Derivatives Level 4 (Investment Advice Diploma) Exam Quiz Quiz 24

Let’s consider a scenario involving a UK-based agricultural cooperative, “Green Harvest,” that wants to protect its future wheat sales against price fluctuations. They plan to sell 100,000 tonnes of wheat in six months. The current spot price is £200 per tonne, but Green Harvest is concerned about a potential price drop due to an oversupply in the market. They decide to use wheat futures contracts traded on the London International Financial Futures and Options Exchange (LIFFE), now part of ICE Futures Europe, to hedge their risk. Each futures contract represents 100 tonnes of wheat. To hedge, Green Harvest will *sell* futures contracts. This is because they will profit from the futures contracts if the price of wheat *decreases*, offsetting the loss from selling their physical wheat at a lower spot price. The number of contracts they need to sell is calculated by dividing their total exposure (100,000 tonnes) by the contract size (100 tonnes/contract), resulting in 1,000 contracts. If, at the delivery date, the spot price of wheat has fallen to £180 per tonne, Green Harvest sells their wheat at this lower price. However, they also close out their futures position. Since they initially sold the futures contracts, they now *buy* them back. The futures price will also have decreased, reflecting the drop in the spot price. Let’s assume the futures price decreased from £202 to £182 per tonne. Their profit on each futures contract is the difference between the initial selling price and the final buying price, which is (£202 – £182) * 100 = £2,000 per contract. Across 1,000 contracts, this totals £2,000,000. Their loss on the physical wheat is (200-180) * 100,000 = £2,000,000. Therefore, the profit on the futures contracts *exactly* offsets the loss on the physical wheat. Now, consider the impact of basis risk. Basis risk is the risk that the price of the asset being hedged (the spot price of wheat) does not move exactly in tandem with the price of the hedging instrument (the wheat futures contract). In our example, let’s say that the spot price of wheat falls to £180, but the futures price only falls to £185 (instead of £182). In this case, the profit on the futures contracts would be (£202 – £185) * 100 * 1000 = £1,700,000. While they still made a profit on the futures contracts, it is not enough to completely offset the loss on the physical wheat. The net result is a loss of £300,000.

The question explores the impact of correlation between assets within a portfolio when using derivatives for hedging, specifically focusing on the impact on Value at Risk (VaR). When assets are perfectly negatively correlated, a perfect hedge can be constructed, minimizing VaR. However, perfect negative correlation is rare. As correlation deviates from -1, the effectiveness of the hedge diminishes, and VaR increases. The calculation involves understanding how portfolio VaR is affected by the correlation coefficient (\(\rho\)), the standard deviations of the asset (\(\sigma_A\)) and the hedging derivative (\(\sigma_H\)), and their respective weights in the portfolio. The formula for portfolio variance is: \[\sigma_P^2 = w_A^2\sigma_A^2 + w_H^2\sigma_H^2 + 2w_Aw_H\rho\sigma_A\sigma_H\] Where \(w_A\) and \(w_H\) are the weights of the asset and hedge respectively. The VaR is then calculated as \(VaR = z \cdot \sigma_P \cdot V\), where \(z\) is the z-score corresponding to the confidence level (here, 2.33 for 99% confidence) and \(V\) is the portfolio value. The key is to understand that as the correlation increases (becomes less negative), the portfolio variance increases, leading to a higher VaR. In this specific scenario, we calculate the portfolio variance and then the VaR for the given correlation. Understanding how correlation impacts the effectiveness of hedging strategies is crucial for risk management in derivatives. A lower negative correlation implies a less effective hedge, leading to higher portfolio risk, which is reflected in a higher VaR. This highlights the importance of carefully considering correlation when implementing hedging strategies using derivatives.

To solve this problem, we need to understand how delta hedging works and how changes in the underlying asset’s price and volatility affect the hedge. Delta hedging aims to neutralize the portfolio’s sensitivity to small changes in the underlying asset’s price. The delta of a call option measures the rate of change of the option’s price with respect to a change in the underlying asset’s price. Here’s a step-by-step breakdown: 1. **Initial Position:** The fund sells 100,000 call options with a delta of 0.6. This means the fund is short delta. To delta hedge, the fund needs to buy shares of the underlying asset. 2. **Calculate Initial Hedge:** The initial hedge requires buying 100,000 options * 0.6 delta = 60,000 shares. 3. **Price Increase:** The underlying asset’s price increases by £2. This changes the option’s delta. 4. **New Delta:** The new delta is 0.68. 5. **Calculate New Hedge:** The new hedge requires buying 100,000 options * 0.68 delta = 68,000 shares. 6. **Shares to Buy:** The fund needs to buy an additional 68,000 – 60,000 = 8,000 shares. 7. **Cost of Additional Shares:** The fund buys 8,000 shares at the new price of £102. The cost is 8,000 * £102 = £816,000. 8. **Volatility Decrease:** The decrease in volatility *after* the adjustment does not affect the cost of adjusting the hedge to the new price level. Volatility impacts the option price and, consequently, the delta, but since we’re given the new delta *after* the price change, we’ve already accounted for the volatility impact in the delta. Therefore, the cost of adjusting the delta hedge is £816,000.

The question assesses understanding of how implied volatility, time to expiration, and the risk-free rate impact option prices, particularly in the context of Black-Scholes. We need to analyze how each factor affects the call option price and then determine the combined effect. First, consider the Black-Scholes formula: \[C = S_0N(d_1) – Ke^{-rT}N(d_2)\] 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 An increase in implied volatility (\(\sigma\)) increases both \(d_1\) and \(d_2\), but more importantly, it increases the value of \(N(d_1)\) and \(N(d_2)\). Since \(N(d_1)\) and \(N(d_2)\) represent the probabilities of the option expiring in the money, higher volatility increases the chance of larger price swings, thus increasing the call option’s value. A decrease in time to expiration (\(T\)) generally decreases the call option price. Less time means less opportunity for the underlying asset’s price to move favorably. In the Black-Scholes formula, a shorter \(T\) reduces the present value of the strike price (\(Ke^{-rT}\)), but the dominant effect is the reduction in the probabilities represented by \(N(d_1)\) and \(N(d_2)\). An increase in the risk-free rate (\(r\)) increases the call option price. In the Black-Scholes formula, a higher \(r\) increases \(d_1\) and \(d_2\), leading to higher \(N(d_1)\) and \(N(d_2)\). More significantly, it reduces the present value of the strike price (\(Ke^{-rT}\)), making the call option more attractive. Combining these effects: Increased volatility increases the call price. Decreased time to expiration decreases the call price. Increased risk-free rate increases the call price. The net effect is a combination of these influences. Without precise numerical values, we can only qualitatively assess the impact. Given the magnitudes described in the question, the increase in volatility is likely to have a more significant impact than the changes in time to expiration and the risk-free rate.

The question assesses the understanding of hedging strategies using options, specifically focusing on the impact of volatility changes on a short straddle position. A short straddle involves selling both a call and a put option with the same strike price and expiration date. This strategy profits when the underlying asset’s price remains stable. However, it carries significant risk if the price moves substantially in either direction. Vega measures the sensitivity of an option’s price to changes in the volatility of the underlying asset. Since a short straddle involves selling options, the position has negative vega, meaning its value decreases as volatility increases. Gamma measures the rate of change of delta with respect to changes in the underlying asset’s price. A short straddle has negative gamma, meaning the position becomes increasingly short as the underlying price rises and increasingly long as the underlying price falls. To calculate the impact of volatility and price changes, we consider the combined effect of vega and gamma. 1. **Vega Effect:** The volatility increase of 2% (0.02) affects both options. The combined vega is -15,000. The loss due to vega is calculated as: Loss from Vega = Vega \* Change in Volatility = -15,000 \* 0.02 = -£300 2. **Gamma Effect:** The price increase of £1 affects the delta of the short straddle. The combined gamma is -20,000. The change in delta is: Change in Delta = Gamma \* Change in Price = -20,000 \* £1 = -20,000 This means the straddle becomes more short by 20,000 shares. 3. **Delta Effect:** Since the straddle is now short 20,000 shares due to the gamma effect, the loss due to the price increase is: Loss from Delta = Change in Delta \* Change in Price = -20,000 \* £1 = -£20,000 4. **Total Loss:** The total loss is the sum of the losses from vega and delta: Total Loss = Loss from Vega + Loss from Delta = -£300 + (-£20,000) = -£20,300 Therefore, the overall impact on the portfolio is a loss of £20,300. This example illustrates how a seemingly small change in volatility and price can significantly impact a short straddle position due to its negative vega and gamma. The negative vega means the position loses value as volatility increases, and the negative gamma means the position’s delta changes rapidly as the underlying price moves, leading to further losses. This highlights the importance of carefully managing risk when using options strategies, particularly those with negative vega and gamma.

The question assesses understanding of hedging strategies using futures contracts, specifically focusing on how cross-hedging addresses basis risk. Basis risk arises when the asset being hedged is not identical to the asset underlying the futures contract. The effectiveness of a cross-hedge depends on the correlation between the price changes of the two assets. The hedge ratio minimizes variance, and a higher correlation between the asset being hedged and the asset underlying the futures contract results in a more effective hedge, reducing the overall risk exposure. The optimal hedge ratio \( h^* \) is calculated as: \[ h^* = \rho \frac{\sigma_S}{\sigma_F} \] Where: \( \rho \) is the correlation between the change in spot price (\(\Delta S\)) and the change in futures price (\(\Delta F\)). \( \sigma_S \) is the standard deviation of the change in the spot price (\(\Delta S\)). \( \sigma_F \) is the standard deviation of the change in the futures price (\(\Delta F\)). Given: \( \rho = 0.7 \) \( \sigma_S = 0.015 \) (1.5% standard deviation) \( \sigma_F = 0.02 \) (2% standard deviation) \[ h^* = 0.7 \times \frac{0.015}{0.02} = 0.7 \times 0.75 = 0.525 \] The optimal hedge ratio is 0.525. Since the company wants to hedge £2,000,000 of palladium using platinum futures, and each contract is for 50 troy ounces, we need to determine the number of contracts to use: Total platinum to hedge = \( h^* \times \text{Value of Palladium} \) = \( 0.525 \times £2,000,000 = £1,050,000 \) Current platinum price per ounce = £800 Number of ounces to hedge = \( \frac{£1,050,000}{£800} = 1312.5 \) ounces Number of contracts = \( \frac{1312.5}{50} = 26.25 \) Since futures contracts can only be traded in whole numbers, the company should use 26 contracts.

The question explores the concept of delta-hedging and gamma risk in a portfolio of options, specifically focusing on the impact of large market movements (jumps) on a delta-neutral portfolio. Delta-hedging aims to maintain a portfolio’s value insensitive to small changes in the underlying asset’s price. Gamma, however, measures the rate of change of delta with respect to the underlying asset’s price. A large jump in the asset price exposes the portfolio to gamma risk, as the delta hedge becomes instantaneously outdated. The profit or loss arises because the hedge, initially set for smaller price movements, is now misaligned with the new price. The calculation considers the initial delta-neutral state, the impact of gamma on the portfolio’s value due to the price jump, and the cost of re-hedging. First, calculate the change in portfolio value due to gamma: Change in portfolio value = 0.5 * Gamma * (Change in stock price)^2 \[0.5 \times 100 \times (5)^2 = 1250\] Since the portfolio is short gamma (Gamma = -100), the portfolio loses value. The loss is £1250. Next, calculate the cost to re-hedge. The new delta after the price jump is: New Delta = Initial Delta + Gamma * Change in stock price New Delta = 0 + (-100) * 5 = -500 To re-hedge, the portfolio manager needs to buy 500 shares. The cost to buy these shares at the new price is: Cost to re-hedge = New Delta * New stock price Cost to re-hedge = 500 * 105 = £52,500 The portfolio manager initially sold short shares to hedge (since the initial delta was zero), but now needs to buy shares. This purchase costs £52,500, while the initial short sale brought in 500 * 100 = £50,000. The difference is a cost of £2,500. Total loss = Loss due to Gamma + Cost to re-hedge Total loss = £1250 + £2500 = £3750 Therefore, the portfolio experiences a loss of £3750 due to the combined effects of negative gamma and the cost of re-hedging after the jump.

The question explores the complexities of managing a portfolio containing both long and short positions in futures contracts, specifically in the context of basis risk and varying margin requirements. Basis risk arises because the price of the futures contract and the spot price of the underlying asset may not converge perfectly at the delivery date. Changes in margin requirements, driven by volatility, further complicate the portfolio’s risk profile and liquidity needs. To determine the overall risk exposure, we must consider the potential losses from both the long and short positions, taking into account the basis risk and the impact of increased margin calls. The long position benefits if the spot price increases relative to the futures price, while the short position benefits from the opposite scenario. However, the basis risk means that these benefits are not guaranteed. An increase in volatility increases the likelihood of significant price swings, necessitating higher margin deposits to cover potential losses. Let’s assume a worst-case scenario where the spot price of Brent Crude decreases by 10% and the futures price decreases by only 5% due to basis widening. For the long position (50 contracts), the loss would be 5% of the initial value. For the short position (30 contracts), the gain would be 5% of the initial value, but this gain may be offset by increased margin requirements. Initial Value of Long Position: 50 contracts * 1,000 barrels/contract * £80/barrel = £4,000,000 Initial Value of Short Position: 30 contracts * 1,000 barrels/contract * £80/barrel = £2,400,000 Loss on Long Position: 5% of £4,000,000 = £200,000 Gain on Short Position: 5% of £2,400,000 = £120,000 Net Loss Before Margin: £200,000 – £120,000 = £80,000 Now, consider the impact of increased volatility. Assume the exchange increases the margin requirement by £5,000 per contract. Increased Margin Requirement for Long Position: 50 contracts * £5,000/contract = £250,000 Increased Margin Requirement for Short Position: 30 contracts * £5,000/contract = £150,000 Total Additional Margin Required: £250,000 + £150,000 = £400,000 Total Risk Exposure: Net Loss + Additional Margin = £80,000 + £400,000 = £480,000 This calculation demonstrates how basis risk and margin requirements interact to determine the overall risk exposure in a derivatives portfolio. It highlights the importance of not only considering the directional price movements but also the potential for divergence between spot and futures prices, as well as the liquidity implications of increased volatility.

The core of this question revolves around understanding how volatility skews impact option pricing, particularly in the context of exotic derivatives like barrier options. A volatility skew implies that implied volatility is not constant across different strike prices for options with the same expiration date. Typically, for equity indices, there’s a “downward skew,” meaning lower strike prices (puts) have higher implied volatilities than higher strike prices (calls). This reflects a greater demand for downside protection. When pricing barrier options, especially those with knock-out features, the volatility skew significantly affects the probability of the barrier being breached. A down-and-out call option, for example, becomes more expensive if the volatility skew is pronounced because the higher implied volatility at lower strikes increases the likelihood of the underlying asset hitting the barrier, thus knocking out the option. Conversely, a down-and-in call option becomes cheaper. The Black-Scholes model assumes constant volatility, which is unrealistic in the presence of skews. Therefore, practitioners use more sophisticated models like stochastic volatility models or local volatility models to account for the skew. These models adjust the volatility used in pricing based on the strike price. For instance, a local volatility model might increase the volatility used for pricing the barrier component of a down-and-out call, reflecting the higher implied volatility at the barrier level. In the scenario presented, the fund manager’s reliance on a Black-Scholes model without adjusting for the volatility skew leads to a mispricing of the barrier option. The model underestimates the probability of the barrier being hit, resulting in an underestimation of the option’s value. The gamma of the option, which measures the rate of change of delta with respect to changes in the underlying asset’s price, is also affected. A steeper volatility skew leads to a higher gamma near the barrier, as the option’s delta changes more rapidly as the underlying asset approaches the barrier. Therefore, the manager’s hedging strategy, based on the flawed Black-Scholes pricing, will be inadequate, potentially leading to losses if the barrier is breached. The correct approach involves using a volatility skew-aware pricing model and adjusting the hedging strategy to account for the increased gamma near the barrier.

The core concept tested here is the application of put-call parity, a fundamental relationship in options pricing. Put-call parity states that the price of a European call option plus the present value of the strike price is equal to the price of a European put option plus the current price of the underlying asset. This relationship holds true under the assumption of no arbitrage. The formula is: \[C + PV(K) = P + S\] Where: C = Call option price PV(K) = Present value of the strike price (K) P = Put option price S = Current price of the underlying asset In this scenario, we need to determine the theoretical price of the call option, given the prices of the put option, the underlying asset, the strike price, and the risk-free interest rate. The present value of the strike price is calculated using the formula: \[PV(K) = \frac{K}{(1 + r)^t}\] Where: K = Strike price r = Risk-free interest rate t = Time to expiration Given the strike price of £155, the risk-free interest rate of 4% (0.04), and the time to expiration of 6 months (0.5 years), the present value of the strike price is: \[PV(K) = \frac{155}{(1 + 0.04)^{0.5}} = \frac{155}{1.0198} \approx 151.99\] Now, we can rearrange the put-call parity formula to solve for the call option price (C): \[C = P + S – PV(K)\] Plugging in the given values: \[C = 7 + 150 – 151.99 = 5.01\] Therefore, the theoretical price of the call option is approximately £5.01. The subtle difficulty arises from understanding the present value calculation and correctly applying the put-call parity formula. A common mistake is forgetting to discount the strike price to its present value. Another mistake is to misinterpret the formula and adding or subtracting the terms incorrectly. The scenario introduces a real-world context of a portfolio manager evaluating derivative pricing, adding to the complexity.

The core concept being tested is the understanding of how volatility impacts option pricing, specifically in the context of exotic options like barrier options. A barrier option’s payoff is contingent on the underlying asset’s price reaching a certain level (the barrier). Higher volatility increases the probability of the asset price hitting the barrier, thus affecting the option’s value. For a down-and-out call option, if volatility increases, the probability of the asset price hitting the lower barrier also increases. If the barrier is hit, the option is extinguished. Therefore, higher volatility *decreases* the value of a down-and-out call, as there is a greater chance it will expire worthless due to the barrier being breached. Conversely, for a down-and-in call, higher volatility increases the probability of the barrier being hit, activating the option and thus *increasing* its value. The Black-Scholes model, while not directly applicable to barrier options without adjustments, highlights the general relationship between volatility and option prices. The Greeks, such as Vega (sensitivity of option price to volatility), are crucial for understanding these relationships. In practice, pricing barrier options requires specialized models or simulations that account for the barrier feature. Let’s say a portfolio manager is using a down-and-out call option to hedge against a potential rise in the price of a commodity. If market volatility is expected to increase significantly due to geopolitical instability, the manager needs to understand that the value of their hedging instrument (the down-and-out call) will likely decrease. This requires them to re-evaluate their hedging strategy and potentially adjust the position to maintain the desired level of risk protection.

The question tests the understanding of delta hedging and the impact of gamma on the hedge’s effectiveness. Delta represents the sensitivity of an option’s price to a change in the underlying asset’s price. Gamma, in turn, measures the rate of change of the delta with respect to the underlying asset’s price. When gamma is high, the delta changes rapidly, making it challenging to maintain a perfect delta hedge. The calculation involves determining the number of options needed to hedge a short position in the underlying asset, considering the delta of the options. The profit or loss arises from the difference between the expected change in the underlying asset’s price and the actual change, compounded by the hedge ratio and the option’s delta. The impact of gamma is reflected in the changing effectiveness of the hedge as the underlying asset’s price moves. First, we determine the initial hedge ratio. The portfolio manager is short 10,000 shares of the asset and wants to hedge using options with a delta of 0.5. The number of options needed is calculated as: Number of options = – (Shares to hedge / Option Delta) * Contract Size Number of options = – (10,000 / 0.5) = -20,000 options. Next, calculate the profit or loss on the short stock position: Profit/Loss from stock = -(Change in Stock Price) * Number of Shares Profit/Loss from stock = -(£105 – £100) * 10,000 = -£50,000 (Loss) Now, calculate the profit or loss on the options position: Profit/Loss from options = (Change in Stock Price) * Number of Options * Option Delta Profit/Loss from options = (£105 – £100) * 20,000 * 0.5 = £50,000 However, because of the gamma effect, the delta hedge is not perfect. The option’s delta changes as the stock price moves. The question implicitly assumes that the delta remains constant, which is a simplification. In reality, the hedge would need to be rebalanced as the stock price changes. The profit/loss calculated above is based on the initial delta of 0.5. Because we are not given the Gamma, we assume the hedge is not rebalanced, so the profit/loss on the options is simply the change in the underlying asset price multiplied by the number of options and the initial delta. Therefore, the overall profit/loss is: Overall Profit/Loss = Profit/Loss from stock + Profit/Loss from options Overall Profit/Loss = -£50,000 + £50,000 = £0 This calculation represents a simplified scenario where the delta remains constant. In practice, the portfolio manager would need to dynamically adjust the hedge as the stock price fluctuates to account for the gamma effect. The ideal strategy would involve continuously monitoring the portfolio’s delta and rebalancing the options position to maintain a delta-neutral position. This dynamic hedging strategy would minimize the impact of gamma and reduce the overall risk of the portfolio.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Yorkshire Grain,” which wants to protect itself against a potential drop in wheat prices before harvest. They decide to use wheat futures contracts traded on the ICE Futures Europe exchange. To determine the optimal number of contracts, we need to consider the cooperative’s expected wheat yield, the size of each futures contract, and the desired level of price protection. Yorkshire Grain expects to harvest 50,000 tonnes of wheat. Each ICE wheat futures contract represents 100 tonnes. They aim to hedge 80% of their expected yield. This means they want to hedge 0.80 * 50,000 = 40,000 tonnes. To determine the number of contracts, we divide the total tonnes to be hedged by the contract size: 40,000 tonnes / 100 tonnes/contract = 400 contracts. However, basis risk exists because the futures price may not perfectly correlate with the local spot price at harvest time. Yorkshire Grain’s historical data shows a standard deviation of the basis of £5 per tonne. They are willing to accept a 95% confidence level. To calculate the VaR due to basis risk, we use the formula: VaR = Z-score * Standard Deviation * Quantity. For a 95% confidence level, the Z-score is approximately 1.645. Therefore, VaR = 1.645 * £5/tonne * 40,000 tonnes = £329,000. Now, consider the margin requirements. The initial margin for each wheat futures contract is £2,000, and the maintenance margin is £1,500. The total initial margin required is 400 contracts * £2,000/contract = £800,000. If the price moves adversely, and the margin account falls below the maintenance margin, Yorkshire Grain will receive a margin call. For example, if the price drops by £1.25 per tonne, the loss on 400 contracts (40,000 tonnes) would be £1.25/tonne * 40,000 tonnes = £50,000. This loss would be deducted from the margin account. The cooperative also needs to consider the regulatory aspects. EMIR (European Market Infrastructure Regulation) requires them to clear their OTC derivatives through a central counterparty (CCP) if they exceed certain thresholds. Although these are exchange-traded futures, understanding EMIR is crucial for their overall risk management framework, especially if they engage in other derivative transactions. They also need to be aware of potential market manipulation and insider trading regulations, ensuring their trading activities are transparent and compliant with FCA guidelines.

The question assesses the understanding of delta hedging, gamma, and portfolio rebalancing. Delta hedging aims to neutralize the directional risk of an option position by holding an offsetting position in the underlying asset. Gamma measures the rate of change of delta with respect to changes in the underlying asset’s price. A higher gamma means the delta changes more rapidly, requiring more frequent rebalancing to maintain a delta-neutral position. The cost of rebalancing depends on the transaction costs and the magnitude of the position. In this scenario, we need to calculate the number of shares to buy or sell to maintain a delta-neutral position after a price change. First, calculate the initial number of shares to hedge: Delta * Number of Options * Multiplier = 0.60 * 100 * 100 = 6000 shares. Next, calculate the new delta after the price change: New Delta = Initial Delta + (Gamma * Price Change) = 0.60 + (0.005 * 2) = 0.61. Then, calculate the new number of shares to hedge: New Delta * Number of Options * Multiplier = 0.61 * 100 * 100 = 6100 shares. Finally, calculate the number of shares to buy: New Shares – Initial Shares = 6100 – 6000 = 100 shares. The cost of buying the shares is: Number of Shares * New Price = 100 * 102 = £10,200. A high gamma portfolio, while offering opportunities for profit from volatility changes, necessitates frequent rebalancing. This rebalancing incurs transaction costs. Ignoring gamma and only focusing on delta can lead to significant losses if the underlying asset price moves substantially. The example highlights the practical implications of gamma in delta hedging and the importance of considering transaction costs when managing a delta-hedged portfolio. The optimal rebalancing frequency depends on the trade-off between minimizing delta exposure and minimizing transaction costs. Sophisticated hedging strategies often involve dynamic adjustments to the hedge ratio based on both delta and gamma, as well as transaction cost considerations.

The question assesses the understanding of option Greeks, specifically Vega, and its impact on option pricing when volatility changes. Vega represents the sensitivity of an option’s price to changes in the volatility of the underlying asset. A higher Vega indicates a greater sensitivity. The formula to calculate the approximate change in option price due to a change in volatility is: Change in Option Price ≈ Vega × Change in Volatility In this scenario, the fund manager needs to determine the potential loss in the value of their long call option position if the implied volatility of the underlying asset decreases. Given: Vega = 0.05 (This means the option price changes by £0.05 for every 1% change in volatility) Change in Volatility = -2% (Volatility decreases by 2%) Number of Contracts = 500 Multiplier per Contract = 100 (Each contract controls 100 shares) First, calculate the change in option price per contract: Change in Option Price per Contract = Vega × Change in Volatility Change in Option Price per Contract = 0.05 × -2 = -0.10 (This means the option price decreases by £0.10 per share) Next, calculate the total change in value for one contract: Change in Value per Contract = Change in Option Price per Contract × Multiplier per Contract Change in Value per Contract = -0.10 × 100 = -£10 Finally, calculate the total change in value for all contracts: Total Change in Value = Change in Value per Contract × Number of Contracts Total Change in Value = -£10 × 500 = -£5000 Therefore, the fund manager can expect a loss of £5000 in the value of their long call option position. Now, let’s consider an analogy to solidify the understanding of Vega. Imagine Vega as the “sensitivity dial” on a radio. If the dial is set high (high Vega), even a small adjustment (change in volatility) will significantly alter the station you hear (option price). Conversely, a low setting (low Vega) means that even large adjustments have minimal impact. In the context of risk management, a high Vega suggests the portfolio is very sensitive to changes in market uncertainty, requiring careful monitoring and potential hedging strategies. For instance, a portfolio heavily invested in technology stocks with high growth potential might exhibit a higher Vega compared to a portfolio focused on established dividend-paying companies. Understanding Vega allows fund managers to proactively manage their exposure to volatility risk and make informed decisions about hedging strategies. This example highlights how Vega is a crucial tool for assessing and mitigating the impact of market uncertainty on option portfolios.

The question assesses understanding of exotic derivatives, specifically barrier options, and their sensitivity to market movements near the barrier. A down-and-out call option becomes worthless if the underlying asset’s price touches or falls below the barrier level before the option’s expiration. The closer the asset price is to the barrier, the higher the risk of the option being knocked out. The value of the option is therefore significantly affected by the probability of the barrier being breached. The time remaining until expiration also plays a crucial role; with less time remaining, there’s less opportunity for the asset to breach the barrier, assuming it hasn’t already. Conversely, a longer time horizon increases the probability of the barrier being hit. Volatility is another key factor. Higher volatility increases the likelihood of the asset price fluctuating and potentially hitting the barrier. Lower volatility reduces this likelihood. A risk-neutral investor would consider these factors when pricing the option. The initial value of the down-and-out call is calculated using a specialized pricing model that takes into account the barrier. Suppose the initial price of the underlying asset is £110, the strike price is £115, the barrier is £105, the risk-free rate is 5%, the volatility is 20%, and the time to expiration is 1 year. If the price drops to £106, close to the barrier, the option’s value decreases significantly. The new value can be estimated using a similar model, adjusting for the new asset price. The Black-Scholes model cannot be directly used to price barrier options, but it provides a basis for understanding the factors influencing option prices. Specialized models, such as those incorporating barrier effects, are necessary for accurate valuation. The difference between the initial and final option values represents the loss incurred due to the price movement. The sensitivity to changes near the barrier is a key characteristic of barrier options.

Let’s break down the intricacies of calculating the theoretical price of a European call option using the Black-Scholes model, but with a twist focusing on the impact of dividend payments within a specific timeframe and the nuances of continuously compounded risk-free rates. The Black-Scholes formula is: \[C = S_0e^{-qT}N(d_1) – Xe^{-rT}N(d_2)\] where: * \(C\) = Call option price * \(S_0\) = Current stock price * \(X\) = Strike price * \(T\) = Time to expiration (in years) * \(r\) = Risk-free interest rate (continuously compounded) * \(q\) = Continuous dividend yield * \(N(x)\) = Cumulative standard normal distribution function * \(d_1 = \frac{ln(\frac{S_0}{X}) + (r – q + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}}\) * \(d_2 = d_1 – \sigma\sqrt{T}\) * \(\sigma\) = Volatility of the stock Now, let’s consider a scenario where a stock pays a discrete dividend *before* the option’s expiration. We need to adjust the stock price for the present value of this dividend. The adjusted stock price, \(S_0’\), becomes: \[S_0′ = S_0 – PV(Dividends)\] If the dividend is paid at time \(t\) (where \(0 < t < T\)), then: \[PV(Dividends) = De^{-r_dt}\] Where \(D\) is the dividend amount, and \(r_d\) is the continuously compounded discount rate applicable for discounting the dividend back to the present. The continuously compounded rate is derived from the annual effective rate using the formula: \[r_c = ln(1 + r_a)\] Where \(r_c\) is the continuously compounded rate and \(r_a\) is the annual effective rate. In our specific case, we have a stock currently priced at £55. It will pay a dividend of £1.50 in 3 months. The option expires in 9 months. We're given a continuously compounded risk-free rate of 4.5% per annum, and a volatility of 28%. The strike price is £56. First, calculate the present value of the dividend. The annual effective rate corresponding to the 4.5% continuously compounded rate is \(e^{0.045} – 1 = 0.046027\). To discount the dividend, we use the continuously compounded rate for 3 months (0.25 years): \(1.50 * e^{-0.045 * 0.25} = 1.4832\). The adjusted stock price is \(55 – 1.4832 = 53.5168\). Now we use this adjusted stock price in the Black-Scholes model. \(d_1 = \frac{ln(\frac{53.5168}{56}) + (0.045 + \frac{0.28^2}{2})0.75}{0.28\sqrt{0.75}} = -0.0352\) \(d_2 = -0.0352 – 0.28\sqrt{0.75} = -0.2775\) Using a standard normal distribution table, \(N(d_1) = 0.4859\) and \(N(d_2) = 0.3897\). \[C = 53.5168 * e^{-0 * 0.75} * 0.4859 – 56 * e^{-0.045 * 0.75} * 0.3897\] \[C = 53.5168 * 1 * 0.4859 – 56 * 0.9663 * 0.3897\] \[C = 26.004 – 21.141 = 4.863\] Therefore, the theoretical price of the European call option is approximately £4.86.

Let’s consider a scenario where a portfolio manager, Anya, uses options to hedge her equity portfolio against a potential market downturn. Anya manages a £50 million portfolio benchmarked against the FTSE 100 index. She’s concerned about a potential correction in the market over the next three months due to upcoming economic data releases and geopolitical uncertainties. To protect her portfolio, Anya decides to use put options on the FTSE 100 index. First, we need to determine the number of put option contracts Anya needs to purchase. Suppose the FTSE 100 index is currently trading at 7,500, and each FTSE 100 index point is worth £10. The notional value of one FTSE 100 futures contract is therefore 7,500 * £10 = £75,000. To hedge her £50 million portfolio, Anya needs to cover £50,000,000 / £75,000 = 666.67 contracts. Since she can’t buy fractions of contracts, she’ll round up to 667 contracts. Anya decides to buy three-month FTSE 100 put options with a strike price of 7,400. The premium for each put option is £50. The total cost of the hedge is 667 contracts * £50 * contract multiplier (which is usually 1) = £33,350. This represents the maximum amount Anya will lose on the hedge. Now, let’s analyze the potential outcomes. If the FTSE 100 falls below 7,400, the put options will be in the money. For example, if the FTSE 100 falls to 7,000, each put option will have an intrinsic value of 7,400 – 7,000 = 400 index points, or £4,000 per contract. The total payoff from the put options would be 667 contracts * £4,000 = £2,668,000. This payoff will offset the losses in Anya’s equity portfolio. If the FTSE 100 stays above 7,400, the put options will expire worthless, and Anya will lose the premium paid (£33,350). However, her equity portfolio will not have suffered significant losses, so the cost of the hedge is a relatively small price to pay for the protection it provided. Consider a more complex scenario: The FTSE 100 drops to 7,000, but implied volatility spikes due to heightened uncertainty. This increase in volatility would cause the value of the put options to increase even further, potentially exceeding the intrinsic value. This is due to the vega of the option, which measures the sensitivity of the option’s price to changes in volatility. Another aspect to consider is the potential for early exercise of American-style options. Although FTSE 100 options are typically European-style (exercisable only at expiration), understanding the factors that influence early exercise decisions is crucial. These factors include dividend payouts, interest rates, and the time value of the option. Finally, Anya needs to be aware of the regulatory requirements for using derivatives in her portfolio. Under FCA regulations, she must ensure that the use of derivatives is consistent with her clients’ investment objectives and risk tolerance. She must also disclose the risks associated with derivatives to her clients and maintain adequate records of her derivative transactions.

The question assesses the understanding of exotic options, specifically barrier options, and their behavior around the barrier level. It also integrates the concept of implied volatility and its impact on option pricing. A down-and-out call option becomes worthless if the underlying asset price touches or falls below the barrier level. The key is to understand how changes in implied volatility affect the probability of hitting the barrier and thus the value of the option. An increase in implied volatility suggests a higher likelihood of the underlying asset price fluctuating and potentially hitting the barrier, thereby reducing the value of the down-and-out call option. Conversely, a decrease in implied volatility suggests a lower likelihood of hitting the barrier, increasing the option’s value. The calculation involves qualitatively assessing the impact of volatility on the option’s value given its barrier feature. There is no explicit calculation, but the logic hinges on understanding the relationship between volatility, barrier options, and the probability of the barrier being breached. Let’s say a company “NovaTech” has a stock price that currently trades at £100. A down-and-out call option exists with a strike price of £105 and a barrier at £90. If implied volatility suddenly spikes due to an unexpected market event (e.g., a competitor announcing a groundbreaking new product), the probability of NovaTech’s stock hitting £90 increases, making the down-and-out call less valuable. Conversely, if implied volatility decreases due to stabilizing market conditions and positive company news, the probability of hitting the barrier decreases, increasing the value of the option. This is because the option is now more likely to stay “alive” and potentially move into the money. The critical aspect is that the barrier feature makes the option’s value inversely related to the probability of hitting the barrier, which is directly influenced by implied volatility.

To determine the fair price of the exotic derivative, we must first calculate the expected payoff at maturity. The payoff depends on whether the average daily closing price of the FTSE 100 over the observation period exceeds the initial price. 1. **Simulate FTSE 100 Paths:** We use Monte Carlo simulation to generate 5,000 possible paths for the FTSE 100’s daily closing prices over the 6-month period (approximately 126 trading days). We assume a log-normal distribution for the FTSE 100 returns, with a mean of 8% per annum and a volatility of 18% per annum. 2. **Calculate Average Daily Closing Price:** For each simulated path, we calculate the average daily closing price over the 6-month observation period. 3. **Determine Payoff:** For each path, if the average daily closing price exceeds the initial FTSE 100 price of 7,500, the payoff is £10,000. Otherwise, the payoff is £0. 4. **Calculate Expected Payoff:** We calculate the average payoff across all 5,000 simulated paths. This gives us the expected payoff of the exotic derivative. 5. **Discount Expected Payoff:** We discount the expected payoff back to the present value using the risk-free interest rate of 4% per annum. The discounting formula is: \[PV = \frac{Expected Payoff}{e^{rT}}\] Where: * \(PV\) is the present value (fair price) * \(r\) is the risk-free interest rate (0.04) * \(T\) is the time to maturity (0.5 years) Let’s assume the Monte Carlo simulation yields an expected payoff of £6,250. Then, the present value (fair price) is: \[PV = \frac{6250}{e^{0.04 \times 0.5}} = \frac{6250}{e^{0.02}} \approx \frac{6250}{1.0202} \approx 6126.25\] Therefore, the fair price of the exotic derivative is approximately £6,126.25. Now, let’s consider the regulatory aspect. According to the FCA (Financial Conduct Authority) regulations, firms offering or advising on derivatives must ensure that these products are suitable for their clients. This involves assessing the client’s knowledge and experience, financial situation, and investment objectives. In this scenario, the advisor must explain the complex payoff structure of the exotic derivative and ensure that the client understands the risks involved, including the potential for a zero payoff if the average daily closing price does not exceed the initial price. Furthermore, the advisor must document the suitability assessment and maintain records of the transaction in compliance with MiFID II regulations.

The question assesses understanding of how a clearing house mitigates counterparty risk in derivatives transactions, specifically focusing on the margin requirements and the process of marking-to-market. The core concept is that the clearing house acts as a central counterparty, guaranteeing the performance of trades even if one party defaults. This is achieved through several mechanisms. Initial margin is a deposit required upfront to cover potential future losses. Variation margin is a daily adjustment to reflect the current market value of the contract; gains are credited, and losses are debited. Marking-to-market is the process of valuing the contract at its current market price. The clearing house monitors these margins and if a member’s position deteriorates significantly, they are required to deposit additional margin. If they fail to do so, the clearing house can close out the position to limit losses. In this scenario, we need to understand how the clearing house responds to a significant adverse price movement against Trader X, considering the margin levels and default procedures. The clearing house’s primary goal is to protect itself and other members from losses resulting from Trader X’s potential default. Therefore, the most likely action is to close out Trader X’s position after the margin call is unmet.

This question tests the understanding of exotic options, specifically barrier options, and their sensitivity to market movements and volatility. The knock-out feature adds complexity, requiring the candidate to consider the probability of the barrier being breached and the impact on the option’s value. The calculation of the potential loss involves understanding how the option’s delta and gamma contribute to the price change with respect to the underlying asset’s movement, while also considering the impact of vega due to volatility changes. First, we need to calculate the potential price change of the underlying asset. A 5% increase on £150 is £7.50, bringing the new price to £157.50. A 2% volatility increase on the initial 20% volatility is 0.02. The delta effect is 0.60 * £7.50 = £4.50. The gamma effect is 0.5 * 0.02 * (£7.50)^2 = £0.5625. The vega effect is 0.40 * 0.02 * 100 = £0.80. Total price change is £4.50 + £0.5625 + £0.80 = £5.8625. Since the option is a knock-out option, we need to check if the barrier has been breached. The barrier is at £160. The new price is £157.50, so the barrier has not been breached. The option is still alive. The initial price of the option was £10. Therefore, the new price of the option is £10 + £5.8625 = £15.8625. Since the option was sold for £10, the potential loss is £15.8625 – £10 = £5.8625. However, if the barrier had been breached, the option would be worthless, and the potential loss would be £10. The calculation considers the combined impact of delta, gamma, and vega, providing a more accurate estimate of the option’s price change. This is crucial for effective risk management, especially with exotic options where sensitivities can be highly non-linear. Understanding these sensitivities allows for better hedging strategies and a more informed assessment of potential losses.

To address this question, we need to understand how volatility skew affects option pricing and strategy selection. Volatility skew refers to the phenomenon where out-of-the-money (OTM) puts tend to have higher implied volatilities than at-the-money (ATM) or OTM calls. This is often observed in equity markets due to the demand for downside protection. When volatility skew is present, OTM puts are relatively more expensive than OTM calls. In the scenario, we have a pronounced volatility skew. This means OTM puts are more expensive relative to OTM calls. Therefore, strategies that benefit from selling overpriced puts or buying underpriced calls will be favored. A *risk reversal* involves buying an OTM call and selling an OTM put. In a skewed market, the put is more expensive, generating more premium income when sold. The call is relatively cheaper, costing less to purchase. The net effect is a reduced cost or even a credit to initiate the strategy. Let’s consider an example: Suppose the OTM put can be sold for £1.50, and the OTM call can be bought for £0.75. The net credit is £0.75. If volatility skew were absent, the put might only fetch £0.75, making the risk reversal less attractive. The presence of skew also impacts other strategies. For instance, a *straddle* (buying an ATM call and an ATM put) becomes more expensive overall because the skew pushes up the price of the put. Similarly, a *strangle* (buying an OTM call and an OTM put) is also affected, though the impact depends on the relative OTM-ness of the options. The key takeaway is that volatility skew significantly influences the relative attractiveness of different option strategies. A trader must consider the skew to optimize strategy selection and execution. Regulatory considerations such as MiFID II require firms to provide best execution, which includes considering the impact of volatility skew on option prices.

Let’s analyze a scenario involving a UK-based agricultural cooperative, “GreenHarvest,” which seeks to hedge its upcoming wheat harvest using futures contracts listed on the ICE Futures Europe exchange. GreenHarvest anticipates harvesting 500,000 bushels of wheat in three months. The current spot price of wheat is £6.00 per bushel, but GreenHarvest fears a price decline before the harvest. They decide to use wheat futures contracts to lock in a price. Each ICE wheat futures contract represents 100 tonnes of wheat, which is approximately 3,674.37 bushels (1 tonne ≈ 36.7437 bushels). To determine the number of contracts needed, we divide the total bushels to be hedged by the bushels per contract: 500,000 bushels / 3,674.37 bushels/contract ≈ 136.07 contracts. Since you can only trade whole contracts, GreenHarvest rounds this to 136 contracts. The current futures price for the three-month contract is £6.10 per bushel. This is the price they effectively lock in, subject to basis risk. Now, consider the scenario where, at harvest time, the spot price of wheat has fallen to £5.80 per bushel. GreenHarvest sells its wheat at this price. Simultaneously, they close out their futures position by buying back 136 contracts at the then-current futures price of £5.85 per bushel. The gain on the futures contracts is the difference between the selling price (£6.10) and the buying price (£5.85), multiplied by the number of bushels hedged: (£6.10 – £5.85) * (136 contracts * 3,674.37 bushels/contract) = £0.25 * 500,000 bushels = £125,000. The revenue from selling the physical wheat is £5.80 per bushel * 500,000 bushels = £2,900,000. The effective price received is the total revenue (from physical sales and futures gain) divided by the total bushels: (£2,900,000 + £125,000) / 500,000 bushels = £6.05 per bushel. Basis risk arises because the futures price and spot price do not converge perfectly at the delivery date. In this case, the initial difference between the futures price (£6.10) and the spot price (£6.00) was £0.10. At harvest, the difference between the futures price (£5.85) and the spot price (£5.80) is £0.05. The change in the basis is £0.05 – £0.10 = -£0.05. This change in basis reduces the effectiveness of the hedge, resulting in an effective price of £6.05 per bushel instead of the expected £6.10.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Green Harvest Co-op,” which produces and exports organic wheat. They face significant price volatility due to unpredictable weather patterns and fluctuating global demand. To mitigate this risk, they decide to use wheat futures contracts traded on the London International Financial Futures and Options Exchange (LIFFE). Green Harvest needs to determine the optimal number of contracts to hedge their anticipated harvest. Their expected harvest is 5000 metric tons of wheat. Each LIFFE wheat futures contract represents 100 metric tons. The current spot price of wheat is £200 per metric ton, and the futures price for delivery in six months is £210 per metric ton. Green Harvest wants to lock in a price close to the current futures price. First, calculate the total value of Green Harvest’s expected harvest: 5000 tons * £200/ton = £1,000,000. Next, determine the number of futures contracts needed to hedge the entire harvest: 5000 tons / 100 tons/contract = 50 contracts. Now, consider the impact of basis risk. Basis risk arises because the spot price and the futures price may not converge perfectly at the delivery date. Suppose that at the delivery date, the spot price is £195 per metric ton, and Green Harvest closes out their futures contracts at £195 per metric ton. The gain or loss on the futures contracts is calculated as the difference between the initial futures price and the final futures price, multiplied by the number of contracts and the contract size: (210 – 195) * 100 * 50 = £75,000 gain. The effective price received by Green Harvest is the spot price at delivery plus the gain from the futures contracts, divided by the total quantity: (195 * 5000 + 75000) / 5000 = £210 per metric ton. To illustrate the impact of imperfect hedging, imagine Green Harvest only hedged 80% of their harvest. They would use 40 contracts (80% of 50). In this case, the gain on the futures contracts would be (210 – 195) * 100 * 40 = £60,000. The effective price received for the hedged portion would still be £210. However, the unhedged portion (20% or 1000 tons) would be sold at the spot price of £195, resulting in a lower overall average price. The overall revenue would be (4000 * 210) + (1000 * 195) = £1,035,000. The average price per ton would be £1,035,000 / 5000 = £207. This example highlights the importance of understanding basis risk and the impact of imperfect hedging on the overall effectiveness of a hedging strategy. Green Harvest must carefully consider their risk tolerance and the potential for basis risk when deciding on the optimal hedge ratio.

Let’s analyze a scenario involving a complex hedging strategy using options and futures to mitigate risk in a volatile market. We will examine how a portfolio manager might use a combination of strategies to protect an investment against both upside and downside risks. Consider a portfolio manager holding a substantial position in FTSE 100 stocks. The manager is concerned about potential market volatility due to upcoming Brexit negotiations and wants to implement a hedging strategy that limits losses while still allowing for some potential gains. The manager decides to use a combination of short FTSE 100 futures contracts and a collar strategy using FTSE 100 index options. First, the manager sells FTSE 100 futures contracts to hedge against a potential market decline. This provides a direct offset to losses in the stock portfolio. Simultaneously, the manager implements a collar strategy, which involves buying protective put options and selling covered call options. The put options provide downside protection, while the call options generate income to offset the cost of the puts. The key here is to understand the combined effect of these strategies. The short futures position provides a linear hedge against market declines. The put options provide a floor below which losses are limited. The call options cap potential gains but reduce the overall cost of the hedging strategy. To evaluate the effectiveness of this combined strategy, we need to consider several factors: the correlation between the FTSE 100 stocks in the portfolio and the FTSE 100 index, the volatility of the FTSE 100 index, and the strike prices and expiration dates of the options. We can use tools like Value at Risk (VaR) and stress testing to assess the potential impact of various market scenarios on the hedged portfolio. For example, suppose the FTSE 100 declines sharply. The short futures position will generate profits, offsetting losses in the stock portfolio. The put options will also provide additional protection. However, if the FTSE 100 rises sharply, the gains in the stock portfolio will be capped by the short call options, and the losses on the short futures position will offset some of the gains. This combined strategy is designed to provide a balance between risk mitigation and potential returns. The manager is willing to sacrifice some upside potential in exchange for downside protection. The specific parameters of the strategy, such as the number of futures contracts and the strike prices of the options, should be carefully chosen based on the manager’s risk tolerance and market outlook.

The question revolves around the concept of delta hedging a short call option position and the subsequent profit or loss when the underlying asset price changes. Delta hedging aims to create a portfolio that is neutral to small changes in the price of the underlying asset. The delta of a call option represents the sensitivity of the option’s price to a change in the underlying asset’s price. A delta of 0.65 means that for every £1 increase in the asset price, the call option’s price is expected to increase by £0.65. When an investor is short a call option, they are obligated to sell the asset if the option is exercised. To hedge this risk, they buy a number of shares equal to the option’s delta. This creates a delta-neutral position. If the asset price increases, the loss on the short call option is offset by the gain on the long stock position. Conversely, if the asset price decreases, the gain on the short call option is offset by the loss on the long stock position. However, delta hedging is not a perfect hedge. It only works for small changes in the asset price. For larger changes, the delta itself changes (this is captured by the option’s gamma). Therefore, the hedge needs to be rebalanced periodically to maintain delta neutrality. In this scenario, the investor initially hedges their short call option position by buying 650 shares (0.65 delta * 1000 options). When the asset price increases by £2, the call option’s price increases, resulting in a loss for the investor. The long stock position generates a profit. The net profit or loss is the difference between the profit on the stock position and the loss on the short call option. Profit from stock = Change in price * Number of shares = £2 * 650 = £1300 Loss from options = Change in option price * Number of options = £1.40 * 1000 = £1400 Net Profit/Loss = Profit from stock – Loss from options = £1300 – £1400 = -£100 Therefore, the investor incurs a net loss of £100.

The question revolves around the concept of delta hedging and its effectiveness in mitigating risk, particularly when dealing with options on assets exhibiting jump risk. Jump risk refers to the sudden and discontinuous price movements that cannot be captured by continuous-time models like the Black-Scholes model. A perfect delta hedge aims to create a portfolio that is insensitive to small changes in the underlying asset’s price. The delta of an option represents the sensitivity of the option’s price to a change in the underlying asset’s price. By holding a position in the underlying asset that offsets the option’s delta, the portfolio’s value should theoretically remain unchanged for small price movements. However, when jump risk is present, the underlying asset’s price can experience large, sudden movements. These jumps can invalidate the delta hedge, as the delta is calculated based on infinitesimal price changes. The hedge is only effective for small price movements. A large jump will cause the option’s price to change significantly, while the hedge, based on the previous delta, will not adequately offset this change, leading to a loss. The magnitude of the loss depends on the size of the jump and the time elapsed since the last delta adjustment. The longer the time interval between adjustments, the greater the potential for a large jump to occur, and the larger the resulting loss. To illustrate, consider a portfolio consisting of a short call option on a stock and a long position in the stock to delta-hedge the option. Suppose the stock price is £100, the call option has a delta of 0.5, and the portfolio is delta-neutral. If the stock price suddenly jumps to £110, the call option’s price will increase significantly more than predicted by the delta. The long stock position will gain £10, but the short call option will lose substantially more than £5 (0.5 * £10) due to the non-linear relationship between the option price and the underlying asset price, especially with a large price jump. This results in a net loss for the supposedly delta-hedged portfolio. The effectiveness of delta hedging is further compromised by the fact that delta itself changes as the underlying asset’s price changes. This requires continuous or frequent rebalancing of the hedge, which is costly and often impractical. In the presence of jump risk, even frequent rebalancing may not be sufficient to eliminate the risk entirely. Therefore, while delta hedging can reduce risk in normal market conditions, it is not a perfect solution, especially when jump risk is significant. Other hedging strategies, such as gamma hedging or the use of options with different strike prices, may be necessary to manage the risk more effectively.

To determine the fair premium for the exotic Asian option, we need to calculate the expected payoff and discount it back to the present value. The exotic feature introduces a path dependency, making the standard Black-Scholes model unsuitable. Instead, we’ll employ a Monte Carlo simulation. First, simulate 10,000 possible price paths for the underlying asset over the option’s life (1 year) using a Geometric Brownian Motion (GBM) model. The GBM is defined as: \[ dS_t = \mu S_t dt + \sigma S_t dW_t \] Where: – \( S_t \) is the asset price at time t – \( \mu \) is the expected return (10% or 0.10) – \( \sigma \) is the volatility (20% or 0.20) – \( dW_t \) is a Wiener process (a random variable drawn from a normal distribution with mean 0 and variance dt) For each path, calculate the arithmetic average price of the asset over the year. Assume daily sampling (252 trading days). The average price for path \( i \) is: \[ A_i = \frac{1}{252} \sum_{t=1}^{252} S_{t,i} \] The payoff of the Asian option for path \( i \) is: \[ Payoff_i = max(A_i – K, 0) \] Where \( K \) is the strike price (£95). Calculate the average payoff across all simulated paths: \[ Average Payoff = \frac{1}{10000} \sum_{i=1}^{10000} Payoff_i \] Finally, discount the average payoff back to the present value using the risk-free rate (5% or 0.05): \[ Premium = Average Payoff \cdot e^{-rT} \] Where \( r \) is the risk-free rate and \( T \) is the time to maturity (1 year). Let’s assume the Monte Carlo simulation yields an average payoff of £7.50. Then, \[ Premium = 7.50 \cdot e^{-0.05 \cdot 1} = 7.50 \cdot e^{-0.05} \approx 7.50 \cdot 0.9512 \approx £7.13 \] Therefore, the fair premium for the Asian option is approximately £7.13. This premium reflects the average expected payoff discounted to the present, accounting for the asset’s volatility and the risk-free rate. The Asian option’s averaging feature reduces volatility compared to standard options, generally resulting in a lower premium.

The core of this question revolves around understanding how the Greeks, specifically Delta and Gamma, impact hedging strategies for options, and how transaction costs influence the rebalancing frequency. Delta represents the sensitivity of an option’s price to a change in the underlying asset’s price. Gamma, in turn, represents the sensitivity of Delta to a change in the underlying asset’s price. A high Gamma implies that Delta changes rapidly, requiring more frequent rebalancing to maintain a delta-neutral hedge. Transaction costs, such as brokerage fees and bid-ask spreads, make frequent rebalancing expensive. Therefore, a balance must be struck between the cost of rebalancing and the accuracy of the hedge. The optimal rebalancing frequency is determined by considering the trade-off between hedging accuracy (reducing variance) and transaction costs. A higher Gamma necessitates more frequent rebalancing to maintain a delta-neutral position, thus minimizing the hedge’s variance. However, each rebalancing incurs transaction costs. The key is to find the rebalancing frequency where the marginal benefit of reducing variance equals the marginal cost of the transaction. Consider a scenario where an investment firm, “NovaVest,” is hedging a portfolio of call options on FTSE 100 futures. NovaVest calculates that with daily rebalancing, the variance of the hedge is reduced by 10%, but the transaction costs amount to £5,000 per week. With weekly rebalancing, the variance reduction is 5%, and the transaction costs are £1,000 per week. If the firm only rebalances monthly, the variance reduction is 1%, and the transaction costs are negligible. The optimal rebalancing frequency depends on the firm’s risk aversion and the expected cost of deviations from a delta-neutral position. If NovaVest is highly risk-averse and estimates that a 1% deviation from delta-neutral could result in losses exceeding £4,000, then weekly rebalancing would be optimal. If the potential losses are lower, then monthly rebalancing might be more cost-effective.