Consider The Stock Flying Kites: The Mean Of Its Daily Retur

Consider The Stock Flying Kites The Mean Of Its Daily Returns Is 0

Consider the stock Flying Kites. The mean of its daily returns is 0%, and the volatility of its daily returns is 2%. An at-the-money (ATM) call option on this stock is priced at $100. The question involves calculating the 1-day 95% percentile Value at Risk (VaR) of this call option, understanding the strategy with the lowest VaR%, and evaluating the use of Monte Carlo Simulation and the delta of a put option.

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The problem set focuses on risk assessment of options and the application of financial modeling strategies. First, understanding the calculation of VaR for an at-the-money call on Flying Kites requires an understanding of the distribution of returns and the associated risk measure. Since the mean daily return is zero, and the standard deviation is 2%, the VaR can be estimated assuming a normal distribution approach, with some adjustments if alternative distribution assumptions are considered.

Calculating the 1-day 95% VaR for the Call Option

The 95% VaR essentially indicates the maximum loss not exceeded with 95% confidence over one day. Given that the options’ value is directly influenced by underlying stock returns, the randomness of the stock’s returns must be translated into potential changes in the option’s value.

For a simple normal distribution, the 95th percentile z-value is approximately 1.645. Since the daily return volatility is 2%, the potential change corresponding to this percentile is:

\[ \Delta S = S \times \sigma \times Z \]

where \( \sigma = 0.02 \), and \( Z = 1.645 \).

Assuming an initial stock price \( S \) (not given explicitly), the expected decrease in the stock price at the 95% percentile is:

\[ \Delta S = S \times 0.02 \times 1.645 \approx S \times 0.0329 \]

The maximum loss in the call’s value would relate to the change in the stock price, and at-the-money options are sensitive to delta and gamma of the option, but for estimation purposes, the change in the option’s value approximates the change in the stock price multiplied by the delta, which is roughly 0.5 at-the-money options.

Given the initial option price of $100, a delta of approximately 0.5 leads to a change in the option value of:

\[ \Delta C \approx \text{Delta} \times \Delta S = 0.5 \times S \times 0.0329 \]

However, without the explicit stock price, we approximate the VaR in dollar terms directly, recognizing that the question offers multiple options.

Answer: The closest approximation is USD 3.3. Therefore, the correct choice here is I. USD 3.3.

Understanding VaR% for Different Strategies

Define VaR% as \( \mathrm{VaR\%} = \frac{\mathrm{VaR}}{\text{Initial investment}} \). Comparing two options: an OTM (out-of-the-money) put on GOOG with a one-week maturity and another with a one-month maturity, the longer maturity generally entails a higher potential risk, but also higher potential payout. The strategy with the shortest maturity tends to have a lower VaR%, assuming similar notional sizes, due to lesser exposure to adverse moves over time.

Answer: The strategy with the lowest VaR% is likely I. an OTM put option on GOOG with one week left to maturity.

Monte Carlo Simulation in Return Modeling

Monte Carlo Simulation is a powerful tool for modeling returns because it allows for incorporating complex features of return distributions without assuming normality. Traditional models often assume returns follow a normal distribution, which underestimates tail risks—scenarios with extreme outcomes. Monte Carlo methods generate simulated paths by sampling from specified distributions or calibrating models based on historical data, capturing skewness, kurtosis, and other real-world behaviors.

True or False: The statement "The true advantage of using Monte Carlo Simulation techniques for simulating returns is that we do not need to assume any distribution for the underlying returns" is False.

While Monte Carlo simulation reduces reliance on strict distributional assumptions, it still requires specifying or calibrating a distribution (e.g., historical data, stochastic processes). It does not inherently eliminate underlying assumptions but provides flexibility in modeling complex behaviors beyond normality.

Delta of a Put Option

Delta measures an option’s sensitivity to changes in the underlying asset’s price. For a put option, delta is always negative because the value of a put increases as the underlying price decreases. The delta of a put ranges from 0 to -1, depending on whether it's deep in-the-money or out-of-the-money.

Answer: The statement "The delta of a put option is negative" is True. This is because the value of a put option generally moves inversely to the price of the underlying.

Conclusion

This set of questions emphasizes understanding risk metrics, option sensitivities, and the benefits and limitations of modeling techniques in financial risk management. Approximations like VaR calculations are crucial for practical risk assessment, and the strategic decisions outlined depend on understanding the time horizon and potential losses involved. Similarly, comprehension of option Greeks, especially delta, plays a vital role in options trading and hedging strategies.

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