State-Dependent Stochastic Models for Fill Probability Estimation

Introduction

In financial markets, the ability to predict the likelihood that a limit order will be filled is essential for optimizing execution strategies. Fill probability determines whether a trader's order—placed at a particular price—will be executed within a set timeframe. While market participants increasingly turn to algorithmic trading to automate and optimize such decisions, understanding and predicting fill probabilities remains a significant challenge. Traditional models have limitations, often assuming static conditions in the order book or overlooking the dynamic factors that influence order execution. However, recent advancements in state-dependent stochastic modeling offer a more accurate framework for estimating fill probabilities by incorporating randomness, liquidity, and order flow dynamics.

This article explores the work of Lokin and Yu (2024), which introduces a state-dependent stochastic order flow model for estimating fill probabilities in limit order books (LOB). The authors present a more realistic approach by modeling order flows as stochastic processes, taking into account the evolving state of the LOB. We will examine the model's mathematical foundation, its computational methods, and its practical applications in algorithmic trading.

This article is inspired by the work of Lokin, F., & Yu, F. (2024). Fill Probabilities in a Limit Order Book with State-Dependent Stochastic Order Flows .

The Limit Order Book: Structure and Dynamics

At the core of many financial markets is the limit order book, a mechanism that matches buy and sell orders from market participants. A limit order book contains limit orders, where traders specify the price at which they are willing to buy or sell an asset, and market orders, which are executed at the best available price. The LOB determines the price and liquidity at any given moment, depending on the orders placed by traders.

In a typical order book, orders are ranked by price—highest bid (buy order) and lowest ask (sell order)—and remain in the book until executed or canceled. The balance of buy and sell orders creates the market’s liquidity, and the closer an order is placed to the best price, the higher its chances of being filled.

However, LOB dynamics are not static. Market orders arrive, cancel orders are updated, and the order book evolves constantly. This constant fluctuation makes predicting fill probabilities a complex task.

Fill Probability: A Vital Trading Metric

Fill probability is the likelihood that a limit order will be executed at the desired price before it is canceled. Understanding and predicting fill probability helps traders refine their strategies by managing execution risk—the risk that an order will not be filled or will be filled at a less favorable price.

Several factors influence fill probability, including:

• Order size: Larger orders are less likely to be filled, as they may exceed available liquidity at the chosen price level.
• Proximity to market: Orders closer to the best bid or ask price have a higher probability of execution.
• Market liquidity: Deep order books with significant liquidity offer a higher chance of execution.
• Market volatility: In volatile markets, the likelihood of fill changes rapidly as orders get consumed by incoming market orders.
• Time horizon: Orders with longer validity periods are more likely to be filled, but this comes with the cost of holding the order open longer.

Estimating fill probabilities requires complex modeling techniques that account for these variables. Traditional models often assume a constant probability of execution based on order distance from the market, but they fail to capture the nuanced, time-varying dynamics that affect order flows.

Approaches to Estimating Fill Probabilities

Traditional modelsy, such as exponential decay functions, treat fill probabilities as decreasing exponentially with increasing distance from the market price. While simple, these models fail to account for real-world complexities such as changes in liquidity or market volatility.

Econometric models like survival analysis have been used to model the probability of order execution over time. These models estimate the likelihood of order fills based on historical data and statistical techniques but struggle to incorporate the randomness of order flow and market conditions.

Machine learning models can handle complex relationships in order book data, learning from historical data to predict fill probabilities. While these models are flexible and powerful, they require large amounts of high-frequency data and can often act as “black boxes,” providing predictions without clear explanations of how they arrived at them.

The state-dependent stochastic model proposed by Lokin and Yu offers a promising alternative by incorporating the randomness of order flows and adapting to the changing state of the LOB. This model uses queueing theory to simulate the evolution of orders in the book, treating order arrivals and cancellations as stochastic processes that depend on the state of the system.

State-Dependent Stochastic Order Flow Model: Mathematical Foundation

Lokin and Yu’s model applies queueing theory to represent the LOB as a system of waiting orders. It treats the arrival rates of both market and limit orders as state-dependent, meaning the rate at which new orders arrive changes based on the current state of the order book. For example, in a market with high bid-ask imbalance, market orders may arrive at a faster rate.

The model uses Laplace transforms to simplify the calculation of fill probabilities. Laplace transforms convert the time-dependent equations governing the order book into a frequency domain, where they can be more easily solved. By applying the first-passage time method, the model estimates the time it takes for an order to be filled, based on the current state of the system.

This approach offers a more dynamic and flexible representation of the order book compared to traditional models, as it captures the evolving nature of order flows and liquidity.

Computational Methods

To solve the state-dependent stochastic model, Lokin and Yu utilize numerical methods like the Euler method for approximating solutions to differential equations and the COS method to efficiently invert Laplace transforms. These methods are essential for calculating fill probabilities in real-time, as they allow for the processing of large amounts of high-frequency order book data.

However, real-time computation of fill probabilities can be challenging due to the need for high-quality, high-frequency data and the computational intensity of the model. Despite these challenges, the model provides accurate predictions that can be used to inform algorithmic trading strategies.

Applications in Algorithmic Trading

The state-dependent stochastic model has significant practical applications in algorithmic trading. By providing accurate estimates of fill probabilities, the model helps traders optimize order placement, minimize execution costs, and manage risk. Some key applications include:

• Order Placement: Traders can use the model to determine the optimal price level for placing orders. If the model predicts a high fill probability at a certain price, traders can place larger orders at those levels, maximizing execution efficiency.
• Dynamic Strategy Adjustments: The model adapts to changing market conditions. In volatile markets, traders can adjust their orders based on real-time estimates of fill probabilities.
• Execution Risk Management: Traders can use the model to assess execution risk and avoid placing orders in market conditions where the probability of fill is low. This helps minimize the chances of executing orders at unfavorable prices or missing fills altogether.
• Portfolio Management: By predicting how orders will be filled, traders can optimize the execution of large portfolios of assets, ensuring that trades are executed efficiently without significantly impacting market prices.

Real-World Use Cases

The state-dependent stochastic model has been shown to provide significant advantages in real-world trading environments:

• High-Frequency Trading (HFT): In high-frequency trading, where execution speed is critical, the model helps traders predict fill probabilities and execute orders faster, reducing transaction costs.
• Institutional Trading: Large institutional traders can use the model to determine the best price levels for executing large orders, avoiding significant market impact.
• Market Making: Market makers, who provide liquidity by continuously quoting prices, can use the model to optimize their bids and asks, improving their chances of executing trades at favorable prices.
• Low-Liquidity Markets: In markets with lower liquidity, the model helps traders assess whether orders are likely to be filled, allowing them to adjust order sizes and prices accordingly.

Conclusion

The state-dependent stochastic model developed by Lokin and Yu (2024) offers a significant improvement over traditional methods for estimating fill probabilities in limit order books. By incorporating state-dependent arrival rates and stochastic processes, the model provides more accurate and flexible predictions of order fills, especially in dynamic market conditions. With its ability to adapt to varying liquidity, volatility, and market dynamics, this model has the potential to enhance execution strategies, minimize risk, and reduce transaction costs in algorithmic trading.

As financial markets become increasingly complex and fast-paced, the need for more accurate and adaptable models of market behavior is clear. The state-dependent stochastic model represents a valuable step forward, providing traders with the tools they need to navigate these challenges effectively.

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