Derivatives Level 4 (Investment Advice Diploma) Exam Quiz Quiz 17

The Black-Scholes model is a cornerstone of options pricing, but it relies on several key assumptions, one of the most critical being constant volatility. In reality, volatility is far from constant; it fluctuates based on market conditions, investor sentiment, and a myriad of other factors. This fluctuation is captured by the “volatility smile” or “volatility skew,” which shows that options with different strike prices (especially out-of-the-money puts and calls) have different implied volatilities. When the market anticipates a significant event, such as a company’s earnings announcement or a major economic release, implied volatility tends to increase. This is because the uncertainty surrounding the event raises the potential for large price swings in the underlying asset. Conversely, after the event passes and the uncertainty is resolved, implied volatility often decreases, a phenomenon known as volatility crush. The question explores how a fund manager might use options to profit from an anticipated volatility crush following a significant market event. A short straddle involves selling both a call and a put option with the same strike price and expiration date. The strategy profits if the underlying asset’s price remains relatively stable, allowing both options to expire worthless. The fund manager is essentially betting that the market’s expectation of high volatility is overblown and that the actual price movement will be less dramatic than anticipated. The maximum profit is the premium received from selling the options, while the potential loss is unlimited if the price moves significantly in either direction. The break-even points for a short straddle are calculated as follows: Upper Break-Even = Strike Price + Net Premium Received Lower Break-Even = Strike Price – Net Premium Received In this case: Strike Price = 5000 Call Premium = 300 Put Premium = 200 Net Premium Received = 300 + 200 = 500 Upper Break-Even = 5000 + 500 = 5500 Lower Break-Even = 5000 – 500 = 4500 Therefore, the fund manager will profit if, at expiration, the index value is between 4500 and 5500.

Let’s analyze a complex scenario involving a UK-based agricultural cooperative, “Green Harvest,” facing volatile wheat prices. Green Harvest wants to hedge against a potential price drop before their harvest in six months. They are considering using wheat futures contracts traded on the London International Financial Futures and Options Exchange (LIFFE). We need to determine the optimal number of contracts to use, taking into account the basis risk and the minimum tick size. First, calculate the hedge ratio. The cooperative expects to harvest 500,000 bushels of wheat. Each LIFFE wheat futures contract covers 100 tonnes of wheat. We need to convert bushels to tonnes. Assuming 1 bushel of wheat is approximately 0.0272155 metric tonnes, 500,000 bushels is equal to 500,000 * 0.0272155 = 13,607.75 tonnes. The number of contracts needed is 13,607.75 / 100 = 136.0775 contracts. Since you can’t trade fractions of contracts, they’ll need to trade 136 contracts. Next, consider the basis risk. Basis risk arises because the spot price and futures price might not converge perfectly at the delivery date. Let’s assume the current spot price is £200 per tonne and the six-month futures price is £210 per tonne. Green Harvest anticipates that the basis (spot price – futures price) might widen by £5 per tonne. This means that the futures price might fall more than the spot price, reducing the effectiveness of the hedge. Finally, consider the minimum tick size. The minimum tick size for LIFFE wheat futures is £0.05 per tonne. This means that the futures price can only move in increments of £0.05. This affects the precision of the hedge and potential profit or loss due to rounding. The total value of the futures contracts is 136 contracts * 100 tonnes/contract * £210/tonne = £2,856,000. A one-tick movement would result in a change of 136 contracts * 100 tonnes/contract * £0.05/tonne = £680. The optimal hedging strategy balances the desire to protect against price declines with the costs and risks associated with using futures contracts, including basis risk and the minimum tick size. Green Harvest should also consider alternative hedging strategies, such as using options or forward contracts, and consult with a qualified advisor to determine the best course of action. They must adhere to the FCA’s regulations regarding derivatives trading and suitability for their specific circumstances.

This question tests the understanding of delta hedging, transaction costs, and the impact of discrete hedging adjustments. The delta of an option measures its sensitivity to changes in the underlying asset’s price. A delta-neutral portfolio is constructed to be insensitive to small price movements in the underlying. However, maintaining delta neutrality requires periodic adjustments, which incur transaction costs. The key is to understand how the magnitude of price movements, the delta of the option, the number of options, and the transaction cost per share interact to affect the profitability of the hedging strategy. We calculate the profit or loss from the option, the cost of hedging, and then subtract the transaction costs to determine the net profit or loss. First, calculate the profit from the short option position: The stock price increased from £100 to £106, so the option expires worthless. The profit from the short option is the premium received, which is £8. Next, calculate the cost of hedging: Initial hedge: Buy 6,000 shares at £100. Cost = 6,000 * £100 = £600,000 Hedge adjustment: Sell 6,000 shares at £106. Revenue = 6,000 * £106 = £636,000 Cost of hedging = £600,000 – £636,000 = -£36,000 (a profit, but we consider it a cost in terms of hedging the option liability) Now, calculate the transaction costs: Transaction cost per share = £0.10 Total shares traded = 6,000 (initial buy) + 6,000 (sell to adjust) = 12,000 shares Total transaction costs = 12,000 * £0.10 = £1,200 Finally, calculate the net profit/loss: Profit from short option = £8 * 1,000 = £8,000 Hedging cost = -£36,000 Transaction costs = £1,200 Net profit/loss = £8,000 + £36,000 – £1,200 = £42,800 Therefore, the net profit from writing the options and delta hedging is £42,800.

The question tests 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 is the sensitivity of an option’s price to a change in the price of the underlying asset. Delta hedging involves adjusting the position in the underlying asset to offset changes in the option’s delta. First, calculate the initial cost of the hedge: * The investor is short 1,000 call options. * Each option has a delta of 0.60. * The investor needs to buy 1,000 * 0.60 = 600 shares to delta hedge. * Initial cost of shares = 600 shares * £50/share = £30,000. Next, calculate the new delta after the price change: * New delta = 0.65. * Shares needed to maintain the hedge = 1,000 * 0.65 = 650 shares. * Additional shares to buy = 650 – 600 = 50 shares. * Cost of additional shares = 50 shares * £52/share = £2,600. Then, calculate the impact of theta (time decay): * Theta = -£0.05 per option per day. * Time passed = 5 days. * Total theta impact = 1,000 options * -£0.05/option/day * 5 days = -£250 (a loss). Finally, calculate the profit or loss on the short options position: * Initial option price is not given, but it’s irrelevant for calculating the *change* in the option value due to the underlying price change and time decay. * However, we can infer the change in the option price from the delta and theta. The delta hedge seeks to neutralize the impact of price changes, and the theta represents the time decay. * Since we are looking for the profit/loss on the short option position, we need to consider the impact of the underlying price change on the options and the theta decay. Approximate change in option value due to stock price increase: Delta * change in stock price * number of options = 0.60 * (£52-£50) * 1000 = £1200. This is an approximate loss on the short option position. Total Profit/Loss on the short options position: The approximate loss from the stock price increase is £1200, and the loss from theta is £250. Therefore, the total loss on the short options position is £1200 + £250 = £1450. The total cost of maintaining the hedge is £2,600. However, the question asks for the profit/loss on the *short options position* itself, not the overall profit/loss of the hedging strategy. Therefore, we only consider the impact of delta (price change) and theta (time decay) on the option value. Therefore, the profit/loss on the short option position is approximately -£1450.

This question delves into the intricacies of hedging strategies using options, specifically focusing on delta-neutral portfolios and the challenges posed by gamma. Delta represents the sensitivity of an option’s price to a change in the underlying asset’s price. A delta-neutral portfolio is constructed to have a delta of zero, making it theoretically immune to small price movements in the underlying asset. However, delta itself changes as the underlying asset’s price fluctuates; this change in delta is known as gamma. Gamma represents the rate of change of delta with respect to the underlying asset’s price. A high gamma implies that the delta of the portfolio is highly sensitive to changes in the underlying asset’s price, requiring frequent rebalancing to maintain delta neutrality. The cost of this rebalancing, due to transaction costs and potential market impact, is a significant consideration. The calculation involves determining the number of options needed to hedge a short position in the underlying asset, considering the option’s delta. Then, we analyze the impact of a price change on the portfolio’s delta and calculate the cost of rebalancing to restore delta neutrality. Let’s assume an investor holds a short position of 100 shares of a company. The investor wants to hedge this position using call options. The current share price is £50. The call option has a delta of 0.5 and costs £2. Transaction costs are £0.10 per share. The share price increases to £51, and the option’s delta increases to 0.55. 1. **Initial Hedge:** To hedge 100 short shares with options having a delta of 0.5, the investor needs to buy 100 / 0.5 = 200 call options. 2. **Impact of Price Change:** The portfolio’s delta is initially zero. After the price increase, the option delta changes to 0.55. The total delta of the options becomes 200 * 0.55 = 110. The portfolio delta is now 110 (options) – 100 (shares) = +10. 3. **Rebalancing:** To restore delta neutrality, the investor needs to sell shares to reduce the portfolio delta back to zero. The investor needs to sell 10 shares. 4. **Rebalancing Cost:** The transaction cost is £0.10 per share. Selling 10 shares costs 10 * £0.10 = £1. 5. **Option Cost:** The initial cost of buying 200 options is 200 * £2 = £400. 6. **Total Cost:** The total cost includes the initial option cost and the rebalancing cost: £400 + £1 = £401. Therefore, the cost of implementing and maintaining the hedge, considering the change in delta and transaction costs, is £401. This illustrates the practical challenges of delta hedging, where gamma and transaction costs can significantly impact the overall cost and effectiveness of the hedging strategy. The example shows that delta-neutral hedging is not a static process but requires continuous monitoring and adjustment, especially when dealing with options with high gamma.

The question assesses understanding of hedging strategies using options, specifically a ratio spread, and how changes in volatility (vega) affect the overall position’s profitability. A ratio spread involves buying and selling different numbers of options with the same expiration date but different strike prices. The key is to understand the combined delta, gamma, and vega of the position. In this case, the investor is short more options than they are long, making them net short vega. Here’s how to break down the scenario and arrive at the correct answer: 1. **Initial Position:** The investor buys 100 call options with a strike price of £50 and sells 200 call options with a strike price of £55. This is a bearish strategy because the investor profits if the stock price stays below £55. 2. **Volatility Impact:** The investor is short vega because they sold more options than they bought. An increase in volatility will hurt a short vega position. 3. **Calculating Vega Effect:** The vega of a single £50 call is 0.05, and the vega of a single £55 call is 0.03. * Long 100 £50 calls: 100 * 0.05 = 5 vega * Short 200 £55 calls: 200 * -0.03 = -6 vega * Net vega: 5 – 6 = -1 vega 4. **Profit/Loss Calculation:** Vega represents the change in the option’s price for a 1% change in implied volatility. In this case, volatility increases by 5% (from 20% to 25%). * Total loss due to vega: -1 vega * 5% * £100 (contract size) = -£500 5. **Initial Premium:** The investor received £2 per option for the £55 calls and paid £5 per option for the £50 calls. * Premium received: 200 * £2 = £400 * Premium paid: 100 * £5 = £500 * Net premium: £400 – £500 = -£100 6. **Total Profit/Loss:** The investor lost £500 due to the increase in volatility and had an initial net premium of -£100. * Total profit/loss: -£500 – £100 = -£600 Therefore, the investor experiences a loss of £600. The key is to understand that being net short vega means a rise in volatility leads to losses.

This question tests the candidate’s understanding of delta hedging, specifically how to rebalance a delta-neutral portfolio after a change in the underlying asset’s price and volatility. Delta represents the sensitivity of an option’s price to a change in the price of the underlying asset. Delta hedging involves adjusting the number of shares held to offset the option’s delta, creating a portfolio that is initially insensitive to small price movements in the underlying asset. The key is to understand that delta changes as the underlying asset’s price and volatility change. When the underlying asset’s price increases, the delta of a call option increases (becomes more positive), and the delta of a put option decreases (becomes more negative). The opposite happens when the underlying asset’s price decreases. Increased volatility generally increases the delta of options that are away from being at-the-money (OTM) and decreases the delta of options that are near-the-money (ATM). In this scenario, the portfolio is delta-neutral, meaning the initial delta is zero. When the asset price increases and volatility increases, the call option’s delta will increase, and the put option’s delta will decrease. To maintain delta neutrality, the portfolio manager needs to reduce the overall delta exposure. Since the call option’s delta has increased (becoming more positive) and the put option’s delta has decreased (becoming more negative), the portfolio manager needs to sell shares of the underlying asset. This action will offset the increased positive delta from the call option and the decreased negative delta from the put option, bringing the portfolio back to a delta-neutral position. The amount of shares to sell depends on the magnitude of the delta changes, which are not provided in the question. However, the qualitative action is to sell shares.

Let’s analyze the pricing of a European call option using the Black-Scholes model, and then assess the impact of an unexpected dividend payment. The Black-Scholes formula is: \[C = S_0N(d_1) – Ke^{-rT}N(d_2)\] Where: * \(C\) = Call option price * \(S_0\) = Current stock price * \(K\) = Strike price * \(r\) = Risk-free interest rate * \(T\) = Time to expiration (in years) * \(N(x)\) = Cumulative standard normal distribution function * \(d_1 = \frac{ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}\) * \(d_2 = d_1 – \sigma\sqrt{T}\) * \(\sigma\) = Volatility of the stock First, we calculate \(d_1\) and \(d_2\). \[d_1 = \frac{ln(45/48) + (0.05 + 0.2^2/2)0.5}{0.2\sqrt{0.5}} = \frac{-0.0645 + 0.03}{0.1414} = -0.244\] \[d_2 = -0.244 – 0.2\sqrt{0.5} = -0.244 – 0.1414 = -0.3854\] Now, we find \(N(d_1)\) and \(N(d_2)\). Using standard normal distribution tables, we approximate: \(N(-0.244) \approx 0.4038\) \(N(-0.3854) \approx 0.3500\) Next, we calculate the call option price: \[C = 45 \times 0.4038 – 48e^{-0.05 \times 0.5} \times 0.3500\] \[C = 18.171 – 48 \times e^{-0.025} \times 0.3500\] \[C = 18.171 – 48 \times 0.9753 \times 0.3500\] \[C = 18.171 – 16.385 = 1.786\] The initial call option price is approximately £1.79. Now, consider the impact of a £1.50 dividend paid out unexpectedly three months (0.25 years) before expiration. This reduces the stock price immediately before the dividend. We adjust the stock price: Adjusted \(S_0 = 45 – 1.50 = 43.50\) We recalculate \(d_1\) and \(d_2\) using the adjusted stock price and the remaining time to expiration (0.25 years): \[d_1 = \frac{ln(43.5/48) + (0.05 + 0.2^2/2)0.25}{0.2\sqrt{0.25}} = \frac{-0.0988 + 0.0125}{0.1} = -0.863\] \[d_2 = -0.863 – 0.2\sqrt{0.25} = -0.863 – 0.1 = -0.963\] We find \(N(d_1)\) and \(N(d_2)\): \(N(-0.863) \approx 0.1942\) \(N(-0.963) \approx 0.1677\) Recalculate the call option price: \[C = 43.5 \times 0.1942 – 48e^{-0.05 \times 0.25} \times 0.1677\] \[C = 8.4417 – 48 \times e^{-0.0125} \times 0.1677\] \[C = 8.4417 – 48 \times 0.9876 \times 0.1677\] \[C = 8.4417 – 7.935 = 0.5067\] The call option price after the dividend payment is approximately £0.51. The change in the call option price is £1.79 – £0.51 = £1.28.

The core of this question lies in understanding how implied volatility, dividends, and time to expiration interact to influence option prices, specifically within the context of a binomial tree model. The binomial tree model is a numerical method used to value options. It works by constructing a tree of possible future stock prices, and then working backward from the expiration date to calculate the option’s value at each node. The initial stock price is £50. The dividend yield impacts the expected growth rate of the stock price in the binomial tree. The risk-free rate is 5% and the dividend yield is 2%. This means the stock price is expected to grow at a rate of 5% – 2% = 3%. The formula for calculating the up and down factors in a binomial model is: Up factor (u) = \(e^{(\sigma \sqrt{\Delta t})}\) Down factor (d) = \(e^{(-\sigma \sqrt{\Delta t})}\) Where: \(\sigma\) = Implied volatility \(\Delta t\) = Time step (in years) In this case, \(\sigma\) = 25% = 0.25, and \(\Delta t\) = 1 year / 2 = 0.5 years (since it’s a two-step tree). Up factor (u) = \(e^{(0.25 \sqrt{0.5})}\) = \(e^{0.1768}\) = 1.1934 Down factor (d) = \(e^{(-0.25 \sqrt{0.5})}\) = \(e^{-0.1768}\) = 0.8379 Next, we calculate the risk-neutral probability (p): p = \(\frac{e^{(r-q)\Delta t} – d}{u – d}\) Where: r = Risk-free rate = 5% = 0.05 q = Dividend yield = 2% = 0.02 p = \(\frac{e^{(0.05-0.02)0.5} – 0.8379}{1.1934 – 0.8379}\) = \(\frac{e^{0.015} – 0.8379}{0.3555}\) = \(\frac{1.0151 – 0.8379}{0.3555}\) = \(\frac{0.1772}{0.3555}\) = 0.4985 Now, we work backward through the binomial tree. At expiration (Step 2), the call option value is: Call Option Value = max(Stock Price – Strike Price, 0) Stock Price at UU = £50 * 1.1934 * 1.1934 = £71.21 Call Option Value at UU = max(£71.21 – £55, 0) = £16.21 Stock Price at UD = £50 * 1.1934 * 0.8379 = £50.00 Call Option Value at UD = max(£50.00 – £55, 0) = £0 Stock Price at DD = £50 * 0.8379 * 0.8379 = £35.14 Call Option Value at DD = max(£35.14 – £55, 0) = £0 At Step 1, we calculate the expected option value, discounted back one period: Call Option Value at U = \(\frac{p * Call Option Value at UU + (1-p) * Call Option Value at UD}{e^{r\Delta t}}\) Call Option Value at U = \(\frac{0.4985 * £16.21 + (1-0.4985) * £0}{e^{0.05 * 0.5}}\) = \(\frac{0.4985 * £16.21}{e^{0.025}}\) = \(\frac{£8.08}{1.0253}\) = £7.88 Call Option Value at D = \(\frac{p * Call Option Value at UD + (1-p) * Call Option Value at DD}{e^{r\Delta t}}\) Call Option Value at D = \(\frac{0.4985 * £0 + (1-0.4985) * £0}{e^{0.05 * 0.5}}\) = £0 Finally, at Step 0 (today), we calculate the option’s present value: Call Option Value Today = \(\frac{p * Call Option Value at U + (1-p) * Call Option Value at D}{e^{r\Delta t}}\) Call Option Value Today = \(\frac{0.4985 * £7.88 + (1-0.4985) * £0}{e^{0.05 * 0.5}}\) = \(\frac{£3.93}{1.0253}\) = £3.83 Therefore, the approximate value of the call option is £3.83.

The core concept being tested is the understanding of delta hedging and how it dynamically adjusts to maintain a neutral position as the underlying asset’s price changes. The question requires the candidate to calculate the necessary adjustment in the number of short call options to maintain a delta-neutral portfolio after a change in the underlying asset’s price and a change in the option’s delta. First, we need to determine the initial portfolio delta. The portfolio consists of 1000 shares of stock, each with a delta of 1, so the initial portfolio delta from the stock is 1000. The short call options initially contribute a negative delta to the portfolio. To find the initial number of short call options, we divide the stock delta by the absolute value of the initial call option delta: 1000 / 0.4 = 2500 call options. Next, we calculate the new desired number of short call options after the asset price change. The new call option delta is 0.6. To maintain delta neutrality, we need to find the number of options that, when multiplied by -0.6, equals -1000 (to offset the stock’s delta). So, we divide 1000 by 0.6: 1000 / 0.6 = 1666.67 call options. Finally, we determine how many options need to be bought back. The initial number of short call options was 2500, and the new desired number is 1666.67. Therefore, the number of options to buy back is 2500 – 1666.67 = 833.33. Since options contracts are typically for 100 shares, we need to buy back approximately 8.33 contracts. Given that we can only trade in whole contracts, we round to the nearest whole number. In this case, we round to 8 contracts. Therefore, the fund manager needs to buy back 8 contracts (800 options) to re-establish a delta-neutral position. This dynamic adjustment is crucial for managing risk in a portfolio using derivatives. The example illustrates how delta hedging isn’t a static strategy but requires continuous monitoring and adjustment based on market movements and changes in option characteristics. It highlights the practical challenges of implementing delta hedging in a real-world trading environment.

The question assesses the understanding of how delta hedging works in practice, particularly its limitations when volatility changes unexpectedly. 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 a change in the underlying asset’s price. By holding a number of shares equal to the option’s delta (or shorting if the option is short), the portfolio’s value should remain relatively stable for small price movements. However, delta is not constant; it changes as the underlying asset’s price changes, as time passes, and, critically, as volatility changes. This change in delta is measured by gamma. When volatility increases unexpectedly, the option’s price changes more rapidly than anticipated by the initial delta. This means the hedge becomes imperfect. To re-establish the delta-neutral position, the trader must adjust the hedge, typically by buying or selling more of the underlying asset. The magnitude of this adjustment depends on the gamma of the option and the size of the volatility shock. In this scenario, the trader is short options, meaning they benefit from a decrease in volatility and suffer from an increase. The initial hedge was designed for a lower volatility environment. The unexpected increase in volatility causes the short options to become more sensitive to price changes in the underlying asset. Since the trader is short options, an increase in volatility means the options’ value increases more than expected, leading to a loss if the hedge isn’t adjusted. To correct for this, the trader needs to sell more of the underlying asset to reduce the portfolio’s sensitivity to further price increases. Here’s the calculation: 1. **Initial Delta:** The trader is short 100 call options, each with a delta of 0.5. The total delta exposure from the options is -100 \* 0.5 = -50. This means the trader initially bought 50 shares to hedge. 2. **Volatility Increase:** Volatility increases unexpectedly. Since the trader is short call options, this increase in volatility hurts their position. 3. **Delta Adjustment:** To re-establish the delta hedge after the volatility increase, the trader needs to reduce their exposure to the underlying asset. Because they are short call options, the increase in volatility makes the call options more sensitive to the underlying asset’s price. Therefore, the trader needs to sell shares to reduce their long delta position and offset the increased delta of the short options.

Let’s consider a scenario where a UK-based agricultural cooperative, “Green Harvest,” wants to protect its future wheat sales from potential price drops. Green Harvest anticipates selling 500,000 bushels of wheat in six months. They decide to use wheat futures contracts traded on the ICE Futures Europe exchange to hedge their price risk. Each contract covers 5,000 bushels of wheat. To determine the number of contracts needed, Green Harvest divides their total expected sales by the contract size: 500,000 bushels / 5,000 bushels/contract = 100 contracts. They sell 100 wheat futures contracts. Now, let’s assume the initial futures price is £200 per bushel. If the price drops to £180 per bushel at the delivery date, Green Harvest will lose £20 per bushel on their physical wheat sales. However, they will profit from their futures position. The profit per contract is (£200 – £180) * 5,000 bushels = £100,000. Across 100 contracts, the total profit is £10,000,000. The hedge isn’t perfect because of basis risk. Basis risk is the difference between the spot price (the price Green Harvest receives for their actual wheat) and the futures price. Let’s say the spot price is actually £175 per bushel when the futures settle at £180. Green Harvest sells their wheat for £175 per bushel, gaining £10,000,000 on the future contracts, which effectively offset the loss. The initial revenue without hedging would be 500,000 * £200 = £100,000,000. The final revenue with hedging is (500,000 * £175) + £10,000,000 = £87,500,000 + £10,000,000 = £97,500,000. Basis risk arises because the futures price and spot price don’t always move in perfect lockstep due to factors like local supply and demand conditions, transportation costs, and storage costs. The hedge ratio, ideally 1:1 in this case (one futures contract for every 5,000 bushels), might need adjustment if Green Harvest anticipates a strong correlation, say 0.8, between the futures price and their local spot price. This adjustment would involve using a hedge ratio of 1.25 (1/0.8) to increase the number of contracts to 125, potentially over-hedging to account for the imperfect correlation.

This question tests the understanding of exotic derivatives, specifically barrier options, and their sensitivity to volatility changes (vega). A down-and-out barrier option becomes worthless if the underlying asset price hits a pre-defined barrier level. The vega of a down-and-out option is complex. Initially, as volatility increases, the option’s value might increase because there’s a higher probability of the underlying asset price fluctuating enough to generate a profit before potentially hitting the barrier. However, beyond a certain volatility level, the probability of the asset hitting the barrier and knocking out the option increases significantly, causing the option’s value to decrease. The specific calculation isn’t about a precise number but about understanding the directional impact. Because the barrier has *not* been breached, the increased volatility initially *increases* the option value because it increases the *chance* of the option going into the money *before* the barrier is hit. However, the *risk* of hitting the barrier has also increased. Because the barrier is relatively close to the current price, the barrier risk dominates. A crucial concept is that vega is not always positive. For standard options, it usually is. But for barrier options, especially near the barrier, it can be negative. This is because increased volatility near the barrier significantly increases the probability of the option being knocked out. The correct answer captures this nuanced understanding of vega for down-and-out barrier options. The incorrect answers represent common misunderstandings, such as assuming vega is always positive or incorrectly assessing the dominant effect of volatility near the barrier.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Yorkshire Grain,” which seeks to hedge against potential price declines in their upcoming wheat harvest using futures contracts. Yorkshire Grain anticipates harvesting 5,000 tonnes of wheat in three months. The current spot price of wheat is £200 per tonne, but they are concerned about a potential oversupply in the market due to favorable weather conditions across Europe. They decide to use wheat futures contracts traded on the London International Financial Futures and Options Exchange (LIFFE) to mitigate this risk. Each LIFFE wheat futures contract represents 100 tonnes of wheat. To determine the number of contracts required, Yorkshire Grain divides their total expected harvest (5,000 tonnes) by the contract size (100 tonnes per contract): 5,000 / 100 = 50 contracts. The December wheat futures contract is currently trading at £195 per tonne. Yorkshire Grain decides to sell 50 December wheat futures contracts to hedge their exposure. This locks in a price of £195 per tonne for their expected harvest. In three months, when Yorkshire Grain harvests their wheat, the spot price has indeed fallen to £180 per tonne due to the oversupply. Yorkshire Grain sells their wheat in the spot market for £180 per tonne, receiving £180 * 5,000 = £900,000. Simultaneously, they close out their futures position by buying back 50 December wheat futures contracts at £180 per tonne. Their profit on the futures contracts is (£195 – £180) * 100 tonnes/contract * 50 contracts = £75,000. Their effective selling price is the spot market price plus the futures profit: £900,000 + £75,000 = £975,000. The effective price per tonne is £975,000 / 5,000 = £195 per tonne. However, basis risk exists because the futures price and spot price might not converge perfectly. Let’s assume that the basis (difference between spot and futures price) narrows less than anticipated. Instead of converging to zero, the December futures contract settles at £182 per tonne when Yorkshire Grain closes out their position. The profit on the futures contracts now becomes (£195 – £182) * 100 tonnes/contract * 50 contracts = £65,000. Their effective selling price is now the spot market price plus the futures profit: £900,000 + £65,000 = £965,000. The effective price per tonne is £965,000 / 5,000 = £193 per tonne. The difference between the initial locked-in price (£195) and the actual effective price (£193) is due to basis risk. The cooperative received £2 less per tonne than initially anticipated due to the basis not converging as expected. This illustrates that hedging with futures reduces price risk but does not eliminate it entirely because of basis risk.

The core of this problem lies in understanding how delta changes with respect to changes in the underlying asset’s price (gamma) and the passage of time (theta), and how these changes impact a hedged portfolio. A delta-neutral portfolio is constructed to be insensitive to small changes in the underlying asset’s price. However, this neutrality is not static. Gamma measures the rate of change of delta with respect to the underlying asset’s price, and theta measures the rate of change of the option’s price with respect to time. To maintain delta neutrality, the portfolio needs to be rebalanced. The cost of rebalancing is directly affected by gamma. A higher gamma means the delta changes more rapidly as the underlying asset’s price moves, requiring more frequent and potentially larger rebalancing trades. Theta, on the other hand, erodes the value of the option as time passes. If an option is sold to create the delta-neutral hedge, theta represents a cost to the hedger. In this scenario, the fund manager sells options to create the delta-neutral hedge. As time passes, the options lose value (theta decay), which benefits the fund manager. However, the fund manager must also consider the impact of gamma. If the underlying asset price moves significantly, the delta of the options will change, and the fund manager will need to rebalance the portfolio to maintain delta neutrality. This rebalancing will incur transaction costs. The profit or loss on the delta-neutral strategy is approximately equal to the theta earned minus the cost of rebalancing due to gamma. The approximate profit/loss can be calculated as: Profit/Loss ≈ (Theta * Time) – 0.5 * Gamma * (Change in Underlying Price)^2. In this specific case: Theta = £-0.05 per day (benefit as it’s negative for the seller) Gamma = 0.002 Change in Underlying Price = 1.5% of £4500 = £67.50 Time = 1 day Profit/Loss ≈ (-0.05 * 1) – (0.5 * 0.002 * (67.50)^2) Profit/Loss ≈ -0.05 – (0.001 * 4556.25) Profit/Loss ≈ -0.05 – 4.56 Profit/Loss ≈ -4.61 Therefore, the approximate profit or loss for the fund on that day is a loss of £4.61.

To determine the net profit/loss, we need to calculate the profit/loss from each leg of the strategy: buying the call, selling the put, and the change in the asset’s price. 1. **Call Option:** The investor buys a call option with a strike price of 1550 at a premium of 65. The maximum loss is the premium paid, which is 65. If the asset price is below 1550 at expiration, the call option expires worthless. If the asset price is above 1550, the profit is the difference between the asset price and the strike price, minus the premium paid. 2. **Put Option:** The investor sells a put option with a strike price of 1450 and receives a premium of 50. The maximum profit is the premium received, which is 50. If the asset price is above 1450 at expiration, the put option expires worthless. If the asset price is below 1450, the loss is the difference between the strike price and the asset price, plus the premium received. 3. **Asset Price Change:** The asset price starts at 1500 and ends at 1480, resulting in a loss of 20 (1500 – 1480). Now, let’s analyze the scenario where the asset price is 1480 at expiration: * **Call Option:** The call option with a strike price of 1550 expires worthless, resulting in a loss of the premium paid, which is 65. * **Put Option:** The put option with a strike price of 1450 is in the money. The payoff is the strike price minus the asset price, which is 1450 – 1480 = -30. Since the investor sold the put, they are liable for this payoff. However, they received a premium of 50, so their net profit from the put is 50 – 30 = 20. * **Asset Price Change:** The asset price decreased from 1500 to 1480, resulting in a loss of 20. The total profit/loss is the sum of the profit/loss from each leg: -65 (call) + 20 (put) – 20 (asset price change) = -65. Therefore, the net result of this strategy is a loss of 65. This example illustrates how combining options with direct asset holdings can create complex payoff profiles. The investor was attempting to generate income from premiums while holding the underlying asset, but the decline in the asset price and the call option expiring worthless resulted in an overall loss. This highlights the importance of carefully considering the potential outcomes and risks associated with such strategies.

The question assesses the understanding of volatility smiles and skews, specifically how they are affected by supply and demand dynamics in the options market, and how implied volatility changes across different strike prices relative to the current market price of the underlying asset. The volatility smile/skew represents the implied volatility of options with different strike prices for the same underlying asset and expiration date. In a normal distribution, implied volatility would be the same for all strike prices. However, in reality, implied volatility often varies. A volatility smile shows higher implied volatility for out-of-the-money (OTM) calls and puts compared to at-the-money (ATM) options. A volatility skew, more common, shows a steeper slope on one side, typically with OTM puts having higher implied volatility than OTM calls. Increased demand for OTM puts, often used as downside protection, drives up their prices, and consequently, their implied volatility. Conversely, increased demand for OTM calls would raise their implied volatility as well. The magnitude of these changes relative to the ATM options is what defines the shape of the smile or skew. The correct answer must accurately reflect the impact of increased demand on the volatility smile/skew. The incorrect options present plausible but flawed interpretations of these dynamics. Let’s consider a hypothetical scenario. A fund manager, Amelia, is managing a large UK equity portfolio. She is concerned about a potential market correction due to upcoming Brexit negotiations. To protect her portfolio, she starts buying a large number of OTM put options on the FTSE 100 index. This increased demand for OTM puts will drive up their prices, causing their implied volatility to rise. Simultaneously, if there’s no significant change in demand for OTM calls, their implied volatility might remain relatively stable or even decrease slightly due to reduced perceived upside risk. This situation would steepen the volatility skew, making it more pronounced.

The core of this question revolves around understanding how a delta-neutral portfolio is constructed and maintained, specifically when dealing with options. Delta represents the sensitivity of an option’s price to a change in the underlying asset’s price. A delta-neutral portfolio aims to have a net delta of zero, meaning that small changes in the underlying asset’s price should not significantly impact the portfolio’s value. Gamma, on the other hand, measures the rate of change of delta with respect to changes in the underlying asset’s price. A positive gamma indicates that the delta will increase as the underlying asset’s price increases and decrease as the underlying asset’s price decreases. To maintain delta neutrality in a portfolio with positive gamma, adjustments must be made as the underlying asset’s price changes. If the underlying asset’s price increases, the delta of the options will increase (due to positive gamma), and to offset this, the portfolio manager needs to sell some of the underlying asset. Conversely, if the underlying asset’s price decreases, the delta of the options will decrease, and the portfolio manager needs to buy some of the underlying asset. The amount of the underlying asset to buy or sell depends on the magnitude of the gamma and the change in the underlying asset’s price. In this specific scenario, the portfolio has a positive gamma of 500. This means that for every £1 change in the underlying asset’s price, the portfolio’s delta will change by 500. The underlying asset’s price decreases by £2. Therefore, the portfolio’s delta will decrease by 500 * 2 = 1000. To restore delta neutrality, the portfolio manager needs to buy 1000 units of the underlying asset.

The core concept being tested is the understanding of how various factors influence option prices, particularly the time value component. The Black-Scholes model provides a framework for understanding these relationships. While a direct calculation isn’t necessary here, understanding the *direction* of the impact of these factors is crucial. A shorter time to expiration generally *decreases* the value of an option. This is because there is less time for the underlying asset to move favorably for the option holder. Volatility increases option prices because it increases the probability of the underlying asset moving significantly in either direction. Interest rates also impact option prices. Higher interest rates generally increase call option prices and decrease put option prices. Dividend payments tend to decrease call option prices and increase put option prices because they reduce the expected future price of the underlying asset. The question requires integrating these concepts to determine the net effect of multiple simultaneous changes. The key is to recognize that the volatility increase will likely outweigh the time decay and dividend effect, especially given the relatively modest changes in time and dividends compared to the substantial volatility increase. The interest rate increase will further contribute to the call option’s price increase.

The question assesses the understanding of put-call parity and its application in identifying arbitrage opportunities, specifically in the context of dividend-paying assets. Put-call parity is a fundamental concept in options pricing theory, stating a relationship between the prices of a European call option, a European put option, the underlying asset, and a risk-free bond. The formula is: \(C + PV(K) = P + S – PV(Div)\), where C is the call option price, PV(K) is the present value of the strike price, P is the put option price, S is the spot price of the asset, and PV(Div) is the present value of expected dividends. The scenario introduces a dividend-paying stock, requiring the adjustment of the put-call parity formula to account for the present value of these dividends. An arbitrage opportunity exists when the observed market prices deviate from the parity relationship. To exploit this, we identify whether the left-hand side (LHS) or right-hand side (RHS) of the equation is greater. If LHS > RHS, we buy the RHS (put and stock) and sell the LHS (call and bond). If RHS > LHS, we buy the LHS (call and bond) and sell the RHS (put and stock). In this case, we first calculate the present value of the dividend: \[PV(Div) = \frac{2}{1.05} = 1.9048\]. Then, we calculate the present value of the strike price: \[PV(K) = \frac{105}{1.05} = 100\]. Now, we plug the given values into the put-call parity formula: \[C + PV(K) = P + S – PV(Div)\] becomes \[12 + 100 = 6 + 108 – 1.9048\], which simplifies to \[112 = 112.0952\]. Since the RHS is greater than the LHS, we buy the call and bond and sell the put and stock. Specifically, the steps are: 1. Buy the call option for £12. 2. Lend £100 at the risk-free rate of 5% (equivalent to buying a risk-free bond). 3. Sell (short) the put option for £6. 4. Sell (short) the stock for £108. The initial cash flow is: -12 – 100 + 6 + 108 = £2. At expiration, if the stock price is above £105, the call option is exercised, and we deliver the stock we shorted. If the stock price is below £105, the put option is exercised, and we receive the stock, which covers our short position. The dividends received from shorting the stock are used to pay the dividend payment. The arbitrage profit is the initial cash flow of £2.

The question assesses the understanding of volatility smiles and skews, particularly their implications for option pricing and trading strategies. A volatility smile indicates that out-of-the-money (OTM) and in-the-money (ITM) options have higher implied volatilities than at-the-money (ATM) options. A volatility skew, on the other hand, shows a consistent upward or downward slope in implied volatility as the strike price changes. In this scenario, the trader observes a volatility skew where OTM puts are more expensive than OTM calls. This suggests a higher demand for downside protection, which could be due to concerns about potential market crashes or negative economic news. The trader believes this skew is overblown and expects the market to correct, leading to a flattening of the skew. To profit from this expected flattening, the trader should implement a strategy that benefits from a decrease in the implied volatility of OTM puts and an increase in the implied volatility of OTM calls. This can be achieved by selling OTM puts (to profit from the decrease in their implied volatility) and buying OTM calls (to profit from the increase in their implied volatility if the skew flattens). This strategy is effectively a risk reversal, but specifically designed to capitalize on the anticipated change in the volatility skew. The trader is essentially betting that the market’s fear of a downside move is overdone and that the skew will normalize. The profit comes from the difference between the premium received from selling the puts and the premium paid for buying the calls, adjusted for the change in implied volatilities.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Harvest Yield Co-op,” which needs to hedge against potential price declines in their upcoming wheat harvest. They are considering using wheat futures contracts traded on the ICE Futures Europe exchange. To determine the optimal number of contracts, we need to calculate the hedge ratio, which is the ratio of the value of the asset being hedged (the wheat harvest) to the value of one futures contract. The cooperative expects to harvest 50,000 tonnes of wheat. One ICE wheat futures contract represents 100 tonnes of wheat. The current spot price of wheat is £200 per tonne, and the futures price for the delivery month corresponding to their harvest is £210 per tonne. First, we calculate the total value of the wheat harvest: 50,000 tonnes * £200/tonne = £10,000,000. Next, we calculate the value of one futures contract: 100 tonnes * £210/tonne = £21,000. The hedge ratio is then calculated as: £10,000,000 / £21,000 = 476.19. Since the cooperative can only trade whole contracts, they should round this number to the nearest whole number, which is 476 contracts. Now, let’s analyze the impact of basis risk. Basis risk arises because the spot price and the futures price do not always move in perfect correlation. Suppose that at the time of harvest, the spot price of wheat has fallen to £180 per tonne, while the futures price has fallen to £190 per tonne. The cooperative sells their wheat at the spot price and simultaneously closes out their futures position. The loss on the wheat sale is: 50,000 tonnes * (£200/tonne – £180/tonne) = £1,000,000. The profit on the futures contracts is: 476 contracts * 100 tonnes/contract * (£210/tonne – £190/tonne) = £952,000. The net outcome is a loss of £1,000,000 – £952,000 = £48,000. This remaining loss is due to basis risk. This example illustrates how futures contracts can be used to hedge price risk, but also highlights the importance of understanding and managing basis risk. The cooperative successfully mitigated a significant portion of their potential losses, but the basis risk prevented a perfect hedge. Understanding these dynamics is crucial for effective risk management in derivatives trading.

Let’s consider a scenario involving a UK-based agricultural cooperative, “BritCrops,” that wants to protect its future wheat sales against price volatility. BritCrops plans to sell 100,000 tonnes of wheat in six months. To hedge this exposure, they decide to use wheat futures contracts traded on the London International Financial Futures and Options Exchange (LIFFE). Each LIFFE wheat futures contract covers 100 tonnes of wheat. Therefore, BritCrops needs to sell 1000 futures contracts (100,000 tonnes / 100 tonnes per contract). The current futures price for wheat with six months to expiry is £200 per tonne. To understand the impact of basis risk, we need to consider the spot price of wheat at the delivery date and compare it to the futures price at that time. Basis risk arises because the futures price and the spot price may not converge perfectly at the delivery date due to factors like transportation costs, storage costs, and local supply and demand conditions. Suppose that in six months, the spot price of wheat is £190 per tonne, and the futures price is £192 per tonne. BritCrops closes out its futures position by buying back 1000 futures contracts at £192 per tonne. The gain on the futures position is calculated as follows: Initial futures price: £200 per tonne Final futures price: £192 per tonne Gain per tonne: £200 – £192 = £8 per tonne Total gain on futures position: £8 per tonne * 100,000 tonnes = £800,000 However, BritCrops sells its wheat in the spot market at £190 per tonne, receiving: £190 per tonne * 100,000 tonnes = £19,000,000 Without hedging, BritCrops would have received £19,000,000. With hedging, they receive £19,000,000 from the spot market sale plus £800,000 from the futures position, totaling £19,800,000. The effective price received is £19,800,000 / 100,000 tonnes = £198 per tonne. The basis is the difference between the spot price and the futures price at the time of delivery: £190 – £192 = -£2 per tonne. The initial expected price was £200, but the effective price received was £198, a difference of £2 per tonne, reflecting the basis risk. Now, let’s calculate the percentage impact of the basis risk. The difference between the initial futures price and the effective price received is £2 per tonne. The percentage impact of the basis risk is (£2 / £200) * 100% = 1%. This represents the degree to which the hedge was imperfect due to the non-convergence of spot and futures prices.

The question assesses understanding of how volatility smiles and skews impact option pricing, particularly when using the Black-Scholes model, which assumes constant volatility. In reality, implied volatility varies across different strike prices for options with the same expiration date, forming a volatility smile or skew. A volatility smile indicates that out-of-the-money (OTM) and in-the-money (ITM) options are relatively more expensive than at-the-money (ATM) options, suggesting higher demand and therefore higher implied volatility for these options. A volatility skew, common in equity markets, shows that OTM puts (downside protection) are more expensive than OTM calls. The Black-Scholes model, which assumes constant volatility, will misprice options when a volatility smile or skew is present. Using ATM volatility for all options will undervalue OTM and ITM options in a smile, and OTM puts in a skew. To account for this, traders adjust the volatility input based on the strike price of the option. Using the ATM volatility will undervalue the OTM put option because the market is pricing in a higher implied volatility due to the skew. Calculating the theoretical price using Black-Scholes with ATM volatility: Given: S = £100 (Current Stock Price) K = £90 (Strike Price) r = 5% (Risk-Free Rate) T = 0.5 years (Time to Expiration) σ = 25% (ATM Volatility) Using the Black-Scholes formula for a put option: \[P = Ke^{-rT}N(-d_2) – Se^{-qT}N(-d_1)\] Where: \[d_1 = \frac{ln(\frac{S}{K}) + (r – q + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}}\] \[d_2 = d_1 – \sigma\sqrt{T}\] q = 0 (Dividend Yield) \[d_1 = \frac{ln(\frac{100}{90}) + (0.05 + \frac{0.25^2}{2})0.5}{0.25\sqrt{0.5}} = \frac{0.1054 + 0.03125}{0.1768} = 0.773\] \[d_2 = 0.773 – 0.25\sqrt{0.5} = 0.773 – 0.1768 = 0.596\] N(-d1) = N(-0.773) = 1 – N(0.773) = 1 – 0.7802 = 0.2198 N(-d2) = N(-0.596) = 1 – N(0.596) = 1 – 0.7244 = 0.2756 \[P = 90e^{-0.05*0.5} * 0.2756 – 100 * 0.2198\] \[P = 90 * 0.9753 * 0.2756 – 21.98\] \[P = 24.19 – 21.98 = 2.21\] The Black-Scholes model calculates a theoretical price of £2.21. Since the market price is £3.50, the option is overpriced relative to the model’s output using ATM volatility. This discrepancy arises from the volatility skew, which the Black-Scholes model fails to capture when using only ATM volatility.

Let’s analyze the potential profit or loss from a covered call strategy, considering the impact of dividends and early exercise. The covered call strategy involves holding an asset (in this case, shares of “TechForward”) and selling a call option on that same asset. The investor profits from the option premium received and any potential price appreciation of the underlying asset, up to the strike price. However, the investor forgoes any potential gains above the strike price, as the option buyer will exercise their right to purchase the shares at the strike price. Dividends received during the option period add to the overall profit. Early exercise, while less common for options on non-dividend paying stocks, can occur if the dividend payment exceeds the time value of the option, making it beneficial for the option holder to exercise early to capture the dividend. In this scenario, the investor owns 1000 shares of TechForward, currently trading at £85. They sell ten call options (each covering 100 shares) with a strike price of £90, receiving a premium of £4 per share. The company declares a dividend of £6 per share, leading the option holder to exercise the options just before the ex-dividend date. First, calculate the total premium received: 10 options * 100 shares/option * £4/share = £4000. Since the options are exercised, the investor must sell their shares at the strike price of £90. The initial cost of the shares is not relevant for calculating the profit from this specific covered call strategy over the option period. The profit from selling the shares at the strike price is calculated based on the difference between the strike price and the initial share price, but only up to the strike price: (£90 – £85) * 1000 shares = £5000. The investor also receives the dividend payment: £6/share * 1000 shares = £6000. The total profit is the sum of the premium received, the profit from selling the shares at the strike price, and the dividend received: £4000 + £5000 + £6000 = £15000. Therefore, the investor’s total profit from the covered call strategy, considering the dividend and early exercise, is £15000.

The question assesses understanding of hedging strategies using options, specifically a ratio spread, and requires calculating the profit or loss at expiration. The ratio spread involves buying a certain number of options at one strike price and selling a different number of options at another strike price. The calculation involves determining the payoff from each leg of the spread (long calls and short calls) and summing them to find the overall profit or loss. First, calculate the payoff from the purchased call options: Strike Price = £50, Premium Paid = £2.50, Number of Options = 2 If the asset price at expiration (ST) is less than or equal to £50, the payoff is zero. If ST is greater than £50, the payoff is (ST – £50). The total cost for buying the options is 2 * £2.50 * 100 = £500 (assuming each option contract covers 100 shares). Next, calculate the payoff from the sold call options: Strike Price = £55, Premium Received = £1.00, Number of Options = 4 If the asset price at expiration (ST) is less than or equal to £55, the payoff is zero. If ST is greater than £55, the payoff is -(ST – £55). The total premium received for selling the options is 4 * £1.00 * 100 = £400. Now, calculate the total profit/loss at expiration when ST = £57: Payoff from purchased calls (strike £50): 2 * (£57 – £50) * 100 = £1400 Payoff from sold calls (strike £55): 4 * -(£57 – £55) * 100 = -£800 Initial cost: (£500 – £400) = £100 Total Profit/Loss: £1400 – £800 – £100 = £500 An alternative approach is to consider the breakeven points and maximum profit/loss scenarios. The maximum profit occurs when the asset price is at the short strike price (£55). The maximum loss is limited due to the purchased calls. This question tests the ability to combine multiple option positions and assess the net result, incorporating the initial premiums paid and received. It goes beyond simple definitions and forces candidates to apply their knowledge in a practical, quantitative setting. Understanding the impact of different strike prices and the number of contracts is crucial. The plausible incorrect answers are designed to trap candidates who miscalculate the payoffs or fail to account for the initial premiums.

This question tests understanding of option pricing models, specifically the Black-Scholes model, and how dividends affect option prices. The core concept is that dividends reduce the stock price, thereby decreasing the value of call options and increasing the value of put options. The Black-Scholes model needs to be adjusted to account for the present value of these dividends. First, we need to calculate the present value of the dividends. We have two dividends: £1.50 payable in 3 months (0.25 years) and £1.50 payable in 9 months (0.75 years). The risk-free rate is 5%. Present Value of Dividend 1: \[PV_1 = \frac{1.50}{e^{0.05 \times 0.25}} = \frac{1.50}{e^{0.0125}} \approx \frac{1.50}{1.01258} \approx 1.4814\] Present Value of Dividend 2: \[PV_2 = \frac{1.50}{e^{0.05 \times 0.75}} = \frac{1.50}{e^{0.0375}} \approx \frac{1.50}{1.03814} \approx 1.4448\] Total Present Value of Dividends: \[PV_{total} = PV_1 + PV_2 = 1.4814 + 1.4448 = 2.9262\] Adjusted Stock Price: \[S_{adjusted} = S – PV_{total} = 50 – 2.9262 = 47.0738\] Now we use the Black-Scholes formula with the adjusted stock price: \[C = S_{adjusted}N(d_1) – Xe^{-rT}N(d_2)\] Where: * \(S_{adjusted} = 47.0738\) * \(X = 50\) * \(r = 0.05\) * \(T = 1\) * \(\sigma = 0.25\) First, calculate \(d_1\) and \(d_2\): \[d_1 = \frac{ln(\frac{S_{adjusted}}{X}) + (r + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}} = \frac{ln(\frac{47.0738}{50}) + (0.05 + \frac{0.25^2}{2})1}{0.25\sqrt{1}} = \frac{ln(0.941476) + (0.05 + 0.03125)}{0.25} = \frac{-0.0603 + 0.08125}{0.25} = \frac{0.02095}{0.25} = 0.0838\] \[d_2 = d_1 – \sigma\sqrt{T} = 0.0838 – 0.25\sqrt{1} = 0.0838 – 0.25 = -0.1662\] Next, find \(N(d_1)\) and \(N(d_2)\): \(N(0.0838) \approx 0.5334\) \(N(-0.1662) \approx 0.4339\) Finally, calculate the call option price: \[C = 47.0738 \times 0.5334 – 50 \times e^{-0.05 \times 1} \times 0.4339 = 25.091 – 50 \times 0.9512 \times 0.4339 = 25.091 – 20.644 = 4.447\] Therefore, the call option price is approximately £4.45.

The core of this problem lies in understanding how delta hedging works and how transaction costs erode profits. 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 a change in the underlying asset’s price. A delta-neutral portfolio is constructed by holding a number of shares of the underlying asset equal to the negative of the option’s delta. When the asset price changes, the portfolio is rebalanced to maintain delta neutrality. This rebalancing involves buying or selling shares, incurring transaction costs. The more volatile the asset, the more frequent the rebalancing, and the higher the transaction costs. In this case, the investor initially sells options, implying a short position in options and a long position in the underlying asset to hedge. As the asset price rises, the delta of the short options position increases (becomes more negative), requiring the investor to buy more of the underlying asset to maintain delta neutrality. Conversely, when the asset price falls, the delta decreases (becomes less negative), requiring the investor to sell shares. The profit or loss is determined by the difference between the option premium received and the cost of hedging, including transaction costs. The transaction costs are calculated based on the number of shares traded during each rebalancing and the commission per share. Finally, the overall profit/loss is the initial premium received minus the total transaction costs. Initial Short Put Premium = £2.50 * 100 contracts * 100 shares/contract = £25,000 1. Initial Hedge: Delta = -0.40. To hedge 10,000 short puts (100 contracts * 100 shares), buy 4,000 shares at £48. 2. Price rises to £49: Delta changes to -0.20. Reduce hedge to 2,000 shares. Sell 2,000 shares. Transaction Cost: 2,000 shares * £0.05 = £100 3. Price rises to £50: Delta changes to 0. Sell remaining 2,000 shares. Transaction Cost: 2,000 shares * £0.05 = £100 4. Price falls to £49: Delta changes to -0.20. Buy 2,000 shares. Transaction Cost: 2,000 shares * £0.05 = £100 5. Price falls to £48: Delta changes to -0.40. Buy 2,000 shares. Transaction Cost: 2,000 shares * £0.05 = £100 Total Transaction Costs = £100 + £100 + £100 + £100 = £400 Final Profit/Loss = £25,000 – £400 = £24,600

The question concerns the application of put-call parity, a fundamental concept in options pricing. Put-call parity states that the price of a European call option plus the present value of the strike price should equal the price of a European put option plus the current price of the underlying asset. This relationship holds true under certain assumptions, including no arbitrage opportunities, European-style options (exercisable only at expiration), and identical strike prices and expiration dates for the put and call options. The formula for put-call parity is: \[C + PV(K) = P + S\] Where: C = Price of the European call option PV(K) = Present value of the strike price (K) P = Price of the European put option S = Current price of the underlying asset In this scenario, we are given the prices of the call and put options, the strike price, the time to expiration, and the risk-free interest rate. We need to determine if the put-call parity relationship holds and, if not, identify the arbitrage opportunity and calculate the profit. First, calculate the present value of the strike price: \[PV(K) = \frac{K}{(1 + r)^t}\] Where: K = Strike price = £155 r = Risk-free interest rate = 3.5% = 0.035 t = Time to expiration = 0.5 years \[PV(K) = \frac{155}{(1 + 0.035)^{0.5}} = \frac{155}{1.01737} \approx 152.35\] Now, check if put-call parity holds: \[C + PV(K) = P + S\] \[12.50 + 152.35 = 9.75 + 156\] \[164.85 \neq 165.75\] Since the equation does not hold, there is an arbitrage opportunity. The left side (call + present value of strike) is less than the right side (put + stock). This indicates that the call option is relatively undervalued compared to the put option and the stock. To exploit this arbitrage, one should buy the undervalued side (call and bond) and sell the overvalued side (put and stock). Arbitrage Strategy: 1. Buy the call option for £12.50 2. Buy a zero-coupon bond that will pay £155 at expiration. The cost today is the PV(K) = £152.35 3. Sell the put option for £9.75 4. Sell the stock for £156 Initial Cash Flow: \[-12.50 – 152.35 + 9.75 + 156 = £0.90\] Therefore, the arbitrage profit is £0.90.

Let’s consider a scenario where a portfolio manager, tasked with hedging a UK-based equity portfolio against a potential market downturn, is contemplating the use of FTSE 100 index put options. The manager needs to decide on the appropriate strike price and number of contracts to effectively protect the portfolio’s value. The FTSE 100 currently stands at 7500. The portfolio’s beta relative to the FTSE 100 is 1.2, and its current market value is £15 million. The manager decides to use put options with a strike price of 7400, expiring in 3 months. The put option premium is £50 per contract, and each contract covers 1 index point (i.e., a movement of 1 point in the FTSE 100). First, determine the number of index points the portfolio represents: Portfolio Value * Beta / Index Level = £15,000,000 * 1.2 / 7500 = 2400 index points Next, calculate the number of put option contracts needed to hedge the portfolio: Number of Contracts = Portfolio Index Points / Contract Size = 2400 / 1 = 2400 contracts Now, let’s analyze a scenario where the FTSE 100 falls to 7200 at expiration. The intrinsic value of each put option is Max(Strike Price – Final Index Level, 0) = Max(7400 – 7200, 0) = £200. Total payoff from put options: 2400 contracts * £200/contract = £480,000. The initial cost of the hedge is the premium paid for the options: Total Premium Paid = Number of Contracts * Premium per Contract = 2400 * £50 = £120,000. Net Payoff = Total Payoff – Total Premium Paid = £480,000 – £120,000 = £360,000. Now, let’s calculate the percentage decline in the FTSE 100: Percentage Decline = (Initial Index Level – Final Index Level) / Initial Index Level = (7500 – 7200) / 7500 = 0.04 or 4%. Expected decline in portfolio value (without hedge): Portfolio Value * Beta * Percentage Decline = £15,000,000 * 1.2 * 0.04 = £720,000. The hedged portfolio value is the original value less the decline, plus the net payoff from the hedge: Hedged Portfolio Value = Original Portfolio Value – Portfolio Decline + Net Payoff = £15,000,000 – £720,000 + £360,000 = £14,640,000. The percentage return on the hedged portfolio is: Percentage Return = (Hedged Portfolio Value – Original Portfolio Value) / Original Portfolio Value = (£14,640,000 – £15,000,000) / £15,000,000 = -0.024 or -2.4%. Therefore, the percentage return on the hedged portfolio is -2.4%.