250,000 – £250,000 = £0 * Period 2: £250,000 – £275,000 = -£25,000 * Period 3: £250,000 – £300,000 = -£50,000 * Period 4: £250,000 – £325,000 = -£75,000 To calculate the present value of these cash flows, we use the spot rates provided. Note that since cash flows are semi-annual, we need to adjust the spot rates accordingly. * 6-month spot rate: 4.5% / 2 = 2.25% * 12-month spot rate: 4.75% / 2 = 2.375% * 18-month spot rate: 5.0% / 2 = 2.5% * 24-month spot rate: 5.25% / 2 = 2.625% Now, we discount each cash flow to its present value: * Period 1: £0 / (1 + 0.0225)^1 = £0 * Period 2: -£25,000 / (1 + 0.02375)^2 = -£23,834.32 * Period 3: -£50,000 / (1 + 0.025)^3 = -£46,306.76 * Period 4: -£75,000 / (1 + 0.02625)^4 = -£68,092.83 Finally, we sum the present values of the cash flows to find the value of the swap: Value = £0 – £23,834.32 – £46,306.76 – £68,092.83 = -£138,233.91 Therefore, the value of the swap to the investor is approximately -£138,233.91.

Let’s break down how to approach valuing an exotic derivative: a barrier option with a knock-out feature tied to a geometric average. We’ll use a simplified binomial model over two periods to illustrate the core concepts. First, understand the barrier. A knock-out barrier means the option ceases to exist if the underlying asset’s price hits a predefined level. The geometric average is calculated by multiplying the asset prices at each time step and taking the nth root (where n is the number of time steps). This averaging mechanism reduces volatility compared to using the spot price directly, influencing the option’s value. Assume the initial asset price (\(S_0\)) is £100. The up factor (\(u\)) is 1.1, and the down factor (\(d\)) is 0.9. The risk-free rate (\(r\)) is 5% per period. The strike price (\(K\)) is £100. The barrier level (\(B\)) is £115. This is a European call option, meaning it can only be exercised at maturity. Period 1: The asset price can either go up to \(S_0 * u = £110\) or down to \(S_0 * d = £90\). Period 2: * Up-Up: \(S_0 * u * u = £121\). Geometric average: \(\sqrt{100 * 110 * 121} = £110.45\). The option is still alive (barrier not breached). Payoff: max(£110.45 – £100, 0) = £10.45 * Up-Down: \(S_0 * u * d = £99\). Geometric average: \(\sqrt{100 * 110 * 99} = £104.40\). The option is still alive (barrier not breached). Payoff: max(£104.40 – £100, 0) = £4.40 * Down-Up: \(S_0 * d * u = £99\). Geometric average: \(\sqrt{100 * 90 * 99} = £94.45\). The option is still alive (barrier not breached). Payoff: max(£94.45 – £100, 0) = £0 * Down-Down: \(S_0 * d * d = £81\). Geometric average: \(\sqrt{100 * 90 * 81} = £85.26\). The option is still alive (barrier not breached). Payoff: max(£85.26 – £100, 0) = £0 Calculate the risk-neutral probabilities: \(q = (e^r – d) / (u – d) = (e^{0.05} – 0.9) / (1.1 – 0.9) = (1.0513 – 0.9) / 0.2 = 0.7565\). Therefore, \(1-q = 0.2435\). Work backward through the binomial tree: * At the Up node in period 1: Value = \((0.7565 * £10.45 + 0.2435 * £4.40) / e^{0.05} = £8.84\). Since £110 is below the barrier of £115, the option is still active. * At the Down node in period 1: Value = \((0.7565 * £0 + 0.2435 * £0) / e^{0.05} = £0\). Since £90 is below the barrier of £115, the option is still active. Finally, the value of the option today is: \((0.7565 * £8.84 + 0.2435 * £0) / e^{0.05} = £6.36\). Therefore, the present value of this exotic option is approximately £6.36.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their payoff structures under different market conditions. A down-and-out call option becomes worthless if the underlying asset’s price touches or falls below the barrier level before the expiration date. The calculation involves determining if the barrier has been breached and then calculating the payoff if the option is still active at expiration. First, we need to check if the barrier was breached. The barrier level is £90. The lowest price recorded during the option’s life was £85, which is below the barrier. Therefore, the option is knocked out and becomes worthless. Therefore, the option expires worthless because the barrier was breached during the option’s lifetime. The scenario is designed to mimic real-world complexities where market fluctuations can significantly impact derivative values. It tests the candidate’s ability to analyze market data, interpret barrier conditions, and determine the appropriate payoff (or lack thereof) for an exotic derivative. The plausible incorrect answers are crafted to reflect common misunderstandings, such as ignoring the barrier condition altogether or misinterpreting the impact of the barrier breach on the option’s value. The question emphasizes the practical application of derivative knowledge in a volatile market environment, moving beyond textbook definitions to assess true comprehension.

The question explores the pricing of a European call option using a binomial tree model. The core concept is that the option’s price is derived from the possible future stock prices and the associated probabilities, discounted back to the present. The binomial tree simplifies the continuous price movement into discrete up and down steps. First, calculate the up and down factors: Up factor (u) = \(e^{\sigma \sqrt{\Delta t}}\) = \(e^{0.25 \sqrt{0.25}}\) = \(e^{0.25 * 0.5}\) = \(e^{0.125}\) ≈ 1.1331 Down factor (d) = \(e^{-\sigma \sqrt{\Delta t}}\) = \(e^{-0.25 \sqrt{0.25}}\) = \(e^{-0.25 * 0.5}\) = \(e^{-0.125}\) ≈ 0.8825 Next, calculate the risk-neutral probability (p): \[p = \frac{e^{r \Delta t} – d}{u – d} = \frac{e^{0.05 * 0.25} – 0.8825}{1.1331 – 0.8825} = \frac{e^{0.0125} – 0.8825}{0.2506} \approx \frac{1.0126 – 0.8825}{0.2506} \approx \frac{0.1301}{0.2506} \approx 0.5191\] Now, construct the binomial tree for the stock price: Initial Stock Price: £100 Up Price (S_u) = £100 * 1.1331 = £113.31 Down Price (S_d) = £100 * 0.8825 = £88.25 Calculate the option values at the final nodes: Call Option Value at Up Node (C_u) = max(£113.31 – £105, 0) = £8.31 Call Option Value at Down Node (C_d) = max(£88.25 – £105, 0) = £0 Finally, discount the expected option value back to the present: \[C = e^{-r \Delta t} [p C_u + (1-p) C_d] = e^{-0.05 * 0.25} [0.5191 * 8.31 + (1-0.5191) * 0] = e^{-0.0125} [0.5191 * 8.31] \approx 0.9876 * 4.3137 \approx 4.2604\] Therefore, the estimated price of the European call option is approximately £4.26. Imagine a scenario where a fund manager uses a binomial tree to price options on a volatile tech stock. The up and down factors represent the potential percentage changes in the stock price over a short period. The risk-neutral probability is crucial because it allows the fund manager to value the option as if there were no arbitrage opportunities in the market. This probability isn’t the actual probability of the stock going up or down, but rather a calculated probability that makes the present value of the expected future payoff of the option equal to its current price, assuming a risk-free rate of return. The binomial tree provides a simplified yet effective way to model the complex dynamics of option pricing, especially when dealing with path-dependent or exotic options. By adjusting the number of steps in the tree, the fund manager can increase the accuracy of the option price estimate.