Performative Market Making

Abstract

Financial models do not merely analyse markets, but actively shape them. This effect, known as performativity, describes how financial theories and the subsequent actions based on them influence market processes, by creating self-fulfilling prophecies. Although discussed in the literature on economic sociology, this deeply rooted phenomenon lacks mathematical formulation in financial markets. Our paper closes this gap by breaking down the canonical separation of diffusion processes between the description of the market environment and the financial model. We do that by embedding the model in the process itself, creating a closed feedback loop, and demonstrate how prices change towards greater conformity to the prevailing financial model used in the market. We further show, with closed-form solutions and machine learning, how a performative market maker can reverse engineer the current dominant strategies in the market and effectively arbitrage them while maintaining competitive quotes and superior P&L.

Figures (4)

• Figure 1: Comparison of optimal bid and ask quotes under classical optimality ($p^b_*,p^a_*$) and performative optimality ($p^b_{\mathrm{PO}},p^a_{\mathrm{PO}}$) for different values of the performativity parameter $\epsilon$. In the left panel, $p^b_{\mathrm{PO}}>p^b_*$ indicates a stronger incentive to buy under performative optimality; in the right panel, $p^a_{\mathrm{PO}}>p^a_*$ indicates a stronger incentive to sell. The intersection point $p^{a,b}_{\mathrm{PO}}=p^{a,b}_*$ marks a phase transition between buying- and selling-dominant behaviour. Parameters: $s=100$, $k=1.6$, $\mu=0.5$, $T=1$.
• Figure 2: The area above the horizontal dashed line indicates the decisions of the Performative Aware MM strategy and the area below it the decisions of the original A&S strategy. When the areas below and above have the same colour, the decisions align and when not, the performative MM is arbitraging the prevailing strategy. Her behaviour changes not only based on the sign of her inventory (like the A&S strategy) but also on its relative value with respect to the critical points shown on the x-axis.
• Figure 3: The shaded red price paths are the result of the performative effect that the MM model of A&S has to the price due to its inventory management ($\epsilon=10$). The blue price paths are the result of pure Brownian noise ($\epsilon=0$). The highlighted dashed and solid lines for each case showcase approximately 10% of the most price paths, i.e., those with maximum and minimum final prices. The price paths were produced for $s_0=100$, $T=10$, $\gamma=0.5$, $\sigma=0.4$, $\Delta_t=0.05$.
• Figure 4: The PnL of all strategies across all $\epsilon$ values for $\gamma=0.2, 0.5$ and $0.8$. Every point is the average of 1000 runs and the shaded area is $\pm$ 1 standard deviation from the mean. Performative strategies consistently outperform the A&S strategy with higher PnL across all $\gamma$'s and the symmetric strategy after $\epsilon$ surpasses a break-off point.

Theorems & Definitions (12)

• definition 1: Performative Price Process
• Remark 1: Martingale Property
• definition 2: Performative Optimality
• definition 3: Performative Stability
• Remark 2: Market-Making Optimisation under a Performative Price Process
• lemma 1: Optimal Reservation Price under Directional Beliefs
• proposition 1: Performatively Optimal Strategy
• theorem 1: Performatively Stable Strategy under Endogenous Beliefs
• theorem 2: Optimal Reservation Price under Inventory Constraints avellaneda2008
• proposition 2: Performatively Optimal Strategy