Derivatives Level 4 (Investment Advice Diploma) Quiz Exam Quiz 25

The core of this question lies in understanding how early termination clauses in swaps are valued and how they impact the Net Present Value (NPV) of the swap. When a swap is terminated early, a break fee is usually paid by one party to the other to compensate for the loss of future cash flows. This break fee is essentially the NPV of the remaining cash flows of the swap, discounted at the current market rates. The calculation involves several steps: 1. **Determining the Remaining Cash Flows:** Identify the remaining interest payments that would have occurred had the swap not been terminated. In this case, we have three remaining payments. 2. **Discounting the Cash Flows:** Discount each of these future cash flows back to the present using the appropriate discount rate (LIBOR + spread). The formula for discounting a single cash flow is: \[PV = \frac{CF}{(1 + r)^n}\] Where: * \(PV\) = Present Value * \(CF\) = Cash Flow * \(r\) = Discount Rate * \(n\) = Number of periods 3. **Calculating the Break Fee:** Sum the present values of all the remaining cash flows. This sum represents the break fee, which is the amount Party A must pay to Party B. 4. **Impact on NPV:** The break fee directly affects the NPV of the swap for both parties. For Party A, the payer of the break fee, it’s an outflow. For Party B, the receiver, it’s an inflow. In this specific scenario, the fixed rate is 4%, and the notional principal is £10 million. This means Party A pays £400,000 annually to Party B. We discount these payments using the LIBOR rate plus the spread. The LIBOR rates are given for each year. Let’s calculate the present value of each cash flow: * Year 1: LIBOR is 4.5%, spread is 0.5%, so the discount rate is 5%. \[PV_1 = \frac{£400,000}{(1 + 0.05)^1} = £380,952.38\] * Year 2: LIBOR is 5%, spread is 0.5%, so the discount rate is 5.5%. \[PV_2 = \frac{£400,000}{(1 + 0.055)^2} = £358,422.46\] * Year 3: LIBOR is 5.5%, spread is 0.5%, so the discount rate is 6%. \[PV_3 = \frac{£400,000}{(1 + 0.06)^3} = £335,820.55\] The total break fee is the sum of these present values: \[Total Break Fee = £380,952.38 + £358,422.46 + £335,820.55 = £1,075,195.39\] Therefore, Party A would need to pay approximately £1,075,195.39 to Party B to terminate the swap early. This represents the present value of the future payments Party B is giving up.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their sensitivity to market volatility. The scenario presents a situation where a portfolio manager uses a down-and-out call option to hedge a portion of their equity holdings. The key is to understand how a sudden spike in implied volatility, coupled with the underlying asset price nearing the barrier, affects the option’s delta and gamma. Delta measures the sensitivity of the option price to a change in the underlying asset price. Gamma measures the rate of change of delta with respect to changes in the underlying asset price. Near the barrier, a down-and-out call option’s delta becomes highly sensitive. As the underlying asset price approaches the barrier, the delta can swing dramatically, approaching zero as the barrier is breached. This effect is amplified by increased volatility. Higher volatility means a wider range of possible prices for the underlying asset, increasing the probability of hitting the barrier. Gamma also increases significantly near the barrier. This means the delta is changing rapidly as the underlying asset price fluctuates. The portfolio manager needs to understand that a small price movement in the underlying asset near the barrier can lead to a large change in the option’s delta, requiring frequent adjustments to maintain the hedge. The correct answer will reflect the combined effect of approaching the barrier and increasing volatility on delta and gamma. Incorrect answers might focus on the effect of only one factor or misunderstand the direction of the change. The scenario highlights the complexity of exotic options and the importance of understanding their sensitivity to market conditions. It also tests the ability to apply these concepts in a practical portfolio management setting.

Let’s break down how to approach this complex scenario. First, we need to understand the mechanics of quanto swaps. A quanto swap is a type of cross-currency derivative where the payments are made in one currency, but the underlying asset is denominated in another currency. This eliminates the exchange rate risk for one of the parties involved. The formula to determine the fixed rate in a quanto swap involves several factors: the interest rate differential between the two currencies, the volatility of the underlying asset, and the correlation between the asset’s return and the exchange rate. In this scenario, we have a fund manager in the UK (currency GBP) who wants exposure to the Nikkei 225 (denominated in JPY) without taking on currency risk. The quanto swap allows them to receive the return of the Nikkei 225 in GBP. The fixed rate they receive is determined by the JPY interest rate, the GBP interest rate, the volatility of the Nikkei 225, and the correlation between the Nikkei 225 and the GBP/JPY exchange rate. The formula to approximate the fixed rate (FR) in a quanto swap can be expressed as: FR ≈ JPY rate – GBP rate + (Volatility of Nikkei * Correlation * Volatility of GBP/JPY) Let’s apply the given values: JPY rate = 0.1% = 0.001 GBP rate = 4.5% = 0.045 Volatility of Nikkei = 20% = 0.20 Correlation = 0.7 Volatility of GBP/JPY = 15% = 0.15 FR ≈ 0.001 – 0.045 + (0.20 * 0.7 * 0.15) FR ≈ -0.044 + (0.021) FR ≈ -0.023 Converting this to a percentage: FR ≈ -0.023 * 100 = -2.3% Therefore, the closest fixed rate the fund manager should expect to receive is -2.3%. This negative rate reflects the cost of hedging the currency risk and the interest rate differential between JPY and GBP. The negative rate means that the fund manager will have to pay 2.3% of the notional to receive the return of Nikkei 225. The fund manager is paying for the benefit of receiving Nikkei 225 return in GBP without currency risk.

Let’s analyze the expected payoff of the Asian option. The arithmetic average strike price is calculated as follows: 1. Calculate the sum of the asset prices at each observation point: 105 + 108 + 112 + 110 + 107 + 103 = 645 2. Divide the sum by the number of observations to get the average: 645 / 6 = 107.5 3. Calculate the payoff, which is the maximum of (Average Price – Strike Price, 0): max(107.5 – 105, 0) = 2.5 4. Discount the payoff back to the present value using the risk-free rate. The formula for present value is: \[PV = \frac{FV}{(1 + r)^n}\] where FV is the future value (payoff), r is the risk-free rate, and n is the time to expiration. In this case, FV = 2.5, r = 0.05, and n = 0.5 years. 5. Calculate the present value: \[PV = \frac{2.5}{(1 + 0.05)^{0.5}} = \frac{2.5}{1.0247} \approx 2.44\] The correct answer is approximately £2.44. The key here is understanding how the arithmetic average Asian option payoff is calculated. Unlike a standard European option that depends on the asset price at expiration, the Asian option depends on the average price over a period. This averaging reduces volatility and makes the option cheaper. For instance, consider a farmer hedging the price of wheat. The farmer isn’t as concerned with the daily fluctuations as they are with the average price they’ll receive over the harvest season. An Asian option on wheat prices would provide a more stable and predictable hedge than a standard option. Furthermore, the discounting step is crucial. We’re finding the present value of the expected payoff, reflecting the time value of money. A higher interest rate would result in a lower present value, and vice versa. This highlights the importance of risk-free rates in derivative pricing.

To determine the breakeven point for the combined strategy, we need to consider the initial cost of the options and the potential profit or loss at different stock prices. The investor buys one call option and sells two put options. Let \(S_T\) be the stock price at expiration. * **Call Option:** The investor buys a call option with a strike price of £150 for £10. * **Put Options:** The investor sells two put options with a strike price of £140 for £5 each (total £10). The total initial cost is £10 (call) – £10 (puts) = £0. We need to find the stock price \(S_T\) at which the investor’s profit/loss is zero. **Scenario 1: \(S_T \le 140\)** * The call option expires worthless. * Both put options are exercised. The investor must buy two shares at £140 each. The investor’s loss is \(2 \times (140 – S_T)\). To breakeven, \(2 \times (140 – S_T) = 0\), so \(S_T = 140\). **Scenario 2: \(140 < S_T < 150\)** * The call option expires worthless. * Both put options may or may not be exercised, depending on the price. If \(S_T\) is slightly above 140, the puts will still be exercised. If both puts are exercised, the loss is \(2 \times (140 – S_T)\). To breakeven, \(2 \times (140 – S_T) = 0\), so \(S_T = 140\). **Scenario 3: \(S_T \ge 150\)** * The call option is exercised. The investor's profit is \(S_T – 150\). * The put options expire worthless. To breakeven, the profit from the call option must equal the initial cost: \(S_T – 150 = 0\), so \(S_T = 150\). However, since the initial cost was zero, the breakeven point is when the profit/loss is zero. If \(S_T < 140\), the investor loses \(2 \times (140 - S_T)\). If \(S_T > 150\), the investor gains \(S_T – 150\). To find the breakeven point below 140, we solve \(2 \times (140 – S_T) = 0\), which gives \(S_T = 140\). To find the breakeven point above 150, we solve \(S_T – 150 = 0\), which gives \(S_T = 150\). Since the net initial cost is zero, the investor breaks even at \(S_T = 140\) and \(S_T = 150\). The question asks for the stock price at which the investor’s *maximum* potential loss is capped. This occurs when the stock price goes to zero. In this case, the call expires worthless, and both puts are exercised, forcing the investor to buy two shares at £140 each. The maximum loss is therefore \(2 \times 140 = £280\). However, we also have to account for the premium received, so the net loss is \(2 \times 140 – 10 = £270\). Since the initial cost was £0, the loss would be £280. However, the question asks for the stock price at which the loss is capped, not the amount of the loss. The loss is capped at stock price zero. The stock price where the investor’s profit/loss is zero is £140 and £150. The stock price where the investor’s loss is maximized is £0.

The core of this question revolves around understanding how different derivative instruments respond to changes in volatility, specifically in the context of hedging a portfolio exposed to a specific commodity. We need to consider the delta and gamma of each instrument, as well as the specific characteristics that make them suitable or unsuitable for managing volatility risk. A forward contract locks in a price today for future delivery. It has a linear payoff and no optionality. Its delta is relatively stable, but its gamma is zero, meaning it does not adjust to changing volatility. A futures contract is similar to a forward but is marked-to-market daily, introducing margin requirements and potential for early closeout. Like forwards, futures have limited ability to adapt to volatility changes. Options, on the other hand, offer non-linear payoffs and sensitivity to volatility (vega). A long call option benefits from increasing volatility if the price moves favorably, while a long put option benefits if the price declines. The gamma of an option is highest when it’s at-the-money and decreases as it moves in-the-money or out-of-the-money. A straddle (long call and long put with the same strike price and expiration) is a volatility play, benefiting from large price swings in either direction. Swaps are agreements to exchange cash flows based on different indices. They are primarily used for managing interest rate or currency risk and are less directly suited for hedging volatility in a commodity portfolio, unless the commodity price is strongly correlated with an underlying index. Exotic derivatives are complex instruments with customized features. Examples include barrier options, Asian options, and lookback options. While some exotic derivatives can be tailored to manage specific volatility profiles, their complexity and illiquidity often make them less attractive for general volatility hedging compared to standard options strategies. In the scenario presented, the fund manager needs to reduce the portfolio’s sensitivity to sudden and unexpected price movements. A simple forward or futures contract will only lock in a price, but won’t provide protection if volatility increases significantly. A swap is also not ideal, as it’s designed for different types of risk. Therefore, an option strategy, specifically one that benefits from increased volatility, is the most appropriate choice. A straddle offers protection regardless of the direction of the price movement, making it the best option.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Green Harvest,” that wants to hedge against potential price fluctuations in their wheat harvest. They are considering using futures contracts traded on the ICE Futures Europe exchange. The current spot price of wheat is £200 per tonne. The December wheat futures contract is trading at £210 per tonne. Green Harvest expects to harvest 5,000 tonnes of wheat in December. They decide to hedge 80% of their expected harvest using futures contracts. Each futures contract represents 100 tonnes of wheat. First, determine the number of futures contracts Green Harvest needs: 5,000 tonnes * 80% = 4,000 tonnes. Then, divide by the contract size: 4,000 tonnes / 100 tonnes/contract = 40 contracts. Green Harvest will *sell* 40 December wheat futures contracts at £210 per tonne. Now, imagine that in December, the spot price of wheat falls to £190 per tonne. Green Harvest sells their wheat in the spot market for £190 per tonne. Simultaneously, they close out their futures position by *buying* 40 December wheat futures contracts. The futures price has also fallen, now trading at £195 per tonne. Calculate the loss in the spot market: £200 (initial expected price) – £190 (actual selling price) = £10 loss per tonne. Total loss on 4,000 tonnes = £10/tonne * 4,000 tonnes = £40,000. Calculate the gain in the futures market: £210 (initial selling price) – £195 (closing buying price) = £15 gain per tonne. Total gain on 40 contracts (4,000 tonnes) = £15/tonne * 4,000 tonnes = £60,000. The net effect of the hedge is: £60,000 (gain on futures) – £40,000 (loss on spot market) = £20,000 gain. Green Harvest effectively locked in a price close to their initial expectation by using the futures contract. Now, consider the regulatory aspects. As a UK-based entity using derivatives, Green Harvest must comply with regulations like EMIR (European Market Infrastructure Regulation) and MiFID II (Markets in Financial Instruments Directive II). EMIR mandates clearing of certain OTC derivatives and reporting of all derivatives contracts to trade repositories. MiFID II governs the provision of investment services and aims to increase transparency and investor protection. If Green Harvest were dealing in derivatives beyond hedging commercial risks, they might need to be authorized as an investment firm. The Financial Conduct Authority (FCA) oversees these regulations in the UK. Green Harvest would need to ensure their derivative trading activities are compliant with these regulations, including reporting requirements and potentially clearing obligations.

To determine the value of the European call option using a two-step binomial tree, we need to work backward from the final nodes to the initial node. First, calculate the possible stock prices at the end of the second period (S_uu, S_ud, S_dd). Then, calculate the option values at those final nodes (C_uu, C_ud, C_dd) based on the payoff max(S – K, 0), where S is the stock price and K is the strike price. Next, calculate the risk-neutral probability (p) using the formula p = (e^(r*Δt) – d) / (u – d), where r is the risk-free rate, Δt is the length of each time step, u is the up factor, and d is the down factor. Use this probability to discount back the expected option values from the second period to the first period (C_u and C_d). Finally, discount back the expected option value from the first period to the initial node (C_0). Given: Initial stock price (S_0) = £80 Strike price (K) = £85 Risk-free rate (r) = 5% per annum Time to expiration (T) = 6 months = 0.5 years Number of steps (n) = 2 Time step (Δt) = T/n = 0.5/2 = 0.25 years Up factor (u) = 1.10 Down factor (d) = 0.90 1. Calculate stock prices at each node: S_uu = S_0 * u * u = £80 * 1.10 * 1.10 = £96.80 S_ud = S_0 * u * d = £80 * 1.10 * 0.90 = £79.20 S_dd = S_0 * d * d = £80 * 0.90 * 0.90 = £64.80 2. Calculate option values at final nodes: C_uu = max(S_uu – K, 0) = max(£96.80 – £85, 0) = £11.80 C_ud = max(S_ud – K, 0) = max(£79.20 – £85, 0) = £0 C_dd = max(S_dd – K, 0) = max(£64.80 – £85, 0) = £0 3. Calculate risk-neutral probability (p): p = (e^(r*Δt) – d) / (u – d) = (e^(0.05*0.25) – 0.90) / (1.10 – 0.90) = (1.01258 – 0.90) / 0.20 = 0.5629 4. Calculate option values at the first period nodes: C_u = e^(-r*Δt) * (p * C_uu + (1-p) * C_ud) = e^(-0.05*0.25) * (0.5629 * £11.80 + (1-0.5629) * £0) = 0.9875 * (0.5629 * £11.80) = £6.601 C_d = e^(-r*Δt) * (p * C_ud + (1-p) * C_dd) = e^(-0.05*0.25) * (0.5629 * £0 + (1-0.5629) * £0) = 0 5. Calculate the option value at the initial node: C_0 = e^(-r*Δt) * (p * C_u + (1-p) * C_d) = e^(-0.05*0.25) * (0.5629 * £6.601 + (1-0.5629) * £0) = 0.9875 * (0.5629 * £6.601) = £3.665 Therefore, the value of the European call option is approximately £3.67. This example illustrates how the binomial model breaks down the time to expiration into discrete steps, allowing for the valuation of options by considering all possible price paths and discounting back the expected payoffs using risk-neutral probabilities. The risk-neutral probability is a crucial concept, as it allows us to value derivatives without needing to know investors’ actual risk preferences. The up and down factors reflect the volatility of the underlying asset, and the more steps used in the binomial tree, the more accurate the option value will be. This method is particularly useful for valuing options on assets that do not follow a strict log-normal distribution or when early exercise is possible (for American options).

To determine the profit or loss on the option positions and then calculate the overall profit or loss for the portfolio, we need to analyze each option separately and then combine the results. First, consider the call option. The investor bought the call option with a strike price of 1500 for a premium of £5. The index settled at 1570. Since the settlement price (1570) is above the strike price (1500), the call option is in the money. The intrinsic value of the call option is the difference between the settlement price and the strike price, which is \(1570 – 1500 = 70\). The profit from the call option is the intrinsic value minus the premium paid, which is \(70 – 5 = 65\) per contract. Since the investor bought 10 contracts, the total profit from the call options is \(65 \times 10 \times 10 = £6500\). The multiplication by 10 is due to the contract multiplier. Next, consider the put option. The investor sold the put option with a strike price of 1450 for a premium of £3. The index settled at 1570. Since the settlement price (1570) is above the strike price (1450), the put option is out of the money. Therefore, the put option expires worthless, and the investor keeps the premium received. The profit from selling the put option is £3 per contract. Since the investor sold 10 contracts, the total profit from the put options is \(3 \times 10 \times 10 = £300\). The multiplication by 10 is due to the contract multiplier. Finally, to calculate the overall profit or loss for the portfolio, we add the profit from the call options and the profit from the put options: \(£6500 + £300 = £6800\). Therefore, the overall profit for the portfolio is £6800.

Let’s analyze the combined impact of margin requirements and contract specifications on a trader’s potential losses in a futures contract. Consider a trader, Anya, who enters a short position in a cocoa futures contract. The initial margin is £4,000, and the maintenance margin is £3,500. The contract size is 10 tonnes, and each tick size is £1 per tonne. Anya is required to deposit the initial margin. If the market moves against her, and her account balance falls below the maintenance margin, she receives a margin call. She must then deposit funds to bring the account balance back to the initial margin level. If she fails to meet the margin call, her position is liquidated, and she bears the loss. Now, let’s calculate the maximum loss Anya could incur before her position is liquidated, assuming she doesn’t respond to the margin call. The difference between the initial margin and the maintenance margin is £500. This represents the amount the market can move against her before she receives a margin call. Since each tick is £1 per tonne, and the contract size is 10 tonnes, each tick movement represents a £10 change in the contract value. Therefore, the market can move \( \frac{£500}{£10} = 50 \) ticks against Anya before she receives a margin call. If Anya fails to respond to the margin call and the market continues to move against her, her position will be liquidated. The maximum loss she can incur before liquidation is the initial margin plus the amount the market moves against her before liquidation. Let’s say the market moves another 20 ticks against her before liquidation. This represents an additional loss of \( 20 \times £10 = £200 \). Therefore, the maximum loss Anya could incur is the initial margin of £4,000 plus the £500 drop to the maintenance margin plus the additional £200 loss before liquidation, totaling £4,700. This example illustrates the importance of understanding margin requirements and contract specifications in managing risk in futures trading. Traders must be aware of the potential losses they can incur and have a plan for responding to margin calls to avoid liquidation and minimize losses. It also demonstrates how seemingly small tick movements can accumulate and result in significant losses, especially when trading leveraged instruments like futures. This emphasizes the need for robust risk management strategies and careful monitoring of market conditions.

The core of this question revolves around understanding how margin requirements work in futures contracts, specifically when considering initial margin, maintenance margin, and variation margin. A key concept is that when the account balance falls below the maintenance margin, a margin call is triggered. The investor must then deposit enough funds to bring the account balance back up to the initial margin level. In this scenario, the investor starts with an initial margin of £8,000. The maintenance margin is £6,000. The futures contract experiences a loss of £2,500 on the first day and a further loss of £2,000 on the second day. * **Day 1:** Account balance = £8,000 – £2,500 = £5,500 * **Day 2:** Account balance = £5,500 – £2,000 = £3,500 Since the account balance of £3,500 is below the maintenance margin of £6,000, a margin call is triggered. The investor needs to deposit enough funds to bring the account balance back up to the initial margin of £8,000. Amount to deposit = Initial margin – Current balance = £8,000 – £3,500 = £4,500 The investor must deposit £4,500 to meet the margin call. This ensures that the account is adequately collateralized against further potential losses. If the investor fails to meet the margin call, the broker has the right to liquidate the position to cover the losses. The futures market operates on a marked-to-market basis, meaning profits and losses are settled daily. This daily settlement, through variation margin, is crucial for managing risk and ensuring the integrity of the market. The initial margin acts as a performance bond, while the maintenance margin serves as a trigger for additional funds to be deposited when losses erode the initial margin.

Let’s consider a scenario where a UK-based agricultural cooperative, “GreenHarvest,” wants to hedge against potential price declines in their upcoming wheat harvest. They decide to use wheat futures contracts traded on the ICE Futures Europe exchange. GreenHarvest anticipates harvesting 5,000 tonnes of wheat in three months. Each futures contract represents 100 tonnes of wheat. The current futures price for delivery in three months is £200 per tonne. To hedge, GreenHarvest sells 50 futures contracts (5,000 tonnes / 100 tonnes per contract = 50 contracts). Now, let’s say that in three months, the spot price of wheat is £180 per tonne. GreenHarvest sells their physical wheat at this price. Simultaneously, they close out their futures position by buying back 50 futures contracts at £180 per tonne. Here’s the calculation: * **Gain/Loss on Futures:** (Selling Price – Buying Price) * Contract Size * Number of Contracts = (£200 – £180) * 100 * 50 = £100,000 gain. * **Sale of Physical Wheat:** 5,000 tonnes * £180/tonne = £900,000. * **Effective Price Received:** Sale of Physical Wheat + Gain on Futures = £900,000 + £100,000 = £1,000,000. * **Effective Price per Tonne:** £1,000,000 / 5,000 tonnes = £200/tonne. This demonstrates how futures contracts can be used to hedge price risk. GreenHarvest locked in a price of £200 per tonne, regardless of the actual spot price at harvest time. The gain on the futures contract offset the lower price received for the physical wheat. Now, consider a slightly different scenario. Instead of hedging, GreenHarvest speculates that wheat prices will rise. They buy 50 futures contracts at £200 per tonne. If the price rises to £220 per tonne, they close out their position, making a profit of (£220 – £200) * 100 * 50 = £100,000. However, if the price falls to £180 per tonne, they would incur a loss of (£200 – £180) * 100 * 50 = £100,000. This highlights the speculative nature of futures trading and the potential for both gains and losses. The FCA requires firms to categorize clients based on their knowledge and experience, and to ensure that speculative trading is only undertaken by clients who understand the risks involved. Finally, consider the impact of margin requirements. If the initial margin requirement is £5,000 per contract, GreenHarvest would need to deposit £250,000 (50 contracts * £5,000) with their broker. If the price moves against them, they may receive margin calls, requiring them to deposit additional funds to maintain their position. Failure to meet margin calls could result in the forced liquidation of their position.

To solve this problem, we need to understand the concept of delta-hedging, the gamma of an option, and how changes in the underlying asset’s price affect the hedge. Delta-hedging involves adjusting the position in the underlying asset to offset changes in the option’s value due to small price movements in the underlying. Gamma represents the rate of change of the delta with respect to changes in the underlying asset’s price. A higher gamma means the delta is more sensitive to price changes, requiring more frequent adjustments to maintain a delta-neutral position. In this scenario, the fund initially delta-hedges its short call option position. When the underlying asset’s price moves significantly, the delta changes, and the hedge needs to be rebalanced. The key is to calculate how many additional shares are needed to maintain the delta hedge after the price change. 1. **Initial Delta:** The fund is short a call option with a delta of 0.45. This means for every £1 increase in the underlying asset’s price, the option’s value is expected to decrease by £0.45 per option. To delta-hedge, the fund initially buys 0.45 shares for each option sold. With 10,000 options, the initial hedge requires 0.45 * 10,000 = 4,500 shares. 2. **Price Change and New Delta:** The underlying asset’s price increases by £5, and the delta increases to 0.60. This means the fund now needs to hold 0.60 shares for each option to maintain the delta hedge. 3. **New Hedge Requirement:** The new hedge requires 0.60 * 10,000 = 6,000 shares. 4. **Shares to Purchase:** The fund needs to purchase additional shares to adjust the hedge. The number of additional shares required is 6,000 – 4,500 = 1,500 shares. 5. **Gamma Implication:** The gamma of the option indicates how much the delta changes for each £1 change in the underlying asset’s price. In this case, the delta increased by 0.15 (from 0.45 to 0.60) for a £5 price increase. This implies a gamma of approximately 0.03 per £1 change. The higher the gamma, the more frequently the hedge needs to be adjusted to maintain delta neutrality. If the fund had ignored the change in delta, it would have been under-hedged, exposing it to losses if the underlying asset’s price continued to increase. Therefore, the fund must purchase 1,500 additional shares to maintain its delta-neutral position.

The core of this question lies in understanding how delta hedging aims to neutralize the price sensitivity of an option portfolio. The delta of a call option measures how much the option price is expected to change for every £1 change in the underlying asset’s price. A delta of 0.6 indicates that for every £1 increase in the share price, the call option’s value should increase by £0.60. To delta hedge, the portfolio manager needs to take an offsetting position in the underlying asset. In this case, the manager is short 10,000 call options, each with a delta of 0.6. This means the portfolio is effectively short 10,000 * 0.6 = 6,000 shares. To neutralize this exposure, the manager needs to buy 6,000 shares. However, the question introduces a twist: transaction costs. Buying the shares incurs a cost of £0.05 per share. Therefore, the total cost of delta hedging is 6,000 shares * £0.05/share = £300. Now, let’s consider the gamma of the option. Gamma measures the rate of change of the delta with respect to changes in the underlying asset’s price. A gamma of 0.0002 per option means that for every £1 change in the share price, the delta of each option changes by 0.0002. Since the manager is short 10,000 options, the total gamma exposure is 10,000 * 0.0002 = 2. This means that for every £1 increase in the share price, the portfolio’s delta becomes 2 units more negative (or 2 units more positive if the share price decreases by £1). The vega of the option is 0.04, which means that for every 1% increase in implied volatility, the option price increases by £0.04. Since the manager is short 10,000 options, the total vega exposure is 10,000 * 0.04 = £400. A decrease of 0.5% in implied volatility will therefore decrease the value of the option portfolio by £400 * 0.5 = £200. The theta of the option is -0.02, meaning the option loses £0.02 in value each day due to time decay. For 10,000 options, the total theta effect is 10,000 * -0.02 = -£200 per day. Over 5 days, this amounts to a loss of -£200 * 5 = -£1000. Combining these effects: – Delta hedging cost: £300 – Vega effect: -£200 – Theta effect: -£1000 – Gamma is not immediately relevant to the initial hedging cost. Therefore, the net cost is £300 – £200 – £1000 = -£900. This means the strategy has a net gain of £900.

The question assesses the understanding of how different option strategies are affected by volatility changes, specifically focusing on a short strangle. A short strangle involves selling both a call and a put option with the same expiration date but different strike prices (one above and one below the current market price). The maximum profit is limited to the net premium received, and the risk is unlimited. Volatility plays a crucial role in option pricing. An increase in volatility generally increases the value of both call and put options because it increases the probability of the underlying asset’s price moving significantly in either direction. For a short strangle, this is detrimental because the investor profits most when the underlying asset price remains within a narrow range between the strike prices of the sold options. If volatility increases, the prices of both the call and put options sold in the strangle will rise. To close the position, the investor must buy back these options at a higher price than they were initially sold for, resulting in a loss. The magnitude of the loss depends on the extent of the volatility increase and the sensitivity of the options to volatility (their vega). In the scenario, initially, the investor receives a combined premium of £600 (£300 for the call and £300 for the put). If volatility increases significantly, the call option price increases by £200 and the put option price increases by £150. To close the position, the investor must spend £200 to buy back the call and £150 to buy back the put, totaling £350. The profit or loss is calculated as: Initial premium received – Cost to close the position = £600 – (£200 + £150) = £600 – £350 = £250. Therefore, the investor makes a profit of £250. This contrasts with a long strangle where the investor buys both a call and a put. In a long strangle, an increase in volatility is beneficial because the investor profits if the underlying asset price moves significantly in either direction. The increase in the value of the options outweighs the cost of holding them, leading to a profit. The question highlights the importance of understanding the risks associated with selling options, particularly the impact of volatility. Investors selling strangles must carefully monitor volatility and have a plan for managing potential losses if volatility increases.

Let’s break down the valuation of this exotic derivative. The key here is understanding the knock-out feature and how it affects the payoff. This is a barrier option, specifically a down-and-out call. A standard call option gives the holder the right, but not the obligation, to buy an asset at a specified price (the strike price) on or before a specified date (the expiration date). The payoff is the maximum of zero and the difference between the asset price at expiration and the strike price, or max(0, S_T – K), where S_T is the asset price at expiration and K is the strike price. However, a down-and-out call option becomes worthless if the underlying asset price touches or falls below a certain barrier level before the expiration date. In this case, the barrier is £85. If the asset price never hits £85 during the life of the option, then the option behaves like a standard call option. First, we need to determine if the barrier was breached. The lowest price reached was £82, which is below the £85 barrier. Therefore, the knock-out condition was met, and the option expires worthless, regardless of the final asset price. The value of the option at expiration is therefore £0. The client’s profit/loss is simply the negative of the premium paid, as the option is now worthless. So, the profit/loss is -£3.50. Now, let’s consider some analogies to solidify understanding. Imagine a high-wire walker who has a safety net set at a certain height. This safety net represents the barrier. If the walker falls and touches the net (the asset price hits the barrier), the act is over, regardless of whether they could have completed the walk successfully (the asset price exceeding the strike price at expiration). Similarly, a down-and-out option becomes worthless if the barrier is hit, even if the asset price later rises above the strike price. Another analogy is a time-sensitive coupon. Imagine a coupon that expires not only on a specific date but also becomes invalid if you visit the store more than twice before that date. The number of visits acts as a barrier. If you exceed the visit limit, the coupon is void, even if the expiration date hasn’t passed. The key takeaway is that barrier options introduce an additional layer of complexity to standard option valuation. The possibility of the barrier being breached significantly affects the option’s price and potential payoff. In this scenario, the barrier was breached, rendering the option worthless and resulting in a loss equal to the premium paid.

The core concept tested here is understanding how various factors influence the price of a European call option, specifically within the context of the Black-Scholes model, and more importantly, how regulatory constraints might modify the theoretical outcomes. The Black-Scholes model posits a direct relationship between the underlying asset’s price and the call option price. An increase in the underlying asset’s price generally increases the call option’s price because the option holder has a greater chance of exercising the option profitably. Volatility also has a direct relationship; higher volatility means a greater potential for price swings, increasing the option’s value. Time to expiration is similar; a longer time horizon allows more opportunity for the option to become in-the-money. The risk-free rate has a more nuanced effect; higher rates tend to increase call option prices (because the present value of the strike price decreases). However, the question introduces a regulatory constraint. The FCA (Financial Conduct Authority) mandates that firms offering complex derivatives to retail clients must incorporate a “complexity charge” into the option price. This charge is designed to offset potential mis-selling risks and ensure clients understand the product’s complexity. This charge reduces the theoretical gain from the option, therefore reducing its value. This charge acts as a direct deduction from the option’s theoretical value, effectively reducing the investor’s potential profit and impacting the attractiveness of the investment. Therefore, even though the underlying asset price, volatility, time to expiration, and risk-free rate all suggest an increase in the call option price, the complexity charge acts as a counterbalancing force, potentially mitigating the overall price increase. The magnitude of this charge relative to the gains from other factors determines the final outcome.

To determine the theoretical forward price, we use the cost of carry model. This model considers the spot price, the risk-free rate, and any storage costs or convenience yields associated with holding the underlying asset. In this case, the asset is palladium, which incurs storage costs but provides no convenience yield. The formula for the theoretical forward price (F) is: \[F = S_0 * e^{(r + c)T}\] Where: \(S_0\) is the spot price of the asset. \(r\) is the risk-free interest rate. \(c\) is the storage cost as a percentage of the spot price. \(T\) is the time to maturity in years. Given: \(S_0 = £2,500\) \(r = 4\%\) or 0.04 \(c = 2\%\) or 0.02 \(T = 9\) months, which is \(9/12 = 0.75\) years Substituting these values into the formula: \[F = 2500 * e^{(0.04 + 0.02) * 0.75}\] \[F = 2500 * e^{(0.06 * 0.75)}\] \[F = 2500 * e^{0.045}\] \[F = 2500 * 1.0460276\] \[F = 2615.069\] The theoretical forward price is approximately £2,615.07. Now, we need to compare this to the actual forward price of £2,650. The basis is the difference between the actual forward price and the theoretical forward price: Basis = Actual Forward Price – Theoretical Forward Price Basis = £2,650 – £2,615.07 Basis = £34.93 A positive basis indicates that the actual forward price is higher than the theoretical forward price. This situation might arise due to market expectations of higher future spot prices, supply constraints, or other market-specific factors not fully captured by the cost of carry model. The presence of a positive basis creates an arbitrage opportunity. An arbitrageur could simultaneously buy palladium at the spot price and sell a forward contract, locking in a profit of £34.93 per unit (before transaction costs). This is a risk-free profit because the arbitrageur knows both the purchase price and the sale price in advance. The arbitrageur would store the palladium for nine months, incurring storage costs, and then deliver it at the forward price. The profit would be the difference between the forward price and the spot price, less the storage costs and financing costs.

The core of this question revolves around understanding how gamma, delta, and theta interact to affect option pricing, particularly in a portfolio context. Gamma measures the rate of change of delta with respect to changes in the underlying asset’s price. Delta measures the sensitivity of the option price to changes in the underlying asset’s price. Theta measures the rate of decline in the value of an option due to the passage of time (time decay). A portfolio’s overall gamma exposure dictates how much the portfolio’s delta will change for a given move in the underlying asset. Positive gamma means the delta will increase as the underlying asset price increases and decrease as the underlying asset price decreases. Negative gamma means the opposite. Theta, generally negative for long option positions, erodes the value of the option over time, but its impact is lessened if the underlying asset moves favorably (in the direction that increases the option’s value). The interplay between these Greeks determines the net effect on the portfolio’s value. To solve this, we need to consider the impact of each Greek independently and then combine them. A positive gamma benefits from large price swings in either direction, as it makes the delta more positive when the price goes up and less negative when the price goes down (for a long option position). However, negative theta constantly detracts from the value, especially if the underlying asset price remains stable. The optimal scenario depends on the magnitude of the price movement and the relative sizes of gamma and theta. The question is designed to test understanding beyond simple definitions and requires integrating the combined effects of these Greeks.

To determine the value of the European call option using a two-step binomial tree, we need to work backward from the final nodes to the initial node. First, calculate the possible stock prices at each node. Then, determine the option values at expiration (final nodes) based on the intrinsic value (max(0, Stock Price – Strike Price)). Next, calculate the option values at the earlier nodes by discounting the expected option value using the risk-neutral probability. The risk-neutral probability (p) is calculated as \( p = \frac{e^{(r-q)\Delta t} – D}{U – D} \), where \(r\) is the risk-free rate, \(q\) is the dividend yield, \(\Delta t\) is the time step, \(U\) is the up factor, and \(D\) is the down factor. Given: – Initial stock price (\(S_0\)): £50 – Strike price (\(K\)): £52 – Risk-free rate (\(r\)): 5% per annum – Time to expiration (\(T\)): 6 months (0.5 years) – Number of steps (\(n\)): 2 – Up factor (\(U\)): 1.1 – Down factor (\(D\)): 0.9 – Dividend yield (\(q\)): 0% Time step (\(\Delta t\)) = \(T/n\) = 0.5/2 = 0.25 years Risk-neutral probability (p) = \( \frac{e^{(0.05-0) \times 0.25} – 0.9}{1.1 – 0.9} \) = \( \frac{e^{0.0125} – 0.9}{0.2} \) ≈ \( \frac{1.012578 – 0.9}{0.2} \) ≈ 0.5629 Now, calculate the stock prices at each node: – S_uu = 50 * 1.1 * 1.1 = £60.50 – S_ud = 50 * 1.1 * 0.9 = £49.50 – S_dd = 50 * 0.9 * 0.9 = £40.50 Option values at expiration: – C_uu = max(0, 60.50 – 52) = £8.50 – C_ud = max(0, 49.50 – 52) = £0 – C_dd = max(0, 40.50 – 52) = £0 Option values at the previous nodes: – C_u = \( e^{-r\Delta t} \) * (p * C_uu + (1-p) * C_ud) = \( e^{-0.05 \times 0.25} \) * (0.5629 * 8.50 + 0.4371 * 0) = \( e^{-0.0125} \) * (4.78465) ≈ 0.98758 * 4.78465 ≈ £4.725 – C_d = \( e^{-r\Delta t} \) * (p * C_ud + (1-p) * C_dd) = \( e^{-0.05 \times 0.25} \) * (0.5629 * 0 + 0.4371 * 0) = 0 Finally, the option value at the initial node (C_0): – C_0 = \( e^{-r\Delta t} \) * (p * C_u + (1-p) * C_d) = \( e^{-0.05 \times 0.25} \) * (0.5629 * 4.725 + 0.4371 * 0) = \( e^{-0.0125} \) * (2.6599) ≈ 0.98758 * 2.6599 ≈ £2.627 Therefore, the value of the European call option is approximately £2.63.

Let’s analyze a complex scenario involving a quanto swap and its sensitivity to various market parameters. A quanto swap is a type of interest rate swap where the interest rate of one currency is exchanged for the interest rate of another currency, and the notional principal is fixed in one currency. The payoff is then converted to the other currency at a pre-agreed exchange rate. This example will focus on how changes in the correlation between interest rates and exchange rates affect the fair value of a GBP/USD quanto swap. The pricing of a quanto swap involves several factors: the interest rate differential between the two currencies, the volatility of the exchange rate, and the correlation between the interest rates and the exchange rate. When the correlation between the GBP interest rate and the GBP/USD exchange rate is positive, it implies that as the GBP interest rate increases, the GBP tends to appreciate against the USD. This positive correlation can impact the expected payoff of the quanto swap. Consider a scenario where a UK-based pension fund enters into a GBP/USD quanto swap to hedge its USD-denominated liabilities. The pension fund receives a fixed GBP interest rate and pays a floating USD interest rate, with the notional principal fixed in GBP and the payments converted to USD at a fixed exchange rate. If the correlation between GBP interest rates and the GBP/USD exchange rate increases, the expected payoff of the swap changes. A higher positive correlation means that when GBP interest rates rise, the GBP is more likely to appreciate against the USD. This makes the fixed GBP payments more valuable in USD terms, increasing the fair value of the swap for the pension fund. Conversely, if the correlation becomes more negative, the fair value decreases. The fair value adjustment due to correlation can be approximated using a convexity adjustment. This adjustment accounts for the non-linear relationship between the swap’s payoff and the underlying market variables. The adjustment is proportional to the product of the correlation, the volatilities of the interest rates and the exchange rate, and the time to maturity. In practice, pricing models incorporate these correlation effects to accurately value quanto swaps.

Let’s consider a scenario where a fund manager, Anya, uses options to hedge her portfolio against a potential market downturn. Anya manages a UK-based equity fund with a significant holding in FTSE 100 companies. She’s concerned about an upcoming Brexit referendum result and its potential negative impact on the UK stock market. To protect her fund, she decides to purchase put options on the FTSE 100 index. The FTSE 100 index is currently trading at 7500. Anya buys 100 put option contracts with a strike price of 7400, expiring in three months. The premium for each put option contract is £50. Each contract represents 1 index point, so each contract covers 7400 * £1 = £7400. The total cost of the hedge is 100 contracts * £50/contract = £5000. Now, let’s analyze two scenarios: Scenario 1: The FTSE 100 drops to 7000 at expiration. Anya exercises her put options. Her profit per contract is (Strike Price – Index Value) = (7400 – 7000) = 400 index points. Her total profit is 100 contracts * 400 index points/contract = 40,000 index points. Since each index point is worth £1, her profit is £40,000. Subtracting the initial premium cost of £5000, her net profit is £35,000. Scenario 2: The FTSE 100 rises to 7600 at expiration. Anya’s put options expire worthless. She loses the initial premium paid of £5000. Now, let’s introduce a twist: Anya’s fund also holds a significant position in a small-cap technology company listed on the AIM. This company’s performance is heavily correlated with investor sentiment and risk appetite, which are also affected by macroeconomic events like Brexit. Anya anticipates that the AIM-listed company will be *more* volatile than the FTSE 100. To account for this higher volatility, Anya decides to use a *ratio hedge*. Instead of simply buying put options to cover the full value of her FTSE 100 holdings, she adjusts the number of contracts based on the relative volatilities of the FTSE 100 and the AIM-listed company. She estimates that the AIM-listed company is 1.5 times more volatile than the FTSE 100. This means that for every £1 of exposure in the AIM-listed company, she needs 1.5 times the hedging protection compared to £1 of exposure in the FTSE 100. If Anya holds £1,000,000 in FTSE 100 stocks and £500,000 in the AIM-listed company, she needs to adjust her hedge accordingly. The FTSE 100 component requires a hedge equivalent to its full value, but the AIM component requires a hedge equivalent to 1.5 * £500,000 = £750,000. The question will focus on calculating the appropriate number of put option contracts needed for the FTSE 100 portion of the portfolio, considering the strike price, premium, and the total value of the FTSE 100 holdings.

Let’s analyze the expected profit from this exotic Asian option. The option’s payoff depends on the *average* price of the underlying asset over the observation period. This makes it path-dependent and distinct from standard European or American options. The key is to simulate the potential average prices and then calculate the option’s payoff for each simulated path. The investor’s profit is the payoff minus the initial cost of the option. First, we need to determine the payoff for each scenario. The payoff of an Asian call option is given by max(Average Price – Strike Price, 0). The average price is calculated as the sum of all observed prices divided by the number of observations. * **Scenario 1:** Average Price = £102. Payoff = max(102 – 100, 0) = £2. Profit = £2 – £1.50 = £0.50 * **Scenario 2:** Average Price = £98. Payoff = max(98 – 100, 0) = £0. Profit = £0 – £1.50 = -£1.50 * **Scenario 3:** Average Price = £105. Payoff = max(105 – 100, 0) = £5. Profit = £5 – £1.50 = £3.50 * **Scenario 4:** Average Price = £95. Payoff = max(95 – 100, 0) = £0. Profit = £0 – £1.50 = -£1.50 Now, calculate the expected profit by averaging the profits from each scenario: Expected Profit = (0.50 – 1.50 + 3.50 – 1.50) / 4 = 1 / 4 = £0.25 Therefore, the expected profit from this Asian option, considering these four scenarios, is £0.25. The complexities of Asian options stem from the averaging mechanism, which reduces volatility compared to standard options. This makes them attractive in markets where price stability is desired. However, pricing these options accurately requires sophisticated models, such as Monte Carlo simulations, especially when the number of averaging periods is large. In our simplified example, we only considered four scenarios. In practice, a Monte Carlo simulation might involve thousands or even millions of simulated price paths to achieve a robust estimate of the option’s expected payoff. Furthermore, factors like interest rates and dividends would also influence the pricing and hedging strategies for such exotic derivatives. The regulatory landscape, particularly MiFID II in the UK, mandates careful suitability assessments for clients trading in complex instruments like Asian options, ensuring they understand the risks involved.

The core of this question lies in understanding how different derivatives contracts respond to changes in volatility, particularly in the context of a firm managing its risk exposure. A forward contract, being an agreement to buy or sell an asset at a predetermined price on a future date, is largely insensitive to volatility *after* the contract is initiated. The price is locked in. However, increased volatility *before* the contract is initiated will increase the uncertainty in the price the firm will receive, which will increase the premium demanded by the counterparty. Futures contracts, while similar to forwards, are marked-to-market daily. This means that gains and losses are realized throughout the contract’s life. While the initial margin requirements may increase with volatility, the contract’s value itself is less directly affected by subsequent volatility changes than an option. Increased volatility would increase the margin requirements but would also increase the potential for gains and losses. Options, on the other hand, are highly sensitive to volatility. A call option gives the holder the right, but not the obligation, to buy an asset at a specified price (the strike price) on or before a specified date. The value of a call option increases with volatility because greater volatility increases the probability that the asset’s price will rise significantly above the strike price, making the option profitable to exercise. Similarly, the value of a put option, which gives the holder the right to sell an asset, also increases with volatility because greater volatility increases the probability that the asset’s price will fall significantly below the strike price. Swaps are contracts where two parties exchange cash flows. While the underlying asset of a swap (e.g., interest rates, currency exchange rates) may be volatile, the swap contract itself is structured to manage this volatility through predetermined payment schedules. The value of a swap is affected by the *level* of the underlying rate or price, and the *correlation* between rates, but is not directly impacted by the volatility itself. Therefore, when seeking a derivative instrument to *benefit* from increased volatility in the price of a commodity the firm needs to purchase, options (specifically, either buying calls or buying puts, depending on the firm’s risk exposure and market view) are the most appropriate choice. The firm could use a long straddle or strangle position to profit from large price movements in either direction.

The core concept being tested is the understanding of how exotic derivatives, specifically barrier options, are priced and how their value is affected by the presence of a barrier. A standard approach to valuing options involves risk-neutral pricing, where we discount the expected payoff at the risk-free rate. However, barrier options introduce a level of complexity because the payoff is contingent on the underlying asset’s price crossing the barrier. In this scenario, the probability of hitting the barrier before expiration significantly impacts the option’s value. To properly assess the value, one must consider the probability of the option expiring in the money AND not hitting the barrier. This requires simulating potential price paths or using specialized pricing models that account for the barrier. The incorrect answers represent common misunderstandings. Option b) incorrectly assumes that the barrier has no impact if the current price is far from it. Option c) suggests that the barrier option will always be worth less than a standard option, which isn’t always true depending on the specific parameters and market conditions. Option d) proposes a simple adjustment to the standard option price without accounting for the probabilistic nature of the barrier event. The correct approach involves understanding that the barrier option’s price is a function of: the underlying asset’s price, volatility, time to expiration, strike price, barrier level, and risk-free interest rate. A simplified, conceptual calculation would involve: 1. Calculate the theoretical price of a standard European call option using a model like Black-Scholes. 2. Estimate the probability of the asset price hitting the barrier before expiration. This often requires simulation techniques or specialized barrier option pricing formulas. 3. Adjust the standard option price based on the probability of the barrier being hit. If the barrier is likely to be hit, the call option price will be significantly reduced. If the barrier is very unlikely to be hit, the price will be closer to the standard option price. The precise calculation of the probability of hitting the barrier is complex and beyond the scope of a simple exam question, but the understanding of the factors involved is crucial. The key is to recognize that the barrier introduces a path-dependent element to the option’s value.

The core concept being tested is the impact of volatility on option pricing, specifically for exotic options like barrier options. The key here is understanding how the presence of a barrier affects the option’s sensitivity to volatility changes compared to a standard vanilla option. A down-and-out call option becomes worthless if the underlying asset’s price hits the barrier before expiration. Higher volatility increases the probability of hitting this barrier, thus decreasing the value of the down-and-out call. Conversely, a vanilla call option benefits from increased volatility because it increases the potential for a larger payoff at expiration. Let’s consider two scenarios: Scenario 1: A down-and-out call option with a strike price of £100 and a barrier at £90. The underlying asset is currently trading at £105. A sudden increase in market volatility makes it more likely that the asset price will fluctuate significantly, potentially hitting the £90 barrier and rendering the option worthless. Therefore, the option’s value decreases. Scenario 2: A standard vanilla call option with a strike price of £100. The underlying asset is also trading at £105. An increase in volatility makes it more likely that the asset price will rise significantly above £100 before expiration, increasing the potential payoff and, consequently, the option’s value. This difference in behavior is due to the barrier feature. The barrier introduces a path dependency: the option’s value depends not only on the final asset price but also on the path the asset price takes during the option’s life. The higher the volatility, the greater the chance of the barrier being triggered, which reduces the value of the down-and-out call. The correct answer will reflect this inverse relationship between volatility and the value of a down-and-out call option. The incorrect options will likely either confuse the effect of volatility on vanilla options with that on barrier options or misinterpret the impact of the barrier on the option’s value. They might also suggest that volatility has no impact, which is clearly incorrect.

The core of this question lies in understanding the mechanics of a swaption, particularly a payer swaption, and its valuation sensitivity to interest rate movements. A payer swaption gives the holder the right, but not the obligation, to enter into a swap where they pay the fixed rate and receive the floating rate. Therefore, if interest rates rise, the fixed rate on a newly issued swap would also rise, making the right to *pay* a lower fixed rate (as specified in the swaption) more valuable. Conversely, if interest rates fall, the value of the payer swaption decreases. The tricky part is calculating the *approximate* change in value. We’re given a notional principal of £50 million and a DV01 (Dollar Value of a Basis Point) of 50. DV01 represents the change in the swaption’s value for a one basis point (0.01%) change in interest rates. The interest rate movement is 25 basis points (0.25%). Therefore, the approximate change in value is calculated as: Change in Value = DV01 * Change in Interest Rates * Notional Principal Since DV01 is 50, this implies that for every basis point movement, the swaption changes by £50. Therefore for a 25 basis point movement: Change in Value = 50 * 25 = 1250 Since the interest rates have increased, the value of payer swaption will increase. The concept of DV01 is crucial here. It’s a measure of interest rate risk, specifically quantifying the change in the present value of a financial instrument for a one basis point change in yield. It’s essential to distinguish DV01 from other measures like duration, which are more commonly used for bonds and less precise for derivatives with complex cash flows. The example here illustrates a simplified, linear approximation of the swaption’s sensitivity. In reality, swaption values exhibit convexity, meaning the actual change in value may deviate from the linear approximation, especially for larger interest rate movements. Furthermore, the DV01 itself is not static; it changes as interest rates and time to expiration change.

To determine the profit or loss on the short strangle, we need to analyze the payoff at the expiration date based on the underlying asset’s price. The investor profits if the underlying asset’s price stays between the two strike prices, and the maximum profit is the sum of the premiums received. The investor incurs a loss if the underlying asset’s price moves outside the range defined by the strike prices plus or minus the premiums. First, calculate the total premium received: £3.50 (call) + £2.00 (put) = £5.50. This is the maximum profit. Next, determine the breakeven points. The upper breakeven point is the strike price of the call option plus the total premium: £110 + £5.50 = £115.50. The lower breakeven point is the strike price of the put option minus the total premium: £90 – £5.50 = £84.50. The underlying asset’s price at expiration is £118. Since this is above the upper breakeven point, the investor will incur a loss. Calculate the loss on the call option: Underlying Price – Strike Price = £118 – £110 = £8. The investor sold the call, so they have to pay out £8. Calculate the net loss: Loss on call – Total premium received = £8 – £5.50 = £2.50. Therefore, the investor has a net loss of £2.50 per share. Consider an analogy: Imagine you’re running a carnival game where people guess the number of jelly beans in a jar. You offer prizes for guesses within a certain range. Selling a strangle is like betting that most people will guess within your set range. The premiums you collect are like the entry fees. If everyone guesses within your range, you keep the entry fees (premiums). However, if someone guesses far outside the range, you have to pay out a large prize, potentially more than the entry fees you collected, resulting in a loss. The key here is understanding the combined effect of both options and how the premium influences the breakeven points. A common mistake is to only consider one option or to forget to factor in the initial premium received. Another misconception is thinking that the profit is unlimited, which is not the case with a short strangle; the maximum profit is capped at the premium received.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their behavior near the barrier level. It requires calculating the probability of the underlying asset breaching the barrier during the option’s life, and then factoring in the rebate. The probability of the asset price reaching the barrier is estimated using a simplified model. The expected rebate is then calculated by multiplying the rebate amount by the probability of the barrier being breached. Here’s the step-by-step calculation: 1. **Estimate Probability of Hitting the Barrier:** Assume a simplified model where the probability of hitting the barrier is estimated as 60%. This is a simplification for illustrative purposes; a more complex model would be used in practice. 2. **Calculate Expected Rebate:** Expected Rebate = Rebate Amount * Probability of Hitting Barrier = £5,000 * 0.60 = £3,000 3. **Consider the “Knock-Out” Feature:** The key is understanding that the investor only receives the rebate *if* the barrier is breached. The expected value is thus the rebate amount times the probability of breaching. The scenario is designed to highlight the risk associated with barrier options. While they may offer attractive premiums compared to standard options, the “knock-out” feature introduces a significant risk of losing the entire investment, or in this case, only receiving a pre-defined rebate. The probability of the barrier being breached is influenced by factors such as the volatility of the underlying asset and the time remaining until expiration. A higher volatility or longer time horizon would generally increase the probability of breaching the barrier. The incorrect options are designed to reflect common misunderstandings about barrier options. Option b) incorrectly assumes the rebate is guaranteed. Option c) misinterprets the rebate as a risk-free return, failing to account for the probability of the barrier not being breached. Option d) introduces a calculation error by adding the rebate to the initial investment, which is not relevant to the expected value of the rebate itself. The correct answer focuses on the probabilistic nature of the rebate payment, highlighting the core risk-reward trade-off of barrier options.

The question examines the understanding of exotic derivatives, specifically barrier options, and their behaviour under different market conditions. A down-and-out barrier option becomes worthless if the underlying asset’s price hits the barrier level before expiration. The key to solving this question is to understand the interplay between volatility, the barrier level, and the time remaining until expiration. High volatility increases the probability of the barrier being hit. A barrier level close to the current price also increases the likelihood of it being breached. The time remaining is crucial because the longer the time, the greater the chance of the barrier being hit. In this scenario, we need to assess how a simultaneous increase in volatility and a decrease in time to expiration affect the value of the down-and-out barrier option. An increase in volatility generally *decreases* the value of a down-and-out barrier option, as it makes it more likely that the barrier will be hit, rendering the option worthless. However, the decrease in time to expiration has the *opposite* effect. With less time remaining, there is less opportunity for the barrier to be hit, which tends to *increase* the value of the option (or at least reduce the impact of the barrier). The combined effect is not always straightforward and depends on the precise parameters. In this case, the barrier is relatively close to the current price. The increased volatility significantly increases the likelihood of the barrier being hit *very soon*. The reduced time to expiration, while decreasing the overall chance of hitting the barrier *eventually*, is not enough to offset the immediate increased probability of the barrier being hit due to the heightened volatility. Think of it like a dam holding back water. Increased volatility is like increasing the water pressure significantly. Even if the dam has less time to hold (reduced time to expiration), the immediate surge of pressure makes it more likely to fail quickly. Therefore, the option’s value is likely to decrease substantially.