Derivatives Level 4 (Investment Advice Diploma) Quiz Exam Quiz 13

Let’s consider a scenario involving a UK-based agricultural cooperative, “Green Harvest,” that exports organic wheat. Green Harvest faces price volatility in the global wheat market and seeks to hedge its future sales using derivatives. They enter into a forward contract to sell 1000 tonnes of wheat at £200 per tonne in six months. Simultaneously, a severe drought hits a major wheat-producing region, causing wheat prices to surge. Spot prices rise to £250 per tonne. To analyze the cooperative’s position, we need to calculate the profit or loss on the forward contract and understand the impact of the drought on their overall financial outcome. The forward contract locks in a selling price of £200,000 (1000 tonnes * £200). If Green Harvest sold the wheat on the spot market at £250 per tonne, they would receive £250,000. The difference represents the opportunity cost of entering into the forward contract. In this case, the opportunity cost is £50,000 (£250,000 – £200,000). This highlights the trade-off between hedging and potential upside. While the forward contract protected Green Harvest from potential price declines, it also prevented them from fully benefiting from the unexpected price increase. Furthermore, consider the implications for Green Harvest’s reputation and future contracts. While they technically “lost” £50,000 compared to the spot market, they fulfilled their contractual obligation, demonstrating reliability to their counterparty. This could lead to more favorable terms in future derivative contracts. If they had reneged on the contract due to the price increase, they would face legal repercussions and damage their reputation, making it harder to secure future hedging arrangements. This example showcases how derivatives management involves not just financial calculations but also strategic considerations about relationships and long-term sustainability. Also, the impact of regulations such as EMIR (European Market Infrastructure Regulation) should be considered, ensuring that the cooperative reports its derivative transactions and manages its counterparty risk effectively.

The question assesses the understanding of option pricing sensitivity, specifically Delta and Gamma, and their combined impact on portfolio rebalancing. Delta represents the change in option price for a unit change in the underlying asset’s price, while Gamma represents the rate of change of Delta. The scenario involves a portfolio manager using options to hedge a stock position. The initial portfolio consists of 10,000 shares of stock XYZ, and the manager uses put options to hedge against potential downside risk. The put options have a Delta of -0.40 and a Gamma of 0.02. The stock price increases by £2. The change in the option’s Delta due to the stock price movement is calculated using Gamma. The new Delta is then used to determine the number of options needed to maintain a delta-neutral position. Initial Delta of the put options = -0.40 Number of put options = 5,000 Total Delta of the put options = 5,000 * -0.40 = -2,000 Delta of the stock position = 10,000 The portfolio is initially not delta neutral, as the stock position’s delta is 10,000 and the put options’ delta is -2,000. The portfolio manager’s intention is to create a delta neutral position. Change in stock price = £2 Change in Delta = Gamma * Change in stock price = 0.02 * £2 = 0.04 New Delta of the put options = -0.40 + 0.04 = -0.36 New total Delta of the put options = 5,000 * -0.36 = -1,800 To maintain a delta-neutral position, the total Delta of the portfolio should be zero. The current total Delta is 10,000 (stock) – 1,800 (options) = 8,200. The manager needs to reduce the portfolio Delta by 8,200. Since each put option has a Delta of -0.36, the number of options needed to offset the remaining Delta is 8,200 / 0.36 ≈ 22,778 options. Therefore, the manager needs to buy an additional 22,778 – 5,000 = 17,778 options. The closest answer is 17,778 put options.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their sensitivity to changes in volatility and the underlying asset price relative to the barrier. A down-and-out call option becomes worthless if the underlying asset price touches or falls below the barrier level during the option’s life. The value of such an option is highly dependent on the volatility of the underlying asset and its current price relative to the barrier. A decrease in volatility generally decreases the value of a standard option because the probability of the option moving into the money decreases. However, for a down-and-out call option, a decrease in volatility can *increase* the option’s value because it reduces the probability of the asset price hitting the barrier and knocking the option out. The closer the asset price is to the barrier, the more sensitive the option’s price is to changes in volatility. If the asset price is significantly above the barrier, the option behaves more like a standard call option, and a decrease in volatility will decrease the option’s value. Conversely, if the asset price is very close to the barrier, the dominant effect is the reduced risk of the option being knocked out, increasing its value. Consider an analogy: imagine a tightrope walker. The “barrier” is the ground. The “volatility” is how much the tightrope walker sways. If the walker is very high up (asset price far from the barrier), a little less swaying (lower volatility) doesn’t change much – they’re still unlikely to fall. But if the walker is very close to the ground (asset price near the barrier), less swaying significantly reduces the risk of falling (being knocked out), making their performance (the option) more valuable. In this specific scenario, the asset price is only slightly above the barrier. Therefore, the dominant effect of decreased volatility is the reduced probability of the option being knocked out, leading to an increase in the option’s value.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their sensitivity to the underlying asset’s volatility. The scenario involves a knock-out put option, where the option becomes worthless if the underlying asset’s price touches the barrier level before the expiration date. The key to answering this question correctly lies in understanding the relationship between volatility, barrier proximity, and the option’s delta. Higher volatility increases the probability of the underlying asset reaching the barrier, thus decreasing the option’s value and potentially making its delta more negative (for a knock-out put). The proximity of the barrier also plays a crucial role; the closer the barrier, the more sensitive the option’s value becomes to changes in volatility. Let’s consider a hypothetical example. Imagine a tech stock, “InnovTech,” currently trading at £100. An investor holds a knock-out put option on InnovTech with a strike price of £95 and a knock-out barrier at £105. Initially, the implied volatility of InnovTech is 20%. Now, suppose there’s an unexpected announcement causing InnovTech’s implied volatility to jump to 40%. This sudden increase in volatility dramatically increases the likelihood of InnovTech’s price hitting the £105 barrier, thereby knocking out the put option and rendering it worthless. The option’s delta, which initially might have been moderately negative, would become even more negative as the option’s value plummets due to the increased probability of being knocked out. Contrast this with a scenario where the barrier is much further away, say at £120. In this case, even with the increase in volatility, the probability of hitting the barrier remains relatively low, and the option’s value and delta would be less affected. The reference to “Gamma hedging” is a distractor. While Gamma hedging is a volatility-related strategy, it is not the primary driver of the delta change in this specific scenario. The dominant factor is the increased probability of the barrier being hit due to the heightened volatility, directly impacting the option’s value and delta. Therefore, the correct answer is the one that accurately reflects this understanding of volatility’s impact on a knock-out put option’s delta when the barrier is close.

The investor’s break-even point on a covered call strategy is calculated by subtracting the premium received from selling the call option from the initial purchase price of the underlying asset. This represents the price at which the investor neither makes nor loses money on the combined position. In this case, the investor bought 500 shares at £22 per share, costing £11,000 (500 * £22). They then sold five call option contracts (each covering 100 shares) at a premium of £2.50 per share, generating a total premium of £1,250 (5 * 100 * £2.50). The break-even point is calculated as follows: Break-even Point = (Total Cost of Shares – Total Premium Received) / Number of Shares Break-even Point = (£11,000 – £1,250) / 500 Break-even Point = £9,750 / 500 Break-even Point = £19.50 Therefore, the break-even point for the covered call strategy is £19.50 per share. This means that if the share price is above £19.50 at expiration, the investor will make a profit. If the share price is below £19.50, the investor will experience a loss. The covered call strategy limits the upside profit potential but provides downside protection up to the amount of the premium received. Consider a scenario where the share price skyrockets to £40. The investor’s profit is capped because the call options will be exercised, and they will have to sell their shares at the strike price. However, if the share price plummets to £10, the investor will still own the shares, but the premium received will offset some of the loss. The break-even point is a crucial metric for understanding the risk-reward profile of this strategy. A lower break-even point indicates a more favorable risk-reward profile, as it provides a larger buffer against potential losses.

The correct answer is (a). This question tests the understanding of how margin requirements work in futures contracts, specifically focusing on the impact of price fluctuations and the maintenance margin. The initial margin is the amount required to open the position, and the maintenance margin is the level below which the account cannot fall. When the account balance drops below the maintenance margin, a margin call is issued, requiring the investor to deposit funds to bring the account back to the initial margin level. In this scenario, the investor starts with an initial margin of £6,000. A price decrease of 3 ticks per day for 5 days results in a total loss of £1,500 (3 ticks/day * 5 days * £100/tick). The account balance after these losses is £4,500 (£6,000 – £1,500). Since this is below the maintenance margin of £5,000, a margin call is triggered. To meet the margin call, the investor must deposit enough funds to bring the account back to the initial margin level of £6,000. Therefore, the investor needs to deposit £1,500 (£6,000 – £4,500). Options (b), (c), and (d) are incorrect because they miscalculate either the total loss due to price fluctuations or the amount needed to cover the margin call, or both. Option (b) only calculates the loss but doesn’t consider the need to replenish the margin to the initial level. Option (c) incorrectly calculates the loss and the deposit amount. Option (d) only calculates the deposit amount to meet the maintenance margin, not the initial margin.

The question explores the complexities of valuing an exotic derivative, specifically a barrier option with a time-dependent barrier, in a real-world scenario involving a UK-based manufacturing company hedging its currency risk. The core challenge lies in understanding how the changing barrier level affects the option’s probability of being triggered and, consequently, its value. The valuation requires a nuanced understanding of option pricing models, barrier option mechanics, and the implications of time-varying parameters. To solve this, we need to consider the probability of the spot rate breaching the barrier at any point during the option’s life. The time-dependent barrier introduces complexity because the probability calculation changes with each time step. While a closed-form solution for a time-dependent barrier option is complex and often requires numerical methods, we can approximate the impact. The initial barrier is 1.25, moving to 1.28 after three months. This means the option is initially further out-of-the-money, decreasing the probability of being knocked out early. However, the barrier increases, making it easier to breach later. The overall effect is a complex interplay between time to maturity and barrier proximity. Since the barrier is moving upwards, it increases the probability of the option *not* being knocked out. The higher barrier level, especially later in the option’s life, makes it less likely the spot rate will reach and breach it. This increases the option’s value, relative to an option with a fixed barrier at the initial level. The most accurate answer will reflect this increased value due to the upward-moving barrier. The other options reflect common misunderstandings: a decrease in value (incorrect, as the higher barrier makes a knockout less likely), no change in value (incorrect, as the barrier movement definitely impacts the probability of knockout), or a value solely determined by the initial barrier (incorrect, as the time-dependent nature is crucial).

The correct answer is (a). This question assesses the understanding of exotic derivatives, specifically barrier options, and their behavior in relation to the underlying asset’s price and the knock-out barrier. The key is to analyze the delta of the option as it approaches the barrier. A standard call option’s delta typically ranges from 0 to 1. However, a knock-out call option’s delta behaves differently near the barrier. As the underlying asset’s price approaches the knock-out barrier from below, the option’s delta increases because a small upward movement in the underlying price can lead to the option being knocked out, making its value zero. This means the option’s value becomes highly sensitive to price changes near the barrier. Once the barrier is breached, the option ceases to exist, and its delta instantaneously becomes zero. The concept can be analogized to a dam holding back water. As the water level rises close to the dam’s crest (the barrier), the pressure on the dam (the option’s sensitivity to price changes) increases dramatically. A small increase in the water level can cause the dam to overflow or break (the option being knocked out), leading to a sudden release of pressure (the delta dropping to zero). The incorrect options present plausible but flawed understandings. Option (b) incorrectly assumes the delta decreases as the price approaches the barrier. Option (c) misunderstands the effect of breaching the barrier, suggesting the delta remains unchanged, and Option (d) provides an unrealistic scenario where the delta becomes infinite, which is not possible in a real-world option scenario.

Let’s break down the pricing of a European call option using a binomial tree model over two periods. This scenario is designed to test the understanding of risk-neutral valuation and the mechanics of option pricing in a multi-period setting. **Step 1: Calculate the Up and Down Factors** Given the volatility of 25% and two periods of 6 months each (0.5 years), we calculate the up (u) and down (d) factors using the following formulas: \[u = e^{\sigma \sqrt{\Delta t}}\] \[d = e^{-\sigma \sqrt{\Delta t}} = \frac{1}{u}\] Where: * \(\sigma\) = Volatility = 0.25 * \(\Delta t\) = Time step = 0.5 \[u = e^{0.25 \sqrt{0.5}} = e^{0.25 \times 0.7071} = e^{0.1768} \approx 1.1933\] \[d = \frac{1}{1.1933} \approx 0.8380\] **Step 2: Calculate the Risk-Neutral Probability (q)** The risk-neutral probability \(q\) is calculated as: \[q = \frac{e^{r \Delta t} – d}{u – d}\] Where: * \(r\) = Risk-free rate = 5% = 0.05 \[q = \frac{e^{0.05 \times 0.5} – 0.8380}{1.1933 – 0.8380} = \frac{e^{0.025} – 0.8380}{0.3553} = \frac{1.0253 – 0.8380}{0.3553} = \frac{0.1873}{0.3553} \approx 0.5271\] **Step 3: Calculate the Stock Prices at Each Node** * **Node S_uu:** \(S_0 \times u \times u = 100 \times 1.1933 \times 1.1933 \approx 142.40\) * **Node S_ud:** \(S_0 \times u \times d = 100 \times 1.1933 \times 0.8380 \approx 100.00\) * **Node S_dd:** \(S_0 \times d \times d = 100 \times 0.8380 \times 0.8380 \approx 70.22\) **Step 4: Calculate the Option Values at Expiry (Final Nodes)** The call option payoff is max(Stock Price – Strike Price, 0): * **C_uu:** max(142.40 – 105, 0) = 37.40 * **C_ud:** max(100.00 – 105, 0) = 0 * **C_dd:** max(70.22 – 105, 0) = 0 **Step 5: Backward Induction to Calculate Option Values at Earlier Nodes** * **C_u:** \(\frac{q \times C_{uu} + (1-q) \times C_{ud}}{e^{r \Delta t}} = \frac{0.5271 \times 37.40 + 0.4729 \times 0}{e^{0.05 \times 0.5}} = \frac{19.71}{1.0253} \approx 19.22\) * **C_d:** \(\frac{q \times C_{ud} + (1-q) \times C_{dd}}{e^{r \Delta t}} = \frac{0.5271 \times 0 + 0.4729 \times 0}{e^{0.05 \times 0.5}} = 0\) **Step 6: Calculate the Option Value at Time 0 (C_0)** \[C_0 = \frac{q \times C_u + (1-q) \times C_d}{e^{r \Delta t}} = \frac{0.5271 \times 19.22 + 0.4729 \times 0}{e^{0.05 \times 0.5}} = \frac{10.13}{1.0253} \approx 9.88\] Therefore, the price of the European call option is approximately 9.88. This example demonstrates the core principles of risk-neutral valuation. We use the risk-free rate to discount expected payoffs, reflecting the idea that in a well-arbitraged market, investors should not be able to earn risk-free profits above the risk-free rate. The binomial tree model is a discrete-time approximation of the continuous-time Black-Scholes model, and it allows for a step-by-step visualization of how option prices are derived based on possible future stock prices. The backward induction process is crucial, as it starts from the known option payoffs at expiration and works backward to determine the fair price today.

The investor’s profit or loss is calculated by comparing the final index value at expiration with the strike price, considering the premium paid. Since the investor bought a call option, they profit if the index exceeds the strike price. The profit is the difference between the index value at expiration and the strike price, minus the premium paid. If the index value at expiration is below the strike price, the option expires worthless, and the investor loses only the premium. Here, the strike price is 7500, the premium is 350, and the index value at expiration is 7900. Profit = (Index Value at Expiration – Strike Price) – Premium Profit = (7900 – 7500) – 350 Profit = 400 – 350 Profit = 50 The investor’s profit is 50. Now, consider a scenario involving a currency swap. A UK-based company, “BritCo,” has borrowed USD 10 million at a fixed rate of 5% per annum but prefers to have its liabilities in GBP. Simultaneously, a US-based company, “YankCorp,” has borrowed GBP 8 million (equivalent to USD 10 million at the initial exchange rate of 1.25 USD/GBP) at a fixed rate of 4% per annum but prefers USD liabilities. They enter into a currency swap agreement facilitated by a financial institution. The notional principal amounts are USD 10 million and GBP 8 million. BritCo pays YankCorp GBP 320,000 annually (4% of GBP 8 million), and YankCorp pays BritCo USD 500,000 annually (5% of USD 10 million). At maturity, the principal amounts are re-exchanged. This arrangement allows both companies to effectively convert their liabilities into their preferred currencies, hedging against exchange rate fluctuations. If, at maturity, the exchange rate has moved to 1.30 USD/GBP, the re-exchange of principals becomes critical. BritCo would need to provide GBP 7,692,307.69 (USD 10 million / 1.30) to YankCorp, which is less than the original GBP 8 million. YankCorp would still provide USD 10 million to BritCo. This illustrates how currency swaps manage both interest rate and exchange rate risks.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Green Harvest,” which wants to protect itself against fluctuations in the price of wheat. Green Harvest anticipates selling 5,000 metric tons of wheat in six months. They are considering using futures contracts traded on the London International Financial Futures and Options Exchange (LIFFE), now part of ICE Futures Europe, to hedge their price risk. The current futures price for wheat with delivery in six months is £200 per metric ton. To determine the number of contracts needed, we first calculate the total amount of wheat Green Harvest wants to hedge: 5,000 metric tons. Let’s assume each wheat futures contract on LIFFE covers 100 metric tons. Then the number of contracts required is 5,000 / 100 = 50 contracts. Green Harvest decides to sell 50 wheat futures contracts at £200 per ton. This locks in a selling price of £200 per ton for the wheat covered by the futures contracts. Now, let’s consider two scenarios: Scenario 1: In six months, the spot price of wheat is £180 per ton. Green Harvest sells its wheat in the spot market at £180 per ton. Simultaneously, they close out their futures position by buying back 50 contracts. Since they initially sold the futures at £200 and now buy them back at £180, they make a profit of £20 per ton on the futures contracts. The total profit on the futures is 50 contracts * 100 tons/contract * £20/ton = £100,000. The total revenue is the spot market sale (5,000 tons * £180/ton = £900,000) plus the futures profit (£100,000), which equals £1,000,000. The effective price received is £1,000,000 / 5,000 tons = £200 per ton. Scenario 2: In six months, the spot price of wheat is £220 per ton. Green Harvest sells its wheat in the spot market at £220 per ton. They close out their futures position by buying back 50 contracts. Since they initially sold the futures at £200 and now buy them back at £220, they incur a loss of £20 per ton on the futures contracts. The total loss on the futures is 50 contracts * 100 tons/contract * £20/ton = £100,000. The total revenue is the spot market sale (5,000 tons * £220/ton = £1,100,000) minus the futures loss (£100,000), which equals £1,000,000. The effective price received is £1,000,000 / 5,000 tons = £200 per ton. In both scenarios, the effective price received by Green Harvest is £200 per ton, demonstrating how futures contracts can be used to hedge price risk.

The question focuses on the impact of early exercise on American options, particularly in the context of dividend-paying assets. American options can be exercised at any time before expiration, which introduces complexities not present in European options. The key concept here is that an investor might choose to exercise an American call option early if the present value of the dividends forgone by not owning the underlying asset exceeds the time value of the option. The time value of an option reflects the probability that the option will become more valuable before expiration due to favorable price movements in the underlying asset. When dividends are involved, the decision becomes more nuanced. If the dividend yield is high enough, the investor might prefer to capture the dividends immediately by exercising the option and owning the stock. In this scenario, we need to consider the present value of the future dividends compared to the potential upside of the option. If the stock price is significantly above the strike price, the option is deep in the money, and the time value is relatively low. However, if the dividends are substantial, the investor may be better off exercising early. Let’s consider an example. Suppose a stock is trading at £120, and an American call option with a strike price of £100 is about to expire in 3 months. The company is expected to pay a dividend of £5 per share in one month. The risk-free interest rate is 5% per annum. The present value of the dividend is \( \frac{5}{1 + \frac{0.05}{12}} \approx £4.98 \). The intrinsic value of the option is £20 (£120 – £100). If the investor expects the stock price to remain relatively stable, they might prefer to exercise the option early to capture the dividend. However, if there is a significant chance that the stock price could increase substantially, the investor might prefer to hold the option. The decision to exercise early depends on a comparison of the dividend income with the potential gain from holding the option. If the dividend income exceeds the time value of the option, early exercise may be optimal. This is especially true when the option is deep in the money and the time value is low. In the case of put options, early exercise is more likely when the option is deep in the money and interest rates are high. By exercising the put option early, the investor receives the strike price and can invest it at the risk-free rate, earning interest income.

The question explores the complexities of valuing a European-style swaption, specifically focusing on the impact of interest rate volatility and correlation between different forward rates within the yield curve. The swaption grants the holder the right, but not the obligation, to enter into a swap at a predetermined future date (the expiration date). The value of a swaption is derived from the potential future value of the underlying swap. To accurately value a swaption, we need to model the evolution of interest rates over time. This is often done using models like the Black-Scholes model (adapted for interest rates) or more sophisticated models like the Libor Market Model (LMM). The Black-Scholes model assumes constant volatility, which is a simplification. The LMM, on the other hand, allows for time-varying volatility and correlation between different forward rates. The key challenge in this scenario is the correlation between the 1-year forward rate (1f1) and the 5-year forward rate (5f1). If these rates are perfectly correlated (correlation = 1), they move in lockstep, simplifying the modeling process. If they are negatively correlated (correlation = -1), they move in opposite directions, potentially creating more complex payoff scenarios for the swaption. A correlation of 0 implies no linear relationship between the movements of the two rates. High volatility in interest rates generally increases the value of options, including swaptions. This is because higher volatility increases the potential for the underlying swap to be “in the money” at expiration. However, the correlation between different forward rates can either amplify or dampen this effect. If rates are highly positively correlated, the impact of volatility is more predictable. If they are negatively correlated, the overall impact on the swaption value becomes less certain and requires careful modeling. The impact of regulations, such as MiFID II, on swaption trading is primarily through increased transparency and reporting requirements. This affects the liquidity and pricing of swaptions, as market participants have access to more information. The EMIR regulation requires central clearing of standardized OTC derivatives, which impacts counterparty risk and margin requirements for swaptions. These regulatory changes can indirectly influence the valuation of swaptions by affecting the cost of trading and hedging them. The correct answer will reflect the combined impact of volatility, correlation, and regulatory factors on the swaption’s value, acknowledging that high volatility generally increases option value, while the correlation structure influences the magnitude and uncertainty of that increase.

The question explores the application of put-call parity in a complex scenario involving transaction costs and early exercise considerations, demanding a nuanced understanding beyond the basic formula. Put-call parity, in its simplest form, states: \(C + PV(X) = P + S\), where \(C\) is the price of a European call option, \(PV(X)\) is the present value of the strike price, \(P\) is the price of a European put option, and \(S\) is the price of the underlying asset. However, this relationship is affected by transaction costs and the possibility of early exercise, especially for American options. The investor faces a situation where the observed market prices deviate from the theoretical put-call parity due to transaction costs. The strategy to exploit this mispricing involves simultaneously buying the undervalued side and selling the overvalued side of the parity equation. In this case, the put option is relatively expensive. 1. **Identify the Mispricing:** The put is overpriced relative to the call. This means the right side of the equation (\(P + S\)) is too high compared to the left side (\(C + PV(X)\)). 2. **Exploit the Mispricing:** To profit, we buy the underpriced assets and sell the overpriced assets. We buy the call and the present value of the strike price (essentially borrowing to buy the strike price at expiry) and sell the put and the stock. 3. **Consider Transaction Costs:** The strategy is viable only if the profit from the mispricing exceeds the total transaction costs. Let’s denote the transaction cost per trade as \(T\). Since we’re making four trades (buying the call, selling the put, buying the risk-free bond equivalent to the present value of the strike price, and selling the stock), the total transaction cost is \(4T\). 4. **Calculate the Profit:** The profit (\(\Pi\)) is the difference between the overvalued side and the undervalued side, minus the transaction costs. \[\Pi = (P + S) – (C + PV(X)) – 4T\] Substituting the given values: \[\Pi = (9.50 + 48) – (5.25 + 42.50) – 4(0.15)\] \[\Pi = 57.50 – 47.75 – 0.60\] \[\Pi = 9.75 – 0.60 = 9.15\] 5. **Early Exercise Risk:** The question mentions the put option is American, introducing the risk of early exercise. Early exercise is most likely when the stock price is significantly below the strike price, making the put deeply in the money. If the put is exercised early, the investor would need to buy back the stock immediately at the then-current market price, which could be higher than the price at which it was initially sold. This risk must be factored into the decision. However, in this calculation, we’re determining the *initial* arbitrage profit, assuming no early exercise. The risk of early exercise would influence the *decision* to enter the trade, but doesn’t change the initial calculated profit. Therefore, the initial arbitrage profit, considering transaction costs, is £9.15.

The correct answer is (a). This question assesses the understanding of exotic derivatives, specifically barrier options, and their sensitivity to volatility changes, also known as vega. A knock-out barrier option ceases to exist if the underlying asset’s price touches the barrier level. In this scenario, the call option is a down-and-out call, meaning it is knocked out if the price falls *below* the barrier. Here’s why the other options are incorrect: * **(b)** While volatility generally increases option prices, the effect on barrier options is not straightforward. Increased volatility *increases* the chance of hitting the barrier. For a knock-out option, this increases the probability that the option will expire worthless. Thus, increased volatility decreases the option’s value. The statement that the option’s value will increase is therefore incorrect. * **(c)** A knock-in option would increase in value with increased volatility. However, the question states that it is a knock-out option. * **(d)** While delta hedging aims to neutralize directional price risk, it does not eliminate the risk of volatility changes (vega risk). Delta hedging only works for small price movements in the underlying asset. Large or rapid price movements, especially those that trigger the barrier, can result in significant losses even with delta hedging. Delta is the change in option price with respect to the change in underlying asset price, and it is a first order derivative, therefore it only provides a local approximation. Gamma is the second order derivative which represents the rate of change of delta with respect to the change in the underlying asset price.

Let’s analyze the hedging strategy and its impact on the final profit. The company initially hedges 75% of its expected production using futures contracts. The unhedged portion is exposed to the spot market price at the time of sale. The hedging gain or loss is calculated by comparing the initial futures price with the final futures price and multiplying by the number of contracts and the contract size. The spot market revenue is determined by the spot price at the time of sale multiplied by the unhedged quantity. The final profit is the sum of the hedging gain/loss and the spot market revenue, minus the initial production cost. Here’s the breakdown of the calculation: 1. **Hedged Quantity:** 75% of 100,000 barrels = 75,000 barrels 2. **Unhedged Quantity:** 25% of 100,000 barrels = 25,000 barrels 3. **Number of Futures Contracts:** 75,000 barrels / 1,000 barrels per contract = 75 contracts 4. **Hedging Gain/Loss:** (Final Futures Price – Initial Futures Price) * Number of Contracts * Contract Size = (\(79 – 82\)) * 75 * 1000 = \(-3 * 75 * 1000 = -\)£225,000 5. **Spot Market Revenue:** Spot Price * Unhedged Quantity = \(81 * 25,000 = \)£2,025,000 6. **Total Revenue:** Hedging Gain/Loss + Spot Market Revenue = \(-225,000 + 2,025,000 = \)£1,800,000 7. **Total Production Cost:** £75 per barrel * 100,000 barrels = £7,500,000 8. **Final Profit:** Total Revenue – Total Production Cost = \(1,800,000 – 7,500,000 = -\)£5,700,000 Now, let’s consider an alternative scenario where the company fully hedges its production. If the company had hedged 100% of its production (100,000 barrels), it would have used 100 futures contracts. The hedging gain/loss would then be (\(79 – 82\)) * 100 * 1000 = \(-3 * 100 * 1000 = -\)£300,000. The spot market revenue would be zero because all production is hedged. Therefore, the total revenue would be -£300,000, and the final profit would be \(-300,000 – 7,500,000 = -\)£7,800,000. This illustrates that while hedging can reduce price risk, it does not guarantee a profit. The key takeaway is that hedging strategies are designed to mitigate price volatility but do not eliminate the risk of losses, especially when production costs are high relative to market prices. The effectiveness of a hedging strategy depends on the specific price movements and the proportion of production that is hedged.

Let’s consider a scenario where a UK-based airline, “Skylar Airways,” aims to hedge its jet fuel costs against potential price increases over the next six months. Skylar uses a combination of futures and options to manage this risk. They enter into a futures contract to purchase 1,000,000 gallons of jet fuel six months from now at a price of £2.50 per gallon. Simultaneously, to limit potential losses if fuel prices unexpectedly decrease, they purchase put options on jet fuel with a strike price of £2.40 per gallon, paying a premium of £0.05 per gallon. Now, let’s analyze a specific outcome. Suppose that at the expiration of the futures contract and the options, the spot price of jet fuel is £2.30 per gallon. First, consider the futures contract. Skylar Airways is obligated to buy fuel at £2.50 per gallon, while the market price is £2.30. This results in a loss of £0.20 per gallon on the futures contract, totaling a loss of \(1,000,000 \times £0.20 = £200,000\). Next, examine the put options. Since the spot price (£2.30) is below the strike price (£2.40), Skylar Airways will exercise the put options. Exercising the options allows them to sell fuel at £2.40 per gallon, effectively offsetting some of the losses from the futures contract. The profit from the put options is £0.10 per gallon (strike price – spot price). However, we must subtract the premium paid for the options (£0.05 per gallon). Therefore, the net profit per gallon from the options is \(£0.10 – £0.05 = £0.05\). Across 1,000,000 gallons, this yields a profit of \(1,000,000 \times £0.05 = £50,000\). Finally, the net effect is the sum of the loss on the futures contract and the profit from the put options: \(-£200,000 + £50,000 = -£150,000\). Therefore, Skylar Airways experiences a net loss of £150,000 despite using a hedging strategy. This example highlights the importance of understanding that hedging doesn’t guarantee profits; it primarily aims to reduce risk, and in this case, it mitigated a potentially larger loss had they not used any hedging instruments. The options strategy acted as an insurance policy, limiting the downside but also reducing the potential upside.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their behavior around the barrier level. It requires calculating the probability of the underlying asset breaching the barrier before the option’s expiration, impacting the option’s payoff. The scenario involves a knock-out barrier option, where the option becomes worthless if the barrier is hit. First, we need to calculate the probability of the asset price reaching the barrier. This involves understanding the concept of reflection principle in Brownian motion. If the asset price reaches the barrier, it’s as if the path is reflected across the barrier. Given the current asset price \( S_0 = 100 \), the barrier level \( B = 120 \), volatility \( \sigma = 0.3 \), time to maturity \( T = 1 \) year, and risk-free rate \( r = 0.05 \), we can calculate the probability of hitting the barrier. The probability \( P \) of hitting the barrier is given by: \[ P = N\left(\frac{\ln(S_0/B) + (r – 0.5\sigma^2)T}{\sigma\sqrt{T}}\right) + N\left(\frac{\ln(S_0/B) – (r – 0.5\sigma^2)T}{\sigma\sqrt{T}}\right) \] Where \( N(x) \) is the cumulative standard normal distribution function. Let’s calculate the values inside the \( N \) functions: \[ \frac{\ln(100/120) + (0.05 – 0.5 \times 0.3^2) \times 1}{0.3 \times \sqrt{1}} = \frac{\ln(0.833) + (0.05 – 0.045)}{0.3} = \frac{-0.182 + 0.005}{0.3} = \frac{-0.177}{0.3} = -0.59 \] \[ \frac{\ln(100/120) – (0.05 – 0.5 \times 0.3^2) \times 1}{0.3 \times \sqrt{1}} = \frac{\ln(0.833) – (0.05 – 0.045)}{0.3} = \frac{-0.182 – 0.005}{0.3} = \frac{-0.187}{0.3} = -0.623 \] So, \( P = N(-0.59) + N(-0.623) \) Using standard normal distribution tables or a calculator, we find: \( N(-0.59) \approx 0.2776 \) \( N(-0.623) \approx 0.2666 \) Therefore, \( P = 0.2776 + 0.2666 = 0.5442 \) The probability of *not* hitting the barrier is \( 1 – P = 1 – 0.5442 = 0.4558 \) If the barrier is not hit, the option pays out the difference between the final asset price and the strike price, if positive. The expected final asset price is \( S_0 \cdot e^{rT} = 100 \cdot e^{0.05 \times 1} = 100 \cdot 1.0513 = 105.13 \) Since the strike price is 110, the option would only pay out if the asset price exceeds 110. However, since the probability of hitting the barrier is considered, we calculate the probability of the option *not* being knocked out and then consider the discounted expected payoff. The expected payoff is zero because the expected final asset price (105.13) is less than the strike price (110). Therefore, the expected payoff of the option is \( 0.4558 \times \text{max}(105.13 – 110, 0) = 0.4558 \times 0 = 0 \). However, the question asks for the probability that the option will *have* a payoff, meaning the barrier is not hit, *and* the final asset price is above the strike price. We have already calculated the probability of not hitting the barrier as 0.4558. We need to estimate the probability of the asset price being above 110, given that it didn’t hit the barrier at 120. This is a complex calculation involving conditional probabilities and is beyond a simple analytical solution without simulation. The closest answer is therefore the probability of not hitting the barrier, adjusted for the likelihood of being in the money. Given the expected asset price is below the strike, and without further complex calculations, we can approximate that the probability of the option having a payoff is low, but not zero. 0.15 is the most plausible estimate.

The core of this question revolves around understanding how different market dynamics influence the pricing of exotic derivatives, specifically barrier options. A barrier option’s value is highly sensitive to the underlying asset’s volatility and its proximity to the barrier level. The question probes the interaction between implied volatility skew, a sudden market crash, and the subsequent impact on a down-and-out call option. Here’s a breakdown of why each option is correct or incorrect: * **Option a (Correct):** This option correctly identifies the combined impact. The initial increase in implied volatility across all strikes increases the option’s price. The market crash, while seemingly detrimental, *decreases* the probability of the barrier being hit, thus *increasing* the option’s value since it’s a down-and-out. The increased skew further benefits the down-and-out call, as the increased implied volatility at lower strikes (relative to higher strikes) makes it less likely the underlying asset will reach the barrier. * **Option b (Incorrect):** This option incorrectly assumes that the market crash would automatically render the option worthless. While a crash increases the risk of hitting the barrier, it doesn’t guarantee it, especially if the barrier is set significantly lower than the pre-crash price. The implied volatility and skew effects are also ignored. * **Option c (Incorrect):** This option incorrectly assumes that the increased implied volatility would be offset by the crash. While increased volatility increases the chance of hitting the barrier, the crash itself provides a cushion, moving the underlying asset further away from the barrier. Additionally, it doesn’t account for the impact of the skew. * **Option d (Incorrect):** This option incorrectly assumes that the increased skew would automatically decrease the option’s value. For a down-and-out *call*, increased skew (higher implied volatility at lower strikes) *decreases* the probability of the barrier being hit, thus *increasing* the option’s value. It also doesn’t fully account for the initial volatility increase and the crash’s impact. In summary, the correct answer requires a nuanced understanding of how volatility, skew, and market shocks interact to affect the value of a barrier option. It tests the candidate’s ability to go beyond simple textbook definitions and apply their knowledge to a complex, real-world scenario.

The core of this question lies in understanding how different option strategies behave under varying market conditions, specifically focusing on volatility and time decay (theta). A short strangle profits when the underlying asset’s price remains within a defined range. The investor sells both a call and a put option with different strike prices. The maximum profit is limited to the net premium received. The maximum loss is unlimited on the call side if the price rises significantly and substantial on the put side if the price falls significantly. Volatility plays a crucial role. Increased volatility increases the value of both the call and put options, potentially leading to losses for the short strangle position. Conversely, decreased volatility reduces the value of the options, increasing the likelihood of profit. Time decay (theta) is beneficial for a short strangle. As time passes, the value of both the call and put options decreases, especially as they move further away from their expiration date. This decay in value contributes to the profitability of the strategy, assuming the underlying asset’s price remains within the desired range. The calculation of the net profit/loss involves summing the initial premiums received and subtracting any losses incurred if either option is exercised or requires a buy-back at a higher price due to market movements. In this scenario, we need to analyze how the change in volatility and the passage of time affect the overall profitability of the short strangle position. Here’s how we approach the problem: 1. **Initial Premium Received:** £3.50 (call) + £2.75 (put) = £6.25 per share 2. **Call Option Outcome:** The price increased to £106. The investor needs to buy back the call option. The loss on the call option is (£106 – £105) + £1.50 (buy back premium) = £2.50. 3. **Put Option Outcome:** The price remained above the put strike price, so the put option expires worthless. The profit on the put option is the initial premium of £2.75. 4. **Net Profit/Loss:** £6.25 (total premium) – £2.50 (call loss) + £2.75 (put profit) = £6.50 per share. 5. **Total Profit/Loss:** £6.50 * 1000 shares = £6,500. Therefore, the investor made a profit of £6,500.

To determine the profit or loss from the close-out of the short strangle, we need to consider the initial premiums received and the final prices at which the options were closed. The investor initially sold a call option with a strike price of 160 for a premium of £4 and a put option with a strike price of 140 for a premium of £3. This means the investor initially received a total premium of £7 (£4 + £3). At expiry, the call option (strike 160) is bought back at £2, and the put option (strike 140) is bought back at £1. Therefore, the total cost to close out the position is £3 (£2 + £1). The profit or loss is calculated as the initial premium received minus the cost to close out the position. In this case, it is £7 (initial premium) – £3 (close-out cost) = £4. Therefore, the investor made a profit of £4. Now, let’s consider a unique analogy: Imagine you’re a street performer. You promise to juggle flaming torches (call option) and chainsaws (put option) for a certain price (premium). Someone pays you £4 to juggle the torches (call premium) and £3 to juggle the chainsaws (put premium). Later, you decide to hire someone else to do the juggling for you. You pay them £2 to juggle the torches and £1 to juggle the chainsaws. Your profit is the initial money you received (£7) minus what you paid the other juggler (£3), which leaves you with £4 profit. Another example: Consider a farmer who sells crop insurance (short strangle) to protect against extreme weather. He receives premiums for insuring against both drought (call) and flood (put). If the weather is moderate, he buys back the insurance policies at a lower price than he sold them for, making a profit. The key is understanding the initial income and the subsequent cost of fulfilling (or, in this case, avoiding fulfilling) the obligations.

The core of this question revolves around understanding how margin requirements work for futures contracts, particularly when considering potential losses and the investor’s risk tolerance. The initial margin is the amount required to open a futures position, and the maintenance margin is the level below which the account cannot fall. A margin call is triggered when the account equity drops below the maintenance margin, requiring the investor to deposit additional funds to bring the account back to the initial margin level. In this scenario, we need to calculate the maximum price decline the investor can withstand before receiving a margin call. The calculation involves finding the difference between the initial margin and the maintenance margin. This difference represents the buffer the investor has before a margin call is issued. Then, we divide this buffer by the contract size multiplier to determine the maximum allowable price decrease per contract. Here’s the calculation: 1. Calculate the margin buffer: Initial Margin – Maintenance Margin = £6,000 – £4,500 = £1,500 2. Calculate the allowable price decrease: Margin Buffer / Contract Multiplier = £1,500 / 10 = £150 Therefore, the maximum price decline the investor can tolerate before a margin call is triggered is £150. Now, let’s consider why the other options are incorrect. Option B incorrectly calculates the allowable price decrease by subtracting the maintenance margin from the contract price and dividing by the contract multiplier, which is not relevant to the margin call trigger. Option C calculates the additional margin required to meet the initial margin after a price decline of £100, but it doesn’t represent the maximum allowable decline before a margin call. Option D calculates the price decline based on the initial margin only, ignoring the maintenance margin, which is the key factor in triggering a margin call. This example highlights the practical application of margin requirements in futures trading and the importance of understanding the relationship between initial margin, maintenance margin, and contract multipliers. It also showcases how risk tolerance is linked to the ability to withstand price fluctuations before facing a margin call. A risk-averse investor would likely maintain a higher margin level to avoid the risk of margin calls.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their pricing sensitivities (Greeks) under different market conditions. It tests the candidate’s ability to analyze how the ‘knock-in’ feature of a barrier option affects its delta and gamma as the underlying asset price approaches the barrier. Delta measures the sensitivity of the option’s price to a change in the underlying asset’s price. Gamma measures the rate of change of delta with respect to changes in the underlying asset’s price. A standard call option has a positive delta, meaning its price increases as the underlying asset price increases. A standard call option also has a positive gamma, meaning its delta increases as the underlying asset price increases. For a knock-in call option, the option only becomes active if the underlying asset price reaches the barrier. Before the barrier is reached, the option has little to no value, and therefore a delta close to zero. As the underlying asset price approaches the barrier from below, the delta increases rapidly as the probability of the option knocking in increases. This rapid change in delta means that the gamma is also high as the asset price nears the barrier. If the underlying asset price exceeds the barrier, the knock-in option behaves like a regular call option. The delta will be positive and relatively stable. The gamma will decrease as the asset price moves further away from the barrier. Therefore, the delta is highest when the underlying asset price is slightly below the barrier, and the gamma is highest when the underlying asset price is close to the barrier. The correct answer is therefore a) because it accurately describes the delta and gamma behavior of a knock-in call option as the underlying asset price approaches and exceeds the barrier.

The core of this question revolves around understanding how different exotic options can be combined to create a payoff profile that mimics a more standard option, and the arbitrage opportunities that arise when pricing discrepancies exist. Specifically, it explores the concept of a butterfly spread constructed using one-touch options and compares its payoff to a vanilla call option. Let’s break down the construction of the butterfly spread using one-touch options: * **One-Touch Options:** These options pay a fixed amount if the underlying asset touches a pre-determined barrier price at any point during the option’s life. * **Butterfly Spread Construction:** The butterfly spread is constructed by buying one one-touch option with a barrier slightly below the strike price of the target vanilla call option, selling two one-touch options with a barrier at the strike price, and buying one one-touch option with a barrier slightly above the strike price. The payoff profile mimics a vanilla call option because: * If the underlying asset stays below the lower barrier, all one-touch options expire worthless, similar to a call option where the asset price is far below the strike. * If the underlying asset touches the middle barrier (the strike price), the short one-touch options pay out, but this is offset by the potential payout from the long one-touch options if the asset moves further. * If the underlying asset moves significantly above the strike price, the upper one-touch option pays out, mirroring the increasing payoff of a vanilla call option. Now, consider the arbitrage opportunity. If the price of this synthetic butterfly spread is significantly lower than the price of an equivalent vanilla call option, an arbitrageur can profit by: 1. Buying the synthetic butterfly spread (the one-touch options). 2. Selling the vanilla call option. The arbitrageur locks in a risk-free profit equal to the price difference, assuming the payoff profiles are truly equivalent. However, real-world factors like transaction costs, liquidity, and the imperfect replication of the payoff profile can affect the profitability of such an arbitrage strategy. The question emphasizes understanding the theoretical underpinnings and potential pitfalls of such a strategy. The key is recognizing that even a near-perfect replication strategy can be impacted by market microstructure issues.

Let’s consider a scenario involving a UK-based agricultural cooperative, “GreenHarvest Co-op,” which wants to hedge its future wheat sales using futures contracts traded on the ICE Futures Europe exchange. GreenHarvest anticipates harvesting 500 tonnes of wheat in three months. The current price of wheat futures for delivery in three months is £200 per tonne. The co-op decides to hedge 75% of its expected harvest to protect against a potential price decline. The co-op sells 375 tonnes (75% of 500 tonnes) of wheat futures contracts. Each ICE wheat futures contract represents 100 tonnes of wheat. Therefore, GreenHarvest sells 3.75 contracts. Since you cannot trade fractions of contracts, the co-op sells 4 contracts (400 tonnes). This is an over-hedge, but it’s the closest they can get. Now, let’s assume that in three months, the spot price of wheat is £180 per tonne, and the futures price converges to the spot price (as expected at delivery). GreenHarvest sells its 500 tonnes of wheat in the spot market for £180 per tonne, receiving £90,000 (500 tonnes * £180/tonne). Simultaneously, GreenHarvest closes out its futures position by buying back 4 wheat futures contracts at £180 per tonne. The initial sale price was £200 per tonne, so the profit on the futures contracts is £20 per tonne (£200 – £180). Since they hedged 400 tonnes (4 contracts * 100 tonnes/contract), the total profit on the futures contracts is £8,000 (400 tonnes * £20/tonne). The effective price received by GreenHarvest is the sum of the revenue from the spot market sale and the profit from the futures contracts. The revenue from the spot market is £90,000. The profit from the futures contracts is £8,000. Therefore, the effective price is £98,000. The effective price per tonne for the entire 500 tonnes is £196 (£98,000 / 500 tonnes). However, because GreenHarvest over-hedged, they sold 400 tonnes of futures contracts when they only needed to hedge 375 tonnes to cover 75% of their production. This means they had an extra 25 tonnes hedged that wasn’t strictly necessary for their risk management strategy. If GreenHarvest had perfectly hedged only 75% of their production, they would have sold 3.75 contracts, which is impossible. By over-hedging, they locked in a price closer to the original futures price for a larger portion of their production, but also exposed themselves to the risk of missing out on potential gains if the spot price had increased. The key concept here is that futures contracts can be used to hedge against price risk, but the effectiveness of the hedge depends on the accuracy of the hedge ratio and the convergence of the futures price to the spot price at delivery. Over-hedging and under-hedging can both lead to deviations from the desired outcome.

The question assesses the understanding of the impact of different delta hedging strategies on portfolio performance under varying market conditions, specifically focusing on gamma exposure and its implications for rebalancing costs and profitability. Let’s break down the concepts and then address the specific scenario: * **Delta Hedging:** This strategy aims to neutralize the directional risk (delta) of an option position. A delta of 0.5 means that for every £1 move in the underlying asset, the option’s value changes by £0.50. To hedge, you would typically short 50 shares for every long option contract (assuming one contract represents 100 shares). * **Gamma:** Gamma measures the rate of change of delta with respect to changes in the underlying asset’s price. High gamma means the delta changes rapidly as the underlying price moves. This necessitates frequent rebalancing of the hedge, incurring transaction costs. * **Theta:** Theta measures the time decay of an option. It represents how much the option’s value decreases each day as it approaches expiration, all else being equal. * **Volatility:** The implied volatility of an option reflects the market’s expectation of future price fluctuations in the underlying asset. Higher volatility generally increases option prices. In this scenario, the portfolio manager is using a delta-neutral strategy. High gamma implies that the delta changes rapidly, requiring frequent rebalancing. When volatility increases unexpectedly, the value of the options in the portfolio will increase. However, the increased volatility also increases the cost of rebalancing the delta hedge. A perfect hedge is impossible in practice due to transaction costs and the discrete nature of trading. Gamma represents the “imperfection” of the delta hedge. The portfolio manager must balance the cost of rebalancing with the desire to maintain a delta-neutral position. If the portfolio manager rebalances too infrequently, the portfolio’s delta will deviate significantly from zero, exposing the portfolio to directional risk. If the portfolio manager rebalances too frequently, the transaction costs will erode profits. The optimal rebalancing frequency depends on the trade-off between these two factors. The portfolio manager must consider the portfolio’s gamma, the volatility of the underlying asset, and the transaction costs when deciding how often to rebalance. In a high volatility environment, the portfolio manager might choose to widen the acceptable delta range to reduce rebalancing costs. Alternatively, they might use a more sophisticated hedging strategy, such as a gamma-neutral strategy, which aims to neutralize both delta and gamma risk.

To determine the profit or loss on the swap, we need to calculate the net cash flows exchanged over the life of the swap. The company receives fixed payments and makes floating payments. The net profit/loss is the sum of these differences. Year 1: Receive 4.5%, Pay 4.2% => Net Receive = 0.3% Year 2: Receive 4.5%, Pay 4.7% => Net Pay = 0.2% Year 3: Receive 4.5%, Pay 5.1% => Net Pay = 0.6% Year 4: Receive 4.5%, Pay 4.8% => Net Pay = 0.3% Year 5: Receive 4.5%, Pay 4.6% => Net Pay = 0.1% Total Net Flow = 0.3% – 0.2% – 0.6% – 0.3% – 0.1% = -0.9% Since the notional principal is £10 million, the total loss is -0.9% of £10 million, which is -£90,000. Now, let’s think of an analogy. Imagine a farmer who agrees to swap his harvest (fixed amount) for the market price of grain (floating amount) over five years. In the first year, the market price is lower than expected, so he benefits. However, in the following years, the market price rises above the fixed amount he agreed to, resulting in losses. The total profit or loss is the sum of these differences each year. This is similar to the swap agreement where the company is either receiving more or paying more based on the difference between the fixed and floating rates. The key here is to understand that a swap is a series of cash flows exchanged over time. The profit or loss is simply the sum of the net cash flows. In this case, the floating rates were generally higher than the fixed rate, resulting in a net loss for the company. It’s also important to note that the notional principal is only used to calculate the cash flows and is not actually exchanged. This is a critical aspect of understanding how swaps work. Furthermore, the regulations surrounding swaps, particularly those impacting investment advice, require firms to accurately assess and disclose these potential gains and losses to clients, ensuring they understand the risks involved in these derivative contracts, as per FCA guidelines and MiFID II regulations.

Let’s break down how to value a European call option using a two-step binomial tree, incorporating risk-neutral probabilities. The core principle is that the option’s value today is the discounted expected payoff at expiration, calculated using risk-neutral probabilities. Risk-neutral probabilities are crucial because they allow us to value derivatives without needing to know investors’ actual risk preferences. We assume all investors are risk-neutral, meaning they don’t require a risk premium. First, calculate the up (u) and down (d) factors. Given a volatility of 20% (0.20) and two steps (n=2) over one year (T=1), the time step is T/n = 1/2 = 0.5 years. \[u = e^{\sigma \sqrt{\Delta t}} = e^{0.20 \sqrt{0.5}} = e^{0.1414} \approx 1.1514\] \[d = \frac{1}{u} = \frac{1}{1.1514} \approx 0.8685\] Next, calculate the risk-neutral probability (p): \[p = \frac{e^{r \Delta t} – d}{u – d} = \frac{e^{0.05 \times 0.5} – 0.8685}{1.1514 – 0.8685} = \frac{1.0253 – 0.8685}{0.2829} \approx 0.5542\] \[1-p = 1 – 0.5542 = 0.4458\] Now, build the binomial tree. The initial stock price is £100. * **Step 1:** * Up node: £100 * 1.1514 = £115.14 * Down node: £100 * 0.8685 = £86.85 * **Step 2:** * Up-Up node: £115.14 * 1.1514 = £132.57 * Up-Down node: £115.14 * 0.8685 = £100 (approximately, due to rounding) * Down-Down node: £86.85 * 0.8685 = £75.43 Calculate the option payoffs at expiration (strike price of £105): * Up-Up: max(£132.57 – £105, 0) = £27.57 * Up-Down: max(£100 – £105, 0) = £0 * Down-Down: max(£75.43 – £105, 0) = £0 Now, work backward through the tree, calculating the option values at each node: * **At the Up node (Step 1):** \[C_u = e^{-r \Delta t} [p \times C_{uu} + (1-p) \times C_{ud}] = e^{-0.05 \times 0.5} [0.5542 \times 27.57 + 0.4458 \times 0] = 0.9753 \times 15.279 \approx 14.89\] * **At the Down node (Step 1):** \[C_d = e^{-r \Delta t} [p \times C_{ud} + (1-p) \times C_{dd}] = e^{-0.05 \times 0.5} [0.5542 \times 0 + 0.4458 \times 0] = 0\] Finally, calculate the option value today: \[C_0 = e^{-r \Delta t} [p \times C_u + (1-p) \times C_d] = e^{-0.05 \times 0.5} [0.5542 \times 14.89 + 0.4458 \times 0] = 0.9753 \times 8.251 \approx 8.05\] Therefore, the value of the European call option is approximately £8.05.

Let’s analyze a complex scenario involving a customized exotic derivative. This derivative, a “Cliquet Range Accrual Swap,” links payments to the performance of the FTSE 100 index, but with unique twists. Firstly, the accrual only occurs if the FTSE 100 stays within a specific range (defined as 6800-7800) for more than 75% of the trading days in a quarter. Secondly, there’s a “ratchet” feature: if the FTSE 100 ends any quarter above 7500, the lower bound of the range for the *next* quarter increases by 50 points. However, if the FTSE 100 ends below 7000, the upper bound decreases by 50 points for the next quarter. The swap’s notional principal is £10 million, and the accrual rate is 3% per annum, paid quarterly *only* if the range condition is met. Now, consider the regulatory implications under MiFID II and its impact on classifying this instrument for different client types. A key aspect is assessing the derivative’s complexity and transparency, which dictates whether it can be offered to retail clients or is restricted to professional clients and eligible counterparties. The ratchet feature introduces path dependency, making valuation and risk assessment significantly more complex. Furthermore, the specific percentage of trading days within the range adds another layer of intricacy not typically found in standard derivatives. The suitability assessment under COBS (Conduct of Business Sourcebook) requires advisors to understand the client’s knowledge and experience, financial situation, and investment objectives. This exotic derivative poses a challenge because its performance is highly sensitive to market conditions and the specific range parameters, which change dynamically. The advisor must demonstrate a clear understanding of how the ratchet mechanism affects potential payouts and the overall risk profile. The key is whether the client truly comprehends these complexities and can bear the potential losses if the FTSE 100 consistently falls outside the range. If the client incorrectly believes that *any* movement of the FTSE 100 will result in some payment, regardless of the range and percentage criteria, that’s a crucial misunderstanding that renders the derivative unsuitable.

To determine the fair value of the swap, we need to discount the expected future cash flows. The fixed leg pays a fixed rate of 3.5% annually on a notional principal of £10 million. The floating leg pays SONIA + 100 bps (1%). We need to project the SONIA rates for the next three years and discount the cash flows using the appropriate discount factors. We will use the provided forward rates to estimate future SONIA rates. Year 1 SONIA forecast: 4.0% Year 2 SONIA forecast: 4.0% + 0.5% = 4.5% Year 3 SONIA forecast: 4.5% + 0.5% = 5.0% The floating rate payments are therefore 4.0% + 1.0% = 5.0% for Year 1, 4.5% + 1.0% = 5.5% for Year 2, and 5.0% + 1.0% = 6.0% for Year 3. The fixed rate payment is always 3.5%. The cash flows are calculated as follows (on £10 million notional): Year 1: Floating = £10,000,000 * 5.0% = £500,000; Fixed = £10,000,000 * 3.5% = £350,000; Net = £500,000 – £350,000 = £150,000 Year 2: Floating = £10,000,000 * 5.5% = £550,000; Fixed = £10,000,000 * 3.5% = £350,000; Net = £550,000 – £350,000 = £200,000 Year 3: Floating = £10,000,000 * 6.0% = £600,000; Fixed = £10,000,000 * 3.5% = £350,000; Net = £600,000 – £350,000 = £250,000 Now, discount these cash flows using the spot rates: Year 1: £150,000 / (1 + 0.03)^1 = £145,631.07 Year 2: £200,000 / (1 + 0.035)^2 = £186,834.13 Year 3: £250,000 / (1 + 0.04)^3 = £222,246.62 Sum of present values: £145,631.07 + £186,834.13 + £222,246.62 = £554,711.82 Since the net present value is positive, it indicates the swap is an asset to the party receiving the floating rate (and paying the fixed rate). This means the fair value is £554,711.82. A negative NPV would indicate the swap is a liability. The forward rates help project future floating rate payments, while spot rates are used to discount future cash flows to present value. The difference between fixed and floating payments at each period is what gets discounted.