Dynamic slippage control and rejection feedback in spot FX market making

This paper proposes a dynamic market-making framework for spot FX that jointly optimizes size-dependent quotes and trade rejection decisions to manage latency-driven adverse selection, while incorporating an endogenous feedback mechanism where past rejections modulate client order flow, ultimately solved via dynamic programming and approximated by a tractable adiabatic-quadratic method.

The Gist

Imagine you are a high-stakes currency trader (a "Market Maker") running a busy booth at a global exchange. Your job is to constantly shout out prices for buying and selling currencies to a crowd of clients.

The problem? You are shouting your prices from a booth, but the clients are in a different city. By the time your shout reaches them, the market might have moved. This delay is called latency.

If the market moves against you while your client is thinking about your price, you could lose money. This paper is a guide on how to manage that risk without losing your customers.

Here is the breakdown of the paper's ideas using simple analogies:

1. The Problem: The "Stale Price" Dilemma

Imagine you are selling a pizza. You shout, "Large Pepperoni for $10!"

• The Delay: It takes 0.5 seconds for the customer to hear you and run to the counter.
• The Risk: In that half-second, the price of cheese spikes. If the customer still wants the pizza at $10, you just lost money on the deal.
• The Old Solution: You could just widen your price gap (charge $12 instead of $10) to protect yourself. But this makes you less competitive, and customers might go to a cheaper booth.

2. The New Tool: "Last Look" (The Rejection Button)

In the real world, traders use a feature called "Last Look." It's like a tiny pause button.

• When a client says, "I'll take that $10 pizza," you get a split second to check the current price.
• If the price has moved against you too much, you can say, "Sorry, that price is no longer valid," and reject the trade.
• The Catch: If you reject too many orders, customers get annoyed. They feel you are being unfair or "gaming the system." They might stop calling you or go to a competitor. This is called Reputation Risk.

3. The Paper's Big Idea: The "Smart Refusal" Strategy

The author, Alexander Barzykin, built a mathematical model to answer a complex question: "How do I decide exactly when to reject a trade to protect my wallet, without ruining my reputation?"

He treats this not just as a rule, but as a dynamic game with two moving parts:

A. The "Slippage" Threshold (The Bouncer)

Instead of having a fixed rule like "Reject if the price moves more than 1 cent," the model suggests the threshold should change based on:

• How much inventory you have: If you are already holding too much of a currency, you become pickier.
• How toxic the order is: If the client seems to be betting against you (adverse selection), you reject faster.
• The "Option" Analogy: The paper views the decision to reject as writing an insurance policy. When you accept a trade, you are selling an option to the client. If the market moves badly, you "exercise" your option to cancel the deal. The model calculates the perfect price for that option.

B. The "Reputation Score" (The Karma Meter)

This is the most creative part. The model introduces a Reputation Score (let's call it the "Karma Meter").

• How it works: Every time you reject a client, your Karma Meter goes down slightly. Every time you accept a fair trade, it slowly recovers.
• The Feedback Loop: The model assumes that if your Karma Meter is low, fewer clients will call you. They sense you are being difficult.
• The Strategy: The model shows that if your Karma Meter is low, you should actually tighten your spreads (make your prices more competitive) and reject fewer trades to win back trust. It's a "repair and exploit" strategy: be nice to fix your reputation, then be smart to make money.

4. The "Adiabatic" Shortcut (The Fast-Forward Button)

Solving the math for this game is incredibly hard because the market moves fast (milliseconds), but reputation changes slowly (hours).

• The Analogy: Imagine driving a car (fast inventory changes) while looking at a map that updates very slowly (reputation).
• The Solution: The author developed a "shortcut" method. He assumes the reputation score stays "frozen" for a moment while you make your quick trading decisions. This turns a super-complex, impossible-to-solve equation into a manageable one that a computer can solve instantly.

5. The "Fair" Protocol (The Honest Bouncer)

The paper also tests a "Fair" version of rejection.

• Unfair: You reject when the market moves against you, but you keep the trade when the market moves in your favor. (This is like a bouncer who only kicks out people who are late, but lets everyone else in).
• Fair: You set a symmetric rule. If the price moves against you by X, you reject. But if the price moves in your favor by X, you give that extra profit back to the client.
• The Result: The model shows that while the "Unfair" method makes slightly more money in a vacuum, the "Fair" method is much better for long-term business because it keeps the Reputation Meter high and clients happy.

Summary: What's the Takeaway?

This paper teaches dealers that rejection is not just a risk control; it's a strategic lever.

1. Don't just widen spreads: Instead, use smart, dynamic rejection rules to filter out bad trades.
2. Watch your Karma: If you reject too many people, you lose future business. The model helps you find the "Goldilocks" zone where you reject enough to stay safe, but not so much that you lose your customers.
3. Be Fair: The most profitable long-term strategy often involves being transparent and fair, because it keeps the "Karma Meter" high, ensuring a steady stream of customers.

In short, it's a guide on how to be a smart, tough, but fair trader in a world where milliseconds matter and reputation is everything.

Technical Summary

Here is a detailed technical summary of the paper "Dynamic slippage control and rejection feedback in spot FX market making" by Alexander Barzykin.

1. Problem Statement

The paper addresses the Over-the-Counter (OTC) Foreign Exchange (FX) market-making problem, specifically focusing on the tension between latency risk and adverse selection.

• The Context: Dealers stream quotes to clients. Due to network latency, a client's request to hit a quote may arrive after the market price has moved. This creates "stale quotes."
• The Mechanism: Dealers use a "Last Look" (LL) or "Price Validation" protocol, where they can accept or reject a trade request after observing the price move (slippage) during the latency window.
• The Gap: Existing literature often treats rejection as a static protocol or an exogenous constraint. It fails to model:
  1. Endogenous Rejection Policy: Treating the rejection threshold as a dynamic control variable optimized alongside quotes, rather than a fixed rule.
  2. Reputation Feedback: Ignoring the fact that frequent rejections damage client relationships, reducing future order flow intensity.
• The Objective: To derive an optimal stochastic control framework where the dealer simultaneously optimizes quote offsets (spreads) and rejection rules (slippage tolerance), while accounting for the feedback loop where rejections degrade a "reputation score," thereby lowering future order flow.

2. Methodology

A. Model Setup

The problem is formulated as a finite-horizon stochastic optimal control problem based on the Avellaneda-Stoikov tradition.

• State Variables:
  • $S_t$: Mid-price (Brownian motion).
  • $q_t$: Inventory.
  • $X_t$: Cash.
  • $R_t$: Rejection Score (a reputation proxy), modeled as an Exponentially Weighted Moving Average (EMA) of past rejections. $R_t \in [0, 1]$.
• Dynamics:
  • Order Flow: Arrivals follow Poisson processes with intensity $\lambda_{n,i} = g(R_t)\Lambda_0 e^{-\kappa \delta}$. The function $g(R)$ is decreasing, meaning a higher rejection score (more past rejections) reduces future flow.
  • Latency & Adverse Selection: Upon request arrival, the dealer observes a "latency mark" $Y$ (price move). $Y$ is modeled as Gaussian: $Y \sim \mathcal{N}(m(\delta), \nu^2)$. The mean $m(\delta)$ captures adverse selection (e.g., clients hitting quotes when prices move against the dealer).
  • Control: The dealer chooses a binary decision $\ell \in \{0, 1\}$ (Reject/Accept) and quote offsets $\delta$.
• Objective: Maximize expected terminal wealth penalized by inventory risk:
  $$\sup_{\delta, \ell} \mathbb{E} \left[ X_T + q_T S_T - \frac{\gamma}{2} \int_0^T q_t^2 dt \right]$$

B. Mathematical Framework

• Hamilton-Jacobi-Bellman (HJB) Equation: The authors derive the HJB equation for the value function $U(t, x, q, R, S)$.
  • Using an affine ansatz ($U = x + qS + V(t, q, R)$), the problem reduces to finding $V$.
  • The HJB involves a maximization over both quote offsets ($\delta$) and the rejection decision ($\ell$).
  • The rejection decision is shown to be equivalent to exercising an embedded option on the latency price move. The dealer accepts if the realized slippage exceeds a threshold determined by the "continuation loss" $J$ (the value loss from worsening reputation).

C. The Adiabatic-Quadratic Approximation

To make the problem tractable for real-time implementation, the authors propose a two-scale approximation:

1. Adiabatic Assumption: The reputation score $R$ evolves slowly compared to inventory $q$. $R$ is treated as quasi-static (frozen) over the inventory control horizon.
2. Quadratic Ansatz: The value function is approximated as $V(t, q) \approx -A(t)q^2 - C(t)$.
3. Riccati ODE: Substituting the ansatz into the HJB yields a Riccati Ordinary Differential Equation for the coefficient $A(t)$.
4. Self-Consistent Calibration: The "shadow cost" of rejection ($J$) is calibrated such that the theoretical rejection rate matches the current reputation level $R$. This creates a closed-loop system linking the fast control (quotes/rejection) to the slow state (reputation).

D. Fair Execution Protocols

The paper extends the model to "fair" protocols (inspired by Oomen, 2017b) where the dealer cannot arbitrarily reject adverse moves without consequence.

• Symmetric Tolerance: The dealer rejects only if slippage $Y < -\epsilon$. If $Y > \epsilon$ (favorable to dealer), the dealer accepts but rebates the excess to the client.
• This transforms the rejection decision into a choice of a symmetric tolerance band $\epsilon$, balancing fairness against risk.

3. Key Contributions

1. Endogenous Rejection Control: The paper is the first to treat the rejection threshold as a dynamic, state-dependent control variable optimized jointly with quotes, rather than a fixed protocol parameter.
2. Reputation Feedback Loop: It introduces a novel state variable ($R_t$) that couples current rejection decisions to future order flow intensity, mathematically formalizing the "reputation cost" of aggressive rejection.
3. Analytical Approximation: The development of the adiabatic-quadratic approximation provides closed-form expressions for optimal quotes and rejection thresholds. This allows for fast policy evaluation and calibration, which is crucial for high-frequency trading systems.
4. Fair Protocol Integration: The framework successfully integrates "fair" execution protocols (symmetric tolerance with rebates) into the stochastic control setting, showing how fairness constraints alter optimal behavior.

4. Results and Numerical Findings

• Impact of Latency: As latency increases, the optimal strategy shifts from widening spreads to increasing rejection rates. However, the spread often tightens because the dealer can filter out toxic flow via rejection.
• Reputation Dynamics:
  • Counter-intuitive Finding: When the reputation score ($R$) is high (indicating a history of rejections), the dealer tightens spreads (to attract more flow and dilute the bad reputation) but becomes more selective (higher rejection threshold) to avoid further damaging the reputation. This is a "repair-exploit" mechanism.
  • Stationary State: In the long run, the presence of reputation feedback significantly reduces the rejection probability compared to a model without feedback. Dealers are incentivized to avoid rejection to preserve future flow.
• Unconstrained vs. Fair Protocols:
  • Unconstrained: Allows the highest utility but results in high rejection rates (up to 33% in simulations), which is unsustainable in practice due to client friction.
  • Symmetric (Fair): Results in a lower utility gain compared to unconstrained but drastically reduces rejection rates (e.g., from 33% to ~17% in the example). The optimal symmetric threshold is found to be roughly half the top-of-book spread.
• Inventory Dependence: Rejection thresholds are relatively insensitive to inventory in constant-toxicity models but become more sensitive when toxicity depends on the quote size.

5. Significance and Implications

• Practical Implementation: The derived closed-form expressions and Riccati ODEs provide a computationally efficient method for dealers to implement dynamic slippage control in real-time systems, moving beyond static "Last Look" rules.
• Regulatory & Ethical Alignment: By modeling fairness and reputation, the paper supports industry codes (e.g., GFXC 2024) that view Last Look as a risk-control mechanism rather than a profit-extraction tool. It quantifies the trade-off between short-term risk mitigation and long-term client retention.
• Market Microstructure Insight: The study highlights that "fair" protocols do not necessarily eliminate the dealer's optionality but shift the nature of the embedded option, leading to more stable market-making behavior and lower systemic rejection rates.
• Calibration: The self-consistent calibration method allows dealers to back-test and tune their rejection policies against historical flow data to find the optimal balance between risk management and business growth.

In summary, Barzykin provides a rigorous mathematical framework that transforms the "Last Look" decision from a static rule into a dynamic, reputation-aware optimization problem, offering both theoretical insights and practical algorithms for modern FX market making.