In this module we'll walk through formulating and solving the optimum

execution problem that we introduced in the previous video module.

In order to set up the problem, I arbitrarily took that the asset we are

trying to sell is the first asset. So the volatility of this one was read

off from the previous worksheet. And it was the worksheet of volatility

was given in percentage so I divided by a 100 to get what the volatility numbers

are. The volume is the typical daily volume

that I'm going to be using, and I need these two numbers in order to compute the

[INAUDIBLE] glance's temporary price impact function.

For the Permanent price function I took it to be gamma times n and the gamma

number was taken to be .001. The alphas and the deltas, these

correspond to the alpha here, corresponds to alpha two times B1 which is the

volatility plus alpha 3. This one corresponds to basically the

constant term in the temporary price impact function delta.

I took I took it to be alpha1 plus the power divided plus beta.

This is the other term that is involved in the temporary price impact function.

And these 2 are used to compute what a temporary price impact function is going

to be. Okay, I have a total of a thousand shares

that I want to sell. These are the various trades so, in trade

1, in this particular setup, whatever I have, I sell 849 shares and in trade 2, I

sell 104 shares and so on. And, should add up, and so this 1,000,

this is a constraint that I'm going to put.

In my portfolio selection, or execution problem.

X is the inventory, so before anything gets sold the inventory is a 1,000.

After I sell whatever is necessary, this is going to be simply initial inventory

minus the trade. Similarly over here it's going to be now.

This inventory minus the trade and so on. So by the end, I must have inventory

equal to 0. In the beginning I have inventory equal

to a 1000. All of these trades must add up to a

1000. So, given the trade amount, I can compute

what the temporary cost is going to be, and what the permanent cost is going to

be. The temporary cost is simply alpha, which

is a linear term, times the absolute value of the trade, plus delta times the

trade to the power 1 plus beta. And the beta turned out to be 1.5, in

this particular case. And that has been read from the worksheet

mean variance liquidity. What about the permanent cost?

Permanent cost is going to be the linear term, so it's gamma times the trade,

times the inventory, because any trade that I do, it's permanent price impact,

effects all the inventory that is there at the end of the day.

So that's what that term is going to be. The variance is simply going to be.

Inventory squared times volatility squared, and that's what I'm going to

sum. So the total trading cost is going to be

the sum of the temporary cost plus the sum of the permanent cost.