Derivatives Level 4 (Investment Advice Diploma) Exam Quiz Quiz 25

To determine the profit or loss from the combined options strategy, we need to analyze the outcomes at the expiration date based on the stock price. The strategy involves selling a call option and buying a put option with different strike prices. 1. **Break-even Calculation**: The investor profits when the stock price is between the two strike prices. The profit is capped by the difference in strike prices minus the net premium received. The investor incurs a loss if the stock price moves outside this range. 2. **Scenario Analysis**: We evaluate the profit/loss at different stock prices: * **Stock Price at £140**: The call option is in the money and will be exercised. The put option expires worthless. The investor loses £10 (150-140) on the call option but gains the initial premium of £15. The net profit is £5. * **Stock Price at £160**: The call option is in the money and will be exercised. The put option expires worthless. The investor loses £10 (160-150) on the call option but gains the initial premium of £15. The net profit is -£5. * **Stock Price at £150**: Both options expire worthless. The investor keeps the initial premium of £15. * **Stock Price at £130**: The call option expires worthless. The put option is in the money and will be exercised. The investor gains £20 (150-130) on the put option but loses the initial premium of £15. The net profit is £5. 3. **Profit/Loss Calculation at £160**: * Call Option: Strike Price = £150, Stock Price = £160. The investor loses £10 (£160 – £150) because they have to sell the stock at £150 when it’s worth £160. * Put Option: Strike Price = £130, Stock Price = £160. The put option expires worthless. * Net Premium: The investor received £15 initially. * Total Profit/Loss: -£10 + £15 = £5. 4. **Example**: Consider a similar strategy involving selling a call option with a strike price of £100 and buying a put option with a strike price of £80. If the stock price at expiration is £110, the call option results in a loss of £10, while the put option expires worthless. If the initial premium received was £5, the net loss would be £5. 5. **Analogy**: Imagine a farmer who sells a contract to deliver wheat at £5 per bushel (selling a call) and buys insurance to sell wheat at £4 per bushel if the market price drops (buying a put). If the market price rises to £6, the farmer loses £1 on the contract but keeps the insurance premium.

To solve this problem, we need to understand how delta hedging works and how changes in the underlying asset’s price and volatility affect the hedge. Delta hedging aims to neutralize the portfolio’s sensitivity to small changes in the underlying asset’s price. The delta of a call option measures this sensitivity. Gamma, on the other hand, measures the rate of change of the delta with respect to changes in the underlying asset’s price. Vega measures the sensitivity of the option’s price to changes in volatility. Initially, the portfolio is delta neutral, meaning the portfolio’s value is not affected by small changes in the underlying asset’s price. However, as the underlying asset’s price and volatility change, the delta changes, and the portfolio is no longer delta neutral. The fund manager needs to rebalance the hedge to maintain delta neutrality. First, calculate the change in the portfolio’s delta due to the change in the underlying asset’s price. This is given by the gamma multiplied by the change in the asset price: \[ \text{Change in Delta due to Price} = \text{Gamma} \times \text{Change in Price} = 0.002 \times (155 – 150) = 0.01 \] Next, calculate the change in the portfolio’s delta due to the change in volatility. This is given by the vega multiplied by the change in volatility: \[ \text{Change in Delta due to Volatility} = \text{Vega} \times \text{Change in Volatility} = -0.005 \times (0.18 – 0.20) = 0.0001 \] The total change in delta is the sum of these two changes: \[ \text{Total Change in Delta} = 0.01 + 0.0001 = 0.0101 \] Since the portfolio was initially delta neutral, the new delta of the portfolio is 0.0101. This means the portfolio’s value will increase by 0.0101 for every $1 increase in the underlying asset’s price. To rebalance the hedge, the fund manager needs to sell call options to reduce the portfolio’s delta back to zero. The number of call options to sell is determined by dividing the total change in delta by the delta of each call option: \[ \text{Number of Call Options to Sell} = \frac{\text{Total Change in Delta}}{\text{Delta of Each Call Option}} = \frac{0.0101}{0.50} = 0.0202 \] Since the fund manager needs to trade in lots of 10,000 options, we multiply the result by 10,000: \[ \text{Number of Options to Sell} = 0.0202 \times 10,000 = 202 \] Therefore, the fund manager should sell 202 call options to rebalance the hedge.

Let’s consider a scenario involving a UK-based agricultural cooperative, “GreenHarvest,” which produces organic wheat. GreenHarvest wants to protect itself against a potential drop in wheat prices over the next six months. They decide to use wheat futures contracts traded on the ICE Futures Europe exchange to hedge their price risk. The cooperative plans to harvest 500,000 bushels of wheat. Each futures contract covers 5,000 bushels. To determine the number of contracts needed, GreenHarvest divides their total production by the contract size: 500,000 bushels / 5,000 bushels/contract = 100 contracts. Now, let’s assume the current futures price for wheat for delivery in six months is £6.00 per bushel. GreenHarvest sells 100 futures contracts at this price, effectively locking in a price of £6.00 per bushel for their wheat. This is a short hedge. Six months later, at the delivery date, the spot price of wheat has fallen to £5.50 per bushel. GreenHarvest sells their wheat in the spot market for £5.50 per bushel, receiving £2,750,000 (500,000 bushels * £5.50/bushel). Simultaneously, they close out their futures position by buying back 100 futures contracts at £5.50 per bushel. This generates a profit on the futures contracts: (£6.00 – £5.50) * 5,000 bushels/contract * 100 contracts = £250,000. The total revenue for GreenHarvest is the sum of the spot market revenue and the futures profit: £2,750,000 + £250,000 = £3,000,000. This is equivalent to selling all their wheat at the initial futures price of £6.00 per bushel (500,000 bushels * £6.00/bushel = £3,000,000). However, hedging isn’t without risks. Basis risk arises because the spot price and futures price may not move perfectly in tandem. Also, GreenHarvest forgoes the opportunity to profit if the spot price of wheat rises above £6.00 per bushel. Margin calls also need to be considered. If the futures price rises instead of falls, GreenHarvest would face margin calls, requiring them to deposit additional funds to maintain their position. In summary, hedging with futures contracts allows GreenHarvest to mitigate the risk of falling wheat prices, providing price certainty. The calculation involves determining the number of contracts, executing the hedge, and calculating the profit or loss on the futures position to offset the spot market price fluctuations.

The core of this question lies in understanding how delta hedging works in practice, especially when transaction costs are involved. Delta hedging aims to neutralize the directional risk of an option position by continuously adjusting the underlying asset holding. The delta of an option represents the sensitivity of the option’s price to a change in the price of the underlying asset. For a call option, the delta ranges from 0 to 1. A delta of 0.6 means that for every £1 increase in the underlying asset’s price, the call option’s price is expected to increase by £0.60. Without transaction costs, the hedging process is straightforward: if you are short a call option, you buy delta shares of the underlying asset. As the asset price changes, you adjust your holdings to maintain delta neutrality. However, transaction costs complicate this process. Each adjustment incurs a cost, which reduces the profitability of the hedge. The key is to determine the optimal rebalancing frequency, which balances the cost of frequent trading with the risk of being unhedged for too long. A common strategy is to set a threshold for the acceptable deviation from delta neutrality. This threshold is often expressed in monetary terms. When the potential profit from rebalancing exceeds the transaction costs, rebalancing is justified. In this scenario, we need to calculate the profit/loss from the option and the underlying asset, considering the transaction costs. The initial delta hedge requires buying 60 shares at £50 each, costing £3,000. When the price increases to £51, the delta changes to 0.65, requiring an additional 5 shares to be bought, costing £51 plus a transaction fee. The profit from the option is £0.60 for each £1 increase in the underlying asset price. The profit from the underlying asset is £1 per share for the initial 60 shares and £1 per share for the additional 5 shares. The total transaction costs are £10. By comparing the profit from the option and the underlying asset with the transaction costs, we can determine the net profit or loss. Calculation: Initial hedge: Buy 60 shares at £50 = £3000 Price increases to £51, delta changes to 0.65, so we need 65 shares. Buy additional 5 shares at £51 = £255 Transaction cost = £10 Total cost of hedge = £3000 + £255 + £10 = £3265 Profit from the option: £0.60 per share * 65 shares = £39 Profit from the underlying asset: £1 per share * 65 shares = £65 Total profit = £39 + £65 = £104 Net profit/loss = Total profit – Total cost of hedge = £104 – £3265 = -£3161 However, this is not the correct approach. Correct Approach: Initial hedge: Buy 60 shares at £50 = £3000 Price increases to £51, delta changes to 0.65, so we need 65 shares. Buy additional 5 shares at £51 = £255 Transaction cost = £10 Total cost of hedge = £255 + £10 = £265 Profit from the option: £0.60 per share * 60 shares = £36 Profit from the underlying asset: £1 per share * 60 shares = £60 Profit from the underlying asset: £1 per share * 5 shares = £5 Total profit = £36 + £60 + £5 = £101 Net profit/loss = Total profit – Total cost of hedge = £101 – £265 = -£164 Profit from the option: 0.60 * (51 – 50) = £0.60 Profit from the 60 shares: 60 * (51 – 50) = £60 Cost of buying 5 shares: 5 * 51 = £255 Transaction costs: £10 Total cost: £255 + £10 = £265 Total profit: £60 + £0.60 = £60.60 Net: £60.60 – £265 = -£204.40 New Calculation: Delta change: 0.65 – 0.60 = 0.05 Shares to buy: 5 Cost of buying shares: 5 * £51 = £255 Transaction cost: £10 Option Profit: 0.60 * £1 = £0.60 Share Profit: 60 * £1 = £60 Total Profit: £60.60 Total Cost: £255 + £10 = £265 Net: £60.60 – £265 = -£204.40

The core of this question lies in understanding how delta hedging works and how transaction costs erode the profits from rebalancing a portfolio. Delta, in this context, represents the sensitivity of the option’s price to changes in the underlying asset’s price. A delta-neutral portfolio is designed to be insensitive to small price movements in the underlying asset. This is achieved by holding a number of shares of the underlying asset that offsets the delta of the option position. However, maintaining delta neutrality requires continuous rebalancing as the underlying asset’s price changes and as time passes (delta changes with time – this is often referred to as “gamma”). Each rebalancing incurs transaction costs, such as brokerage fees or bid-ask spreads. The problem requires us to calculate the profit from the option and the offsetting stock position, then subtract the total transaction costs incurred during the rebalancing process. The initial delta of 0.5 implies that for every £1 increase in the share price, the call option’s price will increase by £0.50. The portfolio is constructed to be delta-neutral by shorting 500 shares initially (since 1000 options * delta of 0.5 = 500 shares). When the share price increases from £10 to £12, the call option’s price increases by £2, and the delta increases to 0.7. The profit on the options is £2 * 1000 = £2000. The loss on the short stock position is (£12 – £10) * 500 = £1000. The initial profit before rebalancing is £2000 – £1000 = £1000. To rebalance, the portfolio needs to short an additional (0.7 – 0.5) * 1000 = 200 shares. The cost of this transaction is 200 shares * £0.10 = £20. When the share price decreases from £12 to £11, the call option’s price decreases by £1, and the delta decreases to 0.3. The loss on the options is £1 * 1000 = £1000. The profit on the short stock position is (£12 – £11) * 700 = £700. The profit before rebalancing is -£1000 + £700 = -£300. To rebalance, the portfolio needs to buy back (0.7 – 0.3) * 1000 = 400 shares. The cost of this transaction is 400 shares * £0.10 = £40. When the share price decreases from £11 to £9, the call option’s price decreases by £2, and the delta decreases to 0. The loss on the options is £2 * 1000 = £2000. The profit on the short stock position is (£11 – £9) * 700 = £1400. The profit before rebalancing is -£2000 + £1400 = -£600. To rebalance, the portfolio needs to buy back 300 shares. The cost of this transaction is 300 shares * £0.10 = £30. Final Calculation: Initial Profit: £1000 Rebalancing cost 1: £20 Loss: -£300 Rebalancing cost 2: £40 Loss: -£600 Rebalancing cost 3: £30 Total profit = £1000 – £300 – £600 – £20 – £40 – £30 = £7

The core of this question revolves around understanding how macroeconomic announcements impact derivative pricing, specifically focusing on the interplay between inflation expectations, interest rate futures, and currency swaps. The scenario involves a surprise announcement that significantly deviates from market consensus, creating a shock to the yield curve and subsequent adjustments in derivative valuations. The calculation involves several steps. First, we need to determine the change in the implied forward rate derived from the interest rate futures contract. The futures price change translates into a basis point change in the implied yield. Since the futures contract is based on a 3-month interest rate, we need to annualize this change. The formula to calculate the change in implied forward rate is: Change in Implied Forward Rate = (Change in Futures Price) * (100 basis points / 1 point) * (Annualization Factor) In this case, the futures price increased by 0.75 points, which translates to 75 basis points. Since the contract is based on a 3-month rate, the annualization factor is 4 (12 months / 3 months). Change in Implied Forward Rate = 0.75 * 100 * 4 = 300 basis points or 3.00% Next, we need to understand how this change in the implied forward rate affects the currency swap. In a currency swap, the parties exchange principal and interest payments in different currencies. The change in the implied forward rate will primarily affect the interest rate leg denominated in the currency linked to the interest rate futures (in this case, GBP). The present value of the future cash flows of this leg will decrease due to the higher discount rate. To approximate the impact on the swap’s present value, we can use the concept of duration. The duration of a swap leg represents the sensitivity of its present value to changes in interest rates. We are given a modified duration of 3.5 years for the GBP leg. Change in Present Value ≈ – (Modified Duration) * (Change in Yield) * (Initial Present Value of GBP Leg) Change in Present Value ≈ – (3.5) * (0.03) * (£50 million) = – £5.25 million The negative sign indicates a decrease in the present value of the GBP leg. The overall impact on the swap’s value will depend on the relative changes in the present values of both currency legs (GBP and USD). Since we are only given information about the GBP leg and assuming the USD leg remains relatively stable, the change in the swap’s value is primarily driven by the GBP leg. Therefore, the value of the swap decreases by approximately £5.25 million. This calculation highlights the interconnectedness of different derivative markets. A shock in one market (interest rate futures) can propagate to other markets (currency swaps) through changes in expectations and discount rates. Understanding these relationships is crucial for managing risk and making informed investment decisions in a derivatives portfolio. The use of duration provides a practical way to estimate the sensitivity of swap values to interest rate changes, enabling portfolio managers to quickly assess the impact of macroeconomic events on their positions.

Let’s consider a scenario where a UK-based agricultural cooperative, “BritCrops,” needs to hedge against potential price declines in their upcoming wheat harvest. They are considering using wheat futures contracts traded on the London International Financial Futures and Options Exchange (LIFFE), now part of ICE Futures Europe. BritCrops plans to harvest 500,000 bushels of wheat in three months. The current futures price for wheat with delivery in three months is £5.00 per bushel. BritCrops decides to sell 100 futures contracts, each representing 5,000 bushels of wheat (100 contracts * 5,000 bushels/contract = 500,000 bushels). Hedging with futures involves offsetting the risk of price fluctuations in the spot market (the actual market for physical wheat) by taking an opposite position in the futures market. BritCrops is concerned that the price of wheat will fall before their harvest, so they “short” (sell) futures contracts. If the spot price of wheat declines, the loss in the physical wheat sale will be partially or fully offset by the gain in the futures market as they can buy back the futures contracts at a lower price. Conversely, if the spot price rises, the gain in the physical wheat sale will be reduced by the loss in the futures market. Basis risk is the risk that the price relationship between the spot market and the futures market will change over time. The basis is calculated as the spot price minus the futures price. Basis risk arises because the spot and futures prices are not perfectly correlated due to factors such as transportation costs, storage costs, and local supply and demand conditions. Let’s assume that in three months, when BritCrops harvests and sells their wheat, the spot price has fallen to £4.75 per bushel. At the same time, the futures price has fallen to £4.80 per bushel. BritCrops sells their 500,000 bushels of wheat in the spot market for £4.75 per bushel, receiving £2,375,000 (500,000 bushels * £4.75/bushel). In the futures market, BritCrops buys back their 100 futures contracts at £4.80 per bushel. They initially sold the contracts at £5.00 per bushel, so they make a profit of £0.20 per bushel on each contract. This profit totals £100,000 (100 contracts * 5,000 bushels/contract * £0.20/bushel). The effective price BritCrops receives for their wheat is the spot price plus the profit from the futures market: £2,375,000 + £100,000 = £2,475,000. This is equivalent to an effective price of £4.95 per bushel (£2,475,000 / 500,000 bushels). Without hedging, BritCrops would have received only £4.75 per bushel. The hedge was not perfect, as they aimed to lock in £5.00 per bushel, but it mitigated the impact of the price decline. The difference between the intended price (£5.00) and the actual effective price (£4.95) is due to basis risk. The basis narrowed from -£0.25 (£4.75 – £5.00) initially to -£0.05 (£4.80 – £4.75) at the time of harvest. This narrowing of the basis reduced the effectiveness of the hedge.

This question explores the complexities of managing a portfolio with derivatives, focusing on the interplay between Value at Risk (VaR), Expected Shortfall (ES), and scenario analysis. VaR estimates the potential loss in value of a risky asset or portfolio over a defined period for a given confidence level. However, VaR has limitations. It does not describe the magnitude of losses exceeding the VaR level. Expected Shortfall (ES), also known as Conditional VaR (CVaR), addresses this by estimating the expected loss given that the VaR level has been exceeded. Scenario analysis involves creating hypothetical future events and analyzing their potential impact on a portfolio. Stress testing is a form of scenario analysis that examines the impact of extreme, but plausible, events. The correct approach involves recognizing that derivatives can be used to hedge against specific risks identified through scenario analysis. Reducing the portfolio’s exposure to these risks will lower both VaR and ES. However, the *relative* change in VaR and ES depends on the shape of the loss distribution. If the hedging strategy primarily targets the tail risk (extreme losses), ES will decrease more than VaR. If the hedging strategy reduces overall volatility, VaR and ES will decrease proportionally. The question requires understanding the *relationship* between VaR, ES, and scenario analysis, not just their individual definitions. It tests the ability to connect hedging strategies to specific risk measures and to interpret the implications of changes in these measures. The key is that scenario analysis informs the hedging strategy, and the effectiveness of that strategy will be reflected differently in VaR and ES depending on the risk profile of the initial portfolio and the specific risks targeted by the hedge. Let’s consider a portfolio with a significant exposure to geopolitical risk. Scenario analysis reveals that a major political event could lead to a sharp decline in specific asset classes. A hedging strategy using options could be implemented to protect against this tail risk. This strategy would likely reduce ES more significantly than VaR, as it specifically addresses the potential for extreme losses identified in the scenario analysis. Conversely, a strategy that diversifies the portfolio across different asset classes might reduce overall volatility, leading to a more proportional decrease in VaR and ES.

The question assesses understanding of hedging strategies using futures contracts, specifically focusing on managing basis risk. Basis risk arises because the price of the asset being hedged (in this case, copper) may not move perfectly in tandem with the price of the futures contract. The formula to determine the number of futures contracts needed to hedge is: Number of contracts = (Portfolio Value / Futures Contract Value) * Hedge Ratio The hedge ratio attempts to adjust for the imperfect correlation. In this case, the refining company wants to hedge against a price decrease, so they will short (sell) futures contracts. We must calculate the optimal number of contracts considering the basis risk and contract specifications. The example highlights the practical challenges of hedging in real-world scenarios where perfect hedges are rarely achievable. The hedge ratio is calculated as: Hedge Ratio = Change in Spot Price / Change in Futures Price A hedge ratio of 0.8 indicates that for every £1 change in the spot price of copper, the futures price changes by £0.8. This is crucial for determining the effective hedge. The calculation then incorporates the contract size to determine the specific number of contracts required. Number of contracts = (10,000 tonnes * £6,000/tonne) / (25 tonnes/contract * £6,000/tonne) * 0.8 = 320 contracts The company needs to short 320 futures contracts to minimize their exposure to price fluctuations. The incorrect options highlight common mistakes, such as neglecting the hedge ratio, using the wrong sign (buying instead of selling), or misinterpreting the contract size. The scenario illustrates how derivatives are used to manage risk in commodity markets, particularly for companies with significant exposure to price volatility.

The question assesses the understanding of delta hedging in a portfolio context, specifically when the underlying asset’s volatility changes unexpectedly. Delta hedging aims to neutralize the portfolio’s sensitivity to small changes in the underlying asset’s price. However, delta is itself sensitive to changes in the underlying asset’s volatility (vega). When volatility increases, the delta of an option changes, requiring a rebalancing of the hedge. The initial delta hedge is designed to offset the price movement of the underlying asset. An increase in volatility will cause the delta of the options to change. Since the portfolio is short options, an increase in volatility will cause the delta to become more negative (for puts) or more positive (for calls). To maintain a delta-neutral position, the portfolio manager must adjust the hedge by buying or selling the underlying asset. In this scenario, the portfolio manager is short options. When volatility increases, the absolute value of the options’ delta increases. To rebalance the hedge and maintain delta neutrality, the portfolio manager needs to reduce the portfolio’s exposure to the underlying asset. Since the portfolio is short options, and the absolute value of the delta has increased, the manager needs to sell the underlying asset to offset the increased delta exposure. Calculation: 1. Initial Portfolio: Short options, Delta-hedged. 2. Volatility Increase: Options’ delta becomes more sensitive. 3. Portfolio Adjustment: To remain delta-neutral, the manager must offset the increased delta. 4. Hedge Adjustment: Since short options, the manager must sell the underlying asset. Therefore, the portfolio manager should sell shares of the underlying asset to maintain a delta-neutral position.

The question revolves around calculating the theoretical price of a European call option using the Black-Scholes model and then assessing the impact of a dividend payment before the option’s expiration. The Black-Scholes model is a cornerstone of options pricing, but it assumes no dividends are paid during the option’s life. When dividends are expected, the stock price needs to be adjusted downward to reflect the present value of those dividends. This adjustment directly impacts the ‘S’ (current stock price) input in the Black-Scholes formula. First, we calculate the initial option price without considering the dividend. The Black-Scholes formula is: \[C = S \cdot N(d_1) – X \cdot e^{-rT} \cdot N(d_2)\] Where: * \(C\) = Call option price * \(S\) = Current stock price * \(X\) = Strike price * \(r\) = Risk-free interest rate * \(T\) = Time to expiration (in years) * \(N(x)\) = Cumulative standard normal distribution function * \(e\) = the exponential constant (approximately 2.71828) * \(d_1 = \frac{ln(\frac{S}{X}) + (r + \frac{\sigma^2}{2})T}{\sigma \sqrt{T}}\) * \(d_2 = d_1 – \sigma \sqrt{T}\) * \(\sigma\) = Volatility of the stock Given: \(S = 55\), \(X = 50\), \(r = 0.05\), \(T = 0.5\), \(\sigma = 0.30\) \[d_1 = \frac{ln(\frac{55}{50}) + (0.05 + \frac{0.30^2}{2})0.5}{0.30 \sqrt{0.5}} = \frac{0.0953 + (0.05 + 0.045)0.5}{0.30 \cdot 0.7071} = \frac{0.0953 + 0.0475}{0.2121} = 0.6738\] \[d_2 = 0.6738 – 0.30 \sqrt{0.5} = 0.6738 – 0.2121 = 0.4617\] Using standard normal distribution tables (or a calculator with the function), \(N(0.6738) \approx 0.7498\) and \(N(0.4617) \approx 0.6778\). \[C = 55 \cdot 0.7498 – 50 \cdot e^{-0.05 \cdot 0.5} \cdot 0.6778 = 41.239 – 50 \cdot e^{-0.025} \cdot 0.6778 = 41.239 – 50 \cdot 0.9753 \cdot 0.6778 = 41.239 – 33.068 = 8.171\] Next, we adjust the stock price for the present value of the dividend. The dividend of £2 is paid in 3 months (0.25 years). Present Value of Dividend = \[2 \cdot e^{-0.05 \cdot 0.25} = 2 \cdot e^{-0.0125} = 2 \cdot 0.9876 = 1.9752\] Adjusted Stock Price = \(55 – 1.9752 = 53.0248\) Now, recalculate \(d_1\) and \(d_2\) using the adjusted stock price: \[d_1 = \frac{ln(\frac{53.0248}{50}) + (0.05 + \frac{0.30^2}{2})0.5}{0.30 \sqrt{0.5}} = \frac{0.0585 + 0.0475}{0.2121} = \frac{0.106}{0.2121} = 0.500\] \[d_2 = 0.500 – 0.2121 = 0.2879\] \(N(0.500) = 0.6915\) and \(N(0.2879) = 0.6134\) \[C = 53.0248 \cdot 0.6915 – 50 \cdot e^{-0.025} \cdot 0.6134 = 36.666 – 50 \cdot 0.9753 \cdot 0.6134 = 36.666 – 29.935 = 6.731\] The difference between the initial call option price and the dividend-adjusted call option price is: \(8.171 – 6.731 = 1.44\) Therefore, the dividend is expected to decrease the call option price by approximately £1.44.

The core concept here revolves around understanding how macroeconomic indicators, specifically inflation expectations, influence the pricing of inflation-protected bonds (linkers) and, consequently, the breakeven inflation rate. Breakeven inflation (BEI) is the difference between the yield of a nominal bond and the yield of an inflation-linked bond of the same maturity. It represents the market’s expectation of average inflation over the bond’s life. An increase in inflation expectations generally leads to a decrease in the real yield of inflation-linked bonds. Investors demand a lower real yield because the inflation adjustment component of the bond’s return is expected to be higher, compensating them for inflation. This decreased real yield directly affects the BEI calculation. Consider a scenario where a 10-year nominal bond yields 4.0% and a 10-year inflation-linked bond yields 1.5%. The breakeven inflation rate is initially 2.5% (4.0% – 1.5%). Now, suppose inflation expectations rise by 0.5%. This increase will likely cause the real yield on the inflation-linked bond to decrease. Let’s assume it decreases to 1.0%. The new breakeven inflation rate would then be 3.0% (4.0% – 1.0%). This demonstrates the direct relationship between rising inflation expectations and an increasing breakeven inflation rate. The scenario also introduces the concept of the “flight to safety.” In times of economic uncertainty, investors often seek safe-haven assets, such as government bonds. This increased demand for nominal bonds can drive their yields down. However, if inflation expectations are simultaneously rising, the decrease in nominal yields might be less pronounced than the decrease in real yields of inflation-linked bonds, further widening the breakeven inflation rate. The calculation involves subtracting the new real yield of the inflation-linked bond from the nominal yield of the government bond to arrive at the new breakeven inflation rate. This difference directly reflects the market’s updated inflation expectations.

The question assesses the understanding of exotic options, specifically barrier options, and their sensitivity to market volatility, particularly in the context of regulatory capital requirements for investment firms under UK regulations like the Capital Requirements Regulation (CRR) as implemented by the Prudential Regulation Authority (PRA). The core concept is that increased volatility can dramatically affect the probability of a barrier being breached, thereby impacting the value of the barrier option and, consequently, the capital an investment firm needs to hold against potential losses. The calculation involves understanding that a knock-out barrier option loses its value if the underlying asset price hits the barrier. Higher volatility increases the likelihood of this happening. An investment firm holding such an option as part of a structured product offering to clients must account for this increased risk. The capital charge is calculated as a percentage of the notional value, adjusted for the delta (sensitivity to price changes) and gamma (sensitivity to delta changes) of the option. In this scenario, the initial capital charge is 5% of the £10 million notional value, which is £500,000. The volatility increase from 15% to 25% raises the probability of the barrier being breached. This necessitates a recalculation of the capital charge, considering the increased risk. The gamma increases, reflecting the heightened sensitivity to price fluctuations near the barrier. The new capital charge is estimated to be 8% of the notional value due to the volatility increase and its impact on the option’s risk profile, resulting in £800,000. The additional capital required is the difference between the new and initial capital charges: £800,000 – £500,000 = £300,000. The analogy here is like driving a car near a cliff edge (the barrier). Higher volatility is like driving in fog or on ice; the risk of going over the edge (breaching the barrier) increases significantly, requiring more caution (higher capital reserves). The unique application is understanding how regulatory bodies like the PRA in the UK mandate capital adequacy based on derivative risk profiles, which are dynamically affected by market conditions. The novel problem-solving approach involves recognizing the direct link between volatility, barrier option valuation, and regulatory capital requirements, demonstrating an understanding beyond textbook definitions.

The question assesses the understanding of delta hedging, specifically in the context of portfolio management and risk mitigation. Delta, a measure of an option’s price sensitivity to changes in the underlying asset’s price, is crucial in hedging strategies. A delta-neutral portfolio is constructed to be insensitive to small movements in the underlying asset’s price. This is achieved by balancing the portfolio’s delta (the sum of the deltas of all assets within the portfolio, including options) to zero. The calculation involves determining the number of shares required to offset the delta of the options position. In this scenario, a portfolio manager holds call options with a specific delta and needs to calculate the number of shares required to create a delta-neutral portfolio. The formula used is: Number of Shares = – (Delta of Options Position) * (Number of Options Contracts) * (Shares per Contract) A negative sign is used because the shares are used to offset the delta of the options. For instance, if the options have a positive delta, the shares must have a negative delta (short position) to neutralize the portfolio. In our example, the portfolio manager holds 100 call option contracts, each representing 100 shares. The delta of each call option is 0.6. Therefore, the total delta of the options position is 0.6 * 100 * 100 = 6000. To neutralize this, the portfolio manager needs to short 6000 shares. The concept can be visualized using a seesaw analogy. The options position, with its positive delta, is like a weight on one side of the seesaw. To balance the seesaw (achieve delta neutrality), we need to place an equal and opposite weight (the short shares) on the other side. The question also touches on the regulatory aspects. According to the FCA (Financial Conduct Authority) guidelines, portfolio managers have a responsibility to manage risks effectively. Failing to implement appropriate hedging strategies, especially when managing client funds, can lead to regulatory scrutiny and potential penalties. The FCA emphasizes the importance of understanding and managing risks associated with derivatives, including delta, gamma, and vega. Portfolio managers must demonstrate that they have the necessary expertise and resources to manage these risks effectively.

The question assesses understanding of the impact of macroeconomic announcements on derivative pricing, specifically focusing on interest rate futures. The key is to recognize how unexpected changes in inflation data influence expectations about future central bank policy (Bank of England in this case) and, consequently, interest rates. An unexpected increase in inflation typically leads to expectations of tighter monetary policy (higher interest rates) by the central bank. This expectation will decrease the price of interest rate futures, as these futures contracts are essentially agreements to lend or borrow money at a future date. Higher expected interest rates make borrowing less attractive and lending more attractive, decreasing the future’s price. The calculation of the impact involves understanding that interest rate futures prices are quoted as 100 minus the implied interest rate. A change in the implied interest rate translates directly into an inverse change in the futures price. The formula to calculate the change in the futures price is: Change in Futures Price = – (Change in Implied Interest Rate) * Contract Value * Duration In this scenario, the change in the implied interest rate is the difference between the expected inflation and the actual inflation, which is 0.5%. The contract value is £500,000, and the duration is assumed to be 1 year (as it is a short-term interest rate future). Change in Futures Price = – (0.005) * £500,000 * 1 = -£2,500 Therefore, the futures price will decrease by £2,500. To further illustrate, imagine a scenario where a farmer uses interest rate futures to hedge against rising borrowing costs. The farmer anticipates taking out a loan in six months to finance the next planting season. To protect against an increase in interest rates, the farmer sells interest rate futures contracts. If inflation unexpectedly rises, as in the question, the futures contracts decrease in value. The farmer can then buy back these contracts at a lower price, offsetting the potential increase in borrowing costs when the loan is actually taken out. The profit from the futures contracts helps to mitigate the impact of higher interest rates on the loan. This demonstrates how interest rate futures are used to manage interest rate risk in real-world situations. The unexpected inflation acts as a catalyst, forcing the farmer to realize the hedging benefit sooner than anticipated.

The core of this question lies in understanding how delta hedging works and how transaction costs impact its effectiveness. Delta hedging aims to create a portfolio that is insensitive to small changes in the underlying asset’s price. The delta of an option measures the sensitivity of the option’s price to changes in the underlying asset’s price. To delta hedge, one takes an offsetting position in the underlying asset. However, real-world trading involves transaction costs, which erode the profits from hedging and necessitate more frequent rebalancing. The optimal rebalancing frequency depends on the trade-off between the cost of rebalancing and the risk of being unhedged. In this scenario, the transaction cost is £0.02 per share. Let’s calculate the profit/loss with and without transaction costs. Initial Delta: 0.6 Initial Stock Price: £100 Number of Shares to Short: 600 (0.6 * 1000) Option Premium Received: £6,000 (6 * 1000) Scenario 1: Stock Price Increases to £101 Without Hedge: Option Loss: Delta * Change in Stock Price * Number of Options = 0.6 * 1 * 1000 = £600 Net Loss: £600 (loss) – £6,000 (premium) = -£5,400 With Initial Hedge: Profit from Short Position: 600 * 1 = £600 Net Profit/Loss: £600 (profit) – £600 (option loss) + £6,000 (premium) = £6,000 Scenario 2: Stock Price Decreases to £99 Without Hedge: Option Profit: 0.6 * 1 * 1000 = £600 Net Profit: £600 (profit) + £6,000 (premium) = £6,600 With Initial Hedge: Loss from Short Position: 600 * 1 = £600 Net Profit/Loss: -£600 (loss) + £600 (option profit) + £6,000 (premium) = £6,000 Now, let’s consider the transaction costs. If the stock price moves to £101, the delta changes, and the portfolio needs rebalancing. Assume the delta increases to 0.62. To rebalance, the trader needs to short an additional 20 shares (0.02 * 1000). The transaction cost is 20 shares * £0.02/share * 2 = £0.8 (buying and selling). If the stock price moves to £99, the delta changes, and the portfolio needs rebalancing. Assume the delta decreases to 0.58. To rebalance, the trader needs to cover 20 shares (0.02 * 1000). The transaction cost is 20 shares * £0.02/share * 2 = £0.8 (buying and selling). The key takeaway is that transaction costs reduce the effectiveness of delta hedging. The optimal strategy involves balancing the cost of frequent rebalancing against the risk of being unhedged. Sophisticated models can be used to determine the optimal rebalancing frequency. In practice, traders often use a “tolerance band” around the target delta, only rebalancing when the actual delta drifts outside this band. This helps to minimize transaction costs while maintaining an acceptable level of risk control.

This question delves into the complexities of hedging a portfolio with options, specifically focusing on the impact of implied volatility changes on the effectiveness of a protective put strategy. A protective put involves buying a put option on an asset already held in a portfolio. The purpose is to limit downside risk. The effectiveness of this hedge is directly tied to the accuracy of implied volatility estimations. When implied volatility increases *after* the protective put is purchased, the value of the put option will increase. This is because higher implied volatility suggests a greater expectation of price fluctuations, making the right to sell the asset at the strike price (the put option) more valuable. Conversely, if implied volatility decreases, the value of the put option will decrease, reducing the effectiveness of the hedge. The key is to understand the relationship between implied volatility and option prices. Implied volatility is essentially the market’s expectation of future price volatility, derived from the option’s price. It is not a direct measure of actual volatility but rather an input into option pricing models like Black-Scholes. Consider a scenario where a portfolio manager anticipates a market correction but underestimates the potential volatility. They buy protective puts based on a lower implied volatility. If a significant market event occurs, and implied volatility spikes dramatically, the put options will increase in value more than initially projected, providing a more substantial hedge than anticipated. Conversely, if the market remains relatively stable and implied volatility declines, the put options will provide less downside protection than expected. The calculation of the hedge’s effectiveness requires understanding how the change in implied volatility affects the put option’s delta (sensitivity to price changes) and gamma (sensitivity of delta to price changes). A higher implied volatility generally increases both delta and gamma for out-of-the-money puts, making the hedge more responsive to price movements. The payoff from the put option must then be compared to the losses in the underlying asset to determine the overall effectiveness of the hedge.

The core of this question lies in understanding how delta changes as the underlying asset’s price moves and how this impacts the hedge ratio in a delta-neutral portfolio. Gamma measures the rate of change of delta with respect to changes in the underlying asset’s price. A higher gamma means the delta is more sensitive to price changes, requiring more frequent rebalancing to maintain a delta-neutral position. The cost of rebalancing is directly related to the transaction costs incurred each time the hedge is adjusted. Here’s how we calculate the approximate change in the number of futures contracts needed: 1. **Calculate the change in delta:** Delta change = Gamma * Change in asset price. In this case, Delta change = 0.04 * £2.50 = 0.1. 2. **Calculate the number of contracts needed per share:** This is the change in delta per share, which we calculated as 0.1. 3. **Calculate the total change in contracts:** Total change = Delta change per share * Number of shares / Contract size. In this case, Total change = 0.1 * 10,000 / 100 = 10 contracts. 4. **Calculate the total transaction cost:** Total cost = Number of contracts * Cost per contract. In this case, Total cost = 10 * £50 = £500. Therefore, the approximate transaction cost is £500. Analogy: Imagine you’re balancing a scale (representing a delta-neutral portfolio). Gamma is like the sensitivity of the scale. A high gamma means even a small breeze (change in asset price) will cause the scale to tip significantly (delta changes rapidly). To keep the scale balanced (delta-neutral), you need to constantly adjust the weights (rebalance the hedge). Each adjustment costs you a small fee (transaction cost). The more sensitive the scale (higher gamma), the more frequently you need to adjust, and the higher your total fees will be. Another example: Consider a tightrope walker (portfolio manager) using a balancing pole (derivatives hedge). Gamma represents how quickly the tightrope walker loses balance. A high gamma is like walking on a very wobbly tightrope – even a small shift in weight (asset price) causes a large imbalance. The walker must constantly make small adjustments with the pole (rebalance the hedge) to avoid falling. Each adjustment requires effort (transaction costs).

The core of this question revolves around understanding how delta changes as an option approaches expiration, particularly when the underlying asset’s price is near the strike price. Delta, representing the sensitivity of the option price to changes in the underlying asset’s price, behaves differently for in-the-money (ITM), at-the-money (ATM), and out-of-the-money (OTM) options as expiration nears. As an option nears expiration, its delta becomes more sensitive to price movements in the underlying asset. An ATM option’s delta tends to swing wildly between 0 and 1 (or -1 for puts) close to expiration. This is because even a small price movement in the underlying can push the option into or out of the money. For an ITM option nearing expiration, the delta approaches 1 (or -1 for puts) as the option’s price becomes almost entirely dictated by the underlying asset’s price. Conversely, for an OTM option, the delta approaches 0, as the option is increasingly unlikely to become profitable before expiration. The scenario presented involves a short call option, meaning the investor has sold the call. A delta of 0.5 indicates that for every £1 increase in the underlying asset’s price, the option’s price will increase by approximately £0.50. However, because the investor is short the call, they are *short* delta. Therefore, the investor needs to *buy* delta to hedge their position. As the expiration date approaches and the underlying asset price hovers near the strike price, the delta of the short call option will become increasingly sensitive. If the underlying price increases even slightly above the strike, the delta will rapidly approach 1, requiring the investor to buy more of the underlying asset to maintain the hedge. If the underlying price decreases slightly below the strike, the delta will rapidly approach 0, allowing the investor to sell their hedge. This dynamic requires constant monitoring and adjustment of the hedge, a process known as dynamic hedging. The investor must continuously rebalance their position to maintain a delta-neutral hedge, buying when the price rises and selling when it falls.

The question assesses understanding of the impact of volatility on option pricing, specifically in the context of exotic options like barrier options. Barrier options have a payoff that depends on whether the underlying asset’s price reaches a predetermined barrier level during the option’s life. Increased volatility generally increases the value of standard options because it increases the probability of the option finishing in the money. However, for barrier options, the relationship is more complex. If volatility increases significantly, the probability of hitting the barrier also increases. In the case of a down-and-out put option, if the barrier is hit, the option becomes worthless. Therefore, extremely high volatility can *decrease* the value of a down-and-out put because the increased probability of hitting the barrier outweighs the increased probability of a larger payoff if the barrier is not hit. The holder is essentially short volatility near the barrier. The Black-Scholes model is used to price European options, but its assumptions (constant volatility, no early exercise) are violated by barrier options. Monte Carlo simulation is a more appropriate method for pricing barrier options, as it can handle the path-dependency and non-constant volatility often observed in real markets. The precise impact of a volatility shift depends on the specific parameters (strike price, barrier level, time to maturity, current price of the underlying asset). However, the general principle is that very high volatility near the barrier increases the likelihood of the option being knocked out, thus reducing its value. A modest increase in volatility might increase the option’s value, but a substantial increase can have the opposite effect.

The question assesses understanding of the impact of correlation on portfolio Value at Risk (VaR). VaR measures the potential loss in value of a portfolio over a specific time period for a given confidence level. When assets are perfectly correlated (correlation coefficient = 1), the portfolio VaR is simply the sum of the individual asset VaRs. When assets are uncorrelated (correlation coefficient = 0), the portfolio VaR is less than the sum of individual VaRs due to diversification benefits. A negative correlation provides even greater diversification benefits, further reducing portfolio VaR. The formula for calculating portfolio VaR with two assets is: Portfolio VaR = \[\sqrt{VaR_1^2 + VaR_2^2 + 2 \cdot \rho \cdot VaR_1 \cdot VaR_2}\] Where: \(VaR_1\) is the VaR of asset 1 \(VaR_2\) is the VaR of asset 2 \(\rho\) is the correlation coefficient between asset 1 and asset 2 In this case, \(VaR_1 = £50,000\), \(VaR_2 = £30,000\), and \(\rho = -0.5\). Plugging these values into the formula: Portfolio VaR = \[\sqrt{50000^2 + 30000^2 + 2 \cdot (-0.5) \cdot 50000 \cdot 30000}\] Portfolio VaR = \[\sqrt{2500000000 + 900000000 – 1500000000}\] Portfolio VaR = \[\sqrt{1900000000}\] Portfolio VaR = £43,589 (approximately) Therefore, the portfolio VaR is approximately £43,589. This is less than the sum of the individual VaRs (£80,000), demonstrating the risk reduction benefit of negative correlation. A portfolio manager should understand how correlation impacts overall portfolio risk and VaR to make informed decisions about asset allocation and hedging strategies. Neglecting correlation can lead to a significant overestimation or underestimation of portfolio risk, potentially resulting in inadequate risk management and unexpected losses. The negative correlation between the assets acts as a natural hedge, reducing the overall portfolio’s vulnerability to market fluctuations.

The question revolves around calculating the fair price of a forward contract on a zero-coupon bond, considering storage costs (analogous to physical commodity storage) and the risk-free rate. The key is to understand that the forward price should reflect the future value of the bond’s current price, adjusted for any costs incurred in holding the asset until the forward contract’s maturity. This is a fundamental concept in derivatives pricing, ensuring no arbitrage opportunities exist. First, we need to calculate the future value of the zero-coupon bond’s current price at the risk-free rate. This represents the cost of financing the purchase of the bond today and holding it until the forward contract’s expiration. The formula for future value is: \[FV = PV * (1 + r)^t\] Where: * FV = Future Value * PV = Present Value (current price of the bond) * r = risk-free rate * t = time to maturity (in years) In this case: * PV = £85 * r = 4% or 0.04 * t = 1 year \[FV = 85 * (1 + 0.04)^1 = 85 * 1.04 = £88.40\] Next, we must consider the storage costs. These costs are directly added to the future value of the bond, as they represent an additional expense incurred by holding the asset. Total Future Cost = Future Value + Storage Costs Total Future Cost = £88.40 + £1.50 = £89.90 Therefore, the fair price of the forward contract is £89.90. Any deviation from this price would create an arbitrage opportunity. If the forward price were lower, an investor could buy the bond, store it, and sell it forward at the higher calculated price, guaranteeing a profit. Conversely, if the forward price were higher, an investor could short the bond, short the forward contract, and earn a risk-free profit. The analogy to physical commodity storage is crucial. Imagine storing gold bars. You have the cost of the gold itself (the PV), the interest you could have earned on that money (the risk-free rate), and the cost of storing the gold in a vault (the storage costs). The forward price must cover all these expenses to be fair. This example highlights the core principle of no-arbitrage pricing in derivatives. The forward price is not just a prediction of the future spot price; it’s a price that eliminates risk-free profit opportunities. Understanding these cost-of-carry relationships is fundamental for anyone advising on or trading derivatives.

This question tests the candidate’s understanding of exotic options, specifically barrier options, and how their payoff structures are affected by the underlying asset’s price breaching a predetermined barrier level. The calculation involves determining whether the barrier has been breached and, if not, calculating the standard European option payoff. The key is to recognize the knock-out feature and its impact on the option’s value. We need to understand the concept of path dependency and how it influences the option’s final value. The barrier option’s value is contingent on the underlying asset’s price behavior during the option’s life. Let’s consider an analogy: Imagine you’re insuring a fragile vase. The insurance policy has a ‘breakage barrier.’ If the vase is moved from its display shelf (barrier), the insurance is voided, regardless of whether it actually breaks. If it stays put, you receive compensation based on its replacement value if it breaks before the policy ends. Similarly, a barrier option becomes worthless if the barrier is breached, irrespective of the asset’s final price, or pays out like a regular option if the barrier isn’t breached. The barrier feature drastically alters the option’s risk profile and valuation. The calculation is as follows: 1. **Barrier Breach Check:** The barrier is set at 155. The highest observed price during the option’s life was 158. Since 158 > 155, the barrier has been breached. 2. **Knock-Out Feature:** Because the barrier was breached, the down-and-out call option is knocked out and expires worthless. 3. **Payoff:** Therefore, the payoff is £0.

The question tests the understanding of delta hedging a short call option position. Delta is the sensitivity of the option price to changes in the underlying asset price. A short call option has a negative delta, meaning its value decreases as the underlying asset price increases. To delta hedge, one needs to buy shares of the underlying asset to offset this negative delta. The number of shares to buy is equal to the absolute value of the option’s delta. The delta changes as the underlying asset price changes; this is called gamma. As the underlying asset price increases, the delta of a short call option becomes more negative. Therefore, the hedger needs to buy *more* shares to maintain a delta-neutral position. In this case, the initial delta is -0.45, so the initial hedge requires buying 450 shares. The underlying asset price increases, and the delta becomes -0.70. This means the hedger needs to increase the number of shares held. The increase in shares needed is the difference between the new delta (in absolute value) and the old delta (in absolute value), multiplied by the contract size: (0.70 – 0.45) * 1000 = 250 shares. The total cost of adjusting the hedge is the number of shares purchased multiplied by the new share price: 250 shares * £105 = £26,250.

To address this question, we need to understand how changes in implied volatility affect option prices, specifically in the context of a short straddle position. A short straddle involves selling both a call and a put option with the same strike price and expiration date. This strategy profits when the underlying asset’s price remains relatively stable. However, it carries significant risk if the price moves substantially in either direction. Implied volatility represents the market’s expectation of future price volatility. When implied volatility increases, the prices of both call and put options generally increase. This is because higher volatility increases the probability of the underlying asset’s price moving significantly, making the options more valuable to their holders. In a short straddle, the investor has sold the options. Therefore, an increase in implied volatility leads to a loss, as the investor would need to buy back the options at a higher price to close the position. The magnitude of the loss depends on the sensitivity of the option prices to changes in volatility, which is measured by Vega. Let’s calculate the impact. We are given that the combined Vega of the short straddle is -10. This means that for every 1% increase in implied volatility, the value of the straddle (and thus the investor’s loss) increases by £10. The implied volatility increases from 20% to 25%, which is a 5% increase. Therefore, the total loss is calculated as follows: Loss = Vega * Change in Implied Volatility Loss = -10 * 5 Loss = -£50 Therefore, the investor would experience a loss of £50. Now, let’s consider a unique analogy. Imagine you are running a car insurance company. You sell policies (similar to selling options). If news comes out that there’s going to be a massive ice storm (increased volatility), the risk of accidents (price movement) goes up. Therefore, your existing policies (short straddle) become more valuable to the policyholders (option buyers), and you, the insurer (option seller), are now at a higher risk and would need to set aside more capital (loss). This is precisely what happens with a short straddle when implied volatility increases. The options you sold become more valuable, and you face a potential loss.

The question assesses understanding of exotic derivatives, specifically barrier options, and their behavior around the barrier level. A down-and-out call option becomes worthless if the underlying asset price touches or goes below the barrier. Therefore, the option’s value is path-dependent and sensitive to the asset price approaching the barrier. The problem requires understanding how volatility and time to maturity influence the probability of hitting the barrier and, consequently, the option’s price. The closer the current asset price is to the barrier, the higher the likelihood of the barrier being hit. Increased volatility also increases the probability of hitting the barrier. Shorter time to maturity reduces the chance of the barrier being hit, assuming the asset price is currently far from the barrier. To calculate the approximate impact, we must consider the ‘Greeks’ conceptually, although a precise calculation isn’t required. A significant drop in volatility would decrease the probability of hitting the barrier, increasing the option’s value. A decrease in time to maturity, if the barrier hasn’t been hit, also increases the option’s value, as there’s less time for the barrier to be reached. Conversely, a slight increase in the underlying asset price, further away from the barrier, also increases the option’s value, as it reduces the probability of hitting the barrier. The combined effect needs careful consideration. The correct answer reflects the most likely outcome given the interplay of these factors.

The core of this question revolves around understanding how a delta-neutral portfolio reacts to changes in volatility (vega) and time decay (theta), particularly when those factors are interconnected. A delta-neutral portfolio is designed to be insensitive to small changes in the underlying asset’s price. However, it remains susceptible to other factors, notably vega and theta. Vega measures the portfolio’s sensitivity to changes in the underlying asset’s volatility. Theta measures the portfolio’s sensitivity to the passage of time. In this scenario, the portfolio manager anticipates a decrease in volatility, which would negatively impact the portfolio’s value if it has positive vega (i.e., benefits from increasing volatility). Simultaneously, the passage of time erodes the value of options, especially those near expiration, which is reflected in a negative theta. The manager’s decision to buy futures contracts aims to offset the combined impact of these two factors. The key is to recognize that buying futures introduces delta. Since the portfolio was initially delta-neutral, this action creates a delta exposure. If volatility decreases as expected, the options’ value decreases, and the portfolio loses money due to its positive vega. However, the futures position gains value if the underlying asset’s price increases. The intention is to balance the losses from vega and theta with potential gains from the introduced delta. The success of this strategy depends on the magnitude of the volatility decrease, the time decay, and the actual price movement of the underlying asset. It’s a complex interplay of sensitivities that requires careful calculation and monitoring. This example highlights the dynamic nature of managing a delta-neutral portfolio and the need to constantly rebalance to maintain the desired risk profile. This differs from a simple hedge because it actively seeks to profit from a specific market view (decreasing volatility) while managing the associated risks.

The question revolves around the concept of hedging a portfolio with futures contracts, specifically focusing on adjusting the hedge ratio based on changing market conditions and correlation. The key is to understand how correlation impacts the effectiveness of a hedge. A lower correlation means the price movements of the asset being hedged and the futures contract are less aligned, requiring a larger hedge ratio to achieve the desired risk reduction. The initial hedge ratio is calculated as \( \text{Hedge Ratio} = \beta \times \frac{\text{Portfolio Value}}{\text{Futures Contract Value} \times \text{Number of Contracts}} \). The beta represents the systematic risk of the portfolio relative to the index. The adjustment for correlation involves dividing the hedge ratio by the correlation coefficient. This adjustment increases the hedge ratio when the correlation is less than 1, reflecting the increased uncertainty and need for a larger position to effectively hedge. In this scenario, the initial hedge ratio is calculated using the portfolio value (£5,000,000), futures contract value (£100,000), number of contracts (50), and portfolio beta (1.2). The initial hedge ratio is \( 1.2 \times \frac{5,000,000}{100,000 \times 50} = 1.2 \). Since the correlation between the portfolio and the index futures decreases from 1 to 0.8, we need to adjust the hedge ratio. The adjusted hedge ratio is \( \frac{1.2}{0.8} = 1.5 \). This means the fund manager needs to increase the hedge ratio to 1.5. To find the new number of contracts needed, we solve for the number of contracts in the hedge ratio formula: \( 1.5 = 1.2 \times \frac{5,000,000}{100,000 \times \text{Number of Contracts}} \). Rearranging the formula, we get \( \text{Number of Contracts} = \frac{1.2 \times 5,000,000}{100,000 \times 1.5} = 40 \). This means the fund manager should short 40 contracts to achieve the adjusted hedge ratio. Since the fund manager initially shorted 50 contracts and now needs to short 40 contracts, they should buy back 10 contracts (50 – 40 = 10). This reduces the short position, reflecting the lower correlation and the need for a smaller hedge position overall.

The core of this question lies in understanding how delta hedging works and how changes in the underlying asset’s price impact the hedge’s effectiveness, particularly when dealing with a non-linear instrument like an option. Delta represents the sensitivity of the option price to a change in the underlying asset’s price. A delta-neutral portfolio aims to have a delta of zero, meaning that small changes in the underlying asset’s price should not significantly impact the portfolio’s value. However, delta itself changes as the underlying asset’s price changes (this is captured by Gamma). To maintain a delta-neutral hedge, the portfolio must be rebalanced periodically. The amount of adjustment needed depends on the change in the underlying asset’s price and the option’s delta. If the underlying asset’s price increases, the option’s delta typically increases for a call option (and decreases for a put option). Therefore, to maintain delta neutrality, the investor needs to buy more of the underlying asset (or sell less if hedging a short position in the option). In this scenario, the investor is short a call option. The negative delta indicates that the investor needs to short the underlying asset to hedge. The initial delta hedge is achieved by shorting 40 shares. When the stock price increases, the call option’s delta increases to -0.60, indicating a greater sensitivity to the stock price. To re-establish delta neutrality, the investor needs to short more shares. The change in delta is -0.60 – (-0.40) = -0.20. This means the investor needs to short an additional 20 shares (0.20 * 100 shares represented by the option contract). Therefore, the investor must sell an additional 20 shares to maintain the delta-neutral position. This rebalancing ensures the portfolio remains hedged against small price movements in the underlying asset. If the investor fails to rebalance, the portfolio becomes exposed to directional risk.

Let’s consider a scenario involving a UK-based agricultural cooperative, “GreenHarvest Co-op,” which aims to protect its future revenue from wheat sales. They decide to use wheat futures contracts traded on the Euronext exchange. GreenHarvest anticipates harvesting 5,000 tonnes of wheat in six months. The current futures price for wheat with a six-month delivery is £200 per tonne. To hedge their risk, they sell 50 futures contracts (each contract representing 100 tonnes). Over the next six months, various factors influence the wheat market. Unforeseen weather events in Eastern Europe damage crops, leading to a supply shortage. Simultaneously, increased demand from emerging markets further drives up prices. Consequently, the spot price of wheat at harvest time jumps to £250 per tonne. GreenHarvest closes out their futures position by buying back 50 contracts at the new price of £250 per tonne. This results in a loss on the futures contracts: 50 contracts * 100 tonnes/contract * (£250 – £200) = £250,000 loss. However, they sell their physical wheat harvest at the spot price of £250 per tonne, generating revenue of 5,000 tonnes * £250/tonne = £1,250,000. Without hedging, GreenHarvest would have been exposed to the risk of wheat prices falling below £200. The hedge allowed them to lock in a price close to their expected revenue, mitigating potential losses. The effective price they received, considering the futures loss, is (£1,250,000 – £250,000) / 5,000 tonnes = £200 per tonne. Basis risk is the risk that the price of a futures contract does not move exactly in tandem with the price of the underlying asset. This can occur due to differences in location, quality, or timing between the futures contract and the physical commodity being hedged. In this scenario, basis risk could arise if the wheat variety GreenHarvest produces is slightly different from the standard wheat specified in the Euronext futures contract. For example, if GreenHarvest’s wheat has a higher protein content, it might command a premium in the spot market. If this premium widens over the hedging period, GreenHarvest would benefit from the basis movement. Conversely, if the premium narrows, they would be worse off. The hedging strategy reduces price risk, but basis risk remains a factor.