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.