Avellaneda-Stoikov High-Frequency Market Making

Scenario

A high-frequency market maker quotes bid and ask limit orders in an order book where the mid-price $S_t$ follows arithmetic Brownian motion: $dS_t = \sigma dW_t$ The market maker has current inventory $q$, risk aversion parameter $\gamma$, and remaining trading horizon $\tau = T - t$. Order execution intensities decay exponentially with distances to mid-price: $\lambda(\delta) = A e^{-\kappa \delta}$.

Your Task

1. Derive the market maker's reservation (indifference) price $r(s, q, t)$ in terms of mid-price $s$, inventory $q$, risk aversion $\gamma$, and asset variance $\sigma^2 \tau$.
2. Derive the optimal half-spreads $\delta^a, \delta^b$ and the optimal total bid-ask spread $s(q)$.
3. Compute the reservation price and optimal bid/ask quotes when $s = 100.00, q = +5, \gamma = 0.1, \sigma = 0.20, \tau = 1.0, \kappa = 1.5$.

Constraints

• Constant inventory risk aversion parameter $\gamma > 0$.
• Limit order fill intensity governed by Poisson point process.

Examples When inventory $q = +5 > 0$ (long inventory), the reservation price drops below mid-price ($r < s$), skewing quotes downwards to attract selling flow and deter buying flow.