We study how trading fees and continuous-time arbitrage affect the profitability of liquidity providers (LPs) in Geometric Mean Market Makers (G3Ms). We use stochastic reflected diffusion processes to analyze the dynamics of a G3M model under the arbitrage-driven market [Milionis et al. 2022a. “Automated Market Making and Loss-Versus-Rebalancing.” arXiv e-prints]. Our research focuses on calculating LP wealth and extends the findings of Tassy and White [Tassy and White. 2020. “Growth Rate of a Liquidity Provider's Wealth in xy = c Automated Market Makers.”] for the constant product market maker (Uniswap v2) to a broader range of G3Ms, including Balancer. This allows us to calculate the long-term expected logarithmic growth of LP wealth, offering new insights into the complex dynamics of AMMs and their implications for LPs in decentralized finance.

The objectives of option hedging/trading extend beyond mere protection against downside risks, with a desire to seek gains also driving agent's strategies. In this study, we showcase the potential of robust risk-aware reinforcement learning (RL) in mitigating the risks associated with path-dependent financial derivatives. We accomplish this by leveraging a policy gradient approach that optimizes robust risk-aware performance criteria. We specifically apply this methodology to the hedging of barrier options, and highlight how the optimal hedging strategy undergoes distortions as the agent moves from being risk-averse to risk-seeking. As well as how the agent robustifies their strategy. We further investigate the performance of the hedge when the data generating process (DGP) varies from the training DGP, and demonstrate that the robust strategies outperform the non-robust ones.

We present a computationally tractable method for simulating arbitrage-free implied volatility surfaces. We illustrate how our method may be combined with a data-driven model based on historical SPX implied volatility data to generate dynamic scenarios for arbitrage-free implied volatility surfaces. Our approach conciliates static arbitrage constraints with a realistic representation of statistical properties of implied volatility co-movements.

Dynamic mean-variance (MV) optimal strategies are inherently contrarian. Following periods of strong equity returns, there is a tendency to de-risk the portfolio by shifting into risk-free investments. On the other hand, if the portfolio still has some equity exposure, the weight on equities will increase following stretches of poor equity returns. This is essentially due to using variance as a risk measure, which penalizes both upside and downside deviations relative to a satiation point. As an alternative, we propose a dynamic trading strategy based on an expected wealth (EW), expected shortfall (ES) objective function. ES is defined as the mean of the worst β fraction of the outcomes, hence the EW-ES objective directly targets left tail risk. We use stochastic control methods to determine the optimal trading strategy. Our numerical method allows us to impose realistic constraints: no leverage, no shorting, infrequent rebalancing. For 5 year investment horizons, this strategy generates an annualized alpha of 180 bps compared to a 60:40 stock-bond constant weight policy. Bootstrap resampling with historical data shows that these results are robust to parametric model misspecification. The optimal EW-ES strategy is generally a momentum-type policy, in contrast to the contrarian MV optimal strategy.

We investigate the portfolio execution problem under a framework in which volatility and liquidity are both uncertain. In our model, we assume that a multidimensional Markovian stochastic factor drives both of them. Moreover, we model indirect liquidity costs as temporary price impact, stipulating a power law to relate it to the agent's turnover rate. We first analyse the regularized setting, in which the admissible strategies do not ensure complete execution of the initial inventory. We prove the existence and uniqueness of a continuous and bounded viscosity solution of the Hamilton–Jacobi–Bellman equation, whence we obtain a characterization of the optimal trading rate. As a byproduct of our proof, we obtain a numerical algorithm. Then, we analyse the constrained problem, in which admissible strategies must guarantee complete execution to the trader. We solve it through a monotonicity argument, obtaining the optimal strategy as a singular limit of the regularized counterparts.

Solving optimal stopping problems by backward induction in high dimensions is often very complex since the computation of conditional expectations is required. Typically, such computations are based on regression, a method that suffers from the curse of dimensionality. Therefore, the objective of this paper is to establish dimension reduction schemes for large-scale asset price models and to solve related optimal stopping problems (e.g., Bermudan option pricing) in the reduced setting, where regression is feasible. The proposed algorithm is based on an error measure between linear stochastic differential equations. We establish optimality conditions for this error measure with respect to the reduced system coefficients and propose a particular method that satisfies these conditions up to a small deviation. We illustrate the benefit of our approach in several numerical experiments, in which Bermudan option prices are determined.

ABSTRACT We analyse high-frequency realized volatility dynamics and spillovers between centralized crypto exchanges that offer spot and derivative contracts for bitcoin against the US dollar or the stable coin tether. The tether-margined perpetual contract on Binance is clearly the main source of volatility, continuously transmitting strong flows to all other instruments and receiving very little volatility from other sources. We also find that crypto exchanges exhibit much higher interconnectedness when traditional Western stock markets are open. Especially during the US time zone, volatility outflows from Binance are much higher than at other times, and Bitcoin traders are more attentive and reactive to prevailing market conditions. Our results highlight that market regulators should pay more attention to the tether-margined derivatives products available on most self-regulated exchanges, most importantly on Binance.

ABSTRACT We propose a new approach for trading VIX futures. We assume that the term structure of VIX futures follows a Markov model. Our trading strategy selects a position in VIX futures by maximizing the expected utility for a day-ahead horizon given the current shape and level of the term structure. Computationally, we model the functional dependence between the VIX futures curve, the VIX futures positions, and the expected utility as a deep neural network with five hidden layers. Out-of-sample backtests of the VIX futures trading strategy suggest that this approach gives rise to reasonable portfolio performance, and to positions in which the investor will be either long or short VIX futures contracts depending on the market environment.

ABSTRACT In this paper, we apply stochastic maximum principles to derive representations for exponential utility indifference prices. We also obtain the related optimal portfolio processes and utility indifference hedging strategies. To illustrate our theoretical results, we present several concrete examples and study the limit behaviour of utility indifference prices for vanishing and infinite risk aversion. We further investigate how the optimal trading strategies and utility indifference prices alter if one assumes that an investor has some additional information on the future behaviour of the underlying stock price process available. In this regard, we propose a customized enlarged filtration approach and deduce a formula for the utility indifference price in this extended setup. We finally provide a representation for the information premium in our utility indifference pricing framework.

ABSTRACT The Sato process associated with self decomposable laws at unit time is further generalized to an additive process with arbitrary innovation term structures. A second generalization to additive processes consistent with bilateral gamma marginal distributions is also made. The Sato process is a parametric special case of the two generalizations. This feature is exploited in defining calibration starting values. Calibration results are presented for days of daily data on SPY options. The deterministic innovation variance model makes a median improvement of in root-mean-square error over the Sato process. The comparable value for the general additive process is The Sato process relative to the general additive process overprices negative moves and underprices positive ones. The underpricing of negative moves decreases with maturity. On the positive side, the overpricing decreases with maturity. For negative moves, the overpricing is larger for smaller moves, while for positive moves the underpricing is larger for the larger moves.