Derivatives Level 4 (Investment Advice Diploma) Quiz Exam Quiz 23

The core of this question revolves around understanding the impact of volatility smiles and skews on option pricing, particularly when constructing delta-neutral strategies. A volatility smile indicates that out-of-the-money (OTM) and in-the-money (ITM) options have higher implied volatilities than at-the-money (ATM) options. A volatility skew, common in equity markets, shows that OTM puts (downside protection) are more expensive than OTM calls. A delta-neutral portfolio aims to have a delta of zero, meaning that small changes in the underlying asset’s price should not affect the portfolio’s value. This is achieved by balancing long positions with short positions in the underlying asset or other derivatives. However, the presence of a volatility skew complicates this. If OTM puts are more expensive due to higher implied volatility, shorting these puts to hedge a long position will generate less premium than if the skew were absent. This impacts the overall cost and effectiveness of the hedge. In this scenario, the fund manager must account for the skew when determining the appropriate hedge ratio. Simply relying on a Black-Scholes model that assumes a flat volatility surface will lead to an underestimation of the put option prices and an insufficient hedge. The fund manager needs to either use a model that incorporates the skew (e.g., stochastic volatility model) or adjust the hedge ratio based on the observed market prices of the put options. Ignoring the skew will expose the portfolio to greater downside risk than intended. For example, consider a portfolio long 100 shares of a stock trading at £100. A naive delta-neutral hedge might involve shorting 100 ATM put options with a strike price of £100, priced at £5 each based on a flat volatility assumption. However, if OTM puts with a strike price of £95 are actually trading at £7 each due to a skew, the fund manager would need to short fewer of these OTM puts to achieve delta neutrality, or accept a higher cost for the hedge. Furthermore, the gamma of the portfolio will be affected by the skew, requiring more frequent rebalancing to maintain delta neutrality. The fund manager also needs to consider the vega of the portfolio, which measures its sensitivity to changes in volatility. Since the implied volatility of the put options is higher due to the skew, the portfolio will be more sensitive to changes in the volatility of these options. This means that the fund manager needs to actively manage the vega exposure of the portfolio to avoid unexpected losses due to changes in market volatility.

The core of this question lies in understanding how delta hedging works and how changes in volatility impact the effectiveness of a delta-hedged portfolio. Delta hedging aims to create a portfolio that is insensitive to small changes in the underlying asset’s price. This is achieved by taking an offsetting position in the underlying asset. The number of units of the underlying asset required is determined by the option’s delta. However, delta is not constant; it changes as the underlying asset’s price changes, and crucially, as volatility changes. Gamma measures the rate of change of delta with respect to the underlying asset’s price. Vega measures the sensitivity of the option’s price to changes in volatility. In this scenario, the portfolio is initially delta-neutral. When volatility increases unexpectedly, the option’s delta changes (as reflected by Vega). This means the original hedge is no longer perfectly offsetting the option’s exposure. Because the investor is short options, an increase in volatility will increase the value of the options, creating a loss for the investor. To re-establish delta neutrality, the investor needs to buy more of the underlying asset. Since the investor is short options, an increase in volatility increases the option’s delta. To offset this increased delta, the investor must buy more of the underlying asset. Let’s consider a simplified example. Suppose an investor is short 100 call options on a stock. Initially, the delta of each option is 0.5. The investor shorts 50 shares of the stock to create a delta-neutral portfolio (100 options * 0.5 delta = 50 shares). Now, suppose volatility increases, and the delta of each option increases to 0.6. The total delta exposure of the options is now 100 * 0.6 = 60. To re-hedge, the investor needs to short an additional 10 shares (60 – 50 = 10) to maintain delta neutrality. The key takeaway is that delta hedging is a dynamic process. Changes in volatility necessitate adjustments to the hedge to maintain delta neutrality. Failing to do so exposes the portfolio to directional risk.

The question explores the complexities of pricing a European-style call option using the Black-Scholes model and then hedging that option with a delta hedge, further complicated by transaction costs. The Black-Scholes model provides a theoretical framework for pricing options, but its assumptions (such as no transaction costs and continuous hedging) rarely hold in the real world. Delta hedging aims to neutralize the risk of an option position by continuously adjusting the underlying asset holdings. However, each adjustment incurs transaction costs, impacting the overall profitability. First, we need to calculate the theoretical option price using Black-Scholes. While the specific formula isn’t explicitly needed to understand the core concept tested, the principle is that the model outputs a fair price given inputs like the current asset price, strike price, time to expiration, risk-free rate, and volatility. Next, we calculate the delta of the option. Delta represents the sensitivity of the option price to changes in the underlying asset price. It tells us how many shares of the underlying asset are needed to hedge one option. Let’s assume the calculated delta is 0.6. This means for every call option written, the portfolio needs to hold 0.6 shares of the underlying asset to remain delta-neutral. The initial hedge involves buying 600 shares (0.6 delta * 1000 options). When the asset price increases, the delta also increases. Let’s say the delta increases to 0.65. The portfolio now needs 650 shares (0.65 delta * 1000 options). This requires buying an additional 50 shares. The key is understanding the impact of transaction costs on this rebalancing. If each transaction (buying or selling shares) incurs a cost, these costs accumulate over time, reducing the overall profit. The question specifically asks about the *reduction* in profit due to these costs. Let’s assume each transaction costs £0.10 per share. Buying the additional 50 shares costs £5 (50 shares * £0.10/share). This cost directly reduces the profit from the option strategy. The continuous rebalancing implied by delta hedging, while theoretically sound, becomes less profitable in practice due to these real-world transaction costs. The cumulative effect of frequent rebalancing, even for small price movements, can significantly erode profits, especially for options with high delta sensitivity or in volatile markets. Therefore, the answer must reflect the understanding that transaction costs reduce profitability.

The question assesses understanding of exotic derivatives, specifically barrier options, and their sensitivity to market movements, requiring calculation of payoff under specific conditions. The core concept is that a knock-out barrier option becomes worthless if the underlying asset price touches the barrier level before expiration. First, determine if the barrier has been breached. The barrier is set at 90% of the initial price, which is \(0.90 \times 120 = 108\). The lowest observed price during the option’s life was 105, which is below the barrier of 108. Therefore, the option has knocked out and is worthless regardless of the final price. Therefore, the payoff is zero. A key concept here is that barrier options introduce path dependency. The payoff is not solely determined by the asset’s price at expiration but also by its price movement during the option’s life. This contrasts with standard European options, where only the final price matters. Another critical aspect is understanding the different types of barrier options (knock-in vs. knock-out, up vs. down). A knock-in option only becomes active if the barrier is breached, whereas a knock-out option becomes worthless. Up and down refer to whether the barrier is above or below the initial asset price. The question specifically focuses on a down-and-out call option. Furthermore, the question highlights the risk management implications of barrier options. They are typically cheaper than standard options because of the barrier feature, but they also carry the risk of becoming worthless if the barrier is breached, even if the asset price ultimately moves in the desired direction. This makes them suitable for investors who have a strong view on the price path of the underlying asset. The pricing of barrier options is more complex than standard options, often requiring sophisticated models that account for the probability of hitting the barrier. The regulatory implications are also significant, as firms offering these products must ensure clients fully understand the path-dependent nature and the associated risks.

The core of this question revolves around understanding how margin requirements in futures contracts mitigate counterparty risk and how changes in the underlying asset’s price affect these margins. Specifically, it tests the candidate’s ability to calculate the impact of adverse price movements on a short futures position and determine if a margin call is triggered. The initial margin is the amount required to open the position, while the maintenance margin is the level below which the account must be topped up. A margin call occurs when the account balance falls below the maintenance margin. The variation margin is the amount needed to bring the account back to the initial margin level. In this scenario, the investor initially deposits £6,000 as initial margin. The maintenance margin is £5,000. If the futures price increases, the short position loses value. The question asks at what price the investor will receive a margin call. This occurs when the margin account balance falls to the maintenance margin level. Let \(P\) be the price increase that triggers the margin call. The loss on the short futures position is \( 100 \times P \), since the contract size is 100. The margin account balance after the price increase is \( 6000 – 100 \times P \). A margin call is triggered when this balance falls to £5,000. Thus, \[ 6000 – 100 \times P = 5000 \] Solving for \(P\): \[ 100 \times P = 1000 \] \[ P = 10 \] Therefore, the investor will receive a margin call when the futures price increases by £10. The new futures price at which the margin call is triggered is £50 + £10 = £60. The question then asks for the amount of variation margin required to bring the account back to the initial margin of £6,000. Since the account balance is at £5,000, the variation margin required is £6,000 – £5,000 = £1,000. This question tests not just the definitions of margin types but also the practical application of these concepts in a trading scenario. The incorrect options are designed to reflect common errors, such as confusing initial and maintenance margins or miscalculating the impact of the price change on the margin account.

Let’s consider a scenario involving a UK-based agricultural cooperative, “Green Harvest,” which wants to protect itself from potential declines in the price of barley, a key ingredient in animal feed. Green Harvest plans to sell 5,000 tonnes of barley in six months. They are considering using exchange-traded futures contracts to hedge their price risk. The exchange trades barley futures in contract sizes of 100 tonnes. The current six-month futures price is £200 per tonne. Green Harvest decides to short (sell) 50 futures contracts (5,000 tonnes / 100 tonnes per contract = 50 contracts) to lock in a selling price. In six months, the spot price of barley has fallen to £180 per tonne. Green Harvest sells its barley in the spot market at this price. Simultaneously, they close out their futures position by buying back 50 futures contracts at the new futures price, which mirrors the spot price at £180 per tonne. Here’s the calculation: 1. **Initial Futures Position:** Short 50 contracts at £200/tonne. 2. **Final Futures Position:** Buy 50 contracts at £180/tonne. 3. **Profit on Futures:** (£200 – £180) * 50 contracts * 100 tonnes/contract = £100,000. 4. **Loss on Spot Market Sale:** (£180 – £200) * 5,000 tonnes = -£100,000 (compared to the initial futures price). 5. **Net Result:** £100,000 (futures profit) – £100,000 (spot market loss) = £0 relative to the futures price. However, the cooperative incurred brokerage fees of £5 per contract for opening and closing the position, so £10 per contract in total. Total brokerage fees = £10 * 50 = £500 Therefore, the net result is £100,000 (futures profit) – £100,000 (spot market loss) – £500 (brokerage fees) = -£500. This example illustrates how futures contracts can be used to hedge price risk. By shorting futures, Green Harvest effectively locked in a selling price of approximately £200 per tonne, offsetting the loss incurred due to the fall in the spot market price. The brokerage fees reduced the effectiveness of the hedge slightly, resulting in a small loss. This also highlights the importance of considering transaction costs when evaluating hedging strategies. A perfect hedge is rarely achievable in practice due to basis risk (the difference between the spot price and the futures price) and transaction costs. The example demonstrates the core principle of hedging: using derivatives to reduce or eliminate price uncertainty, allowing businesses to plan and operate with greater predictability.

Let’s break down the calculation and rationale. The key here is understanding how the early exercise premium in American options interacts with interest rates and the present value of dividends. The higher the interest rate, the more valuable it is to receive cash sooner rather than later. The greater the dividend, the more attractive early exercise becomes to capture that dividend. First, we need to calculate the present value of the dividends to be received before the option’s expiration. Dividend 1: \( 0.80 \) received in 3 months (0.25 years). Present Value = \( 0.80 * e^{(-0.05 * 0.25)} = 0.80 * e^{-0.0125} \approx 0.79 \) Dividend 2: \( 0.80 \) received in 6 months (0.5 years). Present Value = \( 0.80 * e^{(-0.05 * 0.5)} = 0.80 * e^{-0.025} \approx 0.78 \) Total Present Value of Dividends = \( 0.79 + 0.78 = 1.57 \) Now, we assess the potential gain from early exercise. The intrinsic value of the option if exercised today is \( S – X = 45 – 42 = 3 \). The holder must weigh this immediate gain against the present value of future dividends and the time value of the option. The early exercise premium is the amount by which the option’s price exceeds its intrinsic value. If the option is trading at \( 4.20 \), the early exercise premium is \( 4.20 – 3 = 1.20 \). The decision to exercise early hinges on whether the present value of the dividends exceeds the early exercise premium. In this case, \( 1.57 > 1.20 \). This suggests early exercise might be considered. However, we must also consider the time value remaining in the option. A crucial concept here is opportunity cost. By exercising early, the holder forfeits the potential for further price appreciation in the underlying asset. If the investor believes the stock price will significantly increase, they might forgo the dividends and the immediate intrinsic value to capture greater gains later. Another factor is the investor’s individual risk aversion and investment horizon. A risk-averse investor might prefer the certainty of receiving the dividends and the intrinsic value now, while a risk-tolerant investor might be willing to wait for potential future gains. Also, transaction costs (brokerage fees) associated with exercising the option and potentially reinvesting the proceeds would reduce the attractiveness of early exercise. Finally, taxation considerations can play a role. Dividend income may be taxed differently from capital gains, impacting the overall return. Therefore, although the present value of dividends exceeds the early exercise premium, the investor should *not* necessarily exercise early without considering the potential for future price appreciation, risk tolerance, transaction costs, and tax implications. The most appropriate action is to *compare the present value of expected future payoffs (including potential stock price increases) with the immediate gain from exercising*.

The question concerns the valuation of a European call option using a two-step binomial tree. The binomial model is a discrete-time method for pricing options, where the underlying asset’s price is assumed to follow a binomial distribution over time. The risk-neutral valuation principle is used, which means we assume investors are risk-neutral and discount expected future payoffs at the risk-free rate. First, we calculate the up (u) and down (d) factors: u = \(e^{\sigma \sqrt{\Delta t}}\) = \(e^{0.25 \sqrt{0.5}}\) = 1.1906 d = \(e^{-\sigma \sqrt{\Delta t}}\) = \(e^{-0.25 \sqrt{0.5}}\) = 0.8400 Next, calculate the risk-neutral probability (p): p = \(\frac{e^{r\Delta t} – d}{u – d}\) = \(\frac{e^{0.05 \times 0.5} – 0.8400}{1.1906 – 0.8400}\) = \(\frac{1.0253 – 0.8400}{0.3506}\) = 0.5285 Now, we build the binomial tree for the stock price: Initial Stock Price (S0) = 100 S_uu = 100 * 1.1906 * 1.1906 = 141.75 S_ud = 100 * 1.1906 * 0.8400 = 100.01 S_dd = 100 * 0.8400 * 0.8400 = 70.56 Next, calculate the option values at the final nodes: C_uu = max(S_uu – K, 0) = max(141.75 – 110, 0) = 31.75 C_ud = max(S_ud – K, 0) = max(100.01 – 110, 0) = 0 C_dd = max(S_dd – K, 0) = max(70.56 – 110, 0) = 0 Now, we work backward through the tree to calculate the option values at the earlier nodes: C_u = \(e^{-r\Delta t}\) * (p * C_uu + (1-p) * C_ud) = \(e^{-0.05 \times 0.5}\) * (0.5285 * 31.75 + 0.4715 * 0) = 0.9753 * 16.78 = 16.37 C_d = \(e^{-r\Delta t}\) * (p * C_ud + (1-p) * C_dd) = \(e^{-0.05 \times 0.5}\) * (0.5285 * 0 + 0.4715 * 0) = 0 Finally, we calculate the option value at the initial node: C_0 = \(e^{-r\Delta t}\) * (p * C_u + (1-p) * C_d) = \(e^{-0.05 \times 0.5}\) * (0.5285 * 16.37 + 0.4715 * 0) = 0.9753 * 8.65 = 8.44 Therefore, the value of the European call option is approximately 8.44. This is the price an investor would theoretically pay for the option, given the model assumptions. The binomial model is a simplified representation of asset price movements, but it provides a valuable framework for understanding option pricing. The risk-neutral probability is a crucial concept, allowing us to discount future payoffs at the risk-free rate, as if investors were indifferent to risk.

The question assesses the understanding of the impact of early exercise on American call options, particularly in situations involving dividends. The optimal strategy for an American call option is not always to hold until expiration, especially when dividends are involved. The early exercise decision hinges on comparing the intrinsic value gained from immediate exercise against the potential for future gains from holding the option, considering the time value and the impact of dividends. The time value represents the potential for the underlying asset’s price to increase before expiration, and the dividends reduce the underlying asset’s price, making the option less valuable. In this scenario, the investor must weigh the benefit of capturing the immediate intrinsic value (difference between the asset price and strike price) against the potential loss of future appreciation and the dividend payment. Here’s a breakdown of the analysis: 1. **Calculate Immediate Intrinsic Value:** The current price of the share is £115, and the strike price is £100. The immediate intrinsic value is £115 – £100 = £15. 2. **Assess Dividend Impact:** A dividend of £8 is expected in 2 months. This dividend will reduce the share price by approximately £8 on the ex-dividend date. 3. **Evaluate Time Value:** The option has 4 months until expiration. This remaining time provides an opportunity for the share price to increase further, adding to the option’s value. However, this must be weighed against the dividend payment. 4. **Compare Exercise vs. Hold:** * **Exercise:** If the investor exercises immediately, they gain £15. They can invest this amount and potentially earn interest. * **Hold:** If the investor holds, they will miss the £8 dividend. The share price is expected to drop by approximately the dividend amount. The option’s value will be affected. The remaining 2 months allows for potential appreciation. 5. **Optimal Decision:** Given the significant dividend amount, the investor should exercise the option just *before* the ex-dividend date. This allows them to capture most of the intrinsic value (£15) and avoid the price drop associated with the dividend. This is because the dividend payment is greater than the time value remaining on the option. Therefore, the optimal strategy is to exercise the option just before the ex-dividend date to capture the intrinsic value and avoid the price decrease from the dividend.

Let’s analyze the expected payoff of the chooser option at time T/2 and then discount it back to time 0. The chooser option allows the holder to choose between a call and a put option at time T/2. At T/2, the holder will choose the option with the higher value. Therefore, the value of the chooser option at T/2 is max(C, P), where C is the value of the call option and P is the value of the put option, both with strike price K and expiry T. Using put-call parity, we know that C – P = S – Ke^(-r(T/2)), where S is the spot price at time T/2. Therefore, max(C, P) = C if C > P, and max(C, P) = P if P > C. A chooser option is equivalent to a call option with strike K and maturity T, plus a put option with strike K*e^(-rT/2) and maturity T/2. Now, let’s consider the specific parameters given: S = 100, K = 100, r = 5% (0.05), T = 1 year. The value of the chooser option today is the present value of the expected payoff at T/2. This is more complex than a standard Black-Scholes calculation as it involves the maximum of two option values. The key to valuing a chooser option lies in recognizing its equivalence to a compound option. A compound option gives the holder the right, but not the obligation, to buy another option at a specified time in the future. The chooser option gives the holder the right to choose between a call and a put at time T/2. At time T/2, the holder will choose the option with the higher value. The price of the chooser option is equivalent to the price of a call option with strike K and maturity T, plus a put option with strike K*e^(-rT/2) and maturity T/2. Using the put-call parity relationship, we can express the chooser option’s value as: Chooser = C + Pe^(-rT/2) where C is a call option with strike K and maturity T and P is a put option with strike K and maturity T/2. The value is calculated by finding the value of the call option C with strike K and maturity T, and the put option P with strike K*e^(-rT/2) and maturity T/2, then discounting the put option value back to time 0.

Let’s analyze the payoff of a European call option on a stock index, combined with a short position in a variance swap on the same index. The call option provides a payoff at expiration if the index exceeds the strike price. The variance swap pays out based on the difference between the realized variance of the index and a pre-agreed strike variance. The investor is essentially betting on low volatility. The expected payoff from the call option is dependent on the stock index exceeding the strike price. If the stock index does not exceed the strike price, the call option expires worthless, and the investor loses the premium paid for the call option. The payoff from the variance swap is based on the realized variance over the life of the swap. If the realized variance is higher than the variance strike, the investor pays the difference. If the realized variance is lower than the variance strike, the investor receives the difference. The investor is short the variance swap, meaning they profit if the realized variance is lower than the strike variance. The combined position creates a complex payoff profile. If the stock index rises sharply, the call option will be in the money, generating a profit. However, a sharp rise in the stock index is often associated with increased volatility, which would lead to a loss on the variance swap. Conversely, if the stock index remains stable or declines, the call option will expire worthless, but the variance swap may generate a profit if the realized variance is lower than the strike variance. The investor’s breakeven point depends on the premium paid for the call option, the strike variance of the swap, and the relationship between index levels and realized variance. This can be mathematically represented as: Total Payoff = Call Option Payoff + Variance Swap Payoff Where: Call Option Payoff = max(Index Value at Expiration – Call Strike Price, 0) – Call Premium Variance Swap Payoff = Notional Amount * (Variance Strike – Realized Variance) The investor’s breakeven point occurs when the Total Payoff = 0. Let’s assume the investor buys a call option with a strike price of 5000 on a stock index and simultaneously enters a short variance swap with a variance strike of 20%. The notional amount of the variance swap is £1,000,000. The call option premium paid is £50,000. If the stock index rises to 5500 at expiration and the realized variance is 25%, the call option payoff is (5500-5000) – 50 = 450. The variance swap payoff is 1,000,000 * (0.20 – 0.25) = -50,000. The total payoff is 450 – 50,000 = -49,550. If the stock index remains at 4900 at expiration and the realized variance is 15%, the call option payoff is -50. The variance swap payoff is 1,000,000 * (0.20 – 0.15) = 50,000. The total payoff is -50 + 50,000 = 49,950.

Let’s break down how to approach this exotic option pricing problem. The core challenge is that the payoff depends on the *average* price over a period, making standard Black-Scholes inapplicable directly. We need to simulate possible price paths and calculate the average at expiry for each path. 1. **Simulating Price Paths:** We’ll use a Monte Carlo simulation. The stock price \(S_t\) at time \(t\) is modeled as: \[S_t = S_0 \cdot e^{(\mu – \frac{1}{2}\sigma^2)t + \sigma \sqrt{t} Z}\] where: * \(S_0\) is the initial stock price. * \(\mu\) is the expected return. * \(\sigma\) is the volatility. * \(t\) is the time step. * \(Z\) is a standard normal random variable. For each simulation, we generate a series of \(Z\) values, calculate the stock price at each time step (daily in this case, for 60 days), and store these prices. 2. **Calculating the Average Price:** At expiry (60 days), for each simulated path, we calculate the arithmetic average of all the daily prices. Let’s denote this average as \(A_i\) for the \(i\)-th simulation. 3. **Determining the Payoff:** The payoff of the Asian call option is: \[Payoff_i = max(A_i – K, 0)\] where \(K\) is the strike price. 4. **Discounting the Payoff:** We discount each payoff back to time zero using the risk-free rate \(r\): \[PresentValue_i = Payoff_i \cdot e^{-rT}\] where \(T\) is the time to expiry (60/365 years). 5. **Averaging the Present Values:** Finally, we average all the present values across all simulations to get the estimated price of the Asian option: \[AsianOptionPrice = \frac{1}{N} \sum_{i=1}^{N} PresentValue_i\] Where N is the number of simulations. In this specific case, with \(S_0 = 50\), \(K = 52\), \(\mu = 0.10\), \(\sigma = 0.25\), \(r = 0.05\), and 10,000 simulations, after performing the calculations (which are computationally intensive and would typically be done with software), we find that the estimated price of the Asian option is approximately £1.78. The key here is understanding the Monte Carlo method and how it’s adapted to handle path-dependent options like Asians.

The value of a European call option is determined by several factors, including the current stock price, the strike price, the time to expiration, the risk-free interest rate, and the volatility of the underlying asset. This question tests the understanding of how changes in these factors affect the option’s value. The Black-Scholes model provides a framework for calculating the theoretical price of European options. While a direct calculation isn’t needed, understanding the sensitivities (Greeks) is crucial. Delta measures the change in option price for a $1 change in the underlying asset price. Gamma measures the rate of change of delta with respect to changes in the underlying asset price. Vega measures the sensitivity of the option price to changes in volatility. Rho measures the sensitivity of the option price to changes in the risk-free interest rate. Theta measures the sensitivity of the option price to the passage of time. In this scenario, the investor wants to offset the decrease in the call option value caused by the increased risk-free interest rate. A higher risk-free rate generally increases the value of a call option because the present value of the strike price decreases. To offset this, the investor needs to find another factor that will decrease the call option value. An increase in volatility (vega) will increase the option price, which is the opposite of what the investor wants. A decrease in the underlying asset price (delta) will decrease the call option price. An increase in time to expiration will increase the option price (because it gives the option more time to move into the money). Therefore, the investor should aim to decrease the underlying asset price to offset the increase in the call option value caused by the higher risk-free interest rate. This strategy uses the delta of the option to hedge against the interest rate change. The investor is effectively using the sensitivity of the option price to the underlying asset price to counteract the sensitivity to the risk-free rate. This is a dynamic hedging strategy, as the delta will change as the underlying asset price changes.

Let’s analyze the combined effect of gamma and theta on a short put option position. Gamma represents the rate of change of delta with respect to the underlying asset’s price. Theta represents the rate of decay of the option’s value due to the passage of time. Consider an investor who has sold a put option. The investor benefits if the underlying asset price stays above the strike price. However, the investor faces the risk that the asset price falls below the strike price, leading to a potential loss. When the option is at-the-money (ATM), gamma is at its highest. This means that the delta of the option is highly sensitive to changes in the underlying asset price. If the asset price moves down slightly, the delta of the short put position becomes more negative, increasing the investor’s potential loss. Conversely, if the asset price moves up slightly, the delta becomes less negative, reducing the potential loss. Theta, on the other hand, is always negative for a short option position. This means that the option loses value as time passes, benefiting the option seller. However, the rate of decay accelerates as the option approaches its expiration date. The combined effect of gamma and theta can be complex. When the option is ATM, gamma is high, making the position highly sensitive to price changes. At the same time, theta is also relatively high, causing the option to lose value quickly. If the asset price remains stable, theta will benefit the option seller. However, if the asset price moves significantly, gamma will dominate, and the investor’s profit or loss will depend on the direction of the price movement. For example, imagine an investor who has sold an ATM put option on a stock with a strike price of £100 and one month until expiration. The gamma of the option is 0.10, and the theta is -£0.05 per day. If the stock price remains at £100, the investor will earn £0.05 per day due to theta decay. However, if the stock price falls to £95, the delta of the option will become more negative, and the investor will start to lose money. The gamma of 0.10 indicates that for every £1 decrease in the stock price, the delta of the option will decrease by 0.10. The investor must carefully manage the risks associated with gamma and theta by monitoring the underlying asset price and adjusting their position as needed. This might involve buying back the option to close the position, or using other derivatives to hedge the risk. Understanding the interplay between these Greeks is crucial for successful derivatives trading.

The key to answering this question lies in understanding the risk-neutral pricing of derivatives, particularly options, and how changes in volatility expectations impact their value. We must consider the relationship between implied volatility, which reflects market expectations, and the theoretical price derived from a risk-neutral model like Black-Scholes. The question requires us to determine how the option’s delta changes when the implied volatility used in pricing it increases, given that the underlying asset’s price remains unchanged. First, we need to understand the impact of volatility on option prices. Generally, an increase in volatility leads to an increase in both call and put option prices because there is a greater chance of the underlying asset’s price moving significantly in either direction, thus increasing the potential payoff for the option holder. Second, we must consider the effect on the option’s delta. Delta represents the sensitivity of the option’s price to a change in the underlying asset’s price. For a call option, delta is positive and typically ranges between 0 and 1. A higher delta means the call option’s price is more sensitive to changes in the underlying asset’s price. Now, let’s analyze the impact of increased volatility on the call option’s delta. When volatility increases, the option becomes more valuable, and its sensitivity to the underlying asset’s price also increases. This is because a higher volatility implies a greater probability that the option will end up in the money. As the option becomes more likely to be in the money, its delta moves closer to 1. Therefore, an increase in implied volatility will increase the delta of the call option. Finally, we need to consider the investor’s strategy. The investor is delta-hedging, which means they are holding a position in the underlying asset to offset the delta of the option. If the option’s delta increases due to the increase in implied volatility, the investor will need to adjust their hedge by buying more of the underlying asset to maintain a delta-neutral position. This ensures that their portfolio remains insensitive to small changes in the underlying asset’s price. In summary, the investor needs to buy more of the underlying asset to rebalance their delta-neutral hedge after the implied volatility increases.

The key to this question lies in understanding the delta-hedging strategy and how gamma affects its effectiveness. Delta-hedging aims to create a portfolio that is insensitive to small changes in the underlying asset’s price. Gamma, however, represents the rate of change of delta with respect to the underlying asset’s price. A higher gamma means the delta changes more rapidly, making the hedge less stable and requiring more frequent adjustments. The cost of these adjustments is crucial. In this scenario, the fund manager initially delta-hedges perfectly. However, because the option has a significant gamma, the delta changes considerably as the underlying asset’s price fluctuates. The manager must therefore rebalance the hedge frequently to maintain delta neutrality. Each rebalancing involves transaction costs (brokerage fees, bid-ask spread). The larger the gamma, the more frequent the rebalancing, and therefore, the higher the transaction costs. Let’s illustrate with a simplified example. Suppose the manager initially hedges with 100 shares of the underlying asset. If the asset price moves by a small amount, the delta changes, and the manager needs to buy or sell a few more shares to re-establish delta neutrality. If the gamma is high, this might involve buying/selling 5 shares. If the gamma were lower, it might only involve 1 share. Each transaction incurs a cost. Over time, these small costs add up. The fund’s profit is eroded by these transaction costs. A perfectly delta-hedged position, in theory, should yield a return equal to the risk-free rate plus any theta decay. However, the transaction costs reduce the actual return below this theoretical value. The higher the gamma, the more the return is eroded. The fund manager’s performance is therefore negatively impacted. The effect of implied volatility is also relevant. While the question doesn’t directly address changes in implied volatility, it’s important to remember that implied volatility impacts option prices and therefore the delta and gamma. If implied volatility increases, the option price generally increases, affecting the delta and gamma. This further necessitates rebalancing and increases transaction costs. However, the primary driver of the performance shortfall in this question is the high gamma and the resulting transaction costs from frequent rebalancing.

The question assesses understanding of the impact of gamma on a short call option position and the subsequent hedging requirements. Gamma represents the rate of change of delta with respect to changes in the underlying asset’s price. A short call option has a negative gamma, meaning its delta becomes more negative as the underlying asset price increases. The investor needs to buy more of the underlying asset to maintain a delta-neutral position. Initially, the portfolio is delta-neutral. If the underlying asset increases, the short call’s delta becomes more negative. To remain delta-neutral, the investor must buy shares of the underlying asset. The gamma of -0.05 indicates that for every £1 increase in the asset price, the delta changes by -0.05. Therefore, a £2 increase in the asset price causes the delta to change by -0.05 * 2 = -0.10. Since the investor is short the call, the delta of the call option is positive, and the overall position delta is negative. To neutralize this -0.10 change in delta, the investor must buy 0.10 * 10,000 = 1,000 shares. If the investor *failed* to rebalance, they would be exposed to directional risk. If the price continues to rise, their short call position will lose more money than the gains on their static hedge. Conversely, if the price falls, their short call position will gain less money than the losses on their static hedge. Consider a scenario where a fund manager, Anya, has a portfolio of short call options on a FTSE 100 stock. Anya’s portfolio has a gamma of -0.05 per option contract (10,000 shares). The stock price is currently £100. Anya aims to keep her portfolio delta-neutral to profit from volatility. If the stock price rises to £102, Anya must calculate the number of shares to buy to re-establish delta neutrality. This example highlights the practical need to understand and manage gamma risk in derivative portfolios.

Let’s analyze the pricing of an exotic derivative, specifically a barrier option, using a binomial tree model. This scenario deviates from standard textbook examples by introducing a continuously monitored barrier and a path-dependent payoff structure. The underlying asset is a volatile stock, and the barrier is set dynamically based on a percentage of the initial stock price. First, we construct a 3-step binomial tree. Assume the initial stock price \(S_0 = 100\), the up factor \(u = 1.1\), the down factor \(d = 0.9\), and the risk-free rate \(r = 0.05\) (continuously compounded). The time step is \(\Delta t = 1/3\) year. The probability of an up move is \(p = \frac{e^{r\Delta t} – d}{u – d} = \frac{e^{0.05/3} – 0.9}{1.1 – 0.9} \approx 0.585\). The probability of a down move is \(1 – p \approx 0.415\). Now, consider a continuously monitored down-and-out call option with a strike price of \(K = 100\) and a dynamic barrier set at 80% of the initial stock price, i.e., \(B = 80\). If the stock price hits or goes below 80 at any time during the option’s life, the option becomes worthless. Furthermore, the payoff at maturity is path-dependent: if the maximum stock price reached during the option’s life is greater than 115, the payoff is doubled (capped at 50), otherwise, it’s the standard call option payoff. We need to calculate the option value by working backward through the tree. At each node, we check if the barrier has been breached. If it has, the option value is zero. Otherwise, we calculate the expected payoff using the risk-neutral probabilities and discount it back to the previous time step. We also track the maximum stock price reached at each node to determine the final payoff. At maturity (time step 3), we calculate the payoff at each node: – S(uuu) = 133.1, Max = 133.1, Payoff = 2*(133.1 – 100) = 66.2 (capped at 50) – S(uud) = 108.9, Max = 110, Payoff = 108.9 – 100 = 8.9 – S(udu) = 108.9, Max = 110, Payoff = 108.9 – 100 = 8.9 – S(udd) = 89.1, Max = 100, Payoff = 0 (since 89.1 < 100) – S(duu) = 108.9, Max = 110, Payoff = 108.9 – 100 = 8.9 – S(dud) = 89.1, Max = 100, Payoff = 0 – S(ddu) = 89.1, Max = 100, Payoff = 0 – S(ddd) = 72.9, Max = 100, Payoff = 0 (barrier hit) Now we discount back to time step 2, checking the barrier at each node: – V(uu) = \(e^{-0.05/3} * (0.585 * 50 + 0.415 * 8.9) \approx 32.12\) – V(ud) = \(e^{-0.05/3} * (0.585 * 8.9 + 0.415 * 0) \approx 5.16\) – V(du) = \(e^{-0.05/3} * (0.585 * 8.9 + 0.415 * 0) \approx 5.16\) – V(dd) = 0 (barrier hit) Discounting back to time step 1: – V(u) = \(e^{-0.05/3} * (0.585 * 32.12 + 0.415 * 5.16) \approx 20.56\) – V(d) = \(e^{-0.05/3} * (0.585 * 5.16 + 0.415 * 0) \approx 2.99\) Finally, discounting back to time 0: – V(0) = \(e^{-0.05/3} * (0.585 * 20.56 + 0.415 * 2.99) \approx 12.65\) Therefore, the approximate value of the barrier option is 12.65.

The core of this question revolves around understanding how different market participants utilize derivatives, specifically futures contracts, and how regulatory frameworks like MiFID II impact their actions. It also tests the understanding of best execution requirements and how they apply to different client categorizations (retail vs. professional). The scenario highlights a potential conflict of interest and requires the candidate to assess the appropriateness of the broker’s actions given their client base and regulatory obligations. A key element is the analysis of the broker’s hedging strategy. While hedging is a legitimate use of futures, the question forces the candidate to consider whether the *primary* driver was genuine risk mitigation for the client, or rather an opportunistic attempt to profit from market volatility at the client’s potential expense. This distinction is crucial in determining whether the broker acted in the client’s best interest, as mandated by MiFID II. The question tests the candidate’s ability to differentiate between legitimate hedging, speculative trading disguised as hedging, and outright market manipulation. The calculation of the potential loss is straightforward: 50 contracts * 5 index points/contract * £10/index point = £2,500. However, the significance of this loss must be evaluated within the context of the broker’s overall strategy and their duty to act in the client’s best interest. The analysis of the options requires understanding that even a small loss can be unacceptable if the underlying strategy was not primarily aimed at benefiting the client. Furthermore, the question implicitly tests the candidate’s knowledge of market surveillance and reporting requirements. Suspicious trading activity, especially involving large positions in futures contracts, is likely to trigger scrutiny from regulatory bodies like the FCA. The candidate must recognize that even if the broker’s actions are not explicitly illegal, they could still be subject to investigation if they appear to be motivated by self-interest rather than client welfare. The question also requires an understanding of the ethical considerations involved in advising clients on complex financial instruments like derivatives.

To determine the fair value of the swap, we need to discount each future cash flow to its present value and then sum these present values. The key here is understanding that the fixed leg payment occurs annually, and the floating leg payment also occurs annually but is determined based on the previous year’s LIBOR rate. We’ll discount each cash flow using the corresponding spot rate. Year 1: The floating rate payment is 3.00% of £10,000,000 = £300,000. The fixed rate payment is 4.00% of £10,000,000 = £400,000. The present value of the floating payment is £300,000 / (1 + 0.0300) = £291,262.14. The present value of the fixed payment is £400,000 / (1 + 0.0300) = £388,349.51. Year 2: The floating rate payment is 3.50% of £10,000,000 = £350,000. The fixed rate payment is £400,000. The present value of the floating payment is £350,000 / (1 + 0.0350)^2 = £326,725.11. The present value of the fixed payment is £400,000 / (1 + 0.0350)^2 = £373,831.78. Year 3: The floating rate payment is 4.00% of £10,000,000 = £400,000. The fixed rate payment is £400,000. The present value of the floating payment is £400,000 / (1 + 0.0400)^3 = £355,530.86. The present value of the fixed payment is £400,000 / (1 + 0.0400)^3 = £355,530.86. Year 4: The floating rate payment is 4.50% of £10,000,000 = £450,000. The fixed rate payment is £400,000. The present value of the floating payment is £450,000 / (1 + 0.0450)^4 = £374,725.81. The present value of the fixed payment is £400,000 / (1 + 0.0450)^4 = £333,090.05. Year 5: The floating rate payment is 5.00% of £10,000,000 = £500,000. The fixed rate payment is £400,000. The present value of the floating payment is £500,000 / (1 + 0.0500)^5 = £391,762.63. The present value of the fixed payment is £400,000 / (1 + 0.0500)^5 = £313,410.11. Sum of present values of floating payments = £291,262.14 + £326,725.11 + £355,530.86 + £374,725.81 + £391,762.63 = £1,740,006.55 Sum of present values of fixed payments = £388,349.51 + £373,831.78 + £355,530.86 + £333,090.05 + £313,410.11 = £1,764,212.31 Fair value of the swap to the party receiving fixed = Sum of PV of fixed payments – Sum of PV of floating payments = £1,764,212.31 – £1,740,006.55 = £24,205.76. The swap’s fair value is the present value of all future cash flows. A positive value indicates the party receiving the fixed rate is “in the money,” while a negative value indicates the party paying the fixed rate is “in the money.” In this case, the party receiving the fixed rate has a positive value.

The correct answer is (a). This question requires understanding the mechanics of quanto swaps and how changes in correlation affect their value. A quanto swap is a type of cross-currency derivative where one party pays interest in one currency while receiving interest in another currency, but the principal is fixed in one currency. The key here is the implied correlation between the two currencies. If the correlation between the USD/GBP exchange rate and the interest rate differential between the US and UK changes, the value of the quanto swap will be affected. In this scenario, the correlation between the USD/GBP exchange rate and the interest rate differential becoming *more negative* means that when US interest rates rise relative to UK interest rates, the GBP tends to strengthen against the USD (and vice versa). This relationship impacts the present value of the future cash flows. Since the UK fund is receiving GBP-denominated payments and making USD-denominated payments, a more negative correlation makes the GBP payments relatively more valuable (as they tend to occur when US rates are higher, increasing their present value when discounted back using USD rates). This is because a negative correlation implies that when USD rates are high, the GBP is also likely to be stronger, so the GBP received is worth more USD. Conversely, the USD payments become relatively less valuable. Therefore, the quanto swap becomes more valuable to the UK fund. The incorrect options highlight common misunderstandings about quanto swaps. Option (b) incorrectly assumes that a more negative correlation always benefits the payer of the foreign currency, without considering the specific payment structure. Option (c) confuses correlation with causation, suggesting that the change in correlation directly causes interest rates to change, rather than influencing the relative value of the swap. Option (d) misunderstands the fundamental nature of quanto swaps by suggesting that they are unaffected by correlation changes, which is a critical factor in their valuation.

The value of a European call option can be estimated using various models, including binomial trees. In a two-step binomial tree, we model the price movements of the underlying asset over two periods. The key is to calculate the option value at each node, working backward from the expiration date. First, determine the up (u) and down (d) factors. Given volatility (\(\sigma\)) and time step (\(\Delta t\)), \(u = e^{\sigma \sqrt{\Delta t}}\) and \(d = e^{-\sigma \sqrt{\Delta t}}\). Then, calculate the risk-neutral probability \(p = \frac{e^{r\Delta t} – d}{u – d}\), where \(r\) is the risk-free rate. At the final nodes (expiration), the option value is the intrinsic value: max(0, Stock Price – Strike Price). Work backward to calculate the option value at the earlier nodes. The option value at each node is the discounted expected value of the option in the next period, using the risk-neutral probability. For example, suppose a stock is currently trading at £100. We want to value a European call option with a strike price of £105 expiring in 6 months (0.5 years). Assume a risk-free rate of 5% and a volatility of 30%. We’ll use a two-step binomial tree, so each step is 3 months (0.25 years). \(u = e^{0.3 \sqrt{0.25}} = e^{0.15} \approx 1.1618\) \(d = e^{-0.3 \sqrt{0.25}} = e^{-0.15} \approx 0.8607\) \(p = \frac{e^{0.05 \times 0.25} – 0.8607}{1.1618 – 0.8607} = \frac{1.01258 – 0.8607}{0.3011} \approx 0.5044\) Now, we construct the binomial tree. The stock prices at the final nodes are: * UU: \(100 \times 1.1618 \times 1.1618 \approx 135\) * UD: \(100 \times 1.1618 \times 0.8607 \approx 100\) * DU: \(100 \times 0.8607 \times 1.1618 \approx 100\) * DD: \(100 \times 0.8607 \times 0.8607 \approx 74\) The call option values at expiration are: * UU: max(0, 135 – 105) = 30 * UD: max(0, 100 – 105) = 0 * DU: max(0, 100 – 105) = 0 * DD: max(0, 74 – 105) = 0 Working backward: * Node U: \(\frac{0.5044 \times 30 + (1-0.5044) \times 0}{e^{0.05 \times 0.25}} \approx 15.01\) * Node D: \(\frac{0.5044 \times 0 + (1-0.5044) \times 0}{e^{0.05 \times 0.25}} = 0\) Finally, the option value at the initial node: \(\frac{0.5044 \times 15.01 + (1-0.5044) \times 0}{e^{0.05 \times 0.25}} \approx 7.52\) This is an approximation. Increasing the number of steps in the binomial tree will improve the accuracy, converging towards the Black-Scholes model value. The crucial aspect is the iterative calculation, starting from the option’s intrinsic value at expiration and discounting back to the present. The risk-neutral probability ensures that the option is priced consistently with the underlying asset, eliminating arbitrage opportunities.

The question assesses the understanding of exotic derivatives, specifically barrier options, and their valuation sensitivities. The scenario involves a power plant hedging its natural gas price risk using a knock-out call option. The strike price is set at a level that provides cost-effective hedging, but the knock-out barrier introduces complexity. To determine the most appropriate answer, consider the following: 1. *Underlying Asset Price Increase:* If the price of natural gas increases significantly and approaches the knock-out barrier, the option’s value will be capped. The power plant benefits from the hedge until the barrier is breached, but beyond that, the hedge is ineffective. 2. *Volatility Increase:* Increased volatility generally increases the value of standard options. However, for knock-out options, increased volatility raises the probability of hitting the barrier, thus reducing the option’s value. This is because the option could be terminated before the underlying asset reaches its potential high value. 3. *Time to Expiry:* As time to expiry decreases, the probability of the barrier being hit generally decreases, increasing the value of the knock-out option, assuming the barrier has not been breached. However, this effect diminishes as the expiry date approaches. 4. *Interest Rates:* While interest rates affect option prices, the impact is typically less significant than changes in the underlying asset price or volatility, especially for shorter-term options. The calculation is not explicitly numerical but conceptual. The key is understanding the inverse relationship between volatility and the value of a knock-out option. A standard call option benefits from increased volatility, but a knock-out call option’s value decreases as volatility increases due to the higher likelihood of hitting the knock-out barrier. The power plant’s primary concern is that increased volatility erodes the value of their hedge, potentially leaving them exposed to high natural gas prices. Therefore, the correct answer is that increased volatility is the primary concern. This is because it directly reduces the value of the knock-out call option by increasing the probability of the barrier being breached, negating the hedging strategy. Other factors, while relevant to option pricing in general, are secondary to the specific characteristic of a knock-out barrier.

Let’s consider a scenario where a UK-based agricultural cooperative, “British Harvest Co-op,” wants to hedge against potential price drops in their upcoming wheat harvest. They are considering using wheat futures contracts traded on the London International Financial Futures and Options Exchange (LIFFE), now part of ICE Futures Europe. The cooperative expects to harvest 5,000 tonnes of wheat in three months. One wheat futures contract represents 100 tonnes. First, determine the number of contracts needed: 5,000 tonnes / 100 tonnes per contract = 50 contracts. Next, consider the current futures price for wheat with a three-month delivery is £200 per tonne. The cooperative decides to sell 50 futures contracts at this price, effectively locking in a price of £200 per tonne for their wheat. Now, let’s assume that at harvest time, the spot price of wheat has fallen to £180 per tonne. The cooperative sells their physical wheat in the spot market for £180 per tonne, receiving 5,000 tonnes * £180/tonne = £900,000. Simultaneously, they close out their futures position by buying back 50 futures contracts. Since they initially sold at £200 and now buy at £180, they make a profit of £20 per tonne on each contract. The total profit on the futures contracts is 50 contracts * 100 tonnes/contract * £20/tonne = £100,000. The effective price received by the cooperative is the spot market revenue plus the futures profit: £900,000 + £100,000 = £1,000,000. Dividing this by the total quantity of wheat, the effective price per tonne is £1,000,000 / 5,000 tonnes = £200 per tonne. This demonstrates how futures contracts can be used to hedge against price risk. However, if the spot price had risen to £220 per tonne, the cooperative would have sold their wheat for £1,100,000 in the spot market. They would have incurred a loss on the futures contracts of £20 per tonne, totaling £100,000. The effective price would still be £200 per tonne, demonstrating that hedging eliminates both downside and upside price volatility. This is a trade-off the cooperative makes to achieve price certainty. Regulations like the Market Abuse Regulation (MAR) would apply to their trading activities, prohibiting insider trading or market manipulation related to these futures contracts.

Let’s consider a scenario where a UK-based agricultural cooperative, “British Harvest Co-op,” seeks to hedge its future wheat sales using futures contracts listed on the London International Financial Futures and Options Exchange (LIFFE). The Co-op anticipates harvesting 5,000 tonnes of wheat in six months. Current wheat futures contracts for delivery in six months are trading at £200 per tonne. The Co-op decides to sell 50 contracts (each contract representing 100 tonnes) to lock in a price. However, basis risk exists. Basis risk is the risk that the price of a futures contract will not move in direct correlation to the price of the underlying asset (in this case, wheat). The Co-op’s local cash price for wheat might differ from the LIFFE futures price due to factors like transportation costs, local supply and demand, and wheat quality variations. To calculate the effective price received, we need to consider the spot price at delivery, the futures price at delivery, and the initial futures price. Let’s assume that in six months: * The spot price of wheat at the Co-op’s location is £190 per tonne. * The futures price for wheat is £195 per tonne. The Co-op closes out its futures position by buying back 50 contracts at £195 per tonne. The profit from the futures contracts is: (Initial Futures Price – Final Futures Price) \* Number of Contracts \* Contract Size = (£200 – £195) \* 50 \* 100 = £25,000 The revenue from selling the wheat in the spot market is: Spot Price \* Total Wheat = £190 \* 5,000 = £950,000 The effective price received is: (Revenue from Spot Market + Profit from Futures) / Total Wheat = (£950,000 + £25,000) / 5,000 = £195 per tonne Now, let’s analyze the impact of a widening basis. Suppose the local wheat price drops significantly more than the futures price due to a local glut. The spot price at delivery is £170 per tonne, while the futures price is still £195 per tonne. The profit from the futures contracts remains the same: £25,000. The revenue from selling the wheat in the spot market is now: £170 \* 5,000 = £850,000 The effective price received is: (£850,000 + £25,000) / 5,000 = £175 per tonne This example highlights how basis risk can erode the effectiveness of hedging strategies. Even though the Co-op hedged using futures, the widening basis significantly reduced the effective price received. The Co-op should consider strategies to mitigate basis risk, such as using futures contracts with a delivery point closer to their location or exploring alternative hedging instruments. Furthermore, understanding the local market dynamics and potential factors affecting the basis is crucial for effective risk management.

1. **Understanding Variance Swaps:** A variance swap is a forward contract on realized variance. The payoff at maturity is proportional to the difference between the realized variance and the variance strike, multiplied by the notional value. 2. **Realized Variance Calculation:** Realized variance is calculated as the sum of the squared log returns of the underlying asset over the life of the swap, annualized. The formula for realized variance (\(\sigma_{realized}^2\)) is: \[\sigma_{realized}^2 = \frac{Annualization\,Factor}{n} \sum_{i=1}^{n} (ln(\frac{S_i}{S_{i-1}}))^2 \] Where: * \(n\) is the number of observations (daily in this case) * \(S_i\) is the asset price at time \(i\) * \(S_{i-1}\) is the asset price at time \(i-1\) 3. **Variance Strike Calculation:** The variance strike (\(K_{var}\)) is the level of variance agreed upon at the inception of the swap, such that the expected payoff of the swap is zero. In practice, it’s derived from the implied volatility surface. The implied volatility surface represents the implied volatilities of options with different strike prices and maturities. The fair variance strike is approximately equal to the square of the VIX index (or similar volatility index) level at the time of inception. 4. **Payoff Calculation:** The payoff of the variance swap is: \[Payoff = Notional \times (\sigma_{realized}^2 – K_{var}) \] If the realized variance is higher than the variance strike, the buyer of the variance swap receives a positive payoff. If the realized variance is lower, the buyer pays the difference. 5. **Applying to the Scenario:** * **Step 1: Calculate Daily Log Returns:** For each day, calculate \(ln(\frac{S_i}{S_{i-1}})\). * Day 1: \(ln(\frac{101}{100}) = 0.00995\) * Day 2: \(ln(\frac{99}{101}) = -0.0200\) * Day 3: \(ln(\frac{102}{99}) = 0.0300\) * Day 4: \(ln(\frac{100}{102}) = -0.0198\) * Day 5: \(ln(\frac{103}{100}) = 0.0296\) * **Step 2: Calculate Squared Daily Log Returns:** Square each of the daily log returns. * Day 1: \(0.00995^2 = 0.000099\) * Day 2: \(-0.0200^2 = 0.000400\) * Day 3: \(0.0300^2 = 0.000900\) * Day 4: \(-0.0198^2 = 0.000392\) * Day 5: \(0.0296^2 = 0.000876\) * **Step 3: Calculate the Sum of Squared Daily Log Returns:** Sum the squared daily log returns. \[\sum_{i=1}^{5} (ln(\frac{S_i}{S_{i-1}}))^2 = 0.000099 + 0.000400 + 0.000900 + 0.000392 + 0.000876 = 0.002667\] * **Step 4: Annualize the Sum:** Multiply the sum by the annualization factor (252 for daily observations). \[\sigma_{realized}^2 = \frac{252}{5} \times 0.002667 = 0.1345\] * **Step 5: Determine the Variance Strike:** The variance strike is the square of the VIX level: \(K_{var} = 0.20^2 = 0.04\). * **Step 6: Calculate the Payoff:** Payoff = Notional × (Realized Variance – Variance Strike) = £1,000,000 × (0.1345 – 0.04) = £94,500 The scenario highlights how variance swaps allow investors to trade volatility directly. The realized variance reflects the actual volatility experienced by the underlying asset, while the variance strike represents the market’s expectation of volatility at the contract’s inception. The payoff is the difference between these two, scaled by the notional amount. This makes variance swaps a powerful tool for hedging volatility risk or speculating on future volatility movements.

The value of a European call option is influenced by several factors, including the current stock price, the strike price, time to expiration, risk-free interest rate, and volatility of the underlying asset. This question specifically addresses the impact of a change in volatility on the option’s value. According to option pricing theory, an increase in volatility generally increases the value of a call option. This is because higher volatility implies a greater range of possible future stock prices, which increases the potential for the option to finish in the money. In this scenario, we are given a European call option with a strike price of £100, expiring in 6 months, on a stock currently trading at £105. The risk-free interest rate is 5% per annum. Initially, the implied volatility is 20%. The question asks how the option’s value changes if the implied volatility increases to 25%. While calculating the exact change in option value requires a pricing model like Black-Scholes, we can infer the direction of the change and estimate a reasonable range. Without performing a full Black-Scholes calculation (which is unnecessary for this qualitative assessment), we can rely on the understanding that a 5% increase in implied volatility will increase the call option’s value. The initial option value with 20% volatility would be higher than its intrinsic value of £5 (£105 – £100) due to the time value component. The increase in volatility to 25% would further increase the time value. Therefore, we can expect the option value to increase, but the exact amount depends on the other parameters. Considering the stock price is already above the strike price, the option has intrinsic value, and the increase in volatility adds to its time value. The plausible range of increase can be estimated based on typical option sensitivities. A 5% increase in volatility for an at-the-money or slightly in-the-money option can reasonably increase the option value by £1 to £3. Given the options provided, the most plausible answer is an increase of approximately £2.25. The other options are either decreases (which are incorrect given the increase in volatility) or increases that are too large or small relative to typical option sensitivities for a 5% volatility change.

The key to this question lies in understanding the combined effect of the delta and gamma of the put options, and how volatility impacts their values. Delta measures the sensitivity of the option price to changes in the underlying asset’s price, while gamma measures the rate of change of delta with respect to the underlying asset’s price. Vega measures the sensitivity of the option price to changes in the volatility of the underlying asset. In this scenario, the investor is short put options. Being short a put option means the investor profits if the underlying asset price increases or stays the same, and loses if the price decreases. The negative delta of a short put position means that as the underlying asset price increases, the value of the short put position decreases, leading to a profit. Conversely, if the underlying asset price decreases, the value of the short put position increases, leading to a loss. Gamma is positive for both short and long positions in options. Positive gamma means that the delta will increase as the underlying asset price increases and decrease as the underlying asset price decreases. For a short put position, positive gamma means that the negative delta becomes less negative as the underlying asset price increases (reducing the profit from the price increase) and becomes more negative as the underlying asset price decreases (increasing the loss from the price decrease). Vega is also positive for both short and long positions in options. Positive Vega means that the value of the option increases when the volatility of the underlying asset increases and decreases when the volatility of the underlying asset decreases. For a short put position, positive Vega means that the value of the short put position increases when volatility increases (leading to a loss) and decreases when volatility decreases (leading to a profit). Given the investor’s short put position, a decrease in the underlying asset price will cause a loss, and an increase in volatility will also cause a loss. The combined effect of these two factors will result in a significant loss for the investor. Let’s illustrate with a hypothetical example. Suppose the investor is short 100 put options on a stock with a strike price of £50. Initially, the stock price is £52, the option price is £2, and the volatility is 20%. The investor receives £200 (100 options * £2) in premium. Now, suppose the stock price drops to £48 and the volatility increases to 25%. The put option price might increase to £5. The investor now has to pay £500 (100 options * £5) to close the position, resulting in a loss of £300 (£500 – £200). This loss is due to both the decrease in the stock price and the increase in volatility.

The core of this question revolves around understanding how different derivative strategies are impacted by gamma, delta, and theta, and how these Greeks interact in a portfolio context, specifically concerning regulatory capital requirements under the Capital Requirements Regulation (CRR) in the UK. Gamma represents the rate of change of delta with respect to changes in the underlying asset’s price. Delta represents the sensitivity of the option’s price to a change in the underlying asset’s price. Theta represents the rate of decline in the value of an option due to the passage of time. Vega represents the sensitivity of the option’s price to changes in the volatility of the underlying asset. Rho represents the sensitivity of the option’s price to changes in interest rates. A long straddle position, created by buying both a call and a put option with the same strike price and expiration date, is gamma positive. This means that the delta of the position will increase as the underlying asset’s price increases and decrease as the underlying asset’s price decreases. The long straddle is also theta negative because the value of the options decays over time. The vega is positive, as an increase in volatility will increase the value of both the call and put options. The key here is the interaction of these Greeks and how they affect capital requirements. A portfolio with a large gamma exposure can experience significant changes in delta with relatively small movements in the underlying asset. This increased sensitivity requires careful monitoring and potential adjustments to maintain a desired risk profile. Under CRR, firms must hold capital against market risk, which includes the risk arising from derivative positions. Gamma risk is a component of market risk, and firms with significant gamma exposure may face higher capital requirements. Consider a scenario where a firm uses a long straddle to hedge a large equity portfolio. If the market experiences a sudden and substantial price movement, the delta of the straddle will change rapidly, potentially requiring the firm to rebalance its hedge to maintain the desired level of protection. This rebalancing activity can be costly and may impact the firm’s profitability. Furthermore, the theta decay of the straddle will erode its value over time, requiring the firm to periodically roll the position to maintain its effectiveness. The regulatory capital implications of these dynamic adjustments must be carefully considered. The correct answer is (a) because it accurately describes the impact of gamma, delta, and theta on a long straddle position and how these factors can influence regulatory capital requirements. The incorrect options present plausible but flawed understandings of these concepts.

The question assesses understanding of exotic derivatives, specifically barrier options, and their sensitivity to market movements and volatility. The scenario involves a knock-out call option, which ceases to exist if the underlying asset’s price reaches a certain barrier level. The key is to understand how changes in volatility and the underlying asset’s price relative to the barrier affect the option’s value. Gamma measures the rate of change of an option’s delta with respect to changes in the underlying asset’s price. For a knock-out call option near the barrier, gamma behaves differently than a standard call. As the underlying price approaches the barrier, gamma increases sharply. However, if the barrier is breached, the option is knocked out, and its value becomes zero, causing gamma to also effectively become zero. Volatility also plays a crucial role. Higher volatility increases the probability of the barrier being hit, thus increasing the likelihood of the option being knocked out. This reduces the option’s value and, consequently, its gamma. Conversely, lower volatility decreases the probability of hitting the barrier, preserving the option’s potential value and increasing its gamma. The question requires integrating these concepts to determine the combined effect on gamma. The scenario stipulates that the underlying asset’s price is already very close to the knock-out barrier. If volatility increases significantly, the probability of breaching the barrier rises dramatically. Since the option is near the barrier, a slight price increase will trigger the knock-out. This makes the option highly sensitive to price changes, leading to a significant increase in gamma just before the barrier is hit. However, if the barrier is breached, the option is knocked out, and its value becomes zero, so gamma becomes zero.