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

The core of this question lies in understanding how delta hedging works in practice and the impact of transaction costs. Delta hedging aims to neutralize the directional risk of an option position by continuously adjusting the underlying asset holding. Gamma represents the rate of change of delta with respect to changes in the underlying asset’s price. When gamma is high, delta changes rapidly, requiring more frequent rebalancing. Transaction costs erode the profits from delta hedging, especially when rebalancing is frequent. The investor needs to determine the optimal rebalancing frequency, balancing the cost of imperfect hedging (due to infrequent rebalancing) against the transaction costs of frequent rebalancing. In this specific scenario, we calculate the profit/loss (P/L) from delta hedging with and without considering transaction costs. Without transaction costs, the P/L is simply the change in option value minus the cost of hedging. With transaction costs, each rebalancing incurs a cost, reducing the overall profit. 1. **Calculate the initial hedge ratio (Delta):** The delta of the call option is given as 0.5. This means for every £1 increase in the underlying asset’s price, the option price increases by £0.5. 2. **Calculate the cost of the initial hedge:** The investor sells the call option for £5 and buys 0.5 shares at £100 each, costing 0.5 * £100 = £50. 3. **Calculate the value of the hedge after the price change:** The underlying asset price increases to £102. The option price increases to £6. The investor’s 0.5 shares are now worth 0.5 * £102 = £51. 4. **Calculate the profit/loss without rebalancing:** The investor’s profit from the shares is £51 – £50 = £1. The investor’s loss from the option is £6 – £5 = £1. The net P/L is £1 – £1 = £0. 5. **Calculate the new hedge ratio (Delta):** The delta of the call option changes to 0.6. 6. **Calculate the cost of rebalancing:** The investor needs to increase their holding by 0.1 shares (0.6 – 0.5). This costs 0.1 * £102 = £10.20. 7. **Calculate the value of the hedge after the second price change:** The underlying asset price increases to £103. The option price increases to £7. The investor’s 0.6 shares are now worth 0.6 * £103 = £61.80. 8. **Calculate the profit/loss without transaction costs:** The investor’s profit from the shares is £61.80 – £50 – £10.20 = £1.60. The investor’s loss from the option is £7 – £5 = £2. The net P/L is £1.60 – £2 = -£0.40. 9. **Calculate the profit/loss with transaction costs:** Each trade costs £0.10 per share. The initial hedge costs 0.5 * £0.10 = £0.05. The rebalancing costs 0.1 * £0.10 = £0.01. The total transaction costs are £0.06. The net P/L is -£0.40 – £0.06 = -£0.46. Therefore, the overall impact of transaction costs on the delta hedge is -£0.46.

Let’s analyze a scenario involving exotic derivatives and regulatory compliance under EMIR (European Market Infrastructure Regulation). EMIR aims to reduce systemic risk in the OTC derivatives market by mandating central clearing, reporting, and risk management standards. A “snowball” swap is a complex interest rate swap where the interest accrual increases (or decreases) over time, based on a predetermined formula. This increasing accrual can lead to significant risks if not properly managed. Imagine a UK-based investment firm, “Alpha Investments,” enters into a snowball swap with a counterparty in Germany. The swap’s notional principal is £50 million. The interest rate on Alpha’s side starts at 1% and increases by 0.5% every quarter, contingent on the 3-month GBP LIBOR remaining below 2%. If GBP LIBOR exceeds 2%, the accrual resets to 1%. This structure creates a non-linear risk profile. Under EMIR, Alpha Investments must report this transaction to a registered trade repository. Furthermore, given the complexity of the snowball swap, Alpha needs to ensure adequate risk mitigation techniques are in place, including margin requirements and daily valuation. Let’s assume the initial margin requirement, calculated using a model approved by the PRA (Prudential Regulation Authority), is £2.5 million. However, Alpha’s internal risk management team, using stress testing and scenario analysis, determines that a sudden spike in GBP LIBOR to 3% could result in a mark-to-market loss of £4 million. The question assesses whether Alpha Investments is compliant with EMIR’s risk mitigation requirements, considering the potential for significant losses due to the exotic nature of the derivative and the impact of macroeconomic factors on GBP LIBOR. The key is understanding that EMIR requires firms to have adequate risk management procedures that go beyond simply meeting the initial margin requirements. Stress testing and scenario analysis are crucial for identifying potential vulnerabilities and ensuring sufficient capital is allocated to cover potential losses.

The core of this question lies in understanding how delta hedging works in practice, especially when faced with transaction costs. Delta hedging aims to maintain a delta-neutral portfolio, meaning the portfolio’s value is theoretically insensitive to small changes in the underlying asset’s price. The delta of an option represents the sensitivity of the option’s price to a change in the underlying asset’s price. When an investor sells an option, they typically hedge by buying the underlying asset to offset the negative delta of the short option position. Transaction costs erode the profitability of delta hedging, requiring more precise calculations and potentially less frequent rebalancing. Each time the hedge is adjusted (i.e., buying or selling the underlying asset), a transaction cost is incurred. This cost reduces the overall profit and can even turn a profitable strategy into a loss-making one if the underlying asset price fluctuates rapidly, necessitating frequent rebalancing. The optimal hedging strategy balances the cost of hedging against the risk of not hedging. A higher transaction cost makes frequent rebalancing less attractive. The trader must then decide on a tolerable range of delta exposure before rebalancing. This tolerance range is crucial, as it dictates how far the portfolio’s delta can deviate from zero before action is taken. The wider the tolerance range, the less frequent the rebalancing, but the greater the potential for losses if the underlying asset price moves significantly. In this specific scenario, the trader’s profit is reduced by the transaction costs incurred during the rebalancing. To calculate the final profit, we must account for the initial premium received, the changes in the underlying asset price, the number of shares bought and sold, and the transaction costs associated with each trade. The profit is calculated as: Initial Premium + (Final Asset Price – Initial Asset Price) * Number of Shares Initially Short – (Number of Shares Bought * Purchase Price + Number of Shares Sold * Sale Price) – (Total Number of Transactions * Transaction Cost per Transaction) Therefore, the trader’s profit is: £500 + (£108 – £100) * (-50) – (25 * £102 + 25 * £98) – (2 * £25) = £500 – £400 – £5100 – £2450 – £50 = -£7500

The question revolves around the concept of delta hedging a short call option position and the subsequent profit or loss based on the stock price movement. Delta hedging involves adjusting the number of shares held to offset the changes in the option’s value due to changes in the underlying asset’s price. The delta of a call option represents the sensitivity of the option’s price to a change in the underlying asset’s price. A delta of 0.6 indicates that for every £1 increase in the stock price, the call option’s price is expected to increase by £0.60. To delta hedge a short call option, an investor buys shares of the underlying stock. In this case, with a delta of 0.6, the investor initially buys 60 shares for every 100 call options sold. The cost of this initial hedge is 60 shares * £50/share = £3000. If the stock price increases to £55, the delta changes to 0.8. This means the investor needs to increase their hedge. The investor needs to buy an additional 20 shares (80 – 60) to maintain the delta hedge. The cost of these additional shares is 20 shares * £55/share = £1100. Now, the investor closes out the hedge by selling all 80 shares at £55 each, receiving 80 shares * £55/share = £4400. The total cost of implementing the hedge was £3000 (initial purchase) + £1100 (additional purchase) = £4100. The profit/loss from the hedging activity is the difference between the amount received from selling the shares and the total cost of purchasing them: £4400 – £4100 = £300. Finally, the call option expires in the money. As the investor sold the call, they lose £500 (the difference between the final stock price of £55 and the strike price of £50, multiplied by 100 options). The net profit/loss is the profit from hedging minus the loss from the call option expiring in the money: £300 – £500 = -£200.

Let’s consider a scenario where a portfolio manager, Eleanor, is tasked with hedging a GBP 10 million equity portfolio against potential market downturns using FTSE 100 futures. Eleanor believes that the market is overvalued and wants to protect her portfolio’s gains. The current FTSE 100 index is trading at 7,500, and each FTSE 100 futures contract represents £10 per index point. Eleanor decides to use a short hedge. First, we need to determine the number of futures contracts required. The portfolio value is GBP 10,000,000. The value of one futures contract is the index level multiplied by the contract multiplier: 7,500 * £10 = £75,000. The number of contracts needed is the portfolio value divided by the contract value: GBP 10,000,000 / £75,000 ≈ 133.33. Since you can’t trade fractions of contracts, Eleanor will short 133 contracts. Now, let’s assume that over the hedging period, the FTSE 100 index falls to 7,000, and Eleanor closes out her futures position. Her profit on each futures contract is (7,500 – 7,000) * £10 = £5,000. Her total profit on the futures contracts is 133 * £5,000 = £665,000. However, during the same period, Eleanor’s equity portfolio declines in value by 5%, resulting in a loss of GBP 10,000,000 * 0.05 = GBP 500,000. The net effect of the hedge is the profit from the futures contracts minus the loss in the equity portfolio: GBP 665,000 – GBP 500,000 = GBP 165,000. This shows how derivatives can be used to offset losses in an underlying portfolio, although perfect hedging is rarely achievable due to factors like basis risk (the imperfect correlation between the futures price and the spot price of the asset being hedged). Now, let’s introduce a twist involving margin requirements and regulatory considerations. Assume the initial margin requirement for each FTSE 100 futures contract is £5,000, and the maintenance margin is £4,000. Eleanor must deposit 133 * £5,000 = £665,000 as initial margin. If the index rises instead of falling, and the margin account falls below the maintenance margin level, Eleanor will receive a margin call and need to deposit additional funds to bring the account back to the initial margin level. The FCA (Financial Conduct Authority) regulates the margin requirements and trading practices to protect investors and ensure market integrity. Failing to meet margin calls can lead to the forced liquidation of the futures position, potentially undermining the hedging strategy.

The question revolves around the concept of delta hedging and its effectiveness in various market conditions, specifically focusing on the impact of gamma. Delta represents the sensitivity of an 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 is constructed to be insensitive to small changes in the underlying asset’s price. However, this neutrality is not static; gamma measures how quickly the delta changes. The scenario presented involves a portfolio manager using options to hedge against potential losses in a large equity holding. The manager initially establishes a delta-neutral position. However, the underlying equity experiences a significant price movement, leading to a change in the portfolio’s delta. The question explores the implications of this change, considering the portfolio’s gamma and the manager’s subsequent actions to rebalance the hedge. A key aspect is understanding that a positive gamma means the delta will increase as the underlying asset’s price increases and decrease as the underlying asset’s price decreases. Conversely, a negative gamma means the opposite. The manager’s decision to rebalance the hedge depends on the magnitude of the price movement, the portfolio’s gamma, and the desired level of risk exposure. The correct answer requires calculating the new delta exposure after the price movement, considering the gamma. It also involves evaluating the cost implications of rebalancing the hedge, taking into account transaction costs and the potential for further price movements. The incorrect options present plausible scenarios but fail to accurately account for the combined effects of delta, gamma, and rebalancing costs. The calculation proceeds as follows: 1. **Calculate the change in delta:** Change in Delta = Gamma \* Change in Underlying Price = 0.005 \* 15 = 0.075 2. **Calculate the new portfolio delta:** New Delta = Initial Delta + Change in Delta = 0 + 0.075 = 0.075 3. **Calculate the number of options to trade:** Number of Options = New Delta \* Portfolio Size / Option Delta = 0.075 \* 1,000,000 / 0.5 = 150,000 4. **Calculate the cost of rebalancing:** Cost = Number of Options \* Option Price = 150,000 \* 0.55 = £82,500

Let’s consider a scenario where a UK-based agricultural cooperative, “BritCrops,” seeks to hedge its upcoming wheat harvest using futures contracts listed on the London International Financial Futures and Options Exchange (LIFFE). BritCrops anticipates harvesting 5,000 tonnes of wheat in six months. The current spot price of wheat is £200 per tonne. The six-month wheat futures contract is trading at £210 per tonne. BritCrops decides to hedge 80% of its harvest using futures contracts, each representing 100 tonnes of wheat. This equates to hedging 4,000 tonnes (5,000 * 0.80). They will therefore need to purchase 40 futures contracts (4,000 / 100). Six months later, the spot price of wheat has fallen to £190 per tonne due to an unexpectedly large global harvest. The futures contract settles at £192 per tonne. BritCrops sells its wheat in the spot market for £190 per tonne. The gain or loss on the futures contracts is calculated as follows: Initial futures price: £210 per tonne Settlement futures price: £192 per tonne Profit per tonne: £210 – £192 = £18 per tonne Total profit on futures contracts: £18/tonne * 4,000 tonnes = £72,000 The effective price received by BritCrops is calculated as follows: Revenue from spot market sale: £190/tonne * 5,000 tonnes = £950,000 Profit from futures contracts: £72,000 Total revenue: £950,000 + £72,000 = £1,022,000 Effective price per tonne: £1,022,000 / 5,000 tonnes = £204.40 per tonne This example illustrates how futures contracts can be used to hedge against price risk. BritCrops locked in a price close to their expected price, mitigating the impact of the price decline. The remaining 20% of their harvest was unhedged, and sold at the spot price. The final effective price is a weighted average of the hedged and unhedged portions.

The question assesses understanding of delta hedging, gamma, and the impact of volatility changes on option positions. Delta hedging aims to neutralize the directional risk of an option by holding an offsetting position in the underlying asset. Gamma measures the rate of change of delta with respect to changes in the underlying asset’s price. A positive gamma means the delta will increase as the underlying price increases and decrease as the underlying price decreases. Vega measures the sensitivity of an option’s price to changes in volatility. A positive vega means the option’s price increases with increasing volatility. The trader initially delta hedges to neutralise directional risk. However, the trader’s portfolio has a positive gamma, meaning the delta will change as the underlying asset’s price moves. To maintain a delta-neutral position, the trader needs to dynamically adjust their hedge. If the underlying asset’s price increases, the delta of the option position increases, requiring the trader to sell more of the underlying asset to maintain neutrality. Conversely, if the underlying asset’s price decreases, the delta of the option position decreases, requiring the trader to buy more of the underlying asset. The profit or loss from delta hedging depends on the gamma and the magnitude of the price movements. Volatility changes also impact the value of the option position. Since the trader’s portfolio has a positive vega, an increase in volatility will increase the value of the option position, and a decrease in volatility will decrease the value of the option position. This impact is separate from the profit or loss from delta hedging, which depends on the gamma and price movements. In this case, the trader is short options, which means they will have negative vega. Therefore, a decrease in implied volatility will benefit the trader. The profit will be equal to the change in vega multiplied by the change in implied volatility.

The question assesses understanding of put-call parity, a fundamental concept in options pricing. Put-call parity describes the relationship between the price of a European call option, a European put option, the underlying asset’s price, and the risk-free rate. Specifically, it states that a portfolio consisting of a long European call option and a short European put option, both with the same strike price and expiration date, should have the same value as a forward contract on the same underlying asset. The formula for put-call parity is: \[C + PV(X) = P + S\] Where: \(C\) = Price of the European call option \(P\) = Price of the European put option \(S\) = Current price of the underlying asset \(X\) = Strike price of the options \(PV(X)\) = Present value of the strike price, calculated as \(X e^{-rT}\), where \(r\) is the risk-free interest rate and \(T\) is the time to expiration. Given: \(S = £155\) \(X = £150\) \(r = 3\%\) (or 0.03) \(T = 0.5\) years \(C = £12\) We need to find the theoretical price of the put option (P) that satisfies the put-call parity. First, calculate the present value of the strike price: \[PV(X) = X e^{-rT} = 150 \times e^{-0.03 \times 0.5} = 150 \times e^{-0.015} \approx 150 \times 0.98511 = 147.7665\] Now, rearrange the put-call parity formula to solve for P: \[P = C + PV(X) – S\] \[P = 12 + 147.7665 – 155 = 4.7665\] Therefore, the theoretical price of the put option is approximately £4.77. The question also probes understanding of arbitrage opportunities. If the market price of the put option deviates significantly from its theoretical price as calculated by put-call parity, an arbitrageur can exploit this mispricing to generate risk-free profit. For instance, if the put option is overpriced, an arbitrageur could sell the put option, buy the call option and the underlying asset, and borrow an amount equal to the present value of the strike price. This creates a synthetic short forward position, which offsets the long forward position implied by the call and asset. At expiration, the arbitrageur can use the proceeds from the asset to cover the short put obligation and repay the loan, realizing a profit equal to the initial mispricing. Conversely, if the put option is underpriced, the arbitrageur could buy the put option, sell the call option and the underlying asset, and lend an amount equal to the present value of the strike price. This creates a synthetic long forward position, which offsets the short forward position implied by the call and asset. At expiration, the arbitrageur can exercise the put option if it is in the money, use the proceeds to buy back the asset, and repay the loan, realizing a profit equal to the initial mispricing. Understanding put-call parity and its implications for arbitrage is crucial for derivatives trading and risk management. It ensures that options prices are consistent with each other and with the underlying asset price, preventing risk-free profit opportunities from persisting in the market.

The core of this question lies in understanding how a structured note, specifically a reverse convertible, behaves under different market conditions and how its payoff is calculated. A reverse convertible is essentially a short put option combined with a bond. The investor receives a higher coupon payment than a regular bond, but in exchange, they may have to take delivery of the underlying asset if its price falls below the knock-in barrier at maturity. The payoff at maturity depends on the relationship between the final asset price and the strike price (which is the initial price in this case). If the asset price is above the strike price, the investor receives the full principal. If the asset price is below the strike price, the investor receives the equivalent value of the shares that the principal could purchase at the initial price. In this scenario, the initial price is £100, and the knock-in barrier is 70% of the initial price, which is £70. The final price is £60, which is below the knock-in barrier. Therefore, the investor will receive shares. The number of shares is calculated as Principal / Initial Price = £1000 / £100 = 10 shares. The value of these shares at maturity is 10 shares * £60/share = £600. The total return is then calculated as the final value plus the coupon payment minus the initial investment: £600 + £80 – £1000 = -£320. The percentage return is -£320 / £1000 = -32%. A key consideration is the impact of the knock-in barrier. If the final price had been above £70, the investor would have received the full £1000 principal, resulting in a positive return. The reverse convertible offers higher income (the coupon) but exposes the investor to the downside risk of the underlying asset. It’s crucial to understand this trade-off when advising clients on such structured products. A misunderstanding of the knock-in feature and its potential impact on the final payoff is a common pitfall.

The question assesses the understanding of delta hedging, gamma, and the impact of market movements on a delta-hedged 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 positive gamma means the delta increases as the underlying asset price increases and decreases as the underlying asset price decreases. The initial portfolio value is irrelevant to the hedging strategy itself; it is the *change* in value that matters. We are given a portfolio with a delta of -5000 and a gamma of 15. This means that for every £1 increase in the underlying asset’s price, the portfolio’s delta will increase by 15. Conversely, for every £1 decrease, the delta will decrease by 15. To maintain a delta-neutral position, the trader must dynamically adjust the hedge. In this scenario, the underlying asset’s price increases by £2. First, calculate the change in delta: Change in Delta = Gamma * Change in Price = 15 * 2 = 30 The new delta of the portfolio becomes: New Delta = Initial Delta + Change in Delta = -5000 + 30 = -4970 To re-establish a delta-neutral position, the trader needs to offset this new delta of -4970. This means the trader needs to *sell* 4970 units of the underlying asset. Selling offsets the negative delta. Therefore, the trader needs to sell 4970 units of the underlying asset.

To determine the fair value of the swaption, we need to calculate the present value of the expected payoff at the expiry of the swaption. The payoff depends on the difference between the fixed rate of the underlying swap and the market swap rate at expiry, if this difference is positive. The expected swap rate can be estimated using the forward rates implied by the yield curve. First, we need to determine the forward swap rate for a 5-year swap starting in 2 years. We can approximate this using the formula: Forward Swap Rate = \[\frac{((1 + S_7)^{7} * N) – ((1 + S_2)^{2} * N)}{\sum_{i=1}^{5} \frac{N}{(1 + S_{2+i})^{2+i}}}\] Where \(S_t\) is the spot rate for maturity \(t\), and \(N\) is the notional amount. Plugging in the given spot rates: Forward Swap Rate = \[\frac{((1 + 0.045)^{7} * 1) – ((1 + 0.04)^{2} * 1)}{\sum_{i=1}^{5} \frac{1}{(1 + S_{2+i})^{2+i}}}\] The denominator becomes: \[\frac{1}{(1.045)^3} + \frac{1}{(1.05)^4} + \frac{1}{(1.055)^5} + \frac{1}{(1.06)^6} + \frac{1}{(1.065)^7}\] \[= 0.87629 + 0.82270 + 0.77215 + 0.72475 + 0.67995 = 3.87584\] The numerator becomes: \[(1.045)^7 – (1.04)^2 = 1.3609 – 1.0816 = 0.2793\] Forward Swap Rate = \[\frac{0.2793}{3.87584} = 0.07206\] or 7.206% Since the fixed rate of the underlying swap (6%) is less than the forward swap rate (7.206%), the swaption will be in the money at expiry. The payoff will be the present value of the difference between these rates over the 5-year period. Payoff per year = (7.206% – 6%) * Notional = 1.206% * £10 million = £120,600 Present value of this annuity (using the spot rates as discount rates): \[\sum_{i=1}^{5} \frac{120,600}{(1 + S_{2+i})^{2+i}}\] \[\frac{120,600}{(1.045)^3} + \frac{120,600}{(1.05)^4} + \frac{120,600}{(1.055)^5} + \frac{120,600}{(1.06)^6} + \frac{120,600}{(1.065)^7}\] \[= 105,658 + 99,268 + 93,211 + 87,489 + 82,032 = 467,658\] Finally, discount this back to today (2 years) using the 2-year spot rate: \[\frac{467,658}{(1.04)^2} = \frac{467,658}{1.0816} = 432,376\] Therefore, the approximate fair value of the swaption is £432,376. This calculation involves several steps that highlight key concepts in derivatives valuation: (1) Deriving forward rates from the yield curve is essential for pricing instruments with future payoffs. (2) The swaption’s value is contingent on the difference between the fixed rate and the expected future swap rate. (3) Present value calculations, incorporating the time value of money, are crucial for determining the fair value of future cash flows. (4) Understanding the relationship between spot rates and forward rates is fundamental for pricing and hedging interest rate derivatives.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Green Harvest Co-op,” which produces organic wheat. They want to protect themselves from potential price drops in the wheat market over the next year. They decide to use wheat futures contracts traded on the ICE Futures Europe exchange. The current spot price of organic wheat is £200 per tonne. The December wheat futures contract is trading at £210 per tonne. Green Harvest Co-op plans to hedge 5,000 tonnes of wheat. To determine the effectiveness of their hedge, we need to consider basis risk. Basis risk arises because the spot price and the futures price may not move perfectly in tandem. Several factors can influence basis risk in this scenario. First, the quality difference between the organic wheat produced by Green Harvest Co-op and the standard grade wheat underlying the futures contract. Organic wheat typically commands a premium, but this premium can fluctuate. Second, transportation costs from Green Harvest Co-op’s location to the delivery point specified in the futures contract. Increased transportation costs can widen the basis. Third, local supply and demand conditions for organic wheat in the UK market. A local surplus of organic wheat could depress the spot price relative to the futures price. Let’s assume that Green Harvest Co-op implements its hedge by selling 25 December wheat futures contracts (each contract covers 200 tonnes, so 25 contracts cover 5,000 tonnes). At the time of settlement in December, the spot price of Green Harvest Co-op’s organic wheat is £195 per tonne, and the December futures contract settles at £202 per tonne. The initial basis was £10 per tonne (£210 – £200), and the final basis is £7 per tonne (£202 – £195). The change in basis is £3 per tonne (£10 – £7). The gain on the futures contracts is £8 per tonne (£210 – £202). The effective price received by Green Harvest Co-op is the final spot price plus the gain on the futures contracts, minus the change in basis due to hedging. The effective price is calculated as follows: £195 (spot price) + £8 (futures gain) – £3 (change in basis) = £200 per tonne. Without hedging, Green Harvest Co-op would have received only £195 per tonne. The hedge increased their revenue by £5 per tonne, demonstrating the risk mitigation benefit. This example illustrates how basis risk can affect the outcome of a hedging strategy and how it needs to be carefully considered when using futures contracts for risk management.

The core of this question lies in understanding how a delta-neutral portfolio reacts to changes in volatility (vega) and time decay (theta) when nearing an earnings announcement. A delta-neutral portfolio is constructed to be insensitive to small changes in the underlying asset’s price. However, delta neutrality does not eliminate exposure to other risks, particularly volatility and time decay. Vega measures the sensitivity of the portfolio’s value to changes in the implied volatility of the options. A positive vega means the portfolio’s value increases with increasing volatility, and vice versa. Theta, on the other hand, measures the rate at which the portfolio’s value decreases due to the passage of time. Typically, options lose value as they approach their expiration date, and this decay accelerates closer to expiration. Earnings announcements are significant events that usually cause a spike in implied volatility due to increased uncertainty about the company’s future performance. After the announcement, volatility typically collapses, a phenomenon known as “volatility crush.” In this scenario, the portfolio is vega positive and theta negative. The increase in implied volatility before the earnings announcement benefits the portfolio (positive vega). However, the time decay works against it (negative theta). The key is to understand the magnitude of the volatility increase versus the time decay. Post-announcement, the volatility crush will hurt the portfolio, and time decay continues to erode value. To determine the outcome, we need to consider the interplay of these factors. If the volatility increase is substantial enough to outweigh the time decay before the announcement, the portfolio will initially gain value. However, the subsequent volatility crush will likely lead to a significant loss, potentially offsetting any initial gains and resulting in an overall loss. A crucial concept here is the understanding that delta hedging, while mitigating price risk, exposes the portfolio to other risks like volatility and time decay, especially around events like earnings announcements. Risk management involves not only hedging delta but also managing vega and theta exposures based on market expectations and event risks. The scenario tests the understanding of how these “Greeks” interact and influence portfolio performance in a real-world context.

Let’s consider a scenario involving a UK-based agricultural cooperative, “GreenHarvest,” which aims to protect its future revenue from potential fluctuations in wheat prices. GreenHarvest plans to use futures contracts to hedge its exposure. The cooperative anticipates harvesting 5,000 tonnes of wheat in six months. The current spot price of wheat is £200 per tonne. Wheat futures contracts, each representing 100 tonnes of wheat, are trading on the ICE Futures Europe exchange. The six-month futures price is £210 per tonne. To determine the number of contracts GreenHarvest needs to hedge its exposure, we divide the total quantity of wheat to be hedged by the contract size: 5,000 tonnes / 100 tonnes/contract = 50 contracts. GreenHarvest sells 50 wheat futures contracts at £210 per tonne. This locks in a price of £210 per tonne for their wheat. If, at the time of harvest, the spot price of wheat has fallen to £190 per tonne, GreenHarvest will sell its wheat in the spot market for £190 per tonne. However, they will simultaneously close out their futures position by buying back 50 wheat futures contracts. Since they initially sold at £210 and now buy back at £190, they make a profit of £20 per tonne on the futures contracts. The total profit on the futures contracts is 50 contracts * 100 tonnes/contract * £20/tonne = £100,000. The total revenue from selling wheat in the spot market is 5,000 tonnes * £190/tonne = £950,000. The effective price received by GreenHarvest is the total revenue from the spot market plus the profit from the futures contracts, divided by the total quantity of wheat: (£950,000 + £100,000) / 5,000 tonnes = £210 per tonne. Now consider the impact of basis risk. Basis risk is the risk that the price of the asset being hedged (the spot price of wheat) does not move exactly in tandem with the price of the futures contract. Suppose that at harvest, the spot price of wheat is £195 per tonne, and the futures price is £200 per tonne. GreenHarvest sells its wheat in the spot market for £195 per tonne, generating revenue of 5,000 tonnes * £195/tonne = £975,000. They close out their futures position by buying back 50 contracts at £200 per tonne, realizing a profit of £10 per tonne (selling at £210 and buying back at £200). The total profit on the futures contracts is 50 contracts * 100 tonnes/contract * £10/tonne = £50,000. The effective price received by GreenHarvest is (£975,000 + £50,000) / 5,000 tonnes = £205 per tonne. This is less than the £210 per tonne they initially aimed to lock in, demonstrating the impact of basis risk. Finally, let’s examine the role of clearing houses. Clearing houses, such as ICE Clear Europe, act as intermediaries in futures transactions, guaranteeing the performance of both parties. They require participants to post margin, which is a form of collateral that protects the clearing house against losses if a participant defaults. This reduces counterparty risk and ensures the integrity of the futures market.

The question assesses understanding of how macroeconomic factors influence derivative pricing, specifically focusing on interest rate swaps and inflation expectations. An interest rate swap involves exchanging fixed-rate interest payments for floating-rate interest payments, or vice versa. Changes in inflation expectations significantly impact the pricing of these swaps. When inflation is expected to rise, fixed-rate payments become less attractive to the payer (who is paying fixed), as the real value of those payments decreases over time. Conversely, the receiver of fixed payments benefits. The calculation involves determining the present value (PV) impact of an unexpected inflation increase on the fixed leg of the swap. We use the following logic: 1. **Inflation Impact on Discount Rate:** An increase in expected inflation directly increases the nominal discount rate used to calculate the present value of future cash flows. We assume a direct one-to-one relationship for simplicity. 2. **Present Value Calculation:** The present value of the fixed leg is calculated by discounting each fixed payment by the adjusted discount rate (original rate + inflation increase). The formula for the present value of a single payment is: \[PV = \frac{CF}{(1 + r)^n}\], where CF is the cash flow, r is the discount rate, and n is the number of years. 3. **Summation of Present Values:** The present values of all fixed payments are summed to find the total present value of the fixed leg. 4. **Change in Present Value:** The difference between the original present value (before the inflation increase) and the new present value (after the inflation increase) represents the impact on the fixed leg’s value. In this scenario, the original discount rate is 5% (0.05), and inflation increases by 2% (0.02), increasing the discount rate to 7% (0.07). The fixed payments are £100,000 per year for 5 years. The original PV is calculated as: \[PV_{original} = \sum_{n=1}^{5} \frac{100,000}{(1 + 0.05)^n} = 432,947.67\] The new PV, after the inflation increase, is: \[PV_{new} = \sum_{n=1}^{5} \frac{100,000}{(1 + 0.07)^n} = 410,020.35\] The change in present value is: \[\Delta PV = PV_{new} – PV_{original} = 410,020.35 – 432,947.67 = -22,927.32\] Therefore, the present value of the fixed leg decreases by £22,927.32. The party receiving the fixed payments would experience a loss in the value of their position.

To determine the profit or loss from a covered call strategy, we need to consider the initial cost of the stock, the premium received from selling the call option, and the final outcome (whether the option is exercised or not). In this scenario, the investor buys shares of BetaTech at £150, sells a call option with a strike price of £160 for a premium of £8, and the option is exercised when BetaTech’s price rises to £170. First, calculate the net cost of the stock after receiving the premium: £150 (stock cost) – £8 (premium) = £142. Next, determine the outcome if the option is exercised. The investor is obligated to sell the stock at the strike price of £160. The profit from selling the stock is £160 (strike price) – £142 (net cost) = £18. Therefore, the overall profit is £18 per share. Consider another scenario: an investor holds shares of GammaCorp, purchased at £80. They sell a covered call with a strike price of £85, receiving a premium of £5. If GammaCorp’s price rises to £95 and the option is exercised, the investor’s profit is calculated as follows: Net cost = £80 – £5 = £75. Profit from selling at strike price = £85 – £75 = £10. Now, consider a scenario with a loss. An investor buys Delta Industries shares at £200 and sells a covered call with a strike price of £210, receiving a premium of £12. If Delta Industries’ price remains at £190 and the option expires worthless, the investor’s loss is calculated as follows: Net cost = £200 – £12 = £188. Loss = £188 – £190 = -£8. Covered call strategies limit upside potential but provide downside protection through the premium received. The maximum profit is achieved when the stock price rises to the strike price, and the option is exercised. The premium helps offset potential losses if the stock price declines. The strategy is most effective in stable or slightly bullish markets.

The question assesses the understanding of put-call parity, specifically in a scenario involving transaction costs and dividends. Put-call parity is a fundamental concept in options pricing that describes the relationship between the prices of European put and call options with the same strike price and expiration date. The basic put-call parity formula is: \(C + PV(X) = P + S\), where C is the call option price, PV(X) is the present value of the strike price, P is the put option price, and S is the current stock price. In this scenario, we need to adjust the formula for transaction costs and dividends. Transaction costs affect the arbitrage opportunity, creating a band within which no arbitrage is possible. The dividend payment reduces the stock price, affecting the call option price. Here’s the breakdown of the calculation: 1. **Basic Put-Call Parity:** \(C + PV(X) = P + S\) 2. **Adjusted for Dividends:** \(C + PV(X) = P + S – PV(Div)\), where PV(Div) is the present value of the dividends. 3. **Adjusted for Transaction Costs:** To determine the no-arbitrage range, we consider the costs of both buying and selling the assets. * *Upper Bound:* \(C + PV(X) + Cost_{buy} \ge P + S – PV(Div) – Cost_{sell}\) * *Lower Bound:* \(C + PV(X) – Cost_{sell} \le P + S – PV(Div) + Cost_{buy}\) Where \(Cost_{buy}\) and \(Cost_{sell}\) are the transaction costs associated with buying and selling the respective assets. Let’s calculate the present value of the strike price and the dividend: * Strike Price (X) = £50 * Risk-free rate (r) = 5% * Time to expiration (t) = 6 months = 0.5 years * Dividend (Div) = £2, payable in 3 months (0.25 years) \(PV(X) = \frac{X}{e^{rt}} = \frac{50}{e^{0.05 \times 0.5}} = \frac{50}{1.0253} \approx 48.76\) \(PV(Div) = \frac{Div}{e^{rt}} = \frac{2}{e^{0.05 \times 0.25}} = \frac{2}{1.0126} \approx 1.975\) Now, consider the costs: * Cost to buy stock = 0.5% of £52 = £0.26 * Cost to sell stock = 0.5% of £52 = £0.26 * Cost to buy call = 0.5% of £4 = £0.02 * Cost to sell call = 0.5% of £4 = £0.02 * Cost to buy put = 0.5% of £3 = £0.015 * Cost to sell put = 0.5% of £3 = £0.015 Upper Bound: \(4 + 48.76 + 0.02 \ge 3 + 52 – 1.975 – 0.26 – 0.015\) \(52.78 \ge 52.75\) Lower Bound: \(4 + 48.76 – 0.02 \le 3 + 52 – 1.975 + 0.26 + 0.015\) \(52.74 \le 53.39\) Thus, the theoretical put price range is calculated as: * Lower Put Price = \(C + PV(X) – S + PV(Div) – Cost_{sell\,call} – Cost_{buy\,stock} – Cost_{buy\,put} = 4 + 48.76 – 52 + 1.975 – 0.02 – 0.26 – 0.015 = 2.44\) * Upper Put Price = \(C + PV(X) – S + PV(Div) + Cost_{buy\,call} + Cost_{sell\,stock} + Cost_{sell\,put} = 4 + 48.76 – 52 + 1.975 + 0.02 + 0.26 + 0.015 = 3.01\) Therefore, the theoretical put price should fall between £2.44 and £3.01 to prevent arbitrage.

Let’s consider a scenario where a portfolio manager uses options to hedge against potential losses in a volatile market. The portfolio manager wants to protect their portfolio from a sharp decline but also wants to participate in potential upside. They decide to implement a collar strategy. A collar involves buying protective puts and selling covered calls simultaneously. The cost of the puts is partially offset by the premium received from selling the calls. First, we need to determine the net premium paid or received for establishing the collar. This is calculated by subtracting the call premium received from the put premium paid. In this case, the put premium is 3.50 per share, and the call premium is 2.00 per share, resulting in a net premium paid of 1.50 per share. Next, we need to calculate the maximum loss and maximum gain. The maximum loss occurs if the stock price falls below the strike price of the put option, minus the net premium paid. The maximum gain occurs if the stock price rises above the strike price of the call option, plus the net premium received (or minus the net premium paid). In this case, the put strike price is 95, and the call strike price is 105. The maximum loss is calculated as: Maximum Loss = Put Strike Price – Initial Stock Price + Net Premium Paid Maximum Loss = 95 – 100 + 1.50 = -3.50 The maximum gain is calculated as: Maximum Gain = Call Strike Price – Initial Stock Price – Net Premium Paid Maximum Gain = 105 – 100 – 1.50 = 3.50 Now, let’s calculate the breakeven point. The breakeven point for a collar strategy is the initial stock price plus the net premium paid or minus the net premium received. Since the net premium is paid, we subtract it from the initial stock price. Breakeven Point = Initial Stock Price + Net Premium Paid Breakeven Point = 100 + 1.50 = 101.50 The portfolio manager’s breakeven point is at $101.50. This means that if the stock price rises above $101.50, the portfolio manager will start making a profit. If the stock price falls below $101.50, the portfolio manager will incur a loss. The collar strategy is designed to limit both the upside and downside potential. The put option protects against significant losses if the stock price declines, while the call option caps the potential gains if the stock price rises. The net premium paid or received affects the breakeven point and the maximum gain or loss.

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. However, real-world trading isn’t frictionless. Each trade incurs costs, which directly impact the profitability of the hedging strategy. The breakeven point is crucial: it’s the point where the gains from the option offset the hedging costs. Let’s break down the calculation. First, determine the total cost of hedging. This includes the initial cost of establishing the hedge (buying or selling the underlying asset) and the subsequent costs of rebalancing the hedge as the delta changes. The number of shares to buy or sell depends on the option’s delta. The higher the delta, the more shares are needed to hedge. As the option moves closer to being in-the-money or out-of-the-money, the delta changes, necessitating adjustments to the hedge. Each adjustment incurs transaction costs. The profit or loss from the option position is determined by the difference between the option’s final value and its initial price. If the option expires worthless, the loss is simply the initial option premium paid. The breakeven point is reached when the profit from the option (or the avoided loss) equals the total cost of hedging. In this scenario, we need to consider the impact of transaction costs on the overall profitability of a delta-hedged position. The breakeven calculation becomes more complex because the transaction costs reduce the net profit realized from the hedge. Therefore, a larger movement in the underlying asset price is required to offset both the initial cost of the option and the cumulative transaction costs. Let’s imagine a simplified analogy: you’re trying to keep a boat stable in choppy waters. Delta hedging is like constantly adjusting the ballast in the boat to keep it level. Each adjustment requires energy (transaction costs). If the waves are small, the energy spent adjusting the ballast might outweigh the benefit of staying perfectly level. The breakeven point is when the waves are big enough that the energy spent on ballast adjustment is justified by the stability gained. To determine the exact breakeven point, one would typically use a simulation or a more complex model that incorporates the expected volatility of the underlying asset and the frequency of rebalancing. However, the core concept remains the same: transaction costs increase the range within which the delta-hedged position remains unprofitable. Only a substantial price movement outside this range will generate a profit sufficient to cover all costs.

The question assesses understanding of how a sudden, unexpected shift in market volatility affects option positions, specifically straddles, and how the trader should react based on their views. The key is recognizing that a straddle profits from large price movements, regardless of direction. An increase in implied volatility increases the value of both the call and put options in the straddle, even if the underlying asset’s price hasn’t moved yet. However, the trader’s *belief* about future volatility is crucial. If they believe volatility will revert to its mean, they might take profits by closing the position. If they think this is the start of a larger trend, they might hold or even add to the position. The scenario introduces uncertainty about the cause of the volatility spike (economic data vs. technical glitch) to force a deeper consideration of the underlying market dynamics. We will use a simplified Black-Scholes model intuition: Option Price ≈ Intrinsic Value + Time Value. Time Value is heavily influenced by volatility. Let’s consider a hypothetical straddle: A trader holds a straddle on shares of UK Oil PLC, with a strike price of £50, consisting of one call and one put option, each costing £2. Initially, the implied volatility is 20%. The total cost of the straddle is £4. Suddenly, news of unexpected sanctions causes the implied volatility to jump to 30%. The price of both the call and put options increases due to the higher volatility. Let’s say each option’s price increases by £1.50. Now, the call and put are each worth £3.50. The total value of the straddle is now £7, resulting in a profit of £3. If the trader believes the sanctions are a short-term issue and volatility will decrease soon, they might close the position, realizing the £3 profit. However, if they believe the sanctions will escalate, leading to further price uncertainty, they might hold onto the straddle, anticipating even larger price swings in UK Oil PLC’s shares and, therefore, further increases in the value of the options. Alternatively, if the trader had a strong conviction that the volatility jump was a glitch and that the underlying UK Oil PLC price would remain stable, they might even consider writing additional straddles to profit from the expected volatility decrease.

The question focuses on understanding how different market conditions and investor sentiment can influence the implied volatility of options, and subsequently, their prices. A ‘volatility smile’ indicates that out-of-the-money (OTM) and in-the-money (ITM) options have higher implied volatilities than at-the-money (ATM) options. This often reflects a market expectation of larger price swings or increased uncertainty about the underlying asset’s future price. Several factors can contribute to a volatility smile, including: * **Fear of Large Downward Moves (Crash Risk):** Investors often purchase OTM puts as insurance against significant market declines. This increased demand for OTM puts drives up their prices and, consequently, their implied volatilities. This is a common explanation for the skew typically observed in equity markets. * **Supply and Demand Imbalances:** If there’s a higher demand for certain strike prices due to hedging activity or speculative interest, it can distort the implied volatility curve. For example, large institutional investors might need to hedge specific positions, leading to increased demand for particular options. * **Market Sentiment:** Overall market sentiment can influence the shape of the volatility smile. During periods of uncertainty or heightened risk aversion, investors tend to bid up the prices of OTM puts, creating or exacerbating the smile. * **Liquidity:** Options with very high or very low strike prices may have lower liquidity. The lack of active trading can lead to price distortions and higher implied volatilities. * **Model Limitations:** The Black-Scholes model assumes constant volatility, which is unrealistic. The volatility smile is, in part, a reflection of the model’s limitations and the market’s recognition that volatility is not constant. The correct answer reflects an understanding of how these factors interact to shape the volatility smile. The incorrect answers present scenarios that are less likely to cause or exacerbate a volatility smile, or misinterpret the relationship between market sentiment and option prices. The question requires the candidate to apply their knowledge of option pricing theory and market dynamics to a practical scenario.

The question assesses understanding of the impact of correlation between assets in a portfolio when using derivatives for hedging. Specifically, it explores how imperfect correlation affects the effectiveness of a hedging strategy using futures contracts. The core concept is that a hedge is most effective when the asset being hedged and the hedging instrument (in this case, a futures contract) move in a perfectly negatively correlated manner. A correlation of less than -1 (which is impossible) or greater than -1 introduces basis risk, meaning the hedge won’t perfectly offset losses in the underlying asset. The effectiveness of the hedge is measured by how much it reduces the portfolio’s volatility (standard deviation). Here’s how to approach the problem: 1. **Calculate the unhedged portfolio volatility:** This is given as 15%. 2. **Understand the impact of correlation:** A perfect negative correlation (-1) would theoretically eliminate all volatility if the hedge ratio were perfectly calibrated. However, in reality, correlations are rarely perfect. 3. **Determine the hedge effectiveness:** The question states the correlation is -0.8. This means the hedge will reduce volatility, but not eliminate it entirely. The lower the absolute value of the correlation (closer to 0), the less effective the hedge. A correlation of -0.8 suggests a substantial, but not complete, reduction in volatility. We can’t calculate the exact reduced volatility without more information about the hedge ratio (the amount of futures contracts used to hedge the portfolio). 4. **Analyze the options:** We are looking for an option that reflects a reduction in volatility, but not a complete elimination, and acknowledges that the hedge will not be perfect. 5. **Choose the most plausible answer:** Option a) is the most likely answer, as the hedge will not be perfect due to the correlation being -0.8. The hedge will reduce volatility, but not completely eliminate it.

The question assesses understanding of the impact of volatility on option prices, specifically focusing on how changes in implied volatility affect the value of a delta-neutral portfolio consisting of options and the underlying asset. A delta-neutral portfolio is constructed to be insensitive to small changes in the price of the underlying asset. However, it remains sensitive to other factors, most notably volatility (vega). The vega of a portfolio measures the change in the portfolio’s value for a 1% change in the implied volatility of the options in the portfolio. A positive vega indicates that the portfolio’s value will increase if implied volatility increases, while a negative vega indicates the portfolio’s value will decrease if implied volatility increases. The initial portfolio consists of short call options. Being short options means the portfolio has negative vega. When volatility increases, the value of the short options increases, leading to a loss in the portfolio. To hedge against this volatility risk, the portfolio manager can buy options (either calls or puts). Buying options introduces positive vega to the portfolio, offsetting the negative vega of the short call options. The calculation involves determining the number of options needed to neutralize the portfolio’s vega. The portfolio’s initial vega is -5,000. The manager wants to add long put options with a vega of 25 per option. To neutralize the portfolio, the number of put options needed is calculated as follows: Number of put options = – (Portfolio Vega / Vega per put option) = – (-5,000 / 25) = 200 Therefore, the manager needs to purchase 200 put options to neutralize the portfolio’s vega. This ensures that the portfolio’s value is less sensitive to changes in implied volatility. The concept highlights the dynamic nature of hedging and the need to adjust positions as market conditions change. This question emphasizes practical application of derivatives knowledge, going beyond textbook definitions to real-world risk management scenarios.

To determine the fair premium for a credit default swap (CDS), we need to calculate the present value of expected future payouts and equate it to the present value of the premium payments. The calculation involves several steps: 1. **Calculate the Expected Payout:** The expected payout is the product of the probability of default and the loss given default (LGD). In this case, the annual default probability is 3%, and the LGD is 60%. Therefore, the expected payout each year is \(0.03 \times 0.60 = 0.018\) or 1.8% of the notional amount. 2. **Determine the Present Value of Expected Payouts:** We discount the expected payout for each year back to the present using the risk-free rate. * Year 1: \(\frac{0.018}{1 + 0.04} = 0.017307692\) * Year 2: \(\frac{0.018}{(1 + 0.04)^2} = 0.016642012\) * Year 3: \(\frac{0.018}{(1 + 0.04)^3} = 0.015998088\) * Year 4: \(\frac{0.018}{(1 + 0.04)^4} = 0.015371238\) * Year 5: \(\frac{0.018}{(1 + 0.04)^5} = 0.01476079\) The total present value of expected payouts is the sum of these values: \(0.017307692 + 0.016642012 + 0.015998088 + 0.015371238 + 0.01476079 = 0.08007982\) 3. **Calculate the Present Value of Premium Payments:** The premium is paid annually, so we need to find the premium rate (CDS spread) that equates the present value of premium payments to the present value of expected payouts. Let the annual premium rate be *P*. The present value of the premium payments is: * Year 1: \(\frac{P}{1 + 0.04}\) * Year 2: \(\frac{P}{(1 + 0.04)^2}\) * Year 3: \(\frac{P}{(1 + 0.04)^3}\) * Year 4: \(\frac{P}{(1 + 0.04)^4}\) * Year 5: \(\frac{P}{(1 + 0.04)^5}\) Factoring out *P*, we get: \[P \times \left( \frac{1}{1.04} + \frac{1}{1.04^2} + \frac{1}{1.04^3} + \frac{1}{1.04^4} + \frac{1}{1.04^5} \right)\] \[P \times (0.9615 + 0.9246 + 0.8890 + 0.8548 + 0.8219) = P \times 4.4518\] 4. **Equate Present Values and Solve for P:** We set the present value of expected payouts equal to the present value of premium payments: \[0.08007982 = P \times 4.4518\] \[P = \frac{0.08007982}{4.4518} = 0.017988\] Thus, the fair premium (CDS spread) is approximately 1.80%. This calculation ensures that the protection buyer compensates the protection seller adequately for the credit risk assumed. Understanding the underlying mathematics is vital for any investment advisor dealing with derivatives, as it provides a solid foundation for assessing risk and return. It goes beyond mere memorization and enables informed decision-making.

The question assesses understanding of exotic derivatives, specifically barrier options, and how their payoff structure is affected by market volatility and correlation. A down-and-out call option becomes worthless if the underlying asset’s price touches or falls below the barrier level before the option’s expiration date. The investor must understand the impact of increasing correlation between assets on the probability of hitting the barrier. The calculation involves understanding that higher correlation between the assets in the portfolio means that if one asset decreases in value, the other is more likely to also decrease. This increases the probability that the portfolio value as a whole will decrease and hit the barrier level of the down-and-out call option. Therefore, the correct answer is that the option is more likely to expire worthless, and the investor has effectively implemented a view that the correlation between their assets will remain low, as high correlation would negatively impact the option’s value. A lower correlation would decrease the likelihood of hitting the barrier.

The question assesses understanding of the impact of volatility on option prices, particularly in the context of earnings announcements. Increased volatility typically leads to higher option premiums because it increases the probability of the underlying asset’s price moving significantly in either direction, making both calls and puts more valuable. The skewness of volatility, where implied volatility for out-of-the-money puts is higher than for out-of-the-money calls, reflects a market perception of greater downside risk. The trader’s strategy exploits the expected volatility crush after the earnings announcement by selling options (both calls and puts) with high premiums and buying them back at lower premiums once volatility subsides. The Black-Scholes model is used to estimate the fair value of the options, taking into account factors like the current stock price, strike price, time to expiration, risk-free interest rate, and implied volatility. The trader’s profit is derived from the difference between the initial selling price of the options and the subsequent buying price, less any transaction costs. Here’s the breakdown of the calculation: 1. **Calculate the initial value of the straddle:** * Call option premium: £5.50 * Put option premium: £4.50 * Total initial premium received: £5.50 + £4.50 = £10.00 per share 2. **Calculate the value of the straddle after the earnings announcement:** * Call option premium: £2.00 * Put option premium: £1.50 * Total premium paid to close the position: £2.00 + £1.50 = £3.50 per share 3. **Calculate the profit per share:** * Profit per share: Initial premium received – Premium paid to close * Profit per share: £10.00 – £3.50 = £6.50 4. **Calculate the total profit for 10 contracts (1,000 shares per contract):** * Total shares: 10 contracts \* 1,000 shares/contract = 10,000 shares * Total profit: £6.50/share \* 10,000 shares = £65,000 5. **Calculate the profit after brokerage fees:** * Brokerage fees: £5 per contract \* 10 contracts \* 2 (open and close) = £100 * Net Profit = £65,000 – £100 = £64,900 Therefore, the trader’s net profit is £64,900.

The question focuses on the impact of correlation on hedging strategies using futures contracts, specifically in the context of basis risk. Basis risk arises because the price of the asset being hedged and the price of the futures contract are not perfectly correlated. Changes in correlation directly affect the effectiveness of the hedge. A lower correlation means the hedge will be less effective, and a higher correlation means the hedge will be more effective. To determine the effectiveness of a hedge, we look at the hedge ratio, which can be estimated using the correlation coefficient (ρ) between the asset’s price changes and the futures contract’s price changes, multiplied by the ratio of the standard deviation of the asset’s price changes (σ_asset) to the standard deviation of the futures contract’s price changes (σ_futures). This can be represented as: Hedge Ratio = ρ * (σ_asset / σ_futures) The effectiveness of the hedge is measured by how well it reduces the variance of the hedged position. The variance reduction is proportional to the square of the correlation coefficient (ρ^2). If the correlation is 0.8, the variance reduction is 0.8^2 = 0.64 or 64%. If the correlation drops to 0.6, the variance reduction is 0.6^2 = 0.36 or 36%. Therefore, the percentage decrease in the effectiveness of the hedge is calculated as: [(Original Variance Reduction – New Variance Reduction) / Original Variance Reduction] * 100 In this case: Original Variance Reduction = 0.8^2 = 0.64 New Variance Reduction = 0.6^2 = 0.36 Percentage Decrease = [(0.64 – 0.36) / 0.64] * 100 = (0.28 / 0.64) * 100 = 43.75% The hedge becomes 43.75% less effective.

The question explores the application of put-call parity in a scenario involving transaction costs. Put-call parity is a fundamental relationship in options pricing that links the prices of a European call option, a European put option, the underlying asset, and a risk-free bond. The basic formula is: \(C + PV(K) = P + S\), where C is the call option price, PV(K) is the present value of the strike price, P is the put option price, and S is the spot price of the underlying asset. In the presence of transaction costs, the parity relationship can be violated, creating potential arbitrage opportunities. However, these opportunities are not risk-free in practice, as the transaction costs need to be accounted for. The key is to identify the direction of the inequality and the actions needed to exploit the mispricing. If \(C + PV(K) > P + S + \text{Transaction Costs}\), the strategy involves selling the call, buying the put, short selling the underlying asset, and lending the present value of the strike price. The profit is the difference between the left and right sides, minus the transaction costs. If \(C + PV(K) < P + S - \text{Transaction Costs}\), the strategy involves buying the call, selling the put, buying the underlying asset, and borrowing the present value of the strike price. The profit is the difference between the right and left sides, minus the transaction costs. In this specific question, we need to calculate the present value of the strike price using the risk-free rate and then compare the two sides of the put-call parity equation, considering the transaction costs. The correct arbitrage strategy is the one that exploits the mispricing after accounting for all costs. The profit is calculated as the difference between the proceeds from the sold instruments and the cost of the purchased instruments, less any transaction costs. The present value of the strike price (K) is calculated as \( PV(K) = \frac{K}{(1+r)^t} \) where r is the risk-free rate and t is the time to expiration. In this case, \( PV(K) = \frac{105}{(1+0.05)^{0.5}} = \frac{105}{1.0247} \approx 102.47 \). Now, we compare \(C + PV(K)\) with \(P + S\): \(C + PV(K) = 8 + 102.47 = 110.47\) \(P + S = 6 + 100 = 106\) The difference is \(110.47 - 106 = 4.47\). The transaction cost is 0.50. Since \(C + PV(K) > P + S + \text{Transaction Costs}\) or \(110.47 > 106 + 0.50\), the arbitrage strategy is to sell the call, buy the put, short sell the stock, and lend the present value of the strike price. The profit is the difference minus the transaction costs, \(4.47 – 0.50 = 3.97\).

Let’s analyze a scenario involving a UK-based agricultural cooperative, “Golden Harvest,” which seeks to protect its future wheat sales from fluctuating market prices using futures contracts. Understanding basis risk is crucial. Basis risk arises because the price of a futures contract might not perfectly correlate with the spot price of the underlying asset (in this case, Golden Harvest’s wheat) at the time of delivery. This difference, known as the basis, can fluctuate due to factors like transportation costs, storage fees, local supply and demand imbalances, and quality variations. To quantify basis risk, we need to understand how the basis changes over time. Suppose Golden Harvest enters into a short hedge using wheat futures contracts traded on the ICE Futures Europe exchange. The cooperative plans to deliver its wheat in November. In August, the November wheat futures contract is trading at £200 per tonne, and Golden Harvest anticipates selling its wheat locally for £195 per tonne. The initial basis is £195 – £200 = -£5 per tonne. As November approaches, the spot price in Golden Harvest’s local market rises to £210 per tonne due to unexpected local demand. Simultaneously, the November wheat futures contract settles at £212 per tonne. The final basis is £210 – £212 = -£2 per tonne. Golden Harvest’s effective selling price can be calculated as follows: Initial futures price (£200) + (Final basis – Initial basis) = £200 + (-£2 – (-£5)) = £200 + £3 = £203 per tonne. However, if the final basis had been, say, -£10 per tonne (spot price £202, futures price £212), the effective selling price would have been £200 + (-£10 – (-£5)) = £200 – £5 = £195 per tonne. This illustrates how unfavorable basis changes can erode the benefits of hedging. Now, consider a more complex scenario. Golden Harvest also uses options on wheat futures to manage price volatility. They implement a collar strategy, buying put options to protect against price declines and selling call options to offset the cost. If the futures price moves significantly outside the strike prices of the options, the collar’s effectiveness is reduced. The combined impact of basis risk and the limitations of the option collar can lead to unexpected outcomes. Let’s assume Golden Harvest also faces regulatory constraints imposed by the Financial Conduct Authority (FCA) regarding the use of derivatives. These regulations might limit the cooperative’s ability to dynamically adjust its hedging strategy in response to changing market conditions, further exacerbating the impact of basis risk. The regulations aim to protect the cooperative from excessive speculation but can also hinder effective risk management.