Pricing Algorithm Research Articles

Option pricing with the control variate technique beyond Monte Carlo simulation

Although mostly used alongside Monte Carlo simulation, the control-variate (CV) technique can be applied to other numerical algorithms in option pricing. This paper studies the conditions under which a numerical method (simulation-based or not) can benefit from the CV technique and what approximators can serve as CVs. We demonstrate the ideas with Carr and Madan’s Fourier transform-based algorithm, convolution-based pricing algorithms, and classic binomial trees. Numerical results are provided to show that the CV-enhanced versions are more efficient than the original algorithms.

Wirkung der Energiesteuersenkung auf Kraftstoff preise

AbstractThe German federal government intended to alleviate the burden of increasing fuel prices by introducing a temporary reduction of energy taxes on gasoline and diesel. In order to evaluate the impact of this measure on consumer prices at the filling stations the development of procurement costs for crude oil as well as the downstream development of refinery and distribution margins have to be taken into account. It turns out that about 80 % of the tax reduction has been passed on to end consumers on and around the effective date of the tax relief. However, within the first month the impact of the tax reduction has been wiped out for diesel completely as the gross margin of the mineral oil groups have substantially improved since then. On the other hand, for gasoline (E10) at least part of the impact can still be observed as the initial margin improvement has come down in the meantime. For a detailed analysis the German antitrust authority should look into the pricing algorithms of all 14,000 filling stations in Germany.

Dynamic air ticket pricing using reinforcement learning method

This paper studies a dynamic air ticket pricing problem in a strategic and myopic passengers co-existence market. The strategic or myopic passengers can be further divided into high-valuation and low-valuation groups according to how they evaluate their purchases. The strategic passengers have different strategic levels. When the airline sets a ticket price, every passenger makes his or her purchase decision according to his or her type and the strategic level, or might select “wait” or “leave (the market)”. The paper firstly proposes a dynamic pricing algorithm in which the utilities of both the airline and passengers are considered. The reinforcement learning (RL) is employed to deal with the progressive or dynamic decision-making framework, in which the dynamic pricing problem is formulated as a discrete finite Markov decision process (MDP) and the Q-learning is adopted to solve the problem. By using this method, the airline can adaptively decide the ticket price based on passengers strategic behaviors and the time-varying demand. The effects of the passenger type proportion and strategic level are analyzed. The computational results show the higher proportion of strategic passengers is, the smaller price increase the airline can adopt, and the higher proportion of high-valuation strategic passengers is, the larger price increase the airline can put to use under the same strategic level. If the proportion of low-valuation strategic passengers is higher, the price increase should be gentle and step by step when the price increase strategy is adopted. If the airline uses price-cut policy, the adjustment should be small. In addition, the high-valuation passenger mainly affects high-price periods and the low-valuation passenger mainly affects low-price periods. When the proportion of strategic passengers is fixed, the lower the passenger strategic level is, the larger the price slope is. These findings can provide some references for the airline to make more precise and flexible pricing decisions.

Mergers, Acquisitions and Merger Control in an Algorithmic Pricing World

Abstract This paper considers whether pricing algorithms present novel issues that existing merger control frameworks and practices are inadequate to address, particularly in relation to pre-merger disclosure of pricing algorithms and the suitability of current tests and remedies for addressing coordinated effects. Through a comparative and critical analysis of the merger control regimes and practices of the European Union, United Kingdom, and Australia, this paper finds that whilst such regimes and practices are broadly adequate for dealing with algorithmic transactions, there are nonetheless potential areas for improvement. Disclosure of a pricing algorithm may contravene prohibitions on the sharing of competitively sensitive information. As such, merger parties may need to rethink aspects of their usual due diligence procedures. Pricing algorithms may also increase the potential for coordinated effects to arise in some markets that would ordinarily have been considered too complex, asymmetric, opaque, or insufficiently concentrated for tacit coordination to occur, or be used to retaliate more effectively against deviations from a coordinated equilibrium, or to raise the height of barriers to entry. Competition authorities may therefore need to amend their standard approach to investigating and assessing coordinated effects, as well as their traditional approach to remedies.

Distributed asynchronous column generation

We propose a revision of the classical column generation algorithm for solving Dantzig–Wolfe decompositions of mixed integer programs. It is meant to fully exploit the availability of distributed computing resources, making optimization algorithms in general purpose solvers to scale better.The main idea is to trigger massive parallelism by fully decoupling the computing flow of each component, including the resolution of the master problem, thus allowing different pricing algorithms to concurrently work on different sets of dual variables, and the master algorithm to asynchronously update dual information as soon as new columns are available.Our algorithms ensure the same optimality convergency properties of the classical method. Experiments on mixed integer programs for three benchmark problems from the combinatorial optimization literature prove our approach to be one order of magnitude faster than state-of-the-art general purpose solvers in computing high quality root node dual bounds. Even if devised to exploit clusters of machines which do not share memory space, our algorithms show to be faster than earlier attempts from the literature also when run on virtual machines hosted on a single physical one, proving this improvement to derive from our algorithmic methodology rather than technological factors.

Optimal deployment of virtual network functions for securing telecommunication networks against distributed denial of service attacks: A robust optimization approach

Distributed Denial of Service (DDoS) cyberattacks represent a major security risk for network operators and internet service providers. They thus need to invest in security solutions to protect their network against DDoS attacks. The present work focuses on deploying a network function virtualization based architecture to secure a network against an on-going DDoS attack. We assume that the target, sources and volume of the attack have been identified. However, due to 5G network slicing, the exact routing of the illegitimate flow in the network is not known by the internet service provider. We seek to determine the optimal number and locations of virtual network functions in order to remove all the illegitimate traffic while minimizing the total cost of the activated virtual network functions. We propose a robust optimization framework to solve this problem. The uncertain input parameters correspond to the amount of illegitimate flow on each path connecting an attack source to the target and can take values within a predefined uncertainty set. In order to solve this robust optimization problem, we develop an adversarial approach in which the adversarial sub-problem is solved by a Branch & Price algorithm. The results of our computational experiments, carried out on medium-size randomly generated instances, show that the proposed solution approach is able to provide optimal solutions within short computation times.

Efficient and Robust Combinatorial Option Pricing Algorithms on the Trinomial Lattice for Polynomial and Barrier Options

Options can be priced by the lattice model, the results of which converge to the theoretical option value as the lattice’s number of time steps n approaches infinity. The time complexity of a common dynamic programming pricing approach on the lattice is slow (at least O n 2 ), and a large n is required to obtain accurate option values. Although O n -time combinatorial pricing algorithms have been developed for the classical binomial lattice, significantly oscillating convergence behavior makes them impractical. The flexibility of trinomial lattices can be leveraged to reduce the oscillation, but there are as yet no linear-time algorithms on trinomial lattices. We develop O n -time combinatorial pricing algorithms for polynomial options that cannot be analytically priced. The commonly traded plain vanilla and power options are degenerated cases of polynomial options. Barrier options that cannot be stably priced by the binomial lattice can be stably priced by our O n -time algorithm on a trinomial lattice. Numerical experiments demonstrate the efficiency and accuracy of our O n -time trinomial lattice algorithms.

The burgeoning role of iBuyers in the housing market

AbstractWe examine the iBuyers’ business model and their impact on housing markets. We find that iBuyers tend to enter neighborhoods that have more easily priced and homogeneous homes, as price discovery is simpler and more consistent with their pricing algorithm in those areas. iBuyers purchase homes at lower prices than individual owner‐occupiers, and this acquisition discount reflects the benefits iBuyers offer to motivated sellers rather than distressed home purchases or unobserved lower‐quality housing characteristics. Last, a greater presence of iBuyers results in a higher volume of local housing transactions and encourages more home sellers to sell without listing.

Identifying Algorithmic Pricing Technology Adoption in Retail Gasoline Markets

While recent theoretical literature shows that algorithmic pricing (AP) may increase retail prices by facilitating collusive behavior, there has been no empirical evidence relating the adoption of AP technology to higher prices and/or collusion. One key reason is that information about firms' pricing technologies and their adoption of new pricing software is rarely available. In this paper, we use detailed price data from the German retail gasoline market, where AP was supposedly introduced in 2017. We show how to identify changes in pricing technology using structural breaks in pricing behaviors associated with AP.

A Branch and Price Algorithm for the Heterogeneous Fleet Multi-Depot Multi-Trip Vehicle Routing Problem with Time Windows

The multi-trip vehicle routing problem (MTVRP) extends the well-known VRP by allowing vehicles to perform several trips in a workday. The motivation arises from the new challenges in city logistics that push companies to use smaller and cleaner vehicles such as cargo bikes. With the integration of small vehicles into the fleet, many companies start to operate with a heterogeneous fleet and use multiple depots located in the city centers to reload the small vehicles. Inspired by these new challenges the companies face, we study the heterogeneous fleet multi-depot MTVRP with time windows under shared depot resources where small and large vehicles have different travel times in certain areas. We formulate this problem using workday variables and propose a branch and price algorithm that exhibits an enhanced performance by a new heuristic algorithm based on the reduction in the graph size. The proposed algorithm introduces a new way to compute the completion bounds using the iterative structure of the state-space augmenting algorithm and eliminates the need for solving a separate relaxation. We conduct experiments on modified small- and medium-size instances from Solomon’s benchmark set. The results of our computational experiments show that the proposed algorithm is very effective and can solve instances with up to 40 customers, three depots, and two types of vehicles.

Pricing cancellable American put options on the finite time horizon

AbstractThe purpose of this paper is to present a numerical approach for pricing cancellable American put options, also known as game or Israeli options, on the finite time horizon. These options generalize the concept of American derivatives adding an early exercise right for the option's writer to the existing holder's right. The writer has to pay a penalty amount above the usual option payment to use this right. We first obtain the shape of the optimal regions for both participants. Then we approximate the optimal exercise boundaries maximizing the option's writer and holder financial expectations using some first exit properties of the Brownian motion. We also construct an efficient pricing algorithm based on these boundaries. A semiclosed form formula is derived when the underlying asset starts above the strike.

Artificial intelligence and consumer manipulations: from consumer's counter algorithms to firm's self-regulation tools

The growing use of artificial intelligence (A.I.) algorithms in businesses raises regulators' concerns about consumer protection. While pricing and recommendation algorithms have undeniable consumer-friendly effects, they can also be detrimental to them through, for instance, the implementation of dark patterns. These correspond to algorithms aiming to alter consumers' freedom of choice or manipulate their decisions. While the latter is hardly new, A.I. offers significant possibilities for enhancing them, altering consumers' freedom of choice and manipulating their decisions. Consumer protection comes up against several pitfalls. Sanctioning manipulation is even more difficult because the damage may be diffuse and not easy to detect. Symmetrically, both ex-ante regulation and requirements for algorithmic transparency may be insufficient, if not counterproductive. On the one hand, possible solutions can be found in counter-algorithms that consumers can use. On the other hand, in the development of a compliance logic and, more particularly, in tools that allow companies to self-assess the risks induced by their algorithms. Such an approach echoes the one developed in corporate social and environmental responsibility. This contribution shows how self-regulatory and compliance schemes used in these areas can inspire regulatory schemes for addressing the ethical risks of restricting and manipulating consumer choice.

A Branch-and-Price-and-Cut Algorithm for the Vehicle Routing Problem with Two-Dimensional Loading Constraints

The vehicle routing problem with two-dimensional loading constraints (2L-CVRP) is a practical variant of the classic capacitated vehicle routing problem. A number of algorithms have been developed for the problem, but it is very difficult for the existing exact methods to optimally solve instances featuring with large rectangular items. To address this issue, a branch-and-price-and-cut (BPC) algorithm is proposed in this study. A novel data structure and a new dominance rule are developed to build an exact pricing algorithm that takes the loading constraints into account. Several valid inequalities are used to strengthen the linear relaxation. Extensive computational experiments were conducted on the benchmark instances of the 2L-CVRP, showing that the BPC algorithm outperforms all the existing exact methods for the problem in terms of the solution quality. Fourteen instances are solved to optimality for the first time. In particular, the size of solvable instances with large items is nearly doubled. Moreover, managerial insights about the impact of respecting the last-in-first-out constraint are also obtained.

Demand Learning and Pricing for Varying Assortments

Problem definition: We consider the problem of demand learning and pricing for retailers who offer assortments of substitutable products that change frequently, for example, due to limited inventory, perishable or time-sensitive products, or the retailer’s desire to frequently offer new styles. Academic/practical relevance: We are one of the first to consider the demand learning and pricing problem for retailers who offer product assortments that change frequently, and we propose and implement a learn-then-earn algorithm for use in this setting. Our algorithm prioritizes a short learning phase, an important practical characteristic that is only considered by few other algorithms. Methodology: We develop a novel demand learning and pricing algorithm that learns quickly in an environment with varying assortments and limited price changes by adapting the commonly used marketing technique of conjoint analysis to our setting. We partner with Zenrez, an e-commerce company that partners with fitness studios to sell excess capacity of fitness classes, to implement our algorithm in a controlled field experiment to evaluate its effectiveness in practice using the synthetic control method. Results: Relative to a control group, our algorithm led to an expected initial dip in revenue during the learning phase, followed by a sustained and significant increase in average daily revenue of 14%–18% throughout the earning phase, illustrating that our algorithmic contributions can make a significant impact in practice. Managerial implications: The theoretical benefit of demand learning and pricing algorithms is well understood—they allow retailers to optimally match supply and demand in the face of uncertain preseason demand. However, most existing demand learning and pricing algorithms require substantial sales volume and the ability to change prices frequently for each product. Our work provides retailers who do not have this luxury a powerful demand learning and pricing algorithm that has been proven in practice. History: This paper has been accepted as part of the 2021 Manufacturing & Service Operations Management Practice-Based Research Competition. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.1080 .

Exact solution of network flow models with strong relaxations

We address the solution of Mixed Integer Linear Programming (MILP) models with strong relaxations that are derived from Dantzig–Wolfe decompositions and allow a pseudo-polynomial pricing algorithm. We exploit their network-flow characterization and provide a framework based on column generation, reduced-cost variable-fixing, and a highly asymmetric branching scheme that allows us to take advantage of the potential of the current MILP solvers. We apply our framework to a variety of cutting and packing problems from the literature. The efficiency of the framework is proved by extensive computational experiments, in which a significant number of open instances could be solved to proven optimality for the first time.

Surge Pricing and Short-Term Wage Elasticity of Labor Supply in Real-Time Ridesharing Markets

The prominence of real-time ridesharing services, such as Uber and Lyft, has dramatically changed the landscape of traditional industries. This study provides a comprehensive analysis of the short-term wage elasticity of labor supply in real-time ridesharing markets using data from a major ridesharing platform in China. By exploiting an exogenous shock from uneven driving restrictions as an instrumental variable, we find a negative labor supply elasticity for ridesharing drivers, suggesting that drivers tend to drive less during days with a higher average hourly wage. Specifically, a percent increase in hourly wage will lead to a 0.931 percent decrease in daily working hours. This surprising finding is consistent with the behavioral income-targeting model based on the theory of reference-dependent preferences: Drivers have heuristic daily targets for total earnings and are more motivated to supply labor when they are below their income target than when they are above it. Therefore, they work less on days when earnings per hour are high and quit the market once their income target is reached. In addition, we find that taxi drivers are more rational and have positive labor supply elasticity, which implies that drivers are more rational when they have repeated opportunities for learning. Estimating labor supply elasticity is critical to understanding the economic efficiency of various surge pricing algorithms and driver subsidization programs for ridesharing platforms and policymakers. Our research suggests that a uniform price surging or driver subsidization approach for all ridesharing drivers may not incentivize the labor supply of drivers effectively.

Learning to Collude in a Pricing Duopoly

Problem definition: This paper addresses the question whether or not self-learning algorithms can learn to collude instead of compete against each other, without violating existing competition law. Academic/practical relevance: This question is practically relevant (and hotly debated) for competition regulators, and academically relevant in the area of analysis of multi-agent data-driven algorithms. Methodology: We construct a price algorithm based on simultaneous-perturbation Kiefer–Wolfowitz recursions. We derive theoretical bounds on its limiting behavior of prices and revenues, in the case that both sellers in a duopoly independently use the algorithm, and in the case that one seller uses the algorithm and the other seller sets prices competitively. Results: We mathematically prove that, if implemented independently by two price-setting firms in a duopoly, prices will converge to those that maximize the firms’ joint revenue in case this is profitable for both firms, and to a competitive equilibrium otherwise. We prove this latter convergence result under the assumption that the firms use a misspecified monopolist demand model, thereby providing evidence for the so-called market-response hypothesis that both firms’ pricing as a monopolist may result in convergence to a competitive equilibrium. If the competitor is not willing to collaborate but prices according to a strategy from a certain class of strategies, we prove that the prices generated by our algorithm converge to a best-response to the competitor’s limit price. Managerial implications: Our algorithm can learn to collude under self-play while simultaneously learn to price competitively against a ‘regular’ competitor, in a setting where the price-demand relation is unknown and within the boundaries of competition law. This demonstrates that algorithmic collusion is a genuine threat in realistic market scenarios. Moreover, our work exemplifies how algorithms can be explicitly designed to learn to collude, and demonstrates that algorithmic collusion is facilitated (a) by the empirically observed practice of (explicitly or implicitly) sharing demand information, and (b) by allowing different firms in a market to use the same price algorithm. These are important and concrete insights for lawmakers and competition policy professionals struggling with how to respond to algorithmic collusion.

Bermudan option pricing by quantum amplitude estimation and Chebyshev interpolation

Pricing of financial derivatives, in particular early exercisable options such as Bermudan options, is an important but heavy numerical task in financial institutions, and its speed-up will provide a large business impact. Recently, applications of quantum computing to financial problems have been started to be investigated. In this paper, we first propose a quantum algorithm for Bermudan option pricing. This method performs the approximation of the continuation value, which is a crucial part of Bermudan option pricing, by Chebyshev interpolation, using the values at interpolation nodes estimated by quantum amplitude estimation. In this method, the number of calls to the oracle to generate underlying asset price paths scales as widetilde{O}(epsilon ^{-1}), where ϵ is the error tolerance of the option price. This means the quadratic speed-up compared with classical Monte Carlo-based methods such as least-squares Monte Carlo, in which the oracle call number is widetilde{O}(epsilon ^{-2}).

The Effect of Outsourcing Pricing Algorithms on Market Competition

A third party developer designs and sells a pricing algorithm that enhances a firm’s ability to tailor prices to a source of demand variation, whether high-frequency demand shocks or market segmentation. The equilibrium pricing algorithm is characterized that maximizes the third party’s profit given firms’ optimal adoption decisions. Outsourcing the pricing algorithm does not reduce competition but does make prices more sensitive to the demand variation, and this is shown to decrease consumer welfare and increase industry profit. This effect is larger when products are more substitutable. This paper was accepted by Joshua Gans, business strategy.

Multiproduct Pricing with Discrete Price Sets

Dynamic pricing is a well-known revenue management technique used by firms to maximize their revenues. In industries such as fashion retail and airlines, it is observed in practice that firms vary the prices of their products, through time, only among certain pre-fixed discrete price points; for example, in fashion retail, products are first sold at a base price and then later at, say, 10%, 20%, 30% discounted prices. In such discrete-price practices, when the number of products offered by a firm is large, it is mathematically challenging to determine the optimal pricing decisions. In “Multiproduct Pricing with Discrete Price Sets,” Manchiraju, Dawande, and Janakiraman design pricing algorithms that are fast and result in near-optimal revenues.