This paper builds a mathematical model that links what a single market maker does when trading option contracts to how an exchange should set fee rebates to attract liquidity. The authors study a market maker who trades several European call options and who can both post limit orders (wait for others to trade with her) and use market orders (trade immediately). The goal is to find trading rules and rebate policies that balance profit, trading risk, and the exchange’s desire for tighter prices.

To represent trading behaviour they use two kinds of controls. Posting limit orders is treated as a continuous action — the market maker can keep quoting prices over time. Using market orders is treated as occasional impulse actions — one-off trades to adjust positions immediately. The model runs in a local–stochastic volatility setting, a standard way to describe option prices when volatility changes over time and with the price. The market maker’s objective is to maximize net profit from the option portfolio plus the total rebate income, while paying penalties when she holds unwanted exposures: delta (sensitivity to the underlying price) and vega (sensitivity to volatility). These penalties capture the practical cost of carrying inventory that could suffer large losses.

On the exchange side, the paper asks how to design make-take fees — the system that rebates liquidity providers and charges liquidity takers — so that the market maker posts tighter quotes rather than just doing more trades. The authors assume a “first‑best” case where the exchange can observe the market maker’s quoting actions and can pay rebates continuously based on those actions. Under that assumption they propose a three-step rebate design scheme that can be tuned to meet specific liquidity targets the exchange wants.

Mathematically, the problem becomes a nested optimisation: the market maker solves a stochastic control problem under the rebate rule, and the exchange then chooses the rebate rule knowing how the market maker will respond. The authors derive optimal strategies and solve the resulting control equations, and they report numerical experiments that show the rebate scheme can lead the market maker to narrow spreads and improve quoted liquidity.