Derivatives Level 4 (Investment Advice Diploma) Exam Quiz Quiz 36

The question assesses the understanding of the impact of correlation on portfolio variance when using derivatives for hedging. Portfolio variance is a measure of the overall risk in a portfolio. The lower the variance, the less volatile the portfolio. When two assets are perfectly positively correlated (correlation coefficient = +1), the variance of the portfolio is simply the weighted average of the individual asset variances. When assets are uncorrelated (correlation coefficient = 0), the portfolio variance is reduced. The greatest risk reduction (lowest variance) occurs when the assets are perfectly negatively correlated (correlation coefficient = -1). In this scenario, the derivative is used to hedge the underlying asset. The effectiveness of the hedge depends on the correlation between the asset and the hedging instrument (derivative). A perfect hedge (correlation of -1) would theoretically eliminate all risk (variance = 0). In reality, perfect hedges are rare due to basis risk and other market imperfections. The formula for portfolio variance with two assets is: \[ \sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho_{1,2}\sigma_1\sigma_2 \] where: \(\sigma_p^2\) is the portfolio variance, \(w_1\) and \(w_2\) are the weights of asset 1 and asset 2 in the portfolio, \(\sigma_1\) and \(\sigma_2\) are the standard deviations of asset 1 and asset 2, and \(\rho_{1,2}\) is the correlation between asset 1 and asset 2. In our case, asset 1 is the original portfolio and asset 2 is the derivative used for hedging. The goal is to minimize \(\sigma_p^2\). When the correlation (\(\rho_{1,2}\)) is -1, the portfolio variance is minimized. The hedge ratio is the ratio of the size of the hedge position to the size of the underlying asset position. An optimal hedge ratio minimizes portfolio variance.

The question assesses the understanding of credit default swaps (CDS) and their role in managing credit risk, particularly in the context of a bespoke portfolio. The crucial aspect is to determine the breakeven spread, which represents the point where the expected cost of protection equals the expected payout from the CDS in case of default. This involves calculating the expected loss from the bond portfolio, considering the probabilities of default and the loss given default (LGD), and then annualizing this expected loss to find the breakeven CDS spread. First, calculate the expected loss for each bond: Bond A: Expected Loss = Default Probability * LGD * Notional = 0.02 * 0.4 * £5,000,000 = £40,000 Bond B: Expected Loss = Default Probability * LGD * Notional = 0.05 * 0.6 * £3,000,000 = £90,000 Bond C: Expected Loss = Default Probability * LGD * Notional = 0.01 * 0.5 * £2,000,000 = £10,000 Total Expected Loss = £40,000 + £90,000 + £10,000 = £140,000 The total notional value of the portfolio is £5,000,000 + £3,000,000 + £2,000,000 = £10,000,000 Breakeven CDS Spread = (Total Expected Loss / Total Notional) * 10,000 (to convert to basis points) Breakeven CDS Spread = (£140,000 / £10,000,000) * 10,000 = 140 basis points The breakeven spread represents the annualized cost of protection that equates to the expected loss from defaults within the portfolio. In simpler terms, imagine you’re insuring a collection of valuable items. The breakeven “spread” is like the annual insurance premium you’d need to charge to cover the expected cost of replacing those items if they were damaged or stolen. If you charge less, you’ll likely lose money. If you charge more, you’ll make a profit, but might lose customers to cheaper insurers. A higher breakeven spread indicates a riskier portfolio, implying a greater likelihood of defaults and/or higher potential losses given default. Conversely, a lower breakeven spread suggests a safer portfolio with lower default probabilities and/or smaller losses in the event of default. The concept of breakeven spread is crucial for understanding the pricing of credit protection and for making informed decisions about hedging credit risk. It provides a quantitative measure of the underlying credit risk within a portfolio, allowing investors and risk managers to compare the cost of protection against the expected benefits.

The question assesses understanding of delta hedging, gamma, and their impact on portfolio rebalancing. Delta represents the sensitivity of an option’s price to a change in the underlying asset’s price. Gamma represents the rate of change of delta with respect to the underlying asset’s price. A delta-neutral portfolio is constructed to be insensitive to small changes in the underlying asset’s price. However, gamma introduces complexity. As the underlying asset’s price moves, the delta changes, requiring the portfolio to be rebalanced to maintain delta neutrality. The cost of rebalancing depends on gamma and the magnitude of the price change. The formula to approximate the change in portfolio value due to gamma is: Change in Portfolio Value ≈ 0.5 * Gamma * (Change in Underlying Asset Price)^2 * Number of Options. In this scenario, we need to calculate the profit or loss from the option position alone, then factor in the cost of rebalancing to maintain delta neutrality due to the option’s gamma. First, calculate the profit/loss from the options: The underlying asset increased from £150 to £157. The option has a delta of 0.60, so the option value changes by 0.60 * £7 = £4.20 per option. The portfolio contains 1000 options, so the profit from the options is 1000 * £4.20 = £4200. Next, calculate the rebalancing cost due to gamma: The gamma is 0.05. The change in the underlying asset price is £7. The rebalancing cost is approximately 0.5 * 0.05 * (£7)^2 * 1000 = £1225. Finally, calculate the net profit/loss: The profit from the options is £4200, and the rebalancing cost is £1225. The net profit is £4200 – £1225 = £2975.

This question delves into the complexities of exotic options, specifically focusing on barrier options and their sensitivity to market volatility and knock-out events. The key is understanding how the presence of a barrier significantly alters the option’s payoff profile compared to a standard vanilla option. A down-and-out call option becomes worthless if the underlying asset’s price touches or falls below the barrier level before expiration. Therefore, its value is inherently tied to the probability of the asset price remaining above the barrier. To solve this, we need to consider the impact of increased volatility on the probability of the barrier being breached. Higher volatility increases the likelihood of the asset price fluctuating significantly, thereby increasing the probability of the asset price hitting the barrier. If the barrier is hit, the option is knocked out and becomes worthless. This contrasts with a standard call option, where increased volatility generally increases the option’s value because it widens the potential range of the underlying asset’s price at expiration, increasing the chance of a profitable outcome. In this scenario, the initial assessment of the down-and-out call option was based on a specific volatility level. When the market experiences a sudden spike in volatility, the probability of the asset price reaching the barrier increases. This increased probability of the knock-out event reduces the value of the down-and-out call option. Therefore, the fund manager needs to adjust their hedging strategy to account for the reduced value of the option. This might involve increasing the hedge ratio to compensate for the greater sensitivity of the option to price movements near the barrier. The fund manager’s initial strategy likely involved selling the down-and-out call option to generate income or hedge against a specific market scenario. The spike in volatility necessitates a reassessment of the risk profile and an adjustment to maintain the desired level of protection or income generation. They may need to purchase back some of the sold options to reduce their exposure to the increased risk of the barrier being breached, or implement a more complex hedging strategy involving other derivatives.

The question assesses the understanding of put-call parity and how dividends affect it. Put-call parity is a fundamental concept linking the prices of European call and put options with the same strike price and expiration date. The basic formula is: Call Price – Put Price = Stock Price – Present Value of Strike Price. However, when dividends are involved, the formula needs adjustment. The stock price must be reduced by the present value of the dividends to be paid during the option’s life. This adjustment reflects the fact that the option holder will not receive these dividends. In this scenario, calculating the present value of the dividends is crucial. Dividend 1: £1.00 payable in 3 months. Discounted back to today at a continuously compounded risk-free rate of 5% per annum: \( PV_1 = 1.00 \times e^{-0.05 \times 0.25} = 1.00 \times e^{-0.0125} \approx 0.9876 \) Dividend 2: £1.50 payable in 9 months. Discounted back to today at a continuously compounded risk-free rate of 5% per annum: \( PV_2 = 1.50 \times e^{-0.05 \times 0.75} = 1.50 \times e^{-0.0375} \approx 1.4443 \) Total present value of dividends: \( PV_{div} = 0.9876 + 1.4443 = 2.4319 \) Adjusted Put-Call Parity Formula: Call Price – Put Price = Stock Price – Present Value of Strike Price – Present Value of Dividends \[ C – P = S_0 – Ke^{-rT} – PV_{div} \] Where: \( C \) = Call Price \( P \) = Put Price \( S_0 \) = Current Stock Price = £50 \( K \) = Strike Price = £52 \( r \) = Risk-free rate = 5% \( T \) = Time to expiration = 1 year \( PV_{div} \) = Present Value of Dividends = £2.4319 Substituting the values: \[ 6 – P = 50 – 52 \times e^{-0.05 \times 1} – 2.4319 \] \[ 6 – P = 50 – 52 \times e^{-0.05} – 2.4319 \] \[ 6 – P = 50 – 52 \times 0.9512 – 2.4319 \] \[ 6 – P = 50 – 49.4624 – 2.4319 \] \[ 6 – P = -1.8943 \] \[ P = 6 + 1.8943 \] \[ P = 7.8943 \] Therefore, the fair price of the put option is approximately £7.89.

Let’s analyze how Gamma hedging works in a portfolio, and how changes in volatility impact the effectiveness of that hedge. Gamma measures the rate of change of an option’s delta with respect to changes in the underlying asset’s price. A portfolio with a high gamma is very sensitive to price movements. Gamma hedging aims to neutralize this sensitivity. When volatility increases, the gamma of options typically increases, especially for at-the-money options. This means our hedge needs to be rebalanced more frequently to maintain its effectiveness. The cost of rebalancing is directly related to the size of the gamma and the magnitude of the price movements. Consider a portfolio manager who is short a large number of at-the-money call options. The manager initially hedges this position by buying the underlying asset. However, the manager wants to ensure that the hedge remains effective even if volatility spikes unexpectedly. The initial hedge ratio is calculated based on the initial delta of the options. As volatility increases, the delta changes more rapidly, increasing the need for more frequent rebalancing. The cost of rebalancing is calculated as follows: 1. Calculate the initial gamma (\(\Gamma\)) of the portfolio. 2. Estimate the expected change in volatility (\(\Delta \sigma\)). 3. Estimate the expected price movement of the underlying asset (\(\Delta S\)). 4. Calculate the change in delta (\(\Delta \delta = \Gamma \times \Delta S\)). 5. Determine the number of shares to rebalance (\(\Delta \delta \times \text{Portfolio Size}\)). 6. Estimate the cost per transaction (\(C\)). 7. Calculate the total rebalancing cost (\(\text{Rebalancing Cost} = |\Delta \delta \times \text{Portfolio Size}| \times C\)). In our example: 1. \(\Gamma = 0.05\) 2. \(\Delta \sigma = 0.10\) (Volatility increases by 10%) 3. \(\Delta S = 5\) (Price moves by £5) 4. \(\Delta \delta = 0.05 \times 5 = 0.25\) 5. Portfolio Size = 10,000 options 6. Number of shares to rebalance \( = 0.25 \times 10,000 = 2,500 \) shares 7. \(C = £0.50\) per share 8. \(\text{Rebalancing Cost} = |2,500| \times 0.50 = £1,250\) The cost of rebalancing the hedge due to the increased volatility and price movement is £1,250.

The core of this question revolves around understanding how various factors, particularly interest rate volatility and time to expiration, influence the prices of different types of options (European call and put) and how these sensitivities impact hedging strategies. We will calculate the theoretical price change of the portfolio using the provided sensitivities (Delta, Gamma, Vega) and the given market movements. First, calculate the impact of the interest rate volatility change on the portfolio value: Vega = -£25,000 per 1% change in implied volatility Change in implied volatility = 0.5% Impact of Vega = Vega * Change in volatility = -£25,000 * 0.5% = -£12,500 Second, calculate the impact of the change in the underlying asset’s price: Change in underlying asset price = £2 Delta = 15,000 Gamma = -5,000 The first-order impact (Delta) = Delta * Change in asset price = 15,000 * £2 = £30,000 The second-order impact (Gamma) = 0.5 * Gamma * (Change in asset price)^2 = 0.5 * -5,000 * (£2)^2 = -£10,000 Total impact due to asset price change = £30,000 – £10,000 = £20,000 Finally, calculate the total change in the portfolio value: Total change = Impact of Vega + Total impact due to asset price change = -£12,500 + £20,000 = £7,500 The portfolio’s value is expected to increase by £7,500. Now, let’s consider why the other options are incorrect. Option B incorrectly assumes that Vega has a positive relationship with portfolio value, reversing the sign of the volatility impact. Option C neglects the Gamma effect, providing only the linear approximation from Delta, and also reverses the sign of the volatility impact. Option D correctly calculates the impact of Vega and Delta but incorrectly calculates the impact of Gamma, and thus arrives at an incorrect total change. This question tests the candidate’s ability to combine the effects of Delta, Gamma, and Vega to estimate portfolio value changes, a crucial skill in derivatives risk management. The values used are designed to be realistic and relevant to a large portfolio managed by a derivatives trading firm.

The core concept tested here is understanding the impact of volatility on option prices, specifically focusing on how gamma changes with moneyness and time to expiration. Gamma represents the rate of change of an option’s delta with respect to changes in the underlying asset’s price. Options that are at-the-money (ATM) have the highest gamma because their delta is most sensitive to price changes in the underlying asset. As an option moves further in-the-money (ITM) or out-of-the-money (OTM), its gamma decreases. Time to expiration also affects gamma. As the expiration date approaches, gamma generally increases for ATM options, meaning their delta becomes more sensitive to price changes. This is because the probability of the option ending up ITM or OTM becomes more binary as expiration nears. Conversely, for deeply ITM or OTM options, gamma decreases as expiration approaches because their delta is already close to 1 or 0, respectively, and less sensitive to price changes. In this scenario, we must consider the combined effects of moneyness and time to expiration. The fund manager needs to dynamically adjust their hedging strategy based on how gamma changes to maintain a delta-neutral portfolio. If gamma increases, they need to rebalance their portfolio more frequently to offset the increased sensitivity of the option’s delta to price changes in the underlying asset. If gamma decreases, they can rebalance less frequently. The correct action will depend on whether the options are ATM or away from ATM. If the options are ATM, and the expiration date is approaching, the gamma will increase, and the fund manager will need to rebalance more frequently. If the options are deeply ITM or OTM, and the expiration date is approaching, the gamma will decrease, and the fund manager can rebalance less frequently.

The question assesses the understanding of delta hedging and portfolio rebalancing. Delta (Δ) represents the sensitivity of an option’s price to a change in the underlying asset’s price. Delta hedging involves creating a position in the underlying asset to offset the delta of the option portfolio, aiming for a delta-neutral position. This strategy minimizes the portfolio’s sensitivity to small price movements in the underlying asset. Rebalancing is crucial because delta changes as the underlying asset’s price and time to expiration change. Gamma (Γ) measures the rate of change of delta with respect to the underlying asset’s price. A high gamma indicates that the delta will change rapidly, requiring more frequent rebalancing. Theta (Θ) measures the rate of change of the option’s price with respect to time. As time passes, theta erodes the option’s value, affecting the delta. The calculation involves determining the initial hedge position and then calculating the number of shares needed to rebalance the portfolio after the underlying asset’s price changes. 1. **Initial Hedge:** The portfolio has a delta of -500,000. To neutralize this, the fund needs to buy 500,000 shares of the underlying asset. 2. **Price Change Impact:** The underlying asset’s price increases from £50 to £52. 3. **New Delta:** The portfolio delta changes to -400,000. 4. **Rebalancing:** The fund now needs to reduce its holdings to match the new delta. The reduction required is 500,000 (initial) – 400,000 (new) = 100,000 shares. Therefore, the fund needs to sell 100,000 shares to rebalance the delta-hedged portfolio. This example demonstrates how continuous monitoring and rebalancing are essential in delta hedging to maintain a near-zero delta position, especially in volatile markets. The frequency of rebalancing depends on the portfolio’s gamma and the desired level of risk management. A higher gamma necessitates more frequent rebalancing to maintain the hedge’s effectiveness.

The question assesses the understanding of the Greeks, specifically Gamma, and how it impacts hedging strategies, particularly in scenarios involving earnings announcements and options trading. Gamma measures the rate of change of an option’s delta with respect to changes in the underlying asset’s price. A high Gamma indicates that the delta is highly sensitive to price changes, making hedging more complex and requiring frequent adjustments. The scenario involves an investor holding 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 if the underlying asset price remains stable. However, it carries significant risk if the price moves substantially in either direction. Earnings announcements are typically periods of high volatility, which can significantly impact option prices and the overall position. The investor’s primary concern is to manage the risk associated with the short straddle position, especially given the impending earnings announcement. The question specifically asks about the adjustments needed to maintain a delta-neutral position. A delta-neutral position means the portfolio’s delta is zero, making it insensitive to small changes in the underlying asset’s price. Given the high Gamma of the options, the delta will change rapidly as the underlying asset’s price fluctuates, especially around the earnings announcement. To maintain delta neutrality, the investor needs to dynamically adjust their position by buying or selling the underlying asset. The direction and magnitude of these adjustments depend on the direction and magnitude of the price changes. If the price of the underlying asset increases, the delta of the short call option becomes more negative, and the delta of the short put option becomes less negative (or even positive). To offset this, the investor needs to sell shares of the underlying asset. Conversely, if the price decreases, the delta of the short call option becomes less negative, and the delta of the short put option becomes more negative. To offset this, the investor needs to buy shares of the underlying asset. The frequency and size of these adjustments are critical, especially with high Gamma. Waiting too long or making insufficient adjustments can expose the investor to significant losses if the price moves sharply. The earnings announcement exacerbates this risk due to the potential for large, sudden price movements. The investor’s strategy must therefore involve close monitoring of the underlying asset’s price and rapid, precise adjustments to the position to maintain delta neutrality. The goal is to minimize the impact of price fluctuations on the portfolio’s value, thereby mitigating the risk associated with the short straddle.

The core of this question revolves around understanding how volatility skew affects option pricing and, consequently, hedging strategies involving strangles. Volatility skew refers to the phenomenon where out-of-the-money (OTM) puts and calls have different implied volatilities. Typically, OTM puts exhibit higher implied volatilities than OTM calls, creating a “skew” or “smile” in the implied volatility curve. This skew reflects market participants’ greater demand for downside protection (buying puts) compared to upside participation (buying calls), driven by fear of market crashes. A strangle involves buying both an OTM put and an OTM call with the same expiration date. The profit from a strangle is realized when the underlying asset price moves significantly in either direction, exceeding the combined premiums paid for the options. When volatility skew is present, the put option in the strangle will be relatively more expensive than it would be in a skew-neutral environment. This higher premium reflects the higher implied volatility and greater perceived risk of a downside move. Conversely, the call option will be relatively cheaper. Now, consider the scenario where an advisor believes the market will remain stable. Selling a strangle is a strategy to profit from time decay if the underlying asset price stays within the range defined by the strike prices of the options. However, the presence of volatility skew alters the risk-reward profile. Since the put is more expensive, the initial premium received from selling the strangle is higher than it would be without skew. However, the potential loss if the market crashes is also magnified because the put’s value will increase disproportionately due to both the price movement and the volatility increase. The breakeven points for a strangle are calculated as: Upper Breakeven = Call Strike + Net Premium Received Lower Breakeven = Put Strike – Net Premium Received In a skewed market, the higher premium received (due to the expensive put) widens the breakeven range, making the strategy appear safer initially. However, the increased downside risk due to the higher implied volatility of the put makes the lower breakeven point more vulnerable. Therefore, the advisor needs to carefully consider the magnitude of the skew and its potential impact on the downside risk. A naive application of strangle strategies without considering the volatility skew can lead to unexpected and substantial losses, especially if the market experiences a significant downturn. The advisor should assess whether the additional premium received adequately compensates for the heightened downside risk implied by the volatility skew. This involves analyzing the implied volatility surface, stress-testing the position under various market scenarios, and potentially adjusting the strike prices or using other hedging techniques to mitigate the skew risk.

To determine the fair value of the swaption, we need to calculate the present value of the expected future swap payments. This involves several steps: 1. **Calculate the Forward Swap Rate:** The forward swap rate is the fixed rate that makes the present value of the fixed leg equal to the present value of the floating leg at the initiation of the swap. It is calculated using the formula: \[ \text{Forward Swap Rate} = \frac{1 – DF_n}{\sum_{i=1}^{n} DF_i} \] Where \(DF_i\) is the discount factor for period \(i\), and \(n\) is the number of periods. In this case, the discount factors are 0.9804, 0.9612, 0.9423, 0.9238, and 0.9057 for years 1 through 5 respectively. \[ \text{Forward Swap Rate} = \frac{1 – 0.9057}{0.9804 + 0.9612 + 0.9423 + 0.9238 + 0.9057} = \frac{0.0943}{4.7134} \approx 0.020006 \] So, the forward swap rate is approximately 2.0006%. 2. **Calculate the Expected Payoff:** The swaption gives the holder the right to enter into a swap where they pay a fixed rate of 1.5% and receive a floating rate. The expected payoff is based on the difference between the forward swap rate and the fixed rate of the swaption, multiplied by the notional principal and the swap term. Expected Payoff = Notional Principal \* (Forward Swap Rate – Swaption Fixed Rate) \* Swap Term Expected Payoff = £10,000,000 \* (0.020006 – 0.015) \* 5 = £10,000,000 \* 0.005006 \* 5 = £250,300 3. **Discount the Expected Payoff:** The expected payoff needs to be discounted back to the present value using the appropriate discount factor for the swaption’s expiry (1 year). Present Value = Expected Payoff \* Discount Factor Present Value = £250,300 \* 0.9804 = £245,394.12 Therefore, the fair value of the swaption is approximately £245,394.12. This valuation assumes a simplified model and does not account for volatility or potential early exercise. In a real-world scenario, more sophisticated models like Black-Scholes or Monte Carlo simulations would be employed to account for these factors. Consider a scenario where a fund manager uses a swaption to hedge against rising interest rates. If rates rise above 1.5%, the swaption becomes valuable, offsetting losses in their fixed-income portfolio. Conversely, if rates stay below 1.5%, they only lose the premium paid for the swaption, limiting their downside. This example illustrates the strategic use of swaptions in risk management.

The question explores the complexities of hedging a non-linear payoff profile (specifically, a digital option payout) using a combination of standard options. The core challenge is that a digital option’s payoff is binary: either zero or a fixed amount, making it highly sensitive to price movements near the strike price. This sensitivity requires a dynamic hedging strategy. To replicate a digital call option, one approach involves a *call spread*. A call spread consists of buying a call option at a lower strike price (K1) and selling a call option at a higher strike price (K2), where K2 is slightly above K1. As the underlying asset price approaches K1, the purchased call option gains value. As the price moves past K2, the sold call option limits the profit, effectively mimicking the capped payout of a digital call. The narrower the spread (i.e., K2 is closer to K1), the better the approximation of the digital option’s binary payoff. The *Gamma* of an option measures the rate of change of its Delta with respect to changes in the underlying asset price. For a standard call or put option, Gamma is highest when the option is near the money. A digital option has a very high Gamma right at the strike price, reflecting the sharp change in payoff. The call spread attempts to replicate this Gamma profile. A butterfly spread, involving buying one call at a lower strike, selling two calls at a middle strike, and buying one call at a higher strike, provides a more refined way to match the Gamma profile. The question also involves the concept of *vega*. Vega measures the sensitivity of an option’s price to changes in volatility. Digital options, particularly those near the money, are highly sensitive to volatility. Since volatility is rarely constant, changes in implied volatility can significantly impact the hedging strategy. The correct answer will acknowledge the use of a call spread to approximate the digital option payoff, the dynamic adjustment required due to Gamma exposure, and the impact of changing volatility (Vega) on the hedge.

The core concept here is the application of the Black-Scholes model in a scenario where the underlying asset’s volatility is expected to change *before* the option’s expiration. The standard Black-Scholes model assumes constant volatility over the option’s life. When volatility is expected to shift, we need to calculate the option price using a weighted average of the volatilities for each period. First, we need to calculate the time-weighted average variance. The initial volatility is 20% (0.20) for the first 6 months (0.5 years), and then it shifts to 25% (0.25) for the remaining 6 months (0.5 years). The variance is the square of the volatility. Weighted average variance = \[(0.5 \times 0.20^2) + (0.5 \times 0.25^2) = (0.5 \times 0.04) + (0.5 \times 0.0625) = 0.02 + 0.03125 = 0.05125\] Next, we calculate the weighted average volatility by taking the square root of the weighted average variance. Weighted average volatility = \[\sqrt{0.05125} \approx 0.2264\] or 22.64% We then use this weighted average volatility (22.64%) as the volatility input in the Black-Scholes model. While we don’t explicitly calculate the Black-Scholes price here (as the question asks for the *adjusted* volatility), understanding this adjusted volatility is crucial for accurate option pricing under changing volatility conditions. The adjusted volatility is used *instead* of either the initial or final volatilities in the Black-Scholes formula. The Black-Scholes model is used to determine the fair price of a European-style call or put option. The formula considers the current stock price, the option’s strike price, the time until expiration, the risk-free interest rate, and the volatility of the underlying asset. Since the volatility is changing, we need to adjust it before we apply the formula. If the volatility changes during the life of the option, we use the weighted average volatility in the Black-Scholes model. For instance, consider a farmer using a forward contract to sell wheat. Initially, market volatility is low due to stable weather forecasts. However, a sudden report of potential drought increases volatility. The farmer must adjust their pricing strategy to reflect this increased uncertainty. Similarly, a fund manager hedging a portfolio with options must adjust their volatility expectations based on macroeconomic announcements, as these events can significantly impact market volatility.

The question tests understanding of how regulatory capital requirements, specifically the leverage ratio, impact a bank’s derivative trading activities and overall capital adequacy. The leverage ratio, calculated as Tier 1 capital divided by total exposure (including on- and off-balance sheet items like derivatives), sets a minimum capital standard regardless of risk weighting. The bank’s initial Tier 1 capital is £500 million. The initial leverage ratio is calculated as Tier 1 capital / Total Exposure = £500 million / £10 billion = 5%. The minimum regulatory requirement is 4%. The bank enters into a new derivative transaction that increases its total exposure by £2 billion. The new total exposure is £10 billion + £2 billion = £12 billion. To maintain the minimum 4% leverage ratio, the bank needs to have Tier 1 capital of at least 4% of its total exposure. Therefore, required Tier 1 capital = 0.04 * £12 billion = £480 million. The bank’s current Tier 1 capital is £500 million, which is greater than the required £480 million. Therefore, the bank does not need to raise additional capital. The excess Tier 1 capital is £500 million – £480 million = £20 million. The key concept here is that the leverage ratio acts as a constraint on balance sheet expansion, even if risk-weighted assets are well-capitalized. The question requires calculating the impact of a new derivative transaction on the leverage ratio and determining if additional capital is needed to comply with regulations. Understanding the formula and its implications for capital management is essential.

The question revolves around the concept of delta hedging a short call option position and the subsequent adjustments required due to changes in the underlying asset’s price and the passage of time. Delta, Gamma, and Theta are key risk measures for options. Delta represents the sensitivity of the option’s price to changes in the underlying asset’s price. Gamma represents the rate of change of the delta with respect to changes in the underlying asset’s price. Theta represents the sensitivity of the option’s price to the passage of time. Initially, the fund sells call options and hedges by buying shares of the underlying asset to offset the negative delta of the short call position. As the asset price changes, the delta of the call option also changes (Gamma effect), requiring the fund to rebalance its hedge by buying or selling more shares. As time passes, the option’s value decays (Theta effect), impacting the overall hedge. The calculation involves several steps: 1. **Initial Hedge:** Calculate the initial number of shares to buy to hedge the short call options using the initial delta. 2. **Price Change Impact:** Determine the new delta of the call option after the price increase, considering the Gamma. 3. **Hedge Adjustment:** Calculate the number of shares to buy or sell to adjust the hedge to the new delta. 4. **Theta Impact:** Assess how the passage of time (Theta) affects the option’s value and the overall hedge. Since Theta reduces the value of the option, it effectively reduces the hedge requirement. 5. **Net Adjustment:** Combine the adjustments due to price change and time decay to determine the final number of shares to buy or sell. In this case, the fund initially sells 100,000 call options with a delta of 0.40. The fund buys 40,000 shares to hedge. The price increases, and the delta increases to 0.45. The fund needs to buy an additional 5,000 shares. However, one day passes, and the Theta is -0.02. This means the delta decreases by 0.02 to 0.43. The fund only needs to hedge for a delta of 0.43. The fund needs to buy 3,000 additional shares. Calculation: 1. Initial shares: 100,000 * 0.40 = 40,000 2. Delta increase: 0.45 – 0.40 = 0.05 3. Shares due to delta increase: 100,000 * 0.05 = 5,000 4. Delta decrease due to theta: 0.45 – 0.02 = 0.43 5. Shares due to theta: 100,000 * (0.43 – 0.40) = 3,000 6. Shares needed to buy: 3,000

The question assesses the understanding of hedging strategies using futures contracts, specifically in the context of basis risk. Basis risk arises because the price of the asset being hedged (in this case, Grade A Arabica coffee beans) and the price of the futures contract (ICE Coffee “C” futures) may not move perfectly in tandem. This imperfect correlation can lead to hedging imperfections. The optimal hedge ratio minimizes the variance of the hedged portfolio. The hedge ratio is calculated as: Hedge Ratio = (Correlation between spot and futures price changes) * (Standard deviation of spot price changes) / (Standard deviation of futures price changes). In this scenario, the correlation is 0.75, the standard deviation of spot price changes is £0.06/lb, and the standard deviation of futures price changes is £0.08/lb. Therefore, the hedge ratio is (0.75 * 0.06) / 0.08 = 0.5625. This means that for every pound of coffee beans the roaster wants to hedge, they should short 0.5625 futures contracts. Since the roaster wants to hedge 200,000 lbs of coffee beans, they should short 200,000 * 0.5625 = 112,500 futures contracts. Given that each ICE Coffee “C” futures contract represents 37,500 lbs of coffee, the roaster needs to short 112,500 / 37,500 = 3 contracts. The closest answer is 3 contracts. This demonstrates a practical application of calculating the hedge ratio and determining the number of futures contracts needed for hedging, considering basis risk. The concept of basis risk is crucial as it highlights that even with hedging, there is still some residual risk due to the imperfect correlation between the spot and futures prices. Understanding this allows for more informed risk management decisions.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their behavior near the barrier. A down-and-out call option becomes worthless if the underlying asset price hits the barrier before expiration. Therefore, as the price approaches the barrier, the option’s value diminishes, reflecting the increased probability of the barrier being breached. The calculation involves understanding how the option’s value changes incrementally as the underlying asset price moves closer to the barrier, considering the volatility and time to expiration. We are looking for the approximate change in option value for a small movement in the underlying asset price when it is near the barrier. The initial option value is £5. The barrier is at £90, and the current price is £91.5. If the price falls to £90.5, it is very close to the barrier. The option is a down-and-out call, so if the barrier is hit, the option expires worthless. Since the price is very close to the barrier, a small downward movement significantly increases the probability of the barrier being hit. Thus, the option value will decrease substantially. We can estimate the change in value by considering the risk-neutral probability of hitting the barrier. Let’s assume that the option value decreases linearly as it approaches the barrier. The price moved £1 (from £91.5 to £90.5) closer to the barrier. If the barrier is hit, the option becomes worthless (value = £0). The initial value was £5. A reasonable estimate is that the option value decreases by approximately £3.00, reflecting the increased likelihood of the barrier being hit.

The question explores the complexities of pricing a bespoke exotic derivative, specifically a barrier option with a time-dependent barrier level, incorporating stochastic interest rates. The calculation requires understanding of risk-neutral valuation, Monte Carlo simulation, and the application of the Hull-White model for interest rate dynamics. First, we need to simulate the underlying asset price and the interest rate path. The underlying asset price follows a geometric Brownian motion: \[ dS_t = \mu S_t dt + \sigma S_t dW_t \] where \(S_t\) is the asset price, \(\mu\) is the drift, \(\sigma\) is the volatility, and \(dW_t\) is a Wiener process. The Hull-White model for the short rate \(r_t\) is: \[ dr_t = a(b(t) – r_t)dt + \sigma_r dZ_t \] where \(a\) is the mean reversion rate, \(b(t)\) is the time-dependent mean reversion level, \(\sigma_r\) is the volatility of the short rate, and \(dZ_t\) is another Wiener process (correlated with \(dW_t\)). We simulate these processes using Euler discretization: \[ S_{t+\Delta t} = S_t \exp\left( (\mu – \frac{1}{2}\sigma^2)\Delta t + \sigma \sqrt{\Delta t} Z_1 \right) \] \[ r_{t+\Delta t} = r_t + a(b(t) – r_t)\Delta t + \sigma_r \sqrt{\Delta t} Z_2 \] where \(Z_1\) and \(Z_2\) are standard normal random variables with correlation \(\rho\). The time-dependent barrier is given by \(B(t) = B_0 e^{\alpha t}\), where \(B_0\) is the initial barrier level and \(\alpha\) is the barrier growth rate. For each simulated path, we check if the asset price \(S_t\) hits the barrier \(B(t)\) at any time \(t\) before maturity \(T\). If the barrier is hit, the option expires worthless. If the barrier is not hit, the payoff at maturity is \( \max(S_T – K, 0) \), where \(K\) is the strike price. We discount the payoff back to time 0 using the simulated interest rate path. The discount factor is: \[ DF = \exp\left( -\int_0^T r_t dt \right) \approx \exp\left( -\sum_{i=1}^{N} r_{t_i} \Delta t \right) \] where \(N\) is the number of time steps. The option price is the average of the discounted payoffs over all simulated paths: \[ C_0 = \frac{1}{M} \sum_{i=1}^{M} DF_i \cdot \max(S_{T,i} – K, 0) \cdot \mathbb{I}_{\{\text{barrier not hit}\}} \] where \(M\) is the number of simulated paths and \(\mathbb{I}\) is an indicator function. Given the parameters: \(S_0 = 100\), \(K = 105\), \(T = 1\), \(\sigma = 0.2\), \(r_0 = 0.05\), \(a = 0.1\), \(\sigma_r = 0.01\), \(\rho = 0.7\), \(B_0 = 80\), \(\alpha = 0.02\), and using 10,000 simulations with 100 time steps, we obtain an estimated option price. Assume the simulation yields an average discounted payoff of 6.23.

The question assesses understanding of hedging strategies using futures contracts, particularly in the context of basis risk. Basis risk arises because the price of the asset being hedged (spot price) and the price of the futures contract do not always move perfectly in tandem. The hedge ratio is crucial for minimizing this risk. 1. **Calculate the Naive Hedge Ratio:** This is simply the ratio of the value of the exposure being hedged to the value of one futures contract. In this case, it’s £9,000,000 / (£1,800 x 500) = 10. 2. **Calculate the Optimal Hedge Ratio:** This takes into account the correlation between the changes in the spot price and the changes in the futures price. The formula for the optimal hedge ratio is: Optimal Hedge Ratio = Correlation Coefficient \* (Standard Deviation of Spot Price Changes / Standard Deviation of Futures Price Changes) Given values: Correlation Coefficient = 0.8, Standard Deviation of Spot Price Changes = 0.015, Standard Deviation of Futures Price Changes = 0.02 Optimal Hedge Ratio = 0.8 \* (0.015 / 0.02) = 0.6 3. **Calculate the Number of Contracts for the Optimal Hedge:** Multiply the naive hedge ratio by the optimal hedge ratio: Number of Contracts = Naive Hedge Ratio \* Optimal Hedge Ratio = 10 \* 0.6 = 6 Therefore, to minimize basis risk, the fund manager should use 6 futures contracts. A crucial aspect of hedging is understanding that it’s about reducing risk, not necessarily maximizing profit. The optimal hedge ratio is designed to minimize the variance of the hedged position, which includes both the asset being hedged and the hedging instrument (futures contracts). The correlation between the spot and futures prices is a critical determinant of the hedge ratio. A lower correlation implies a greater need to reduce the hedge to account for basis risk. Stress testing and scenario analysis are essential to evaluate the effectiveness of the hedge under different market conditions, including extreme scenarios where the correlation may break down. This question highlights the practical application of risk management principles in derivative markets, emphasizing the importance of adjusting hedge ratios based on market dynamics.

The question explores the application of put-call parity in a market where transaction costs exist. Put-call parity is a fundamental concept in options pricing, stating a relationship between the prices of a European call option, a European put option, the underlying asset, and a risk-free bond, all with the same strike price and expiration date. The formula is: \(C + PV(X) = P + S\), where C is the call option price, PV(X) is the present value of the strike price, P is the put option price, and S is the current price of the underlying asset. However, real-world markets aren’t frictionless. Transaction costs, such as brokerage fees and bid-ask spreads, can disrupt this theoretical parity. Arbitrageurs exploit deviations from put-call parity to generate risk-free profits. But with transaction costs, the profit must exceed these costs for the arbitrage to be worthwhile. The question challenges the candidate to determine the arbitrage strategy and the minimum profit required to overcome these costs. In this scenario, the observed market prices deviate from the theoretical put-call parity. To identify the arbitrage, we must first calculate the theoretical put-call parity value and compare it to the market prices. If \(C + PV(X) > P + S\), the call and bond are overpriced relative to the put and stock. The arbitrageur would buy the put and stock and sell the call and bond. Conversely, if \(C + PV(X) < P + S\), the call and bond are underpriced relative to the put and stock, and the arbitrageur would buy the call and bond and sell the put and stock. The transaction costs are crucial. They reduce the profit from each transaction. In this case, the round-trip transaction cost is 0.5% of the stock price. This cost must be factored into the arbitrage profit calculation. The arbitrage profit must exceed this cost for the strategy to be viable. The problem-solving approach involves: 1) Calculating the present value of the strike price using the risk-free rate. 2) Determining the theoretical put-call parity value. 3) Comparing the theoretical value to the market prices to identify the mispricing. 4) Designing the appropriate arbitrage strategy (buying the relatively underpriced assets and selling the relatively overpriced assets). 5) Calculating the gross profit from the arbitrage. 6) Subtracting the transaction costs to find the net profit. 7) Ensuring that the net profit is positive and greater than the transaction costs. The question requires understanding of put-call parity, present value calculations, arbitrage strategies, and the impact of transaction costs on arbitrage opportunities. It goes beyond simple formula application and requires a practical understanding of market dynamics.

The core of this question revolves around understanding how hedging with futures contracts works in a real-world scenario, specifically focusing on basis risk and the impact of contract rollover. Basis risk arises because the spot price of an asset and the futures price are not perfectly correlated. This difference, known as the basis, can change over time, affecting the effectiveness of a hedge. Contract rollover involves closing out an expiring futures contract and simultaneously opening a new one with a later expiration date. This is necessary when the hedging period extends beyond the life of the initial futures contract. To calculate the effective price achieved by the hedge, we need to consider the following: 1. **Initial Hedge:** The farmer sells futures contracts to lock in a price. 2. **Basis at Initiation:** The difference between the spot price at the time of hedging and the futures price. 3. **Contract Rollover:** Closing the initial contract at a loss (or gain) and opening a new contract. 4. **Basis at Harvest:** The difference between the spot price at harvest and the price of the new futures contract. 5. **Effective Price:** The final price received, considering the initial futures sale, the rollover adjustment, and the final spot price received. Let’s break down the calculation: * **Initial Futures Sale:** The farmer sells at £250/tonne. * **Contract Rollover:** Closes at £240/tonne and opens a new contract at £260/tonne. This results in a loss of £10/tonne on the initial contract and a new short position at £260/tonne. * **Harvest:** The farmer sells the wheat at the spot price of £270/tonne. The new futures contract is closed at £280/tonne, resulting in a loss of £20/tonne (£260 – £280). * **Effective Price Calculation:** Effective Price = Spot Price at Harvest + (Initial Futures Price – Rollover Close Price) + (New Futures Open Price – Final Futures Price) Effective Price = £270 + (£250 – £240) + (£260 – £280) Effective Price = £270 + £10 – £20 = £260/tonne Therefore, the farmer effectively receives £260/tonne for the wheat. The basis risk and the rollover process impacted the final hedged price. A perfect hedge would have resulted in the initial futures price, but basis fluctuations and rollover losses reduced the effective price. This highlights the importance of understanding basis risk and rollover strategies when using futures for hedging. The example showcases a typical hedging scenario, emphasizing the complexities introduced by basis risk and contract rollovers. Understanding these nuances is crucial for effective risk management in derivatives trading. It demonstrates that hedging is not a perfect price lock, but rather a risk mitigation strategy that can be affected by market dynamics.

The core of this question revolves around understanding the interplay between macroeconomic indicators, specifically inflation expectations, and their impact on the pricing of interest rate swaps (IRS). An IRS is essentially a contract where two parties agree to exchange interest rate cash flows, one based on a fixed rate and the other on a floating rate (typically linked to a benchmark like LIBOR or SONIA). Inflation expectations are crucial because they influence the future path of interest rates. Higher inflation expectations generally lead to higher nominal interest rates, as lenders demand a premium to compensate for the erosion of purchasing power. In this scenario, we need to assess how a sudden upward revision in the Bank of England’s inflation forecast affects the fixed rate side of an IRS. The fixed rate in an IRS is determined by the market’s expectation of future floating rates over the swap’s tenor. If inflation is expected to be higher, the market will anticipate the central bank raising policy rates (e.g., the bank rate) to combat inflation. This, in turn, will push up the floating rate side of the swap. To compensate for the expected higher floating rates, the fixed rate side of the swap must also increase to maintain equilibrium. The magnitude of the increase in the fixed rate depends on several factors, including the size of the revision in the inflation forecast, the term structure of interest rates, and the market’s risk aversion. Let’s assume the market believes the Bank of England will raise rates by approximately half the increase in inflation expectation over the swap’s tenor. The 5-year IRS will be more sensitive to medium-term inflation expectations than a shorter-term swap. To calculate the approximate change, consider a simplified example: if the Bank of England revises its 5-year inflation forecast upward by 1.00%, the market might expect the average policy rate over the next five years to increase by roughly 0.50% (50 basis points). This increase would be reflected in the fixed rate of the 5-year IRS. We are assuming that the increase in the fixed rate will be roughly half of the increase in inflation expectation.

To solve this problem, we need to understand the implications of early exercise on American options, particularly in the context of dividend-paying stocks. An American call option allows the holder to exercise the option at any time before the expiration date. Early exercise is generally not optimal for call options on non-dividend-paying stocks because the option’s time value erodes more slowly than the interest earned on the strike price. However, when a stock pays a dividend, early exercise might be optimal if the dividend exceeds the time value lost by exercising early. Here’s the breakdown of the decision-making process: 1. **Calculate the present value of the dividends:** The stock pays two dividends: £2.00 in 3 months and £2.50 in 9 months. We need to discount these dividends back to the present to determine their present value. The risk-free interest rate is 5% per annum. * Present Value of Dividend 1 (in 3 months): \[ PV_1 = \frac{2.00}{1 + (0.05 \times \frac{3}{12})} = \frac{2.00}{1.0125} \approx 1.9753 \] * Present Value of Dividend 2 (in 9 months): \[ PV_2 = \frac{2.50}{1 + (0.05 \times \frac{9}{12})} = \frac{2.50}{1.0375} \approx 2.4096 \] * Total Present Value of Dividends: \[ PV_{Total} = PV_1 + PV_2 = 1.9753 + 2.4096 \approx 4.3849 \] 2. **Compare the present value of dividends with the potential gain from delaying exercise:** If Amelia waits until just before the first dividend payment in 3 months, she retains the option’s time value. The question implies that the option currently has a time value of £3.50. 3. **Assess the early exercise decision:** If Amelia exercises now, she receives the stock immediately, allowing her to capture both dividends. The present value of these dividends is £4.3849. If she waits, she retains the option’s time value of £3.50 but forgoes the dividends. Since £4.3849 (PV of dividends) > £3.50 (time value), early exercise appears beneficial. 4. **Consider the strike price and stock price:** Exercising now means paying the strike price of £95 and receiving the stock, which is currently trading at £97. The intrinsic value of the option is £97 – £95 = £2. 5. **Account for the opportunity cost:** By exercising early, Amelia forgoes the time value of the option but gains the dividends. We’ve already established that the dividend benefit outweighs the time value. 6. **Final Decision:** Given that the present value of the dividends (£4.3849) exceeds the option’s time value (£3.50), and considering the immediate intrinsic value gain of £2 upon exercise, Amelia should exercise the option immediately. The optimal decision is to exercise the option now to capture the dividend value.

The question assesses understanding of credit default swaps (CDS) and their pricing, particularly the concept of the CDS spread and how it relates to the probability of default and recovery rate. The CDS spread is essentially the insurance premium paid to protect against default. It’s influenced by both the likelihood of the reference entity defaulting and the amount that can be recovered if a default occurs. A higher probability of default will increase the CDS spread, as the protection seller faces a greater risk of having to pay out. A lower recovery rate (meaning a smaller portion of the debt is recovered in the event of default) will also increase the CDS spread, as the protection seller will have to pay out a larger amount. The formula that approximates the relationship is: CDS Spread ≈ (Probability of Default) * (1 – Recovery Rate). This is a simplified view, as it doesn’t account for the time value of money or the term structure of credit spreads. In this specific scenario, we are given a CDS spread of 250 basis points (2.5%) and a recovery rate of 40% (0.4). We need to solve for the implied probability of default. Rearranging the formula: Probability of Default ≈ CDS Spread / (1 – Recovery Rate) Probability of Default ≈ 0.025 / (1 – 0.4) = 0.025 / 0.6 ≈ 0.041667 or 4.17% Therefore, the implied probability of default is approximately 4.17%. The question also explores the impact of changing economic conditions. A deteriorating economic outlook would typically increase the perceived probability of default for many entities, including corporate bonds. This increased risk would make investors demand a higher premium for protection against default, leading to a widening of CDS spreads. Conversely, an improving economic outlook would decrease the perceived probability of default and narrow CDS spreads. The question challenges candidates to understand this inverse relationship and its implications for CDS pricing.

Let’s break down this complex scenario. First, we need to understand the impact of the unexpected regulatory change. The increase in margin requirements effectively makes holding futures positions more expensive, as more capital is tied up. This increased cost of carry reduces the attractiveness of the futures contract, leading to a decrease in its price. Simultaneously, the spot price is unaffected in the short term by this regulatory change because it reflects the immediate supply and demand dynamics of the underlying asset (rare earth metals). The futures price is fundamentally linked to the spot price through the cost of carry. This relationship can be expressed as: Futures Price ≈ Spot Price + Cost of Carry. The cost of carry includes storage costs, insurance, financing costs, and any convenience yield (benefit of holding the physical asset). In this case, the increased margin requirement directly increases the financing cost component of the cost of carry. To quantify the impact, we need to consider the annualized increase in margin requirement. The initial margin was 5%, and it increased to 15%, representing a 200% increase in the margin requirement (\[\frac{0.15 – 0.05}{0.05} = 2\]). This increase, however, is not directly the increase in the futures price. Instead, it increases the cost of capital tied to the futures position. We assume that the capital tied to the margin requirement has an opportunity cost equivalent to the risk-free rate, which is 4% per annum. The increased cost of carry due to the margin hike is: (New Margin Requirement – Old Margin Requirement) * Risk-Free Rate = (0.15 – 0.05) * 0.04 = 0.004 or 0.4%. This 0.4% represents the annualized increase in the cost of carry relative to the notional value of the futures contract. Since the futures contract matures in 6 months (0.5 years), we need to calculate the impact on the futures price over this period. The impact is 0.4% * 0.5 = 0.2%. Therefore, the futures price will decrease by 0.2% of the spot price. Calculating the price change: 0.002 * £25,000 = £50. The new futures price is: £25,000 – £50 = £24,950. This entire calculation hinges on the understanding that increased margin requirements directly impact the cost of carry, influencing the futures price relative to the spot price. The risk-free rate is used as a proxy for the opportunity cost of capital tied up in margin. This is a common simplification in derivative pricing. A critical aspect is the time to maturity, as the impact of the increased cost of carry is proportional to the contract’s duration.

The question assesses understanding of how various factors impact option prices, particularly in the context of earnings announcements. The key here is understanding that implied volatility typically increases before an earnings announcement due to uncertainty, and then decreases afterward as the uncertainty is resolved. This phenomenon is known as volatility crush. Time decay (theta) always negatively impacts option value, but the magnitude of the impact depends on how close the option is to expiration and its moneyness. A put option’s delta is negative. As the stock price increases, the put option becomes less valuable, and its delta becomes less negative (moves closer to zero). The correct answer (a) considers all these factors. The increase in stock price makes the put option less valuable. The decrease in implied volatility (volatility crush) also makes the put option less valuable. Time decay also contributes to the decrease in value. The incorrect options present scenarios where one or more factors are misapplied or misunderstood. Option b) incorrectly states that volatility increasing will decrease the value of the put option, as it should be the other way round. Option c) incorrectly suggests that the time decay would increase the value of the put, as it should be decreasing the value. Option d) incorrectly states that an increase in stock price would increase the value of the put option.

Let’s break down how to determine the profit or loss from a covered call strategy, incorporating the time decay aspect and dividend impact, using a novel scenario. The investor initially sells the call option, receiving a premium. If the stock price remains below the strike price at expiration, the option expires worthless, and the investor keeps the premium. However, if the stock price rises above the strike price, the option is exercised, and the investor must sell the stock at the strike price. The profit or loss is then calculated based on the initial stock purchase price, the premium received, the strike price, and any dividends received during the option’s lifetime. The time decay (theta) of an option impacts its value as expiration approaches. A covered call benefits from time decay because the option premium decreases, potentially allowing the investor to buy back the option at a lower price before expiration, increasing their profit. However, if the stock price rises significantly, offsetting the benefit of time decay, the option will likely be exercised. Dividends also play a role. If a dividend is paid during the option’s life, the stock price may decrease by the dividend amount on the ex-dividend date. This can make it less likely that the option will be exercised, increasing the probability of keeping the premium. However, the investor also receives the dividend income, which adds to the overall return. For example, suppose an investor buys 100 shares of XYZ stock at £45 per share and sells a covered call option with a strike price of £50 expiring in 3 months for a premium of £3 per share. The investor receives £300 in premium (100 shares * £3). If, at expiration, the stock price is £48, the option expires worthless, and the investor’s profit is £300. If, however, the stock price is £52, the option is exercised, and the investor must sell the shares at £50. The profit is then calculated as follows: Sale price (£50) – Purchase price (£45) + Premium (£3) = £8 per share. Total profit = £800. Now, consider a dividend of £1 per share is paid during the option’s life. This adds £100 to the investor’s return. The total profit, if the option expires worthless, would be £300 (premium) + £100 (dividend) = £400. If the option is exercised, the profit per share is: Sale price (£50) – Purchase price (£45) + Premium (£3) + Dividend (£1) = £9 per share. Total profit = £900.