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# HW2 - Statistics 133 Homework 2 due 11pm Tuesday Oct 6 on...

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Statistics 133 - Homework 2 due 11pm Tuesday, Oct. 6 on bSpace Pairs Trading: Simulation vs. Reality Background Pairs trading is a strategy for buying and selling stocks developed at Morgan Stanley in the 1980s. It was one of the first strategies based on computer-intensive analysis of past stock performance. To illustrate how it works, here is an example based on the Dow Jones and S&P 500 indices. These are actually collections of stocks, but the same principle applies. I’ve chosen to use them because pairs trading requires a pair of stocks (or other equities – I will often just use the word stocks from now on) that are correlated. As you can see for the time interval 1990-1995 in panel (a) of Figure 1, these two indices are highly correlated. Panel (b) shows the ratio of the two indices over the same time period. The ratio seems to be fluctuating around a stable value, at least for most of the time period. 2500 3000 3500 4000 Dow Jones 1990 1991 1992 1993 1994 1995 300 350 400 450 S&P 500 (a) Historical Prices 7.6 7.8 8.0 8.2 8.4 Ratio (DJI/GSPC) 1990 1991 1992 1993 1994 1995 mean +/- 1 sd +/- 2 sd (b) Historical Ratio Figure 1: Historical prices and ratio for the Dow Jones and S&P 500 indices. The key idea is that the movement of the ratio away from its historical average represents an opportunity to make money. For example, if stock 1 is doing better than it usually does, relative to stock 2, then we should sell stock 1 and buy stock 2. This is called “opening a position.” Then, when the ratio returns to its historical average, we should buy stock 1 and sell stock 2. This is called “closing the position.” To implement this strategy, we need rules for when we should open and close positions. (We’ll assume we can always sell a stock, even if we don’t actually own it. The mechanism for this is called “short selling,” but the details aren’t important here.) We can devise rules based on what the ratio of stock 1 to stock 2 has been in the past. I’ll refer to the mean and standard deviation of the ratio for this “training” data as m and s . We’ll consider rules of following form: When the ratio moves above m + ks , sell \$1 worth of stock 1 and buy \$1 worth of stock 2. Then wait until the ratio is less than or equal to m , at which point close the position by buying back however many shares of stock 1 we initially sold, and selling the shares of stock 2 we initially bought.

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When the ratio is less than m - ks , do the same thing but reversing the roles of stock 1 and stock 2. In this case we wait until the ratio comes back up, to be greater than or equal to m .
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