Sharp distribution free lower bounds for spread options and the corresponding optimal subreplicating portfolios

The New York Mercantile Exchange (NYMEX) asked Texas Instruments (TI) to study and propose the implementation of a system to support commodities traders in the exchange trading pits. The system includes a custom hand held computer, an integrated wireless communications system, a wireless-to-wired network bridge, and a fault detection and recovery scheme. This paper summarizes the findings of a site survey on the trading floor at NYMEX both during off hours (“empty room” but with existing equipment) and during trading (hundreds of traders in constant motion present). The goal was to evaluate the system sensitivity and impact of the impairments in terms of friendly and external interferers. Using a spectrum analyzer and a tracking generator, we measured the real-time path loss and frequency response from the temporary access point to the hand-held computer. Measurements of the indoor propagation path loss at NYMEX show good correlation with theory. Due to the large number of metal structures and plasma displays there is significant multi-path throughout the room and, hence, very little shadowing. With a location 100 floors below the World Trade Center broadcast center, the NYMEX commodities floor is a particularly interesting environment from an external interferer standpoint (i.e. intermodulation suppression, adjacent channel rejection). In addition, the excessive trader activity and user density in a hyper-active physical environment places several constraints on communication system design.

This paper addresses the lack of hedging effectiveness that yellow corn 1-month futures of the Chicago Mercantile Exchange (CME) offer for cross-hedging the price of Mexican white corn. For this purpose, the authors tested 1013 combinations (portfolios) of the ten most traded futures on the CME and the New York Mercantile Exchange (NYMEX). The results suggest that using a 51.6741% corn and a 48.3259% wheat portfolio mimics the white corn price with a hedging effectiveness of 0.6180. To test the practical use of such a portfolio, the authors backtested its use from 1 January 2000 to 9 February 2025 as a balancing short position for sale of white corn at t + 1. By using the corn–wheat portfolio, the simulated seller (farmer or intermediary) would have earned MXN 5.7664 per kilo traded. The results in this paper provide the first solution to the Mexican white corn cross-hedging problem with a futures portfolio. This hedge can be used as the balancing (short) position for the strike or minimum buy price that the Mexican Government or a financial institution could offer to farmers and intermediaries to enhance food security.

Futures markets have been used by traders in commodities for hundreds of years. Trading in rice futures was being conducted in Osaka, Japan as early as the 18th century. The New York Mercantile Exchange (NYMEX), the world’s largest regulated energy futures exchange, started life in 1872 as the Butter and Cheese Exchange of New York, before being renamed ten years later.

News, Volatility and Jumps: The Case of Natural Gas Futures

The forward premium anomaly in the energy futures markets: A time-varying approach

Seasonal Stochastic Volatility: Implications for the pricing of commodity options

As a core clean energy source in the global energy transition, natural gas price fluctuations directly affect the energy market supply demand balance, industrial chain cost control, etc. Thus, accurate natural gas price prediction is crucial for market participants’ decision making and policymakers’ regulation. To tackle the issue that traditional single models fail to capture data patterns of the New York Mercantile Exchange (NYMEX) natural gas futures daily prices—due to their nonlinearity, high volatility, and multi-scale features—this study proposes a hybrid model: VMD-CNN-BiLSTM-attention, integrating Variational Mode Decomposition (VMD), Convolutional Neural Network (CNN), Bidirectional Long Short-Term Memory (BiLSTM), and an attention mechanism. A one-step to four-step forecasting comparison was conducted using NYMEX natural gas futures daily closing prices, with the proposed model vs. CNN-BiLSTM-Attention and Autoregressive Integrated Moving Average (ARIMA) models. The empirical results show that the VMD-CNN-BiLSTM-attention model outperforms the comparison models in terms of Mean Absolute Percentage Error (MAPE), Mean Absolute Error (MAE), etc. Specifically, its four-step forecast MAPE stays ≤3.5% and R2 ≥ 98%, demonstrating a stronger ability to capture complex price fluctuations, better accuracy, and stability than traditional single models and deep learning models without VMD, and provides reliable technical support for short-to-medium-term natural gas price prediction.

A story in Aristotle's Politics is often cited as the first published discussion of a financial derivative, an instrument whose value is derived from other assets or events. In Greece of the fifth century B.C., the philosopher Thales, believing six months in advance that the spring weather would be exceptionally favorable to olive growth, negotiated the right to rent oil presses for the next harvest at a bargain rate from press owners who wanted to hedge their bets against the possibility of a poor harvest. In today's parlance, Thales purchased a call option. When spring brought a lush olive harvest, Thales rented the machines to others at a much higher rate, making a tidy profit to support his philosophical inquiries. Options and other financial derivatives have become extremely popular in recent years, for better or worse. Overinvestment in derivatives of questionable value has led to notorious financial disasters such as the bankruptcy of Orange County, California, and the collapse of Barings Bank. Warren Buffett famously described derivatives as "financial weapons of mass destruction" in the Berkshire Hathaway Annual Report of 2002. Nonetheless, derivatives are undoubtedly here to stay; in 2002, trade in derivatives worldwide was estimated at $100 trillion or more, equaling or exceeding the total value of "real" equities. Readers of SIAM Review can take a special interest in derivatives because their analysis is an arena in which mathematics has had a huge impact on finance. In particular, the 1970s papers of Black and Scholes (1973) and Merton (1976) began a flood of research on options pricing that continues to this day. This issue's Survey and Review paper focuses on the special case of a spread option, a derivative that is based on the difference in value of two assets. Suppose you have some money to invest and the choice between two stocks to buy. You suspect that one will do much better than the other over the next few years, but you don't know which one. An ideal situation might be if you could buy one of them and hold it for a certain length of time, and then have the option of trading it in for the other stock if it turns out that that one did better. Another way to achieve the same benefit would be if someone agreed to pay you the difference in value between the two stocks at some specified time if you discover with hindsight that you made the wrong decision. Of course no one will offer you such an option free of charge, and so the mathematical question is how to price this option. This question is even more challenging if the option has a strike price that must be paid in order to exercise it. While spread options aren't publicly traded on arbitrary pairs of stocks, a variety of complex spread options are available in many markets, and are particularly popular in energy markets. Energy prices are extremely volatile compared to other commodities, a fact that most consumers are well aware of and that greatly affects producers, distributors, and speculators. Spread options are an important hedging and speculation tool, and are available in many different forms. For example, options may be based on the difference between the prices of two different forms of oil (the crack spread), between the prices of oil and electricity (the spark spread), or between the prices of a single commodity at two different times (calendar spreads) or locations (locational spreads). In the following article by Carmona and Durrleman, section 2 is devoted to a detailed description of these and other spread options that are used in practice. The authors then summarize the classical Black--Scholes pricing paradigm for options based on a single commodity, and the multivariate extensions needed for the analysis of spread options. The underlying stochastic differential equations can be solved by various approaches, and several approximations are illustrated and compared. Particular difficulties associated with energy markets are discussed throughout and are a main focus of the latter sections. Financial mathematics is as exciting growth area for applied mathematics, and we hope this paper will introduce readers to some challenging problems of current interest in this field.

Analytic Approximations for Spread Options

This paper introduces parallel computation for spread options using two-dimensional Fourier transform. Spread options are multi-asset options whose payoffs depend on the difference of two underlying financial securities. Pricing these securities, however, cannot be done using closed-form methods; as such, we propose an algorithm which employs the fast Fourier Transform (FFT) method to numerically solve spread option prices in a reasonable amount of short time while preserving the pricing accuracy. Our results indicate a significant increase in computational performance when the algorithm is performed on multiple CPU cores and GPU. Moreover, the literature on spread option pricing using FFT methods documents that the pricing accuracy increases with FFT grid size while the computational speed has opposite effect. By using the multi-core/GPU implementation, the trade-off between pricing accuracy and speed is taken into account effectively.

Pricing Spread Options by the Operator Splitting Method

Multiple per-claim reinsurance based on maximizing the Lundberg exponent

Stochastic Correlation in Commodity Spread Options

Reverse pricing has been recognized as an effective tool to handle demand variability and uncertainty in the travel industry (e.g., airlines and hotels). To investigate its viability in mobile communication services, as a benchmark case, we first consider that a single mobile network operator (MNO) adopts (MNO-driven) forward pricing only, taking into account heterogeneous and stochastic user demands. To effectively deal with the drawbacks of forward pricing only, we propose (user-driven) two-dimensional reverse pricing on top of forward pricing and design a ξ-approximate polynomial-time algorithm that can maximize the revenue of the MNO. Through analytical and numerical results, we show that the proposed scheme can achieve solutions: Higher average network capacity utilization, the increase in the average revenue of the MNO, and the increment in the total average payoff of the users. To verify its feasibility in practice, we further implement its real prototype and perform experimental studies. We show that the proposed scheme still creates triple-win solutions in practice. Our findings provide a new outlook on resource allocation, and design guidelines for adopting two-dimensional reverse pricing on top of forward pricing.

The aims of this paper are twofold. Firstly, we present an approximating formula for pricing basket and multi-asset spread options, which genuinely extends Caldana and Fusai's (2013) two-asset spread options formula. Secondly, under the lognormal setting, we show that our formula becomes a Black and Scholes type formula, extending Bjerksund and Stensland's (2011). Numerical experiments and comparison with Monte Carlo simulations and other methods available in the literature are discussed. The main contribution of this paper is to provide practitioners with a pricing formula, which can be used for pricing basket and multi-asset spread options, even under a non-Gaussian framework.