8 – M3L609 Optimization SC PT4 V2

Now, if we compare this result to what we had before, we can see that the implementation shortfall in this case is bigger due to the random fluctuations in stock price. We can think of this more realistic price model as the original price model, but with some noise added to it. So, let’s see if we can write this out mathematically. Here, I have the first implementation shortfall which corresponds to trading at this points, and here is the second implementation shortfall which corresponds to trading at these points. What I want to do now is to express the second implementation shortfall in terms of the first one. To do this, I will start out by writing 97 as 100 minus 3, 77 as an 90 minus 13 and so on. If I simplify and rearrange terms, I can see that this terms over here in blue actually correspond to the capture of the first implementation shortfall, while these terms in yellow actually correspond to the difference in execution price due to noise. For example, this $3 means that the first transaction took place $3 below the first implementation shortfall. Similarly, the minus $2 means that the third transaction took place $2 above the first implementation shortfall. If I simplify further and make the substitution from the first implementation shortfall, then simplifying further I actually get that the second implementation shortfall is actually the first implementation shortfall plus some term that just depends on noise. What I want to do now is actually to write I S1 the first implementation shortfall as E of X, and I will write I S2 as simply the implementation shortfall. We call this first term the expected shortfall, because it is the expected implementation shortfall if we had no noise. This expected shortfall will depend on how we sell our shares. For example, if we had decided to sell our shares in this fashion, by selling five shares in the first trade, two shares on the second trade and so on, we will have resulted in a different implementation shortfall. In this case, a smaller one of $172. The way in which we decide to sell our stocks is called a trade list. The trade list tells us the number of trades and the quantity of shares to be sold at each trade. The second term only depends on the random price fluctuations. The absolute value of this term gives us an idea of how much the stock price is fluctuating. For example, if we had bigger price fluctuations like here, we will get a number with a bigger magnitude. In this case, 253. Also notice that in this case, the number is negative. This term will therefore represent how much money we can gain or lose due to the random price fluctuations. In finance, this random price fluctuations are referred to as price volatility or risk. Risk is usually quantified by the variance of the price fluctuations. So, if the price fluctuations of a particular stock are very big, we say that is a risky investment. Now that you have a better idea of what the financial terms mean, let’s look at the problem once again in this new light. So, given a certain number of stocks and knowing how much the stock price is affected every time we sell a stock, we have to find the trade list that tells us how many shares to sell at each trade such that the implementation shortfall is minimized. However, we’re bound to certain constraints. One of them is that we need to sell theses shares within a given timeframe. So, we need to set a limited number of trades within the liquidation period. The other is that we need to take into account the traders risk aversion. The traders risk aversion that says how much a particular trader is willing to tolerate risk. For example, suppose the stock we’re selling is experiencing a large volatility in the stock price. In this scenario, a trader that doesn’t like to take risks would rather sell his shares as quickly as possible. On the other hand, a trader that does like to take risks will rather sell these shares at a constant rate over the whole liquidation period, even if the price volatility is high.

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