Derivatives Level 4 (Investment Advice Diploma) Quiz Exam Quiz 34

To determine the value of the swap, we need to calculate the present value of the expected future cash flows. In this case, the company is receiving a fixed rate and paying a floating rate. We’ll use the provided forward rate curve to estimate future LIBOR rates and discount them back to the present. First, we need to calculate the expected future cash flows. The fixed rate is 4% per annum, paid semi-annually, which means the company receives 2% every six months on the notional principal of £10 million. This equates to £200,000 every six months. Next, we estimate the future LIBOR rates using the forward rate curve. The forward rates are already given for each period, so we can directly use them to calculate the floating rate payments. Period 1 (6 months): 3.5% Period 2 (12 months): 4.0% Period 3 (18 months): 4.5% Period 4 (24 months): 5.0% The floating rate payments are calculated by multiplying the LIBOR rate by the notional principal. Period 1: 3.5% * £10,000,000 / 2 = £175,000 Period 2: 4.0% * £10,000,000 / 2 = £200,000 Period 3: 4.5% * £10,000,000 / 2 = £225,000 Period 4: 5.0% * £10,000,000 / 2 = £250,000 Now, we calculate the net cash flows for each period by subtracting the floating rate payment from the fixed rate payment. Period 1: £200,000 – £175,000 = £25,000 Period 2: £200,000 – £200,000 = £0 Period 3: £200,000 – £225,000 = -£25,000 Period 4: £200,000 – £250,000 = -£50,000 To calculate the present value of these cash flows, we use the discount factors provided. PV of Period 1: £25,000 * 0.982 = £24,550 PV of Period 2: £0 * 0.965 = £0 PV of Period 3: -£25,000 * 0.948 = -£23,700 PV of Period 4: -£50,000 * 0.932 = -£46,600 Finally, we sum the present values of all cash flows to find the value of the swap. Value of Swap = £24,550 + £0 – £23,700 – £46,600 = -£45,750 Therefore, the value of the swap to the company is -£45,750. This means the company is at a disadvantage, as the present value of their expected payments exceeds the present value of their expected receipts. This would be the price another party would pay to take over the company’s side of the swap.

Let’s consider a scenario involving a UK-based agricultural cooperative, “GreenHarvest,” which produces organic wheat. GreenHarvest wants to protect itself against a potential fall in wheat prices over the next six months. They are considering using exchange-traded wheat futures contracts listed on the ICE Futures Europe exchange. Each contract represents 100 metric tons of wheat. The current spot price of wheat is £200 per metric ton. The six-month futures price is £210 per metric ton. GreenHarvest expects to harvest 500 metric tons of wheat in six months. Their risk management policy allows for hedging up to 80% of their expected production. To determine the number of contracts GreenHarvest should short, we first calculate the amount of wheat they want to hedge: 500 tons * 80% = 400 tons. Then, we divide the amount to be hedged by the contract size: 400 tons / 100 tons/contract = 4 contracts. Therefore, GreenHarvest should short 4 futures contracts. Now, let’s analyze the potential outcomes. If the spot price of wheat falls to £190 per metric ton at the expiration of the futures contract, GreenHarvest will lose £10 per ton on their physical wheat sale (£200 – £190). However, they will profit from their short futures position. The profit per ton on the futures contract is £210 (initial futures price) – £190 (final spot price) = £20 per ton. Their total profit on the futures contracts is 400 tons * £20/ton = £8,000. This profit offsets some of the loss on their physical wheat sale. Conversely, if the spot price of wheat rises to £230 per metric ton, GreenHarvest will gain £30 per ton on their physical wheat sale (£230 – £200). However, they will lose on their short futures position. The loss per ton on the futures contract is £230 (final spot price) – £210 (initial futures price) = £20 per ton. Their total loss on the futures contracts is 400 tons * £20/ton = £8,000. This loss offsets some of the gain on their physical wheat sale. The key concept here is basis risk. Basis risk is the risk that the price of the asset being hedged (spot price) and the price of the hedging instrument (futures price) do not move perfectly together. In this example, even if GreenHarvest perfectly hedges their expected production, they may still experience some gains or losses due to basis risk. For instance, if the futures price converges to a price slightly different than the spot price at expiration, the hedge will not be perfect. Understanding basis risk is crucial for effective hedging strategies, particularly in agricultural markets where local supply and demand conditions can significantly impact spot prices. Furthermore, firms must comply with regulations such as those outlined in the Financial Services and Markets Act 2000 when dealing with derivatives, ensuring fair practices and investor protection.

The core of this question revolves around understanding how margin requirements function in futures contracts, specifically their role in mitigating counterparty risk and ensuring contract performance. Initial margin acts as a performance bond, while variation margin addresses daily mark-to-market losses. The key is to recognize that margin calls are triggered when the account balance falls below the maintenance margin, requiring funds to be added to restore the initial margin level. This prevents losses from accumulating to a point where a party defaults. Let’s analyze the scenario: An investor initiates a short position in a futures contract at £250 per contract. The initial margin is £25 per contract, and the maintenance margin is £20 per contract. This means the investor must initially deposit £25 per contract. If the futures price increases, the investor incurs a loss, which is deducted from their margin account. A margin call occurs when the account balance drops below £20. To determine the price at which a margin call is triggered, we need to calculate the loss that would reduce the margin account to the maintenance margin level. The difference between the initial margin and the maintenance margin is £5 (£25 – £20). Therefore, the investor can withstand a loss of £5 per contract before a margin call is issued. Since the investor has a short position, a price increase will result in a loss. The price increase that triggers the margin call is calculated as follows: Margin Call Trigger Price = Initial Price + (Initial Margin – Maintenance Margin) = £250 + (£25 – £20) = £250 + £5 = £255. Therefore, a margin call will be triggered when the futures price reaches £255 per contract. This calculation illustrates the fundamental principle of margin requirements in futures trading: to protect the exchange and other market participants from losses due to default. The maintenance margin acts as a safety net, ensuring that the investor has sufficient funds to cover potential losses. The margin call forces the investor to replenish the account, preventing further erosion of the margin and reducing the risk of default. In essence, margin requirements are a crucial risk management tool in futures markets, facilitating efficient and secure trading.

Let’s analyze how to determine the arbitrage-free price of a forward contract on an asset that provides a discrete cash dividend during the contract’s life. The core principle is to ensure no riskless profit can be made by simultaneously entering offsetting positions. The formula for the arbitrage-free forward price (F) is derived from the spot price (S), the cost of carry (which includes interest earned and dividends received), and the time to maturity (T). If the asset pays a discrete dividend (D) at time \(t\) before the maturity of the forward contract, the formula becomes: \[F = (S – D \cdot e^{-r(t)})e^{rT}\] Where: * S is the current spot price of the asset. * D is the discrete dividend paid at time \(t\). * r is the risk-free interest rate. * T is the time to maturity of the forward contract. * \(e\) is the base of the natural logarithm. The term \(D \cdot e^{-r(t)}\) represents the present value of the dividend received. We subtract this from the spot price because the forward contract holder will not receive this dividend. The remaining portion, \((S – D \cdot e^{-r(t)})\), is then compounded forward at the risk-free rate to the maturity date of the forward contract using \(e^{rT}\). Consider a stock trading at £50. A dividend of £2 is expected in 3 months (0.25 years). The risk-free rate is 5% per annum. First, we calculate the present value of the dividend: \[D \cdot e^{-r(t)} = 2 \cdot e^{-0.05 \cdot 0.25} = 2 \cdot e^{-0.0125} \approx 2 \cdot 0.9876 = £1.9752\] Next, we subtract the present value of the dividend from the spot price: \[S – D \cdot e^{-r(t)} = 50 – 1.9752 = £48.0248\] Finally, we compound this value forward to the maturity date (let’s assume the forward contract matures in 1 year): \[F = 48.0248 \cdot e^{0.05 \cdot 1} = 48.0248 \cdot e^{0.05} \approx 48.0248 \cdot 1.0513 = £50.49\] Therefore, the arbitrage-free forward price is approximately £50.49. If the market price deviates significantly from this, an arbitrage opportunity exists. For instance, if the market price is £52, an arbitrageur could buy the stock, sell the forward contract, and lock in a riskless profit by short selling the forward. The dividend received would partially offset the cost of carry. Conversely, if the market price is £49, an arbitrageur could short the stock, buy the forward contract, and profit from the mispricing. This mechanism ensures that forward prices converge to their theoretical arbitrage-free values.

To determine the most suitable strategy, we need to calculate the potential profit or loss under different scenarios, considering the costs and payoffs associated with each option. This involves understanding the payoff structure of options and how they interact with the underlying asset’s price movements. The investor’s objective is to maximize potential profit while mitigating risk, considering the initial investment and the potential for the stock price to fluctuate. The analysis must account for the cost of the options, the strike prices, and the potential profit or loss at different stock prices at expiration. The breakeven point for each strategy needs to be calculated to assess the risk-reward profile. Strategy A (Long Call): Maximum profit is unlimited, but the breakeven point is the strike price plus the premium paid (\(£50 + £5 = £55\)). Strategy B (Short Put): Maximum profit is limited to the premium received (\(£5\)), but the potential loss is substantial if the stock price falls below the strike price (\(£50\)). The breakeven point is the strike price minus the premium received (\(£50 – £5 = £45\)). Strategy C (Protective Put): This strategy provides downside protection. The maximum loss is limited to the cost of the put option minus the difference between the initial stock price and the strike price. Strategy D (Covered Call): This strategy generates income from the premium received but limits the upside potential. The maximum profit is the premium received plus the difference between the strike price and the initial stock price. Given the investor’s risk aversion and the desire to generate income, a covered call (Strategy D) is the most suitable option. It provides a limited profit potential but also reduces the overall risk by generating income from the premium received. This aligns with the investor’s goal of generating income while mitigating risk.

The question assesses understanding of exotic derivatives, specifically barrier options, and their sensitivity to market movements. A down-and-out call option becomes worthless if the underlying asset’s price touches or falls below the barrier level before expiration. The investor’s profit or loss depends on whether the barrier is breached and, if not, on the underlying asset’s price at expiration relative to the strike price. In this scenario, the investor buys a down-and-out call option with a strike price of £100 and a barrier at £90. The initial premium paid is £8. Scenario 1: The underlying asset price falls to £85 during the option’s life. The barrier is breached, and the option becomes worthless. The investor loses the entire premium of £8. Scenario 2: The underlying asset price fluctuates but never touches or falls below £90. At expiration, the asset price is £115. The option is now a regular call option because the barrier was never breached. The payoff is the difference between the asset price and the strike price: £115 – £100 = £15. The net profit is the payoff minus the premium: £15 – £8 = £7. Scenario 3: The underlying asset price fluctuates but never touches or falls below £90. At expiration, the asset price is £95. The option expires worthless because the asset price is below the strike price. The investor loses the entire premium of £8. Scenario 4: The underlying asset price fluctuates but never touches or falls below £90. At expiration, the asset price is £100. The option expires worthless because the asset price is equal to the strike price. The investor loses the entire premium of £8. The question tests the ability to determine the profit/loss in different scenarios, considering the impact of the barrier and the asset price at expiration. It goes beyond simple definitions and requires applying the concepts to a practical situation.

Let’s break down this exotic derivative pricing problem. The key is understanding the knock-in barrier and how it affects the option’s value. First, we calculate the probability of the underlying asset (a basket of tech stocks, in this case) breaching the barrier before the maturity date. We’ll use a simplified model assuming a log-normal distribution for the asset price and estimate the probability using Monte Carlo simulation. Assume we run 10,000 simulations of the basket’s price path. In 3,000 of those simulations, the barrier is breached. This gives us an approximate probability of breaching the barrier of 30% (3000/10000). Next, we need to calculate the expected payoff of the vanilla call option *given* that the barrier has been breached. The payoff of a call option is \( max(S_T – K, 0) \), where \( S_T \) is the asset price at maturity and \( K \) is the strike price. Let’s assume that, conditional on breaching the barrier, the average terminal asset price in our simulations is £160. The strike price is £150. Therefore, the average payoff is £10. Finally, we discount this expected payoff back to today’s value using the risk-free rate. Assuming a risk-free rate of 5% and a time to maturity of 1 year, the discount factor is \( e^{-0.05 \cdot 1} \approx 0.9512 \). The present value of the expected payoff, given the barrier is breached, is \( 10 \cdot 0.9512 = 9.512 \). The price of the knock-in option is the probability of breaching the barrier multiplied by the present value of the expected payoff, conditional on breaching the barrier. This is \( 0.30 \cdot 9.512 = 2.8536 \). Therefore, the estimated price of the knock-in call option is approximately £2.85. The complexities lie in the fact that barrier options are path-dependent. The investor is betting on both the direction of the underlying asset and its volatility. Higher volatility increases the likelihood of hitting the barrier. The choice of the barrier level is crucial. A lower barrier increases the probability of the option knocking in, but it also makes the option behave more like a standard call option. The pricing models are also sensitive to the assumptions about the underlying asset’s price distribution and volatility. In reality, one would use more sophisticated volatility models (e.g., stochastic volatility models) to better capture the market dynamics.

Let’s break down this complex scenario. First, we need to understand the payoff structure of the exotic derivative – a Barrier Option. Specifically, this is a *down-and-out* barrier option. This means the option ceases to exist (knocks out) if the underlying asset’s price touches or goes below the barrier level *at any point* during the option’s life. In this case, the barrier is set at 90% of the initial spot price, which is \(0.90 \times 120 = 108\). The option is worthless if the price ever hits or goes below 108. Now, consider the two possible price paths. In Path 1, the price dips to 105 before recovering. Since 105 is *below* the barrier of 108, the option is knocked out and becomes worthless. The final price at expiry is irrelevant because the knock-out event has already occurred. In Path 2, the price fluctuates but never touches or goes below 108. At expiry, the underlying asset’s price is 125. Since this is a call option with a strike price of 120, the payoff is the difference between the spot price at expiry and the strike price: \(125 – 120 = 5\). Finally, we calculate the average payoff across the two paths. Path 1 has a payoff of 0, and Path 2 has a payoff of 5. The average payoff is \(\frac{0 + 5}{2} = 2.5\). A crucial element here is understanding that barrier options are path-dependent. Their payoff depends not only on the final price of the underlying asset but also on the asset’s price movements during the option’s life. This contrasts with standard European or American options, where only the price at expiry matters. Furthermore, the “down-and-out” feature makes it cheaper than a vanilla call option, reflecting the increased risk of the option becoming worthless before expiry. The regulatory implications for advising on such products under COBS (Conduct of Business Sourcebook) require a firm understanding of the client’s risk tolerance and investment objectives.

The question explores the concept of delta hedging a short call option position. Delta hedging involves adjusting the portfolio’s holdings of the underlying asset to offset changes in the option’s value due to fluctuations in the asset’s price. The delta of a call option represents the sensitivity of the option’s price to a change in the underlying asset’s price. A short call option has a positive delta, meaning that as the underlying asset’s price increases, the option’s price also increases, resulting in a loss for the option writer. To hedge this risk, the option writer needs to buy the underlying asset. The amount of the asset to buy is determined by the option’s delta. In this scenario, the portfolio manager initially holds a short position in 500 call options, each representing 100 shares, for a total of 50,000 shares. The initial delta is 0.4, indicating that for every $1 increase in the underlying asset’s price, the option’s price will increase by $0.40 per share. To hedge this position, the manager buys 50,000 * 0.4 = 20,000 shares. When the underlying asset’s price increases and the delta increases to 0.6, the manager needs to adjust the hedge. The new hedge requires 50,000 * 0.6 = 30,000 shares. Since the manager already holds 20,000 shares, they need to buy an additional 30,000 – 20,000 = 10,000 shares. This example demonstrates the dynamic nature of delta hedging. The hedge needs to be continuously adjusted as the underlying asset’s price and the option’s delta change. Failure to adjust the hedge can result in losses if the underlying asset’s price moves significantly. The analogy here is a boat trying to stay in place on a lake with wind. The boat is the option position, and the wind is the movement of the underlying asset. The anchor is the hedge. As the wind changes direction and strength (delta changes), you need to adjust the anchor’s position and strength to keep the boat steady (hedged). This requires constant monitoring and adjustments. The cost of these adjustments (transaction costs) is a key consideration in delta hedging.

The core of this question lies in understanding how gamma scalping profits are affected by transaction costs (bid-ask spread) and the trader’s ability to perfectly hedge. Gamma, the rate of change of delta, indicates how much the hedge (delta) needs to be adjusted as the underlying asset’s price changes. A higher gamma means more frequent adjustments. The profit from gamma scalping comes from buying low and selling high (or vice versa) as the price fluctuates, while continuously re-hedging to maintain a delta-neutral position. However, each trade incurs transaction costs, reducing the overall profit. Imperfect hedging also reduces profit, as the hedge will not perfectly offset the option’s price movement. In this scenario, the investor sells a call option, so they are short gamma (negative gamma). This means they need to buy when the price goes up and sell when the price goes down to remain delta neutral. If the investor can perfectly hedge, the profit from the gamma scalping strategy is: \[ \text{Profit} = \frac{1}{2} \times \text{Gamma} \times (\text{Price Change})^2 \times \text{Number of Contracts} \times \text{Multiplier} – (\text{Number of Trades} \times \text{Bid-Ask Spread}) \] Given the gamma of 0.05 per share, a price movement of £0.10, 10 contracts (each representing 100 shares), and a bid-ask spread of £0.02 per share: \[ \text{Profit per move} = \frac{1}{2} \times 0.05 \times (0.10)^2 \times 10 \times 100 = £0.25 \] With 200 such price moves, the gross profit would be: \[ \text{Gross Profit} = 200 \times £0.25 = £50 \] Each price move requires a hedge adjustment, so there are 200 trades. The total transaction cost is: \[ \text{Transaction Cost} = 200 \times (£0.02 \times 10 \times 100) = £400 \] Net profit before the hedging error is: \[ \text{Net Profit} = £50 – £400 = -£350 \] Since the hedging is only 90% effective, the profit per move is reduced by 10%: \[ \text{Adjusted Profit per move} = £0.25 \times 0.90 = £0.225 \] The adjusted gross profit is: \[ \text{Adjusted Gross Profit} = 200 \times £0.225 = £45 \] The net profit with the hedging error is: \[ \text{Adjusted Net Profit} = £45 – £400 = -£355 \] Therefore, the overall profit is a loss of £355.

To determine the profit or loss from the swap, we need to calculate the net payments made or received by Alpha Corp. First, let’s calculate the fixed rate payments made by Alpha Corp: 2.5% of £10 million = £250,000 per year. Over three years, this totals £750,000. Next, we need to calculate the floating rate payments received by Alpha Corp. For Year 1, the payment is 2.0% of £10 million = £200,000. For Year 2, the payment is 2.7% of £10 million = £270,000. For Year 3, the payment is 3.1% of £10 million = £310,000. The total floating rate payments received are £200,000 + £270,000 + £310,000 = £780,000. The net profit or loss is the floating rate payments received minus the fixed rate payments made: £780,000 – £750,000 = £30,000. Therefore, Alpha Corp made a profit of £30,000 on the swap. A swap is essentially a series of forward contracts. In this instance, Alpha Corp entered into a contract to exchange a fixed rate for a floating rate. The interest rate swap can be seen as a tool for managing interest rate risk. Alpha Corp. swapped a fixed rate liability for a floating rate asset, thus betting that the floating rate would exceed the fixed rate over the swap period. Consider a farmer who agrees to sell his produce at a fixed price in the future. This is similar to paying a fixed rate in a swap. If the market price of the produce increases, the farmer benefits from the swap. Conversely, if the market price decreases, the farmer loses out. The same principle applies to interest rate swaps. If the floating rate exceeds the fixed rate, the party receiving the floating rate benefits, and vice versa. In this case, the floating rate exceeded the fixed rate, resulting in a profit for Alpha Corp.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Green Harvest,” which wants to protect its future wheat sales from price fluctuations. They plan to use exchange-traded wheat futures contracts on the London International Financial Futures and Options Exchange (LIFFE), now part of ICE Futures Europe. Green Harvest expects to harvest 500 tonnes of wheat in three months and wants to hedge against a potential price decrease. Each wheat futures contract on LIFFE represents 100 tonnes of wheat. Therefore, Green Harvest needs to sell 5 contracts (500 tonnes / 100 tonnes per contract = 5 contracts). The current futures price for wheat with delivery in three months is £200 per tonne. Green Harvest sells 5 futures contracts at this price, effectively locking in a selling price of £200 per tonne. Now, let’s analyze two scenarios: Scenario 1: At harvest time, the spot price of wheat has fallen to £180 per tonne. Green Harvest sells its wheat in the spot market for £180 per tonne, receiving £90,000 (500 tonnes * £180). Simultaneously, they close out their futures position by buying back 5 futures contracts. Since the futures price will have converged towards the spot price, they buy back the futures contracts at approximately £180 per tonne. This results in a profit on the futures contracts of £20 per tonne (£200 – £180), or £10,000 in total (5 contracts * 100 tonnes/contract * £20/tonne). The effective selling price is £180 (spot) + £20 (futures profit) = £200 per tonne. The total revenue is £90,000 (spot sales) + £10,000 (futures profit) = £100,000. Scenario 2: At harvest time, the spot price of wheat has risen to £220 per tonne. Green Harvest sells its wheat in the spot market for £220 per tonne, receiving £110,000 (500 tonnes * £220). They close out their futures position by buying back 5 futures contracts at approximately £220 per tonne. This results in a loss on the futures contracts of £20 per tonne (£200 – £220), or £10,000 in total (5 contracts * 100 tonnes/contract * £20/tonne). The effective selling price is £220 (spot) – £20 (futures loss) = £200 per tonne. The total revenue is £110,000 (spot sales) – £10,000 (futures loss) = £100,000. In both scenarios, Green Harvest effectively locked in a selling price close to £200 per tonne for their wheat, mitigating the risk of price fluctuations. This demonstrates how hedging with futures contracts can stabilize revenue for agricultural producers, even though it limits potential gains when prices rise. The key here is understanding basis risk, which is the difference between the spot price and the futures price at the time the hedge is lifted. Basis risk is the reason the effective selling price is not exactly £200 per tonne in reality. Regulations such as the Market Abuse Regulation (MAR) would apply to Green Harvest’s trading activities, ensuring fair market practices and preventing insider trading.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their sensitivity to market movements near the barrier. It requires calculating the probability of a knock-out event based on implied volatility and time to maturity. The formula for approximating the probability of breaching a barrier is complex but can be simplified for illustrative purposes here. We use a simplified approach for demonstration, recognizing that a precise calculation would involve more sophisticated models. Let’s assume a simplified model where the probability of hitting the barrier is proportional to the standard deviation of the asset price movement over the time to maturity. The standard deviation is calculated as \( \sigma \sqrt{T} \), where \( \sigma \) is the implied volatility and \( T \) is the time to maturity. In this case, \( \sigma = 20\% = 0.20 \) and \( T = 0.5 \) years. Therefore, the standard deviation is \( 0.20 \times \sqrt{0.5} \approx 0.1414 \). The distance to the barrier is \( \frac{495 – 490}{490} \approx 0.0102 \) or 1.02%. We can approximate the probability of hitting the barrier by considering the ratio of the distance to the barrier to the standard deviation. A smaller ratio suggests a higher probability. Probability ≈ Distance to Barrier / Standard Deviation = \( \frac{0.0102}{0.1414} \approx 0.0721 \) or 7.21%. However, this is a simplified approximation. A more realistic model would consider the distribution of price movements (e.g., using a normal distribution) and the continuous monitoring of the barrier. Also, the formula is not linear and should be considered as an approximation. The most accurate answer, considering the complexities of barrier option pricing and the proximity to the barrier, is around 15%. This accounts for the increased likelihood of the underlying asset fluctuating and breaching the barrier as time progresses. A simpler analogy is to consider a ball rolling near the edge of a cliff. The closer the ball is to the edge, and the more erratic its movements (higher volatility), the higher the chance it will fall off the cliff (knock-out). The time until a strong gust of wind (maturity) also plays a role – more time means more opportunity for the ball to be blown off.

The optimal hedge ratio minimizes the variance of the hedged portfolio. This is achieved when the hedge ratio equals the correlation between the asset and the hedging instrument, multiplied by the ratio of the standard deviation of the asset’s price changes to the standard deviation of the hedging instrument’s price changes. In this scenario, we’re hedging an equity portfolio with a value of £5,000,000 using FTSE 100 futures contracts. Each contract represents £10 per index point. The current FTSE 100 index is 7,500. 1. **Calculate the contract size:** Each futures contract controls \(7500 \times £10 = £75,000\) worth of the underlying index. 2. **Calculate the number of contracts needed without considering correlation and volatility:** Without considering correlation and volatility, the naive hedge ratio would suggest needing \(\frac{£5,000,000}{£75,000} \approx 66.67\) contracts. This is the number of contracts required to fully hedge the portfolio if the portfolio and the index moved perfectly together with equal volatility. 3. **Adjust for correlation and volatility:** The correlation between the portfolio and the FTSE 100 is 0.75, and the volatility of the portfolio is 1.2 times that of the FTSE 100. Therefore, the optimal hedge ratio is \(0.75 \times 1.2 = 0.9\). This means that for every £1 exposure in the portfolio, we need to hedge £0.9 with the futures contract. 4. **Calculate the adjusted number of contracts:** The adjusted number of contracts is \(66.67 \times 0.9 \approx 60\). Therefore, 60 futures contracts are required to minimize the variance of the hedged portfolio. This calculation demonstrates a critical aspect of hedging: it’s not simply about matching the notional value of the exposure. The correlation and relative volatility between the asset being hedged and the hedging instrument significantly impact the optimal hedge ratio. Ignoring these factors can lead to under-hedging or over-hedging, both of which increase the risk of the hedged portfolio. For instance, if the correlation were lower, say 0.5, the hedge ratio would decrease, requiring fewer contracts. Conversely, higher volatility in the portfolio relative to the index would increase the hedge ratio, necessitating more contracts. The concept of beta in portfolio management is closely related, where beta represents the systematic risk of an asset relative to the market. The optimal hedge ratio can be seen as a refined beta-adjustment for hedging purposes.

The core of this question revolves around understanding how different factors impact the price of an American-style call option, particularly focusing on volatility and time decay. The question specifically tests the understanding of how an American-style call option, which can be exercised at any time before expiration, is affected by changes in implied volatility and time remaining until expiration. While an increase in implied volatility generally increases the option’s price due to the greater potential for the underlying asset’s price to move favorably, the effect of time decay (theta) is more nuanced. As an option approaches its expiration date, its time value erodes, reducing its price. However, this erosion is not linear and is influenced by whether the option is in-the-money (ITM), at-the-money (ATM), or out-of-the-money (OTM). An ATM option experiences the most rapid time decay. To solve this, we need to consider the combined effects. The volatility increase pushes the price up, while the time decay pulls it down. The key is to recognize that the magnitude of these effects depends on the option’s moneyness. The implied volatility increase will have a greater impact on the option’s price if the option is closer to being at-the-money. The time decay will have a more significant impact as the option approaches its expiration, especially if it is at-the-money. Let’s assume the initial option price is \(P_0\). An increase in implied volatility by 5% might increase the option price by, say, \(+2\%\) of \(P_0\), leading to \(P_0 + 0.02P_0 = 1.02P_0\). However, the time decay of 2 weeks could reduce the option price by, say, \(-1\%\) of \(P_0\), leading to \(1.02P_0 – 0.01P_0 = 1.01P_0\). Therefore, the net effect is a slight increase in the option price. However, the precise magnitude will depend on the specific characteristics of the option (strike price, time to expiration, underlying asset price, etc.). The correct answer is most likely to be slightly higher.

The question focuses on understanding the mechanics of a variance swap, specifically how the notional amount is determined and its impact on the payoff. The key is to recognize that the notional is calculated to make the present value of the fixed leg equal to the present value of the floating leg at inception. The formula for the fair variance strike \( K_{var} \) is derived from equating the expected payoff of the variance swap to zero at initiation. The payoff of the variance swap is proportional to \( N \times (Realized Variance – K_{var}) \), where N is the notional and \( K_{var} \) is the variance strike. The fair variance strike is approximated by the square of the implied volatility \( \sigma^2 \). Given a volatility of 20%, the variance strike \( K_{var} \) is \( (0.20)^2 = 0.04 \). The variance notional \( N \) is calculated using the formula: \( N = \frac{V}{2 \times K_{var} \times T} \), where \( V \) is the value of the volatility leg (given as £500,000), and \( T \) is the tenor of the swap (given as 1 year). Therefore, \( N = \frac{500,000}{2 \times 0.04 \times 1} = 6,250,000 \). The variance notional is £6,250,000. The impact on the payoff can be illustrated with a scenario. Suppose the realized variance turns out to be 0.06 (6%). The payoff would be \( 6,250,000 \times (0.06 – 0.04) = 125,000 \). This means the party that sold the variance swap would pay £125,000 to the party that bought it. Now, consider if the volatility increased to 25%. The variance strike would be \( (0.25)^2 = 0.0625 \). If the value of the volatility leg increased to £750,000, the new variance notional would be \( N = \frac{750,000}{2 \times 0.0625 \times 1} = 6,000,000 \). The variance notional decreased because the variance strike increased more than the volatility leg value increased. This demonstrates the inverse relationship between the variance strike and the variance notional when the value of the volatility leg changes. The question requires understanding this relationship and applying the formula correctly.

The question assesses the understanding of how different types of exotic options react to market volatility and specific price movements, particularly in scenarios where the underlying asset exhibits unusual behavior. We need to consider the payoff structures of each option type and how they are affected by the described market conditions. A barrier option’s payoff is contingent on the underlying asset reaching a certain price level (the barrier). A knock-out barrier option ceases to exist if the barrier is hit. In this scenario, the sudden spike and subsequent fall of the asset price suggests the barrier was likely breached, rendering the knock-out option worthless. An Asian option’s payoff is based on the average price of the underlying asset over a specified period. The spike in price, followed by a fall, would influence the average price, but the impact would be diluted by the other prices within the averaging period. It is less sensitive to short-term volatility compared to barrier options. A Cliquet option, also known as a ratchet option, consists of a series of forward start options. Each period, the option resets based on the performance of the underlying asset, locking in gains. In a volatile market, a Cliquet option can perform well if it locks in gains during the upward spike, even if the asset subsequently falls. The local “ratcheting” feature makes it attractive in uncertain markets. A lookback option allows the holder to choose the most favorable price of the underlying asset during the option’s life as the strike price. In a market with a significant price spike, a lookback option would likely benefit significantly, as the holder could use the highest price as the strike price when selling (for a call option) or the lowest price as the strike price when buying (for a put option), maximizing their profit. Therefore, in this highly volatile scenario, the lookback option is most likely to provide the greatest return due to its ability to capitalize on the extreme price movements.

The question explores the concept of gamma risk within a portfolio of options, specifically focusing on how the gamma profile changes with the passage of time (theta) and how this impacts hedging strategies. Gamma represents the rate of change of delta with respect to the underlying asset’s price. A high gamma implies that the delta is highly sensitive to price changes, requiring more frequent rebalancing of the hedge. The ‘gamma exposure’ of a portfolio is the sum of the gammas of all options in the portfolio. Theta, on the other hand, measures the rate of decline in an option’s value due to the passage of time. Options typically lose value as they approach their expiration date. The key here is understanding that gamma is highest when an option is at-the-money (ATM) and closest to expiration. As time passes, the gamma of an ATM option will initially increase, peaking just before expiration, and then rapidly decline to zero at expiration. Conversely, deep in-the-money (ITM) or out-of-the-money (OTM) options have lower gamma, which tends to decay more steadily as time passes. The scenario presented involves a portfolio that is initially gamma-neutral. This means the positive and negative gammas of the constituent options offset each other. However, as time passes, the gamma profile changes due to the differing theta sensitivities of the individual options. Specifically, the ATM options will experience a more pronounced increase in gamma as they approach expiration compared to the ITM/OTM options. This leads to a shift in the overall portfolio gamma. The question tests the understanding of how the gamma profile of a portfolio changes over time, particularly when the portfolio contains options with different moneyness and expiration dates. It requires an understanding of how theta impacts gamma and the implications for maintaining a gamma-neutral hedge. The correct answer will reflect the understanding that the portfolio’s gamma will likely become non-zero as time passes, with the direction of the gamma depending on the specific composition of the portfolio. Furthermore, the frequency of hedge rebalancing must increase as the options approach expiration and their gamma increases.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their valuation sensitivity to volatility. The knock-out feature introduces complexity, as the option’s existence depends on the underlying asset’s price trajectory. A standard Black-Scholes model is insufficient for pricing barrier options accurately, especially when considering volatility smiles. The calculation involves understanding how implied volatility affects the probability of the barrier being hit before expiration. Increased volatility increases the likelihood of the barrier being breached, thus reducing the value of a knock-out option. Conversely, a knock-in option’s value would increase with volatility. The pricing of barrier options also depends on the relationship between the current spot price and the barrier level. If the spot price is near the barrier, the option’s value is highly sensitive to volatility changes. If the spot price is far from the barrier, volatility changes have a lesser impact on the option’s price. Consider a hypothetical scenario: A portfolio manager uses a down-and-out call option on a FTSE 100 index to hedge a portfolio. The FTSE 100 is currently trading at 7500, and the down-and-out barrier is set at 7000. The portfolio manager believes that the FTSE 100 will rise but wants to limit losses if the index falls significantly. If implied volatility increases sharply due to unforeseen economic news, the value of the down-and-out call option will decrease because the probability of the FTSE 100 hitting the 7000 barrier has increased. This highlights the importance of understanding volatility’s impact on barrier option pricing and hedging strategies. Another example involves a corporate treasurer using a knock-out swap to hedge interest rate risk. The treasurer enters into a swap that pays fixed and receives floating, with a knock-out barrier based on a specific interest rate level. If interest rate volatility increases, the value of the knock-out swap will decrease because the probability of the interest rate hitting the barrier increases, causing the swap to terminate. This illustrates how volatility affects the valuation and risk management of knock-out swaps.

To determine the most appropriate action for Alpha Investments, we need to calculate the potential profit/loss from exercising the options and compare it to the cost of the options themselves. Then, we must consider the regulatory implications under COBS 2.3A.4R regarding best execution. First, calculate the intrinsic value of each option: * Call Option: Intrinsic Value = Max(Spot Price – Strike Price, 0) = Max(162 – 155, 0) = 7 * Put Option: Intrinsic Value = Max(Strike Price – Spot Price, 0) = Max(105 – 162, 0) = 0 Next, calculate the total profit/loss from exercising, considering the number of contracts and the multiplier: * Call Option Profit: (Intrinsic Value – Premium) * Multiplier * Number of Contracts = (7 – 3) * 100 * 50 = 20,000 * Put Option Profit: (Intrinsic Value – Premium) * Multiplier * Number of Contracts = (0 – 2) * 100 * 30 = -6,000 Total Profit/Loss = Call Option Profit + Put Option Profit = 20,000 – 6,000 = 14,000 Now, consider COBS 2.3A.4R, which mandates firms to take all reasonable steps to obtain the best possible result for their clients when executing orders. This includes considering factors like price, costs, speed, likelihood of execution and settlement, size, nature, or any other consideration relevant to the execution of the order. In this scenario, even though exercising the options yields a profit of £14,000, Alpha Investments must also consider if they could have achieved a better outcome by selling the options in the market rather than exercising them. For example, if the market value of the call options was higher than their intrinsic value due to time value, selling them could have yielded a better return. Similarly, if the put options had any remaining time value, selling them, even at a loss, might have been preferable to exercising them for zero intrinsic value. The firm must document their decision-making process to demonstrate compliance with best execution requirements. If selling the options in the market would have yielded a better result, exercising them would be a breach of COBS 2.3A.4R.

To determine the fair value of the swap, we need to discount the expected future cash flows. Since the interest rate is continuously compounded, we use the formula for present value under continuous compounding: \( PV = FV \cdot e^{-rt} \), where \( PV \) is the present value, \( FV \) is the future value, \( r \) is the interest rate, and \( t \) is the time in years. First, calculate the expected cash flows. The notional principal is £5,000,000. The fixed rate is 4% per annum, so the fixed payment is \( 0.04 \cdot 5,000,000 = £200,000 \) per year. The floating rate is reset annually and paid annually. The expected floating rates are 4.5% in year 1, 5% in year 2, and 5.5% in year 3. The corresponding floating payments are \( 0.045 \cdot 5,000,000 = £225,000 \) in year 1, \( 0.05 \cdot 5,000,000 = £250,000 \) in year 2, and \( 0.055 \cdot 5,000,000 = £275,000 \) in year 3. Next, determine the net cash flows for each year. In year 1, the net cash flow is \( £225,000 – £200,000 = £25,000 \). In year 2, the net cash flow is \( £250,000 – £200,000 = £50,000 \). In year 3, the net cash flow is \( £275,000 – £200,000 = £75,000 \). Now, discount these cash flows using the continuously compounded interest rates: 4.2% for year 1, 4.5% for year 2, and 4.8% for year 3. The present value of the cash flow in year 1 is \( 25,000 \cdot e^{-0.042 \cdot 1} = 25,000 \cdot e^{-0.042} \approx 25,000 \cdot 0.9588 = £23,970 \). The present value of the cash flow in year 2 is \( 50,000 \cdot e^{-0.045 \cdot 2} = 50,000 \cdot e^{-0.09} \approx 50,000 \cdot 0.9139 = £45,695 \). The present value of the cash flow in year 3 is \( 75,000 \cdot e^{-0.048 \cdot 3} = 75,000 \cdot e^{-0.144} \approx 75,000 \cdot 0.8658 = £64,935 \). The fair value of the swap is the sum of these present values: \( £23,970 + £45,695 + £64,935 = £134,600 \). Therefore, the fair value of the swap to the company is £134,600.

The core of this question revolves around understanding how regulatory bodies like the FCA (Financial Conduct Authority) in the UK view and treat exotic derivatives, particularly concerning their suitability for retail clients. The FCA has specific guidelines and concerns regarding the complexity and risk associated with these instruments. The key is to recognize that while exotic derivatives can offer tailored solutions and potentially higher returns, their complexity often makes it difficult for retail clients to fully understand the risks involved. This lack of transparency and potential for unexpected losses is a major concern for regulators. The question tests the candidate’s ability to apply these regulatory principles to a specific scenario involving a barrier option linked to a volatile emerging market index. The calculation isn’t explicitly numerical, but conceptual. The “knock-out” feature of the barrier option adds another layer of complexity, as the option’s value can disappear entirely if the underlying index breaches a certain threshold. This feature significantly increases the risk profile, making it even less suitable for retail investors who may not have the sophistication to understand and manage this type of risk. Therefore, the most appropriate course of action is to advise the client against investing in the barrier option and instead suggest simpler, more transparent investment products that align with their risk tolerance and investment objectives. This aligns with the FCA’s principles of treating customers fairly and ensuring that financial products are suitable for their target market. The incorrect options highlight common misconceptions about exotic derivatives, such as assuming that high potential returns justify the risks or that a client’s existing investment experience automatically qualifies them to invest in complex instruments. The FCA emphasizes the need for a thorough understanding of the specific risks associated with each product, regardless of the client’s overall investment knowledge. The question is designed to assess the candidate’s ability to apply these regulatory principles in a practical advisory context.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Green Harvest,” that produces organic wheat. They face price volatility in the wheat market and seek to hedge their risk using futures contracts traded on the ICE Futures Europe exchange. Green Harvest anticipates harvesting 5,000 tonnes of wheat in three months. The current spot price of wheat is £200 per tonne, but they are concerned about a potential price drop due to an expected bumper harvest across Europe. The December wheat futures contract (expiring in three months) is trading at £210 per tonne. Green Harvest decides to hedge 4,000 tonnes of their expected harvest by selling 80 December wheat futures contracts (each contract representing 50 tonnes). Now, imagine two scenarios: Scenario 1: At harvest time, the spot price of wheat has fallen to £180 per tonne. The December wheat futures contract settles at £185 per tonne. Green Harvest sells their physical wheat at the spot price of £180 per tonne. Simultaneously, they close out their futures position by buying back 80 December wheat futures contracts at £185 per tonne. Calculation: Loss on physical wheat: (£200 – £180) * 4,000 tonnes = £80,000 Profit on futures: (£210 – £185) * 4,000 tonnes = £100,000 Net effect: £100,000 – £80,000 = £20,000 profit Scenario 2: At harvest time, the spot price of wheat has risen to £230 per tonne. The December wheat futures contract settles at £235 per tonne. Green Harvest sells their physical wheat at the spot price of £230 per tonne. Simultaneously, they close out their futures position by buying back 80 December wheat futures contracts at £235 per tonne. Calculation: Profit on physical wheat: (£230 – £200) * 4,000 tonnes = £120,000 Loss on futures: (£235 – £210) * 4,000 tonnes = £100,000 Net effect: £120,000 – £100,000 = £20,000 profit In both scenarios, the futures hedge helped Green Harvest to stabilize their revenue around the futures price level, regardless of the spot price movement. This demonstrates how futures contracts can be used to mitigate price risk, a key aspect of derivatives trading and risk management.

Let’s consider a scenario involving a bespoke exotic derivative – a Barrier Reverse Convertible Note (BRCN) linked to the performance of a basket of three UK-listed pharmaceutical companies: PharmaCorp, MediHoldings, and BioGenesis. The BRCN has a maturity of 2 years and a capital protection barrier set at 70% of the initial basket value. The initial basket value is calculated as the average of the closing prices of the three stocks on the issue date. If, at any point during the 2-year term, the basket value trades at or below the 70% barrier, the capital protection is breached, and the investor receives the value of the basket at maturity, which could be significantly less than the initial investment. In addition to the barrier feature, the BRCN offers an enhanced coupon of 8% per annum, paid semi-annually, provided the barrier has not been breached. This enhanced coupon is the primary attraction for investors seeking higher yields in a low-interest-rate environment. Now, consider a situation where six months into the term, PharmaCorp experiences a significant setback due to adverse clinical trial results, causing its stock price to plummet. This decline triggers a domino effect, negatively impacting investor sentiment towards the entire pharmaceutical sector, including MediHoldings and BioGenesis. As a result, the basket value declines sharply, approaching the 70% barrier level. The investor, a retiree named Mrs. Eleanor Vance, seeks your advice. She is heavily reliant on the income generated from her investments and is deeply concerned about the potential loss of capital should the barrier be breached. You need to explain the risks associated with the BRCN, the potential implications of a barrier breach, and the strategies available to mitigate her losses. To further complicate matters, the BRCN documentation specifies that the barrier level is monitored continuously throughout the trading day. This means that even a brief intraday breach of the barrier can trigger the loss of capital protection, regardless of the basket’s closing value. This continuous monitoring feature adds an extra layer of risk compared to BRCNs with end-of-day barrier monitoring. You must provide a clear and concise explanation of the derivative’s features, risks, and potential outcomes, enabling Mrs. Vance to make an informed decision about her investment. You should also consider the regulatory implications of advising on complex exotic derivatives, ensuring compliance with FCA guidelines on suitability and client categorization.

The problem requires understanding how delta hedging works and its impact on profitability, especially when the implied volatility changes. A delta-neutral portfolio aims to eliminate directional risk. However, changes in implied volatility affect the option’s delta, requiring adjustments to maintain the delta-neutral position. Gamma represents the rate of change of delta with respect to the underlying asset’s price. Vega represents the sensitivity of the option’s price to changes in implied volatility. When implied volatility increases, the option’s price increases, and the delta changes. If the portfolio is delta-hedged by shorting the underlying asset, an increase in implied volatility necessitates adjusting the hedge. The trader must buy back some of the underlying asset to maintain delta neutrality. If the trader initially shorts shares and then buys them back at a higher price, this leads to a loss. The profit or loss from the hedge adjustment depends on the gamma of the portfolio and the magnitude of the volatility change. In this scenario, the portfolio has a gamma of 5,000. A gamma of 5,000 means that for every £1 change in the underlying asset’s price, the delta of the portfolio changes by 5,000. The implied volatility increases by 1%. Vega is -£25,000 per 1% change in volatility, meaning the portfolio loses £25,000 due to the volatility increase. The change in portfolio value due to the volatility change is: Vega * Change in Volatility = -£25,000 * 0.01 = -£25,000. The portfolio value decreases by £25,000 due to the increase in implied volatility. Therefore, the net effect is a loss of £25,000. This example highlights the risks associated with delta hedging when implied volatility changes, demonstrating how vega can impact the profitability of a delta-neutral portfolio. It also shows the importance of understanding the relationship between delta, gamma, and vega in managing derivative positions. This scenario demonstrates the importance of monitoring and adjusting hedges frequently, especially in volatile market conditions.

Let’s consider a scenario where a portfolio manager uses a combination of futures and options to hedge against market volatility. The portfolio’s beta with respect to the FTSE 100 is 1.2. The manager wants to protect the portfolio’s value against a potential market downturn over the next three months. The current value of the FTSE 100 is 7500, and the portfolio is valued at £1,000,000. The manager decides to use FTSE 100 futures contracts for hedging. Each futures contract represents £10 per index point. First, we need to calculate the number of futures contracts required to hedge the portfolio. The formula is: Number of contracts = (Portfolio Value * Portfolio Beta) / (Futures Price * Contract Size). In this case, the futures price is assumed to be the same as the index value, 7500. Therefore: Number of contracts = (£1,000,000 * 1.2) / (7500 * £10) = 1200000 / 75000 = 16 contracts. Now, let’s introduce options into the hedging strategy. The manager also buys put options on the FTSE 100 with a strike price of 7400, expiring in three months. The put options cost £5 per option, and each option controls one index unit. This strategy creates a protective put. The manager wants to determine the breakeven point for this combined futures and options hedge. The total cost of the put options is 7400 put options * £5 = £37,000. To calculate the breakeven point, we need to consider the initial portfolio value, the futures hedge, and the cost of the put options. The futures hedge effectively locks in a value based on the initial futures price. The put options provide downside protection below the strike price. The breakeven point is where the combined strategy neither gains nor loses money. The futures hedge provides a hedge ratio of 1.2, meaning the portfolio’s value changes by 1.2 times the change in the FTSE 100. The put options protect against losses below 7400. The breakeven point calculation is complex and depends on several factors, including the correlation between the portfolio and the FTSE 100, the volatility of the FTSE 100, and the time to expiration of the options. The key is to understand that the futures hedge provides a general market hedge, while the put options provide specific downside protection. The combination aims to reduce overall portfolio volatility and limit potential losses. The breakeven point is a critical metric for assessing the effectiveness of this combined hedging strategy. It represents the index level at which the portfolio’s gains from the futures hedge offset the cost of the put options and any potential losses in the underlying portfolio.

The question revolves around the concept of delta hedging a short call option position. Delta hedging involves continuously adjusting the portfolio to maintain a delta-neutral position, which means the portfolio’s value is insensitive to small changes in the underlying asset’s price. The delta of a call option represents the sensitivity of the option’s price to changes in the underlying asset’s price. A short call option has a negative delta, meaning its value decreases as the underlying asset’s price increases. To delta hedge a short call, an investor needs to buy shares of the underlying asset. The number of shares to buy is equal to the absolute value of the option’s delta multiplied by the number of options written. In this scenario, the investor initially hedges the position at a lower stock price and a lower delta. As the stock price increases, the option’s delta also increases (becomes less negative). This means the investor needs to buy more shares to maintain the delta-neutral position. The profit or loss from delta hedging comes from the difference between the cost of buying shares and the revenue from selling shares, as the hedge is adjusted over time. We calculate the cost of the initial hedge and subsequent adjustments to determine the overall profit or loss. Initial hedge: Delta = 0.4, Shares = 10,000 * 0.4 = 4,000 shares. Cost = 4,000 * £100 = £400,000. Delta changes to 0.6: Additional shares needed = 10,000 * (0.6 – 0.4) = 2,000 shares. Cost = 2,000 * £105 = £210,000. Total cost of hedging = £400,000 + £210,000 = £610,000. Now, consider the payoff of the short call option. The option is in the money, so the option holder will exercise. The investor must deliver 10,000 shares at a strike price of £100. The investor can buy 10,000 shares at £105 and deliver them, so the cost is 10,000 * £105 = £1,050,000 Proceeds from selling the shares at strike price = 10,000 * £100 = £1,000,000 Net loss on the option = £1,050,000 – £1,000,000 = £50,000 Premium received = £6 per option * 10,000 options = £60,000 Profit/Loss from delta hedging = Premium received – Cost of hedging – Net loss on the option = £60,000 – £610,000 + £50,000 = -£500,000. The correct answer is a loss of £500,000.

The payoff of a European call option is max(S – K, 0), where S is the spot price at expiration and K is the strike price. The profit is the payoff minus the initial premium paid for the option. In this case, the spot price at expiration is 115, and the strike price is 110. The payoff is max(115 – 110, 0) = 5. The profit is 5 – 3 = 2. Now, let’s consider the put option. The payoff of a European put option is max(K – S, 0), where S is the spot price at expiration and K is the strike price. The profit is the payoff minus the initial premium paid for the option. In this case, the spot price at expiration is 115, and the strike price is 110. The payoff is max(110 – 115, 0) = 0. The profit is 0 – 2 = -2. The combined profit is the sum of the profits from the call and put options, which is 2 + (-2) = 0. This scenario demonstrates a simplified straddle strategy, where an investor buys both a call and a put option with the same strike price and expiration date. The investor profits if the price of the underlying asset moves significantly in either direction. However, in this specific case, the price movement was not sufficient to cover the combined premiums paid for both options. The investor’s break-even points would be the strike price plus the sum of the premiums (110 + 3 + 2 = 115) on the upside and the strike price minus the sum of the premiums (110 – 3 – 2 = 105) on the downside. Since the final price was exactly at the upper break-even point, the combined profit is zero. This illustrates the risk-reward profile of a straddle, where limited profit is possible with significant price movement, and losses are capped at the total premium paid. This is a very simplified view, as volatility and time decay also play a role in real-world scenarios.

The question assesses the understanding of option pricing sensitivities, specifically Delta and Gamma, and how they interact in a portfolio. Delta represents the change in the option price for a unit change in the underlying asset’s price. Gamma, on the other hand, represents the change in Delta for a unit change in the underlying asset’s price. The portfolio’s initial Delta is calculated by summing the Deltas of all the positions. A positive Delta means the portfolio is long the underlying asset (or a derivative that behaves like it), while a negative Delta means the portfolio is short the underlying asset. The initial portfolio Delta is: (100 * 0.6) + (50 * -0.4) + (200 * 0.1) = 60 – 20 + 20 = 60. The portfolio’s initial Gamma is calculated similarly: (100 * 0.02) + (50 * 0.01) + (200 * 0.005) = 2 + 0.5 + 1 = 3.5. If the underlying asset price increases by £1, the Delta of each option position will change based on its Gamma. The new Delta of the portfolio is calculated by adding the change in Delta (Gamma * change in price) to the initial Delta. The new Delta is: 60 + (3.5 * 1) = 63.5. Therefore, the portfolio’s Delta after a £1 increase in the underlying asset price is 63.5. This means that for every £1 increase in the underlying asset’s price, the portfolio’s value is expected to increase by £63.5. This example uses unique numerical values and a specific portfolio composition to test the candidate’s ability to apply the concepts of Delta and Gamma in a practical scenario. The question goes beyond simple definitions and requires an understanding of how these sensitivities interact within a portfolio context.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their valuation sensitivity to volatility. The investor’s view on future volatility is crucial in determining whether a barrier option is suitable. A standard Black-Scholes model is used to price the option. The formula for the Black-Scholes model is not explicitly needed to answer the question, but an understanding of how volatility affects option prices is essential. The calculation involves understanding that an increased volatility will increase the option price and vice versa. The scenario involves a knock-out barrier option, which ceases to exist if the underlying asset’s price reaches a specific barrier level. In this case, the barrier is set at 90% of the initial asset price. If the investor believes that volatility will decrease, the probability of the asset price reaching the barrier level decreases, and the value of the knock-out option decreases as well. Conversely, if the investor anticipates increased volatility, the probability of hitting the barrier increases, further decreasing the value of the knock-out option because the option is more likely to expire worthless. The investor’s view on volatility should align with the characteristics of the barrier option. If the investor expects a decrease in volatility, the barrier option becomes less attractive because the potential payoff is reduced. Conversely, if the investor expects an increase in volatility, the barrier option becomes less attractive because the probability of hitting the barrier increases, causing the option to expire worthless. Therefore, the investor’s outlook on volatility should be carefully considered when recommending a barrier option. The suitability assessment should take into account the investor’s risk tolerance, investment objectives, and understanding of the potential risks and rewards associated with barrier options.