Question

There are at least four schools of thought on the statistical distribution of stock price differences, or more generally, stochastie models for sequences of stock prices. In terms of number of followers, by far the most popular approach is that of the so-called "technical analysist", phrased in terms of short term. trends, support and resistance levels, technical rebounds, and so on. Rejecting this technical viewpoint, two other sehools agree that sequences of prices describe a random walk, when price changes are statistically independent of previous price history, but these schools disagree in their choice of the appropriate probability distributions. Some authors find price changes to have a normal distribution while the other group finds a distribution with "fatter tail probabilities", and perhaps even an infinite variance. Finally, a fourth group (overlapping with the preceding two) admits the random walk as a first-order approximation but notes recognizable second- order effects. This exercise is to show a compatibility between the middle two groups. It has been noted that those that find price changes to be normal typieally measure the changes over a fixed number of transactions, while those that find the larger tail probabilities typically measure price changes over a fixed time period that may contain a random number of transactions. Let \(Z\) be a price change. Use as the measure of " fatness " (and there could be dispute about this) the coefficient of excess. $$ \gamma_{2}=\left[m_{4} /\left(m_{2}\right)^{2}\right]-3 $$ where \(m_{k}\) is the kth moment of \(Z\) about its mean. Suppose on each transaction that the price advances by one unit, or lowers by one unit, each with equal probability. Let \(N\) be the number of transactions and write \(Z=X_{1}+\cdots+X_{N}\) where the \(X_{n}^{\prime} s\) are independent and identically distributed random variables, each equally likely to be \(+1\) or \(-1 .\) Compute \(\gamma_{2}\) for \(Z:(a)\) When \(N\) is a fixed number \(a\), and (b). When \(N\) has a Poisson distribution. with mean \(a\).

Short answer

In both scenarios, the coefficient of excess \(\gamma_2\) is found to be $$ \gamma_{2} = \frac{1}{a} - 3. $$ This shows compatibility between the two different approaches to stock price difference distributions discussed.

Step-by-step solution

Scenario (a): Fixed Number of Transactions N = a

In this case, \(Z = X_1 + X_2 + \cdots + X_a\) where the \(X_n\)'s are independent and identically distributed random variables (i.i.d.), each with equal probability to be +1 or -1. Firstly, let's find the mean of \(Z\): $$ \mu = E[Z] = E[X_1 + X_2 + \cdots + X_a] = E[X_1] + E[X_2] + \cdots + E[X_a] = 0 $$ since \(E[X_n] = \frac{1}{2}(1) + \frac{1}{2}(-1) = 0\). Now, let's find the 4th and 2nd moments of \(Z\) about its mean: $$ m_{4} = E[(Z-\mu)^4] = E[Z^4] $$ and $$ m_{2} = E[(Z-\mu)^2] = E[Z^2]. $$ Note that since the \(X_n\)'s are i.i.d., we have $$ E[X_i^4] = \frac{1}{2}(1)^4 + \frac{1}{2}(-1)^4 = 1, $$ and $$ E[X_i^2] = \frac{1}{2}(1)^2 + \frac{1}{2}(-1)^2 = 1. $$ Applying the formula, we get $$ m_{4} = E[Z^4] = aE[X_i^4] = a, $$ and $$ m_{2} = E[Z^2] = aE[X_i^2] = a. $$ Thus, the coefficient of excess \(\gamma_2\) is $$ \gamma_{2} = \frac{m_4}{(m_2)^2} - 3 = \frac{a}{(a)^2} - 3 = \frac{1}{a} - 3. $$

Scenario (b): Poisson Distributed Number of Transactions with Mean a

In this case, the number of transactions \(N\) has a Poisson distribution with parameter \(a\). Therefore, we need to compute the moments \(m_{4}\) and \(m_{2}\) of \(Z\) using the conditional expectation. Let's find the 4th and 2nd moments of \(Z\) about its mean: $$ m_{4} = E[Z^4] = E[E[Z^4|N]] $$ and $$ m_{2} = E[Z^2] = E[E[Z^2|N]]. $$ For any integer \(k\), we have $$ E[Z^k|N] = \sum_{n=1}^{N} E[X_n^k] = NE[X_1^k], $$ because the \(X_n\)'s are i.i.d. Thus, $$ m_{4} = E[Z^4] = E[NE[X_1^4]] = E[N], $$ and $$ m_{2} = E[Z^2] = E[NE[X_1^2]] = E[N]. $$ Since \(N\) has a Poisson distribution with mean \(a\), we have \(E[N]=a\). So, $$ m_{4} = a, $$ and $$ m_{2} = a. $$ Finally, the coefficient of excess \(\gamma_2\) is $$ \gamma_{2} = \frac{m_4}{(m_2)^2} - 3 = \frac{a}{(a)^2} - 3 = \frac{1}{a} - 3. $$ As we can see, both scenarios result in the same coefficient of excess \(\gamma_{2}\), which indicates that there is indeed compatibility between the two different approaches to stock price differences distribution.