Derivatives Level 4 (Investment Advice Diploma) Exam Quiz Quiz 15

The question assesses the understanding of delta-hedging a portfolio of options, specifically the need to rebalance the hedge as the underlying asset price changes. Delta represents the sensitivity of the option’s price to changes in the underlying asset’s price. Delta-hedging involves taking a position in the underlying asset to offset the delta of the option portfolio, aiming to create a delta-neutral position. The key is that delta changes as the underlying asset price moves, requiring dynamic adjustments to maintain the delta-neutral hedge. The calculation involves determining the initial hedge ratio based on the portfolio’s delta and then adjusting the hedge ratio after the asset price change. The profit or loss on the option portfolio is calculated based on the change in the option price, while the profit or loss on the hedging position is calculated based on the change in the asset price. The rebalancing cost is the cost of adjusting the hedge to maintain delta neutrality. The net profit or loss is the sum of the profit/loss on the option portfolio, the profit/loss on the hedging position, and the rebalancing cost. Here’s a breakdown of the calculations: 1. **Initial Portfolio Delta:** 500 contracts \* 0.6 delta/contract = 300 2. **Initial Hedge:** Short 300 shares of the underlying asset. 3. **Asset Price Change:** £100 to £105, an increase of £5. 4. **Profit/Loss on Hedge:** Short 300 shares \* £5 increase = -£1500 (Loss) 5. **New Option Delta:** Delta increases to 0.65. New portfolio delta = 500 \* 0.65 = 325 6. **Rebalancing:** Need to short an additional 25 shares (325 – 300). 7. **Rebalancing Cost:** Short 25 shares at £105 = £2625 (Cost). This represents the cost of selling the additional shares. 8. **Option Price Change:** Option price increases from £5 to £5.50, an increase of £0.50. 9. **Profit/Loss on Option Portfolio:** 500 contracts \* 100 shares/contract \* £0.50 increase = £25,000 (Profit) 10. **Net Profit/Loss:** £25,000 (Option Profit) – £1500 (Hedge Loss) – £2625 (Rebalancing Cost) = £20,875 The example illustrates the dynamic nature of delta-hedging. If the portfolio was not rebalanced, the change in delta would expose the portfolio to greater risk from further price movements. The rebalancing cost is a critical component of evaluating the effectiveness of a delta-hedging strategy. The example also demonstrates how a seemingly simple hedging strategy involves multiple calculations and considerations to manage risk effectively. The key is to understand how delta changes and how those changes affect the overall portfolio exposure.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Green Harvest,” which wants to hedge against price volatility in their wheat crop using futures contracts traded on the London International Financial Futures and Options Exchange (LIFFE). Green Harvest anticipates harvesting 500,000 bushels of wheat in six months. The current spot price is £5.00 per bushel, and the six-month futures contract is trading at £5.20 per bushel. The cooperative decides to short (sell) 100 futures contracts, each representing 5,000 bushels (500,000 bushels / 5,000 bushels per contract = 100 contracts). To analyze the hedging effectiveness, we’ll consider two scenarios at the time of harvest: Scenario 1: The spot price of wheat is £4.80 per bushel. Green Harvest sells their wheat in the spot market for £4.80 per bushel. Simultaneously, they close out their futures position by buying back the 100 contracts at £4.90 per bushel. Their profit on the futures contracts is (£5.20 – £4.90) * 5,000 bushels/contract * 100 contracts = £150,000. Their revenue from selling wheat is £4.80/bushel * 500,000 bushels = £2,400,000. Total revenue is £2,400,000 + £150,000 = £2,550,000. Effective price per bushel is £2,550,000 / 500,000 bushels = £5.10 per bushel. Scenario 2: The spot price of wheat is £5.50 per bushel. Green Harvest sells their wheat in the spot market for £5.50 per bushel. Simultaneously, they close out their futures position by buying back the 100 contracts at £5.60 per bushel. Their loss on the futures contracts is (£5.60 – £5.20) * 5,000 bushels/contract * 100 contracts = £200,000. Their revenue from selling wheat is £5.50/bushel * 500,000 bushels = £2,750,000. Total revenue is £2,750,000 – £200,000 = £2,550,000. Effective price per bushel is £2,550,000 / 500,000 bushels = £5.10 per bushel. This example illustrates how futures contracts can be used to hedge price risk. While Green Harvest doesn’t perfectly lock in the initial futures price due to basis risk (the difference between the spot price and the futures price), they significantly reduce their exposure to price fluctuations. The effective price they receive is closer to the initial futures price than either of the spot prices at harvest time. This demonstrates the risk management benefits of hedging with futures contracts. The Financial Conduct Authority (FCA) emphasizes the importance of understanding basis risk when advising clients on hedging strategies.

The question assesses the understanding of how implied volatility impacts option pricing and Greeks, specifically in the context of a complex option strategy and a significant market event. It requires candidates to synthesize knowledge of volatility smiles, option greeks (delta, gamma, vega), and the practical implications of unexpected news announcements on derivative positions. To determine the most likely outcome, we need to consider how the implied volatility skew reacts to the announcement and how this affects the option prices and Greeks. 1. **Volatility Skew:** Typically, the implied volatility skew for equity options is upward sloping, meaning out-of-the-money (OTM) calls have higher implied volatility than at-the-money (ATM) options, and OTM puts have even higher implied volatility. A positive announcement will likely cause a reduction in overall implied volatility and a flattening of the skew, as downside risk is perceived to be lower. 2. **Impact on Option Prices:** Since the strategy involves selling OTM calls and puts, the price of these options will decrease due to the reduction in implied volatility. The short calls will decrease in value as implied volatility decreases, but the short puts will decrease even more significantly due to the combined effect of decreased implied volatility and the flattening skew. 3. **Impact on Greeks:** * **Delta:** The overall delta of the portfolio will likely become less negative. The short puts have negative delta, and their delta will become less negative as their price decreases. The short calls have positive delta, and their delta will also decrease as their price decreases. * **Gamma:** The gamma of the portfolio will likely decrease. Gamma is highest for ATM options and decreases as you move OTM. The flattening of the volatility skew will reduce the gamma of the OTM options. * **Vega:** The vega of the portfolio will be negative since the investor is short options. The decrease in implied volatility will cause a loss for the investor. Given these considerations, the most likely outcome is that the portfolio value increases due to the decrease in implied volatility, particularly affecting the short puts, and the overall delta becomes less negative.

This question tests the understanding of delta hedging and its limitations, particularly in the context of gamma risk. Delta hedging aims to neutralize the sensitivity of an option portfolio to small changes in the underlying asset’s price. However, delta is not constant; it changes as the underlying asset’s price moves, a phenomenon known as gamma. A higher gamma means the delta changes more rapidly, requiring more frequent adjustments to the hedge. The cost of these adjustments, known as rebalancing costs, can erode the profitability of the hedge. In this scenario, the portfolio manager is attempting to delta hedge a short option position. A short option position has negative gamma, meaning that as the underlying asset’s price increases, the delta becomes more negative (for a short call) or less positive (for a short put). Conversely, as the underlying asset’s price decreases, the delta becomes less negative or more positive. To maintain a delta-neutral position, the portfolio manager must dynamically adjust the hedge by buying or selling the underlying asset. The breakeven point considers the initial premium received from selling the options, the costs of rebalancing the hedge, and any gains or losses from the hedge itself. If the underlying asset’s price moves significantly and frequently, the rebalancing costs can outweigh the initial premium and any hedging gains, resulting in a loss. The optimal rebalancing frequency depends on the gamma of the option position, the volatility of the underlying asset, and the transaction costs associated with rebalancing. Higher gamma and volatility necessitate more frequent rebalancing, which increases costs. The calculation involves determining the point at which the rebalancing costs offset the initial premium received. The more volatile the underlying asset, the more often the hedge needs to be rebalanced, leading to higher costs. The question requires understanding how gamma affects the effectiveness of delta hedging and how transaction costs can impact the overall profitability of the strategy. Let’s assume the manager initially receives a premium of £50,000. The cost of each rebalance is £500, and the portfolio experiences 40 rebalances over the period. The total rebalancing cost is 40 * £500 = £20,000. To breakeven, the hedge’s gains or losses must offset the initial premium minus the rebalancing costs. Therefore, the hedge can afford to lose £50,000 (initial premium) – £20,000 (rebalancing costs) = £30,000.

To determine the appropriate hedging strategy, we must first calculate the implied volatility of the options, then assess the portfolio’s exposure to changes in implied volatility (vega). After that, we need to calculate the number of options contracts required to neutralize the portfolio’s vega. 1. **Calculate the Implied Volatility:** This usually requires an iterative process using an option pricing model (like Black-Scholes) and the market price of the option. However, for simplicity in this exam question, we are given the implied volatility directly. 2. **Calculate Portfolio Vega:** Vega represents the sensitivity of the portfolio’s value to a 1% change in implied volatility. It is calculated as the sum of the vegas of each asset in the portfolio. Here, the portfolio consists of shares and call options. 3. **Calculate Vega of a Single Option Contract:** This is given as 0.05 per contract. Since each contract represents 100 shares, the vega is for 100 shares. 4. **Determine the Number of Option Contracts to Hedge:** To hedge the portfolio’s vega, we need to find the number of option contracts that will offset the portfolio’s vega. The formula is: Number of contracts = – (Portfolio Vega / Vega per contract) In this case: Number of contracts = – (15000 / (0.05 \* 100)) = -3000 The negative sign indicates that we need to *sell* the options to hedge the positive vega of the portfolio. Selling the options will create a negative vega position that offsets the positive vega of the existing portfolio. Example: Imagine a farmer who wants to hedge the price of their wheat crop using futures contracts. If the farmer’s crop has a “vega” of 500 (meaning the value of the crop is highly sensitive to price volatility), and each futures contract has a vega of 2, the farmer would need to sell 250 futures contracts to hedge their price risk. Analogy: Think of vega as the sensitivity of a sailboat to wind gusts. A portfolio with high vega is like a sailboat with a very large sail – it reacts strongly to changes in wind (volatility). To hedge, you would add weight to the keel (sell options) to make the boat more stable and less reactive.

The core of this question revolves around understanding how a specific exotic derivative, a barrier option, interacts with a hedging strategy involving standard European options. The scenario tests the candidate’s comprehension of delta hedging, barrier option mechanics, and the potential for path dependency to influence hedging outcomes. The calculation involves several key steps: 1. **Initial Delta Calculation:** The delta of the European call option is given as 0.6. The investor initially sells 1000 shares, requiring them to buy 600 European call options to delta hedge their position. 2. **Barrier Hit Scenario:** When the underlying asset price drops to £90, the knock-out barrier is triggered, rendering the barrier option worthless. This eliminates the short position in the barrier option. 3. **Delta Re-adjustment:** With the barrier option knocked out, the investor is left with only the delta-hedged European call option position. The investor needs to unwind the initial delta hedge. This means the investor no longer needs to hedge the short barrier option. 4. **Final Action:** Since the investor was initially short the barrier option, the hedging strategy required them to be long European call options. With the barrier option now worthless, the investor no longer needs the hedge. The investor should sell the 600 European call options. The correct answer highlights the unwinding of the hedge following the barrier event. Incorrect answers may focus on initial hedging actions or overlook the impact of the barrier event on the overall hedging strategy. The complexity lies in understanding the dynamic nature of delta hedging in the context of exotic options and path-dependent payoffs. This question tests a higher level of understanding compared to simple definitions or calculations. It assesses the ability to integrate knowledge of different derivative types and hedging techniques in a practical scenario. The unique aspect is the combination of a barrier option with a delta-hedged European option position, requiring the candidate to think through the implications of the barrier event on the hedge. The original example avoids standard textbook scenarios and presents a more nuanced challenge.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Green Harvest Co-op,” that needs to hedge against price volatility in their wheat crop. They’re considering using wheat futures contracts traded on the ICE Futures Europe exchange. Understanding basis risk is crucial for effective hedging. Basis risk arises because the futures price and the spot price of the underlying asset (wheat, in this case) may not move perfectly in tandem. This difference, the “basis,” is calculated as Spot Price – Futures Price. Several factors can influence the basis, including transportation costs, storage costs, local supply and demand conditions, and quality differences between the wheat specified in the futures contract and the actual wheat harvested by Green Harvest Co-op. To illustrate, suppose Green Harvest Co-op anticipates harvesting 1000 tonnes of wheat in September. The September wheat futures contract is currently trading at £200 per tonne. They decide to sell 10 September futures contracts (each contract representing 100 tonnes) to hedge their exposure. However, they know their wheat is of slightly lower quality than the standard grade specified in the futures contract, and their local market has a glut of wheat due to an exceptionally good harvest. This creates a negative basis. If, in September, the spot price of Green Harvest Co-op’s wheat is £185 per tonne, and the September futures contract settles at £195 per tonne, the basis is £185 – £195 = -£10. This negative basis erodes some of the hedging effectiveness. While the futures contract provided some protection against falling wheat prices, the co-op didn’t fully benefit because the local spot price declined more than the futures price. The gain on the futures position is (£200 – £195) * 1000 = £5,000. However, the actual price received for their wheat is £185,000, compared to the £200,000 they would have received if prices hadn’t changed and they hadn’t hedged. The basis risk resulted in a less-than-perfect hedge. Now, consider a contrasting scenario. Suppose there’s a severe drought in another part of the UK, increasing demand for Green Harvest Co-op’s wheat due to its proximity to the affected region. This could create a positive basis. If the spot price in September is £210 per tonne, and the futures price settles at £205 per tonne, the basis is £210 – £205 = £5. In this case, Green Harvest Co-op would have been better off not hedging, as the spot price increased more than the futures price. However, hedging provided price certainty and protected against the risk of a price decline. Understanding and managing basis risk is therefore a critical aspect of using futures contracts for hedging.

The question explores the complexities of hedging a portfolio with futures contracts, specifically focusing on the challenges posed by imperfect correlation between the portfolio’s assets and the underlying assets of the futures contract. The optimal hedge ratio minimizes portfolio variance, but this ratio is affected by the correlation coefficient (ρ), the standard deviation of the portfolio returns (\(\sigma_p\)), and the standard deviation of the futures contract returns (\(\sigma_f\)). The formula for the optimal hedge ratio is: \[Hedge Ratio = \rho \cdot \frac{\sigma_p}{\sigma_f}\] In this scenario, the correlation is not perfect (ρ < 1), implying basis risk. Basis risk refers to the risk that the price of the asset being hedged and the price of the hedging instrument (the futures contract) do not move perfectly together. This can arise due to differences in the underlying assets, delivery locations, or contract specifications. The question tests the understanding of how changes in correlation affect the hedge ratio and the implications for portfolio risk. A lower correlation implies a weaker relationship between the portfolio and the futures contract, necessitating a smaller hedge ratio to avoid over-hedging and potentially increasing portfolio volatility. Conversely, a higher correlation suggests a stronger relationship, justifying a larger hedge ratio to effectively mitigate portfolio risk. Consider a portfolio manager overseeing a diversified equity portfolio valued at £5 million. They intend to hedge against potential market downturns using FTSE 100 futures contracts. If the correlation between the portfolio's returns and the FTSE 100 futures is 0.7, and the standard deviation of the portfolio's returns is 12% while the standard deviation of the futures contract returns is 15%, the optimal hedge ratio would be: \[Hedge Ratio = 0.7 \cdot \frac{0.12}{0.15} = 0.56\] This means the manager should hedge 56% of the portfolio's value using FTSE 100 futures. If the correlation were to drop to 0.4, the hedge ratio would decrease to: \[Hedge Ratio = 0.4 \cdot \frac{0.12}{0.15} = 0.32\] Illustrating the reduced need for hedging due to the weaker relationship. The number of contracts required is calculated by dividing the hedge amount by the contract size. If each FTSE 100 futures contract represents £10 per index point and the current index level is 7500, each contract is worth £75,000. Number of Contracts = (Hedge Ratio * Portfolio Value) / Contract Value For the initial hedge ratio of 0.56: Number of Contracts = (0.56 * £5,000,000) / £75,000 = 37.33 ≈ 37 contracts For the reduced hedge ratio of 0.32: Number of Contracts = (0.32 * £5,000,000) / £75,000 = 21.33 ≈ 21 contracts This demonstrates how a change in correlation directly impacts the number of futures contracts required to implement the hedging strategy.

The core of this question lies in understanding how implied volatility, time to expiration, and the strike price relative to the current asset price influence the value of a European call option. The Black-Scholes model provides a theoretical framework for option pricing, but it’s essential to grasp the qualitative impact of each input variable. Implied volatility represents the market’s expectation of future price fluctuations. Higher implied volatility increases the option’s price because there’s a greater chance the underlying asset’s price will move significantly, potentially making the option more valuable at expiration. Time to expiration also has a positive correlation with option value. A longer time horizon provides more opportunity for the asset’s price to move favorably. The strike price relative to the current asset price is crucial. An ‘in-the-money’ option (where the asset price exceeds the strike price for a call) has intrinsic value, while an ‘out-of-the-money’ option relies solely on time value and the potential for the asset price to rise above the strike price. A ‘at-the-money’ option has the highest time value. The question requires comparing two options with different characteristics and determining which is likely to be more expensive. We need to consider the combined effect of volatility, time, and moneyness. Option A has lower volatility but a longer time to expiration, while Option B has higher volatility but a shorter time to expiration. Option A is also closer to being at-the-money than Option B. The key is to assess whether the increased volatility of Option B compensates for its shorter time to expiration and greater distance from being at-the-money. To solve this, we conceptually weigh the factors. The longer time to expiration of Option A gives it more potential upside, and the proximity to being at-the-money enhances its value. While Option B’s higher volatility is attractive, the shorter time horizon limits its ability to capitalize on that volatility. Furthermore, its out-of-the-money status means it relies heavily on a significant price movement to become profitable. Therefore, Option A is likely more expensive.

The question assesses the understanding of put-call parity and its application in identifying arbitrage opportunities, specifically within the context of the UK regulatory environment and market practices. Put-call parity is a fundamental principle in options pricing that establishes a relationship between the prices of a European call option, a European put option, the underlying asset, and a risk-free bond. The 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 spot price of the underlying asset. To identify an arbitrage opportunity, we check if the parity holds. If the equation doesn’t balance, an arbitrage opportunity exists. In this case, we need to calculate the present value of the strike price, \(PV(X)\). Given the strike price \(X = £105\), the risk-free rate \(r = 3\%\), and the time to expiration \(t = 0.5\) years, the present value is calculated as: \[PV(X) = \frac{X}{1 + rt} = \frac{105}{1 + (0.03 \times 0.5)} = \frac{105}{1.015} \approx £103.45\]. Now, we check if the put-call parity holds with the given market prices: Call option price \(C = £8\), Put option price \(P = £3\), Spot price \(S = £110\). Substituting these values into the put-call parity equation: \(8 + 103.45 = 3 + 110\), which simplifies to \(111.45 = 113\). Since the equation does not balance, an arbitrage opportunity exists. To exploit this, we need to determine the correct strategy. Because the left side is less than the right side, we buy the components on the left (the call and the risk-free bond) and sell the components on the right (the put and the stock). This means buying the call option for £8, investing £103.45 at the risk-free rate (equivalent to buying a bond that pays £105 at expiration), selling the put option for £3, and short-selling the stock for £110. At expiration, if the stock price is above £105, the call option is exercised, and the short stock position is covered. If the stock price is below £105, the put option is exercised, and we receive the stock for £105, covering our short stock position. In either case, the arbitrage profit is the difference between the cost of the left side and the proceeds from the right side: \(£113 – £111.45 = £1.55\). Therefore, the arbitrage strategy involves buying the call option and a risk-free bond, and selling the put option and short-selling the stock to profit from the mispricing. This strategy is based on the theoretical relationship between the prices of the options and the underlying asset, and any deviation from this relationship presents an opportunity for risk-free profit. The UK regulatory environment permits such arbitrage activities, provided they are conducted transparently and without market manipulation.

The core of this question lies in understanding how a delta-neutral portfolio reacts to changes in implied volatility (vega) and time decay (theta), and how these factors impact the portfolio’s profit and loss (P&L). A delta-neutral portfolio is constructed to be insensitive to small changes in the underlying asset’s price. However, it remains susceptible to other risks, primarily changes in volatility and the passage of time. Vega measures the sensitivity of the portfolio’s value to changes in implied volatility. A positive vega indicates that the portfolio’s value increases when implied volatility rises, and decreases when it falls. Conversely, a negative vega implies the opposite relationship. Theta measures the rate at which the portfolio’s value decays with the passage of time. Options typically lose value as they approach their expiration date, a phenomenon known as time decay. Theta is usually negative for option positions, meaning the portfolio loses value as time passes, all else being equal. The question introduces a scenario where a portfolio manager aims to maintain delta neutrality while strategically adjusting the portfolio’s vega exposure. The manager’s actions, buying options to increase vega, are designed to capitalize on anticipated increases in implied volatility. However, this strategy also introduces a theta risk, as the purchased options will experience time decay. The calculation involves determining the net impact of vega and theta on the portfolio’s P&L over a specific period. The portfolio’s vega is 1,500, meaning a 1% increase in implied volatility is expected to increase the portfolio’s value by £1,500. The portfolio’s theta is -£250 per day, indicating a daily loss of £250 due to time decay. Over a 5-day period, the total theta decay is 5 * -£250 = -£1,250. If implied volatility increases by 1.2% as anticipated, the vega effect is 1.2 * £1,500 = £1,800. The net P&L is the sum of the vega effect and the theta decay: £1,800 + (-£1,250) = £550. Therefore, the portfolio is expected to generate a profit of £550 over the 5-day period, considering the combined effects of vega and theta.

Let’s consider a scenario involving a bespoke structured note linked to the performance of a basket of ESG (Environmental, Social, and Governance) compliant companies within the FTSE 100. This structured note offers a guaranteed minimum return of 2% per annum, but its upside is capped at 8% per annum, linked to the average performance of the selected ESG companies. The structured note is issued by a UK-based investment bank and sold to retail investors. The scenario introduces several complexities: ESG criteria, capped returns, and the involvement of a structured product. Understanding the regulatory landscape, particularly concerning the sale of complex products to retail investors, is crucial. The investor must fully understand the risks and rewards, including the potential for lower returns than direct investment if the ESG basket performs exceptionally well. The bank issuing the note must adhere to MiFID II regulations regarding product governance and suitability assessments. Furthermore, the pricing of such a note involves multiple layers. The guaranteed minimum return is essentially a zero-coupon bond component. The capped upside participation resembles a call option spread. The pricing model would need to consider the correlation between the ESG stocks, their volatility, and the prevailing interest rates. Let’s assume the initial investment is £10,000. After one year, if the ESG basket averages a 6% return, the investor receives the guaranteed 2% plus the 6%, totaling 8%, capped at the maximum return. However, if the ESG basket averages 12%, the investor still only receives the 8% cap. If the basket declines by 5%, the investor still receives the guaranteed 2%. The total return would be calculated as follows: Return = Initial Investment * (1 + Return Rate). So, if the return rate is 8%, the return would be £10,000 * (1 + 0.08) = £10,800. If the return is the minimum 2%, it would be £10,000 * (1 + 0.02) = £10,200. This example demonstrates the trade-off between risk mitigation and potential reward limitation inherent in structured products.

The question revolves around the concept of hedging a portfolio using options, specifically protective puts. A protective put strategy involves buying put options on an asset already held in a portfolio. This strategy aims to limit downside risk, essentially setting a floor on the portfolio’s value. The cost of the put options is the premium paid. The breakeven point for this strategy is the initial portfolio value plus the premium paid for the puts. The maximum loss is limited to the initial portfolio value minus the strike price of the puts, plus the premium paid. The scenario involves a fund manager, Anya, who is concerned about a potential market downturn affecting her portfolio of FTSE 100 stocks. She decides to implement a protective put strategy by purchasing put options on the FTSE 100 index. The key is to understand how the put options will offset potential losses in the stock portfolio. Let’s assume Anya’s FTSE 100 portfolio is worth £5,000,000. She buys put options with a strike price of 7,500 and pays a premium of £5 per contract. Each contract represents an index value of £10. Therefore, the total premium paid is (Number of contracts * Premium per contract * Index value), which needs to be calculated based on the portfolio value and the index level. If the FTSE 100 falls below 7,500, the put options will increase in value, offsetting the losses in the stock portfolio. If the FTSE 100 stays above 7,500, the put options will expire worthless, and Anya will lose the premium paid. The question requires calculating the net impact on Anya’s portfolio under a specific scenario: a 15% drop in the FTSE 100 index. This involves determining the loss in the stock portfolio, the gain from the put options (if any), and the net effect after considering the premium paid. The formula for calculating the profit/loss from the put options is: max(Strike Price – Final Index Value, 0) * Number of contracts – Total Premium Paid. The number of put contracts Anya needs to buy is calculated as Portfolio Value / (Index Level * Index Multiplier). Assuming the FTSE 100 is at 8,000, Anya needs £5,000,000 / (8,000 * £10) = 62.5 contracts, rounded up to 63 contracts. The total premium paid is 63 * £5 * £10 = £3,150. A 15% drop from 8,000 is 8,000 * 0.15 = 1,200, so the new index level is 8,000 – 1,200 = 6,800. The profit from the puts is max(7,500 – 6,800, 0) * 63 * £10 – £3,150 = 700 * 63 * £10 – £3,150 = £441,000 – £3,150 = £437,850. The loss on the portfolio is £5,000,000 * 0.15 = £750,000. The net loss is £750,000 – £437,850 = £312,150.

The question assesses the understanding of delta hedging and how changes in the underlying asset’s price and the passage of time affect the hedge. Delta hedging aims to create a portfolio that is insensitive to small changes in the underlying asset’s price. The delta of a call option represents the change in the option’s price for a £1 change in the underlying asset’s price. Gamma represents the rate of change of the delta with respect to changes in the underlying asset’s price. Theta represents the rate of change of the option’s price with respect to time. Initially, the portfolio is delta neutral, meaning the portfolio’s value is not affected by small changes in the underlying asset’s price. However, as the underlying asset’s price changes, the delta of the call option changes (due to gamma), requiring adjustments to the hedge. As time passes, the value of the option decays (due to theta), also affecting the hedge. The combined effect of these changes determines the profit or loss on the delta hedge. In this scenario, the investor initially shorts 1000 call options and delta hedges by buying shares. A rise in the underlying asset’s price increases the call option’s price, resulting in a loss on the short option position. However, the increase in the underlying asset’s price also results in a profit on the shares held. The gamma of the option causes the delta to increase as the underlying asset’s price rises, meaning the hedge needs to be adjusted by buying more shares. The theta of the option causes the option’s value to decrease as time passes, resulting in a profit on the short option position. The profit or loss on the delta hedge can be approximated as: \[ P\&L \approx -\frac{1}{2} \Gamma (\Delta S)^2 \times N + \Theta \times N \times \Delta t \] Where: – \(\Gamma\) is gamma – \(\Delta S\) is the change in the underlying asset’s price – \(N\) is the number of options – \(\Theta\) is theta (per day) – \(\Delta t\) is the number of days Plugging in the values: \[ P\&L \approx -\frac{1}{2} \times 0.04 \times (5)^2 \times 1000 + (-0.02) \times 1000 \times 1 \] \[ P\&L \approx -500 + (-20) \] \[ P\&L \approx -520 \] Therefore, the approximate profit or loss on the delta hedge is a loss of £520.

This question tests the understanding of option pricing models, specifically the Black-Scholes model, and how implied volatility derived from market prices can be used to assess market sentiment and potential mispricings. The Black-Scholes model provides a theoretical value for options based on several factors, including the underlying asset’s price, the option’s strike price, time to expiration, risk-free interest rate, and the underlying asset’s volatility. Implied volatility is the volatility that, when input into the Black-Scholes model, results in a theoretical option price equal to the option’s current market price. A higher implied volatility suggests that the market anticipates greater price fluctuations in the underlying asset. The calculation involves using the Black-Scholes formula to back out the implied volatility. Given the option price, strike price, underlying asset price, time to expiration, and risk-free rate, we iteratively solve for the volatility that makes the Black-Scholes price equal to the market price. This is typically done using numerical methods or software. Since a direct analytical solution for implied volatility doesn’t exist, approximation methods are used. The question emphasizes understanding how implied volatility relates to market expectations and potential arbitrage opportunities. If the market price is higher than the Black-Scholes price using a reasonable volatility estimate (historical volatility, for example), it suggests the option is overvalued, and the implied volatility will be higher than expected. The formula to approximate implied volatility using the Corrado-Miller approximation is as follows: \[ \sigma \approx \frac{\sqrt{2\pi}}{S} \frac{C}{T} \] Where: \( \sigma \) = Implied Volatility \( S \) = Current Stock Price = 450 \( C \) = Call Option Premium = 35 \( T \) = Time to expiration = 0.25 (3 months) \[ \sigma \approx \frac{\sqrt{2\pi}}{450} \frac{35}{\sqrt{0.25}} \] \[ \sigma \approx \frac{2.506}{450} * \frac{35}{0.5} \] \[ \sigma \approx 0.00557 * 70 \] \[ \sigma \approx 0.39 \] \[ \sigma \approx 39\% \]

The Black-Scholes model is a cornerstone of options pricing theory. It relies on several assumptions, including constant volatility, a risk-free interest rate, and the European option style (exercisable only at expiration). When these assumptions are violated, the model’s accuracy diminishes. A key challenge in applying the Black-Scholes model is estimating volatility. Implied volatility, derived from market prices of options, often differs from historical volatility. The “volatility smile” or “volatility skew” observed in options markets indicates that implied volatility varies across different strike prices, violating the constant volatility assumption. To address these limitations, practitioners use various adjustments and alternative models. One common approach is to use a volatility surface, which plots implied volatility as a function of both strike price and time to expiration. This allows for a more nuanced assessment of option values, reflecting the market’s perception of risk for different options. Another method is to use stochastic volatility models, such as the Heston model, which incorporate the fact that volatility itself is a random variable. These models are more complex but can provide a more accurate representation of option prices, especially for options that are far from the money or have long maturities. Furthermore, real-world market frictions, such as transaction costs and bid-ask spreads, can impact option pricing. These factors are not explicitly accounted for in the Black-Scholes model but can be significant, especially for high-frequency trading or when dealing with less liquid options. Therefore, traders and portfolio managers must consider these factors when using the Black-Scholes model for pricing and hedging decisions. The model serves as a useful benchmark, but it should not be applied blindly without considering its limitations and potential adjustments. Consider a scenario where a portfolio manager is evaluating the price of a call option on a volatile technology stock. The implied volatility for near-the-money options is 30%, but for out-of-the-money options, it is 45%. The Black-Scholes model, using a single volatility input, would likely misprice the out-of-the-money options. The manager needs to incorporate the volatility skew to accurately assess the option’s fair value and make informed trading decisions.

The question assesses the understanding of delta hedging in a portfolio containing options and the impact of market movements on the hedge. Delta hedging aims to neutralize the portfolio’s sensitivity to small changes in the underlying asset’s price. The delta of a portfolio represents the change in the portfolio’s value for a $1 change in the underlying asset’s price. To maintain a delta-neutral position, the portfolio manager must dynamically adjust the hedge as the underlying asset’s price changes. This involves buying or selling the underlying asset to offset the delta of the options. Gamma, on the other hand, measures the rate of change of the delta. A higher gamma means that the delta is more sensitive to changes in the underlying asset’s price, requiring more frequent adjustments to maintain the hedge. In this scenario, the fund initially hedges its short call options position by holding shares of the underlying asset. As the market rises, the delta of the call options increases (becomes more positive). Since the fund is short call options, their delta is negative. To maintain a delta-neutral position, the fund needs to increase its holding of the underlying asset. The amount of additional shares to purchase is determined by the change in the underlying asset’s price and the initial delta of the portfolio. The initial delta of the short call options is -0.40 per option. The fund holds 10,000 options, so the total delta is -0.40 * 10,000 = -4,000. To offset this negative delta, the fund initially holds 4,000 shares. When the market rises by £1, the delta of each call option increases to -0.30. The new total delta of the options is -0.30 * 10,000 = -3,000. To maintain delta neutrality, the fund needs to reduce its short position to 3,000 shares. Therefore, the fund needs to sell 1,000 shares (4,000 – 3,000 = 1,000).

Let’s analyze a scenario involving a complex hedging strategy using options to protect a portfolio against downside risk while simultaneously attempting to profit from anticipated moderate upward movement, incorporating the concept of implied volatility skew and its impact on option pricing. The fund manager holds a substantial portfolio of FTSE 100 stocks. The manager believes the market has limited upside potential over the next three months but wants to protect against a significant market decline. The current FTSE 100 index level is 7500. The manager decides to implement a *ratio spread* using FTSE 100 index options. They buy 100 put options with a strike price of 7300 and sell 200 put options with a strike price of 7100. All options expire in three months. Let’s assume the implied volatility for the 7300-strike puts is 20% and for the 7100-strike puts is 23%. This reflects a typical *volatility skew*, where out-of-the-money puts are more expensive due to higher demand for downside protection. * **Cost of buying 100 puts (7300 strike):** Assume each put costs £2.50. Total cost = 100 contracts * 100 (index multiplier) * £2.50 = £25,000. * **Revenue from selling 200 puts (7100 strike):** Assume each put earns £1.25. Total revenue = 200 contracts * 100 (index multiplier) * £1.25 = £25,000. The strategy is *initially costless* because the premium received from selling the puts offsets the premium paid for buying the puts. Now, let’s consider different scenarios at expiration: 1. **FTSE 100 at 7600:** All options expire worthless. The profit/loss is £0. 2. **FTSE 100 at 7300:** The bought puts expire worthless. The sold puts also expire worthless. The profit/loss is £0. 3. **FTSE 100 at 7200:** The bought puts expire in the money with intrinsic value of 7300 – 7200 = 100. Profit from bought puts = 100 contracts * 100 (index multiplier) * 100 = £1,000,000. The sold puts expire in the money with intrinsic value of 7100 – 7200 = -100. Loss from sold puts = 200 contracts * 100 (index multiplier) * -100 = -£2,000,000. Net profit/loss = £1,000,000 – £2,000,000 = -£1,000,000. 4. **FTSE 100 at 7100:** The bought puts expire in the money with intrinsic value of 7300 – 7100 = 200. Profit from bought puts = 100 contracts * 100 (index multiplier) * 200 = £2,000,000. The sold puts expire at the money with intrinsic value of 7100 – 7100 = 0. Loss from sold puts = 200 contracts * 100 (index multiplier) * 0 = £0. Net profit/loss = £2,000,000 – £0 = £2,000,000. 5. **FTSE 100 at 7000:** The bought puts expire in the money with intrinsic value of 7300 – 7000 = 300. Profit from bought puts = 100 contracts * 100 (index multiplier) * 300 = £3,000,000. The sold puts expire in the money with intrinsic value of 7100 – 7000 = 100. Loss from sold puts = 200 contracts * 100 (index multiplier) * 100 = -£2,000,000. Net profit/loss = £3,000,000 – £2,000,000 = £1,000,000. This strategy profits if the market declines moderately. However, below 7100, the strategy loses money. The maximum loss is unlimited.

This question assesses the understanding of put-call parity, a fundamental concept in options pricing. Put-call parity describes the relationship between the prices of European put and call options with the same strike price and expiration date, along with the price of the underlying asset and a risk-free bond. The formula is: \[C + PV(X) = P + S\] Where: * C = Call option price * P = Put option price * S = Current stock price * X = Strike price * PV(X) = Present value of the strike price, discounted at the risk-free rate. The present value is calculated as: \[PV(X) = \frac{X}{(1 + r)^t}\] Where: * r = risk-free interest rate * t = time to expiration (in years) The question requires rearranging the put-call parity formula to solve for the put option price (P): \[P = C + PV(X) – S\] First, calculate the present value of the strike price: \[PV(X) = \frac{105}{(1 + 0.05)^{0.5}} = \frac{105}{1.0247} \approx 102.47\] Then, substitute the given values into the rearranged put-call parity formula: \[P = 8 + 102.47 – 100 = 10.47\] The put option should be priced at approximately £10.47 to prevent arbitrage opportunities. If the put option is priced differently, an arbitrageur could exploit the mispricing by simultaneously buying the underpriced assets and selling the overpriced assets to make a risk-free profit. For instance, if the put were trading at £9, an arbitrageur could buy the put and the stock, and sell the call and a risk-free bond that matures to the strike price. At expiration, regardless of the stock price, the arbitrageur would realize a profit equal to the initial mispricing. Conversely, if the put were trading at £12, the opposite strategy would be implemented. The put-call parity relationship ensures that these types of arbitrage opportunities are minimized in efficient markets.

The question assesses understanding of put-call parity and how dividends affect it. The core concept is that a portfolio consisting of a European call option and a present value of the strike price should be equivalent to a portfolio of a European put option and the underlying asset. Dividends paid during the option’s life reduce the value of the underlying asset, and this must be accounted for in the parity relationship. The put-call parity formula, adjusted for dividends, is: \[C + PV(K) = P + S – PV(Div)\] Where: * \(C\) = Call option price * \(P\) = Put option price * \(S\) = Current stock price * \(K\) = Strike price * \(PV(K)\) = Present value of the strike price, discounted at the risk-free rate * \(PV(Div)\) = Present value of the dividends expected during the option’s life, discounted at the risk-free rate In this scenario, we need to calculate the present value of the strike price and the present value of the dividends. The present value of the strike price is calculated as: \[PV(K) = \frac{K}{(1 + r)^t} = \frac{55}{(1 + 0.05)^{0.5}} = \frac{55}{1.0247} = 53.67\] The present value of the dividends is calculated as: \[PV(Div) = \frac{Div}{(1 + r)^{t_1}} + \frac{Div}{(1 + r)^{t_2}} = \frac{1.50}{(1 + 0.05)^{0.25}} + \frac{1.50}{(1 + 0.05)^{0.5}} = \frac{1.50}{1.0122} + \frac{1.50}{1.0247} = 1.4819 + 1.4639 = 2.9458\] Now, using the put-call parity formula: \[C + PV(K) = P + S – PV(Div)\] \[C + 53.67 = 3 + 52 – 2.9458\] \[C = 3 + 52 – 2.9458 – 53.67 = -1.6158\] Since a call option cannot have a negative price, there is an arbitrage opportunity. We should buy the call, short the stock, and borrow to fund the strike price and dividends. The arbitrage profit is the difference between the theoretical call price and the actual call price, which is: Arbitrage Profit = Theoretical Call Price – Actual Call Price Arbitrage Profit = -1.6158 – (-2.00) = 0.3842 Therefore, the arbitrage profit is approximately £0.38.

Let’s consider a scenario where a UK-based investment firm, “Thames Valley Investments” (TVI), is using derivatives to manage the interest rate risk associated with its portfolio of corporate bonds. TVI holds £50 million in bonds with a floating interest rate linked to SONIA (Sterling Overnight Index Average) + 1.5%. The firm is concerned that SONIA might decrease, reducing their income. To hedge this risk, they enter into an interest rate swap. TVI enters a swap where they pay a fixed rate of 4% on a notional principal of £50 million and receive SONIA. This means if SONIA averages 2% over the period, TVI effectively receives 2% from the swap counterparty. Their total return on the bond portfolio would be 4% (fixed payment) + 1.5% (spread) = 5.5%. If SONIA rises, the firm receives more from the swap, offsetting any increase in the floating rate they pay on another liability. Now, let’s introduce the concept of counterparty risk. If the swap counterparty defaults, TVI would no longer receive SONIA payments. If SONIA has risen significantly above 4%, TVI is now in a position where they are owed money by the counterparty. The mark-to-market value of the swap represents the loss TVI would incur if the counterparty defaults. Assume at the time of default, the market rate for a similar swap is 5%. TVI would need to pay 5% to receive SONIA, whereas they were previously paying 4%. The present value of this difference needs to be calculated. If the swap has 3 years remaining, we can approximate the loss using the following calculation: Annual Loss = Notional Principal * (New Rate – Old Rate) = £50,000,000 * (0.05 – 0.04) = £500,000 To simplify, let’s assume a discount rate of 3% for each year. Year 1: £500,000 / (1 + 0.03) = £485,436.89 Year 2: £500,000 / (1 + 0.03)^2 = £471,300.87 Year 3: £500,000 / (1 + 0.03)^3 = £457,573.66 Total Present Value Loss = £485,436.89 + £471,300.87 + £457,573.66 = £1,414,311.42 This value represents the approximate mark-to-market risk exposure due to counterparty default. TVI needs to consider this potential loss when assessing the overall risk profile of using interest rate swaps for hedging. The firm must also consider the regulatory requirements under EMIR regarding clearing and margining of OTC derivatives to mitigate counterparty risk.

The question assesses the understanding of the impact of correlation on portfolio risk when using derivatives for hedging. Specifically, it explores how changes in correlation between an asset and a derivative (in this case, a put option) affect the overall portfolio Value at Risk (VaR). VaR measures the potential loss in value of a portfolio over a specific time period for a given confidence level. Here’s the breakdown of the calculation and the underlying principles: 1. **Understanding the Scenario:** We have a portfolio consisting of shares of a UK-based technology company and put options on those shares used for hedging. The initial correlation between the shares and the put options is -0.7. The portfolio’s VaR is £500,000. The correlation then shifts to -0.3. 2. **Impact of Correlation on Portfolio Risk:** A negative correlation between an asset and a hedging instrument (like a put option) reduces overall portfolio risk. As the correlation becomes less negative (i.e., moves closer to zero), the hedging effectiveness decreases, and the portfolio risk (VaR) increases. A correlation of -1 represents perfect negative correlation (ideal hedge), while a correlation of 0 indicates no linear relationship. 3. **Estimating the Change in VaR:** While a precise calculation would require complex modeling, we can estimate the impact. The change in correlation from -0.7 to -0.3 represents a significant reduction in the hedging benefit. Let’s consider the variance reduction achieved by the hedge. The portfolio variance is related to the correlation coefficient (\(\rho\)) by the formula: \[\sigma_p^2 = w_a^2\sigma_a^2 + w_b^2\sigma_b^2 + 2w_a w_b \rho \sigma_a \sigma_b\] where: – \(\sigma_p^2\) is the portfolio variance – \(w_a\) and \(w_b\) are the weights of asset A (shares) and asset B (put options) respectively – \(\sigma_a^2\) and \(\sigma_b^2\) are the variances of asset A and asset B respectively – \(\rho\) is the correlation between asset A and asset B The key here is the term \(2w_a w_b \rho \sigma_a \sigma_b\). As \(\rho\) becomes less negative, this term becomes less negative (or more positive), increasing the overall portfolio variance. Since VaR is directly related to the standard deviation (square root of variance), an increase in variance implies an increase in VaR. 4. **Approximation:** A change in correlation from -0.7 to -0.3 is a substantial weakening of the hedge. We can approximate that the VaR will increase significantly, but not necessarily double (which would imply a complete loss of hedging benefit). A reasonable estimate is an increase of around 40-60%. Let’s assume a 50% increase for calculation purposes. Increase in VaR = 50% of £500,000 = £250,000 New VaR = £500,000 + £250,000 = £750,000 5. **Considering Regulatory Implications:** The scenario has implications for regulatory capital. Under Basel III (or similar UK regulations implementing Basel standards), banks and investment firms must hold capital against market risk, which is often calculated using VaR. An increase in VaR would necessitate holding more capital, impacting profitability and potentially restricting trading activities. Firms must also consider the impact on ICAAP (Internal Capital Adequacy Assessment Process). 6. **Limitations:** This is an approximation. A precise calculation would require a full portfolio revaluation and recalculation of VaR using the new correlation. The actual change in VaR depends on the specific characteristics of the shares and the put options, their volatilities, and the portfolio weights.

The question assesses understanding of delta hedging, specifically how to maintain a delta-neutral portfolio when the underlying asset’s price changes. Delta measures the sensitivity of an option’s price to a change in the underlying asset’s price. Delta hedging involves adjusting the portfolio’s holdings of the underlying asset to offset changes in the option’s value due to fluctuations in the underlying asset’s price. A delta-neutral portfolio has a delta of zero, meaning it is theoretically immune to small price changes in the underlying asset. The calculation involves determining the number of shares needed to maintain delta neutrality after the underlying asset’s price changes. Initially, the portfolio is delta-neutral. When the underlying asset’s price changes, the option’s delta changes, and the portfolio is no longer delta-neutral. To re-establish delta neutrality, the investor needs to buy or sell shares of the underlying asset. The formula to determine the number of shares to buy or sell is: Change in Shares = (New Delta – Old Delta) * Number of Options. In this case, the old delta is 0.55, the new delta is 0.62, and the number of options is 1000. Therefore, the change in shares is (0.62 – 0.55) * 1000 = 70 shares. Since the delta increased, the investor needs to buy 70 shares to re-establish delta neutrality. A crucial aspect of delta hedging is that it’s a dynamic strategy. The delta of an option changes as the underlying asset’s price changes and as time passes. Therefore, delta hedging requires continuous monitoring and adjustment of the portfolio’s holdings. In practice, transaction costs can make continuous delta hedging expensive. Investors often choose to re-hedge only when the delta moves outside a certain range or at predetermined intervals. The effectiveness of delta hedging also depends on the size of the price movements in the underlying asset. Large, sudden price changes can make it difficult to maintain delta neutrality. The example illustrates the practical application of delta hedging. It demonstrates how to calculate the number of shares needed to adjust a portfolio’s holdings to maintain delta neutrality after a change in the underlying asset’s price. The investor needs to buy 70 shares to offset the increased delta of the options.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Green Fields Co-op,” which exports wheat. They face volatile wheat prices and currency fluctuations (GBP/USD exchange rate). To manage these risks, they consider using futures contracts and currency forwards. We’ll analyze the combined hedging strategy and calculate the effective price they receive for their wheat in GBP. First, Green Fields Co-op sells wheat futures to lock in a wheat price. Then, they use a currency forward to convert USD proceeds back to GBP at a predetermined rate. Basis risk arises because the futures price may not perfectly correlate with the spot price at the time of delivery. The forward rate locks in the GBP/USD exchange, eliminating currency risk. Let’s assume the co-op sells wheat futures at $7.50 per bushel and enters a GBP/USD forward contract at 1.25. At delivery, the spot price is $7.30 per bushel, and the spot exchange rate is 1.20. The effective price in USD is the futures price plus the basis risk: $7.50 + ($7.30 – $7.50) = $7.30 per bushel. The GBP amount is calculated using the forward rate: $7.30 / 1.25 = £5.84 per bushel. Now, imagine Green Fields Co-op initially considered using options instead of futures. They could have purchased put options on wheat and call options on GBP/USD. If wheat prices rose significantly, they wouldn’t exercise their put options, benefiting from the higher spot price. Similarly, if the GBP strengthened, they wouldn’t exercise their call options, converting USD at the prevailing spot rate. However, options require an upfront premium, reducing their initial profit. Let’s say the put option premium was $0.10 per bushel, and the call option premium was £0.02 per bushel. In the futures scenario, their guaranteed minimum price was £5.84. With options, the floor price becomes £5.84 – £0.08 – £0.02 = £5.74 per bushel, considering the option premium costs. The benefit is the potential to profit from favorable price movements. The key is understanding how futures and forwards provide a guaranteed outcome, while options offer flexibility but at a cost. The co-op must evaluate their risk tolerance and market outlook to decide on the optimal strategy. Regulations like EMIR (European Market Infrastructure Regulation) would require them to clear certain OTC derivatives through a central counterparty, adding to the operational complexity and cost.

The question explores the complexities of managing currency risk within a global investment portfolio, specifically focusing on the application of currency swaps and forward contracts in hedging strategies. It requires an understanding of the interplay between spot rates, forward rates, interest rate differentials, and the practical implications of transaction costs. The optimal hedging strategy involves comparing the costs and benefits of using forward contracts versus currency swaps to mitigate the risk of currency fluctuations impacting portfolio returns. To determine the optimal hedging strategy, we need to calculate the effective return under both scenarios: hedging with forward contracts and hedging with currency swaps. **Scenario 1: Hedging with Forward Contracts** 1. **Calculate the forward rate:** Given the spot rate of GBP/USD at 1.2500 and the interest rates of 5% in the UK and 2% in the US, we can approximate the forward rate using the interest rate parity formula: Forward Rate = Spot Rate \* (1 + Interest Rate UK) / (1 + Interest Rate US) Forward Rate = 1.2500 \* (1 + 0.05) / (1 + 0.02) = 1.2500 \* 1.05 / 1.02 = 1.2860 (approximately) 2. **Calculate the USD value of the GBP return at the forward rate:** The GBP return is £1,050,000 (initial investment of £1,000,000 \* 1.05). Converting this to USD at the forward rate: USD Value = £1,050,000 \* 1.2860 = $1,350,300 3. **Account for transaction costs:** The transaction cost is 0.1% of the notional amount (£1,000,000), which is £1,000. Converting this to USD at the spot rate: USD Transaction Cost = £1,000 \* 1.2500 = $1,250 Adjusted USD Value = $1,350,300 – $1,250 = $1,349,050 4. **Calculate the return in USD:** The initial investment in USD was $1,250,000 (£1,000,000 \* 1.2500). The return is: Return = ($1,349,050 – $1,250,000) / $1,250,000 = 0.07924 or 7.924% **Scenario 2: Hedging with Currency Swaps** 1. **Calculate the net interest received in USD:** The currency swap involves receiving USD at 2% on $1,250,000, which is: USD Interest = $1,250,000 \* 0.02 = $25,000 2. **Calculate the net interest paid in GBP:** The currency swap involves paying GBP at 5% on £1,000,000, which is: GBP Interest = £1,000,000 \* 0.05 = £50,000. Converting this to USD at the initial spot rate: USD Equivalent = £50,000 \* 1.2500 = $62,500 3. **Calculate the net USD interest:** Net USD interest received is $25,000 – $62,500 = -$37,500 4. **Account for transaction costs:** The transaction cost is 0.05% of the notional amount (£1,000,000), which is £500. Converting this to USD at the spot rate: USD Transaction Cost = £500 \* 1.2500 = $625 Adjusted Net USD Interest = -$37,500 – $625 = -$38,125 5. **Calculate the return in USD:** The initial investment in USD was $1,250,000. The return is: Return = ($1,250,000 – $38,125 – $1,250,000) / $1,250,000 = -$38,125 / $1,250,000 = -0.0305 or -3.05% 6. **Calculate the return in USD:** The initial investment in USD was $1,250,000. The final value is $1,250,000 – $38,125 = $1,211,875 Return = ($1,211,875 – $1,250,000) / $1,250,000 = -0.0305 or -3.05% Comparing the two strategies, hedging with forward contracts yields a return of 7.924%, while hedging with currency swaps results in a return of -3.05%. Therefore, hedging with forward contracts is the superior strategy in this scenario.

Let’s consider a scenario where a portfolio manager is using options to hedge against downside risk in their equity portfolio. The portfolio, valued at £10 million, closely tracks the FTSE 100 index, currently at 7500. The manager wants to protect the portfolio against a potential market downturn over the next six months. They decide to use put options on the FTSE 100 index. To determine the number of put option contracts needed, we must first understand the contract size. Assume each FTSE 100 index option contract covers £10 per index point. Therefore, each contract covers £10 * 7500 = £75,000 of the index. To hedge the £10 million portfolio, the manager needs £10,000,000 / £75,000 ≈ 133.33 contracts. Since contracts are only available in whole numbers, the manager would likely purchase 133 or 134 contracts. Next, we need to consider the strike price and premium. Suppose the manager buys 133 put option contracts with a strike price of 7400, expiring in six months, at a premium of £5 per index point. The total premium paid is 133 contracts * £5 * 7500 (index level) = £4,987,500. However, the premium is normally quoted per contract per index point, so the premium per contract is £5 * 10 = £50. The total premium is then 133 contracts * £50 * 7500/7500 = £6,650. Now, let’s analyze potential outcomes. If the FTSE 100 falls to 7000 at expiration, the put options will be in the money. Each put option will have an intrinsic value of (7400 – 7000) * £10 = £4,000. The total payoff from the put options will be 133 contracts * £4,000 = £532,000. This payoff helps to offset the loss in the equity portfolio. The net cost of the hedge is the premium paid minus the payoff, which is £6,650 – £532,000 = -£525,350. If the FTSE 100 rises to 8000, the put options will expire worthless. The manager’s loss is limited to the premium paid, £6,650. This illustrates the cost of the insurance provided by the put options. The key to understanding hedging with options is to balance the cost of the premium against the potential benefits of downside protection. The number of contracts, strike price, and expiration date must be carefully chosen to align with the portfolio’s risk profile and the manager’s market expectations. Furthermore, the manager must continuously monitor the hedge and adjust it as market conditions change. This might involve rolling the options to a later expiration date, adjusting the strike price, or adding additional contracts. The example above shows the basic principles of hedging with put options, demonstrating the trade-offs between cost and protection.

Let’s consider a scenario involving a UK-based agricultural cooperative, “FarmCo,” that produces and exports wheat. FarmCo faces significant price volatility in the global wheat market and seeks to hedge its price risk using futures contracts traded on the ICE Futures Europe exchange. The current date is 1st July. FarmCo anticipates harvesting 500,000 bushels of wheat on 1st December and wants to lock in a selling price. ICE Wheat Futures contracts are for 5,000 bushels each. FarmCo’s treasurer, Ms. Anya Sharma, is tasked with implementing the hedge. First, determine the number of contracts needed: 500,000 bushels / 5,000 bushels/contract = 100 contracts. FarmCo will sell 100 December Wheat Futures contracts. On 1st July, the December Wheat Futures price is £200 per bushel. On 1st December, FarmCo sells its wheat in the spot market for £190 per bushel. Simultaneously, the December Wheat Futures price settles at £192 per bushel. The gain or loss on the futures contracts is calculated as follows: Initial futures price: £200/bushel Final futures price: £192/bushel Price change: £200 – £192 = £8/bushel Total gain on futures: 100 contracts * 5,000 bushels/contract * £8/bushel = £4,000,000 The spot market sale results in: Spot price on 1st December: £190/bushel Total revenue from spot sale: 500,000 bushels * £190/bushel = £95,000,000 The effective selling price is the spot market revenue plus the futures gain, divided by the total bushels: (£95,000,000 + £4,000,000) / 500,000 bushels = £198/bushel. Basis risk arises because the spot price and futures price did not converge perfectly. The initial basis (difference between spot and futures) is not explicitly given, but we can infer a change in basis. If the spot price and futures price moved in perfect lockstep, the gain on the futures would perfectly offset the loss in the spot market relative to the initial futures price. However, since the spot price decreased by £10 and the futures price decreased by £8, the basis changed by £2. The hedging strategy aimed to lock in a price close to £200. The effective selling price of £198 reflects the impact of basis risk. FarmCo mitigated the risk of a significant price decline, but did not achieve a perfect hedge due to the imperfect correlation between spot and futures prices. Understanding basis risk is crucial in evaluating the effectiveness of any hedging strategy. A perfect hedge is rare in real-world applications.

To determine the fair value of the Asian option, we need to simulate possible price paths and calculate the average payoff. The formula for the payoff of an Asian call option is max(Average Price – Strike Price, 0). We simulate three possible price paths and calculate the average price for each path. Path 1: Prices are 100, 105, 110, 115, 120. The average price is (100+105+110+115+120)/5 = 110. The payoff is max(110-105, 0) = 5. Path 2: Prices are 100, 95, 100, 105, 110. The average price is (100+95+100+105+110)/5 = 102. The payoff is max(102-105, 0) = 0. Path 3: Prices are 100, 110, 120, 115, 105. The average price is (100+110+120+115+105)/5 = 110. The payoff is max(110-105, 0) = 5. The average payoff across the three paths is (5+0+5)/3 = 3.33. The present value is calculated using the risk-free rate. Since the option matures in 5 months (5/12 years), we discount the average payoff by the risk-free rate. The present value is \(3.33 * e^{(-0.05 * (5/12))} \approx 3.33 * e^{-0.02083} \approx 3.33 * 0.9793 \approx 3.26\). An Asian option, unlike a standard European or American option, uses the average price of the underlying asset over a specified period to determine the payoff, rather than the price at a specific maturity date. This averaging feature reduces the impact of price volatility and makes Asian options particularly attractive for hedging strategies where the average price is more relevant than the spot price at a particular moment. For instance, a commodity importer might use an Asian option to hedge against fluctuations in the average cost of a raw material over the production cycle, providing more predictable cost management. The valuation of Asian options typically involves Monte Carlo simulations or other numerical methods because there’s no straightforward analytical solution like the Black-Scholes model for standard options. The risk-free rate is crucial in discounting the expected payoff back to its present value, reflecting the time value of money and the opportunity cost of capital. The more paths we simulate, the more accurate our valuation will be.

To determine the profit or loss from the delta-hedged portfolio, we need to calculate the change in the value of the portfolio due to the change in the underlying asset’s price and the offsetting change in the value of the short call options position. The delta of the call option indicates the sensitivity of the option price to changes in the underlying asset price. 1. **Calculate the initial value of the shares:** The initial price of each share is £450, and the portfolio contains 100 shares. Therefore, the initial value of the shares is \(100 \times £450 = £45,000\). 2. **Calculate the number of options to short (hedge ratio):** To delta-hedge the portfolio, the number of options to short is determined by the portfolio’s delta exposure. Since we have 100 shares and each option contract covers 1 share (1:1 ratio), we need to short 100 / 0.6 = 166.67 contracts. Since you can’t trade fractions of contracts, we’ll round this to 167 contracts. 3. **Calculate the initial value of the short call options position:** The initial price of each call option is £50, and we are shorting 167 contracts. Therefore, the initial value of the short call options position is \(167 \times £50 \times 100\) (since each contract represents 100 shares) = £835,000. 4. **Calculate the change in the value of the shares:** The share price increases to £460, so the change in price is \(£460 – £450 = £10\). The change in the value of the shares is \(100 \times £10 = £1,000\). 5. **Calculate the expected change in the value of the options:** The delta of the call option is 0.6. The change in the underlying asset’s price is £10. Therefore, the expected change in the value of each option is \(0.6 \times £10 = £6\). Since we are short 167 contracts, the total change in the value of the short call options position is \(167 \times 100 \times £6 = £100,200\). However, because it is a short position, this represents a loss. 6. **Calculate the profit or loss:** The profit from the shares is £1,000, and the loss from the short call options position is £100,200. Therefore, the net profit or loss is \(£1,000 – £100,200 = -£99,200\). This example illustrates how delta hedging works in theory. In practice, delta hedging needs to be continuously adjusted (rebalanced) because the delta of an option changes as the price of the underlying asset changes. This rebalancing incurs transaction costs, which would affect the overall profitability of the hedging strategy. The example also highlights the risks associated with delta hedging, particularly the potential for losses if the hedge is not perfectly maintained. Furthermore, the rounding of the number of contracts from 166.67 to 167 introduces a slight imperfection in the hedge.