- Liquidity Pool Design on Automated Market Makers (with Xuedong He and Yutian Zhou).
working paper. [Abstract|SSRN|arXiv]
Automated market makers are a popular type of decentralized exchange in which users trade assets with each other directly and automatically through a liquidity pool and a fixed pricing function. The liquidity provider contributes to the liquidity pool by supplying assets to the pool and in return they earn transaction fees from traders who trade through the pool. We propose a model of optimal liquidity provision in which the risk-averse liquidity provider decides the investment proportion of wealth she would like to supply to the pool, trade in a centralized market, and consume in multiple periods. We derive the liquidity provider's optimal strategy by dynamic programming and numerically find the optimal liquidity pool that maximizes the liquidity provider's utility. Our findings indicate that the exchange rate volatility on the centralized market exerts a positive effect on the optimal transaction fee. Moreover, the optimal constant mean pricing formula is found to be related to the relative performance of the underlying assets on the centralized market.
- Calibration of Local Volatility Models under the Implied Volatility Criterion (with Xinfu Chen, Min Dai, and Zhou Yang).
submitted. [Abstract|SSRN]
We study non-parametric calibration of local volatility models, which is formulated as an inverse problem of partial differential equations with Tikhonov regularization. In contrast to the existing literature minimizing the distance between theoretical and market prices of options as a calibration criterion, we instead minimize the distance between theoretical and market implied volatilities, complying with market practices. We prove that our calibration criterion naturally leads to the well-posedness of the calibration problem. In particular, comparing to Jiang and Tao (2001), we obtain a global uniqueness result, where no additional weight functions are required. Numerical results reveal that our method achieves a better trade-off between minimizing calibration errors and reducing overfitting.
- Patience is a Virtue: Optimal Investment in the Presence of Market Resilience (with Nan Chen, Min Dai, and Qiheng Ding).
submitted. [Abstract|SSRN]
This paper investigates an optimal investment problem in an illiquid market, modeling explicitly the effects of three key features of market microstructure --- market tightness, market depth, and finite market resilience --- on the investor's decision. By employing a Bachelier process to model the dynamic of the fundamental value of the asset and assuming CARA-type utility for the investor, we manage to obtain the investor's optimal dynamic trading strategy in closed form by solving the resulting high-dimensional singular control problem. Furthermore, we extend the model to incorporate return-predicting signals and utilize an asymptotic expansion approach to derive approximate optimal trading strategies. The theoretical and numerical results emphasize the vital role of patience. Specifically, rather than dispersing small trades continuously over time as advocated by the existing literature, our findings suggest that investors should strategically time their trading activities to align with the aim portfolio in the presence of market resilience. To quantify this timing decision, we introduce a patience index that enables investors to strike a balance among various competing goals, including achieving currently optimal risk exposure, incorporating signals about future predictions, and minimizing trading costs, by leveraging market resilience.
- Non-Concave Utility Maximization with Transaction Costs (with Shuaijie Qian).
submitted. [Abstract|SSRN|arXiv]
This paper studies a finite-horizon portfolio selection problem with non-concave terminal utility and proportional transaction costs. The commonly used concavification principle for terminal value is no longer valid here, and we establish a proper theoretical characterization of this problem. We first give the asymptotic terminal behavior of the value function, which implies any transaction close to maturity only provides a marginal contribution to the utility. After that, the theoretical foundation is established in terms of a novel definition of the viscosity solution incorporating our asymptotic terminal condition. Via numerical analyses, we find that the introduction of transaction costs into non-concave utility maximization problems can prevent the portfolio from unbounded leverage and make a large short position in stock optimal despite a positive risk premium and symmetric transaction costs.
- Designing Stable Coins (with Yizhou Cao, Min Dai, Steven Kou and Lewei Li).
submitted. [Abstract|SSRN]
Stable coins, which are cryptocurrencies pegged to other stable financial assets such as U.S. dollar, are desirable for payments within blockchain networks, whereby being often called the “Holy Grail of cryptocurrency.” However, existing cryptocurrencies are too volatile for these purposes. By using the option pricing theory, we design several dual-class structures that offer a fixed income crypto asset, a stable coin pegged to a traditional currency, and leveraged investment instruments. To understand the impact of the proposed coins on the speculative and non-speculative demands of cryptocurrencies, we study equilibrium with and without the stable coins. Our investigation of the values of stable coins in presence of jump risk and black-swan type events shows the robustness of the design.
- An Equilibrium Model for the Cross-Section of Liquidity Premia (with Johannes Muhle-Karbe and Xiaofei Shi).
Mathematics of Operations Research,
48(3):1423-1453, 2023. [Abstract|SSRN|arXiv|Article]
We study a risk-sharing economy where an arbitrary number of heterogenous agents trades an arbitrary number of risky assets subject to quadratic transaction costs. For linear state dynamics, the forward-backward stochastic differential equations characterizing equilibrium asset prices and trading strategies in this context reduce to a system of matrix-valued Riccati equations. We prove the existence of a unique global solution and provide explicit asymptotic expansions that allow us to approximate the corresponding equilibrium for small transaction costs. These tractable approximation formulas make it feasible to calibrate the model to time series of prices and trading volume, and to study the cross-section of liquidity premia earned by assets with higher and lower trading costs. This is illustrated by an empirical case study.
- Leveraged Exchange-Traded Funds with Market Closure and Frictions (with Min Dai, Steven Kou and H. Mete Soner).
Management Science, 69(4):2517-2535, 2023. [Abstract|SSRN|Article]
Although leveraged ETFs are popular products for retail investors, how to hedge them poses a great challenge to financial institutions. We develop an optimal rebalancing (hedging) model for leveraged ETFs in a comprehensive setting, including overnight market closure and market frictions. The model allows for an analytical optimal rebalancing strategy.
The result extends the principle of "aiming in front of target" introduced by Gârleanu and Pedersen (2013) from a constant weight between current and future positions to a time-varying weight, because the rebalancing performance is monitored only at discrete time points but the rebalancing takes place continuously. Empirical findings and implications for the weekend effect and the intraday trading volume are also presented.
- A Stochastic Representation for Nonlocal Parabolic PDEs with Applications (with Min Dai and Steven Kou).
Mathematics of Operations Research, 47(3):1707-1730, 2022 [Abstract|SSRN|Article]
We establish a stochastic representation for a class of nonlocal parabolic terminal-boundary value problems, whose terminal and boundary conditions depend on the solution in the interior domain; in particular, the solution is represented as the expectation of functionals of a diffusion process with random jumps from boundaries. We discuss three applications of the representation, the first one on the pricing of dual-purpose funds, the second one on the connection to regenerative processes, and the third one on modeling the entropy on a one-dimensional non-rigid body.
- Inventory Management for High-Frequency Trading with Imperfect Competition (with Sebastian Herrmann, Johannes Muhle-Karbe and Dapeng Shang).
SIAM Journal on Financial Mathematics, 11(1):1-26, 2020. [Abstract|SSRN|arXiv|Article]
We study Nash equilibria for inventory-averse high-frequency traders (HFTs), who trade to exploit information about future price changes. For discrete trading rounds, the HFTs' optimal trading strategies and their equilibrium price impact are described by a system of nonlinear equations; explicit solutions obtain around the continuous-time limit. Unlike in the risk-neutral case, the optimal inventories become mean-reverting and vanish as the number of trading rounds becomes large. In contrast, the HFTs' risk-adjusted profits and the equilibrium price impact converge to their risk-neutral counterparts. Compared to a social-planner solution for cooperative HFTs, Nash competition leads to excess trading, so that marginal transaction taxes in fact decrease market liquidity.
- Optimal Tax-timing with Asymmetric Long-term/short-term Capital Gains Tax (with Min Dai, Hong Liu and Yifei Zhong).
The Review of Financial Studies, 28.9:2687-2721, 2015. [Abstract|SSRN|Article]
We develop an optimal tax-timing model that takes into account asymmetric long-term and short-term tax rates for positive capital gains and limited tax deductibility of capital losses. In contrast to the existing literature, this model can help explain why many investors not only defer short-term capital losses to long term but also defer large long-term capital gains and losses. Because the benefit of tax deductibility of capital losses increases with the short-term tax rates, effective tax rates can decrease as short-term capital gains tax rates increase.
