The Ho-Stoll Dealer Model

A dealer quoting both a bid and an ask is exposed to more than just where the price will go — they're exposed to their own inventory. If they're already sitting long a stock, another buy fills them up further and adds risk; another sell brings them back toward flat. The Ho-Stoll model was one of the first to formalize this: a dealer's optimal quotes should be centered not on the market mid-price, but on a reservation price that shifts away from mid in the direction that reduces their current inventory.

Concretely, a dealer who is long shifts both their bid and ask down: a lower ask makes it more attractive for someone to buy from them (unloading inventory), while a lower bid makes it less attractive for the dealer to buy even more. The size of that skew grows with the size of the position and the dealer's risk aversion — a nervous dealer holding a large long position skews harder than a relaxed one holding a small position.

Worked example. A dealer with no risk aversion would quote symmetrically around a $100.00 mid, say $99.95 / $100.05. Holding a large long position, the Ho-Stoll logic pushes both quotes down, say to $99.85 / $99.95 — still a nickel-wide spread, but centered below the true mid, so incoming random buy/sell flow naturally works the position back toward flat.