Derivatives Level 4 (Investment Advice Diploma) Exam Quiz Quiz 28

The question assesses the understanding of delta hedging and how changes in the underlying asset’s price and the passage of time affect the hedge’s effectiveness. Delta hedging involves continuously adjusting the hedge ratio to maintain a delta-neutral position. Gamma represents the rate of change of delta with respect to changes in the underlying asset’s price, while theta represents the rate of change of the option’s price with respect to time. A long gamma position (resulting from selling options) means the delta will increase as the underlying asset’s price increases and decrease as the underlying asset’s price decreases. In this scenario, the portfolio manager initially establishes a delta-neutral hedge. When the market moves significantly, the gamma effect causes the delta of the option position to change substantially. As the underlying asset’s price rises, the delta of the short option position becomes more negative, requiring the manager to sell more of the underlying asset to maintain the hedge. Conversely, as time passes, theta decay reduces the value of the option, impacting the overall delta of the portfolio. The manager must actively manage both gamma and theta to maintain a delta-neutral position and minimize losses. The calculation involves understanding how changes in the underlying asset’s price and the passage of time affect the delta hedge. The manager initially sells options and hedges by buying the underlying asset. When the underlying asset’s price increases, the delta of the short option position becomes more negative, requiring the manager to sell more of the underlying asset to maintain the hedge. This action results in a loss because the manager is selling an asset that was previously bought at a lower price. The passage of time also affects the value of the option due to theta decay, which further impacts the hedge’s effectiveness. The net effect of these changes determines whether the manager experiences a profit or loss on the hedging strategy.

The core of this question lies in understanding how implied volatility impacts option pricing, particularly when combined with differing views on future asset price movements. A straddle involves simultaneously buying a call and a put option with the same strike price and expiration date. This strategy profits when the underlying asset price moves significantly in either direction. The price of a straddle is directly affected by implied volatility; higher implied volatility leads to higher option prices (both call and put), and thus a more expensive straddle. An investor who believes a company’s stock price will remain stable would typically avoid buying a straddle, as the cost of the straddle erodes potential profits unless a substantial price move occurs. However, if this investor also perceives that the market is underestimating the stock’s volatility (i.e., they believe the implied volatility is too low), they might consider a straddle. This is because if the actual volatility turns out to be higher than the implied volatility priced into the options, the value of the straddle will increase, even if the stock price doesn’t move dramatically. This is a play on volatility itself, rather than a directional bet on the underlying asset. In this scenario, the investor is betting that the implied volatility will increase, causing the value of the straddle to rise. If the investor is correct, they can profit by selling the straddle at a higher price than they paid for it, even if the stock price doesn’t move much. The profit comes from the increase in the options’ prices due to the increase in implied volatility. The investor is essentially exploiting a perceived mispricing of volatility.

The Black-Scholes model is a cornerstone of options pricing theory, but it relies on several key assumptions, including constant volatility, a risk-free interest rate, and no dividends during the option’s life. In reality, volatility is rarely constant and often exhibits a “smile” or “skew,” where out-of-the-money puts and calls have different implied volatilities. Dividends, especially for longer-dated options, can significantly impact the option’s price. Interest rates also fluctuate, though their impact is generally less pronounced for shorter-dated options. To accurately price an option, we need to adjust the Black-Scholes model to account for these real-world factors. Dividend adjustments can be made by subtracting the present value of expected dividends from the stock price. Volatility smiles/skews can be addressed by using different implied volatilities for different strike prices. Interest rate fluctuations can be incorporated by using a term structure model to forecast future interest rates. In this scenario, we have a call option on a stock that pays a known dividend and exhibits a volatility smile. First, calculate the present value of the dividend: \[ PV = \frac{Dividend}{(1 + r)^t} = \frac{2.50}{(1 + 0.05)^{0.5}} = 2.439 \]. Then, adjust the stock price: \[ S_{adj} = S – PV = 52 – 2.439 = 49.561 \]. Now, use the adjusted stock price and the implied volatility to calculate the call option price using the Black-Scholes formula: \[ C = S_{adj}N(d_1) – Ke^{-rT}N(d_2) \], where \[ d_1 = \frac{ln(\frac{S_{adj}}{K}) + (r + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}} \] and \[ d_2 = d_1 – \sigma\sqrt{T} \]. Plugging in the values: \[ d_1 = \frac{ln(\frac{49.561}{50}) + (0.05 + \frac{0.22^2}{2})0.5}{0.22\sqrt{0.5}} = 0.234 \] and \[ d_2 = 0.234 – 0.22\sqrt{0.5} = 0.078 \]. Using a standard normal distribution table (or calculator), N(d1) = 0.5926 and N(d2) = 0.5311. Therefore, \[ C = 49.561 * 0.5926 – 50e^{-0.05*0.5} * 0.5311 = 29.364 – 25.672 = 3.692 \].

The core of this question revolves around understanding how changes in correlation between assets within a portfolio affect the portfolio’s overall Value at Risk (VaR). VaR measures the potential loss in value of a portfolio over a specific time period for a given confidence level. When assets are perfectly correlated (correlation = 1), the portfolio’s risk is simply the sum of the individual asset risks. However, when assets are less than perfectly correlated, diversification benefits arise, reducing the overall portfolio risk. The lower the correlation, the greater the diversification benefit and the lower the portfolio VaR. Conversely, as correlation increases, the diversification benefit decreases, and the portfolio VaR increases. The calculation involves understanding how correlation affects portfolio standard deviation, which is a key input into VaR calculations. We will use the formula for portfolio variance with two assets: \[ \sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2w_A w_B \rho_{AB} \sigma_A \sigma_B \] Where: * \(\sigma_p^2\) is the portfolio variance * \(w_A\) and \(w_B\) are the weights of assets A and B in the portfolio * \(\sigma_A\) and \(\sigma_B\) are the standard deviations of assets A and B * \(\rho_{AB}\) is the correlation between assets A and B The VaR is then calculated as: \[ VaR = Portfolio Value \times z \times \sigma_p \] Where: * z is the z-score corresponding to the desired confidence level (e.g., 1.645 for 95% confidence) First, calculate the portfolio standard deviation when the correlation is 0.2: \[ \sigma_p^2 = (0.6)^2 (0.15)^2 + (0.4)^2 (0.20)^2 + 2(0.6)(0.4)(0.2)(0.15)(0.20) \] \[ \sigma_p^2 = 0.0081 + 0.0064 + 0.00288 = 0.01738 \] \[ \sigma_p = \sqrt{0.01738} = 0.1318 \] Then, calculate the VaR at 95% confidence level: \[ VaR = 500000 \times 1.645 \times 0.1318 = 108458.5 \] Next, calculate the portfolio standard deviation when the correlation is 0.6: \[ \sigma_p^2 = (0.6)^2 (0.15)^2 + (0.4)^2 (0.20)^2 + 2(0.6)(0.4)(0.6)(0.15)(0.20) \] \[ \sigma_p^2 = 0.0081 + 0.0064 + 0.00864 = 0.02314 \] \[ \sigma_p = \sqrt{0.02314} = 0.1521 \] Then, calculate the VaR at 95% confidence level: \[ VaR = 500000 \times 1.645 \times 0.1521 = 125166.75 \] Finally, calculate the change in VaR: \[ Change \ in \ VaR = 125166.75 – 108458.5 = 16708.25 \] Therefore, an increase in correlation from 0.2 to 0.6 increases the portfolio VaR by £16,708.25. This illustrates the principle that lower correlations provide greater diversification benefits, reducing overall portfolio risk as measured by VaR. The change in VaR reflects the reduced diversification benefit as correlation increases.

The question assesses understanding of hedging strategies using options, specifically focusing on creating a synthetic forward contract. A synthetic forward contract replicates the payoff of a forward contract using options. The goal is to lock in a future price for an asset. This is achieved by buying a call option and selling a put option with the same strike price and expiration date. Here’s how the synthetic forward is constructed and its payoff calculated: * **Buy a Call Option:** This gives the holder the right, but not the obligation, to buy the asset at the strike price at expiration. If the asset price at expiration is above the strike price, the call option will be exercised, and the asset will be bought at the strike price. * **Sell a Put Option:** This obligates the seller to buy the asset at the strike price if the option is exercised by the buyer. If the asset price at expiration is below the strike price, the put option will be exercised, and the seller will be forced to buy the asset at the strike price. The combined payoff mimics a forward contract: you are effectively obligated to buy the asset at the strike price, regardless of the market price at expiration. Let’s denote: * \(S_T\) = Asset price at expiration * \(K\) = Strike price of both call and put options Payoff Scenarios: 1. **\(S_T > K\):** * Call Option: Exercised, payoff = \(S_T – K\) * Put Option: Expires worthless, payoff = 0 * Net Payoff: \(S_T – K\) 2. **\(S_T < K\):** * Call Option: Expires worthless, payoff = 0 * Put Option: Exercised, payoff = \(K – S_T\) (since you have to buy the asset at \(K\) when it's worth \(S_T\)) * Net Payoff: \(K – S_T\) The cost of establishing the synthetic forward is the call premium minus the put premium. If the call and put are at-the-money (strike price equals the current market price), and assuming no arbitrage, the call and put premiums should be approximately equal (put-call parity). Therefore, the initial cost is close to zero. The effective forward price is the strike price \(K\). In the given scenario, \(K = 520\). Therefore, the synthetic forward obligates the investor to effectively buy the asset for £520.

The question tests understanding of hedging strategies using futures contracts, specifically focusing on cross-hedging. Cross-hedging involves hedging an asset with a futures contract on a different, but correlated, asset. The hedge ratio minimizes the variance of the hedged position. The optimal hedge ratio (h) is calculated as: \(h = \rho \frac{\sigma_S}{\sigma_F}\), where \(\rho\) is the correlation between the spot asset and the futures contract, \(\sigma_S\) is the standard deviation of the spot asset, and \(\sigma_F\) is the standard deviation of the futures contract. The number of contracts needed is then calculated by adjusting for the contract size: \(N = h \frac{Q_S}{Q_F}\), where \(Q_S\) is the quantity of the spot asset being hedged and \(Q_F\) is the contract size of the futures contract. In this scenario, a coffee roaster needs to hedge against price increases in Arabica beans using Robusta coffee futures. We calculate the optimal hedge ratio using the given correlation and standard deviations. Then, we determine the number of futures contracts needed to hedge the roaster’s exposure, considering the amount of Arabica beans they need and the contract size of the Robusta futures. This requires a practical application of hedging principles in a real-world context. First, calculate the hedge ratio: \[h = 0.85 \times \frac{0.07}{0.05} = 1.19\] Next, calculate the number of contracts: \[N = 1.19 \times \frac{250,000}{5,000} = 59.5\] Since you can’t trade fractional contracts, round to the nearest whole number. In hedging, it’s generally better to slightly over-hedge than under-hedge if precision isn’t critical, but in this case, we choose the closest number. Therefore, the roaster should purchase 60 Robusta coffee futures contracts.

The core of this question revolves around understanding how delta hedging works in practice, particularly when transaction costs are involved. Delta hedging aims to create a portfolio that is insensitive to small changes in the underlying asset’s price. The delta of an option measures the sensitivity of the option’s price to changes in the underlying asset’s price. To hedge, one would typically buy or sell the underlying asset to offset the option’s delta. However, real-world trading incurs transaction costs, which affect the profitability of delta hedging. The optimal hedging frequency balances the cost of frequent rebalancing (high transaction costs) against the risk of infrequent rebalancing (increased exposure to price fluctuations). In this scenario, the fund manager must consider both the volatility of the underlying asset (XYZ shares) and the transaction costs associated with trading those shares. Higher volatility implies that the delta changes more rapidly, requiring more frequent rebalancing. Higher transaction costs make frequent rebalancing less attractive. The fund manager is using the Black-Scholes model to calculate the option’s delta. The initial delta is 0.5, meaning for every £1 increase in the price of XYZ shares, the option’s price is expected to increase by £0.5. To hedge this, the manager sells 5,000 shares (since they are long 10,000 options with a delta of 0.5 each). The share price then increases by £0.50, and the delta increases to 0.55. This means the manager needs to rebalance the hedge. The change in delta is 0.05 per option, or 500 shares in total (10,000 options * 0.05). To rebalance, the manager needs to sell an additional 500 shares. The total transaction cost is calculated as follows: The manager sells 500 shares at a cost of £0.02 per share. The total cost is 500 * £0.02 = £10.

The question assesses the understanding of hedging strategies using options, specifically a ratio spread, and the ability to calculate the profit or loss at expiration based on different underlying asset prices. A ratio spread involves buying and selling different numbers of options with the same expiration date but different strike prices. The profit or loss depends on the price of the underlying asset at expiration. First, we need to calculate the profit or loss from the purchased call option. The investor buys one call option with a strike price of 1500 for a premium of £75. The profit or loss from this option at expiration is given by: * If \(S_T \le 1500\), the option expires worthless, and the loss is the premium paid: -£75. * If \(S_T > 1500\), the profit is \(S_T – 1500 – 75\), where \(S_T\) is the price of the underlying asset at expiration. Next, we need to calculate the profit or loss from the written call options. The investor sells two call options with a strike price of 1550 for a premium of £25 each, totaling £50. The profit or loss from these options at expiration is given by: * If \(S_T \le 1550\), the options expire worthless, and the profit is the premium received: £50. * If \(S_T > 1550\), the loss is \(2 \times (S_T – 1550) – 50\), where \(S_T\) is the price of the underlying asset at expiration. Now, we combine the profit/loss from both the purchased and written call options for each scenario: Scenario 1: \(S_T = 1450\) * Purchased call: -£75 * Written calls: £50 * Total profit/loss: -£75 + £50 = -£25 Scenario 2: \(S_T = 1525\) * Purchased call: \(1525 – 1500 – 75 = -£50\) * Written calls: £50 * Total profit/loss: -£50 + £50 = £0 Scenario 3: \(S_T = 1575\) * Purchased call: \(1575 – 1500 – 75 = £0\) * Written calls: \(2 \times (1575 – 1550) – 50 = 2 \times 25 – 50 = £0\) * Total profit/loss: £0 + £0 = £0 Scenario 4: \(S_T = 1600\) * Purchased call: \(1600 – 1500 – 75 = £25\) * Written calls: \(2 \times (1600 – 1550) – 50 = 2 \times 50 – 50 = £50\) * Total profit/loss: £25 – £50 = -£25 Therefore, the profit or loss at expiration for each scenario is: £-25, £0, £0, £-25.

The core of this question lies in understanding how implied volatility impacts option pricing, particularly when approaching an earnings announcement. Implied volatility reflects the market’s expectation of future price fluctuations. Earnings announcements are typically periods of heightened uncertainty, leading to a surge in implied volatility for options expiring shortly after the announcement. This phenomenon is often referred to as an “earnings volatility crush.” After the announcement, if the actual price movement is less dramatic than anticipated, implied volatility tends to decrease rapidly, impacting the value of options. The calculation requires understanding how changes in implied volatility affect option premiums. While a precise calculation requires an option pricing model (like Black-Scholes), we can approximate the impact. A general rule of thumb is that a 1% change in implied volatility can lead to a change in the option premium proportional to the Vega of the option. However, Vega itself changes with volatility and time to expiration, so we must consider the *relative* change. Let’s assume, for simplicity, that the option’s Vega is approximately 0.5. This means that for every 1% increase in implied volatility, the option price increases by roughly £0.5. The initial implied volatility is 30%, and it increases to 60% before the announcement, a 30% increase. The option premium increases by approximately 30% * £0.5 = £15. Therefore, the premium before the announcement is £5 + £15 = £20. After the announcement, the implied volatility drops to 20%, a decrease of 40% from the pre-announcement level of 60%. The option premium decreases by approximately 40% * £0.5 = £20. This means that the option premium after the announcement is £20 – £20 = £0. Therefore, the investor experiences a loss equal to the initial investment of £5. This scenario highlights the importance of understanding volatility dynamics, especially around significant events like earnings announcements. It’s not enough to simply buy an option anticipating a price move; one must also consider the impact of volatility changes on the option’s value. The “volatility crush” can significantly erode profits, even if the underlying asset moves in the anticipated direction. Furthermore, this example emphasizes the need for more sophisticated strategies, such as volatility trading strategies, which aim to profit from changes in volatility itself, rather than just directional price movements. Understanding the interplay between implied volatility, time decay, and the underlying asset’s price is crucial for successful derivatives trading.

The core of this question lies in understanding how a delta-neutral portfolio is constructed and how its value changes with shifts in implied volatility. A delta-neutral portfolio is designed to be insensitive to small changes in the underlying asset’s price. This is achieved by balancing long and short positions in the underlying asset and its derivatives (typically options). The portfolio’s value, however, is sensitive to other factors, most notably implied volatility (vega) and the passage of time (theta). When implied volatility increases, the value of options generally increases. This is because higher volatility implies a greater probability of the underlying asset’s price moving significantly, which benefits option holders. The vega of a portfolio measures its sensitivity to changes in implied volatility. A positive vega indicates that the portfolio’s value will increase with increasing volatility, while a negative vega indicates the opposite. In this scenario, the portfolio is initially delta-neutral and has a positive vega. This means that the portfolio’s value is expected to increase if implied volatility rises. The magnitude of the increase depends on the size of the vega. The question describes a 2% increase in implied volatility. To estimate the change in portfolio value, we multiply the vega by the change in volatility: Change in portfolio value ≈ Vega * Change in implied volatility Change in portfolio value ≈ £25,000 * 0.02 = £500 Therefore, the portfolio’s value is expected to increase by approximately £500. This calculation is a first-order approximation and does not account for other factors that may affect the portfolio’s value, such as changes in interest rates or the underlying asset’s price (beyond what’s already neutralized by delta). In real-world scenarios, more sophisticated models and stress testing would be used to assess the impact of volatility changes on a portfolio’s value.

Let’s consider a scenario where a portfolio manager, overseeing a UK-based pension fund, uses options to hedge against potential downside risk in their equity holdings. The fund holds a significant position in the FTSE 100 index. The manager decides to implement a protective put strategy using FTSE 100 index options. The protective put strategy involves buying put options on the FTSE 100 index, giving the fund the right, but not the obligation, to sell the index at a specified strike price on or before the expiration date. This strategy protects against a decline in the value of the fund’s equity holdings. To calculate the cost of the hedge, we need to determine the price of the put options. Assume the FTSE 100 index is currently trading at 7,500. The portfolio manager purchases put options with a strike price of 7,400, expiring in three months, at a premium of 100 index points. Each index point represents £10. The total cost of the hedge is calculated as follows: Put option premium = 100 index points Value per index point = £10 Number of contracts = 1 (assuming the fund holds a portfolio whose value mirrors one FTSE 100 index contract) Total cost of hedge = Put option premium * Value per index point * Number of contracts Total cost of hedge = 100 * £10 = £1,000 per contract The protective put strategy provides downside protection. If the FTSE 100 index falls below the strike price of 7,400, the put options become in-the-money, and the fund can exercise the options to offset losses in the equity portfolio. If the index remains above 7,400, the put options expire worthless, and the fund only loses the premium paid for the options. Now, consider the impact of implied volatility on the option premium. If implied volatility increases, the premium of the put options will also increase, making the hedge more expensive. Conversely, if implied volatility decreases, the premium will decrease, making the hedge cheaper. The decision to implement a protective put strategy depends on the portfolio manager’s risk tolerance, market outlook, and the cost of the options. It’s a trade-off between protecting against downside risk and potentially sacrificing upside gains if the market rises. In this example, the cost of the hedge is £1,000 per contract. This cost needs to be weighed against the potential benefits of downside protection. The portfolio manager must also consider the impact of factors such as implied volatility, time to expiration, and interest rates on the option premium.

The question assesses understanding of hedging strategies using options, specifically focusing on gamma, delta, and theta. Gamma measures the rate of change of delta with respect to changes in the underlying asset’s price. Delta measures the sensitivity of the option’s price to changes in the underlying asset’s price. Theta measures the sensitivity of the option’s price to the passage of time. A portfolio manager who is delta-neutral but has negative gamma is vulnerable to large price swings in the underlying asset. To hedge this, the manager needs to implement a strategy that increases gamma. Buying a straddle (buying both a call and a put option with the same strike price and expiration date) increases gamma because the gamma of both the call and put options are positive. The combined position will have a positive gamma, offsetting the negative gamma of the original portfolio. The portfolio manager’s goal is to neutralize the negative gamma while maintaining the delta-neutral position. Selling a straddle would further increase the negative gamma exposure, which is the opposite of what the manager wants to achieve. Buying or selling the underlying asset would affect the delta of the portfolio, moving it away from the desired delta-neutral position. The optimal strategy is to buy a straddle.

Let’s consider a scenario where a UK-based agricultural cooperative, “FarmCo,” wants to hedge against potential fluctuations in wheat prices using futures contracts listed on the London International Financial Futures and Options Exchange (LIFFE). FarmCo anticipates harvesting 500,000 bushels of wheat in six months and is concerned about a price drop. They decide to use short hedging with wheat futures contracts. Each contract represents 5,000 bushels. The current futures price for wheat with a six-month delivery is £200 per bushel. FarmCo sells 100 futures contracts (500,000 bushels / 5,000 bushels per contract = 100 contracts). Six months later, FarmCo harvests the wheat. The spot price of wheat is now £180 per bushel. Simultaneously, the futures price has also decreased to £182 per bushel. FarmCo sells its wheat in the spot market for £180 per bushel. To close out their hedge, they buy back 100 futures contracts at £182 per bushel. Profit/Loss Calculation: * **Spot Market:** FarmCo receives £180 per bushel, a loss of £20 per bushel compared to the initial futures price (£200 – £180 = £20 loss). Total loss in spot market: 500,000 bushels * £20/bushel = £10,000,000 loss. * **Futures Market:** FarmCo initially sold futures at £200 per bushel and bought them back at £182 per bushel, resulting in a profit of £18 per bushel (£200 – £182 = £18 profit). Total profit in futures market: 500,000 bushels * £18/bushel = £9,000,000 profit. * **Net Result:** The net result is a loss of £1,000,000 (£9,000,000 profit – £10,000,000 loss). Basis Risk Consideration: Basis risk arises because the futures price and spot price did not converge perfectly. Initially, the basis (spot price – futures price) was assumed to be zero for simplicity. However, in reality, the basis changed. If the spot price had fallen more than the futures price, the hedge would have been less effective. Conversely, if the spot price had fallen less than the futures price, the hedge would have been more effective. In this case, the basis moved unfavorably for FarmCo, resulting in a less-than-perfect hedge. Factors Affecting Basis: * **Transportation Costs:** Differences in transportation costs between the location of the futures contract delivery point and FarmCo’s location. * **Storage Costs:** Costs associated with storing wheat until the delivery date of the futures contract. * **Quality Differences:** Variations in the quality of wheat that FarmCo produces compared to the standard grade specified in the futures contract. * **Local Supply and Demand:** Regional supply and demand factors that affect the spot price in FarmCo’s specific market. The effectiveness of the hedge depends on how closely the futures price tracks the spot price. Basis risk is inherent in hedging strategies and can either reduce or increase the overall outcome compared to a perfect hedge. FarmCo must continuously monitor the basis and adjust their hedging strategy if necessary. Understanding and managing basis risk is crucial for effective risk management using derivatives.

To determine the most suitable hedging strategy, we need to calculate the potential loss without hedging and compare it to the cost and effectiveness of different hedging instruments. The loss without hedging is the difference between the current market value and the potential future value if the market declines. First, calculate the potential loss without hedging: Current Portfolio Value: £5,000,000 Expected Market Decline: 8% Potential Loss: £5,000,000 * 0.08 = £400,000 Now, let’s evaluate each hedging option: a) Short FTSE 100 Futures Contracts: Each contract covers £10 per index point. The current index level is 7,500. Therefore, each contract covers £75,000 (7,500 * £10). Number of contracts needed to hedge: £5,000,000 / £75,000 ≈ 66.67 contracts. Round up to 67 contracts. Cost per contract: £500 Total cost: 67 * £500 = £33,500 Hedge effectiveness: Assumes a perfect correlation between the portfolio and the FTSE 100, which is unlikely. b) Purchase Put Options on the FTSE 100: Strike price: 7,500 Option premium: 3% of the index level = 0.03 * 7,500 * £10 = £2,250 per contract Number of contracts: £5,000,000 / (£7,500 * £10) ≈ 66.67 contracts. Round up to 67 contracts. Total cost: 67 * £2,250 = £150,750 Hedge effectiveness: Provides downside protection but at a higher upfront cost. c) Credit Default Swaps (CDS) on a basket of similar corporate bonds: Notional amount: £5,000,000 CDS spread: 150 bps = 1.5% per annum Annual cost: £5,000,000 * 0.015 = £75,000 Hedge effectiveness: Protects against credit risk, not market risk. May not be directly correlated to the portfolio’s market movements. d) Variance Swaps on the VIX Index: Notional amount: £5,000,000 Expected increase in VIX: 20% Cost: Assumes a cost proportional to the notional amount and the expected volatility change. This is highly speculative and complex to implement for a broad market hedge. Comparing the options: – Futures: Cheapest but relies on correlation. – Put Options: More expensive but provides guaranteed downside protection. – CDS: Protects against credit risk, not market risk. – Variance Swaps: Highly complex and speculative. Considering the need for downside protection against a market decline, purchasing put options offers the most direct and reliable hedge, despite the higher cost. Therefore, the best strategy is to purchase put options on the FTSE 100 with a strike price of 7,500.

The core of this question lies in understanding how the delta of an option changes as the underlying asset’s price moves, and how this impacts the number of shares needed to maintain a delta-neutral hedge. Delta represents the sensitivity of the option’s price to a change in the underlying asset’s price. Gamma, on the other hand, measures the rate of change of delta with respect to the underlying asset’s price. In simpler terms, gamma tells us how much delta will change for every £1 move in the underlying asset. To maintain a delta-neutral position, the trader must continuously adjust the number of shares held to offset the changes in the option’s delta. When the underlying asset’s price increases, the delta of a call option increases (and decreases for a put option). To remain delta-neutral, the trader must buy more shares of the underlying asset when holding a short call position. Conversely, when the underlying asset’s price decreases, the trader must sell shares. The magnitude of this adjustment is determined by the option’s gamma. The initial delta-neutral position is calculated by determining the number of options contracts multiplied by the delta of each contract and then offsetting that with the appropriate number of shares. The change in the number of shares required is calculated by multiplying the gamma by the number of contracts, the change in the underlying asset price, and the contract multiplier. In this specific case, the trader is short call options. Therefore, when the price of the underlying asset increases, the trader needs to buy shares to maintain delta neutrality. The calculation is as follows: 1. **Initial Delta Exposure:** 100 contracts * 0.55 delta * 100 shares/contract = 5500 shares (short) 2. **Delta Change per Contract:** 0.04 gamma * £2.50 price increase = 0.10 delta increase per contract 3. **Total Delta Change:** 100 contracts * 0.10 = 10 4. **Shares to Buy:** 10 * 100 shares/contract = 1000 shares Therefore, the trader needs to buy 1000 shares to maintain a delta-neutral position. This example highlights the dynamic nature of delta hedging and the importance of gamma in managing risk. A higher gamma implies a more frequent need to rebalance the hedge, increasing transaction costs but also providing better protection against large price movements.

The question assesses understanding of how market psychology, specifically the disposition effect, can influence options trading strategies and portfolio performance. The disposition effect is the tendency for investors to sell assets that have increased in value too early and hold onto assets that have decreased in value for too long. This behavior is often driven by the desire to realize gains quickly and avoid realizing losses. The optimal strategy involves understanding how the disposition effect influences investor behavior and how this, in turn, affects option prices and volatility. The scenario presented requires understanding that investors exhibiting the disposition effect may be more likely to prematurely close out profitable short option positions (e.g., covered calls) and hold onto losing long option positions (e.g., protective puts). In this scenario, if a significant portion of the market is exhibiting the disposition effect, the demand for options that protect against further losses (puts) will be artificially inflated, while the supply of options that allow investors to realize quick gains (calls) will also be artificially inflated as investors look to close out positions. This leads to an overpricing of both calls and puts. An appropriate strategy to take advantage of this is to sell both calls and puts, creating a short straddle position. This strategy profits if the underlying asset price remains relatively stable. The profit from the short straddle comes from the decay of the option premiums. The potential loss is unlimited if the price of the underlying asset moves significantly in either direction. The calculation is conceptual, focusing on identifying the correct trading strategy based on behavioral bias: 1. **Identify the Bias:** The disposition effect is present. 2. **Assess Market Impact:** Overpricing of both call and put options due to the disposition effect. 3. **Choose Strategy:** A short straddle is appropriate to capitalize on the overpricing of both calls and puts, assuming the investor believes the market will remain stable. The other options are incorrect because they either do not align with the disposition effect or involve strategies that would be less effective in this specific scenario. A long straddle would profit from volatility, but the options are already overpriced. A covered call only addresses call option overpricing and a protective put only addresses put option overpricing.

This question tests the understanding of hedging a portfolio with equity index futures, specifically focusing on beta adjustment. Beta represents the systematic risk of a portfolio relative to the market. To hedge a portfolio and reduce its market exposure, one can use equity index futures. The number of contracts required depends on the portfolio’s beta, the future’s beta (usually assumed to be 1), the portfolio value, and the future’s price. The formula used is: Number of contracts = \( \frac{(Portfolio \ Beta – Desired \ Beta) \times Portfolio \ Value}{(Futures \ Price \times Multiplier)} \) In this scenario, the portfolio’s beta is 1.2, and the desired beta is 0.5. The portfolio value is £5,000,000, the future’s price is 4500, and the multiplier is £10 per index point. Number of contracts = \( \frac{(1.2 – 0.5) \times 5,000,000}{4500 \times 10} \) Number of contracts = \( \frac{0.7 \times 5,000,000}{45,000} \) Number of contracts = \( \frac{3,500,000}{45,000} \) Number of contracts ≈ 77.78 Since you can’t trade fractional contracts, the number of contracts is rounded to the nearest whole number. In this case, it is 78 contracts. Because the portfolio’s beta is being reduced (from 1.2 to 0.5), a short hedge is needed, meaning selling the futures contracts.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their sensitivity to market volatility, interest rates, and dividend payouts. It requires the candidate to evaluate how a change in these factors impacts the price of a knock-out call option, considering the barrier effect. A knock-out call option ceases to exist if the underlying asset’s price touches or exceeds the barrier level before the expiration date. * **Volatility:** Increased volatility generally *decreases* the price of a knock-out call. A more volatile underlying asset has a higher probability of hitting the knock-out barrier, thus reducing the option’s value. * **Interest Rates:** Higher interest rates typically *increase* the price of a call option (including a knock-out). The present value of the strike price is lower, making the call option more attractive. * **Dividend Payouts:** Increased dividend payouts generally *decrease* the price of a call option. Dividends reduce the future price of the underlying asset, making the call option less valuable. The combined effect of these changes needs to be considered. Since volatility has the most significant impact on a knock-out option, the overall price will likely decrease. Here’s a simplified, illustrative example using hypothetical values (not for calculation, but for understanding the principle): Assume the initial knock-out call option price is £5. 1. **Volatility Increase:** A significant volatility increase might reduce the option price by £2 due to the higher probability of hitting the barrier. 2. **Interest Rate Increase:** A moderate interest rate increase might increase the option price by £0.50, reflecting the reduced present value of the strike price. 3. **Dividend Payout Increase:** A slight dividend increase might decrease the option price by £0.25, as the underlying asset’s future price is expected to be lower. The net effect would be a decrease of £1.75 (£0.50 – £2 – £0.25), resulting in a final option price of £3.25 (£5 – £1.75). This example demonstrates that the volatility effect outweighs the interest rate and dividend effects in this scenario. The specific magnitude of the changes will depend on the sensitivity of the option to each factor, represented by its “Greeks” (Vega, Rho, and Delta, respectively). However, understanding the directional impact is crucial.

The core of this question lies in understanding how a fund manager utilizes futures contracts to tactically adjust the duration of their fixed-income portfolio, particularly in response to anticipated shifts in the yield curve. Duration, a measure of a bond portfolio’s sensitivity to interest rate changes, is crucial for managing interest rate risk. Shorting bond futures effectively shortens the portfolio’s duration, protecting it from potential losses if interest rates rise. The number of futures contracts needed depends on the portfolio’s current duration, the desired duration, the price sensitivity of the futures contract, and the conversion factor. The formula to calculate the number of futures contracts is: \[N = \frac{(D_{target} – D_{current}) \times P}{D_{futures} \times P_{futures} \times CF}\] Where: \(N\) = Number of futures contracts \(D_{target}\) = Target duration of the portfolio \(D_{current}\) = Current duration of the portfolio \(P\) = Current value of the portfolio \(D_{futures}\) = Duration of the bond futures contract \(P_{futures}\) = Price of the bond futures contract \(CF\) = Conversion factor of the bond futures contract In this scenario, the fund manager wants to reduce the portfolio’s duration from 7 years to 4 years. The portfolio is valued at £100 million. The bond futures contract has a duration of 8 years, a price of £125,000, and a conversion factor of 1.2. Plugging these values into the formula, we get: \[N = \frac{(4 – 7) \times 100,000,000}{8 \times 125,000 \times 1.2} = \frac{-300,000,000}{1,200,000} = -250\] The negative sign indicates that the fund manager needs to short 250 futures contracts to achieve the desired reduction in duration. It’s crucial to understand that the conversion factor adjusts for the difference between the underlying bond in the futures contract and a standardized bond. Failing to account for this conversion factor would lead to an incorrect calculation of the number of contracts needed, potentially leaving the portfolio under- or over-hedged. This tactical adjustment using futures contracts allows the fund manager to proactively manage interest rate risk and optimize portfolio performance in anticipation of changing market conditions. Ignoring transaction costs and margin requirements simplifies the calculation, focusing on the core concept of duration adjustment.

The question assesses the understanding of delta-hedging a portfolio of options, specifically focusing on the rebalancing frequency and its impact on hedging effectiveness and transaction costs. Delta-hedging aims to neutralize the portfolio’s sensitivity to small changes in the underlying asset’s price. However, delta changes as the underlying asset’s price and time to expiration change (gamma and theta, respectively). Therefore, the hedge needs to be rebalanced periodically. Rebalancing too frequently leads to excessive transaction costs (brokerage fees, bid-ask spreads), eroding profits. Rebalancing too infrequently exposes the portfolio to greater risk as the delta deviates significantly from the desired level (usually zero for a delta-neutral portfolio). The optimal rebalancing frequency balances these two opposing forces. In this scenario, we consider the impact of volatility on the optimal rebalancing frequency. Higher volatility implies larger and more rapid price fluctuations in the underlying asset. This, in turn, means the delta of the options portfolio changes more quickly. Consequently, a more frequent rebalancing is required to maintain the delta-neutral position. The calculation involves considering the trade-off between the cost of rebalancing and the cost of imperfect hedging. A simplified model might consider the expected cost of deviations from delta neutrality (related to volatility and gamma) versus the transaction costs of rebalancing. Let’s assume the expected cost of deviations from delta neutrality increases quadratically with the time since the last rebalance, and the transaction cost is a fixed amount per rebalance. Suppose the portfolio has a gamma of 1000. The volatility of the underlying asset is 20% per annum. The transaction cost per rebalance is £50. We aim to minimize the total cost (hedging error + transaction costs). Let \( \Delta S \) be the expected change in the underlying asset price over a period \( \Delta t \). \( \Delta S \approx S \cdot \sigma \cdot \sqrt{\Delta t} \), where \( S \) is the asset price and \( \sigma \) is the volatility. The expected change in delta is \( \Gamma \cdot \Delta S \). The expected hedging error is proportional to \( (\Gamma \cdot \Delta S)^2 \). The total cost \( C \) can be approximated as \[ C = k \cdot (\Gamma \cdot S \cdot \sigma \cdot \sqrt{\Delta t})^2 + \frac{50}{\Delta t} \] where \( k \) is a constant of proportionality. To minimize \( C \), we differentiate with respect to \( \Delta t \) and set the derivative to zero. \[ \frac{dC}{d\Delta t} = k \cdot \Gamma^2 \cdot S^2 \cdot \sigma^2 – \frac{50}{(\Delta t)^2} = 0 \] \[ (\Delta t)^2 = \frac{50}{k \cdot \Gamma^2 \cdot S^2 \cdot \sigma^2} \] \[ \Delta t = \sqrt{\frac{50}{k \cdot \Gamma^2 \cdot S^2 \cdot \sigma^2}} \] Without knowing \( k \) and \( S \), we can still infer that an increase in \( \sigma \) (volatility) will *decrease* \( \Delta t \), meaning more frequent rebalancing. The exact optimal frequency requires a more precise model and parameter values, but the qualitative relationship holds.

The core of this question lies in understanding how delta hedging works in conjunction with option pricing, particularly the Black-Scholes model, and how transaction costs impact the profitability of a delta-hedged portfolio. Delta hedging aims to neutralize the directional risk of an option position by continuously adjusting the underlying asset holding. However, real-world trading incurs transaction costs, which erode the theoretical profit predicted by the Black-Scholes model. The Black-Scholes model assumes a perfect market, meaning no transaction costs, continuous trading, and constant volatility. In reality, these assumptions are violated. Each time the hedge is adjusted (rebalanced), a transaction cost is incurred. The more volatile the underlying asset, the more frequently the hedge needs to be adjusted, and the higher the total transaction costs. The theoretical profit from a delta-hedged short option position comes from the difference between the premium received for selling the option and the cost of maintaining the hedge. This cost includes the initial cost of establishing the hedge and the ongoing costs of rebalancing. Transaction costs directly reduce the net profit. To calculate the expected profit, we need to: 1. Calculate the initial cost of the hedge (Delta * Initial Asset Price). 2. Estimate the number of rebalances required over the option’s life. This is qualitatively related to volatility; higher volatility implies more rebalances. 3. Calculate the total transaction costs (Number of Rebalances * Transaction Cost per Rebalance). 4. Calculate the net profit (Option Premium – Cost of Hedge – Total Transaction Costs). In this scenario, the high volatility necessitates frequent rebalancing, leading to significant transaction costs. These costs can outweigh the theoretical profit from the delta hedge, resulting in an overall loss. The trader must carefully consider the volatility of the asset and the transaction costs before implementing a delta-hedging strategy. Let’s assume the initial delta is 0.5. The initial cost of the hedge is 0.5 * £100 = £50. Let’s estimate that, due to high volatility, the hedge needs to be rebalanced 20 times over the option’s life. Total transaction costs are 20 * £0.25 = £5. Net profit = £55 – £50 – £5 = £0. If the number of rebalances is higher or the transaction cost is higher, then the net profit will be negative.

Let’s analyze the impact of implied volatility skew on option pricing, particularly when considering strategies involving both call and put options. Implied volatility skew refers to the phenomenon where options with different strike prices for the same underlying asset and expiration date have different implied volatilities. Typically, lower strike puts exhibit higher implied volatility than higher strike calls, creating a “skewed” curve. This skew impacts the pricing of option strategies. Consider a butterfly spread, constructed by buying one call option at a lower strike price \(K_1\), selling two call options at a middle strike price \(K_2\), and buying one call option at a higher strike price \(K_3\). If the implied volatility skew is present, the lower strike call option (bought) will have a lower implied volatility than the middle strike call option (sold). Similarly, the higher strike call option (bought) will have an even lower implied volatility than the middle strike. This means the calls that are bought are cheaper and calls that are sold are expensive due to implied volatility skew. Now, let’s consider a reverse butterfly spread using put options. The strategy involves buying one put option at a lower strike price \(K_1\), selling two put options at a middle strike price \(K_2\), and buying one put option at a higher strike price \(K_3\). Given the implied volatility skew, the lower strike put option (bought) will have a *higher* implied volatility than the middle strike put option (sold). The higher strike put option (bought) will have a lower implied volatility than the middle strike put option (sold). Assume the following: \(K_1 = 90\), \(K_2 = 100\), \(K_3 = 110\). The prices are: – Put option at \(K_1 = 90\): £12 (Implied Volatility = 28%) – Put option at \(K_2 = 100\): £6 (Implied Volatility = 22%) – Put option at \(K_3 = 110\): £2 (Implied Volatility = 18%) The cost of the reverse butterfly spread is: Cost = Price(90 Put) – 2 * Price(100 Put) + Price(110 Put) Cost = £12 – 2 * £6 + £2 = £12 – £12 + £2 = £2 Therefore, the initial cost of implementing the reverse butterfly spread with put options is £2. The investor profits if the price moves outside the range of £90 and £110.

The question assesses the understanding of the impact of interest rate volatility on swaption pricing. A swaption is an option to enter into an interest rate swap. The value of a swaption is highly sensitive to interest rate volatility. Higher volatility increases the uncertainty about future interest rates, making the option to enter into a swap more valuable. The Black-Scholes model, although originally designed for equity options, provides a useful analogy. Just as higher stock price volatility increases the value of a stock option, higher interest rate volatility increases the value of a swaption. The question requires calculating the approximate change in swaption premium given a change in implied volatility. We can approximate the change using the vega of the swaption. Vega represents the sensitivity of the swaption’s price to a 1% change in implied volatility. Given: Swaption Premium: 4.5% Swaption Vega: 0.6% per 1% volatility change Volatility Increase: 2% Change in Swaption Premium = Vega * Change in Volatility Change in Swaption Premium = 0.6% * 2 = 1.2% New Swaption Premium = Original Premium + Change in Premium New Swaption Premium = 4.5% + 1.2% = 5.7% Therefore, the swaption premium is expected to increase to approximately 5.7%.

The question assesses understanding of delta hedging, gamma, and the practical implications of these concepts for managing a derivatives portfolio. The calculation involves determining the necessary adjustment to a delta-hedged portfolio to maintain its delta neutrality after a change in the underlying asset’s price. The initial delta is 0. A gamma of -50 indicates that for every £1 change in the underlying asset’s price, the delta of the portfolio changes by -50. A price increase of £2 results in a delta change of -50 * 2 = -100. To re-establish delta neutrality, one must offset this change by buying contracts with an equivalent delta. Since each contract has a delta of 0.5, the number of contracts to buy is -(-100) / 0.5 = 200 contracts. Consider a fund manager, Alice, who manages a large portfolio of options on FTSE 100. She initially delta hedges her portfolio to protect against small price movements. However, Alice is aware that her portfolio has a significant negative gamma. She understands that as the FTSE 100 index moves substantially, the delta of her portfolio will change, requiring her to rebalance the hedge. Suppose Alice’s portfolio has a gamma of -50. This means that for every £1 move in the FTSE 100, the delta of her portfolio changes by -50. Alice observes that the FTSE 100 has unexpectedly risen by £2. To maintain delta neutrality, Alice needs to adjust her position. Each FTSE 100 futures contract has a delta of approximately 0.5. The calculation is as follows: Change in delta = Gamma * Change in asset price = -50 * 2 = -100. Number of contracts to buy = – (Change in delta) / Delta of each contract = -(-100) / 0.5 = 200 contracts. Alice must buy 200 futures contracts to bring her portfolio back to delta neutrality.

This question delves into the intricacies of hedging strategies using options, specifically focusing on the impact of implied volatility on option pricing and the subsequent adjustments needed in a delta-neutral portfolio. A delta-neutral portfolio is constructed to be insensitive to small changes in the underlying asset’s price. However, changes in implied volatility can significantly impact the option’s price and, consequently, the portfolio’s delta. The initial delta of the portfolio is zero. A long position in an asset implies a positive delta, while a short call option position introduces a negative delta. To maintain delta neutrality, the number of short call options needs to be carefully calculated to offset the delta of the underlying asset. The delta of a call option is sensitive to implied volatility; an increase in implied volatility increases the call option’s delta, making the short call position less effective in hedging the long asset position. The problem requires calculating the change in the call option’s delta due to the change in implied volatility and then determining the number of additional call options to sell to re-establish delta neutrality. The formula for the change in delta due to a change in implied volatility is approximately: Change in Delta ≈ Vega * Change in Implied Volatility Where Vega represents the sensitivity of the option’s price to changes in implied volatility. In this scenario, Vega is given as 0.65, and the implied volatility increases by 3% (0.03). Therefore, the change in delta for each call option is: Change in Delta = 0.65 * 0.03 = 0.0195 This means each short call option now provides 0.0195 less negative delta than before. The total reduction in negative delta provided by the existing 100 short call options is: Total Reduction in Negative Delta = 100 * 0.0195 = 1.95 To re-establish delta neutrality, we need to sell additional call options to offset this reduction in negative delta. Since each additional call option sold will add a negative delta of approximately 0.65 (the new delta after the volatility change), the number of additional call options required is: Number of Additional Call Options = Total Reduction in Negative Delta / New Call Option Delta Number of Additional Call Options = 1.95 / 0.65 ≈ 3 Therefore, approximately 3 additional call options need to be sold to restore delta neutrality after the increase in implied volatility.

The question tests the understanding of exotic derivatives, specifically barrier options, and how their payoff structure is affected by market movements and knock-in/knock-out levels. The core concept revolves around calculating the payoff of a down-and-in call option. This requires understanding that the option only becomes active if the underlying asset’s price touches or falls below the barrier level. First, we must determine if the barrier has been breached. The barrier level is 90, and the asset price *did* fall to 85 during the option’s life. Therefore, the option is “knocked-in” and becomes a standard call option with a strike price of 100. Next, we calculate the payoff of this call option at expiration. The payoff is max(S – K, 0), where S is the spot price at expiration and K is the strike price. In this case, S = 110 and K = 100, so the payoff is max(110 – 100, 0) = 10. The option premium is irrelevant to the *payoff* calculation. The investor has already paid the premium. The question asks for the *payoff*, not the *profit*. Finally, we must consider the investor’s initial expectation of the asset price and the option’s behavior. The investor bought the option believing the price would increase, and the fact that the price *did* increase above the strike price after breaching the barrier is what generates the payoff. If the price had remained below the strike price at expiration, the payoff would have been zero, even though the option had been knocked in. This highlights the risk associated with barrier options – the barrier event must occur, *and* the underlying asset must move favorably relative to the strike price *after* the barrier is breached for a payoff to be realized.

The question assesses understanding of the impact of macroeconomic indicators on derivative pricing, specifically focusing on inflation expectations and their influence on interest rate swaps. The scenario involves a pension fund using interest rate swaps to manage its liabilities in an environment of changing inflation expectations. The correct answer requires the candidate to understand how increasing inflation expectations impact fixed and floating rates in a swap, and how the pension fund should adjust its position to benefit from these changes. The pension fund’s liabilities are linked to future pension payments, which are likely to increase with inflation. Therefore, the fund is essentially short inflation. To hedge this risk, the fund wants to benefit from rising inflation expectations. In an interest rate swap, the fixed rate payer benefits when interest rates rise. If inflation expectations increase, the fixed rate on the swap will also increase to compensate the fixed rate payer for the increased risk of higher floating rates. The fund should therefore enter into a receive-fixed, pay-floating interest rate swap. If inflation expectations increase, the fixed rate will increase, and the fund will receive higher fixed payments, offsetting the increase in its liabilities.

The core of this question lies in understanding how changes in volatility expectations affect option prices, particularly in the context of straddles. A straddle consists of buying both a call and a put option with the same strike price and expiration date. The value of a straddle is highly sensitive to changes in volatility. Vega, one of the “Greeks,” measures the sensitivity of an option’s price to changes in the volatility of the underlying asset. A positive Vega means that the option’s price will increase if volatility increases, and decrease if volatility decreases. In this scenario, the investor holds a short straddle, meaning they have *sold* both the call and the put. Therefore, their position has a *negative* Vega. If implied volatility increases, the value of the short straddle position *decreases*, resulting in a loss for the investor. Conversely, if implied volatility decreases, the value of the short straddle position increases, resulting in a profit for the investor. The profit or loss can be estimated by multiplying the Vega of the straddle by the change in implied volatility. In this case, the straddle has a Vega of -0.05 (per contract), and implied volatility increases by 5% (0.05). The change in the value of the straddle is therefore -0.05 * 0.05 = -0.0025 per contract. This means the straddle’s value decreases by £0.0025 per point of the underlying asset. Since each contract covers 100 units of the underlying asset, the total loss per contract is £0.0025 * 100 = £0.25. With 100 contracts, the total loss is £0.25 * 100 = £25. The key takeaway is that a short straddle benefits from decreased volatility and suffers from increased volatility. Understanding the sign of Vega and its impact on the position’s value is crucial for managing the risk associated with straddle strategies. This example highlights how seemingly small changes in implied volatility can significantly impact the profitability of derivative positions, particularly those sensitive to volatility like straddles. This is particularly important given the regulations around suitability and appropriateness when advising clients on derivative strategies under MiFID II.

The question assesses the understanding of exotic derivatives, specifically barrier options, and how their payoff structures are affected by market volatility and the knock-out level. The investor’s strategy of using a down-and-out call option to manage risk and potentially enhance returns is examined. The calculation involves determining the payoff of the option based on whether the underlying asset’s price breaches the barrier during the option’s life and the asset’s price at expiration. The concept of implied volatility and its influence on option pricing is also crucial. First, we need to determine if the barrier has been breached. The barrier is at 90% of the initial asset price, which is \(0.90 \times 200 = 180\). Since the asset price dipped to 175 during the option’s life, the barrier was breached, and the option is knocked out. Therefore, the payoff is zero, regardless of the asset price at expiration. The investor will not receive any payoff from the option, as it ceased to exist when the barrier was breached. This highlights the risk associated with barrier options, where even a temporary breach of the barrier can eliminate the option’s value. The question also touches on the impact of implied volatility. Higher implied volatility increases the probability of the barrier being breached, thereby decreasing the value of a down-and-out call option. This is because a higher volatility suggests a greater likelihood of the asset price reaching the barrier level. Conversely, lower implied volatility would increase the value of the down-and-out call option, as the probability of the barrier being breached is lower. This scenario illustrates how barrier options can be used for cost-effective hedging or speculative strategies, but it also emphasizes the importance of understanding the risks involved, particularly the sensitivity to barrier breaches and implied volatility.

The question assesses understanding of delta hedging, a risk management technique used to neutralize the directional risk of an option position. Delta represents the sensitivity of an option’s price to a change in the underlying asset’s price. A delta-neutral portfolio has a delta of zero, meaning that small changes in the underlying asset’s price will not significantly affect the portfolio’s value. The hedge ratio is calculated to offset the option’s delta with an opposing position in the underlying asset. To calculate the number of shares needed to delta-hedge a short option position, we use the following formula: Number of shares = – (Option Delta) * (Number of Options Contracts * Number of Shares per Contract) In this case, the fund has sold 50 call option contracts, each covering 100 shares, and the delta of each call option is 0.60. Therefore: Number of shares = – (0.60) * (50 * 100) = -3000 The negative sign indicates that the fund needs to buy shares to offset the negative delta of the short call options. If the delta were negative (for example, for a short put option), the fund would need to sell shares. The fund currently holds 1000 shares. To achieve a delta-neutral position, they need to buy an additional number of shares. Additional shares to buy = Number of shares needed for hedge – Current number of shares Additional shares to buy = 3000 – 1000 = 2000 Therefore, the fund needs to buy 2000 shares to delta-hedge their short call option position. A crucial aspect of delta hedging is its dynamic nature. The delta of an option changes as the underlying asset’s price changes and as time passes (theta). Therefore, the hedge needs to be adjusted periodically to maintain a delta-neutral position. This process is known as dynamic hedging. Consider a scenario where the price of the underlying asset increases significantly. The delta of the call options would also increase, requiring the fund to buy more shares to maintain the hedge. Conversely, if the price of the underlying asset decreases, the delta of the call options would decrease, and the fund would need to sell some shares. Delta hedging is not a perfect hedge. It only protects against small price movements in the underlying asset. Large price movements can still result in losses. Also, delta hedging does not protect against changes in volatility (vega) or the passage of time (theta).