Stochastic Methods in Finance Afin270
Measures of Location and Spread
Parameters and Statistics
Many variables of interest are random in nature, e.g. share return, inflation rate, interest rate
Parameters describe the true underlying mechanism and are unknown. Statistics estimate these parameters and are subject to parameter error, model error, data error
Parameters include the mean, median, mode, percentile, interquartile range, mean absolute deviation, variance, standard deviation, coefficient of variation, skewness.
Mean
Suppose X is a (quantitative) random variable:
• Mean (parameter) can be expressed as 𝜇 = 𝐸(𝑋)
• Sample mean (statistic) is calculated as
• E.g. sample mean of All Ords daily returns 𝑥̅ = 0.02%
• Mean is the expected value but incurs some loss of information
• Weighted mean (statistic) is calculated as
Median
• Median (parameter) is the value that at least half of the outcomes are smaller than or equal to it and also that at least half of the outcomes are larger than or equal to it
• If the number of samples n is odd, sample median (statistic) is taken as the middle sample of the sorted ascending data; if n is even, sample median is taken as the average of the middle two samples of the sorted ascending data
• E.g. sample median of All Ords daily returns = 0.05%
Mode
• Mode (parameter) is the value that has the highest chance to occur
• Sample mode (statistic) is taken as the value that occurs most often in the data
• E.g. sample mode of All Ords daily returns = 0.20%
• There can be more than one mode in some situations
Percentile
α-percentile (parameter) is the value that at least α of the outcomes are smaller than or equal to it and also that at least 1 – α of the outcomes are larger than or equal to it.
• If n x α is not an integer, sample α-percentile (statistic) is taken as the [ n x α ]th
Interquartile Range
Interquartile range (parameter) is the difference between 25% percentile and 75%
percentile. The sample interquartile range (statistic) is the difference between sample 25%
percentile and sample 75% percentile.
Interquartile range uses only a fraction of the information and conveys little about the entire variation
E.g. Sample interquartile range of All Ords daily returns = 1.00%
Mean Absolute Deviation
Mean absolution deviation (parameter) can be expressed as Sample mean absolute deviation (statistic) is calculated as
Mean absolution deviation takes every sample into account. E.g. sample mean absolute deviation of All Ords daily returns = 0.69%
Variance
Variance (parameter) can be expressed as Sample variance (statistic) is calculated as
E.g. sample variance of All Ords daily returns s2 = 0.00009726
Compared to mean absolute deviation, variance has even more contribution from larger deviations. Standard deviation is the square root of variance and is in original units of the data. E.g. sample standard deviation of All Ords daily returns s = 0.99%
Coefficient of Variation
Coefficient of variation is a relative measure for comparing different random variables.
- Coefficient of variation (parameter) is defined as 𝜎𝜇 - Sample coefficient of variation (statistic) is calculated as 𝑥̅𝑠
E.g. sample coefficient of variation of All Ords daily returns = 53.83
Skewness
- Skewness (parameter) can be expressed as
- sample skewness (statistic) is calculated as 𝑆3
If skewness is negative, the distribution is skewed to the left; if skewness is positive, the distribution is skewed to the right; if skewness is zero, the distribution is symmetrical.
E.g. sample skewness of All Ords daily returns = -0.44
Discrete Probability Distributions
Bernoulli Distribution
Example:
One can model the credit risk of a firm. If the firm defaults, X = 0; if the firm survives, X = 1 Probability of default is:
Bernoulli Distribution- Extension:
One can extend this distribution to form the binomial share price model
• Suppose the share price at time 0 is $20 (S0)
