In this module, we will focus on a single asset and we will double up a model which takes into account the temporary price impact of a trade and the permanent price impact of a trade. And construct an optimization problem that allows you to select a treating strategy that minimizes the cost of the execution, and as well as the risk associated with this execution. In this module we are going to take the approach that we have decided a portfolio, that we want to trade to words. So this is a new position, we had an original position and now we want to trade to a new position. And we have computed this new position perhaps by taking liquidity into consideration or perhaps not. And in one of the pieces of this over, over all trade, we have to sell a total of capital x shares over at most t traits. I choose a certain number of trades that I can put and over those trades I want to sell these shares in order to have the least amount of trading cost associated with it. And in order to compute the trading cost, I need to have a model for trading cost and that's what we're going to develop over this module. So, in order to model how going to trade, I'm going to introduce this notion of a trading strategy. So, N sub j would denote the number of shares that will be sold on the jth trade. Capital N or bold face N, which is N1 through N capital T will be called the execution sequence. It's just a complete sequence of execution that I will be interested in. Over the entire period, j going from 1 through capital T, the sum of all the trades that I made must be equal to X, capital X, the amount that I wanted to trade. I'm going to introduce a new notion of little x, which is going to be the inventory. That is, the holdings at the end of a particular trade. So x sub zero will be capital X. x sub k will be the holdings at the end of not the jth trade, but kth trade. After the kth trade, I would have n1, n2, n3, up to nk. I subtract that from capital x, that tells me what is the holdings left over, and that's the inventory that I have. In order to understand how to choose these sequences, I have to bring in this notion of what happens to the prices. So let S hat k denote the price per share received for the nk shares sold in the k-th trade. x hat, this S hat nk is going to be a function of all the trades, n1 through nk. Note that it also depends on the trade that you actually put in. And now in the next slide I'm going to set up a particular model for how these prices behave. And using those models, I'm going to try to understand how I can trade off various important quantities that play a role in the execution of a particular stock. So this slide is perhaps the most important one in this module. I'm going to assume that the price impact, the impact on the price of my trade is going to have two components. There's going to be a temporary price impact component, which is the impact of trade nk on its own price per share, S hat k. And a permanent price impact is the impact of trade nk on all the future prices. And here's the model that's going to happen. Remember, I'm selling shares, so whatever I do tries to drag the price down. If I were buying orders, if I were putting in my orders then the, the whole story will be a mirror image and I will start increasing prices because of my trade and not decreasing prices. So let Sk denote the price that I observe in the market, when I am contemplating the k-th trade. When I put in the K-th trade, the average price that I get for this trade is not going to be Sk but a smaller amount S hat k. This is the amount of money that I would get per share when I put in the trade hnk. Now, I'm going to assume that S hat k is going to be Sk minus hnk, it's something hn is a temporary price impact function. It defines what'll happen to my price by what amount the price is going to dip if I decide to sell nk amounts of shares. Now what happens to the price in the future periods? This particular trade, nk, is going to have an impact of what happens to price Sk plus one. And therefore, it's going to start having an impact on Sk plus 1, Sk plus 2, Sk plus 3, all the way up through S capital T. And the model that we are going to say is the following. The price at the next time period is going to be some random walk to its current price, plus a random walk component. So sigma is the variability or the standard deviation, zk is just IID standard Normal random variables. So without any price impact, this particular asset is just doing a random walk. If there was drift in the market, we can simply add another drift term. Typically, when ignores the drift term and claims that this trades are happening over a certain period said that the price is not significantly drifting, but it is varying. So Sk plus sigma zk tells you what happens in the random walk. The expected cost term takes into consideration the fact that if I sell a large chunk it's going to have a price impact, and I won't get the revenue that I want. Rho, the term over here, is the trade off that trades off my concern for variance with my concerns for trying to keep the cost minimized the cost that I end up getting from it. The total revenue from execution, is simply the price per share for every trade times the number of shares sold in that trade. So it's the sum of k going from 1 through capital T of S hat k, which is the price per share times nk, which is the number of shares that we're trading. If you write out the expression for S hat k, it becomes Sk minus the term that corresponds to the temporary price impact times nk. So this term, we have taken it together and kept it over there. Then Sk has an expression. Sk is equal to S1, the initial price when we started trading sum from j going from 1 to k minus 1 of all the random walk terms, minus nj, which is all the permanent price impact terms, times nk. So if you unravel the sum and do it two different ways, you end up getting the first term, which is S1 times the total number that I sold. So this is the revenue that I expect to get if there was no price impact. I could just sh, sell everything all at one go, I get the current price. That's it, I'm done. Plus I get a term which corresponds to random walk. The random walk term starts to show what is going to happen, what the trade off between selling and inventory. So the random walk term has a term z delta k that refers to the random term at time k, the random walk at time k and it affects all the left over inventory at time k. So, this is not nk but xk, it's the inventory at the end of the k-th trade. Similarly the term that corresponds to the permanent price impact also af, affects, is affected by the inventory and not the current trade. So gnk, the trade at time k, is going to affect all the stocks that I have not yet sold, all the shares that are not yet sold, and sitting in inven, inventory. This is the revenue that was expected, that is the revenue that was realized, so the difference between the two of them is the slip edge or the expected cost of the trading strategy. So, the trading strategy Cn gives me a cost, which is gnk times xk, which is the inventory plus hnk times nk. This is the expected cost, and therefore the term that corresponds with the random walk goes away. The risk or the variance of the trading strategy is that all the deterministic terms go away. Since the random walk terms were assumed to be IID standard normal, you get sigma squared times the sum of the inventory from time 1 to time Capital T. Now we can define waht is called the optimal execution frontier, which is a trade off between expected cost and risk. So I want to minimize over all trading strategies, the expected cost, plus rho times the risk term. The time horizon capital T is also a variable, but in this particular case we are going to assume that Capital T is given and just optimizer of the execution strategy. What this objective is basically trying to do is the following. If I completely ignore variability, then the best thing for me to do is to sell equally, across the entire time horizon. That minimizes the amount that I sell on any given day and therefore that minimizes my price impact. And we will see that on the Excel spreadsheet that is exactly what is going to happen. But if I spread it uniformly, what's happening is that I expose my inventory to very high availability. And therefore the, the realized revenue could be very far away from the expected revenue. If I want to control the variability, then I want to sell quickly, but then I pay a cost in the expected slippage, or the, or equal until if my revenue comes down. So I need to balance these two, and rho is sort of exploring the different points on the surface, that is a trade off between the expected cost and variability. What are some typical choices for the price impact function, for the permanent price impact function typically one takes it to be linear. For the temporary price impact function, you can use the Kissell-Glantz function that we introduced in the last module, and we're going to be doing that in the excel module that we, we'll be talking about very soon.