Derivatives Foundation puzzles, solved step by step

Skew the whole quote up, to about 50 at 52, keeping the width of 2. The short is a risk you want to shed, so make your bid attractive to sellers and your offer less attractive to buyers. A simple rule moves the mid 0.05 per lot of inventory, so 20 lots short moves it up 1. Widening to 48 at 52 also stops the buying, but it pushes sellers away too, so the short stays on your book and you earn less while you wait.

What is the position telling you to do?

A fruit seller who has run short of mangoes by mid-morning does not shut the stall; she raises the price she pays suppliers and nudges up the price to customers, so more mangoes arrive and fewer leave. Inventory changes what you want to trade next, not what the contract is worth: with fair value still 50, a short of 20 lots means your next trade should be a buy, so the quote should lean towards buying. You have used 80% of the risk limit, and five more lifts would put you at it. The question is testing whether you separate the two numbers a market maker carries: the fair value, which has not moved, and the price at which you want to trade, which has.

How far should you skew, and what does it cost?

A common rule moves the mid in proportion to the position: here 0.05 per lot, so 20 lots short lifts the mid by 1 and the quote becomes 50 at 52. The bid now sits at fair value, so you buy back with no edge, and the offer is 2 above fair, so a buyer who still lifts it pays you well for adding to the risk. Skewing trades some edge for risk reduction, and the closer the position is to the limit, the more edge you should be willing to give up. In a toy model where a quote d away from fair trades with probability 0.6 x e to the minus d each period, the unchanged quote earns 0.44 a period and never reduces the short; the full skew earns 0.16 but buys back a net 0.52 lots a period, about 39 periods to flatten; the half skew sits between at 0.38 and 0.23. The numbers are illustrative; the direction is not.

Why not simply widen?

Widening to 48 at 52 makes the offer as unattractive as the skew does, but it also moves the bid two points below fair, so sellers go elsewhere. In the toy model both sides fill 0.08 of the time and the expected change in the position is zero: a wider quote stops new risk arriving but does nothing about the risk you already hold, and the 20-lot short keeps moving with the market while you wait. Widening has its place, when you think fair value is uncertain or that the people lifting you know something, because then both sides are dangerous. Say that distinction, and then say the last resort: if skewing does not bring sellers fast enough, hedge the short in a related market, or cross the spread and buy, rather than drift up to the limit.