The author's website (www-2.rotman.utoronto.ca/ hull/data)...

The author's website (www-2.rotman.utoronto.ca/ hull/data) contains daily closing prices for the crude oil futures contract and the gold futures contract. You are required to download the data for crude oil and answer the following: (a) Assuming that daily price changes are normally distributed with zero mean, estimate the standard deviation of daily price changes. Calculate the standard deviation of two-day changes from the standard deviation of one-day changes assuming that changes are independent. (b) Suppose that an exchange wants to set the margin requirement for a member with a long position in one contract so that it is $99 \%$ certain that the margin will not be wiped out by a two-day price move. (It chooses two days because it considers that it can take two days to close out a defaulting member.) How high does the margin have to be when the normal distribution assumption is made? Each contract is on 1,000 barrels of oil. (c) Use the data to determine how often the margin of the member would actually be wiped out by a two-day price move. What do your results suggest about the appropriateness of the normal distribution assumption? (d) Suppose that for retail clients the maintenance margin is equal to the amount calculated in (b) and is $75 \%$ of the initial margin. How frequently would the balance in the account of a client with a long position be negative immediately before a margin payment is due (so that the client has an incentive to default)? Assume that balances in excess of the initial margin are withdrawn by the client.

The author's website (www-2.rotman.utoronto.ca/ hull/data) contains daily closing prices for the crude oil futures contract and the gold futures contract. You are required to download the data for crude oil and answer the following:
(a) Assuming that daily price changes are normally distributed with zero mean, estimate the standard deviation of daily price changes. Calculate the standard deviation of two-day changes from the standard deviation of one-day changes assuming that changes are independent.
(b) Suppose that an exchange wants to set the margin requirement for a member with a long position in one contract so that it is $99 \%$ certain that the margin will not be wiped out by a two-day price move. (It chooses two days because it considers that it can take two days to close out a defaulting member.) How high does the margin have to be when the normal distribution assumption is made? Each contract is on 1,000 barrels of oil.
(c) Use the data to determine how often the margin of the member would actually be wiped out by a two-day price move. What do your results suggest about the appropriateness of the normal distribution assumption?
(d) Suppose that for retail clients the maintenance margin is equal to the amount calculated in (b) and is $75 \%$ of the initial margin. How frequently would the balance in the account of a client with a long position be negative immediately before a margin payment is due (so that the client has an incentive to default)? Assume that balances in excess of the initial margin are withdrawn by the client.

Key Concepts

Empirical Validation

Empirical validation refers to the process of comparing theoretical models, such as those based on the normal distribution, with actual market data. This step is crucial in determining whether the assumptions made in the risk models hold true in real-world trading conditions and if adjustments are needed to better capture the observed behaviors of asset prices.

Risk Management

Risk management involves analyzing and mitigating the potential financial losses from adverse market movements. In derivatives trading, it encompasses strategies like setting margin requirements to ensure that the probability of a margin call or a default is kept within acceptable limits, thereby protecting both the exchange and market participants from systemic risks.

Margin Requirements

Margin requirements are risk management tools used by exchanges to ensure that traders have sufficient collateral to cover potential losses. They are set based on statistical measures of price risk, such as the estimated standard deviation of price movements, and aim to reduce the likelihood that a market position will incur a loss severe enough to wipe out the collateral provided.

Maintenance Margin and Incentives to Default

The maintenance margin is the minimum amount of equity that must be maintained in a margin account after the initial deposit, ensuring that there is a cushion against adverse price movements. This concept is important in understanding when a client might face a negative balance and have an incentive to default, particularly if the account's balance falls below the required maintenance margin threshold.

Standard Deviation

Standard deviation is a statistical measure of variability or volatility that quantifies the spread of a dataset around its mean. In the analysis of daily price changes, it provides an estimate of typical fluctuations in price and is crucial for assessing the risk associated with holding an asset, especially in calculating the potential impact of extreme market moves.

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about the mean, where most outcomes cluster around the central value. In this context, it is assumed that daily price changes follow a normal distribution, which simplifies statistical analysis and risk assessment by allowing the use of standard probabilistic measures, such as standard deviations, to estimate the likelihood of extreme price movements.

Aggregation of Independent Price Changes

When daily price changes are assumed to be independent, the volatility over multiple days can be derived from the one-day volatility. This concept employs the idea that variances (the squares of standard deviations) add over independent time periods, meaning that the volatility over two days is calculated as the square root of the sum of the individual variances, often simply multiplied by the square root of time when variances are equal.

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