Your inventory charge is 0.002*q^2 dollars and crossing the spread to flatten costs 2 cents per lot. You are long 30 lots. How many lots do you cross to sell, and what is the general rule for the band inside which you should not trade?

Your inventory charge is 0.002*q^2 dollars and crossing the spread to flatten costs 2 cents per lot. You are long 30 lots. How many lots do you cross to sell, and what is the general rule for the band inside which you should not trade?

Approach: Write total cost as the quadratic charge on the position you keep plus the crossing cost on the lots you sell, then minimise over the number sold and read off the first order condition.

25. Selling x lots costs 0.02x to cross and leaves an inventory penalty of 0.002*(30 - x)^2, so total cost is f(x) = 0.002*(30 - x)^2 + 0.02x. Setting f'(x) = -0.004*(30 - x) + 0.02 = 0 gives 30 - x = 5, so you sell 25 and keep 5. The general rule is that the marginal cost of carrying the last lot, 2*lambda*|q|, has to exceed the crossing cost c, so the no-trade region is |q| <= c/(2*lambda), here 0.02/0.004, which is 5 lots. Flattening all the way destroys value, because the last five lots cost more to remove than they cost to hold, and a quadratic charge makes the band symmetric around zero.

Follow-up: How does the band change if the inventory charge is linear in the absolute position instead of quadratic?

Key concepts: inventory penalty, no-trade region, marginal cost, quadratic charge.