Search Results (32)

This paper demonstrates that perfectly calibrating a multi-asset model to observed market prices of all basket call options is insufficient to uniquely determine the price of a best-of call option. Previous research on multi-asset option pricing has primarily focused on complete market settings or assumed specific parametric models, leaving fundamental questions about model risk and pricing uniqueness in incomplete markets inadequately addressed. This limitation has critical practical implications: derivatives practitioners who hedge best-of options using basket-equivalent instruments face fundamental distributional uncertainty that compounds the well-recognized non-linearity challenges. We establish this non-uniqueness using convex analysis (extreme ray characterization demonstrating geometric incompatibility between payoff structures), measure theory (explicit construction of distinct equivalent probability measures), and geometric analysis (payoff structure comparison). Specifically, we prove that the set of equivalent probability measures consistent with observed basket prices contains distinct measures yielding different best-of option prices, with explicit no-arbitrage bounds $[a_K,b_K]$ quantifying this uncertainty. Our theoretical contribution provides the first rigorous mathematical foundation for several empirically observed market phenomena: wide bid-ask spreads on extremal options, practitioners’ preference for over-hedging strategies, and substantial model reserves for exotic derivatives. We demonstrate through concrete examples that substantial model risk persists even with perfect basket calibration and equivalent measure constraints. For risk-neutral pricing applications, equivalent martingale measure constraints can be imposed using optimal transport theory, though this requires additional mathematical complexity via Schrödinger bridge techniques while preserving our fundamental non-uniqueness results. The findings establish that additional market instruments beyond basket options are mathematically necessary for robust exotic derivative pricing. Full article

We develop a deep reinforcement learning (RL) framework for an optimal market-making (MM) trading problem, specifically focusing on price processes with semi-Markov and Hawkes Jump-Diffusion dynamics. We begin by discussing the basics of RL and the deep RL framework used; we deployed the state-of-the-art Soft Actor–Critic (SAC) algorithm for the deep learning part. The SAC algorithm is an off-policy entropy maximization algorithm more suitable for tackling complex, high-dimensional problems with continuous state and action spaces, like those in optimal market-making (MM). We introduce the optimal MM problem considered, where we detail all the deterministic and stochastic processes that go into setting up an environment to simulate this strategy. Here, we also provide an in-depth overview of the jump-diffusion pricing dynamics used and our method for dealing with adverse selection within the limit order book, and we highlight the working parts of our optimization problem. Next, we discuss the training and testing results, where we provide visuals of how important deterministic and stochastic processes such as the bid/ask prices, trade executions, inventory, and the reward function evolved. Our study includes an analysis of simulated and real data. We include a discussion on the limitations of these results, which are important points for most diffusion style models in this setting. Full article

Futures trading analysis plays a pivotal role in the development of macroeconomic policies and corporate strategy planning. High-frequency futures data, typically presented as time series, contain valuable historical patterns. To address challenges such as non-stationary in modeling futures prices, we propose a novel architecture called FuturesNet, which uses an InceptionTime module to capture the short-term fluctuations between ask and bid orders, as well as a long-short-term-memory (LSTM) module with skip connections to capture long-term temporal dependencies. We evaluated the performance of FuturesNet using datasets numbered 50, 300, and 500 from the domestic financial market. The comprehensive experimental results show that FuturesNet outperforms other competitive baselines in most settings. Additionally, we conducted ablation studies to interpret the behaviors of FuturesNet. Our code and collected futures datasets are released. Full article

The foreign exchange (FX) market has evolved into a complex system where locally generated information percolates through the dealer network via high-frequency interactions. Information related to major events, such as economic announcements, spreads rapidly through this network, potentially inducing volatility, liquidity disruptions, and contagion effects across financial markets. Yet, research on the mechanics of information flows in the FX market is limited. In this paper, we introduce a novel approach employing conditional transfer entropy to construct networks of information flows. Leveraging a unique, high-resolution dataset of bid and ask prices, we investigate the impact of an announcement by the European Central Bank on the information transfer within the market. During the announcement, we identify key dealers as information sources, conduits, and sinks, and, through comparison to a baseline, uncover shifts in the network topology. Full article

The Constant Leverage covering strategy for the equity momentum portfolio (CLvg) developed in this project cannot mask its shortcomings by increasing leverage. It has to successfully forecast and avoid more losses than profits to perform better than the momentum portfolio. This approach is different from other covering strategies available in the literature that focus on increasing the right tail of the momentum returns distribution at a faster rate than they increase the left tail. The CLvg strategy only depends on past information and uses the daily volatility of the loser portfolio to determine episodes of high and low volatility. The daily volatility of the loser portfolio has a stronger relationship with large negative momentum returns than the daily volatility of the momentum portfolio. The daily volatility of the loser portfolio also has a weaker relationship with larger positive monthly returns, and it is more predictable because it has a higher volatility persistence. The negative effects of transaction costs on the CLvg strategy are measured using bid and ask prices reported by CRSP from 1992 to 2021. During this period, the stock market presented an average excess return of 9.19% and a Sharpe ratio of 0.61, and 9.74% of its returns were crashes, which is a better performance than the momentum portfolio. The CLvg adjusted by transaction costs presented excess returns of 16.93% and a Sharpe ratio of 0.84, and only 8.31% of its returns were crashes. Full article

Shadow prices simplify the derivation of optimal trading strategies in markets with transaction costs by transferring optimization into a more tractable, frictionless market. This paper establishes that a naïve shadow price ansatz for maximizing long-term returns, given average volatility yields a strategy that is, for small bid–ask spreads, asymptotically optimal at the third order. Considering the second-order impact of transaction costs, such a strategy is essentially optimal. However, for risk aversion different from one, we devise alternative strategies that outperform the shadow market at the fourth order. Finally, it is shown that the risk-neutral objective rules out the existence of shadow prices. Full article

This study bridges finance and physics by applying thermodynamic concepts to model the limit order book (LOB) with high-frequency trading data on the Bitcoin spot. We derive the measures of Market Temperature and Market Entropy from the kinetic and potential energies in the LOB to provide a deeper understanding of order activities and market participant behavior. Market Temperature emerges as a robust indicator of market liquidity, correlating with liquidity measures such as Active Quote Volume, bid–ask spread and match volume. Market Entropy, on the other hand, quantifies the degree of disorder or randomness in the LOB, providing insights into the instantaneous volatility of price in the high-frequency trading market. Our empirical findings not only broaden the theoretical framework of econophysics but also enhance comprehensive understanding of the market microstructure and order book dynamics. Full article

We introduce a class of Markov models to describe the bid–ask price dynamics in the presence of liquidity fluctuations. In a highly competitive regime, the spread evolution belongs to a class of Markov processes known as a population process with uniform catastrophes. Our mathematical analysis focuses on establishing the law of large numbers, the central limit theorem, and large deviations for this catastrophe-based model. Large deviation theory allows us to illustrate how huge deviations in the spread and prices can occur in the model. Moreover, our research highlights how these local trends and volatility are influenced by the typical values of the bid–ask spread. We calibrated the model parameters using available high-frequency data and conducted Monte Carlo numerical simulations to demonstrate its ability to reasonably replicate key phenomena in the presence of liquidity fluctuations. Full article

This article deals with the identification of a superior forecasting method for market liquidity using a calibrated Heston model for the bid/ask price path simulation instead of a standard Brownian motion, as well as a compound Poisson process and inverse transform sampling for the generation of the bid/ask volume distribution. We show that the simulated trading volumes converge to one single value, which can be used as a liquidity estimator, and find that the calibrated Heston model as well as the inverse transform sampling are superior to both the use of standard Brownian motion and compound Poisson process. Full article

This paper studies the portfolio selection problem where tradable assets are a bank account, and standard put and call options are written on the S&P 500 index in incomplete markets in which there exist bid–ask spreads and finite liquidity. The problem is mathematically formulated as an optimization problem where the variance of the portfolio is perceived as a risk. The task is to find the portfolio which has a satisfactory return but has the minimum variance. The underlying is modeled by a variance gamma process which can explain the extreme price movement of the asset. We also study how the optimized portfolio changes subject to a user’s views of the future asset price. Moreover, the optimization model is extended for asset pricing and hedging. To illustrate the technique, we compute indifference prices for buying and selling six options namely a European call option, a quadratic option, a sine option, a butterfly spread option, a digital option, and a log option, and propose the hedging portfolios, which are the portfolios one needs to hold to minimize risk from selling or buying such options, for all the options. The sensitivity of the price from modeling parameters is also investigated. Our hedging strategies are decent with the symmetry property of the kernel density estimation of the portfolio payout. The payouts of the hedging portfolios are very close to those of the bought or sold options. The results shown in this study are just illustrations of the techniques. The approach can also be used for other derivatives products with known payoffs in other financial markets. Full article

Most of the canonical macroeconomic models simulate liquidity anomalies by changing the economic fundamentals or adding massive financial shock to firms’ collateral constraints, but a few facts somehow tell a different story. Instead of relying on the exogenous shocks, we introduce uncertainty into an otherwise classical liquidity framework and try to answer what worsens the aggregate liquidity in the absence of exogenous simulations and what a firm dynamics and financing strategy would be. Our analysis shows that (1) uncertainty induces agents to make decisions under the worst-case scenario and hence generates a unique expectation threshold that drags market (or firms) liquidity from sufficiency to insufficiency even without any shock or economic changes. (2) Precautionary saving occurs before the real liquidity shortage as the expectation shifts, causing firms to secure external financing by raising the equity issuing price and hoarding liquid assets, such as fiat money, against liquidity tightening. (3) To achieve liquidity stability and sustainability, an extra mathematical constraint is supplemented for the uniqueness and the existence of equilibrium under uncertainty. Other properties of firms’ intertemporal allocations, such as the bid-ask spread and return of holding of the illiquid asset, are derived. Moreover, some approaches for further empirical research are discussed. Full article

In economics, some transactions are conducted by the bid rate, and some are conducted by the ask rate. The spread between these two rates creates an essential cost and inefficiency for the economy. Taking these problems into account, the purpose of this study was to analyze the effects of macroeconomic and financial variables on the USD/TL exchange rate bid–ask spread for Türkiye using daily data spanning the period between 2 January 1990 and 2 August 2022. The quantile autoregressive distributed lag (QARDL) model was drawn upon to capture possible asymmetry in parameters and distinguish the results between different locations. The results obtained in this study may differ from the linear model and may change by the location, implying that the spread is reduced by the volume while it is increased by volatility and interest rates in the long run for some quantiles. Stock prices stir it in the long run, yet they decline it in the short run, indicating an asymmetry. Following the examples from the literature that analyzed the relationship via linear models, this paper employed a QARDL model for exploring location and sign asymmetry in the results for some quantiles. As the results indicate, efficiency in the bid–ask exchange rate spread can be controlled; therefore, it is our suggestion for policymakers to consider the extreme levels and asymmetry of the bid–ask exchange rate spread while evaluating its penetrating macro-financial variates. Full article

Underlying idiosyncratic and illiquidity risks are suppressed in infrequently reported indexes of house prices and rents. Idiosyncratic risks result from bid–ask spreads for prices and rents. Time series autocovariances generate a distribution of prices and rents. Capital gains and rent-price ratios are transforms of these distributions, generating cross-sectional idiosyncratic volatility. Housing data are infrequent and usually made available every month. The monthly–quarterly volatility ratios of house prices and rents and their spreads estimate unobserved daily fluctuations and illiquidity risks. Including idiosyncratic and illiquidity risks, a U.S. house has a standard deviation in returns of 8.7% annually for three decades after 1990. With a mean excess return of 3.7%, the Sharpe ratio of 0.42 is comparable to the S&P 500. Excluding spreads, the house Sharpe ratio is 0.69. House returns respond to liquidity. A 1% increase in volume raises returns by 0.8%. Full article

This survey starts with a general overview of the strategies for stock price change predictions based on market data and in particular Limit Order Book (LOB) data. The main discussion is devoted to the systematic analysis, comparison, and critical evaluation of the state-of-the-art studies in the research area of stock price movement predictions based on LOB data. LOB and Order Flow data are two of the most valuable information sources available to traders on the stock markets. Academic researchers are actively exploring the application of different quantitative methods and algorithms for this type of data to predict stock price movements. With the advancements in machine learning and subsequently in deep learning, the complexity and computational intensity of these models was growing, as well as the claimed predictive power. Some researchers claim accuracy of stock price movement prediction well in excess of 80%. These models are now commonly employed by automated market-making programs to set bids and ask quotes. If these results were also applicable to arbitrage trading strategies, then those algorithms could make a fortune for their developers. Thus, the open question is whether these results could be used to generate buy and sell signals that could be exploited with active trading. Therefore, this survey paper is intended to answer this question by reviewing these results and scrutinising their reliability. The ultimate conclusion from this analysis is that although considerable progress was achieved in this direction, even the state-of-art models can not guarantee a consistent profit in active trading. Taking this into account several suggestions for future research in this area were formulated along the three dimensions: input data, model’s architecture, and experimental setup. In particular, from the input data perspective, it is critical that the dataset is properly processed, up-to-date, and its size is sufficient for the particular model training. From the model architecture perspective, even though deep learning models are demonstrating a stronger performance than classical models, they are also more prone to over-fitting. To avoid over-fitting it is suggested to optimize the feature space, as well as a number of layers and neurons, and apply dropout functionality. The over-fitting problem can be also addressed by optimising the experimental setup in several ways: Introducing the early stopping mechanism; Saving the best weights of the model achieved during the training; Testing the model on the out-of-sample data, which should be separated from the validation and training samples. Finally, it is suggested to always conduct the trading simulation under realistic market conditions considering transactions costs, bid–ask spreads, and market impact. Full article

Prior studies that examine the relation between market depth and bid–ask spread are often limited to the first level of the limit order book. However, the full limit order book provides important information beyond the first level about the depth and spread, which affects the trading decisions of market participants. This paper examines the intraday behavior of depth and spread in the five-deep limit order book and the relation between depth and spread in a futures market setting. A dummy-variables regression framework is employed and is estimated using the generalized method of moments (GMM). Results indicate an inverse U-shaped pattern for depth and an increasing pattern for spread. After controlling for known explanatory factors, an inverse relation between the limit order book depth and spread is documented. The inverse relation holds for depth and spread at individual levels in the limit order book as well. Results indicate that market participants actively manage both the price (spread) and quantity (depth) dimensions of liquidity along the five-deep limit order book. Full article