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

The question assesses the understanding of the impact of correlation on the value of a spread option. A spread option’s payoff is dependent on the difference between the prices of two underlying assets. The correlation between these assets significantly influences the volatility of the spread and, consequently, the option’s price. Lower correlation generally increases the value of a spread option. To solve this, consider two extreme scenarios: perfect positive correlation (+1) and perfect negative correlation (-1). When the correlation is +1, the prices of the two assets move in perfect lockstep. The spread (difference) between them is relatively stable, leading to low volatility of the spread and a lower option value. When the correlation is -1, the prices move in opposite directions. This causes significant fluctuations in the spread, leading to high volatility and a higher option value. Now, consider the given scenario. The initial correlation is +0.7. If the correlation decreases to +0.2, the prices become less synchronized, and the spread becomes more volatile. This increased volatility increases the probability that the spread will exceed the strike price of the spread option, thus increasing the option’s value. The extent of the increase is non-linear and dependent on the specifics of the option (strike, time to expiry, individual asset volatilities, etc.), but the general direction is an increase in value. Therefore, a decrease in correlation from +0.7 to +0.2 would increase the value of the spread option, as the spread between the assets becomes more volatile.

To determine the breakeven point for the covered call strategy, we need to consider the initial cost of purchasing the shares, the premium received from selling the call option, and how these factors influence the overall profit or loss at different stock prices. First, calculate the net cost of the shares after accounting for the premium received. The investor buys the shares at £45 each and receives a premium of £3 per share. The net cost is therefore £45 – £3 = £42 per share. The breakeven point is the stock price at which the investor neither makes a profit nor incurs a loss. This occurs when the stock price equals the net cost of the shares. If the stock price rises above the strike price (£50), the call option will be exercised, and the investor will be obligated to sell the shares at £50. However, the initial premium received partially offsets the cost of buying the shares. If the stock price remains below the strike price at expiration, the call option expires worthless, and the investor retains the shares. In this scenario, the investor’s profit or loss depends solely on the difference between the purchase price and the final stock price, adjusted for the premium received. The covered call strategy’s breakeven point is calculated as: Breakeven Point = Purchase Price of Shares – Premium Received Breakeven Point = £45 – £3 = £42 Therefore, the breakeven point for this covered call strategy is £42. This means that if the stock price is £42 at expiration, the investor will have neither a profit nor a loss, considering the initial cost of the shares and the premium received. If the stock price is above £42, the investor will make a profit, and if it is below £42, the investor will incur a loss. The covered call strategy provides downside protection up to the amount of the premium received.

The core of this question lies in understanding how margin requirements operate within futures contracts, specifically when considering a short position. A short position obligates the holder to deliver the underlying asset at a future date. As the futures price increases, the short position incurs a loss. The margin account acts as collateral to ensure the holder can cover these potential losses. The maintenance margin is the level below which the margin account cannot fall. When the margin balance drops below this level, a margin call is triggered, requiring the investor to deposit additional funds to bring the balance back to the initial margin level. In this scenario, the initial margin is £8,000, and the maintenance margin is £6,000. The investor receives a margin call when their account balance falls to £6,000. The amount needed to meet the margin call is the difference between the initial margin and the current balance. In this case, the investor needs to deposit £2,000 (£8,000 – £6,000) to restore the margin account to the initial level. Now, let’s consider a slightly more complex, real-world analogy. Imagine a construction company, “SkyHigh Builders,” has entered into a futures contract to purchase steel at a fixed price for a skyscraper project. The initial margin they deposited acts like a performance bond, guaranteeing their ability to fulfill the contract. If the price of steel unexpectedly rises due to global supply chain disruptions, SkyHigh Builders starts incurring losses on their futures contract. If these losses erode their performance bond (margin account) to a certain threshold (maintenance margin), the exchange issues a “rectification notice” (margin call), demanding they replenish the bond to its original level. This ensures SkyHigh Builders can still afford the steel at the agreed-upon price, despite the market fluctuations. This system is designed to prevent defaults and maintain the integrity of the futures market. The key takeaway is that margin calls are not about covering the entire potential loss, but about maintaining a sufficient buffer to absorb further price fluctuations and prevent the investor from defaulting on their obligations. The investor must deposit enough funds to bring the account back to the initial margin level, not just above the maintenance margin.

Let’s consider a scenario where a UK-based agricultural cooperative, “Green Harvest,” is exploring ways to hedge against fluctuating wheat prices. Green Harvest anticipates harvesting 5,000 tonnes of wheat in six months. They are concerned about a potential price drop due to an expected bumper crop in Eastern Europe. They decide to use futures contracts traded on the ICE Futures Europe exchange to hedge their price risk. The current futures price for wheat for delivery in six months is £200 per tonne. Each futures contract is for 100 tonnes. To fully hedge their expected harvest, Green Harvest needs to sell 50 futures contracts (5,000 tonnes / 100 tonnes per contract = 50 contracts). Now, imagine that three months later, the price of wheat futures has indeed fallen to £180 per tonne due to the anticipated large harvest. Green Harvest decides to close out their position. They buy back 50 futures contracts at £180 per tonne. Their profit on the futures contracts is calculated as follows: Profit per contract = (Initial selling price – Final buying price) * Contract size = (£200 – £180) * 100 = £2,000. Total profit = Profit per contract * Number of contracts = £2,000 * 50 = £100,000. However, during those three months, Green Harvest had to post margin. Initially, the margin requirement was £5,000 per contract, totaling £250,000 (50 contracts * £5,000). Let’s say that due to price volatility, they received margin calls totaling an additional £50,000 over the three months. This brings their total margin deposited to £300,000. Now, consider the impact of basis risk. While the futures price decreased by £20 per tonne, the spot price of wheat (the price Green Harvest actually receives when they sell their harvest) only decreased by £15 per tonne. This difference is the basis. The spot price at harvest is £185 per tonne. Without hedging, Green Harvest would have received £185 * 5,000 = £925,000. With hedging, they receive £185 * 5,000 = £925,000 from selling their wheat plus £100,000 profit from the futures contracts, totaling £1,025,000. However, they also had to deposit £300,000 in margin. While the profit from the futures contracts helped offset the price decline, the basis risk and margin requirements affected the overall effectiveness of the hedge. Finally, the question explores the regulatory aspect. Under UK regulations, specifically MiFID II, Green Harvest, as an agricultural cooperative, needs to be classified. If their hedging activity is deemed to exceed what is necessary for genuine commercial purposes, they could be reclassified as a financial counterparty, subjecting them to additional regulatory requirements such as clearing obligations and higher capital requirements. This reclassification depends on a detailed assessment of their trading activity relative to their actual production and storage capacity, as determined by the Financial Conduct Authority (FCA).

The correct answer is (a). The question assesses the understanding of exotic derivatives, specifically barrier options, and their suitability in hedging strategies under specific market conditions. A down-and-out call option becomes worthless if the underlying asset’s price touches or falls below the barrier level. The investor’s belief that the price will stay above \(£90\) is crucial. If the barrier is breached, the option expires worthless, limiting the hedge’s effectiveness. Here’s why the other options are incorrect: * **(b)** A standard European call option would provide upside exposure but no protection against a price decrease. While the investor anticipates the price staying above \(£90\), relying solely on this expectation without hedging is risky. A standard call doesn’t account for the possibility, however small, of the price dipping below \(£90\). * **(c)** A protective put strategy involves buying a put option to protect against downside risk while holding the underlying asset. This strategy is suitable when an investor wants to limit potential losses but still participate in potential gains. However, given the investor’s strong conviction that the price won’t fall below \(£90\), the premium paid for the put option might be considered an unnecessary expense. Moreover, the investor is seeking a cost-effective solution, and a protective put strategy is generally more expensive than a barrier option. * **(d)** A short strangle involves selling both a call and a put option with different strike prices. This strategy profits if the underlying asset’s price remains within a defined range. While it generates income from the premiums, it exposes the investor to potentially unlimited losses if the price moves significantly in either direction. This strategy is unsuitable given the investor’s primary objective of hedging against downside risk and their belief about the price floor. Furthermore, selling options increases risk exposure, which contradicts the hedging objective. The down-and-out call option is the most suitable strategy because it provides upside exposure while being cheaper than a standard call option, reflecting the investor’s belief that the price will remain above the barrier level. The investor is willing to forgo protection if the price falls below \(£90\), making the down-and-out call a cost-effective choice. The premium saved compared to a vanilla call option reflects the risk that the option will be knocked out. The investor accepts this risk in exchange for the lower premium, aligning with their market outlook.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Green Harvest,” which produces organic wheat. Green Harvest anticipates a bumper crop in six months but fears a price drop due to oversupply in the market. They decide to hedge their risk using wheat futures contracts traded on the ICE Futures Europe exchange. Simultaneously, a small bakery chain, “Artisan Breads,” needs to secure a supply of wheat at a predictable price to manage their production costs. They also consider using futures. Green Harvest enters into a short futures contract, agreeing to deliver a specified quantity of wheat at a specified price in six months. Artisan Breads enters into a long futures contract, agreeing to accept delivery of a specified quantity of wheat at a specified price in six months. Now, imagine that three months into the contract, a severe drought hits parts of Europe, significantly reducing the expected wheat yield. This causes the price of wheat futures to rise sharply. Green Harvest, holding a short position, faces a margin call due to the increased value of the futures contract. Artisan Breads, holding a long position, sees their position increase in value. However, Green Harvest also anticipates a higher price for their actual wheat crop due to the drought-induced shortage. They decide to close out their futures position by buying offsetting futures contracts. This locks in a loss on their futures position but allows them to sell their physical wheat crop at a higher spot price. Artisan Breads also decides to close out their futures positions, locking in a profit, and then negotiate a new contract with Green Harvest based on the new spot price. The key is to understand the inverse relationship between the futures position and the physical commodity. Green Harvest’s loss on the futures is offset by the increased value of their physical wheat. Artisan Breads’ profit on the futures is offset by the increased cost of securing the wheat. The futures contract serves as a hedge, mitigating the risk of price fluctuations. The calculation involves determining the profit or loss on the futures contract and comparing it to the change in value of the physical commodity. Let’s say Green Harvest initially sold futures at £200 per tonne and closed out their position at £250 per tonne. Their loss is £50 per tonne. However, the spot price of wheat increased by £60 per tonne due to the drought. Their net gain is £10 per tonne. This example illustrates how futures contracts can be used to hedge price risk and how changes in market conditions can impact the value of futures positions. It also highlights the importance of understanding the relationship between the futures market and the underlying physical commodity market.

The question revolves around the complexities of early termination of a swap agreement, specifically an interest rate swap. It requires calculating the present value of the future cash flows that would have been exchanged under the original swap agreement, considering the prevailing market interest rates at the time of termination. The calculation involves projecting the future cash flows based on the fixed rate of the swap and the expected floating rates derived from the yield curve, then discounting these cash flows back to the present using the appropriate discount rates. The difference between the present value of the fixed and floating legs determines the termination value. Here’s a breakdown of the calculation: 1. **Project Future Cash Flows:** Based on the yield curve, we project the expected floating rates for each period. These rates, along with the fixed rate of the swap, determine the net cash flow for each period. 2. **Discount Cash Flows:** Each net cash flow is discounted back to the present using the corresponding spot rate from the yield curve. This yields the present value of each individual cash flow. 3. **Sum Present Values:** The present values of all future cash flows are summed to arrive at the total present value of the swap. This represents the termination value of the swap. The challenge lies in accurately interpreting the yield curve data, projecting the floating rates, and applying the correct discount rates for each period. Furthermore, understanding the implications of negative present values is crucial. A negative value implies that the party receiving the fixed rate (in this case, the client) would need to pay the counterparty to terminate the swap. For instance, consider a scenario where a company entered into an interest rate swap to hedge against rising interest rates. If interest rates subsequently fall significantly, the fixed rate they are paying under the swap becomes more attractive than the prevailing market rates. Terminating the swap would mean foregoing this advantage, hence the negative termination value. This concept is analogous to breaking a fixed-rate mortgage when interest rates have fallen; the borrower would typically need to compensate the lender for the lost interest income. Another analogy would be a farmer who has entered into a forward contract to sell their crops at a fixed price. If the market price of the crops falls below the forward price, the farmer is in a favorable position. Terminating the forward contract would mean losing this advantage, and the farmer would likely need to compensate the counterparty.

The question revolves around the concept of delta hedging a short call option position and the impact of gamma on the hedge’s effectiveness. Delta represents the sensitivity of the option price to changes in the underlying asset’s price. Gamma, in turn, represents the sensitivity of the delta to changes in the underlying asset’s price. A delta-neutral portfolio is constructed to be insensitive to small price movements in the underlying asset. However, this neutrality is only valid for a small range due to gamma. When gamma is high, the delta changes rapidly, and the hedge needs to be adjusted more frequently to maintain its effectiveness. Transaction costs associated with rebalancing the hedge eat into the profits. Here’s how we calculate the profit/loss: 1. **Initial Hedge:** The investor is short a call option with a delta of 0.4. This means they need to buy 0.4 shares of the underlying asset to delta hedge. 2. **Initial Cost of Hedge:** The initial cost of the hedge is 0.4 shares * £100/share = £40. 3. **Price Increase:** The underlying asset’s price increases by £10, so the new price is £110. 4. **New Delta:** The delta increases to 0.7 due to gamma. 5. **Rebalancing:** The investor needs to buy an additional 0.3 shares (0.7 – 0.4) to rebalance the hedge. 6. **Cost of Rebalancing:** The cost of buying the additional 0.3 shares is 0.3 shares * £110/share = £33. 7. **Total Cost of Hedge:** The total cost of the hedge is £40 + £33 = £73. 8. **Profit on Hedge:** The profit on the hedge is the change in the underlying asset’s price multiplied by the initial delta (since we were short the option) + the cost of rebalancing, so (110-100) * 0.4 – 33 = 4 – 33 = -£29. The hedge lost £29. 9. **Option Value Change:** The option’s value increased. Since the investor is short the call, this represents a loss. The option price increased by £12. 10. **Net Profit/Loss:** The net profit/loss is the profit/loss on the hedge plus the profit/loss on the option. Since we are short the call, the profit is -12, so the total profit/loss is -29 – 12 = -£41. 11. **Transaction Costs:** Transaction costs are £1 per share traded. The investor bought 0.4 shares initially and 0.3 shares to rebalance, for a total of 0.7 shares traded. Total transaction costs are 0.7 * £1 = £0.7. 12. **Final Profit/Loss:** The final profit/loss is the net profit/loss minus transaction costs: -£41 – £0.7 = -£41.7. This example highlights how gamma affects the cost of maintaining a delta hedge and how transaction costs can erode profits. High gamma means more frequent rebalancing, leading to higher transaction costs.

The question explores the impact of volatility on option pricing, specifically focusing on the gamma of a portfolio. Gamma measures the rate of change of delta with respect to changes in the underlying asset’s price. A higher gamma implies greater sensitivity to price fluctuations. The scenario involves a portfolio manager using options to hedge a large equity position. The manager needs to understand how changes in implied volatility, influenced by macroeconomic events, will affect the portfolio’s gamma and, consequently, the effectiveness of the hedge. The calculation involves understanding the relationship between volatility, gamma, and hedging effectiveness. If implied volatility increases, the gamma of the options also increases. This means the hedge becomes more sensitive to price changes in the underlying asset. The manager must then adjust the hedge to maintain the desired level of risk protection. Let’s consider a hypothetical example. Suppose the manager initially has a portfolio with a gamma of 500 (meaning the delta changes by 500 for every £1 change in the underlying asset). If implied volatility rises significantly due to unexpected economic data, the gamma might increase to 750. This higher gamma indicates that the portfolio’s delta is now much more sensitive to movements in the underlying asset. To compensate, the manager needs to reduce the portfolio’s gamma back to the original level, which can be achieved by adjusting the number of options held. The question tests the candidate’s ability to link macroeconomic events, implied volatility, gamma, and hedging strategies. It requires a deep understanding of how derivatives are used in risk management and how market dynamics can impact the effectiveness of these strategies. The correct answer will demonstrate an understanding of the relationship between volatility and gamma, and how this impacts the number of options needed to maintain a hedge.

Let’s break down this complex scenario. First, we need to understand the potential profit from the swap. Company A effectively receives a fixed rate of 4.5% and pays a floating rate of LIBOR + 0.5%. Company B does the opposite, receiving LIBOR + 0.5% and paying 4.5%. The intermediary bank profits from the spread between these rates. The total cost of funds for Company A if it directly issued floating-rate notes would be LIBOR + 0.7%. Through the swap, it pays LIBOR + 0.5% to Company B and 0.15% to the bank (half of the total spread), resulting in a total cost of LIBOR + 0.65%. This represents a saving of 0.05% (0.7% – 0.65%). Similarly, the total cost of funds for Company B if it directly issued fixed-rate notes would be 4.7%. Through the swap, it pays 4.5% to Company A and 0.15% to the bank (half of the total spread), resulting in a total cost of 4.65%. This represents a saving of 0.05% (4.7% – 4.65%). The total saving generated by the swap is 0.1% (0.05% for Company A + 0.05% for Company B). The bank captures the remaining 0.1% spread (0.2% – 0.1%). Now, let’s consider the impact of the credit rating downgrade of Company B. This increases the perceived risk of the swap. If the swap is unwound, the replacement cost reflects the current market conditions and the increased risk premium demanded by counterparties due to Company B’s lower credit rating. If the replacement cost is estimated at 0.3% of the notional principal, this means it would cost 0.3% * £50 million = £150,000 to replace the swap with another counterparty. This cost would typically be borne by the party whose creditworthiness deteriorated, in this case, Company B. However, the question asks for the *maximum* potential loss for the *intermediary bank*. The bank’s loss arises from the possibility that Company B defaults and the replacement cost is higher than anticipated, or that the bank faces difficulty finding a counterparty willing to take on the swap with Company B. The bank faces the risk of not being able to fully recover the replacement cost from Company B. The maximum loss for the bank would occur if Company B defaults and the bank cannot recover any of the £150,000 replacement cost, plus the loss of any future profits it expected to earn from the swap. However, the question specifically asks about the impact of the *unwinding* of the swap due to the downgrade, not a default. Therefore, the most direct loss to the bank is the potential inability to fully recover the replacement cost if Company B cannot pay. The bank might also have internal costs associated with unwinding and finding a new counterparty. The question does not provide enough information to quantify the bank’s internal costs or loss of future profits, so the best answer is the replacement cost.

The question focuses on understanding how different factors affect the price of an American-style call option on a stock index. The core concept is the Black-Scholes model, even though the question doesn’t explicitly state it, the factors influencing option prices are derived from it. Key factors include: the current index level, the strike price, time to expiration, risk-free interest rate, dividend yield, and volatility. * **Index Level:** A higher index level generally increases the call option price as the option is more likely to be in the money at expiration. * **Strike Price:** A higher strike price decreases the call option price because the index needs to reach a higher level for the option to be profitable. * **Time to Expiration:** A longer time to expiration generally increases the call option price. This is because there is more time for the index to move favorably (above the strike price). * **Risk-Free Interest Rate:** A higher risk-free interest rate increases the call option price. This is because the present value of the strike price decreases, making the option relatively more attractive. * **Dividend Yield:** A higher dividend yield decreases the call option price. Dividends reduce the stock price, making it less likely that the option will be in the money. * **Volatility:** Higher volatility increases the call option price. Greater volatility means there’s a higher chance the index will make a large move, either up or down, but the call option benefits only from upward moves. The question presents a scenario where a portfolio manager needs to hedge against downside risk. The best strategy will involve using a call option, and understanding how changes in these factors will affect the option’s price is crucial for effective hedging. Let’s say the manager is concerned about a potential market correction and wants to purchase call options on a stock index to protect against losses. If the manager expects volatility to increase due to upcoming economic data releases, they should be aware that the call option price will likely increase, making the hedge more expensive. Conversely, if the risk-free interest rate is expected to rise, the call option price may also increase. Understanding these relationships allows the manager to make informed decisions about the timing and type of options to purchase for their hedging strategy.

The core of this question lies in understanding how the price of a European call option is affected by the time to expiration, interest rates, and volatility, specifically in the context of a binomial model and the implications of risk-neutral valuation. The binomial model is a simplified way to understand option pricing, where the underlying asset price can only move up or down in discrete steps. In a risk-neutral world, all assets are expected to earn the risk-free rate of return. Therefore, the option price is calculated by discounting the expected payoff at expiration back to the present using the risk-free rate. To solve this problem, we need to consider the impact of each factor individually and then combine them. * **Time to Expiration:** Increasing the time to expiration generally increases the value of a call option. This is because there is more time for the underlying asset to increase in value, making the option more likely to be in the money at expiration. * **Interest Rates:** Higher interest rates typically increase the value of a call option. This is because the present value of the strike price decreases, making the option cheaper to exercise in the future. Also, in a risk-neutral world, higher interest rates imply a higher expected growth rate for the underlying asset. * **Volatility:** Increased volatility increases the value of a call option. Higher volatility means there is a greater chance that the underlying asset will experience a significant price increase, making the option more valuable. In the scenario provided, we have two options: Option A and Option B. Option B has a longer time to expiration, higher interest rates, and higher volatility than Option A. Therefore, Option B should have a higher value than Option A. The binomial model calculation, while not explicitly performed here, implicitly underpins this logic. The risk-neutral probabilities and discounted payoffs would reflect the longer time horizon, higher interest rates, and increased volatility, resulting in a higher overall option value for Option B. This is a complex question that requires a deep understanding of option pricing theory and the factors that influence option values. It tests the candidate’s ability to apply these concepts in a practical scenario.

To determine the profit or loss from the combined strategy, we need to calculate the premium paid and received, and the payoff at expiration. The investor buys a call option with a strike price of 1450 and a premium of 80 and sells a call option with a strike price of 1500 and a premium of 40. The net premium paid is 80 – 40 = 40. Now, we need to analyze the payoff at expiration at the spot price of 1480. Since the spot price (1480) is higher than the strike price of the call option purchased (1450), the call option will be exercised. The payoff from the purchased call option is the spot price minus the strike price, which is 1480 – 1450 = 30. Since the spot price (1480) is lower than the strike price of the call option sold (1500), the call option will not be exercised. The payoff from the sold call option is 0. The total payoff is 30 – 0 = 30. Since the net premium paid is 40 and the total payoff is 30, the profit or loss is the total payoff minus the net premium paid, which is 30 – 40 = -10. Therefore, the investor incurs a loss of 10. Let’s consider a scenario where a farmer uses a forward contract to lock in the price for their wheat crop. This eliminates price uncertainty but also removes the potential for profit if the market price rises significantly. Conversely, a manufacturing company might use a swap to convert floating-rate debt into fixed-rate debt, providing predictability in their financing costs but potentially missing out on lower rates if interest rates decline. An exotic derivative, such as a barrier option, could be used by a hedge fund to speculate on a specific price range for a stock, offering high potential returns but also significant risk if the price moves outside the defined barrier. These examples highlight how derivatives can be used for hedging and speculation, each with its own set of risks and rewards. Understanding these trade-offs is crucial for advisors when recommending derivative strategies to clients, considering their risk tolerance and investment objectives.

The question assesses the understanding of exotic derivatives, specifically a cliquet option, and how its payoff is determined based on a series of capped returns over multiple periods. The key is to calculate the return for each period, apply the cap, and then sum the capped returns. Period 1: Return = (110 – 100) / 100 = 10%. Since 10% < 8%, the capped return is 8%. Period 2: Return = (115 – 110) / 110 = 4.55%. Since 4.55% < 8%, the capped return is 4.55%. Period 3: Return = (112 - 115) / 115 = -2.61%. Since -2.61% > -3%, the capped return is -2.61%. Period 4: Return = (120 – 112) / 112 = 7.14%. Since 7.14% < 8%, the capped return is 7.14%. Period 5: Return = (125 – 120) / 120 = 4.17%. Since 4.17% < 8%, the capped return is 4.17%. Sum of capped returns = 8% + 4.55% – 2.61% + 7.14% + 4.17% = 21.25%. Now consider a contrasting scenario: Imagine a farmer using a series of forward contracts to sell wheat over several months. Each month, the farmer agrees to sell a certain quantity of wheat at a predetermined price. This is similar to a cliquet option, where the "return" is the difference between the spot price and the forward price each month, and the farmer's overall profit is the sum of these differences. However, unlike the cliquet option with caps, the farmer's profit in each month is uncapped, meaning they could potentially earn significantly more or less depending on the actual market price of wheat. Another analogy is a bond with a variable coupon rate linked to an index, but with annual caps and floors on the coupon payment. This bond's overall return is the sum of the capped coupon payments, much like the cliquet option's payoff. However, the bond also has a floor, which guarantees a minimum coupon payment even if the index performs poorly, while the cliquet option only has a floor on each period's return, not on the overall payoff.

A wealth manager constructs a synthetic forward position by buying a call option and selling a put option on the FTSE 100 index, both with a strike price of 7500 and expiring in three months. The call option costs £800, and the put option generates a premium of £500. At expiration, the FTSE 100 index closes at 7650. Calculate the net profit or loss for the wealth manager on this synthetic forward position, ignoring transaction costs and margin requirements.

Let’s break down this exotic derivative pricing scenario. The key is understanding how the barrier affects the option’s payoff and how the correlation between the asset and the reference index impacts the probability of the barrier being hit. First, we need to calculate the Black-Scholes price of a standard European call option on the asset. The Black-Scholes formula is: \[C = S_0N(d_1) – Ke^{-rT}N(d_2)\] where: * \(S_0\) is the initial asset price (£100) * \(K\) is the strike price (£105) * \(r\) is the risk-free rate (5% or 0.05) * \(T\) is the time to maturity (1 year) * \(N(x)\) is the cumulative standard normal distribution function * \[d_1 = \frac{ln(\frac{S_0}{K}) + (r + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}}\] * \[d_2 = d_1 – \sigma\sqrt{T}\] * \(\sigma\) is the volatility of the asset (20% or 0.20) Plugging in the values: \[d_1 = \frac{ln(\frac{100}{105}) + (0.05 + \frac{0.20^2}{2})1}{0.20\sqrt{1}} = \frac{-0.0488 + 0.07}{0.20} = 0.106\] \[d_2 = 0.106 – 0.20\sqrt{1} = -0.094\] Using a standard normal distribution table or calculator: \(N(d_1) = N(0.106) \approx 0.5421\) \(N(d_2) = N(-0.094) \approx 0.4626\) Therefore, the Black-Scholes price of the standard call option is: \[C = 100 \times 0.5421 – 105 \times e^{-0.05 \times 1} \times 0.4626 = 54.21 – 105 \times 0.9512 \times 0.4626 = 54.21 – 46.09 = 8.12\] Now, consider the down-and-out barrier. The correlation between the asset and the reference index is crucial. A negative correlation means that when the asset price decreases, the index tends to increase, making it *less* likely that the barrier will be hit. This increases the value of the down-and-out call compared to a situation with zero correlation, as the protection against hitting the barrier is higher. The standard call option price of £8.12 represents the *upper bound* of the down-and-out call’s value in this scenario. The barrier feature *reduces* the value from this upper bound, but the negative correlation *partially offsets* this reduction. Since the barrier is relatively close to the initial asset price, the barrier effect is significant, and the negative correlation only provides partial compensation. Therefore, the price should be lower than £8.12 but higher than a price calculated assuming zero correlation (which would be significantly lower). Considering these factors, a reasonable price would be £4.75. This price reflects the value erosion due to the barrier, partially offset by the negative correlation providing some protection against the barrier being triggered.

The question revolves around the concept of a chooser option, a type of exotic derivative that gives the holder the right to decide, at a predetermined future date (the choice date), whether the option will become a call or a put option. The strike price and expiration date are usually the same for both potential call and put options. To solve this, we need to understand the payoff structure and the factors influencing the chooser option’s value. The investor will choose whichever option (call or put) is more valuable at the choice date. A higher underlying asset price at the choice date will favor the call option, while a lower price will favor the put option. Here’s how to approach the problem: 1. **Calculate the intrinsic value of the call option:** The call option’s intrinsic value is the maximum of (Asset Price – Strike Price, 0). In this case, it’s max(105 – 100, 0) = 5. 2. **Calculate the intrinsic value of the put option:** The put option’s intrinsic value is the maximum of (Strike Price – Asset Price, 0). In this case, it’s max(100 – 105, 0) = 0. 3. **Determine the investor’s choice:** Since the call option has a higher intrinsic value (5) than the put option (0) at the choice date, the investor will choose the call option. 4. **Calculate the profit/loss:** The investor’s profit is the intrinsic value of the chosen option (call option) minus the initial premium paid for the chooser option. Profit = 5 – 3 = 2. Therefore, the investor’s profit is £2. A good analogy for a chooser option is a restaurant that offers a “soup or salad” option with a meal. You pay a fixed price upfront (the premium), and on a specific date (when your meal arrives), you get to choose whether you want soup or salad, based on which one you prefer at that moment. Similarly, with a chooser option, you pay a premium upfront and get to choose between a call or a put option at a later date, based on market conditions. Another way to think about it is as an insurance policy. You’re unsure if the asset price will go up or down, so you buy the right to choose the best outcome. If the price goes up, you benefit from the call option; if it goes down, you benefit from the put option. The initial premium is the cost of this insurance. This highlights the flexibility and risk management aspect of chooser options. Finally, it’s crucial to remember that the value of a chooser option depends on the volatility of the underlying asset. Higher volatility increases the potential payoff from both the call and put options, making the chooser option more valuable. The time to the choice date also matters. A longer time allows for greater price fluctuations, increasing the value of the option.

Let’s consider a scenario involving a UK-based agricultural cooperative, “GreenHarvest,” which produces and exports organic wheat. GreenHarvest faces price volatility in the global wheat market and seeks to hedge its price risk using futures contracts traded on the ICE Futures Europe exchange. They are considering using short hedge strategy. Here’s how we analyze the optimal hedge ratio: 1. **Understanding the Hedge Ratio:** The hedge ratio minimizes the variance of the hedged position. A perfect hedge is often unattainable in practice due to basis risk (the difference between the spot price and the futures price). 2. **Calculating the Hedge Ratio:** The optimal hedge ratio (HR) is calculated as: \[HR = \rho \cdot \frac{\sigma_s}{\sigma_f}\] where: * \(\rho\) is the correlation coefficient between changes in the spot price of GreenHarvest’s wheat and changes in the futures price of the ICE Futures Europe wheat contract. * \(\sigma_s\) is the standard deviation of changes in the spot price of GreenHarvest’s wheat. * \(\sigma_f\) is the standard deviation of changes in the futures price of the ICE Futures Europe wheat contract. 3. **Applying the Data:** Assume that GreenHarvest’s historical data shows: * \(\rho = 0.8\) (Correlation between GreenHarvest’s wheat spot price and ICE Futures Europe wheat futures price) * \(\sigma_s = 0.05\) (Standard deviation of changes in GreenHarvest’s wheat spot price) * \(\sigma_f = 0.04\) (Standard deviation of changes in the ICE Futures Europe wheat futures price) Therefore, the optimal hedge ratio is: \[HR = 0.8 \cdot \frac{0.05}{0.04} = 0.8 \cdot 1.25 = 1.0\] 4. **Interpreting the Hedge Ratio:** A hedge ratio of 1.0 implies that GreenHarvest should sell one futures contract for each unit of wheat they want to hedge. If GreenHarvest anticipates selling 500 tonnes of wheat, they should sell 500 futures contracts (assuming each contract covers one tonne). 5. **Impact of Basis Risk:** Basis risk is the difference between the spot price and the futures price at the time the hedge is lifted. If the basis weakens (spot price decreases relative to the futures price), GreenHarvest’s hedge will be less effective. Conversely, if the basis strengthens (spot price increases relative to the futures price), the hedge will be more effective, potentially leading to a gain above the expected hedged price. 6. **Practical Considerations:** GreenHarvest also needs to consider margin requirements, counterparty risk, and the liquidity of the futures market. They should also have a clear understanding of their risk tolerance and hedging objectives. For example, if they are extremely risk-averse, they might choose to over-hedge (hedge ratio > 1) to minimize potential losses, even if it means sacrificing some potential gains. 7. **Regulatory Context:** GreenHarvest must comply with relevant UK regulations, including those related to market abuse and reporting requirements under EMIR (European Market Infrastructure Regulation). They should also consult with a qualified advisor to ensure their hedging strategy is appropriate for their specific circumstances and complies with all applicable laws and regulations.

The value of a swap is determined by calculating the present value of the expected future cash flows. In an interest rate swap, these cash flows are the differences between the fixed and floating interest payments. To determine the swap’s value to Company Alpha, we need to discount the future cash flows at the appropriate discount rates. These rates are derived from the spot rates and are used to calculate the present value of each cash flow. First, we calculate the expected future cash flows. Company Alpha receives fixed and pays floating. The expected floating rates for each year are given. We calculate the net cash flow for each year by subtracting the expected floating rate from the fixed rate (5%) and multiplying by the notional principal (£10 million). Year 1: (0.05 – 0.04) * £10,000,000 = £100,000 Year 2: (0.05 – 0.045) * £10,000,000 = £50,000 Year 3: (0.05 – 0.055) * £10,000,000 = -£50,000 Next, we calculate the discount factors using the spot rates. The discount factor for each year is calculated as 1 / (1 + spot rate)^year. Year 1: 1 / (1 + 0.03) = 0.97087 Year 2: 1 / (1 + 0.035)^2 = 0.93242 Year 3: 1 / (1 + 0.04)^3 = 0.88899 Now, we calculate the present value of each cash flow by multiplying the cash flow by the corresponding discount factor. Year 1: £100,000 * 0.97087 = £97,087 Year 2: £50,000 * 0.93242 = £46,621 Year 3: -£50,000 * 0.88899 = -£44,449.5 Finally, we sum the present values of all cash flows to find the value of the swap to Company Alpha. £97,087 + £46,621 – £44,449.5 = £99,258.5 Therefore, the value of the swap to Company Alpha is approximately £99,258.5. This represents the economic benefit or cost to Alpha of being in the swap, given the current market expectations for interest rates. A positive value indicates a benefit, while a negative value would indicate a cost. This valuation is crucial for risk management, accounting, and potential trading decisions related to the swap.

Let’s analyze the situation. A quanto swap is a type of cross-currency derivative where interest rate payments are calculated in one currency but paid in another. This eliminates exchange rate risk for one of the parties. The key here is the fixed exchange rate used for payment conversion. The investor, a UK pension fund, wants to receive USD interest payments but doesn’t want to be exposed to GBP/USD exchange rate fluctuations. They enter a GBP-based quanto swap, receiving USD LIBOR and paying GBP LIBOR. The fixed GBP/USD rate is crucial. If the actual GBP/USD rate appreciates (GBP becomes stronger), the pension fund benefits because they are receiving USD converted at the lower fixed rate. If the GBP/USD rate depreciates (GBP becomes weaker), they are worse off as the fixed rate is now more favorable than the actual spot rate. In this case, the fixed rate is 1.25 GBP/USD. The notional principal is £50 million. The actual GBP/USD spot rate at the payment date is 1.30 GBP/USD. The fund is receiving USD, so the fixed rate of 1.25 means they receive fewer USD for each GBP than they would at the spot rate of 1.30. The interest payment is calculated on the USD notional, which needs to be derived from the GBP notional using the fixed rate. USD notional = £50,000,000 * 1.25 = $62,500,000. The USD LIBOR is 5% per annum, and the payment is semi-annual, so the interest payment is $62,500,000 * 0.05 * 0.5 = $1,562,500. Now, we calculate the GBP equivalent of this USD amount using both the fixed rate and the spot rate. At the fixed rate: $1,562,500 / 1.25 = £1,250,000. At the spot rate: $1,562,500 / 1.30 = £1,201,923.08. The difference is £1,250,000 – £1,201,923.08 = £48,076.92. Since the spot rate is higher than the fixed rate, the pension fund loses out compared to converting at the spot rate. Therefore, there is a loss of £48,076.92 due to the quanto feature.

To determine the most suitable derivative instrument, we must analyze the company’s specific needs and risk profile. In this case, AgriCorp faces the risk of fluctuating soybean prices, impacting its profitability. A forward contract locks in a future price, eliminating uncertainty but also foregoing potential gains if prices rise. A futures contract offers similar price certainty but with standardized terms and daily marking-to-market, which may require AgriCorp to manage margin calls. An option provides the right, but not the obligation, to buy or sell soybeans at a specific price, offering downside protection while allowing participation in favorable price movements. A swap involves exchanging cash flows based on different price benchmarks, suitable for long-term hedging strategies. Exotic derivatives are more complex and tailored to specific needs but may involve higher costs and liquidity risks. Considering AgriCorp’s desire for downside protection while retaining upside potential, an option strategy is the most appropriate. Specifically, buying a put option on soybean futures contracts would provide a floor price for their soybeans, protecting them from price declines. If soybean prices rise, AgriCorp can choose not to exercise the option and sell their soybeans at the higher market price. The cost of the put option is the premium paid, which represents the maximum loss AgriCorp would incur if soybean prices rise significantly. This strategy aligns with AgriCorp’s risk management objectives by providing a balance between price protection and potential profit maximization. The other options are less suitable. A forward or futures contract would eliminate upside potential. A swap is more complex and may not be necessary for AgriCorp’s relatively straightforward hedging needs. Exotic derivatives are typically used for highly specialized situations and are unlikely to be the best choice for managing soybean price risk.

Let’s break down how to determine the most suitable derivative for mitigating specific risks faced by a multinational corporation (MNC) with operations in both the UK and the Eurozone. The MNC, “EuroBrit Industries,” faces a complex web of exposures, including fluctuating exchange rates between GBP and EUR, interest rate volatility in both regions, and commodity price risk due to raw material sourcing. We need to analyze the nuances of each derivative type to identify the optimal solution. A forward contract locks in a specific exchange rate or price for a future transaction. While simple, forwards lack the flexibility to adapt to changing market conditions. If EuroBrit’s outlook changes, unwinding the forward can be costly. Futures contracts, being exchange-traded, offer greater liquidity and standardization. However, they also involve margin calls and daily marking-to-market, which can strain cash flow if the market moves against EuroBrit. Options provide the right, but not the obligation, to buy or sell an asset at a predetermined price. This flexibility comes at the cost of a premium. If EuroBrit’s risk is asymmetric (e.g., they only need protection against a specific adverse scenario), options can be highly efficient. Swaps involve exchanging cash flows based on different underlying assets or indices. For instance, EuroBrit could enter into an interest rate swap to convert floating-rate debt into fixed-rate debt, or a currency swap to manage long-term currency exposure. Exotic derivatives are customized instruments tailored to specific needs. These can be powerful tools, but they also come with higher complexity and potential illiquidity. In EuroBrit’s case, a combination of derivatives might be the most effective strategy. For short-term currency exposure related to trade flows, forwards or futures could be used. For managing long-term currency risk associated with foreign subsidiaries, a currency swap would be more appropriate. To protect against rising commodity prices, EuroBrit could use options, providing upside potential if prices fall. Interest rate swaps could be used to manage the interest rate risk on their debt portfolio. The key is to carefully assess the specific nature of each risk, the desired level of protection, and the cost-benefit trade-off of each derivative type. The regulatory environment in both the UK and Eurozone, including EMIR requirements, must also be considered. Furthermore, EuroBrit must have robust internal controls and expertise to manage the complexities of derivative transactions.

To determine the profit or loss from the covered call strategy, we need to consider the premium received from selling the call option, the cost of purchasing the underlying shares, and the final market price of those shares at expiration. First, calculate the total cost of purchasing the shares: 1000 shares * £10.50/share = £10,500. Next, calculate the total premium received from selling the call options: 10 contracts * 100 shares/contract * £0.75/share = £750. The net cost of the position is the cost of the shares minus the premium received: £10,500 – £750 = £9,750. Now, we analyze the outcome at expiration. Since the market price of the shares (£11.00) is above the strike price (£10.75), the call options will be exercised. This means the investor is obligated to sell their shares at the strike price. The total revenue from selling the shares at the strike price is: 1000 shares * £10.75/share = £10,750. The profit is the total revenue minus the net cost: £10,750 – £9,750 = £1,000. Therefore, the investor makes a profit of £1,000. Consider a different scenario: Suppose the share price remained at £10.50 at expiration. In this case, the call options would expire worthless, and the investor would keep the £750 premium. Their shares would still be worth £10.50 each, so their total asset value would be £10,500. The net profit would be £750 (the premium). Another scenario: If the share price dropped to £9.50, the options would expire worthless, and the investor would keep the £750 premium. However, their shares would now be worth £9.50 each, totaling £9,500. The net loss would be £250 (£10,500 initial cost – £9,500 current value + £750 premium). This covered call strategy is designed to generate income (the premium) and provide some downside protection. However, it limits the upside potential because the investor is obligated to sell the shares if the price rises above the strike price.

The optimal hedge ratio minimizes the variance of the hedged portfolio. In this scenario, we need to calculate the hedge ratio using the correlation coefficient and standard deviations of the asset and the futures contract. The formula for the hedge ratio (h) is: \[h = \rho \cdot \frac{\sigma_S}{\sigma_F}\] Where: – \(\rho\) is the correlation coefficient between the spot asset and the futures contract. – \(\sigma_S\) is the standard deviation of the spot asset. – \(\sigma_F\) is the standard deviation of the futures contract. Given: – \(\rho = 0.75\) – \(\sigma_S = 0.15\) (15% standard deviation of the spot asset) – \(\sigma_F = 0.20\) (20% standard deviation of the futures contract) Plugging in the values: \[h = 0.75 \cdot \frac{0.15}{0.20} = 0.75 \cdot 0.75 = 0.5625\] Since the investor wants to hedge a portfolio worth £2,000,000, we need to determine the number of futures contracts required. Each futures contract is based on £250,000 of the underlying asset. Total futures contract value needed for hedging = Hedge ratio * Portfolio value = \(0.5625 \times £2,000,000 = £1,125,000\). Number of futures contracts = Total futures contract value needed / Contract size = \(\frac{£1,125,000}{£250,000} = 4.5\) Since you can’t trade fractions of contracts, the investor needs to decide whether to round up or down. In this case, rounding to the nearest whole number is appropriate. Rounding 4.5 to the nearest whole number gives 5 contracts. Therefore, the investor should short 5 futures contracts to best hedge their portfolio. This calculation illustrates a practical application of hedging using futures contracts. It involves understanding correlation, standard deviation, and the mechanics of futures contracts. The hedge ratio is a critical concept in risk management, allowing investors to mitigate potential losses from adverse price movements. The example showcases how theoretical concepts are applied in real-world investment decisions. The use of the correlation coefficient and standard deviations allows for a tailored hedge that reflects the specific relationship between the asset and the hedging instrument. The rounding decision highlights the practical constraints in trading futures contracts and the need for careful consideration when implementing a hedge.

The question assesses understanding of how margin requirements function in futures contracts, specifically focusing on the impact of adverse price movements and the margin call process. The initial margin is the amount required to open a futures position, while the maintenance margin is the level below which the account cannot fall. If the account balance drops below the maintenance margin, a margin call is issued, requiring the investor to deposit funds to bring the account back to the initial margin level. In this scenario, an investor opens a short position in orange juice futures. A short position profits when the price of the underlying asset decreases. However, in this case, the price unexpectedly rises, resulting in losses for the investor. The margin call is triggered when the account balance falls below the maintenance margin. The investor must then deposit enough funds to restore the account balance to the initial margin level. Let’s calculate the margin call: 1. Initial Margin: £6,000 2. Maintenance Margin: £5,000 3. Price Increase: 8 pence per pound, and the contract is for 15,000 pounds. Therefore, the total loss is \( 8 \times 15,000 = 120,000 \) pence or £1,200. 4. Account Balance after Price Increase: £6,000 – £1,200 = £4,800. 5. Margin Call Amount: To bring the account back to the initial margin of £6,000, the investor needs to deposit £6,000 – £4,800 = £1,200. A key nuance is understanding that the margin call restores the account to the *initial* margin, not just above the maintenance margin. This ensures the investor has sufficient funds to cover potential further losses. It also tests the understanding of regulatory requirements concerning margin calls and the broker’s responsibility to protect themselves against counterparty risk. The example uses orange juice futures to create a relatable, yet unique, scenario.

The question explores the application of put-call parity in a scenario where transaction costs exist. Put-call parity, in its simplest form, states that for European options with the same strike price and expiration date, the price of a call option plus the present value of the strike price should equal the price of a put option plus the price of the underlying asset. However, real-world markets are not perfect and include transaction costs, which affect the arbitrage-free relationship described by put-call parity. The formula representing put-call parity is: \(C + PV(K) = P + S\), where \(C\) is the call option price, \(PV(K)\) is the present value of the strike price, \(P\) is the put option price, and \(S\) is the current price of the underlying asset. When transaction costs are involved, the arbitrage opportunity is only profitable if the profit exceeds these costs. The scenario involves calculating the maximum transaction cost for which an arbitrage opportunity remains viable. The trader will either execute a “reverse conversion” or a “conversion” strategy, depending on which side of the put-call parity equation is cheaper to replicate. In a conversion, the trader buys the stock and a put option and sells a call option. In a reverse conversion, the trader sells the stock and buys a call option and sells a put option. In this case, \(S = 50\), \(K = 52\), \(r = 0.05\), \(t = 0.25\), \(C = 4\), and \(P = 5\). First, calculate the present value of the strike price: \(PV(K) = \frac{52}{e^{0.05 \times 0.25}} \approx 51.35\). Now, check for arbitrage opportunities: 1. Conversion: \(C + PV(K) = 4 + 51.35 = 55.35\). \(P + S = 5 + 50 = 55\). Here, \(P + S < C + PV(K)\), suggesting a conversion strategy. 2. Reverse Conversion: \(P + S = 55\). \(C + PV(K) = 55.35\). Here, \(C + PV(K) > P + S\), suggesting a reverse conversion strategy. The difference is \(55.35 – 55 = 0.35\). This difference represents the potential profit before transaction costs. The maximum transaction cost must be less than this profit to make the arbitrage viable. Since the trader has to buy and sell assets, the transaction cost applies twice (once for buying and once for selling). Therefore, the maximum transaction cost per transaction is \(0.35 / 2 = 0.175\). Therefore, the maximum transaction cost per share for which an arbitrage opportunity exists is £0.175.

Let’s consider a scenario where a UK-based agricultural cooperative, “Golden Harvest,” needs to hedge against price volatility in the wheat market. They plan to sell 500,000 bushels of wheat in six months. They are considering using wheat futures contracts traded on the ICE Futures Europe exchange to manage their price risk. Each contract represents 5,000 bushels. The current futures price for wheat for delivery in six months is £200 per bushel. Golden Harvest decides to short (sell) 100 wheat futures contracts (500,000 bushels / 5,000 bushels per contract = 100 contracts) at £200 per bushel. Now, let’s assume that over the next three months, the price of wheat falls unexpectedly due to a global surplus. The futures price drops to £180 per bushel. Golden Harvest decides to close out their position to lock in their gains. To do this, they buy back 100 wheat futures contracts at £180 per bushel. The profit on the futures contracts is calculated as follows: * Initial sale price: £200 per bushel * Final purchase price: £180 per bushel * Profit per bushel: £200 – £180 = £20 * Total profit: £20 per bushel * 500,000 bushels = £10,000,000 However, it’s crucial to understand the concept of margin requirements. Futures contracts require investors to deposit an initial margin with their broker. This margin acts as collateral and is adjusted daily based on the market price movements (mark-to-market). Let’s assume the initial margin requirement is £10 per bushel. Therefore, Golden Harvest initially deposited £5,000,000 (100 contracts * 5,000 bushels/contract * £10/bushel). During the three months, the margin account would have fluctuated daily based on the wheat futures price. Since the price fell, Golden Harvest would have received variation margin payments into their account, reflecting their accumulating profit. The total profit of £10,000,000 is available to Golden Harvest upon closing out the position. Now, consider the real-world impact. Golden Harvest has locked in a price of £200 per bushel (effectively) through their hedging strategy. Without hedging, they would have had to sell their wheat at the lower spot price of £180 per bushel, resulting in a significant loss. The futures market allowed them to transfer price risk to speculators willing to take the opposite side of the trade. The profit from the futures market offsets the loss from selling the wheat at a lower price. This example highlights the fundamental purpose of futures contracts: risk management. Finally, remember the regulations surrounding derivatives trading. Golden Harvest must comply with regulations like EMIR (European Market Infrastructure Regulation) and MiFID II (Markets in Financial Instruments Directive II), which aim to increase transparency and reduce systemic risk in the derivatives market. They must report their derivatives trades to a trade repository and may be subject to clearing obligations.

Let’s analyze the combined impact of margin requirements, the daily settlement process (marking-to-market), and the potential for early termination on a short futures position. Imagine a scenario where a client, Amelia, shorts 50 orange juice futures contracts. The initial margin is £4,000 per contract, and the maintenance margin is £3,000 per contract. On day one, the futures price increases by £200 per contract. On day two, it increases again by £300 per contract. On day three, Amelia receives a margin call, and the brokerage firm decides to liquidate her position immediately due to concerns about her ability to meet future obligations. The liquidation occurs at a price £100 higher than the previous day’s close. First, calculate the total initial margin: 50 contracts * £4,000/contract = £200,000. Next, calculate the loss on day one: 50 contracts * £200/contract = £10,000. Amelia’s margin account balance is now £200,000 – £10,000 = £190,000. Calculate the loss on day two: 50 contracts * £300/contract = £15,000. Amelia’s margin account balance is now £190,000 – £15,000 = £175,000. Now, determine the total maintenance margin requirement: 50 contracts * £3,000/contract = £150,000. The margin call is triggered when the account balance falls below £150,000. Since £175,000 > £150,000, no margin call is triggered at the end of day two. Calculate the loss on day three: 50 contracts * £100/contract = £5,000. Amelia’s margin account balance is now £175,000 – £5,000 = £170,000. Although Amelia’s account is above the maintenance margin on day 3, the brokerage firm’s decision to liquidate immediately introduces another layer. The firm anticipates further losses and deems Amelia a high credit risk. The total loss Amelia experiences is the sum of the losses from all three days: £10,000 + £15,000 + £5,000 = £30,000. The brokerage firm returns the remaining balance to Amelia: £200,000 (initial margin) – £30,000 (total loss) = £170,000. The key here is the *brokerage firm’s discretion*. While the maintenance margin wasn’t breached to the point of automatic liquidation, the firm acted preemptively based on its risk assessment. This illustrates the power of the brokerage firm, which can override standard margin rules based on its internal risk parameters. The firm is not obliged to wait for the maintenance margin to be breached if they foresee further losses. This power is derived from the terms of the client agreement and regulatory requirements to manage risk effectively.

The optimal hedge ratio minimizes the variance of the hedged portfolio. This is achieved when the number of futures contracts is calculated as: Hedge Ratio = Correlation between asset and futures \* (Volatility of asset / Volatility of futures) \* (Size of position in asset / Size of one futures contract) In this scenario, the investor wants to hedge a portfolio of UK equities with FTSE 100 futures. The correlation between the portfolio and the futures is 0.75. The volatility of the portfolio is 18% and the volatility of the futures is 22%. The portfolio is worth £5,000,000 and each FTSE 100 futures contract represents £10 per index point. The current FTSE 100 index is 7500. Therefore, each futures contract is worth £10 * 7500 = £75,000. Hedge Ratio = 0.75 \* (0.18 / 0.22) \* (£5,000,000 / £75,000) = 0.75 \* 0.818 \* 66.67 = 40.90. Since futures contracts can only be traded in whole numbers, the investor should use 41 contracts. This will provide the closest hedge to the optimal ratio. Hedging is never perfect. The basis risk is the risk that the price of the asset being hedged and the price of the hedging instrument (in this case, the FTSE 100 futures contract) do not move perfectly together. This can occur because the portfolio is not perfectly correlated with the FTSE 100 index, or because of differences in the way the portfolio and the futures contract are priced. A higher hedge ratio does not eliminate basis risk, but rather aims to minimize the overall variance of the hedged position, considering the correlation and volatility dynamics. The hedge ratio calculation directly addresses the relative price movements and attempts to neutralize the portfolio’s exposure. The hedge ratio is also not static, it needs to be rebalanced periodically as the correlation and volatility change.

To determine the breakeven point, we need to consider the costs and revenues associated with the exotic derivative. In this case, the client pays an upfront premium and receives payouts based on the performance of the underlying asset relative to the knock-in barrier. The breakeven point is where the present value of the expected payouts equals the initial premium paid. First, we need to calculate the present value of the expected payouts. This involves discounting the potential payouts back to the present using the risk-free rate. Next, we need to determine the expected payout. This is done by calculating the expected payoff amount and multiplying it by the probability of the knock-in barrier being breached. Finally, we equate the present value of the expected payouts to the initial premium paid and solve for the underlying asset price at which this occurs. Let’s assume the expected payout is £25,000 if the knock-in barrier is breached. The risk-free rate is 5% per annum. The time to maturity is 2 years. The initial premium paid is £22,675.74. The present value of the expected payout is calculated as: \[PV = \frac{Expected\ Payout}{(1 + Risk-Free\ Rate)^{Time}}\] \[PV = \frac{25000}{(1 + 0.05)^2} = \frac{25000}{1.1025} = £22,675.74\] In this example, the initial premium paid equals the present value of the expected payout. Therefore, the breakeven point is when the underlying asset price reaches the knock-in barrier, which is £110. A real-world analogy could be a farmer buying crop insurance (similar to the premium). The insurance pays out if the crop yield falls below a certain threshold (knock-in barrier). The breakeven point for the farmer is when the insurance payout covers the initial premium paid. Another analogy is a company investing in a new technology (exotic derivative). The company pays an upfront cost (premium) and expects future profits if the technology becomes successful (knock-in barrier breached). The breakeven point is when the discounted future profits equal the initial investment.