## Variance Ratio Test - Lo and MacKinlay (1988)

A simple test for the random walk hypothesis of prices and efficient market.

A simple test for the random walk hypothesis of prices and efficient market.

This note briefly explains what's the **minimum variance hedge ratio** and how
to derive it in a cross hedge, where the asset to be hedged is not the same as
underlying asset.

These are two versions of winsorization in SAS, of which I recommend the first one.

Suppose today the stock price is \(S\) and in one year time, the stock price could be either \(S_1\) or \(S_2\). You hold an European call option on this stock with an exercise price of \(X=S\), where \(S_1<X<S_2\) for simplicity. So you'll exercise the call when the stock price turns out to be \(S_2\) and leave it unexercised if \(S_1\).

Using the CRSP/Compustat Merged Database (CCM) to extract data is one of the
fundamental steps in most finance studies. Here I document several SAS programs
for annual, quarterly and monthly data, inspired by and adapted from several
examples from the WRDS.^{1}

This note is just to show that the different variants of Black-Scholes formula in textbook and tutorial solutions are in fact the same.

Herfindahlâ€“Hirschman (HHI) Index is a well-known market concentration measure determined by two factors:

- the size distribution (variance) of firms, and
- the number of firms.

Intuitively, having a hundred similar-sized gas stations in town means a far less concentrated environment than just one or two available, and when the number of firms is constant, their size distribution or variance determines the magnitude of market concentration.

Since these two properties jointly determine the HHI measure of concentration, naturally we want a decomposition of HHI that can reflects these two dimensions respectively. This is particularly useful when two distinct markets have the same level of HHI measure, but the concentration may result from different sources. Note that here these two markets do not necessarily have to be industry A versus industry B, but can be the same industry niche in two geographical areas, for example.

Thus, we can think of HHI as the sum of the actual market state's deviation from
**1)** all firms having the same size, and the deviation from **2)** a fully
competitive environment with infinite number of firms in the market. Some simple
math can solve our problem.

Bloomberg is developing a new function in the Terminal, called BQuant, BQNT

Computing the weekly returns from the CRSP daily stock data is a common task but may be tricky sometimes. Let's discuss a few different ways to get it done incorrectly and correctly.

**TL;DR** Take me to the final solution!

Surely -> The solution