Derivatives Level 4 (Investment Advice Diploma) Exam Quiz Quiz 19

The question assesses understanding of how a sudden shift in implied volatility impacts a delta-neutral portfolio constructed using options. A delta-neutral portfolio is designed to be insensitive to small changes in the underlying asset’s price. However, it is not immune to changes in other factors, particularly implied volatility, which is measured by Vega. Vega represents the sensitivity of an option’s price to changes in implied volatility. A positive Vega means the option’s price increases as implied volatility increases, and vice versa. The initial portfolio is delta-neutral, meaning its overall delta is zero. This is achieved by balancing long and short positions in options and the underlying asset. However, a change in implied volatility will affect the value of the options, impacting the portfolio’s overall value. In this case, a sudden increase in implied volatility will increase the value of long options and decrease the value of short options (or increase the losses on short options). The magnitude of this change is determined by the portfolio’s Vega. To determine the portfolio’s loss, we use the formula: Portfolio Loss ≈ – Vega * Change in Volatility. In this case, the portfolio has a Vega of -25,000, and implied volatility increases by 3%. So, the Portfolio Loss ≈ -(-25,000) * 0.03 = 750. The negative sign on Vega indicates that the portfolio will lose value if volatility decreases and gain value if volatility increases. Since volatility increased, the portfolio experiences a gain of £750.

Let’s consider a scenario involving a UK-based agricultural cooperative, “BritCrops,” that wants to hedge against potential price declines in their upcoming wheat harvest using futures contracts traded on the ICE Futures Europe exchange. BritCrops anticipates harvesting 500,000 bushels of wheat in three months. The current futures price for wheat with a three-month expiry is £5.00 per bushel. BritCrops decides to short (sell) 100 wheat futures contracts, each representing 5,000 bushels of wheat (totaling 500,000 bushels). To assess the effectiveness of their hedge, we need to calculate the overall profit or loss, considering both the futures market and the physical market. Scenario 1: Wheat prices fall. Assume that at harvest time, the spot price of wheat has fallen to £4.50 per bushel, and the futures price has also converged to £4.50 per bushel. BritCrops sells their wheat in the spot market for £4.50 per bushel, receiving £2,250,000 (500,000 bushels * £4.50). Simultaneously, they close out their futures position by buying back the 100 contracts at £4.50 per bushel. Their profit on the futures contracts is calculated as the difference between the initial selling price (£5.00) and the final buying price (£4.50), multiplied by the total number of bushels covered by the contracts: (£5.00 – £4.50) * 500,000 = £250,000. The total realized value for BritCrops is the sum of the spot market revenue and the futures market profit: £2,250,000 + £250,000 = £2,500,000. Scenario 2: Wheat prices rise. Assume that at harvest time, the spot price of wheat has risen to £5.50 per bushel, and the futures price has also converged to £5.50 per bushel. BritCrops sells their wheat in the spot market for £5.50 per bushel, receiving £2,750,000 (500,000 bushels * £5.50). Simultaneously, they close out their futures position by buying back the 100 contracts at £5.50 per bushel. Their loss on the futures contracts is calculated as the difference between the initial selling price (£5.00) and the final buying price (£5.50), multiplied by the total number of bushels covered by the contracts: (£5.00 – £5.50) * 500,000 = -£250,000. The total realized value for BritCrops is the sum of the spot market revenue and the futures market loss: £2,750,000 – £250,000 = £2,500,000. In both scenarios, the effective price received by BritCrops is £5.00 per bushel (£2,500,000 / 500,000 bushels), demonstrating the hedging effectiveness. Basis risk, which arises from the imperfect correlation between the spot price and the futures price, is assumed to be negligible in this simplified example. A perfect hedge offsets price fluctuations in the physical market with corresponding gains or losses in the futures market, stabilizing the overall revenue.

Let’s consider a scenario where a UK-based manufacturing company, “Precision Engineering Ltd,” needs to hedge against potential fluctuations in the EUR/GBP exchange rate. Precision Engineering imports specialized components from Germany, priced in Euros. They have a large order due in three months and are concerned that a strengthening GBP will make these imports more expensive. The company’s treasurer is evaluating different hedging strategies using currency options. The treasurer considers buying EUR put options (giving them the right to sell EUR and buy GBP) to protect against a fall in the EUR/GBP exchange rate. To determine the appropriate strike price, they analyze the current spot rate, forward rates, and implied volatility. The treasurer also needs to consider the option premium, which will reduce the overall hedging effectiveness. Suppose the current spot rate is EUR/GBP = 1.15. The three-month forward rate is EUR/GBP = 1.14. The treasurer considers buying EUR put options with a strike price of 1.14. The option premium is 0.01 GBP per EUR. Scenario 1: At expiration, the spot rate is EUR/GBP = 1.10. Precision Engineering exercises the put option, selling EUR at 1.14 and buying GBP. Their effective exchange rate, considering the premium, is 1.14 – 0.01 = 1.13. Without hedging, they would have received only 1.10 GBP per EUR. Scenario 2: At expiration, the spot rate is EUR/GBP = 1.16. Precision Engineering does not exercise the put option because the market rate is more favorable. Their effective exchange rate is 1.16 – 0.01 = 1.15. They lose the premium but benefit from the favorable market rate. Scenario 3: The treasurer also considers a collar strategy, buying EUR puts at 1.14 and selling EUR calls at 1.16. This limits their upside but also reduces the cost of hedging. The effectiveness of the hedge depends on the actual exchange rate at expiration and the cost of the options. The treasurer must weigh the cost of the premium against the potential benefits of hedging. The decision to hedge also depends on Precision Engineering’s risk tolerance and their view on the future direction of the EUR/GBP exchange rate. The treasurer must consider the impact of the hedge on the company’s financial statements. Under IFRS 9, the hedging relationship must be documented, and the hedge must be highly effective. Any ineffectiveness must be recognized in profit or loss.

Let’s analyze a scenario involving a UK-based pension fund considering a currency overlay strategy using forward contracts to hedge its Euro-denominated equity investments. The fund needs to understand the implications of covered interest parity (CIP) deviations and their potential impact on hedging effectiveness. CIP suggests that the forward premium or discount should equal the interest rate differential between the two currencies. However, deviations from CIP can arise due to factors such as credit risk, liquidity constraints, and regulatory differences. Suppose the current spot exchange rate is EUR/GBP = 0.85. The UK interest rate is 5% per annum, and the Eurozone interest rate is 3% per annum. According to CIP, the one-year forward rate should be approximately EUR/GBP = 0.85 * (1 + 0.03) / (1 + 0.05) = 0.833. However, due to market frictions, the actual one-year forward rate is quoted at EUR/GBP = 0.84. This deviation presents both an opportunity and a risk. The pension fund needs to assess the potential impact of this CIP deviation on its hedging strategy. If the fund hedges its Euro exposure using the market forward rate of 0.84, it effectively earns a return that deviates from the theoretical CIP level. This deviation could be advantageous if the fund anticipates the deviation to persist or widen. However, it also introduces uncertainty, as the deviation could narrow or even reverse, impacting the hedging outcome. To quantify the impact, consider a scenario where the fund hedges EUR 10 million of equity investments. Using the CIP-implied forward rate of 0.833, the GBP value would be EUR 10,000,000 / 0.833 = GBP 12,000,000. Using the market forward rate of 0.84, the GBP value is EUR 10,000,000 / 0.84 = GBP 11,904,762. The difference of GBP 95,238 represents the initial impact of the CIP deviation. The fund must then consider how this deviation might change over the hedging period and its implications for the overall portfolio risk and return. This requires stress testing and scenario analysis to understand the potential range of outcomes. Furthermore, the fund must consider the regulatory implications of engaging in strategies that exploit CIP deviations, ensuring compliance with relevant regulations and guidelines.

The question assesses the understanding of how macroeconomic announcements impact derivative pricing, specifically focusing on implied volatility. Implied volatility reflects the market’s expectation of future price fluctuations. A higher-than-expected inflation announcement typically leads to increased uncertainty about future interest rate hikes by the Bank of England (BoE). This uncertainty translates to higher implied volatility, especially for options on short-term interest rate futures. The calculation involves understanding how the implied volatility surface shifts. The implied volatility surface represents the implied volatility of options with different strike prices and expiration dates. In this scenario, the implied volatility for short-term options (3-month SONIA futures) increases more than for longer-term options (1-year Gilt futures) because the immediate impact of inflation is on short-term interest rate expectations. Let’s assume the initial implied volatility for 3-month SONIA futures options is 12% and for 1-year Gilt futures options is 9%. If the inflation announcement is significantly higher than expected, the implied volatility for 3-month SONIA futures options might increase by 3 percentage points (a significant jump due to immediate uncertainty), while the implied volatility for 1-year Gilt futures options might increase by only 1 percentage point (less immediate impact, expectations of BoE intervention). Therefore: New implied volatility for 3-month SONIA futures options = 12% + 3% = 15% New implied volatility for 1-year Gilt futures options = 9% + 1% = 10% The percentage increase for 3-month SONIA futures options is \[\frac{15\% – 12\%}{12\%} \times 100\% = 25\%\] The percentage increase for 1-year Gilt futures options is \[\frac{10\% – 9\%}{9\%} \times 100\% \approx 11.11\%\] The correct answer reflects this differential impact, showing a larger percentage increase in implied volatility for short-term interest rate derivatives compared to longer-term ones. This understanding is crucial for traders and portfolio managers who use derivatives to hedge against interest rate risk or to speculate on interest rate movements. The scenario highlights the dynamic relationship between macroeconomic news, market expectations, and derivative pricing.

The core of this question revolves around understanding the relationship between delta, gamma, and option position adjustments to maintain a delta-neutral portfolio. Delta represents the sensitivity of the option’s price to a change in the underlying asset’s price. Gamma, in turn, represents the sensitivity of the delta to a change in the underlying asset’s price. A delta-neutral portfolio aims to have a combined delta of zero, meaning that small changes in the underlying asset’s price should not significantly impact the portfolio’s value. However, because gamma measures how delta changes, a delta-neutral portfolio needs constant adjustment. In this scenario, the fund manager initially establishes a delta-neutral portfolio using call options on a stock. The stock price increases, causing the delta of the call options to increase. This means the portfolio is no longer delta-neutral; it has become delta-positive, meaning it will gain value if the stock price increases further. To re-establish delta neutrality, the fund manager must reduce the overall delta of the portfolio. This can be achieved by selling (shorting) some of the underlying stock. The number of shares to sell is determined by the change in delta caused by the stock price movement. The change in the portfolio delta due to the stock price increase is calculated by multiplying the number of options by the option’s gamma and the change in stock price. The fund manager needs to offset this change by selling shares. The calculation involves dividing the total change in portfolio delta by 1 (since each share has a delta of 1) to determine the number of shares to sell. This adjustment ensures that the portfolio remains as close to delta-neutral as possible, mitigating the impact of further small price movements in the underlying asset. The fund manager is actively managing risk by constantly monitoring and adjusting the portfolio’s delta to maintain its neutral stance. This strategy is crucial for funds aiming to profit from volatility or time decay rather than directional movements in the underlying asset.

The question assesses understanding of hedging strategies using options, specifically focusing on the impact of implied volatility changes on a short strangle position. A short strangle involves selling both a call and a put option with different strike prices (out-of-the-money). The strategy profits if the underlying asset price remains within a defined range until expiration. However, the strategy is highly sensitive to changes in implied volatility. An increase in implied volatility increases the value of both the call and put options, leading to a loss for the short strangle position. The magnitude of the loss depends on the option’s delta and gamma, and the extent of the volatility increase. Here’s the breakdown of the calculation: 1. **Initial position:** Short 1 call and short 1 put. 2. **Volatility increase:** Implied volatility increases by 5%. 3. **Impact on call option:** The call option value increases by £1.20. Since you’re short the call, this results in a loss of £1.20 per option. 4. **Impact on put option:** The put option value increases by £0.80. Since you’re short the put, this results in a loss of £0.80 per option. 5. **Total loss:** £1.20 (call loss) + £0.80 (put loss) = £2.00 per share. 6. **Total loss for 10 contracts (1,000 shares per contract):** £2.00/share * 1,000 shares/contract * 10 contracts = £20,000. Therefore, the short strangle position would experience a loss of £20,000 due to the increase in implied volatility. This highlights the importance of monitoring implied volatility when implementing option strategies, especially those that are short volatility, like strangles and straddles. Managing risk involves understanding the “Greeks” (Delta, Gamma, Vega, Theta, Rho) and their impact on the portfolio. Vega measures the sensitivity of an option’s price to changes in implied volatility. A short strangle has negative Vega, meaning it will lose value when implied volatility increases. Furthermore, regulations such as EMIR require firms to adequately manage counterparty risk and perform stress testing, including scenarios where volatility spikes unexpectedly.

The core of this question lies in understanding how different Greeks (Delta, Gamma, Vega) interact and impact a portfolio’s overall risk profile. We must also consider the effect of regulatory capital requirements. Delta represents the sensitivity of the option’s price to changes in the underlying asset’s price. Gamma represents the rate of change of delta with respect to changes in the underlying asset’s price. Vega represents the sensitivity of the option’s price to changes in the volatility of the underlying asset. A positive Gamma means that the Delta will increase as the underlying asset price increases, and decrease as the underlying asset price decreases. Vega is crucial because increased market uncertainty typically leads to higher volatility, directly affecting option prices. Regulatory capital requirements are influenced by the firm’s overall risk profile, encompassing market risk, credit risk, and operational risk. Higher risk translates to higher capital requirements. In this scenario, the portfolio manager needs to strategically adjust the options positions to reduce the sensitivity to volatility and maintain the overall risk profile within acceptable regulatory parameters. Reducing Vega will directly address the volatility risk. Given the positive Gamma, the manager must be careful about how Delta is adjusted. Simply reducing Delta might increase overall risk if the underlying asset price moves significantly. A negative Delta position would offset the positive Delta of the existing positions, reducing the overall Delta. The optimal strategy involves reducing Vega while carefully managing Delta exposure, considering the impact of Gamma. This can be achieved by selling options with high Vega and adjusting the Delta through other offsetting positions. The capital relief would then be a function of the reduction in overall risk exposure, specifically related to volatility and delta.

The question assesses the understanding of the impact of correlation between assets in a portfolio when using derivatives for hedging, specifically focusing on Value at Risk (VaR). When assets are perfectly correlated, the diversification benefit is minimal, and the VaR of the portfolio is essentially the sum of the individual VaRs. However, when assets are less than perfectly correlated, the diversification effect reduces the overall portfolio VaR. The formula for portfolio VaR with two assets is: \[VaR_p = \sqrt{VaR_A^2 + VaR_B^2 + 2 \cdot \rho \cdot VaR_A \cdot VaR_B}\] Where: * \(VaR_p\) is the portfolio VaR * \(VaR_A\) is the VaR of Asset A * \(VaR_B\) is the VaR of Asset B * \(\rho\) is the correlation between Asset A and Asset B In this scenario, Asset A (UK Equities) has a VaR of £300,000 and Asset B (FTSE 100 Futures) has a VaR of £200,000. The correlation (\(\rho\)) is 0.6. Plugging these values into the formula: \[VaR_p = \sqrt{300,000^2 + 200,000^2 + 2 \cdot 0.6 \cdot 300,000 \cdot 200,000}\] \[VaR_p = \sqrt{90,000,000,000 + 40,000,000,000 + 72,000,000,000}\] \[VaR_p = \sqrt{202,000,000,000}\] \[VaR_p \approx 449,444.10\] Therefore, the portfolio VaR is approximately £449,444.10. This result is lower than the sum of individual VaRs (£300,000 + £200,000 = £500,000) due to the diversification effect arising from the less-than-perfect correlation. If the assets were perfectly correlated (\(\rho = 1\)), the portfolio VaR would equal the sum of the individual VaRs. The example demonstrates how correlation impacts the effectiveness of hedging strategies using derivatives. A lower correlation allows for greater risk reduction in the portfolio. In practical terms, understanding this relationship is crucial for fund managers seeking to minimize portfolio risk while adhering to regulatory requirements and internal risk mandates. It also highlights the importance of accurately estimating correlations, as these estimates directly affect the calculated VaR and, consequently, the risk management decisions. Misestimating correlations can lead to either under- or over-hedging, both of which can have detrimental financial consequences. Furthermore, regulatory bodies like the FCA in the UK scrutinize the risk management practices of financial institutions, emphasizing the need for accurate VaR calculations and a thorough understanding of correlation effects.

The question assesses the understanding of hedging strategies using options, specifically a ratio spread. A ratio spread involves buying and selling different numbers of options with the same expiration date but different strike prices. The profit/loss profile is complex and depends heavily on the underlying asset’s price movement. In this scenario, the investor is trying to hedge a short position in shares of “Innovatech Solutions” using a ratio call spread. The investor buys one call option to protect against a significant upward price movement, but also sells two out-of-the-money call options to offset the cost of the purchased call. The maximum profit is achieved if the stock price stays below the strike price of the call option that was purchased. The maximum loss is potentially unlimited if the stock price rises significantly above the strike price of the call options that were sold, because the investor is obligated to sell shares at the strike price, but must buy the shares at the higher market price. Let’s break down the profit/loss at different price points: * **Stock Price below £150:** The purchased call expires worthless, and the sold calls also expire worthless. The investor makes a profit equal to the net premium received from selling the two calls minus the premium paid for the purchased call. * **Stock Price at £160:** The purchased call is in the money by £10, but the sold calls are worthless. The profit is £10 minus the net premium paid. * **Stock Price at £170:** The purchased call is in the money by £20, and the sold calls are in the money by £10 each. The profit is £20 – 2\*£10 = £0, minus the net premium paid. * **Stock Price above £170:** The purchased call continues to gain value, but the sold calls also gain value. For every £1 increase in the stock price above £170, the purchased call gains £1, but the two sold calls lose £2. The investor loses £1 for every £1 increase in the stock price above £170. The breakeven point is the stock price at which the profit/loss is zero. In this case, the maximum profit is the premium received from selling the two calls minus the premium paid for the purchased call, which is £200 – £100 = £100. The maximum loss is unlimited, but the loss starts to occur when the stock price is above £170. For every £1 increase in the stock price above £170, the investor loses £1. Therefore, the breakeven point is £170 + £100 = £270.

The question tests understanding of the impact of implied volatility on option prices, specifically within the context of earnings announcements. The scenario involves a complex option strategy (short strangle) and requires the candidate to assess how a post-earnings announcement volatility crush would affect the position’s profitability. The correct answer involves recognizing that a decrease in implied volatility benefits a short strangle position. Here’s the breakdown: 1. **Short Strangle:** A short strangle involves selling both an out-of-the-money call option and an out-of-the-money put option on the same underlying asset with the same expiration date. The strategy profits if the underlying asset’s price remains within a defined range between the strike prices of the call and put options. 2. **Implied Volatility (IV):** IV reflects the market’s expectation of the underlying asset’s price fluctuations. Higher IV leads to higher option premiums because there’s a greater chance the option will end up in the money. Conversely, lower IV leads to lower premiums. 3. **Earnings Announcements:** Earnings announcements are significant events that can cause substantial price movements in a company’s stock. Leading up to the announcement, IV tends to increase due to heightened uncertainty. After the announcement, the uncertainty typically decreases, leading to a “volatility crush” – a sharp decline in IV. 4. **Impact on Short Strangle:** When you sell options (as in a short strangle), you *want* IV to decrease. A decrease in IV reduces the value of the options you’ve sold, allowing you to potentially buy them back at a lower price and pocket the difference as profit. 5. **Scenario Analysis:** In this scenario, the investor sold the strangle *before* the earnings announcement, anticipating a volatility crush *after* the announcement. If the stock price remains within the strangle’s range (between the strike prices), and IV decreases significantly, the investor profits. 6. **Calculation (Illustrative):** * Assume the investor collected a premium of £5 per share for the call and £3 per share for the put, totaling £8 per share. * After the earnings announcement, implied volatility plummets. * The call option’s value decreases to £1, and the put option’s value decreases to £0.50. * The investor can now buy back the call and put options for a total of £1.50. * Profit per share: £8 (initial premium) – £1.50 (buyback cost) = £6.50. * If the investor sold 10 contracts (1,000 shares), the total profit would be £6,500 (excluding commissions and other fees). The other options present common misconceptions: assuming that any price movement is detrimental to a short strangle, failing to account for the volatility crush, or incorrectly associating increased volatility with profit for a short position. The correct response recognizes that a decrease in implied volatility, *combined* with the stock price remaining within the defined range, is the ideal scenario for a short strangle position.

The core of this question revolves around understanding how a delta-neutral portfolio is constructed and maintained, specifically focusing on the impact of gamma on the portfolio’s delta as the underlying asset’s price changes. A delta-neutral portfolio aims to have a zero delta, meaning it is initially insensitive to small changes in the underlying asset’s price. However, gamma measures the rate of change of delta with respect to changes in the underlying asset’s price. Therefore, a portfolio with a non-zero gamma will see its delta change as the underlying asset’s price moves. To maintain delta neutrality, the portfolio must be rebalanced. The amount of rebalancing depends on the gamma of the portfolio and the change in the underlying asset’s price. The formula to calculate the change in delta is: Change in Delta ≈ Gamma * Change in Underlying Asset Price. The number of shares to buy or sell to re-establish delta neutrality is then calculated by dividing the change in delta by the delta of a single share (which is 1 for the underlying asset). In this scenario, the portfolio has a gamma of -500. This means that for every £1 change in the underlying asset’s price, the portfolio’s delta changes by -500. If the underlying asset’s price increases by £2, the portfolio’s delta will decrease by -500 * 2 = -1000. Since the portfolio was initially delta-neutral (delta = 0), the new delta will be -1000. To re-establish delta neutrality, the portfolio manager needs to buy 1000 shares of the underlying asset to offset this negative delta. The crucial understanding here is that gamma represents the “curvature” of the portfolio’s value with respect to the underlying asset’s price. A negative gamma implies that the portfolio’s delta becomes more negative as the underlying asset’s price increases and more positive as the price decreases. Rebalancing involves continuously adjusting the portfolio’s holdings to counteract this effect and maintain the desired delta-neutral position. A portfolio with a large absolute gamma requires more frequent rebalancing than a portfolio with a small absolute gamma. The rebalancing strategy aims to keep the portfolio’s delta as close to zero as possible, minimizing its sensitivity to small price movements in the underlying asset.

The question assesses the understanding of put-call parity and how dividends affect option pricing. Put-call parity is a fundamental relationship that defines the theoretical price relationship between European put and call options with the same strike price and expiration date. The formula is: \(C + PV(K) = P + S – PV(Div)\), where C is the call option price, K is the strike price, P is the put option price, S is the current stock price, PV(K) is the present value of the strike price, and PV(Div) is the present value of expected dividends. In this scenario, the dividends introduce a critical adjustment. The present value of the dividends must be subtracted from the stock price side of the equation. This is because the call option holder does not receive the dividends, while the stock holder does. If the put-call parity is violated, an arbitrage opportunity exists. In this case, the dividend payment impacts the parity and the arbitrage strategy. The present value of the dividend is calculated as \(Div / (1 + r)^t\), where Div is the dividend amount, r is the risk-free rate, and t is the time to dividend payment in years. Here, \(Div = £2\), \(r = 0.05\), and \(t = 0.5\) years. So, \(PV(Div) = 2 / (1 + 0.05)^{0.5} = 2 / 1.0247 = £1.952\). The theoretical put price is calculated using the put-call parity formula: \(P = C + PV(K) – S + PV(Div)\). Plugging in the values, \(P = 8 + (100 / (1 + 0.05)^{1}) – 105 + 1.952 = 8 + 95.238 – 105 + 1.952 = £0.19\). If the actual put price is £1.00, it is overpriced compared to the theoretical price of £0.19. The arbitrage strategy involves selling the overpriced put option and buying the call option, and the stock, while borrowing to cover the present value of the strike price. 1. **Sell the overpriced put:** Receive £1.00 2. **Buy the call option:** Pay £8.00 3. **Buy the stock:** Pay £105.00 4. **Borrow the present value of the strike price:** Borrow \(100 / (1 + 0.05) = £95.238\) At expiration: * If the stock price is above £100, the call option is exercised, and the stock is delivered to cover the obligation. * If the stock price is below £100, the put option expires worthless. The borrowed amount is repaid with interest. The arbitrage profit is calculated as the initial cash inflow minus the initial cash outflow: \(1 + 95.238 – 8 – 105 = -16.762\). However, this is not the end of the strategy. The dividend payment must be accounted for. Since you own the stock, you receive a dividend of £2.00. The net profit is the initial arbitrage profit plus the present value of the dividend: \(-16.762 + 1.952 = -14.81\). However, we need to consider the dividend received. Since we own the stock, we receive the dividend of £2 in 6 months. So, the correct arbitrage strategy is: 1. Sell the put option for £1.00 2. Buy the call option for £8.00 3. Buy the stock for £105.00 4. Borrow the present value of the strike price: £95.238 Initial cash flow: 1.00 – 8.00 – 105.00 + 95.238 = -16.762 In 6 months, receive dividend of £2.00. At expiration, if stock price > £100, exercise call option, deliver stock. If stock price < £100, put expires worthless. Arbitrage profit = Initial cash flow + Dividend - PV(Dividend) = -16.762 + 2 - 1.952 = -16.714 This is incorrect. Correct Arbitrage Strategy: 1. Sell the overpriced put for £1.00. 2. Buy the call option for £8.00. 3. Buy the stock for £105.00. 4. Borrow £95.238 (PV of strike price). Initial Cash Flow: 1.00 - 8.00 - 105.00 + 95.238 = -16.762 Receive dividend of £2.00 in 6 months. At expiration: If ST > 100, deliver stock, repay loan of £100. If ST < 100, put expires worthless, repay loan of £100. Arbitrage Profit = -16.762 + 2.00 – 2.00/(1.05)^0.5 = -16.762 + 2 – 1.952 = -16.714 This is still incorrect. The key is to calculate the arbitrage profit correctly, considering the present value of the dividend and the initial cash flows. We need to buy the *underpriced* asset and sell the *overpriced* asset. In this case, the put is overpriced. 1. Sell the put: +£1.00 2. Buy the call: -£8.00 3. Buy the stock: -£105.00 4. Borrow PV(Strike): +£95.24 Initial flow: -£16.76 Receive dividend: +£2.00 Arbitrage Profit = Sell Put + Borrow PV(Strike) – Buy Call – Buy Stock + Dividend – PV(Dividend) = 1 + 95.24 – 8 – 105 + 2 – 1.95 = -16.71 Correct Approach: The put is overpriced, so we SELL the put. We also BUY the call and BUY the stock, and BORROW the present value of the strike. Initial cash flow: 1 – 8 – 105 + 95.24 = -16.76 Dividend received: 2 Arbitrage Profit = 1 – 8 – 105 + 95.24 + 2 – 2/(1.05)^0.5 = -16.71 The correct answer is that there is an arbitrage opportunity, and by executing the correct strategy, the investor can profit.

To determine the value of the forward contract, we need to calculate the present value of the expected future price difference, considering the risk-free rate and the storage costs. First, calculate the expected future price: Expected Future Price = Current Spot Price + (Storage Costs per Unit * Number of Units) Expected Future Price = £85 + (£3 * 100) = £385 Next, calculate the present value of the expected future price difference: Present Value = (Expected Future Price – Current Spot Price) / (1 + Risk-Free Rate)^(Time to Maturity) Present Value = (£385 – £85) / (1 + 0.05)^0.5 Present Value = £300 / (1.05)^0.5 Present Value = £300 / 1.0247 Present Value = £292.77 Now, determine the fair value of the forward contract: Fair Value = Present Value – (Storage Costs per Unit * Number of Units) / (1 + Risk-Free Rate)^(Time to Maturity) Fair Value = £292.77 – (£3 * 100) / (1 + 0.05)^0.5 Fair Value = £292.77 – £300 / 1.0247 Fair Value = £292.77 – £292.77 Fair Value = £0 Finally, determine the value of the forward contract: Value of Forward Contract = Current Forward Price – Fair Value Value of Forward Contract = £90 – £0 = £90 Therefore, the value of the forward contract is £90. In this example, understanding the time value of money and the impact of storage costs on forward contract pricing is crucial. The calculation demonstrates how to discount future values to present values using the risk-free rate and how to incorporate storage costs into the valuation. The final value represents the intrinsic worth of the forward contract based on the given market conditions and contract terms. This approach allows for a comprehensive assessment of the contract’s profitability and risk. The calculation ensures that all relevant factors are considered, providing a sound basis for investment decisions.

The core of this question lies in understanding how delta hedging works in practice, specifically the adjustments needed as the underlying asset’s price changes. Delta represents the sensitivity of the option’s price to a change in the underlying asset’s price. A delta of 0.6 indicates that for every £1 increase in the asset’s price, the option’s price is expected to increase by £0.60. The initial hedge is established by selling short 60 shares (0.6 * 100 shares represented by one contract). As the asset price moves, the delta changes, requiring adjustments to maintain a delta-neutral position. If the asset price increases, the delta typically increases for a call option, meaning the option becomes more sensitive to further price increases. To remain hedged, the portfolio manager needs to buy more shares to offset this increased sensitivity. Conversely, if the asset price decreases, the delta decreases, and the portfolio manager needs to sell shares. The question incorporates transaction costs, which are crucial in real-world hedging strategies. These costs reduce the profitability of hedging and must be considered when determining the optimal re-hedging frequency. Let’s break down the calculations: 1. **Initial Hedge:** Short 60 shares at £50. 2. **Price Increase:** Asset price rises to £52. 3. **New Delta:** Delta increases to 0.65. 4. **Shares to Buy:** To adjust the hedge, the portfolio manager needs to increase their short position to 65 shares. This means buying 5 shares (65 – 60). 5. **Cost of Buying Shares:** 5 shares * £52/share = £260. 6. **Transaction Cost:** £10. 7. **Total Cost:** £260 + £10 = £270. Therefore, the total cost to re-hedge the portfolio after the price increase is £270. This example highlights the dynamic nature of delta hedging and the importance of considering transaction costs in hedging decisions. A portfolio manager must carefully weigh the benefits of maintaining a precise hedge against the costs associated with frequent re-hedging. In practice, managers often use a “tolerance band” around a delta-neutral position, only re-hedging when the delta moves outside this band. This helps to reduce transaction costs and improve overall hedging efficiency. Furthermore, the choice of hedging instrument can also impact the overall cost and effectiveness of the hedge. For instance, using options to hedge a portfolio can provide a more precise hedge but may also be more expensive than using futures or forwards.

The question assesses the understanding of exotic options, specifically barrier options, and their sensitivity to market movements. The scenario involves a “knock-out” barrier option, where the option becomes worthless if the underlying asset’s price touches a predefined barrier level. The key is to understand how the probability of the barrier being hit before the option’s expiration influences the option’s price. To calculate the adjusted probability, we need to consider the initial probability of the barrier being hit (30%) and how the market movement impacts this. The market decline increases the likelihood of hitting the knock-out barrier. We’re given that the probability increases by 50% of the initial probability due to the market movement. 1. **Calculate the increase in probability:** 50% of 30% = 0.50 * 0.30 = 0.15 2. **Calculate the new probability:** Initial probability + increase = 0.30 + 0.15 = 0.45 or 45% 3. **Calculate the adjusted option price:** Initial option price * (1 – New probability) = £10 * (1 – 0.45) = £10 * 0.55 = £5.50 Therefore, the adjusted price of the barrier option, reflecting the increased probability of the knock-out barrier being hit, is £5.50. This problem highlights the importance of continuous monitoring and dynamic adjustments in risk management when dealing with barrier options. It also emphasizes the need to understand how market events can rapidly alter the value of these types of derivatives. A similar analogy would be a dam with a spillway. The barrier option is like the water level of the dam. If the water level (asset price) reaches the spillway (barrier), the dam’s function (option’s value) is nullified. A sudden rainfall (market movement) increases the probability of the water reaching the spillway.

The Black-Scholes model is used to calculate the theoretical price of European-style options. The formula is: \[ C = S_0N(d_1) – Ke^{-rT}N(d_2) \] Where: \( C \) = Call option price \( S_0 \) = Current stock price \( K \) = Strike price \( r \) = Risk-free interest rate \( T \) = Time to expiration (in years) \( N(x) \) = Cumulative standard normal distribution function \( d_1 = \frac{ln(\frac{S_0}{K}) + (r + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}} \) \( d_2 = d_1 – \sigma\sqrt{T} \) \( \sigma \) = Volatility of the stock In this scenario, we’re looking for the impact of a change in the risk-free rate on a call option’s price. The risk-free rate (r) appears in two places within the Black-Scholes formula: directly in the exponent \(e^{-rT}\) and indirectly within \(d_1\) and \(d_2\). A higher risk-free rate increases \(d_1\) and \(d_2\), which in turn increases \(N(d_1)\) and \(N(d_2)\). Simultaneously, the term \(e^{-rT}\) decreases as ‘r’ increases. The overall impact is that the call option price increases with an increasing risk-free rate. Let’s consider a stock trading at £50, with a call option having a strike price of £50, expiring in 1 year, and a volatility of 20%. If the risk-free rate increases from 2% to 3%, the call option price will increase. This is because the present value of the strike price decreases (since we’re discounting it at a higher rate), making the call option more valuable. The calculation of the call option price with a 2% risk-free rate: \( S_0 = 50 \), \( K = 50 \), \( r = 0.02 \), \( T = 1 \), \( \sigma = 0.2 \) \( d_1 = \frac{ln(\frac{50}{50}) + (0.02 + \frac{0.2^2}{2})1}{0.2\sqrt{1}} = \frac{0 + 0.04}{0.2} = 0.2 \) \( d_2 = 0.2 – 0.2\sqrt{1} = 0 \) \( N(d_1) = N(0.2) \approx 0.5793 \) \( N(d_2) = N(0) = 0.5 \) \( C = 50 \times 0.5793 – 50e^{-0.02 \times 1} \times 0.5 = 28.965 – 50 \times 0.9802 \times 0.5 = 28.965 – 24.505 = 4.46 \) Now, with a 3% risk-free rate: \( S_0 = 50 \), \( K = 50 \), \( r = 0.03 \), \( T = 1 \), \( \sigma = 0.2 \) \( d_1 = \frac{ln(\frac{50}{50}) + (0.03 + \frac{0.2^2}{2})1}{0.2\sqrt{1}} = \frac{0 + 0.05}{0.2} = 0.25 \) \( d_2 = 0.25 – 0.2\sqrt{1} = 0.05 \) \( N(d_1) = N(0.25) \approx 0.5987 \) \( N(d_2) = N(0.05) \approx 0.5199 \) \( C = 50 \times 0.5987 – 50e^{-0.03 \times 1} \times 0.5199 = 29.935 – 50 \times 0.9704 \times 0.5199 = 29.935 – 25.225 = 4.71 \) The call option price increased from £4.46 to £4.71 when the risk-free rate increased from 2% to 3%.

To solve this problem, we need to understand how delta hedging works and how changes in volatility affect the hedge. Delta hedging aims to create a portfolio that is insensitive to small changes in the underlying asset’s price. However, the delta of an option changes as the underlying asset’s price and volatility change. This change in delta is measured by gamma. Vega measures the sensitivity of an option’s price to changes in volatility. In this scenario, the fund manager initially delta hedges their short call option position. This means they buy shares of the underlying asset to offset the negative delta of the short call. When volatility increases unexpectedly, the value of the call option increases (because options are more valuable when volatility is higher). Since the fund manager is short the call, they lose money. To re-establish the delta hedge, they need to adjust their position in the underlying asset. An increase in volatility increases the absolute value of the option’s delta. If the call option is out-of-the-money, its delta will move closer to zero as volatility increases (less sensitive to price changes). If the call option is in-the-money, its delta will move closer to 1 as volatility increases (more sensitive to price changes). Since the question does not specify whether the option is in-the-money or out-of-the-money, we must consider both scenarios. However, the key here is understanding the fund manager is short the call, and an increase in volatility will increase the option price. To maintain a delta-neutral position, the fund manager must buy more shares of the underlying asset if the call option moves further in-the-money. If the call option moves further out-of-the-money, the fund manager can sell some shares of the underlying asset. However, the fund manager is short the option. The increase in volatility makes the option more expensive. Since they are short, they lose money. They need to buy shares to hedge the increased delta. Here’s the step-by-step calculation of the required adjustment: 1. **Initial Delta:** -0.45 (Short call option) 2. **Volatility Increase:** From 15% to 20% 3. **New Delta (estimated):** -0.55 (Delta becomes more negative due to increased volatility – assuming the option is approaching being in-the-money or was already in-the-money. The absolute value increases.) 4. **Delta Change:** -0.55 – (-0.45) = -0.10 5. **Number of Options:** 10,000 6. **Shares per Option:** 100 (Standard for equity options) 7. **Total Delta Change:** -0.10 * 10,000 * 100 = -100,000 shares. Since the delta change is negative, the fund manager needs to buy shares to offset the negative delta. Therefore, the fund manager needs to buy 10,000 shares to re-establish the delta hedge.

This question tests understanding of volatility smiles, their implications for option pricing, and the ability to interpret market sentiment reflected in the shape of the smile. It requires applying theoretical knowledge to a practical scenario involving trading decisions. The volatility smile arises because the Black-Scholes model assumes constant volatility across all strike prices for a given expiration date, which is rarely true in reality. Typically, out-of-the-money puts and calls have higher implied volatilities than at-the-money options, creating a “smile” shape when implied volatility is plotted against strike price. This skew reflects market expectations about the likelihood of large price movements in either direction. A steeper skew suggests a greater demand for downside protection (puts) or upside participation (calls), implying a bearish or bullish outlook, respectively. The question requires integrating knowledge of volatility smiles, option pricing, and market sentiment to determine the most appropriate trading strategy. The investor must analyze the relative prices of options at different strike prices, considering the implied volatility skew, to profit from perceived mispricings. The correct answer reflects a strategy that exploits the overpricing of options in the area of the smile where market expectations are strongest. Understanding the regulatory landscape is also important. For example, MAR (Market Abuse Regulation) in the UK prohibits trading on inside information or engaging in market manipulation. A strategy that seeks to exploit market inefficiencies must be carefully considered to ensure compliance with these regulations.

This question tests the candidate’s understanding of delta-hedging, gamma, and the associated costs when managing a derivatives portfolio. Delta-hedging aims to neutralize the portfolio’s sensitivity to small changes in the underlying asset’s price. Gamma, on the other hand, measures the rate of change of delta with respect to the underlying asset’s price. A higher gamma implies that the delta changes more rapidly, requiring more frequent adjustments to maintain a delta-neutral position. These adjustments incur transaction costs. The problem requires calculating the expected cost of maintaining a delta-neutral position over a specific period, considering the portfolio’s gamma, the expected volatility of the underlying asset, and the transaction costs associated with each hedge adjustment. The formula to estimate the cost of delta-hedging due to gamma is: Cost = 0.5 * Gamma * (Change in Asset Price)^2 * Number of Hedge Adjustments * Transaction Cost per Adjustment Where: Change in Asset Price = Asset Price * Volatility * Time Step Number of Hedge Adjustments = Total Time / Time Step In this scenario, we must first calculate the expected change in the asset price during each weekly interval. We then use this value to calculate the cost of rebalancing each week, and finally, we multiply by the number of weeks to find the total cost. 1. **Calculate the expected change in asset price per week:** Change in Asset Price = £1500 * 0.02 * (1/52)^0.5 ≈ £4.16 2. **Calculate the cost of rebalancing each week:** Weekly Rebalancing Cost = 0.5 * 25 * (£4.16)^2 * £1.50 ≈ £325.00 3. **Calculate the total cost over 13 weeks:** Total Cost = £325.00 * 13 ≈ £4225.00 This calculation considers the gamma of the portfolio, the volatility of the underlying asset, and the transaction costs. It emphasizes that while delta-hedging aims to reduce risk, the dynamic adjustments required due to gamma exposure can lead to significant transaction costs.

The core of this question revolves around understanding how implied volatility, as reflected in the “volatility smile,” impacts option pricing and hedging strategies, particularly in the context of exotic options like barrier options. A standard Black-Scholes model assumes constant volatility across all strike prices, which is rarely the case in real markets. The volatility smile shows that out-of-the-money (OTM) puts and calls often have higher implied volatilities than at-the-money (ATM) options. The key here is that barrier options are sensitive to volatility changes, especially near the barrier level. If the market anticipates increased volatility (and the volatility smile steepens), the probability of the underlying asset hitting the barrier increases. This increased probability affects the option’s price. For a knock-out option, a higher probability of hitting the barrier *decreases* the option’s value, as it’s more likely to expire worthless. Conversely, for a knock-in option, a higher probability of hitting the barrier *increases* the option’s value, as it’s more likely to come into existence. Delta hedging aims to neutralize the sensitivity of an option portfolio to small changes in the underlying asset’s price. However, delta is itself sensitive to volatility (vega). If implied volatility changes significantly, the delta hedge needs to be adjusted. In this scenario, the steeper volatility smile indicates a greater need to hedge against volatility risk (vega risk) in addition to price risk (delta risk). Failing to account for the changing volatility skew could lead to significant losses, especially with barrier options. The calculation is conceptual rather than numerical. The change in the volatility smile suggests a higher implied volatility for strikes further away from the current price. This means the probability of the underlying asset reaching the barrier is higher. Since the barrier option is a knock-out, its value will decrease. To hedge, the trader needs to *short* the option to offset the potential loss.

The question assesses the understanding of delta hedging, specifically in the context of a short call option position. A delta hedge aims to neutralize the price sensitivity of an option position to changes in the underlying asset’s price. Delta, representing the change in the option’s price for a £1 change in the underlying asset’s price, is crucial. A short call option has a positive delta, meaning its value decreases as the underlying asset’s price increases. To hedge this, one needs to short the underlying asset in proportion to the option’s delta. The formula for determining the number of shares to short is: Number of shares = Delta * Number of options contracts * Contract size. In this case, the delta is 0.60, the number of contracts is 100, and the contract size is 100 shares. Therefore, the number of shares to short is 0.60 * 100 * 100 = 6000 shares. The scenario introduces a real-world complication: transaction costs. These costs impact the profitability of hedging and must be considered. The question tests the candidate’s ability to integrate these costs into their decision-making process. While the initial delta hedge requires shorting 6000 shares, the question further introduces a scenario where the underlying asset’s price increases, altering the option’s delta. This necessitates rebalancing the hedge. The new delta is 0.75, requiring a new short position of 0.75 * 100 * 100 = 7500 shares. Since the investor already has a short position of 6000 shares, they need to short an additional 1500 shares (7500 – 6000). The transaction cost of £0.05 per share on these 1500 shares is 1500 * £0.05 = £75. This cost reduces the overall effectiveness of the hedge and impacts the investor’s profit or loss. The question evaluates whether the candidate can correctly calculate the required rebalancing and account for the associated transaction costs.

The Black-Scholes model is a cornerstone of options pricing theory, but it relies on several assumptions that are often violated in real-world markets. One crucial assumption is constant volatility. In reality, volatility fluctuates, and the market’s expectation of future volatility is reflected in the volatility “smile” or “skew.” The volatility smile refers to the phenomenon where options with the same expiration date but different strike prices have different implied volatilities. Typically, out-of-the-money puts and calls exhibit higher implied volatilities than at-the-money options. When the volatility smile is present, using a single implied volatility from an at-the-money option to price all other options with the same expiration will lead to mispricing. Specifically, options that are further away from the at-the-money strike price (i.e., deep out-of-the-money puts and calls) will be undervalued by the Black-Scholes model. This is because the model underestimates the probability of large price movements that would make these options profitable. To account for the volatility smile, traders and risk managers use techniques such as interpolating or extrapolating implied volatilities from options with different strike prices to create a volatility surface. They can then use these strike-specific implied volatilities in modified Black-Scholes models or other pricing models to obtain more accurate option prices. Furthermore, risk management strategies, such as hedging, need to consider the volatility smile to ensure adequate protection against market movements. Ignoring the smile can lead to under-hedging and potential losses, especially in volatile market conditions. For example, if a portfolio contains out-of-the-money puts, and the volatility smile steepens (meaning the implied volatility of these puts increases), the portfolio’s value will decrease if the puts were initially priced using an at-the-money volatility.

Let’s consider a scenario involving a UK-based agricultural cooperative, “GreenHarvest Co-op,” which exports organic barley to several European breweries. GreenHarvest faces significant price volatility due to unpredictable weather patterns and fluctuating demand. To mitigate this risk, they decide to use futures contracts. The co-op needs to determine the optimal number of contracts to hedge their exposure. First, we need to calculate the hedge ratio. The hedge ratio minimizes the variance of the hedged position. It is calculated as: Hedge Ratio = \(\frac{\text{Size of Exposure}}{\text{Size of Futures Contract}} \times \text{Correlation between Spot and Futures Prices} \times \text{Volatility Adjustment}\) Let’s assume GreenHarvest expects to export 500,000 bushels of barley. Each futures contract covers 5,000 bushels. The correlation between the spot price of GreenHarvest’s barley and the futures price is 0.8. The volatility of the spot price is 15%, and the volatility of the futures price is 12%. Volatility Adjustment = \(\frac{\text{Volatility of Spot Price}}{\text{Volatility of Futures Price}} = \frac{0.15}{0.12} = 1.25\) Hedge Ratio = \(\frac{500,000}{5,000} \times 0.8 \times 1.25 = 100 \times 0.8 \times 1.25 = 100\) Therefore, GreenHarvest should sell 100 futures contracts to effectively hedge their price risk. Now, consider the impact of basis risk. Basis risk arises because the spot price and futures price may not move perfectly in tandem. For example, if local demand for organic barley surges in the UK while European demand weakens, the spot price may increase while the futures price decreases, eroding the effectiveness of the hedge. To mitigate basis risk, GreenHarvest could explore alternative hedging instruments, such as options, or diversify their export markets. Furthermore, regulatory considerations under EMIR (European Market Infrastructure Regulation) are crucial. GreenHarvest must ensure they comply with reporting requirements for their derivative transactions. If they exceed the clearing threshold, they will be required to centrally clear their futures contracts through a CCP (Central Counterparty). Understanding these regulatory obligations is essential for the co-op to avoid penalties and maintain market access. Finally, let’s consider the initial margin requirements. The exchange will require GreenHarvest to deposit an initial margin to cover potential losses. If the futures price moves against them, they may also be required to post variation margin. Proper cash flow management is critical to meet these margin calls and avoid forced liquidation of their hedging position.

To address this question, we need to understand how changes in volatility affect option prices, particularly in the context of a strangle strategy. A strangle involves buying both an out-of-the-money call and an out-of-the-money put option on the same underlying asset with the same expiration date. The strategy profits when the underlying asset’s price moves significantly in either direction. Vega measures an option’s sensitivity to changes in volatility. Since a strangle consists of both a call and a put, its value is positively correlated with volatility. The trader’s initial analysis suggested a potential for high volatility around the earnings announcement, leading them to implement the strangle. However, the market’s reaction was muted, indicating lower-than-expected volatility. This decrease in volatility directly impacts the value of both the call and put options in the strangle, causing them to lose value. To calculate the approximate loss, we need to consider the combined Vega of the options. The call option has a Vega of 0.06, meaning its price changes by £0.06 for every 1% change in volatility. The put option has a Vega of 0.04, so its price changes by £0.04 for every 1% change in volatility. Together, the strangle has a Vega of 0.06 + 0.04 = 0.10. The volatility decreased by 5%. Therefore, the approximate loss on the strangle is Vega * Change in Volatility * Contract Size = 0.10 * 5 * £100 = £50. Therefore, the trader would experience an approximate loss of £50 on the strangle strategy due to the unexpected decrease in volatility following the earnings announcement. This illustrates the importance of accurately forecasting volatility when using options strategies like strangles.

A portfolio manager is implementing a delta-hedging strategy for a portfolio containing 1000 call options on shares of a UK-listed company. The initial share price is £100, and the delta of the option is -0.6. To implement the hedge, the manager buys the required number of shares. The brokerage charges a transaction cost of £0.05 per share for each buy or sell order. After the initial hedge, the share price increases to £102, and the option’s delta changes to -0.65. The manager rebalances the hedge accordingly. Subsequently, the share price decreases to £98, and the option’s delta changes to -0.55. The manager rebalances the hedge again. What is the portfolio manager’s approximate profit or loss after these hedging activities, considering the impact of transaction costs?

The question assesses understanding of how a sudden, unexpected shift in the yield curve impacts a complex hedging strategy involving Eurodollar futures contracts. Specifically, it tests the ability to analyze the effect of a “twist” – a non-parallel shift where short-term rates rise while long-term rates fall – on a hedge designed to protect against parallel rate movements. The key is recognizing that a twist will create offsetting effects on different parts of the futures position, potentially reducing the effectiveness of the hedge and even leading to unexpected losses. Here’s how we can break down the scenario and arrive at the solution: 1. **Initial Hedge:** The portfolio manager is using Eurodollar futures to hedge against a potential increase in interest rates. Shorting Eurodollar futures profits when interest rates rise (futures prices fall). The manager has hedged duration by shorting contracts representing a notional principal equal to the portfolio’s duration-adjusted exposure. 2. **The Yield Curve Twist:** A yield curve twist, where short-term rates rise and long-term rates fall, presents a challenge. Eurodollar futures contracts with shorter maturities will be more affected by the increase in short-term rates, leading to a larger price decrease (and profit for the short position). Conversely, contracts with longer maturities will be affected by the decrease in long-term rates, leading to a smaller price increase (and a loss for the short position). 3. **Impact on the Hedge:** Because the hedge involves a mix of short- and long-dated Eurodollar futures, the twist will cause some contracts to generate profits while others generate losses. The net effect depends on the magnitude of the rate changes at each maturity and the relative weighting of the contracts. 4. **Calculating the Net Effect:** We need to calculate the profit or loss on each leg of the hedge and then sum them to determine the overall impact. * **Short-Dated Contracts:** The 3-month contracts decrease in value by 0.25% (25 basis points) of their notional value. Profit = 0.0025 * £5,000,000 = £12,500. * **Long-Dated Contracts:** The 2-year contracts increase in value by 0.10% (10 basis points) of their notional value. Loss = 0.0010 * £3,000,000 = £3,000. 5. **Net Profit/Loss:** Net Profit/Loss = Profit from short-dated contracts – Loss from long-dated contracts = £12,500 – £3,000 = £9,500. Therefore, the portfolio will experience a net profit of £9,500 on the Eurodollar futures hedge due to the yield curve twist. This example highlights the complexities of hedging with derivatives and the importance of understanding how different market movements can affect a hedging strategy. It also underscores the need for sophisticated risk management techniques to account for non-parallel yield curve shifts and other unexpected market events.

Let’s consider a scenario involving a UK-based agricultural cooperative, “GreenHarvest,” that needs to hedge against price volatility in the wheat market. GreenHarvest plans to sell 5,000 metric tons of wheat in six months. They are concerned about a potential price drop. To hedge this risk, they decide to use wheat futures contracts traded on the ICE Futures Europe exchange. Each contract represents 100 metric tons of wheat. Therefore, GreenHarvest needs to purchase 50 futures contracts (5,000 metric tons / 100 metric tons per contract). The current futures price for wheat with a six-month expiry is £200 per metric ton. GreenHarvest enters into a short hedge, meaning they sell 50 futures contracts at £200 per metric ton. After three months, the spot price of wheat has fallen to £180 per metric ton, and the futures price has fallen to £185 per metric ton. GreenHarvest decides to close out their hedge. Profit/Loss on Futures Contracts: GreenHarvest sold 50 contracts at £200 and bought them back at £185. Profit per contract = £200 – £185 = £15 Total profit = 50 contracts * 100 metric tons/contract * £15/metric ton = £75,000 Effective Price Received: Price received in the spot market = £180 per metric ton Gain from futures = £15 per metric ton Effective price = £180 + £15 = £195 per metric ton Now, let’s calculate the basis. The basis is the difference between the spot price and the futures price. Initial basis = Spot price – Futures price = Not applicable as we are looking at the effective price after closing out the hedge. Final basis = £180 – £185 = -£5 The effective price received by GreenHarvest is £195 per metric ton. The initial price they expected was £200, and the final price they received was £195 due to the basis change. This example illustrates how hedging with futures can help protect against price declines, but the final price received may differ from the initial expected price due to basis risk. Basis risk arises because the spot and futures prices do not always move in perfect lockstep. In this case, the basis narrowed (became less negative) from the start of the hedge to the close of the hedge, which reduced the effectiveness of the hedge.

The core of this question revolves around understanding how a delta-neutral portfolio is constructed and maintained, specifically when the underlying asset price moves and time decays. Delta-neutrality means the portfolio’s value is initially insensitive to small changes in the underlying asset’s price. However, delta changes as the underlying price changes (gamma) and as time passes (theta). To maintain delta-neutrality, the portfolio needs to be rebalanced. First, calculate the initial portfolio delta: * Shares: 1000 shares \* 1.0 delta = 1000 * Options: 100 options \* -0.5 delta/option \* 100 shares/option = -5000 * Portfolio Delta = 1000 – 5000 = -4000 This means the portfolio is short 4000 shares equivalent in delta. Next, calculate the delta change due to the stock price increase: * Delta change = Gamma \* Price Change \* Portfolio Size * Delta change = 0.001 \* £5 \* 100 options \* 100 shares/option = 50 The portfolio delta is now -4000 + 50 = -3950 Then, calculate the delta change due to time decay: * Delta change = Theta \* Number of Days \* Portfolio Size * Delta change = -0.01 \* 1 \* 100 options \* 100 shares/option = -100 The portfolio delta is now -3950 – 100 = -4050 To rebalance to delta-neutral, we need to offset the -4050 delta. Since we can only trade in blocks of 100 shares, we need to calculate how many shares to trade. * Shares to trade = -Portfolio Delta = 4050 This means we need to buy 4050 shares to become delta-neutral. Since we can only trade in blocks of 100, we divide 4050 by 100 to get the number of blocks to trade. * Blocks to trade = 4050/100 = 40.5 blocks Since we can only trade in whole blocks, we need to buy 41 blocks of 100 shares to get as close as possible to being delta-neutral. The closest delta-neutral position that can be achieved by trading in blocks of 100 shares is buying 4100 shares.

Let’s break down how to determine the appropriate hedging strategy using options in a volatile market environment. The core concept revolves around understanding an investor’s risk tolerance and market outlook, then selecting an options strategy that aligns with those factors. In this scenario, the investor, Ms. Anya Sharma, is risk-averse and believes the market will experience significant fluctuations but isn’t sure about the direction. A long straddle involves buying both a call and a put option with the same strike price and expiration date. This strategy profits from significant price movements in either direction. The maximum profit is unlimited (for the call option) minus the premium paid, and the put option’s profit is also unlimited (for the put option) minus the premium paid. The maximum loss is limited to the total premium paid for both options. A covered call involves selling a call option on a stock you already own. This strategy generates income but limits upside potential. It’s suitable for investors who are neutral to slightly bullish. A protective put involves buying a put option on a stock you already own. This strategy protects against downside risk but reduces potential profit. It’s suitable for investors who are bullish but want to hedge against potential losses. A short strangle involves selling both a call and a put option with different strike prices (one above and one below the current market price) and the same expiration date. This strategy profits from low volatility but has unlimited risk. Given Anya’s risk aversion and uncertain market outlook, a long straddle is the most appropriate strategy. It allows her to profit from significant price movements in either direction while limiting her potential loss to the premium paid. The other strategies are either too risky (short strangle) or limit potential profit (covered call, protective put). The key is that Anya is unsure of the direction of the market, and the long straddle is designed to profit from volatility regardless of direction.