Approximately optimal trade execution strategies under fast mean-reversion (2307.07024v2)

Abstract: In a fixed time horizon, appropriately executing a large amount of a particular asset -- meaning a considerable portion of the volume traded within this frame -- is challenging. Especially for illiquid or even highly liquid but also highly volatile ones, the role of "market quality" is quite relevant in properly designing execution strategies. Here, we model it by considering uncertain volatility and liquidity; hence, moments of high or low price impact and risk vary randomly throughout the trading period. We work under the central assumption: although there are these uncertain variations, we assume they occur in a fast mean-reverting fashion. We thus employ singular perturbation arguments to study approximations to the optimal strategies in this framework. By using high-frequency data, we provide estimation methods for our model in face of microstructure noise, as well as numerically assess all of our results.

• The paper introduces approximately optimal execution strategies that account for fast mean-reverting stochastic liquidity and volatility using asymptotic expansion techniques.
• It develops a two-level approach with leading-order and first-order corrections to optimize trading performance while managing inventory risk.
• Empirical analysis with high-frequency cryptocurrency data validates the model, showing that adaptive strategies outperform constant-parameter benchmarks in various risk settings.

This paper (2307.07024) addresses the problem of executing a large volume of a financial asset within a fixed time horizon, focusing on assets traded in markets with uncertain liquidity and volatility. The core idea is to model these uncertainties as stochastic processes that exhibit fast mean-reversion and leverage asymptotic techniques to find approximately optimal trading strategies.

The authors consider a model where the asset price follows an arithmetic Brownian motion with a stochastic volatility $\sigma_u$. The trader's execution is subject to a temporary price impact proportional to $|\nu_u|^\phi \text{sgn}(\nu_u)$, where $\nu_u$ is the trading rate and $\phi \in (0,1)$ is a fixed exponent. The proportionality coefficient $\kappa_u$ is also stochastic. Both $\kappa_u$ and $\sigma_u$ are modeled as functions of a multi-dimensional Markov diffusion process $\boldsymbol{y}_u$. The central assumption is that $\boldsymbol{y}_u$ mean-reverts very quickly, driven by a process scaled by $1/\epsilon$, where $\epsilon$ is a small parameter. The trader's objective is to maximize a performance criterion related to terminal wealth, penalizing squared inventory risk and terminal inventory holdings. This problem is formulated as a stochastic control problem, leading to a Hamilton-Jacobi-BeLLMan (HJB) partial differential equation for the value function.

To tackle the complexity introduced by the fast mean-reverting factor, the authors employ a formal asymptotic expansion of the value function in powers of $\epsilon$. They derive the equations governing the leading-order ($z_0$) and first-order ($z_1$) terms of this expansion.

The leading-order term, $z_0$, is found to be a function of time only, independent of the current state $\boldsymbol{y}$. Its dynamics are governed by an ordinary differential equation (ODE) that depends on the averages of $\kappa^{-1/\phi}$ and $\sigma^{1+\phi}$ with respect to the invariant distribution of the fast process $\boldsymbol{y}$. This leads to a leading-order optimal trading strategy $\nu^0$ that is adaptive to the current stochastic liquidity $\kappa(\boldsymbol{y}_t)$ but only to the average impact of stochastic volatility $\sigma^{1+\phi}$.

The first-order term, $z_1$, is found by solving a Poisson equation involving the generator of the fast process $\mathcal{L}$ and terms derived from the deviation of $\kappa^{-1/\phi}$ and $\sigma^{1+\phi}$ from their average values. $z_1$ depends on both time and the state $\boldsymbol{y}$, and includes a time-dependent 'boundary layer' term to satisfy the terminal condition. The first-order approximation $\overline{z}_1 = z_0 + \epsilon z_1$ yields a strategy $\nu^1$ that is more fully adaptive to the current state of both $\kappa(\boldsymbol{y}_t)$ and $\sigma(\boldsymbol{y}_t)$.

Practical Implementation and Data Analysis:

The paper provides a data-driven approach to estimate the model parameters using high-frequency financial data.

1. Data: Level 2 order book data for BTCUSDT on Binance is used.
2. Temporary Price Impact Estimation:
   • The exponent $\phi$ is estimated using a bagging methodology applied to empirical impact curves derived from order book snapshots. The paper finds $\widehat{\phi} \approx 0.2833$ for BTCUSDT, supporting the concavity ($\phi < 1$) found in empirical literature.
   • The coefficient $\kappa_t$ is estimated at each time step via linear regression over a lookback window, fitting the power-law impact function.
3. Volatility Estimation: Intraday volatility $\sigma_t$ of the bid price is estimated using the Two-Scale Realized Variance (TSRV) method [zhang2005tale], which is robust to microstructure noise.
4. Fast Mean-Reversion Modeling: $\kappa_t$ and $\log(\sigma_t)$ are modeled as Ornstein-Uhlenbeck (OU) processes. Their parameters (mean-reversion speed $\lambda$, mean $m$, diffusion $\eta$) are estimated from the time series of $\widehat{\kappa}_t$ and $\log(\widehat{\sigma}_t)$ using a regression-based method [holy2018estimation]. The estimated mean-reversion speeds are high ((1905.2180) per day for $\kappa$, (1279.7954) per day for $\log(\sigma)$), validating the fast mean-reversion assumption. A positive correlation ($\approx 21\%$) is observed between $\kappa$ and $\log(\sigma)$, indicating that periods of higher volatility are associated with lower liquidity (higher $\kappa$).
5. Numerical Computation: Implementing the first-order correction $\nu^1$ requires solving Poisson equations for auxiliary functions ($\varphi_0, \varphi_1$). The appendix describes an iterative numerical algorithm for solving such equations under the assumption that $\boldsymbol{y}$ is a multi-dimensional OU process, leveraging spectral properties and Poincaré inequality for Gaussian measures.

Numerical Experiments:

Using the estimated parameters for a 2D OU process driving $\kappa$ and $\log(\sigma)$, the authors conduct Monte Carlo simulations to assess the performance of the derived strategies ($\nu^0, \nu^1$) against a benchmark Almgren-Chriss strategy ($\nu^{AC}$) that assumes constant liquidity and volatility equal to their long-run means.

• Risk-Neutral Setting ($\gamma=0$): The leading-order strategy $\nu^0$ consistently outperforms the constant-parameter $\nu^{AC}$ benchmark, achieving a higher terminal cash position on average and finishing with less remaining inventory. This highlights the practical value of adapting to stochastic liquidity.
• Risk-Averse Setting ($\gamma>0$): Both $\nu^0$ and $\nu^1$ (with the same risk aversion level $\gamma$) outperform the constant-parameter $\nu^{AC}$. The first-order correction $\nu^1$ provides a further marginal improvement over $\nu^0$. This improvement is of order $\epsilon$, which is small based on the parameter estimates, suggesting that $\nu^0$ might be sufficient for practical purposes due to its computational simplicity. While $\nu^1$ might not always yield significantly higher terminal wealth than $\nu^0$ for higher risk aversion, it is shown to be more effective in managing inventory risk by consistently ending with less inventory.

Accuracy Results:

The paper provides theoretical proofs for the accuracy of the approximations. Using Feynman-Kac representations and Gronwall's Lemma, it's shown that:

• The leading-order approximation $\overline{z}_0$ is accurate up to $O(\epsilon)$ pointwise.
• The first-order approximation $\overline{z}_1$ is accurate up to $O(\epsilon^2)$ in $L^p$ norms averaged over time and pointwise away from the terminal time $T$.

Conclusion:

The paper successfully develops and assesses approximately optimal trade execution strategies in markets with fast mean-reverting stochastic liquidity and volatility. The approach provides practical strategies that are adaptive to current market conditions. Empirical analysis using high-frequency cryptocurrency data supports the fast mean-reversion assumption and provides realistic parameters for simulations. The leading-order approximation $\nu^0$, which is simpler to compute, is shown to provide significant performance improvements over traditional constant-parameter benchmarks by adapting to stochastic liquidity. The first-order correction $\nu^1$ offers a theoretically justified, albeit marginal in the tested scenario, further improvement by incorporating more detailed information about the fast market state, particularly beneficial for inventory risk management. The work provides a solid framework for implementing adaptive execution algorithms in volatile and less liquid markets like cryptocurrency spot markets.