Derivatives Level 4 (Investment Advice Diploma) Quiz Exam Quiz 29

Let’s analyze the expected payoff of the Asian option. The arithmetic average is calculated as the sum of the asset prices at the observation points divided by the number of observation points. In this case, the observation points are at the end of each quarter for one year, so there are four observation points. The asset prices at the end of each quarter are given as £98, £102, £105, and £108. The arithmetic average is calculated as follows: Arithmetic Average = \(\frac{98 + 102 + 105 + 108}{4}\) = \(\frac{413}{4}\) = £103.25 The payoff of an Asian call option is the maximum of zero and the difference between the arithmetic average and the strike price. In this case, the strike price is £100. Payoff = max(0, Arithmetic Average – Strike Price) = max(0, 103.25 – 100) = max(0, 3.25) = £3.25 The risk-neutral probability of this scenario occurring is given as 0.3. Therefore, the expected payoff is the product of the payoff and the risk-neutral probability. Expected Payoff = Payoff × Risk-Neutral Probability = 3.25 × 0.3 = £0.975 Now, let’s consider the implications for advising a client. Imagine a client, Mrs. Davies, who is risk-averse and seeks to hedge against potential increases in the price of a specific commodity she uses in her business. An Asian option, with its averaging feature, can be more suitable than a standard European or American option because it reduces the impact of price volatility at specific points in time. If Mrs. Davies is concerned about a sharp price spike right before the expiration date, the Asian option smooths out the price over the observation period, potentially leading to a lower premium and a more predictable hedging cost. However, the expected payoff needs to be carefully assessed against the premium paid for the option. If the premium is significantly higher than the expected payoff, even considering the risk mitigation benefits, it might not be the most efficient hedging strategy. Other strategies, such as using a series of forward contracts or a different type of exotic option, might be more appropriate depending on Mrs. Davies’ specific risk profile and market outlook. The advisor must clearly communicate the potential payoffs, risks, and costs associated with the Asian option, ensuring that Mrs. Davies fully understands the implications before making a decision.

The question assesses the 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 touches or goes below the barrier level. The payoff at expiration is determined by the underlying asset’s price relative to the strike price, but only if the barrier hasn’t been breached. In this scenario, the initial stock price is £110, the strike price is £100, and the barrier is £90. The option is a European down-and-out call option, meaning it can only be exercised at expiration. We need to determine if the barrier was breached at any point during the option’s life and, if not, calculate the payoff at expiration. The stock price fluctuated during the option’s life, reaching a low of £85. Since this is below the barrier of £90, the option is knocked out and becomes worthless, regardless of the price at expiration. Therefore, the investor receives nothing. A standard call option would have paid out the difference between the stock price at expiration and the strike price if the stock price was above the strike price. However, the down-and-out feature makes this option cheaper but also riskier, as a temporary dip below the barrier renders it worthless. Consider an analogy: Imagine you are promised a prize if you can walk across a bridge, but there’s a safety net. This down-and-out option is like having the safety net removed. If you fall at any point, you lose the chance to win the prize, even if you could have made it across safely otherwise. This illustrates the path dependency of barrier options. The value isn’t just determined by the final price but also by the path the price takes during the option’s life. Therefore, the investor receives £0 because the barrier was breached.

The question explores the complexities of hedging a portfolio with options, specifically focusing on gamma and vega neutrality. Gamma measures the rate of change of delta with respect to changes in the underlying asset’s price, and vega measures the sensitivity of the option’s price to changes in the underlying asset’s volatility. Maintaining gamma neutrality helps to stabilize the delta hedge as the underlying asset’s price fluctuates, while vega neutrality protects the portfolio from changes in implied volatility. The scenario involves a portfolio manager who needs to hedge a long position in an asset using options, while simultaneously managing gamma and vega risk. To achieve gamma neutrality, the manager needs to offset the portfolio’s gamma with an opposing gamma position, typically achieved by trading options. The number of options required can be calculated using the formula: Number of options = – (Portfolio Gamma / Option Gamma). Vega neutrality is achieved similarly, by offsetting the portfolio’s vega with an opposing vega position. The number of options required for vega neutrality is calculated as: Number of options = – (Portfolio Vega / Option Vega). In this scenario, the portfolio manager initially hedges the long position with put options, creating a delta-neutral position. However, this initial hedge introduces gamma and vega exposure. To neutralize these risks, the manager needs to trade additional options. The question requires calculating the number of call options needed to simultaneously achieve gamma and vega neutrality. The key is to understand that a single type of option (in this case, call options) can be used to adjust both gamma and vega. The calculations involve solving a system of two equations with one unknown (the number of call options). This requires careful consideration of the signs of gamma and vega for both the portfolio and the options used for hedging. The number of call options required to achieve both gamma and vega neutrality is calculated by solving a system of equations. Let ‘x’ be the number of call options. Gamma neutrality equation: Portfolio Gamma + (x * Call Option Gamma) = 0 Vega neutrality equation: Portfolio Vega + (x * Call Option Vega) = 0 Solving for x involves dividing the negative of the Portfolio Gamma by the Call Option Gamma or the negative of the Portfolio Vega by the Call Option Vega. Given the provided values, the calculation would be: x = -Portfolio Gamma / Call Option Gamma = -(-120) / 0.6 = 200 x = -Portfolio Vega / Call Option Vega = -180 / -0.9 = 200 Therefore, the portfolio manager needs to buy 200 call options to achieve both gamma and vega neutrality.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Green Harvest,” which aims to stabilize its future revenue by hedging its wheat crop using futures contracts listed on the ICE Futures Europe exchange. Green Harvest anticipates harvesting 5,000 tonnes of wheat in six months. The current futures price for wheat with a six-month expiry is £200 per tonne. To hedge against a potential price decline, Green Harvest sells 50 wheat futures contracts (each contract representing 100 tonnes). The initial margin requirement is £5,000 per contract, and the maintenance margin is £4,000 per contract. Three months later, the futures price has fallen to £180 per tonne. Green Harvest decides to close out its position by buying back the 50 futures contracts. At the same time, due to unforeseen weather conditions, Green Harvest’s actual wheat yield is only 4,500 tonnes. The spot price at harvest is £170 per tonne. First, calculate the profit from the futures position: Profit = (Initial Futures Price – Final Futures Price) * Contract Size * Number of Contracts Profit = (£200 – £180) * 100 * 50 = £100,000 Next, calculate the loss from the reduced wheat yield sold at the spot price: Expected Revenue (based on initial futures price): 5,000 tonnes * £200/tonne = £1,000,000 Actual Revenue (reduced yield at spot price): 4,500 tonnes * £170/tonne = £765,000 Revenue Shortfall = £1,000,000 – £765,000 = £235,000 Now, calculate the net outcome of the hedging strategy: Net Outcome = Profit from Futures – Revenue Shortfall Net Outcome = £100,000 – £235,000 = -£135,000 Finally, consider the margin calls. The initial margin was £5,000 per contract, totaling £250,000. The price drop of £20 per tonne translates to a profit, so no margin calls would have been made. The cooperative initially expected to sell 5,000 tonnes at £200/tonne, yielding £1,000,000. Due to lower yield and spot price, they sold 4,500 tonnes at £170/tonne, yielding £765,000. The hedge generated a profit of £100,000, partially offsetting the loss, but overall, they are worse off by £135,000. This demonstrates the basis risk – the difference between the futures price and the spot price at the time the hedge is lifted, and the impact of production volume differing from the hedged amount.

The correct answer is (a). This question tests the understanding of exotic options, specifically barrier options, and their sensitivity to market movements, especially around the barrier level. A knock-out option ceases to exist if the underlying asset price reaches a pre-defined barrier level. The closer the underlying asset price is to the barrier, the more sensitive the option’s value becomes to small price changes. This is because there is an increased probability that the barrier will be hit, causing the option to expire worthless. The delta of an option measures its sensitivity to changes in the underlying asset price. For a knock-out option near its barrier, the delta will be significantly affected as the probability of the option knocking out changes rapidly with small price movements. Gamma measures the rate of change of delta with respect to changes in the underlying asset price. Near the barrier, the gamma of a knock-out option is typically high because the delta is highly sensitive. Vega measures the sensitivity of an option’s price to changes in the volatility of the underlying asset. While volatility does affect the price of a knock-out option, the proximity to the barrier has a more direct and immediate impact on its value. Rho measures the sensitivity of an option’s price to changes in interest rates. This effect is generally less pronounced than the impact of the underlying asset price, especially when the option is near its barrier. Therefore, the delta and gamma are the most affected Greeks when the underlying asset price approaches the barrier level of a knock-out option. Consider a digital knock-out call option on a stock. If the stock price is far below the barrier, the option behaves like a regular call option. However, as the stock price nears the barrier, the option’s delta becomes increasingly negative, reflecting the high probability of the option expiring worthless. This change in delta is captured by a high gamma.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Golden Harvest,” which exports barley to several European countries. Golden Harvest faces price volatility in the barley market and uses forward contracts to hedge against potential losses. The cooperative enters into a forward contract to sell 1000 metric tons of barley at £200 per ton in six months. Simultaneously, a severe drought hits several barley-producing regions, driving up the spot price of barley. Three months into the contract, the spot price has risen to £250 per ton, and the estimated future price in three months is £240 per ton. Golden Harvest is considering unwinding the forward contract. To calculate the profit or loss from unwinding the forward contract, we need to determine the present value of the difference between the forward contract price and the current future price. The difference between the current future price (£240) and the original forward price (£200) is £40 per ton. We need to discount this difference back to the present using a suitable discount rate. Let’s assume a risk-free interest rate of 5% per annum, compounded continuously. Since we are looking at a period of 3 months (0.25 years), the discount factor is \(e^{-0.05 \times 0.25}\) which is approximately 0.9876. The present value of the profit per ton is then \(£40 \times 0.9876 = £39.50\). For 1000 metric tons, the total profit is \(£39.50 \times 1000 = £39,500\). However, unwinding the contract would likely involve a cost. If the counterparty charges £500 to unwind the contract, then the net profit would be \(£39,500 – £500 = £39,000\). The key here is understanding that unwinding a forward contract involves settling the difference between the original contract price and the current market’s expectation of the price at the original delivery date, discounted back to the present. A higher future price than the original forward price will result in a profit for the party selling in the forward contract (Golden Harvest in this case), while a lower future price would result in a loss. The discount rate reflects the time value of money and the risk-free return that could be earned on the profit if it were received today. Transaction costs, such as unwinding fees, must also be considered when evaluating the overall profitability of unwinding a forward contract.

To determine the breakeven point, we need to consider the initial cost of the strategy and the potential payoffs at different stock prices. The investor buys 10 call options at £5 each, totaling an initial cost of £50. They also sell 5 call options at £8 each, receiving £40. The net cost is £50 – £40 = £10. The long call options have a strike price of £100, and the short call options have a strike price of £105. The breakeven point is where the profit from the long calls offsets the net cost. If the stock price at expiration is below £100, both options expire worthless, and the investor loses the net cost of £10. If the stock price is above £100, the long calls start to generate profit. The profit per long call is (Stock Price – £100). Since the investor has 10 long calls, the total profit is 10 * (Stock Price – £100). If the stock price is above £105, the short calls create a loss for the investor. The loss per short call is (Stock Price – £105). Since the investor has 5 short calls, the total loss is 5 * (Stock Price – £105). The breakeven point is where the total profit from the long calls minus the total loss from the short calls equals the net cost. The profit equation is: 10 * (Stock Price – £100) – 5 * (Stock Price – £105) – £10 = 0. Simplifying the equation: 10 * Stock Price – £1000 – 5 * Stock Price + £525 – £10 = 0 5 * Stock Price – £485 = 0 5 * Stock Price = £485 Stock Price = £485 / 5 = £97 Therefore, the breakeven point is £97. However, this is below the strike price of the long calls, meaning the long calls will not be exercised. The investor loses the net cost of £10. The question is looking for the point where the strategy starts to become profitable. Let’s examine the region where the stock price is above £100. In this region, the investor makes money on the long calls and loses money on the short calls if the stock price goes above £105. We can consider the combined position. The investor has a ratio of 2:1 long to short calls. The net cost of the position is £10. The long calls have a strike of £100 and the short calls have a strike of £105. The profit equation is: Profit = 10 * max(0, Stock Price – £100) – 5 * max(0, Stock Price – £105) – £10 If the stock price is between £100 and £105, the equation simplifies to: Profit = 10 * (Stock Price – £100) – £10 Breakeven is when Profit = 0, so 10 * (Stock Price – £100) – £10 = 0 10 * Stock Price – £1000 – £10 = 0 10 * Stock Price = £1010 Stock Price = £101 If the stock price is above £105, the equation is: Profit = 10 * (Stock Price – £100) – 5 * (Stock Price – £105) – £10 Profit = 10 * Stock Price – £1000 – 5 * Stock Price + £525 – £10 Profit = 5 * Stock Price – £485 Breakeven is when Profit = 0, so 5 * Stock Price – £485 = 0 5 * Stock Price = £485 Stock Price = £97 This is impossible because the stock price has to be above £105 for this equation to hold. The breakeven point is £101.

The core of this question lies in understanding how a quanto swap operates and how its valuation is affected by the correlation between two currencies. A quanto swap is a type of cross-currency interest rate swap where the interest rate of one currency is exchanged for the interest rate of another currency, but the principal is fixed in one currency. This eliminates the exchange rate risk on the principal. The key here is that the fixed exchange rate used at inception remains constant throughout the life of the swap, regardless of actual exchange rate fluctuations. In this scenario, the correlation between the GBP/USD exchange rate and the relative interest rates in the UK and US plays a crucial role. If the correlation is positive, it means that as UK interest rates rise relative to US interest rates, the GBP/USD exchange rate also tends to rise. This impacts the present value of the future cash flows. The present value of the future cash flows in GBP is calculated by discounting them back to the present using the GBP interest rate. Similarly, the present value of the future cash flows in USD is calculated by discounting them back to the present using the USD interest rate, and then converting to GBP at the fixed exchange rate. The difference between these present values determines the initial value of the swap. A positive correlation implies that when UK interest rates rise relative to US rates, the GBP strengthens. This makes the future USD payments, when converted back to GBP at the fixed rate, less valuable in GBP terms. Conversely, if the correlation is negative, a rise in UK interest rates relative to US rates would imply a weakening of the GBP, making the future USD payments more valuable in GBP terms. Therefore, the initial value of the quanto swap is highly sensitive to the correlation. If the correlation is positive, the initial value will be different than if the correlation is negative or zero. To calculate the precise initial value, we would need to model the expected future interest rate paths and exchange rate movements, taking into account the correlation. In practice, this is done using complex models.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their sensitivity to volatility. A down-and-out call option becomes worthless if the underlying asset’s price touches or falls below the barrier level. Vega measures the sensitivity of an option’s price to changes in the volatility of the underlying asset. For a standard call option, an increase in volatility generally increases the option’s value because it increases the probability of the option ending in the money. However, for a down-and-out call option, the relationship is more complex. In this scenario, the barrier is close to the current market price. If volatility increases, the probability of the asset price hitting the barrier and the option being knocked out also increases significantly. This negative effect can outweigh the positive effect of increased volatility on the standard call option value, resulting in a negative Vega. If the barrier were very far away from the current market price, the option would behave more like a standard call option, and Vega would be positive. The proximity of the barrier is the key factor. To illustrate, consider two scenarios: In scenario A, the asset price is £100, the barrier is £95, and volatility is 10%. In scenario B, volatility increases to 20%. The probability of hitting £95 increases dramatically in scenario B, likely causing the option to be knocked out. Now, imagine the barrier was at £50. The increase in volatility would still increase the probability of the option ending in the money, with a much smaller increase in the probability of hitting the barrier. Therefore, when the barrier is close to the current asset price, increased volatility can decrease the value of a down-and-out call option, leading to a negative Vega.

The question revolves around the concept of delta hedging and how gamma affects the hedge’s performance. Delta hedging aims to neutralize the sensitivity of an option portfolio to changes in the underlying asset’s price. Gamma, on the other hand, measures the rate of change of the delta with respect to the underlying asset’s price. A higher gamma implies that the delta changes more rapidly, making the hedge more sensitive to price movements and requiring more frequent adjustments. In this scenario, the portfolio manager is using options to hedge a bond portfolio. The delta of the options position is offsetting the delta of the bond portfolio, creating a delta-neutral position. However, because the options have a non-zero gamma, the delta neutrality is only valid for small price movements in the underlying asset. If the underlying asset price moves significantly, the delta of the options position will change, and the portfolio will no longer be delta-neutral. This means that the portfolio will become exposed to price risk again. The manager needs to rebalance the hedge by adjusting the options position to bring the overall delta back to zero. The key is to understand that gamma is a measure of the *curvature* of the option’s price with respect to the underlying asset’s price. Imagine a road: delta is the slope of the road at a particular point, and gamma is how quickly that slope is changing as you drive along the road. A high gamma is like a very winding road, where you constantly need to adjust the steering wheel (the hedge). The calculation involves understanding how gamma impacts the change in delta. The change in delta is approximately equal to gamma multiplied by the change in the underlying asset’s price. In this case, the change in delta is 0.05 (gamma) * 2 (price change) = 0.10. Since the manager needs to bring the overall delta back to zero, they need to adjust the options position to offset this change in delta. The correct answer involves determining the number of options contracts needed to achieve this delta adjustment, considering the delta of each contract. The formula to determine the number of contracts to trade is: Number of contracts = (Target Delta Change) / (Delta per contract). In this case, it’s 0.10 / 0.5 = 0.2 contracts. Since you can’t trade fractions of contracts, the manager would need to trade 1 contract to bring the delta as close to zero as possible. Since the delta of the option position increased, the manager needs to sell 1 contract to reduce the overall delta.

The breakeven point for a covered call strategy is the stock purchase price less the premium received. This is because the premium income offsets the initial cost of purchasing the stock. If the stock price rises above the strike price, the investor’s upside is capped, but the initial premium still provides a cushion. The maximum profit is the premium received plus the difference between the strike price and the stock purchase price. The maximum loss is the stock purchase price less the premium received, which is the breakeven point. In this case, the investor buys the stock for £95 and receives a premium of £8. Therefore, the breakeven point is £95 – £8 = £87. The maximum profit is £8 (premium) + (£100 – £95) = £13. The maximum loss is £87 (the breakeven point). If the stock price at expiration is £85, the investor incurs a loss. The loss is the difference between the purchase price and the final stock price, less the premium received: (£95 – £85) – £8 = £2. If the stock price at expiration is £105, the call option will be exercised, and the investor will be forced to sell the stock at £100. The profit will be the difference between the selling price and the purchase price, plus the premium received: (£100 – £95) + £8 = £13. Therefore, the profit or loss is calculated as follows: If the stock price at expiration is £85, the investor’s profit/loss is (£85 – £95) + £8 = -£2. If the stock price at expiration is £105, the investor’s profit/loss is (£100 – £95) + £8 = £13.

The question assesses understanding of option pricing sensitivities (Greeks), specifically Delta, Gamma, Vega, Theta, and Rho, and how they interact in a portfolio context. The scenario involves a portfolio manager dynamically hedging a short option position, requiring knowledge of how these Greeks change as the underlying asset price and time to expiration fluctuate. To solve this, we need to consider the following: * **Delta:** Measures the sensitivity of the option price to changes in the underlying asset price. A delta of 0.6 means the option price will change by £0.6 for every £1 change in the underlying. * **Gamma:** Measures the rate of change of delta with respect to changes in the underlying asset price. It indicates how stable the delta hedge is. * **Vega:** Measures the sensitivity of the option price to changes in implied volatility. * **Theta:** Measures the sensitivity of the option price to the passage of time (time decay). * **Rho:** Measures the sensitivity of the option price to changes in the risk-free interest rate. In this scenario, the portfolio manager is short options, meaning they profit if the option expires worthless. As the underlying asset price increases, the short option position becomes more exposed (delta increases). The manager must buy more of the underlying asset to maintain a delta-neutral hedge. Gamma indicates how much the delta will change for each unit change in the underlying asset. Vega, Theta, and Rho affect the option price, but the primary concern for dynamic hedging is Delta and Gamma. The manager initially hedges by buying shares to offset the negative delta of the short options. As the underlying asset price rises, the delta of the short options becomes more negative (closer to -1). Therefore, the manager needs to buy *more* shares to maintain the delta-neutral hedge. The exact amount depends on the gamma of the options, which tells us how quickly the delta is changing. The rising implied volatility (reflected by Vega) will increase the value of the options, making the short position more risky. Time decay (Theta) works in favor of the short option position as time passes, reducing the option’s value, but the immediate concern is managing the delta exposure due to the price increase. Rho has a relatively minor impact compared to Delta, Gamma and Vega in this scenario.

The question assesses the understanding of exotic options, specifically barrier options, and their sensitivity to market movements. A down-and-out barrier option becomes worthless if the underlying asset’s price touches or goes below the barrier level before the option’s expiration. The holder loses the premium paid. In this scenario, the investor bought the option expecting the price to remain above the barrier. The question focuses on the implications of the price breaching the barrier just before expiration and subsequently recovering. The key is to understand that once the barrier is breached, the option is extinguished, regardless of any subsequent price recovery. The investor loses the entire premium paid. Therefore, the investor will lose the premium paid of £1.50 per contract, multiplied by 100 shares per contract, totaling £150.00. The recovery in share price after the barrier has been hit is irrelevant. The other options are incorrect because they either ignore the barrier feature, calculate profit based on the final share price as if it were a standard option, or assume that the recovery negates the barrier breach. The scenario highlights the risk associated with barrier options and emphasizes the importance of understanding the specific terms and conditions of exotic derivatives. For example, consider a steel manufacturer who uses a down-and-out put option on iron ore to hedge against price declines, with the barrier set at a level that would make their operations unprofitable. If iron ore prices briefly dip below this barrier due to a temporary market disruption, the hedge is lost, even if prices quickly recover. The manufacturer is now exposed to the risk of a sustained price decline. This illustrates the vulnerability of barrier options to short-term market volatility.

The core of this question lies in understanding how implied volatility, time to expiry, and the strike price relative to the current asset price interact to influence option prices, particularly in the context of exotic options like barrier options. Barrier options, unlike vanilla options, have a payoff that depends on whether the underlying asset’s price reaches a predetermined barrier level during the option’s life. A knock-out barrier option ceases to exist if the barrier is hit, while a knock-in option only comes into existence if the barrier is hit. The Vega of an option measures its sensitivity to changes in implied volatility. Generally, a higher implied volatility increases the value of an option because it reflects greater uncertainty about the future price of the underlying asset. However, for barrier options, the relationship is more complex. If the barrier is close to the current asset price, increased volatility can significantly increase the probability of the barrier being hit. For a knock-out option, hitting the barrier means the option expires worthless, so increased volatility can decrease its value. Conversely, for a knock-in option, increased volatility increases the chance of the option coming into existence, thereby increasing its value. The time to expiry also plays a crucial role. A longer time to expiry generally increases the value of an option because there is more time for the underlying asset to move in a favorable direction. However, for knock-out options, a longer time to expiry also means a greater chance of the barrier being hit, potentially reducing the option’s value. For knock-in options, a longer time to expiry increases the likelihood of the barrier being reached, potentially increasing the option’s value. The moneyness of the option (i.e., whether it is in-the-money, at-the-money, or out-of-the-money) also influences the impact of volatility and time to expiry. For example, an at-the-money knock-out option with a barrier close to the current price will be highly sensitive to changes in volatility because even small price fluctuations can cause the barrier to be hit. The question specifically asks about the impact of a decrease in implied volatility on a down-and-out call option. Since it’s a down-and-out call, the barrier is below the current price. A decrease in implied volatility reduces the probability of the barrier being hit. This is because lower volatility suggests a smaller range of potential price fluctuations. Since hitting the barrier causes the option to expire worthless, reducing the probability of hitting the barrier increases the option’s value. Therefore, the value of the down-and-out call option will increase.

The core of this question lies in understanding how the cost of carry model is adapted for assets that provide a convenience yield, like crude oil. Unlike financial assets that might pay dividends (which reduce the cost of carry), commodities can offer a benefit to the holder simply by possessing them – the convenience yield. This yield represents the value derived from having the physical commodity readily available for use in production or to meet immediate demand. The futures price reflects the spot price, plus the cost of carry (storage, insurance, financing), minus the convenience yield. The formula is: Futures Price = Spot Price + Cost of Carry – Convenience Yield. In this scenario, the refinery’s willingness to pay a premium above the theoretical futures price indicates a perceived higher convenience yield than the market is currently pricing in. This could be due to specific operational needs, anticipated supply disruptions, or proprietary information that suggests a greater value in having the physical oil on hand. First, we calculate the theoretical futures price: Spot Price = £75/barrel Storage Costs = £2/barrel Insurance Costs = £0.5/barrel Financing Costs = Spot Price * Risk-Free Rate = £75 * 0.05 = £3.75/barrel Convenience Yield = £4/barrel Theoretical Futures Price = £75 + £2 + £0.5 + £3.75 – £4 = £77.25/barrel The refinery is willing to pay £78/barrel, which is higher than the theoretical futures price of £77.25/barrel. This difference reflects their implied convenience yield. To find the implied convenience yield: £78 = £75 + £2 + £0.5 + £3.75 – Implied Convenience Yield Implied Convenience Yield = £75 + £2 + £0.5 + £3.75 – £78 = £3.25 The difference between the market’s convenience yield (£4) and the refinery’s implied convenience yield (£3.25) is £0.75. Therefore, the refinery’s implied convenience yield is lower than the market’s perceived convenience yield. The refinery’s decision to pay more than the theoretical futures price suggests they are willing to accept a lower return on the cost of carry, because the benefit they get from having the oil immediately available is greater than the market’s perceived convenience.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their sensitivity to the underlying asset’s volatility. Barrier options are path-dependent, meaning their payoff depends not only on the underlying asset’s price at maturity but also on whether the asset price has crossed a pre-defined barrier level during the option’s life. The key here is to understand how volatility impacts the probability of hitting the barrier. Higher volatility increases the likelihood of the asset price fluctuating significantly, thereby increasing the probability of hitting the barrier. This, in turn, affects the option’s value differently depending on whether it’s a knock-in or knock-out option. For a knock-out option, hitting the barrier renders the option worthless. Therefore, increased volatility makes it more likely the option will expire worthless, decreasing its value. For a knock-in option, hitting the barrier activates the option. Increased volatility makes it more likely the option will become active, increasing its value. In this specific scenario, we are dealing with a down-and-out call option. This means the call option will cease to exist if the underlying asset price hits the barrier level. Therefore, the higher the volatility, the higher the chance the barrier will be hit, and the lower the value of the option. To illustrate, imagine two identical down-and-out call options on shares of “VolatileTech,” a hypothetical tech company. Option A is based on a market expectation of low volatility (10%), while Option B is based on high volatility (40%). The barrier for both options is set at 80% of the initial share price. With low volatility (Option A), the share price is less likely to fluctuate wildly and hit the 80% barrier. However, with high volatility (Option B), the share price is much more likely to swing downwards and hit the barrier, causing the option to expire worthless. The regulatory aspect is crucial because firms offering these complex products must ensure clients understand the risks involved, including the impact of volatility on barrier option values. MiFID II regulations require firms to provide clear and understandable information about the nature and risks of investment products, including derivatives. Failing to do so could result in regulatory penalties. The firm must classify the client appropriately (e.g., retail, professional, eligible counterparty) and ensure the product is suitable for their risk profile and investment objectives.

The core of this question revolves around understanding how delta hedging is applied in practice, particularly when dealing with transaction costs. Delta hedging aims to create a portfolio that is insensitive to small changes in the underlying asset’s price. The delta of a portfolio represents the change in the portfolio’s value for a $1 change in the underlying asset’s price. To maintain a delta-neutral position, traders must continuously adjust their holdings in the underlying asset. However, real-world trading involves transaction costs, which can significantly impact the profitability of delta hedging strategies. The optimal rebalancing frequency is a trade-off: frequent rebalancing reduces delta exposure but increases transaction costs, while infrequent rebalancing reduces transaction costs but increases delta exposure. The question explores how a fund manager, specifically bound by the FCA’s conduct of business rules, must balance these factors. The FCA expects firms to act in the best interests of their clients, which includes minimizing unnecessary costs and managing risks effectively. The fund manager must consider the volatility of the underlying asset, the transaction costs, and the acceptable level of delta exposure when determining the rebalancing frequency. In this scenario, we are given the option’s delta (0.6), the number of options (10,000), the price per share (£100), the transaction cost per share (£0.05), and the volatility (20%). The fund manager needs to determine the optimal rebalancing strategy considering these factors. A higher volatility suggests more frequent rebalancing to maintain delta neutrality, but the transaction costs impose a limit. The FCA regulations require the fund manager to prioritize the client’s best interest, which means finding the rebalancing frequency that minimizes the combined cost of delta exposure and transaction costs. The calculation to determine the cost of rebalancing is as follows: 1. **Shares to hedge:** Delta * Number of Options = 0.6 * 10,000 = 6,000 shares 2. **Transaction cost per rebalance:** Shares to hedge * Transaction cost per share = 6,000 * £0.05 = £300 3. **Annual rebalancing cost:** Transaction cost per rebalance * Rebalancing frequency. The fund manager must weigh this cost against the potential cost of not rebalancing frequently enough, which would lead to a non-delta-neutral portfolio and potential losses due to adverse price movements. The optimal strategy will depend on the fund’s specific risk tolerance and the expected volatility of the underlying asset.

Let’s analyze how a contingent credit default swap (CDS) operates under specific circumstances. A contingent CDS activates only when a pre-defined credit event occurs and a specified trigger condition is met. The trigger is usually linked to the credit rating of the reference entity. In this scenario, the bank’s credit rating deteriorating below a certain level (e.g., BBB-) triggers the swap. The calculation involves determining the payoff to the protection buyer (Alpha Investments) when the CDS is triggered. The payoff is based on the notional amount of the CDS and the recovery rate following the credit event. The recovery rate is the percentage of the notional amount that the debt holders are expected to recover. The payoff is calculated as: Payoff = Notional Amount * (1 – Recovery Rate) In this case, the notional amount is £50 million and the recovery rate is 30%. Therefore: Payoff = £50,000,000 * (1 – 0.30) = £50,000,000 * 0.70 = £35,000,000 However, Alpha Investments only receives this payoff if the trigger event has occurred. In this case, it has. The concept of a contingent CDS is crucial in risk management. Unlike a standard CDS, which provides continuous protection against default, a contingent CDS offers protection only when the reference entity’s creditworthiness deteriorates significantly. This makes it a more targeted and potentially cheaper form of credit protection. For example, consider a pension fund holding bonds issued by a corporation. The fund might use a contingent CDS to protect against a sharp downgrade in the corporation’s credit rating, which could trigger a sell-off and reduce the value of their bond holdings. The contingent CDS acts as insurance against this specific scenario. Another application is in structured finance. Imagine a special purpose vehicle (SPV) issuing asset-backed securities (ABS). The SPV could use a contingent CDS to protect against a downgrade in the credit rating of the underlying assets. This enhances the creditworthiness of the ABS and makes it more attractive to investors. The contingent CDS only pays out if the assets are downgraded below a certain level, providing targeted protection against severe credit deterioration. Contingent CDSs are also valuable in regulatory compliance. Financial institutions are often required to hold capital against credit risk exposures. A contingent CDS can be used to reduce the capital requirements by providing protection against specific credit events. For example, a bank lending to a highly leveraged company might use a contingent CDS to protect against a downgrade in the company’s credit rating. This reduces the bank’s exposure to credit risk and lowers the amount of capital it needs to hold.

The optimal hedge ratio in this scenario minimizes the variance of the hedged portfolio. The formula for the optimal hedge ratio is: \[h = \rho \frac{\sigma_S}{\sigma_F}\] Where: * \(h\) is the hedge ratio * \(\rho\) is the correlation between the spot price changes and the futures price changes * \(\sigma_S\) is the standard deviation of the spot price changes * \(\sigma_F\) is the standard deviation of the futures price changes In this case: * \(\rho = 0.75\) * \(\sigma_S = 0.02\) (2% or 0.02) * \(\sigma_F = 0.025\) (2.5% or 0.025) Therefore, the optimal hedge ratio is: \[h = 0.75 \times \frac{0.02}{0.025} = 0.75 \times 0.8 = 0.6\] This means that for every unit of the underlying asset, the investor should short 0.6 units of the futures contract to minimize risk. Since the portfolio is worth £5,000,000 and each futures contract is on £100,000 of the underlying asset, the number of futures contracts to short is: Number of contracts = (Portfolio Value / Contract Size) * Hedge Ratio Number of contracts = (£5,000,000 / £100,000) * 0.6 = 50 * 0.6 = 30 Therefore, the investor should short 30 futures contracts to minimize the variance of the hedged portfolio. Now, consider a slightly different scenario. Suppose the investor believes that the correlation between the spot and futures prices will decrease significantly due to impending regulatory changes in the derivatives market. This expectation would directly impact the hedge ratio calculation, potentially making the original hedge ratio less effective. Furthermore, imagine that the investor has access to a sophisticated risk management model that incorporates not only historical volatility but also implied volatility derived from options prices. This implied volatility could provide a more forward-looking estimate of price fluctuations, leading to a more refined hedge ratio. The model might also consider the cost of carry, including storage and insurance, for the physical commodity, which affects the relationship between spot and futures prices. Finally, the investor could explore using a dynamic hedging strategy, adjusting the hedge ratio periodically based on changes in market conditions and model outputs, rather than relying on a static hedge for the entire period. This requires careful consideration of transaction costs and potential tracking error.

Let’s break down how to determine the most suitable derivative for mitigating specific risks in a portfolio, considering regulatory constraints. First, understand the portfolio’s risk profile. Imagine a portfolio heavily invested in UK-based small-cap companies, primarily exporters. This portfolio is highly susceptible to fluctuations in the GBP/USD exchange rate. A strengthening GBP would make their exports more expensive, potentially reducing revenue and profitability. Additionally, rising UK interest rates could increase borrowing costs for these companies, further impacting their performance. Next, analyze the derivative options. A forward contract could lock in a specific GBP/USD exchange rate for future export revenues, providing certainty but lacking flexibility if the GBP weakens. A futures contract offers similar hedging but with daily mark-to-market, potentially requiring margin calls. An option gives the right, but not the obligation, to buy or sell GBP at a specific rate, offering downside protection while allowing participation in favorable movements. A swap could exchange floating interest rate payments for fixed payments, hedging against rising interest rates. Consider regulatory constraints. Under MiFID II regulations, complex derivatives like exotic options may only be suitable for sophisticated investors who fully understand their risks. For a retail client, a simpler option or forward contract might be more appropriate. The firm also has a duty to ensure the derivative is “suitable” for the client’s needs and risk tolerance, documented via a suitability report. Finally, assess the cost-benefit of each derivative. A forward contract might have a lower upfront cost but less flexibility. An option has a premium cost but offers potential upside. A swap has ongoing payments but provides long-term interest rate certainty. The best choice depends on the client’s risk appetite, hedging goals, and regulatory constraints. In this scenario, a vanilla GBP/USD put option, giving the right to sell GBP at a specific rate, offers a balance of downside protection and potential upside, and is generally considered suitable for a broader range of investors than more complex derivatives. Swaps are generally not used for currency hedging, and are more appropriate for interest rate hedging. Therefore, considering the portfolio’s exposure to currency risk, the need for downside protection, and regulatory suitability, a vanilla GBP/USD put option is the most appropriate derivative in this scenario.

The question assesses the understanding of the impact of various factors on the price of European options, particularly focusing on implied volatility and time to expiration. The correct answer reflects the combined effect of an increase in implied volatility and a decrease in time to expiration on the option price. Implied volatility is a key determinant of option prices, reflecting the market’s expectation of future price fluctuations of the underlying asset. Higher implied volatility generally leads to higher option prices because it increases the probability of the option ending in the money. Conversely, time to expiration also significantly impacts option prices. As the time to expiration decreases, the option’s price tends to decrease because there is less time for the underlying asset’s price to move favorably for the option holder. However, the magnitude of these effects depends on various factors, including the moneyness of the option and the level of implied volatility. For an at-the-money option, the impact of implied volatility is typically more pronounced than the impact of time decay, especially when the time to expiration is relatively long. The question requires integrating these concepts to determine the net effect on the option price. To illustrate, consider two scenarios: Scenario 1: A call option on a stock trading at £100 with a strike price of £100 and 6 months to expiration. The implied volatility is 20%. If the implied volatility increases to 30%, the option price will increase significantly, reflecting the higher probability of the stock price moving above £100. Scenario 2: The same call option with an implied volatility of 20% but only 1 month to expiration. If the implied volatility remains at 20%, but the time to expiration decreases to 2 weeks, the option price will decrease, reflecting the reduced time for the stock price to move above £100. The question challenges the candidate to synthesize these two effects and determine the overall impact on the option price.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their sensitivity to market movements and volatility, especially near the barrier level. The investor’s view on the underlying asset’s volatility and direction is crucial in selecting the appropriate barrier option strategy. A knock-out barrier option ceases to exist if the underlying asset price reaches the barrier level. If the investor believes the underlying asset price will stay within a certain range and volatility will decrease, selling a knock-out option can be profitable. The premium is collected upfront, and if the barrier is never breached, the option expires worthless, and the investor keeps the premium. However, if the barrier is breached, the option is knocked out, and the investor no longer has any exposure. In this scenario, the investor believes the FTSE 100 will remain range-bound and volatility will decrease. Selling a knock-out call option allows the investor to profit from the premium received if the FTSE 100 stays below the barrier level. The investor is betting that the FTSE 100 will not rise above the barrier level before the option’s expiration. Selling a knock-out put option would be suitable if the investor believed the FTSE 100 would not fall below the barrier level. Buying a knock-out option would be suitable if the investor believed the FTSE 100 would break through the barrier level.

The question assesses understanding of exotic derivatives, specifically barrier options, and their sensitivity to volatility changes as they approach the barrier. A down-and-out barrier option becomes worthless if the underlying asset’s price hits the barrier level before expiration. As the asset price nears the barrier, the option’s value becomes highly sensitive to volatility. Increased volatility raises the probability of the barrier being hit, thus decreasing the option’s value. Conversely, decreased volatility reduces the probability of hitting the barrier, increasing the option’s value. Gamma measures the rate of change of delta with respect to changes in the underlying asset’s price. Vega measures the sensitivity of the option’s price to changes in volatility. Near the barrier, the vega of a down-and-out option increases significantly because small changes in volatility can dramatically alter the probability of the option being knocked out. To illustrate, consider a hypothetical scenario involving a cocoa bean farmer in Côte d’Ivoire who uses a down-and-out put option to hedge against price declines in their upcoming harvest. The barrier is set close to the current market price to minimize the hedging cost. As the harvest time approaches and the cocoa bean price fluctuates near the barrier, the farmer becomes extremely sensitive to market volatility. A sudden spike in volatility due to, say, political instability in the region could trigger the barrier, rendering their hedge worthless. Conversely, a period of calm and stable prices would increase the value of their option, providing better downside protection. This example highlights how the vega of a barrier option becomes crucial for risk management as the underlying asset approaches the barrier. The farmer must carefully monitor volatility and adjust their hedging strategy accordingly to avoid losing their downside protection. Another example could be a tech startup using a down-and-out call option on a specific technology index to leverage potential gains in the sector. The barrier is set below the current index level. As the index price meanders close to the barrier, the startup’s financial team must closely monitor the volatility of the tech sector. Increased volatility, perhaps due to unexpected regulatory changes or a major competitor’s breakthrough, could trigger the barrier, causing the option to expire worthless. This scenario underscores the importance of understanding vega and its impact on barrier options, especially when the underlying asset’s price is near the barrier.

To determine the value of the European call option using a two-step binomial tree, we need to work backward from the final nodes. The stock price can either go up or down at each step. Given the initial stock price of £80, an up factor of 1.15, and a down factor of 0.88, we calculate the stock prices at each node. First Step: Up Node: £80 * 1.15 = £92 Down Node: £80 * 0.88 = £70.40 Second Step: Up-Up Node: £92 * 1.15 = £105.80 Up-Down Node: £92 * 0.88 = £80.96 Down-Down Node: £70.40 * 0.88 = £61.95 Now, we calculate the option values at the final nodes based on the strike price of £85: Up-Up Node: Max(£105.80 – £85, 0) = £20.80 Up-Down Node: Max(£80.96 – £85, 0) = £0 Down-Down Node: Max(£61.95 – £85, 0) = £0 Next, we calculate the risk-neutral probability (p) using the formula: \[p = \frac{e^{r \Delta t} – d}{u – d}\] Where r = risk-free rate (5% or 0.05), Δt = time step (1 year / 2 = 0.5 years), u = up factor (1.15), and d = down factor (0.88). \[p = \frac{e^{0.05 * 0.5} – 0.88}{1.15 – 0.88}\] \[p = \frac{1.0253 – 0.88}{0.27}\] \[p = \frac{0.1453}{0.27} \approx 0.5381\] Now, we discount back the option values at the previous nodes: Up Node Option Value: \[\frac{p * 20.80 + (1-p) * 0}{e^{0.05 * 0.5}} = \frac{0.5381 * 20.80}{1.0253} = \frac{11.1925}{1.0253} \approx 10.916\] Down Node Option Value: \[\frac{p * 0 + (1-p) * 0}{e^{0.05 * 0.5}} = 0\] Finally, we discount back to the initial node to find the option value: \[\frac{p * 10.916 + (1-p) * 0}{e^{0.05 * 0.5}} = \frac{0.5381 * 10.916}{1.0253} = \frac{5.874}{1.0253} \approx 5.73\] The value of the European call option using the two-step binomial tree is approximately £5.73. This calculation relies on the risk-neutral valuation principle, where we use risk-neutral probabilities to discount future cash flows. The binomial tree model simplifies the continuous price movements of the underlying asset into discrete steps, making it easier to calculate option prices. The accuracy of the model increases with the number of steps. The risk-neutral probability is crucial as it allows us to value the option without needing to know the actual expected return of the underlying asset. The exponential discounting accounts for the time value of money.

The key to this question lies in understanding how early exercise affects the value of American options, particularly in relation to dividends. American options allow the holder to exercise the option at any time before the expiration date. The decision to exercise early is influenced by factors such as dividends, interest rates, and the option’s moneyness. A crucial concept is that an American call option on a non-dividend-paying stock should never be exercised early because the option’s time value always exceeds the immediate intrinsic value. However, when dividends are involved, early exercise might be optimal to capture the dividend payment. In this scenario, the investor holds an American call option on a stock that is about to pay a significant dividend. The dividend payment effectively reduces the stock price. If the present value of the expected dividend exceeds the time value of the option, it may be optimal to exercise the call option early to capture the dividend and reinvest the proceeds. Conversely, the holder of an American put option may want to exercise early if the option is deep in the money and interest rates are high enough that the present value of receiving the strike price now exceeds the value of holding the option. The optimal decision depends on comparing the immediate gain from exercising the option and capturing the dividend (or avoiding the loss from a price decrease) with the potential future gain from holding the option. The time value of an option reflects the possibility of the stock price moving favorably before expiration. Early exercise sacrifices this potential upside. The Black-Scholes model, while useful for European options, does not directly address early exercise decisions for American options. Binomial trees or other numerical methods are often used to value American options and determine optimal exercise strategies. The investor must weigh the dividend amount against the time value of the option and the potential for future price appreciation. In this case, the investor is considering whether to exercise the option *just before* the ex-dividend date. This means they would receive the dividend but would forego any further potential gains from holding the option until expiration.

To determine the breakeven point for the combined strategy, we need to consider the initial costs and potential payoffs of each option. The investor buys a call option for £3 and sells a put option for £1. This results in a net cost of £2 (£3 – £1). The investor will only profit if the share price rises above the breakeven point. The sold put option will be exercised if the share price falls below £95. This means the investor is obligated to buy the shares at £95, regardless of the market price. The bought call option will be exercised if the share price rises above £105. This means the investor has the right to buy the shares at £105. The combined strategy has a breakeven point where the profit from the call option equals the net cost of the strategy. Let’s denote the breakeven point as \(x\). The profit from the call option is \(x – 105\). Thus, we have: \[x – 105 = 2\] \[x = 107\] Therefore, the breakeven point is £107. Consider a scenario where a tech company, “Innovatech,” is trading at £100. An investor believes the stock will either rise significantly or fall sharply due to an upcoming product launch. To capitalize on this anticipated volatility without committing substantial capital, the investor implements a strategy involving both call and put options. This approach allows them to profit from either a substantial price increase or decrease, while limiting potential losses to the net premium paid. This is akin to a straddle strategy but with different strike prices. If Innovatech’s stock soars to £120, the call option becomes highly profitable, covering the initial cost and generating a substantial return. Conversely, if the stock plummets to £80, the put option gains value, offsetting the initial cost. However, the investor needs to calculate the exact breakeven point to understand the price levels at which the strategy becomes profitable in either direction. The breakeven point calculation is crucial for understanding the risk-reward profile of the combined options strategy. It helps the investor determine the minimum price movement needed to achieve profitability, considering the premiums paid and received. It also highlights the potential losses if the stock price remains within a narrow range around the strike prices. This is especially important in volatile markets where large price swings can significantly impact the outcome of the strategy.

The core of this question revolves around understanding how margin requirements work for futures contracts, specifically focusing on the impact of price fluctuations on the variation margin and the potential for margin calls. We’ll use a unique example involving a cocoa bean futures contract to illustrate these concepts. First, we need to calculate the change in the futures price: \(1050 – 1000 = 50\). This represents the increase in the price of the futures contract. Next, we determine the total profit or loss based on the contract size. Since each contract represents 10 metric tons of cocoa beans, the total profit is \(50 \times 10 = 500\) GBP. Now, we need to assess the impact on the margin account. The initial margin was 2000 GBP, and the maintenance margin is 1500 GBP. Since the price increased, the investor made a profit, which is added to the margin account. The new margin account balance is \(2000 + 500 = 2500\) GBP. Finally, we determine if a margin call is triggered. Because the new margin account balance (2500 GBP) is above the initial margin (2000 GBP) and significantly above the maintenance margin (1500 GBP), no margin call is necessary. The investor’s position is well-covered by the existing margin. Imagine a scenario where a cocoa bean futures trader, Anya, has a margin account that acts like a buffer against potential losses. The initial margin is like a security deposit, ensuring she can cover some adverse price movements. The maintenance margin is the minimum level this account can reach before she needs to add more funds. If the price of cocoa beans suddenly plummets, Anya’s account balance decreases. If it falls below the maintenance margin, she gets a margin call, which is essentially a demand from her broker to deposit more money to bring the account back up to the initial margin level. This prevents her from owing the broker money if the price continues to fall. Conversely, if the price of cocoa beans rises, Anya’s account balance increases, reflecting her profit. This profit can be withdrawn or used as additional margin to support other trades. This system protects both the investor and the broker from excessive risk. The variation margin is the daily adjustment to the margin account reflecting the daily price changes. It’s a crucial mechanism for managing risk in futures trading.

The key to answering this question lies in understanding how gamma and vega interact within a delta-hedged portfolio, particularly when considering the impact of discrete hedging intervals. Gamma represents the rate of change of delta with respect to the underlying asset’s price, while vega represents the rate of change of the portfolio’s value with respect to changes in the underlying asset’s volatility. A positive gamma position benefits from large price movements (either up or down) because the delta hedge becomes more profitable. However, it also necessitates more frequent rebalancing to maintain the delta-neutral position, incurring transaction costs. Vega, on the other hand, benefits from increased volatility, as option prices generally rise with volatility. In this scenario, the portfolio manager chooses a longer rebalancing interval (weekly instead of daily) to reduce transaction costs. This decision has a direct impact on the portfolio’s gamma and vega exposure. By rebalancing less frequently, the portfolio’s gamma exposure is effectively amplified. The delta deviates further from zero between rebalancing intervals, increasing the potential for larger gains or losses due to price movements. However, the reduced rebalancing frequency also means that the portfolio is less actively managed to capitalize on vega exposure. The portfolio will not be able to capture any movement in volatility if the portfolio is not rebalanced frequently. The question highlights a trade-off: reducing transaction costs by rebalancing less frequently versus potentially missing out on profits from volatility changes and increasing the risk associated with gamma exposure. The optimal rebalancing strategy depends on the specific characteristics of the portfolio, the underlying asset, and the prevailing market conditions. If the market experiences large price swings during the week, the amplified gamma exposure could lead to significant gains or losses. If volatility increases significantly, the portfolio will not be able to capture it due to less frequent rebalancing. Therefore, the portfolio manager’s decision to rebalance weekly, while reducing transaction costs, increases the risk associated with gamma exposure and reduces the potential benefit from vega exposure.

Let’s break down how to value this exotic derivative and why option (a) is the correct approach. The derivative in question is a barrier option with a knock-out feature tied to the average price of a basket of commodities. This requires a sophisticated valuation approach. Standard Black-Scholes models are inadequate because they don’t account for the path-dependent nature of the average price and the correlation between the commodities. Monte Carlo simulation is the most appropriate technique here. The simulation process involves several key steps: 1. **Modeling Commodity Prices:** We need to model the price dynamics of each commodity in the basket. A common approach is to use a geometric Brownian motion (GBM) for each commodity: \[ dS_i = \mu_i S_i dt + \sigma_i S_i dW_i \] where \(S_i\) is the price of commodity *i*, \(\mu_i\) is its drift, \(\sigma_i\) is its volatility, and \(dW_i\) is a Wiener process. 2. **Correlation:** The Wiener processes \(dW_i\) need to be correlated to reflect the real-world relationships between commodity prices. This is achieved using a Cholesky decomposition of the correlation matrix. 3. **Simulating Paths:** We simulate a large number of possible price paths for each commodity over the life of the option. Each path represents a possible future scenario. 4. **Calculating the Average Price:** For each simulated path, we calculate the average price of the commodity basket over the specified period. This average is calculated at discrete time intervals, as specified in the option contract. 5. **Knock-Out Condition:** We check if the average price of the basket ever hits the knock-out barrier during the life of the option. If it does, the option is worthless for that particular simulated path. 6. **Payoff Calculation:** If the option does not knock out, we calculate the payoff at maturity based on the final price of the underlying asset (in this case, the basket of commodities) and the strike price. 7. **Discounting:** We discount the payoff back to the present value using the risk-free rate. 8. **Averaging:** We average the discounted payoffs over all simulated paths to obtain the estimated value of the option. The complexity arises from the path dependency and correlation. The other options suggest simplified approaches that would not accurately capture the value of this derivative. For instance, using a simple Black-Scholes model ignores the averaging and knock-out features. A static hedge would not account for the dynamic changes in the basket’s composition and correlations. A binomial tree, while suitable for some options, becomes computationally intractable with multiple correlated assets and a continuous averaging feature.

The question revolves around the concept of a variance swap and how its fair value is determined. The core idea is that the fair strike \(K\) of a variance swap is set such that the expected payoff at initiation is zero. This means the present value of the fixed leg (strike) equals the present value of the floating leg (realized variance). The realized variance is calculated from observed market data (in this case, the VIX index). The question also tests the understanding of how volatility is quoted (in annualized terms) and how it translates to variance (the square of volatility). The calculation involves converting the annualized volatility (VIX) into variance, averaging the variance over the swap’s life, and then taking the square root to express the result back in volatility terms. This volatility is then squared again to arrive at the fair variance strike \(K\). The calculation proceeds as follows: 1. **Convert VIX to Variance:** Square each VIX observation to get the daily variance. * Day 1: \(20^2 = 400\) * Day 2: \(22^2 = 484\) * Day 3: \(25^2 = 625\) * Day 4: \(23^2 = 529\) * Day 5: \(21^2 = 441\) 2. **Average the Variance:** Calculate the average of the daily variances. * Average Variance = \(\frac{400 + 484 + 625 + 529 + 441}{5} = \frac{2479}{5} = 495.8\) 3. **Annualize the Variance:** Since VIX is annualized, the variance is already annualized. 4. **Calculate the Fair Variance Strike (K):** The fair variance strike \(K\) is simply the average annualized variance. * \(K = 495.8\) Therefore, the fair variance strike for the swap is 495.8. The example uses the VIX index, a well-known measure of market volatility, to provide a real-world context. The variance swap is presented as a tool for hedging volatility risk, which is a common application in the financial industry. The question requires the candidate to understand the relationship between volatility and variance and how they are used in pricing variance swaps. The incorrect options are designed to reflect common mistakes, such as failing to square the volatility, incorrectly averaging the volatility instead of the variance, or misunderstanding the annualization factor. The question tests not just the formula but the underlying concept of how variance swaps are priced to have zero value at inception.