Derivatives Level 4 (Investment Advice Diploma) Exam Quiz Quiz 23

To determine the fair value of the equity swap, we need to calculate the present value of the expected dividend payments and compare it to the present value of the fixed payments. The equity leg pays dividends of £3 per share annually. The fixed leg pays 5% annually on a notional principal of £10 million, which is £500,000 per year. The discount rate is 4%. First, calculate the number of shares: £10,000,000 / £50 = 200,000 shares. Next, calculate the total annual dividend payment: 200,000 shares * £3/share = £600,000. Now, calculate the present value of the dividend payments using the perpetuity formula: PV = Payment / Discount Rate. PV of dividends = £600,000 / 0.04 = £15,000,000. Calculate the present value of the fixed payments: PV of fixed payments = £500,000 / 0.04 = £12,500,000. Finally, determine the fair value of the swap by subtracting the present value of the fixed payments from the present value of the dividend payments: Fair Value = £15,000,000 – £12,500,000 = £2,500,000. Therefore, the equity swap has a fair value of £2,500,000 to the party receiving the equity return. This value represents the difference between the present value of the expected dividend income and the present value of the fixed payments. If the market’s expectation of future dividend performance rises, the value of the equity leg increases, making the swap more valuable to the equity receiver. Conversely, if interest rates rise, the present value of the fixed payments decreases, also making the swap more valuable to the equity receiver. The fair value calculation is crucial for marking-to-market the swap and for understanding the potential gains or losses associated with the position. This valuation also aids in risk management, as it provides a clear indication of the swap’s sensitivity to changes in equity prices and interest rates.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Green Harvest,” which exports organic wheat. Green Harvest faces significant currency risk due to fluctuating exchange rates between the British Pound (GBP) and the US Dollar (USD), as their sales are denominated in USD but their costs are largely in GBP. To mitigate this risk, they enter into a forward contract. Understanding the mechanics of forward contracts, particularly the impact of interest rate differentials between the two currencies, is crucial. The forward rate is determined by the spot rate and the interest rate differential between the two currencies. The formula to calculate the forward rate is: Forward Rate = Spot Rate * (1 + Interest Rate of Price Currency) / (1 + Interest Rate of Base Currency) In this case, GBP is the base currency and USD is the price currency. The spot rate is GBP/USD. Let’s say the current spot rate is 1.25 GBP/USD. The UK interest rate (GBP) is 5% per annum, and the US interest rate (USD) is 2% per annum. The contract is for 6 months (0.5 years). Forward Rate = 1.25 * (1 + 0.02 * 0.5) / (1 + 0.05 * 0.5) Forward Rate = 1.25 * (1.01) / (1.025) Forward Rate = 1.25 * 0.98536585 Forward Rate ≈ 1.2317 GBP/USD Therefore, Green Harvest would enter into a forward contract to sell USD at approximately 1.2317 GBP/USD. This locks in a guaranteed exchange rate, protecting them from adverse movements in the GBP/USD exchange rate. Now, consider what happens if Green Harvest expects to receive $1,000,000 in six months. Without hedging, if the GBP/USD rate falls to 1.20, they would receive £1,200,000. With the forward contract at 1.2317, they are guaranteed £1,231,700. This illustrates how forward contracts can provide certainty and mitigate risk. Finally, understanding the regulatory aspects is crucial. Under EMIR (European Market Infrastructure Regulation), Green Harvest may be required to clear this forward contract through a central counterparty (CCP) if they exceed certain thresholds, adding to the complexity of their risk management strategy. They also need to consider the implications of MiFID II regarding best execution and reporting requirements for their derivative transactions.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their sensitivity to the underlying asset’s volatility. Barrier options have a payoff that depends on whether the underlying asset’s price reaches a certain barrier level during the option’s life. The value of a barrier option is highly sensitive to volatility because the probability of hitting the barrier changes significantly with different volatility levels. A higher volatility increases the likelihood of the asset price hitting the barrier, which can either activate or deactivate the option, depending on whether it’s a knock-in or knock-out option. This is unlike standard European options, where volatility primarily affects the option’s premium. To calculate the adjusted premium, we need to consider the impact of increased volatility on the probability of the barrier being hit. Since the option is a knock-out option, an increase in volatility makes it more likely that the barrier will be hit, thus reducing the value of the option. We are given that the market maker uses a volatility adjustment factor of 0.8 for knock-out options. This means that for every 1% increase in implied volatility, the option premium is reduced by 0.8%. The implied volatility increased from 18% to 22%, a change of 4%. Therefore, the premium adjustment is 4% * 0.8 = 3.2%. The original premium is £7.50. The adjusted premium is calculated as follows: Adjustment = 3.2% of £7.50 = 0.032 * £7.50 = £0.24 Adjusted Premium = Original Premium – Adjustment = £7.50 – £0.24 = £7.26 This example uses a volatility adjustment factor to reflect the specific risk associated with barrier options. The factor accounts for the higher sensitivity of these options to changes in volatility compared to standard options. The market maker needs to adjust the premium to account for the increased probability of the barrier being hit, which would render the option worthless. This adjustment ensures that the market maker is adequately compensated for the risk they are taking. The use of a volatility adjustment factor is a practical application of risk management in the derivatives market. It demonstrates how market makers adapt pricing models to account for the unique characteristics of different derivative products.

The question revolves around the concept of implied volatility, a crucial element in options trading and risk management. Implied volatility is the market’s expectation of future volatility, derived from the prices of options. A volatility smile or skew refers to the situation where out-of-the-money (OTM) puts and calls have higher implied volatilities than at-the-money (ATM) options. This phenomenon is commonly observed in equity markets, reflecting a greater demand for downside protection. The question specifically tests the understanding of how changes in market sentiment, particularly increased risk aversion, impact the volatility skew. When investors become more risk-averse, they tend to buy more OTM puts for protection against potential market declines. This increased demand for OTM puts drives up their prices, which in turn increases their implied volatilities. The calculation of the skew involves comparing the implied volatilities of OTM puts and calls. A steeper skew indicates a greater difference in implied volatilities between OTM puts and calls, suggesting a stronger demand for downside protection. Let’s assume the initial implied volatility of ATM options is 20%. OTM calls have an implied volatility of 18%, and OTM puts have an implied volatility of 25%. The initial skew can be represented as the difference between the OTM put and call volatilities: 25% – 18% = 7%. Now, suppose market sentiment shifts, leading to increased risk aversion. The implied volatility of OTM puts increases to 30%, while the implied volatility of OTM calls remains unchanged at 18%. The new skew is 30% – 18% = 12%. The change in the skew is the difference between the new skew and the initial skew: 12% – 7% = 5%. Therefore, the volatility skew has increased by 5 percentage points. This increase reflects the heightened demand for downside protection due to increased risk aversion. The scenario highlights the dynamic nature of the volatility skew and its sensitivity to market sentiment. Understanding these dynamics is essential for effective options trading and risk management.

Let’s consider a scenario involving Gamma, a second-order derivative measuring the rate of change of an option’s delta with respect to changes in the underlying asset’s price. High Gamma implies that the delta is very sensitive to price changes, leading to potentially significant adjustments needed in a hedging strategy. Conversely, low Gamma indicates a more stable delta, requiring less frequent adjustments. Imagine a portfolio manager, Anya, using options to hedge a large equity position. Anya needs to understand how Gamma affects her hedging strategy, especially given the increased market volatility due to an unexpected announcement by the Bank of England regarding interest rate policy. The formula for Gamma is given by: \[ \Gamma = \frac{\partial \Delta}{\partial S} \] where \( \Delta \) is the option’s delta and \( S \) is the price of the underlying asset. To illustrate, consider a call option with a delta of 0.50 when the underlying asset is priced at £100. If the option has a Gamma of 0.05, a £1 increase in the asset’s price to £101 would cause the delta to increase by approximately 0.05, to 0.55. This means Anya would need to adjust her hedge ratio accordingly. Now, let’s consider Anya’s specific situation. She holds 10,000 shares of a UK-listed company, currently priced at £50 per share. To hedge against a potential downturn, she purchases put options on the company’s stock. Each option contract covers 100 shares. The put options she bought have a delta of -0.40 and a Gamma of 0.02. The initial hedge ratio is calculated as: \[ \text{Number of options} = \frac{\text{Number of shares}}{\text{Shares per contract}} \times |\text{Delta}| = \frac{10,000}{100} \times 0.40 = 40 \text{ contracts} \] Anya initially purchases 40 put option contracts. Now, suppose the stock price falls to £48. This £2 decrease in price will affect the option’s delta due to its Gamma. The change in delta is approximately: \[ \Delta \Delta = \Gamma \times \Delta S = 0.02 \times (-2) = -0.04 \] The new delta is: \[ \Delta_{\text{new}} = \Delta_{\text{old}} + \Delta \Delta = -0.40 – 0.04 = -0.44 \] The revised number of option contracts needed to maintain the hedge is: \[ \text{New number of options} = \frac{10,000}{100} \times 0.44 = 44 \text{ contracts} \] Therefore, Anya needs to buy an additional 4 put option contracts to rebalance her hedge. This example highlights the importance of Gamma in dynamic hedging. Ignoring Gamma could lead to an under- or over-hedged position, exposing Anya’s portfolio to unnecessary risk. The Bank of England’s announcement increased market volatility, making Gamma a critical factor in managing her hedge effectively.

The question tests the understanding of hedging strategies using options, specifically a ratio spread, in the context of managing downside risk in a portfolio. The ratio spread involves buying a certain number of put options at a specific strike price and selling a larger number of put options at a lower strike price. This strategy aims to reduce the cost of hedging (compared to buying puts outright) while still providing downside protection, although it also introduces potential losses if the asset price falls significantly below the lower strike price. The calculation determines the breakeven point, which is the asset price at which the hedge starts to incur losses beyond the initial cost. The initial cost of the hedge is calculated as the premium paid for the purchased puts minus the premium received for the sold puts. The breakeven point is then calculated by subtracting this net cost from the strike price of the purchased puts. This calculation determines the level to which the underlying asset price can fall before the hedging strategy begins to lose money. In this specific scenario, the investor buys 10 put options with a strike price of 150 at a premium of £8 each and sells 20 put options with a strike price of 140 at a premium of £3 each. The net cost of the hedge is (£8 * 10) – (£3 * 20) = £80 – £60 = £20. The breakeven point is calculated as 150 – (20/10) = 150 – 2 = £148. This means that the portfolio is protected against losses until the asset price falls below £148. If the asset price falls below £148, the investor will start incurring losses because the profit from the long puts will be offset by the losses from the short puts. This strategy is useful when the investor believes that a moderate decline in the asset price is possible, but a large decline is unlikely.

The question tests the understanding of hedging strategies using options, specifically a ratio spread. A ratio spread involves buying and selling options of the same type (calls or puts) but with different strike prices and in different ratios. This strategy is used to profit from limited movement in the underlying asset’s price. The payoff is calculated by considering the premiums paid and received, and the intrinsic value of the options at expiration. Here’s how to calculate the payoff: 1. **Initial Investment:** Calculate the net premium paid or received. * Premium paid for buying 1 call option with a strike price of 4800: £180 * Premium received for selling 2 call options with a strike price of 4850: 2 \* £80 = £160 * Net Premium Paid = £180 – £160 = £20 2. **Payoff at Expiration:** Consider three scenarios: * **Scenario 1: FTSE 100 price ≤ 4800:** All options expire worthless. The payoff is the negative of the net premium paid: -£20. * **Scenario 2: 4800 < FTSE 100 price < 4850:** The 4800 call is in the money, and the 4850 calls are out of the money. The payoff is: * (FTSE 100 Price – 4800) – £20 * **Scenario 3: FTSE 100 price ≥ 4850:** The 4800 call and the two 4850 calls are in the money. The payoff is: * (FTSE 100 Price – 4800) – 2 \* (FTSE 100 Price – 4850) – £20 * = FTSE 100 Price – 4800 – 2 \* FTSE 100 Price + 9700 – £20 * = -FTSE 100 Price + 4880 3. **Calculate the Payoff for each given FTSE 100 price:** * **FTSE 100 = 4780:** Payoff = -£20 * **FTSE 100 = 4820:** Payoff = (4820 – 4800) – £20 = £0 * **FTSE 100 = 4860:** Payoff = -4860 + 4880 = £20 * **FTSE 100 = 4900:** Payoff = -4900 + 4880 = -£20 This example illustrates how a ratio call spread can generate profits within a specific range but can also lead to losses if the price moves significantly outside that range. The strategy is particularly sensitive to volatility and the trader's expectations about the magnitude of price movements.

The question assesses understanding of how a sudden shift in the yield curve’s slope impacts swap valuation, particularly concerning the fixed rate payer. A steeper yield curve implies higher longer-term interest rates. This impacts the present value of future fixed rate payments in the swap. The fixed rate payer will now be at a disadvantage because they are committed to paying a lower fixed rate than what the market now dictates (as reflected by the higher end of the yield curve). Therefore, the present value of the fixed payments will decrease, resulting in a loss for the fixed rate payer. To quantify the loss, we need to consider the present value of the difference between the original fixed rate and the rates implied by the new, steeper yield curve. Assume the notional principal is £10,000,000 and the swap has 5 years remaining with annual payments. Let’s say the original fixed rate was 2%. After the yield curve steepens, the implied forward rates for each of the next 5 years are now: 2.5%, 3.0%, 3.5%, 4.0%, and 4.5%. The present value of the *original* fixed payments is: \[ \sum_{t=1}^{5} \frac{0.02 \times 10,000,000}{(1+r_t)^t} \] where \(r_t\) are the original discount rates. The present value of the *new* implied fixed payments (using the new forward rates as discount rates) is: \[ \sum_{t=1}^{5} \frac{0.02 \times 10,000,000}{(1+f_t)^t} \] where \(f_t\) are the new forward rates (2.5%, 3.0%, 3.5%, 4.0%, 4.5%). The *loss* to the fixed rate payer is the difference between the present value calculated using the original yield curve and the present value using the new, steeper yield curve. A close approximation can be found by discounting the difference in rates each year. Approximating the present value calculation, the approximate loss is: \[ \sum_{t=1}^{5} \frac{(0.02 – f_t’) \times 10,000,000}{(1+0.02)^t} \] where \(f_t’\) is the difference between new forward rates and original rate (0.005, 0.01, 0.015, 0.02, 0.025). This calculation will result in a negative value, representing a loss. The closest answer to this negative value is the correct one.

The core of this problem lies in understanding how a delta-neutral portfolio reacts to changes in implied volatility (vega) and time decay (theta), and how these sensitivities can be exploited or mitigated using options strategies. A delta-neutral portfolio is designed to be insensitive to small changes in the underlying asset’s price. However, it is still exposed to other risks, particularly those related to volatility and time. Vega measures the sensitivity of the portfolio’s value to changes in implied volatility. Theta measures the sensitivity of the portfolio’s value to the passage of time. The investor’s strategy involves shorting straddles, which is inherently a bet that implied volatility will decrease or remain stable, and that the underlying asset will not make a large move. Short straddles are profitable when the underlying asset price stays within a narrow range around the strike price of the options. However, short straddles have negative vega and theta, meaning that the portfolio loses value if implied volatility increases or as time passes. To create a vega-neutral position, the investor needs to offset the negative vega of the short straddles. This can be achieved by buying options with positive vega. In this case, the investor buys a call option. The number of call options needed to offset the vega is calculated by dividing the negative vega of the short straddles by the vega of the call option. Once the portfolio is vega-neutral, the investor can assess the overall theta of the portfolio. The theta of the short straddles is negative, and the theta of the call options is also negative. Therefore, the overall theta of the portfolio is negative, meaning that the portfolio loses value as time passes. To calculate the daily profit or loss, we need to consider the theta of the entire portfolio. The theta of the short straddles is -£500 per day, and the theta of the call options is -£100 per day. Therefore, the overall theta of the portfolio is -£600 per day. The formula for calculating the daily profit or loss is: Daily Profit/Loss = (Theta of Short Straddles + Theta of Call Options) In this case: Daily Profit/Loss = (-£500 + -£100) = -£600 Therefore, the portfolio is expected to lose £600 per day due to time decay.

The Black-Scholes model is a cornerstone of options pricing theory, but its practical application requires careful consideration of its underlying assumptions and limitations. One key assumption is constant volatility over the option’s lifetime. In reality, volatility fluctuates, and the *volatility smile* or *volatility skew* effect demonstrates that options with different strike prices (but the same expiration date) have different implied volatilities. The question probes the understanding of how a deviation from constant volatility, specifically a volatility skew, impacts the fair value of options. The volatility skew typically arises from supply and demand imbalances for out-of-the-money (OTM) puts and calls. For instance, a market anticipating a potential sharp decline often sees increased demand for OTM puts, driving up their prices and implied volatilities. Conversely, OTM calls may be less in demand, resulting in lower implied volatilities. This creates a skew where implied volatility is higher for lower strike prices (puts) and lower for higher strike prices (calls). The fair value of an option is directly related to its implied volatility; higher volatility generally means a higher option price. In a market exhibiting a volatility skew, OTM puts will be more expensive than predicted by the Black-Scholes model if it were calibrated using at-the-money (ATM) volatility. Similarly, OTM calls will be cheaper. An investor using the Black-Scholes model with a single volatility input (e.g., ATM volatility) to price options across different strike prices would thus systematically misprice OTM puts and calls. In this scenario, a portfolio manager uses the Black-Scholes model calibrated to the ATM volatility to price and trade options. Understanding the volatility skew and its impact on option prices is crucial for avoiding arbitrage opportunities and managing risk effectively. The correct answer will recognize that OTM puts will be relatively undervalued by the model, leading to potential profit opportunities by buying them.

The question assesses the understanding of option strategies, specifically a butterfly spread, and how changes in implied volatility (the “volatility smile”) affect its profitability. A butterfly spread profits when the underlying asset price remains near the strike price of the short options. The “volatility smile” refers to the phenomenon where out-of-the-money and in-the-money options have higher implied volatilities than at-the-money options. Here’s the breakdown of how the volatility smile impacts the butterfly spread: 1. **Initial Setup:** The butterfly spread is constructed by buying one call option at a lower strike price (K1), selling two call options at a middle strike price (K2), and buying one call option at a higher strike price (K3). K2 is the strike price the investor expects the underlying asset to be near at expiration. 2. **Volatility Smile Impact:** * **Increased Volatility Smile:** If the volatility smile steepens (i.e., OTM and ITM option volatilities increase more than ATM option volatilities), the prices of the K1 and K3 options (the long calls) will increase more than the price of the K2 options (the short calls). Since you are long the K1 and K3 options and short two K2 options, the value of the butterfly spread increases. * **Decreased Volatility Smile (Flattening):** If the volatility smile flattens, the prices of the K1 and K3 options will increase less (or even decrease more) than the price of the K2 options. This decreases the value of the butterfly spread. 3. **Time Decay:** As time passes, all options lose value due to time decay (theta). However, the short options (K2) are more sensitive to time decay when the underlying asset price is near K2. 4. **Profit/Loss Scenarios:** * **Asset Price Stays Near K2:** If the asset price remains close to K2 at expiration, the investor profits. The short options expire worthless, and the long options expire with minimal value. * **Asset Price Moves Significantly Away from K2:** If the asset price moves significantly away from K2, the investor loses money. The long options become more valuable, but the losses on the short options outweigh the gains. In the given scenario, the steepening volatility smile benefits the butterfly spread because the long options (K1 and K3) increase in value more than the short options (K2). Time decay is a factor, but the steepening volatility smile has a more significant positive impact in the short term.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Golden Harvest,” which wants to protect its future wheat sales against price declines. They plan to use wheat futures contracts traded on the ICE Futures Europe exchange. To determine the number of contracts needed, we need to understand the concept of hedge ratio. The hedge ratio minimizes the variance of the hedged position. A simplified approach to calculate the hedge ratio is: Hedge Ratio = (Size of Exposure) / (Size of One Futures Contract) \* Correlation Adjustment First, determine the total wheat Golden Harvest wants to hedge: 25,000 tonnes. Next, determine the size of one ICE Wheat Futures contract: 100 tonnes. The initial hedge ratio is 25,000 / 100 = 250 contracts. However, this assumes a perfect correlation between the spot price of Golden Harvest’s wheat and the futures price. In reality, there’s basis risk. Let’s assume Golden Harvest’s historical data shows that for every £1 change in the futures price, their wheat price changes by £0.8. This is the price sensitivity factor, often referred to as the “delta” in hedging terms, although not directly related to options delta in this context. This delta is the correlation adjustment. Adjusted Hedge Ratio = Initial Hedge Ratio \* Price Sensitivity Factor = 250 \* 0.8 = 200 contracts. Therefore, Golden Harvest should short 200 wheat futures contracts to hedge their exposure effectively.

Let’s consider a scenario involving exotic derivatives and hedging strategies. A UK-based agricultural firm, “HarvestYield PLC,” faces price volatility in barley, a key ingredient for their malt production. They decide to use an Asian option to hedge their exposure. An Asian option’s payoff depends on the average price of the underlying asset (barley) over a specified period, making it suitable for smoothing out price fluctuations. First, we need to understand the potential impact of using an Asian option versus a standard European option. The Asian option reduces the impact of extreme price spikes or drops occurring at a specific date (as with a European option), providing a more stable hedging outcome. Next, let’s examine the calculation of the option’s payoff. Suppose HarvestYield buys an Asian call option on barley with a strike price of £200 per ton, and the averaging period is one month (20 trading days). The daily barley prices are recorded, and their average is calculated. If the average price is above £200, the option pays out the difference; otherwise, it expires worthless. Assume the average barley price over the 20-day period is £215 per ton. The payoff would be £215 – £200 = £15 per ton. The total payoff depends on the number of options contracts HarvestYield purchased, each representing a specific tonnage of barley. Now, consider the risk management implications. The Asian option reduces the risk of a single day’s price volatility severely impacting HarvestYield’s profitability. However, it also means they won’t fully benefit from a significant price increase on a single day. The choice between Asian and European options depends on HarvestYield’s specific risk appetite and hedging goals. Finally, let’s analyze the regulatory aspects. As a UK-based firm, HarvestYield must comply with EMIR (European Market Infrastructure Regulation) regarding derivatives trading. This includes reporting their derivatives positions to a trade repository and potentially clearing certain over-the-counter (OTC) derivatives through a central counterparty (CCP) to reduce counterparty risk. The specific requirements depend on HarvestYield’s classification under EMIR (e.g., financial counterparty or non-financial counterparty) and the nature of the derivatives they trade.

This question delves into the complexities of managing credit risk within a portfolio utilizing Credit Default Swaps (CDS). The core challenge lies in understanding how changes in correlation between reference entities within the CDS portfolio impact the overall portfolio risk. The initial portfolio has a notional value of £5,000,000, equally distributed across five companies. The initial default probability for each company is 2%. To calculate the initial expected loss, we multiply the notional exposure per company (£1,000,000) by the default probability (2%) and the Loss Given Default (LGD) of 60%: Expected Loss per Company = £1,000,000 * 0.02 * 0.60 = £12,000 Total Expected Loss = £12,000 * 5 = £60,000 Now, let’s analyze the impact of increased correlation. When correlation rises, the likelihood of multiple defaults occurring simultaneously increases. This “clustering” effect magnifies the potential for larger losses. To quantify this, we need to consider a scenario where two companies default. The question states that with increased correlation, the probability of at least two companies defaulting increases by 5%. This implies a joint default probability that needs to be factored into our risk assessment. Let’s assume that the 5% increase represents the probability of exactly two defaults occurring. To calculate the incremental loss due to the increased correlation, we consider the scenario where two companies default. The loss from these two defaults would be: Loss from Two Defaults = 2 * £1,000,000 * 0.60 = £1,200,000 However, we need to factor in the probability of this event occurring (the 5% increase). The expected incremental loss is: Incremental Expected Loss = £1,200,000 * 0.05 = £60,000 Therefore, the new total expected loss is the sum of the initial expected loss and the incremental expected loss: New Total Expected Loss = £60,000 + £60,000 = £120,000 The percentage increase in expected loss is: Percentage Increase = ((£120,000 – £60,000) / £60,000) * 100% = 100% This example highlights the importance of correlation in credit risk management. A seemingly small increase in correlation can significantly amplify the potential for losses, especially in portfolios with concentrated exposures. The calculation demonstrates how to quantify this impact and adjust risk management strategies accordingly. It showcases the need for stress testing and scenario analysis to understand the potential impact of adverse market conditions on derivative portfolios.

The question revolves around the impact of unexpected regulatory changes on a sophisticated hedging strategy using options, specifically designed to protect a portfolio from market downturns. The core concept tested is the understanding of how regulatory shifts, particularly those affecting margin requirements and position limits, can drastically alter the effectiveness and cost of a hedging strategy. We need to consider the direct impact of increased margin requirements, which tie up more capital, and position limits, which restrict the size of the hedge. The calculation involves determining the initial cost of the hedge, the additional capital required due to increased margin requirements, and the potential loss due to the hedge being less effective because of position limits. First, calculate the initial cost of the put options: 10,000 contracts * £2.50/contract = £25,000. Next, determine the increased margin requirement per contract: £1.50/contract. The total increase in margin is: 10,000 contracts * £1.50/contract = £15,000. Now, calculate the reduction in hedge effectiveness due to position limits. The fund can only hedge 70% of its original position. This means 30% of the portfolio is now unhedged. Assuming a potential market downturn of 15%, the unhedged portion would lose: 0.30 * £5,000,000 * 0.15 = £225,000. Finally, calculate the total cost, which is the initial option cost + increased margin + potential loss due to reduced hedge effectiveness: £25,000 + £15,000 + £225,000 = £265,000. The key takeaway is that regulatory changes don’t just affect the cost of derivatives directly; they can also impact the effectiveness of risk management strategies, potentially exposing the portfolio to greater losses. This highlights the importance of ongoing monitoring of the regulatory landscape and adapting hedging strategies accordingly. The example illustrates a scenario where a seemingly straightforward hedging strategy is complicated by external factors, demanding a nuanced understanding of both the derivatives themselves and the regulatory environment in which they operate.

To solve this problem, we need to understand how implied volatility affects option prices, especially in the context of exotic options like barrier options. A knock-out barrier option ceases to exist if the underlying asset’s price hits the barrier level. Increased volatility raises the probability of hitting the barrier, thus decreasing the option’s value. Conversely, a knock-in barrier option only comes into existence if the barrier is hit. Higher volatility increases the likelihood of the barrier being triggered, raising the option’s value. In this scenario, the fund manager uses a combination of a knock-out call option and a knock-in put option. The knock-out call option loses value as volatility increases because there’s a higher chance of the barrier being hit, causing the option to expire worthless. The knock-in put option gains value as volatility increases because the barrier is more likely to be triggered, bringing the put option into existence. The net impact on the portfolio depends on the relative sensitivities of the knock-out call and knock-in put to changes in volatility. This sensitivity is commonly referred to as Vega. If the Vega of the knock-out call is higher in absolute terms than the Vega of the knock-in put, the portfolio’s value will decrease as implied volatility increases. This is because the loss in value of the knock-out call outweighs the gain in value of the knock-in put. Let’s consider a numerical example. Suppose the knock-out call has a Vega of -0.05 (meaning for every 1% increase in implied volatility, the option loses £0.05 in value) and the knock-in put has a Vega of 0.03 (meaning for every 1% increase in implied volatility, the option gains £0.03 in value). If implied volatility increases by 1%, the knock-out call loses £0.05, and the knock-in put gains £0.03. The net effect on the portfolio is a loss of £0.02. This illustrates that if the absolute value of the knock-out call’s Vega exceeds the knock-in put’s Vega, an increase in implied volatility will lead to a decrease in the portfolio’s value.

The question assesses understanding of hedging strategies using options, specifically a ratio spread, and the impact of implied volatility on the strategy’s profitability. A ratio spread involves buying and selling different numbers of call or put options with different strike prices but the same expiration date. This strategy is used to profit from a specific view on the underlying asset’s price movement and volatility. The initial setup involves buying 100 call options with a strike price of 150 and selling 200 call options with a strike price of 160. This creates a situation where the investor profits if the price stays below 160, but faces potentially unlimited losses if the price rises significantly above 160. The net premium paid is £500, representing the cost of establishing the position. The key to this question is understanding how implied volatility affects the value of the options and, consequently, the profitability of the ratio spread. Implied volatility is the market’s expectation of future volatility, and it directly impacts option prices. An increase in implied volatility generally increases the value of both the bought and sold options. However, the effect is not linear and depends on the strike prices relative to the current asset price. In this scenario, implied volatility increases from 20% to 25%. This increase will affect the 150 strike calls (bought) and the 160 strike calls (sold) differently. We need to consider the ‘vega’ of each option, which measures the sensitivity of an option’s price to changes in implied volatility. Higher strike calls generally have lower vegas. Since we are short twice as many 160 strike calls as we are long 150 strike calls, the increase in implied volatility will likely result in a net loss due to the short calls increasing in value more than the long calls. To calculate the approximate change in the value of the portfolio, we can estimate the change in option prices due to the volatility change. Assume that the 150 strike calls increase in value by £0.25 each due to the volatility increase, and the 160 strike calls increase by £0.30 each. Change in value of 150 strike calls: 100 * £0.25 = £25 Change in value of 160 strike calls: 200 * £0.30 = £60 Net change in value: £25 – £60 = -£35 The net loss due to the change in implied volatility is £35. Subtracting this from the initial net premium paid of £500 results in a total loss of £535. Final Answer: The final answer is £535 loss.

The question assesses the understanding of hedging strategies using futures contracts, specifically focusing on the concept of basis risk. Basis risk arises because the price of the asset being hedged (in this case, Grade A Arabica coffee beans) may not move perfectly in tandem with the price of the futures contract (ICE Coffee C futures). The optimal hedge ratio minimizes the variance of the hedged position, taking into account the correlation between the spot price and the futures price. The formula for the hedge ratio is: \[ \text{Hedge Ratio} = \rho \frac{\sigma_S}{\sigma_F} \] Where: – \(\rho\) is the correlation coefficient between the spot price change (\(\Delta S\)) and the futures price change (\(\Delta F\)). – \(\sigma_S\) is the standard deviation of the spot price change. – \(\sigma_F\) is the standard deviation of the futures price change. Given: – \(\rho = 0.75\) – \(\sigma_S = 0.04\) (4% standard deviation of spot price) – \(\sigma_F = 0.05\) (5% standard deviation of futures price) Plugging the values into the formula: \[ \text{Hedge Ratio} = 0.75 \times \frac{0.04}{0.05} = 0.75 \times 0.8 = 0.6 \] Therefore, the optimal hedge ratio is 0.6. This means that for every unit of the underlying asset (coffee beans), the roaster should short 0.6 units of the futures contract to minimize risk. Since the roaster needs to hedge 250,000 kg of coffee and each ICE Coffee C futures contract is for 37,500 lbs (approximately 17,009.7 kg), we first calculate how many futures contracts would be needed to hedge the entire exposure without considering the hedge ratio: Number of contracts without hedge ratio = \(\frac{250,000 \text{ kg}}{17,009.7 \text{ kg/contract}} \approx 14.7\) contracts Since you can’t trade fractions of contracts, you would typically round to the nearest whole number, which would be 15 contracts. Now, applying the hedge ratio of 0.6: Number of contracts with hedge ratio = \(15 \times 0.6 = 9\) contracts Therefore, the roaster should short 9 ICE Coffee C futures contracts to optimally hedge their exposure, taking into account the basis risk reflected in the correlation and standard deviations. A crucial aspect of understanding hedging is recognizing that it doesn’t eliminate risk entirely, but rather minimizes it. The hedge ratio of 0.6 indicates that the futures contract provides only partial coverage due to the imperfect correlation between the spot and futures prices. This highlights the ever-present basis risk. Consider a scenario where the spot price of Arabica increases significantly due to a localized weather event in Brazil, while the ICE Coffee C futures, influenced by global supply, doesn’t rise as much. The roaster would benefit from the increased value of their inventory but would experience a loss on their short futures position. The hedge serves to dampen the overall volatility, but not eliminate it completely.

The question focuses on the application of delta hedging in a portfolio management context, specifically when dealing with dividend-paying stocks. The challenge lies in understanding how dividend payments affect the option’s delta and, consequently, the adjustments needed in the hedge. The calculation involves determining the initial hedge ratio, adjusting for the dividend payment’s impact on the stock price, and then recalculating the required hedge adjustment. The initial delta hedge is established by selling the appropriate number of call options to offset the directional risk of the shares. When a dividend is paid, the stock price typically drops by approximately the dividend amount. This price drop impacts the option’s delta. The option becomes less sensitive to further price changes as it moves out-of-the-money (or further out-of-the-money). To maintain a delta-neutral position, the portfolio manager must reduce the short call position. The amount of the adjustment depends on the new delta of the call option after the dividend payment. The goal is to reduce the number of short calls to match the reduced sensitivity of the option to the stock price movement. The calculation involves determining the change in the delta and adjusting the number of options accordingly. The final step is to calculate the number of options that need to be bought back to rebalance the hedge. This is done by comparing the initial number of options shorted with the new number of options required to maintain the delta-neutral position. This scenario requires a deep understanding of option pricing dynamics, delta hedging, and the impact of corporate actions like dividends on derivative positions. The correct answer reflects the accurate adjustment needed to maintain a delta-neutral portfolio after the dividend payment.

The question assesses the understanding of delta hedging, specifically when dealing with discrete hedging intervals and the impact of transaction costs. Delta hedging aims to create a portfolio that is insensitive to small changes in the underlying asset’s price. However, in practice, continuous hedging is impossible. Hedging is done at discrete intervals (e.g., daily, weekly). When the asset price moves between hedging intervals, the hedge becomes imperfect, leading to gains or losses. Transaction costs further erode the profitability of the hedging strategy. The key is to calculate the cost of rebalancing the hedge and compare it to the potential profit from the option. Here’s how to solve the problem: 1. **Calculate the initial hedge ratio (Delta):** The delta of the call option is 0.6. This means for every £1 increase in the share price, the call option’s price is expected to increase by £0.60. 2. **Determine the number of shares to short:** Since the fund is short 10,000 call options, the initial hedge requires shorting 10,000 * 0.6 = 6,000 shares. 3. **Calculate the change in share price:** The share price increased by £0.50 (from £10 to £10.50). 4. **Calculate the new delta:** The delta increased to 0.65. 5. **Calculate the new number of shares to short:** The new hedge requires shorting 10,000 * 0.65 = 6,500 shares. 6. **Calculate the number of shares to buy:** The fund needs to buy 6,500 – 6,000 = 500 shares. 7. **Calculate the total transaction cost:** The transaction cost is £0.02 per share, so buying 500 shares costs 500 * £0.02 = £10. 8. **Calculate the profit from the option position:** The fund is short 10,000 call options. We need to know the change in the call option’s price. While we know the delta, we don’t have enough information to precisely calculate the change in the call option’s price. However, we can approximate it using the delta: 10,000 options * 0.6 * £0.50 = £3,000 increase in the value of the options. Since the fund is short the options, this represents a loss of £3,000 * (1+0.05) = £3,150 9. **Calculate the loss from the share hedge:** The fund shorted 6,000 shares at £10 and the price increased to £10.50. The loss is 6,000 * £0.50 = £3,000. 10. **Calculate the net profit/loss:** The net profit/loss is the loss from the option position, plus the profit from the shorted shares, minus the transaction costs: -£3,150 + £3,000 – £10 = -£160. Therefore, the net result is a loss of £160.

The core concept tested here is the understanding of how volatility skew impacts option pricing and strategy selection, specifically in the context of earnings announcements. Volatility skew refers to the difference in implied volatility between options with the same expiration date but different strike prices. Typically, equity options exhibit a “volatility smile” or “smirk,” where out-of-the-money puts have higher implied volatilities than at-the-money options, reflecting a greater demand for downside protection. Earnings announcements are significant events that often cause a temporary increase in implied volatility across all options, known as an “implied volatility crush” *after* the announcement. This is because the uncertainty surrounding the earnings release leads to increased demand for options, driving up their prices (and thus implied volatility). Once the earnings are released, the uncertainty is resolved, and implied volatility typically decreases sharply. The trader’s strategy selection should consider the skew and the expected volatility crush. A short strangle involves selling both an out-of-the-money call and an out-of-the-money put option. The trader profits if the stock price stays within a defined range between the strike prices of the call and put options. However, the trader is exposed to potentially unlimited losses if the stock price moves significantly beyond these strike prices. Given the skew (puts more expensive) and the expected volatility crush, the trader needs to consider how these factors affect the pricing of the options and the potential profitability of the short strangle. The put side being more expensive due to skew means the trader collects a higher premium for the put. However, the volatility crush will erode this premium. The key is to assess whether the premium collected is sufficient to offset the potential losses if the stock price moves significantly in either direction *and* the impact of the volatility crush. To calculate the potential profit/loss, we need to consider the premiums received and the potential payout if the stock price moves beyond the breakeven points. The breakeven points are calculated as: * **Upper Breakeven:** Call Strike Price + Net Premium Received * **Lower Breakeven:** Put Strike Price – Net Premium Received In this case, the net premium received is £2.50 (£1.75 + £0.75). Therefore: * **Upper Breakeven:** £105 + £2.50 = £107.50 * **Lower Breakeven:** £95 – £2.50 = £92.50 If the stock price ends up at £110, the call option will be in the money by £5 (£110 – £105). Since the trader sold the call, they will lose £5. However, they initially received a premium of £2.50. The net loss is £2.50 (£5 – £2.50). The put option expires worthless, so there is no additional profit or loss from the put side.

To address this question, we need to understand how changes in volatility expectations impact option prices, particularly in the context of earnings announcements. Earnings announcements typically lead to a spike in implied volatility (IV) due to increased uncertainty about the company’s future performance. This volatility crush occurs after the announcement as the uncertainty resolves. The question tests understanding of how different option strategies are affected by these volatility changes. The strangle strategy, which involves buying both a call and a put option with strike prices around the current market price, benefits from large price movements regardless of direction. However, the key here is the volatility component. Before the earnings announcement, the implied volatility is high, making the options expensive. After the announcement, the volatility drops, reducing the value of the options. This volatility crush can offset any gains from the stock price movement, especially if the price movement is not substantial enough to compensate for the volatility decline. A covered call strategy, selling a call option on a stock you own, benefits from stable or slightly increasing prices. The premium received from selling the call provides downside protection. However, in the context of an earnings announcement, the increased implied volatility makes the call option more expensive, increasing the premium received. After the announcement, the volatility crush reduces the value of the sold call option, allowing the covered call writer to keep more of the premium. If the stock price remains relatively stable or increases moderately, the covered call strategy can perform well, capitalizing on both the premium and the price movement. A protective put strategy, buying a put option on a stock you own, acts as insurance against a price decline. Before the earnings announcement, the implied volatility is high, making the put option expensive. After the announcement, the volatility crush reduces the value of the put option. If the stock price does not decline significantly, the protective put strategy loses value due to the volatility crush. A long call option benefits from an increase in the stock price. However, the implied volatility is high before the earnings announcement, making the call option expensive. After the announcement, the volatility crush reduces the value of the call option, potentially offsetting gains from the stock price movement. Considering these factors, the covered call strategy is most likely to benefit from the volatility crush after an earnings announcement, assuming the stock price remains relatively stable or increases moderately.

The core of this question lies in understanding how changes in the underlying asset’s price, time to expiration, and volatility affect the value of a European call option. The Black-Scholes model provides a framework for valuing such options. However, the question deliberately avoids direct calculation and instead focuses on the *relative* impact of these factors. The Black-Scholes formula is: \[C = S_0N(d_1) – Ke^{-rT}N(d_2)\] where: * \(C\) = Call option price * \(S_0\) = Current stock price * \(K\) = Strike price * \(r\) = Risk-free interest rate * \(T\) = Time to expiration * \(N(x)\) = Cumulative standard normal distribution function * \(d_1 = \frac{ln(\frac{S_0}{K}) + (r + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}}\) * \(d_2 = d_1 – \sigma\sqrt{T}\) * \(\sigma\) = Volatility of the stock The question asks for the *greatest* impact. A significant *increase* in the underlying asset’s price generally has the most substantial positive impact on a call option’s value because it directly increases the likelihood that the option will be in the money at expiration. Time decay (Theta) erodes option value, and while increased volatility (Vega) generally increases option value, the effect is less direct and immediate than a change in the underlying asset’s price, especially when close to expiration. A decrease in the risk-free rate (Rho) will have a positive impact on a call option’s value, but it is usually less significant than changes in the underlying asset’s price. Consider a scenario where a biotech company, “GeneSys,” has a breakthrough drug trial. Its stock price jumps dramatically. A call option on GeneSys stock would see a much larger increase in value than if there was simply a slight increase in implied volatility due to general market uncertainty, or a small drop in interest rates. Therefore, a substantial positive price movement in the underlying asset will have the most significant impact on the call option’s value.

The core of this question revolves around understanding how implied volatility, specifically derived from option prices, reflects market sentiment and its potential predictive power regarding future realized volatility. Implied volatility (IV) is essentially the market’s expectation of future volatility over the life of the option. The VIX, or volatility index, is a real-time index that represents the market’s expectation of 30-day volatility derived from the price of S&P 500 index options. A key concept is the volatility risk premium (VRP), which is the difference between implied volatility and realized volatility. A positive VRP suggests that investors are willing to pay a premium for protection against market downturns, implying a fear of increased volatility. Conversely, a negative VRP might suggest complacency. The scenario presents a situation where a fund manager is using VIX as a signal. The VIX level, in isolation, is less important than its relationship to historical levels and other market indicators. An unusually high VIX suggests heightened uncertainty and potential for large market swings. A low VIX suggests the opposite – complacency and a belief in market stability. However, the fund manager needs to understand the nuances. For instance, a high VIX might not always translate into immediate market decline; it could represent a temporary spike followed by a period of consolidation. The fund manager’s strategy of using VIX to allocate to defensive assets hinges on the assumption that a rising VIX signals increased market risk. The fund manager should consider other factors, such as the shape of the volatility smile (the relationship between implied volatility and strike price), skewness, and kurtosis of the implied volatility distribution, to get a more complete picture of market sentiment. Furthermore, the manager needs to backtest the strategy to determine its historical effectiveness and understand its limitations. The strategy should be dynamically adjusted based on changing market conditions and the evolving relationship between implied and realized volatility. Finally, the fund manager should be aware of the potential for VIX manipulation or distortions due to factors unrelated to fundamental market risk. Large option positions, algorithmic trading strategies, and other market microstructure effects can influence VIX levels.

The question assesses the understanding of the impact of implied volatility on option prices and the subsequent hedging requirements for a portfolio. Specifically, it focuses on how a portfolio manager should adjust their delta-neutral hedge when implied volatility unexpectedly changes. Delta-neutral hedging aims to create a portfolio whose value is insensitive to small changes in the underlying asset’s price. However, delta is itself affected by volatility (as reflected by Vega). A change in implied volatility directly impacts the option’s delta, thus requiring an adjustment to the hedge to maintain delta neutrality. The formula for calculating the change in delta due to a change in volatility is approximated by Vega. Vega represents the sensitivity of an option’s price to a 1% change in implied volatility. If a portfolio is delta-neutral but not Vega-neutral, a change in volatility will cause the portfolio’s delta to shift away from zero. The problem requires calculating the change in the portfolio’s delta resulting from the increase in implied volatility, and then determining the number of additional shares needed to re-establish delta neutrality. Here’s how to solve the problem: 1. **Calculate the change in delta:** The portfolio’s Vega is £50,000 per 1% change in implied volatility. Volatility increases by 2% (from 20% to 22%). Therefore, the change in the portfolio’s delta is: Change in Delta = Vega * Change in Volatility = £50,000 * 2% = £1,000 This means the portfolio’s delta has increased by £1,000. Since the portfolio was initially delta-neutral, this change means the portfolio is now equivalent to being long 1,000 shares. 2. **Determine the number of shares to trade:** To re-establish delta neutrality, the portfolio manager needs to offset this increased delta. Since the portfolio is now long 1,000 shares (in delta terms), the manager needs to short 1,000 shares. Therefore, the portfolio manager should sell 1,000 shares to return the portfolio to a delta-neutral position. This scenario uniquely tests the understanding of Vega and its practical implications for dynamic hedging in a portfolio context, going beyond textbook definitions. It requires the candidate to apply the concept of Vega to a real-world hedging problem, demonstrating a deep understanding of derivatives risk management.

The core of this question lies in understanding how delta hedging works in practice, and how transaction costs impact the effectiveness of that hedging strategy. Delta hedging aims to neutralize the directional risk of an option position by dynamically adjusting the underlying asset holding. However, real-world trading incurs transaction costs (brokerage fees, bid-ask spread), which erode the profit from delta adjustments. The optimal hedging frequency balances the cost of frequent adjustments against the risk of a poorly hedged position. A higher transaction cost environment pushes the optimal hedging frequency lower, because the cost of constantly rebalancing outweighs the benefit of a slightly more precise hedge. The question requires calculating the profit/loss from a delta hedging strategy given transaction costs, and comparing it to the theoretical outcome without transaction costs, to determine the impact. The Black-Scholes model provides a theoretical framework for option pricing and delta calculation. The delta (\(\Delta\)) represents the sensitivity of the option price to changes in the underlying asset’s price. The delta is used to determine the number of shares to buy or sell to offset the option’s price movement. Here’s how to solve this problem step-by-step: 1. **Initial Hedge:** The investor sells 100 call options with a delta of 0.4. To delta hedge, the investor needs to buy \(100 \times 0.4 = 40\) shares of the underlying asset. The cost of buying these shares is \(40 \times 50 = £2000\). 2. **Price Increase:** The stock price increases to £51, and the delta increases to 0.45. The investor needs to adjust the hedge by buying an additional \(100 \times (0.45 – 0.4) = 5\) shares. The cost of buying these shares is \(5 \times 51 = £255\). 3. **Price Decrease:** The stock price decreases to £50.50, and the delta decreases to 0.42. The investor needs to adjust the hedge by selling \(100 \times (0.45 – 0.42) = 3\) shares. The revenue from selling these shares is \(3 \times 50.50 = £151.50\). 4. **Final Price Decrease:** The stock price decreases to £49, and the delta decreases to 0.35. The investor needs to adjust the hedge by selling \(100 \times (0.42 – 0.35) = 7\) shares. The revenue from selling these shares is \(7 \times 49 = £343\). 5. **Unwind Hedge:** At the end, the investor unwinds the hedge by selling all remaining shares. The investor sells \(100 \times 0.35 = 35\) shares at £49. The revenue from selling these shares is \(35 \times 49 = £1715\). 6. **Calculate Total Cost of Shares:** The total cost of buying shares is \(£2000 + £255 = £2255\). 7. **Calculate Total Revenue from Selling Shares:** The total revenue from selling shares is \(£151.50 + £343 + £1715 = £2209.50\). 8. **Calculate Profit/Loss from Hedging:** The profit/loss from hedging is \(£2209.50 – £2255 = -£45.50\). 9. **Calculate Total Transaction Costs:** The investor made 4 transactions of 10 GBP each, for a total of 40 GBP. 10. **Calculate Net Profit/Loss:** Subtract transaction costs from the hedging profit/loss: \(-£45.50 – £40 = -£85.50\). 11. **Options P/L:** The investor sold 100 options and the options expired worthless so the profit is 100*5=500 GBP 12. **Total P/L:** The total profit/loss is 500-85.5 = 414.50 GBP

The question tests understanding of how implied volatility, dividends, and time to expiration affect option prices, and specifically how to calculate the theoretical impact on a call option’s price given simultaneous changes in these factors. We’ll use a modified Black-Scholes intuition to approximate the price change. First, we calculate the impact of the volatility change. A volatility increase of 5% generally increases the call option price. A rough approximation is that a 1% change in volatility leads to roughly a 5-10% change in the option price (this is highly dependent on moneyness and time to expiry). For simplicity, let’s assume a vega of 0.4 (meaning a 1% increase in volatility increases the option price by 0.4). Therefore, a 5% increase in volatility increases the option price by approximately 5 * 0.4 = 2.0. Second, we calculate the impact of the dividend payment. Dividends reduce the stock price, which negatively impacts call options. The present value of the dividend is subtracted from the stock price in option pricing models. So, the call option price will decrease by the present value of the dividend. With a risk-free rate of 5% and time to dividend of 0.25 years, the present value is approximately 2.0 / (1 + 0.05 * 0.25) ≈ 1.975. Third, we calculate the impact of the time decay. A decrease in time to expiration generally decreases the call option price. With 0.1 years less to expiry, the option loses some of its time value. Theta, which measures the sensitivity of the option price to time, might be around -0.1 (meaning the option loses 0.1 per day). Therefore, with 0.1 years (approximately 36.5 days), the price decreases by 36.5 * 0.1/365 ≈ 0.1. Combining these effects, the net change is +2.0 (volatility) – 1.975 (dividend) – 0.1 (time decay) = -0.075. Therefore, the option price decreases by approximately 0.075. Given an initial price of 10, the new price is approximately 10 – 0.075 = 9.925.

The question assesses the understanding of volatility smiles and skews in options pricing, particularly in relation to the GBP/USD currency pair. It requires calculating the implied volatility of a put option using the provided data and understanding how the volatility smile affects pricing. First, we need to understand that a volatility smile (or skew) indicates that implied volatilities are not constant across different strike prices. In this case, the at-the-money (ATM) volatility is 10%. The 25-delta call option has an implied volatility of 11%, and the 25-delta put option’s implied volatility is what we need to calculate. Since options are quoted by delta, and the put-call parity relationship links call and put options, we can infer the implied volatility of the 25-delta put option. The volatility skew is often expressed as the difference between the implied volatilities of out-of-the-money (OTM) calls and puts with the same absolute delta. The calculation is as follows: Given: ATM Volatility = 10% 25-Delta Call Volatility = 11% We can assume a linear interpolation or extrapolation. In practice, more complex models are used, but for this question, a linear approach is sufficient. The skew is the difference between the 25-delta call volatility and the ATM volatility: 11% – 10% = 1%. To find the 25-delta put volatility, we can infer that it is symmetrically opposite the call skew from the ATM volatility. 25-Delta Put Volatility = ATM Volatility – Skew = 10% – 1% = 9% Therefore, the implied volatility of the 25-delta put option is 9%. This illustrates how the volatility smile/skew impacts option prices, with OTM puts generally having higher implied volatilities than OTM calls in currency markets due to demand for downside protection. This is a crucial concept for understanding option pricing and risk management in derivatives trading.

The question revolves around the concept of hedging currency risk using futures contracts, specifically focusing on the impact of basis risk and how it affects the effectiveness of the hedge. Basis risk arises because the spot price and the futures price do not always move in perfect lockstep, particularly at the hedge’s termination. This difference between the spot price and the futures price at the delivery date is known as the basis. The formula to calculate the effective hedged price is: Effective Hedged Price = Spot Price at Termination + (Initial Futures Price – Futures Price at Termination) In this scenario, a UK-based investment firm is hedging its USD earnings using USD/GBP futures. The initial futures price reflects the exchange rate at the time the hedge is initiated. As time progresses, the futures price will fluctuate. When the hedge is lifted, the difference between the initial futures price and the final futures price determines the gain or loss on the futures position, which offsets some of the gains or losses on the underlying exposure (USD earnings). Basis risk is the difference between the spot price and the futures price at the time the hedge is closed out. It can either enhance or diminish the effectiveness of the hedge. If the basis narrows (futures price converges toward the spot price), the hedge will be more effective. If the basis widens, the hedge will be less effective. For instance, consider a UK firm expecting USD 1,000,000 in three months. The firm hedges using futures contracts. Initially, the USD/GBP futures price is 0.80. At the hedge’s termination, the spot rate is 0.78, and the futures rate is 0.79. Gain on futures = (0.80 – 0.79) * USD 1,000,000 = GBP 10,000 Effective hedged rate = 0.78 + (0.80 – 0.79) = 0.79 The firm effectively converted USD at 0.79, mitigating some, but not all, of the adverse movement in the spot rate. The basis risk is the difference between the spot rate (0.78) and the futures rate (0.79) at termination, which is 0.01.

The question assesses understanding of delta hedging, particularly the impact of discrete hedging intervals on hedge effectiveness. Delta, a measure of an option’s price sensitivity to changes in the underlying asset’s price, is constantly changing. A perfect delta hedge requires continuous adjustments, which are impossible in practice. Therefore, hedging is performed at discrete intervals (e.g., daily, weekly). The hedge error arises because the delta changes between hedging intervals. A larger price movement between adjustments leads to a greater deviation between the theoretical hedge (based on the initial delta) and the actual price movement of the option. This deviation results in a profit or loss on the hedge, which is not intended in a pure hedging strategy. The formula to approximate the hedge error is: Hedge Error ≈ 0.5 * Gamma * (Change in Underlying Price)^2 Where: * Gamma is the rate of change of the delta with respect to the underlying asset’s price. * Change in Underlying Price is the price movement of the underlying asset between hedging intervals. In this scenario, the investor is short options, meaning they have sold the options. A positive gamma means the delta increases as the underlying asset price increases. Therefore, if the underlying asset price increases significantly between hedging intervals, the delta will be higher than initially calculated, and the short option position will lose more money than the hedge anticipates, resulting in a loss on the hedge (negative hedge error). Calculation: Given: * Gamma = 0.04 * Price Change = £2.50 * Number of Options = 200 Hedge Error per option = 0.5 * 0.04 * (2.50)^2 = 0.5 * 0.04 * 6.25 = £0.125 Total Hedge Error = Hedge Error per option * Number of Options * Multiplier Total Hedge Error = £0.125 * 200 * 100 = £2500 Since the investor is short options and the price increased, the hedge error will be a loss.