Quadratic Payoff Research Articles

Quadratic American Strangle Options in Light of Two-Sided Optimal Stopping Problems

The aim of this paper is to examine some American-style financial instruments that lead to two-sided optimal hitting problems. We pay particular attention to derivatives that are similar to strangle options but have a quadratic payoff function. We consider these derivatives in light of much more general payoff structures under certain conditions which guarantee that the optimal strategy is an exit from a strip. Closed-form formulas for the optimal boundaries and the fair price are derived when the contract has no maturity constraints. We obtain the form of the optimal boundaries under the finite maturity horizon and approximate them by maximizing the financial utility of the derivative holder. The Crank–Nicolson finite difference method is applied to the pricing problem. The importance of these novel financial instruments is supported by several features that are very useful for financial practice. They combine the characteristics of the power options and the ordinary American straddles. Quadratic strangles are suitable for investors who need to hedge strongly, far from the strike positions. In contrast, the near-the-money positions offer a relatively lower payoff than the ordinary straddles. Note that the usual options pay exactly the overprice; no more, no less. In addition, the quadratic strangles allow investors to hedge the positions below and above the strike together. This is very useful in periods of high volatility when large market movements are expected but their direction is unknown.

Maximizing Social Welfare and Agreement via Information Design in Linear-Quadratic-Gaussian Games

Information design in an incomplete information game involves a designer that aims to influence players' actions through signals generated from a designed probability distribution to optimize its objective function. For quadratic design objective functions, if the players have quadratic payoffs that depend on the players' actions and an unknown payoff-relevant state, and signals on the state that follow a Gaussian distribution conditional on the state realization, the information design problem is a semidefinite program (SDP) [1]. In this note, we seek to characterize the optimal information design analytically by leveraging the SDP formulation, when the design objective is to maximize social welfare or the agreement among players' action. We show that full information disclosure maximizes social welfare when there is a common payoff state, the payoff dependencies among players' actions are homogeneous, or when the signals are public. When the objective is to maximize the agreement among players' actions, not revealing any information is optimal. When the objective is a weighted combination of social welfare and agreement terms, we establish a threshold weight below which full information disclosure is optimal under public signals for games with homogeneous payoffs. Numerical results corroborate the analytical results, and identify partial information disclosure structures that are optimal.

Linear Riley equilibria in quadratic signaling games

We study signaling games with quadratic payoffs. As signaling games admit multiple separating equilibria, many equilibrium selection rules are proposed and a well-known solution is Riley equilibria. They are separating equilibria in which the sender achieves the highest equilibrium payoff for all types among all separating equilibria. We analyze the conditions for Riley equilibria to be linear, a common assumption in many applications. We derive a sufficient and necessary condition for the existence and uniqueness of linear Riley equilibria. We apply the result to confirm the dominance of linear equilibria in some classic examples, and we show that, in some other examples, there exist previously unknown nonlinear Riley equilibria.

Duality theory for exponential utility-based hedging in the Almgren–Chriss model

AbstractIn this paper we obtain a duality result for the exponential utility maximization problem where trading is subject to quadratic transaction costs and the investor is required to liquidate her position at the maturity date. As an application of the duality, we treat utility-based hedging in the Bachelier model. For European contingent claims with a quadratic payoff, we compute the optimal trading strategy explicitly.

WEAK ERROR RATES FOR OPTION PRICING UNDER LINEAR ROUGH VOLATILITY

In quantitative finance, modeling the volatility structure of underlying assets is vital to pricing options. Rough stochastic volatility models, such as the rough Bergomi model [C. Bayer, P. K. Friz & J. Gatheral (2016) Pricing under rough volatility, Quantitative Finance 16 (6), 887–904, doi:10.1080/14697688.2015.1099717], seek to fit observed market data based on the observation that the log-realized variance behaves like a fractional Brownian motion with small Hurst parameter, [Formula: see text], over reasonable timescales. Both time series of asset prices and option-derived price data indicate that [Formula: see text] often takes values close to [Formula: see text] or less, i.e. rougher than Brownian motion. This change improves the fit to both option prices and time series of underlying asset prices while maintaining parsimoniousness. However, the non-Markovian nature of the driving fractional Brownian motion in rough volatility models poses severe challenges for theoretical and numerical analyses and for computational practice. While the explicit Euler method is known to converge to the solution of the rough Bergomi and similar models, its strong rate of convergence is only [Formula: see text]. We prove rate [Formula: see text] for the weak convergence of the Euler method for the rough Stein–Stein model, which treats the volatility as a linear function of the driving fractional Brownian motion, and, surprisingly, we prove rate one for the case of quadratic payoff functions. Indeed, the problem of weak convergence for rough volatility models is very subtle; we provide examples demonstrating the rate of convergence for payoff functions that are well approximated by second-order polynomials, as weighted by the law of the fractional Brownian motion, may be hard to distinguish from rate one empirically. Our proof uses Talay–Tubaro expansions and an affine Markovian representation of the underlying and is further supported by numerical experiments. These convergence results provide a first step toward deriving weak rates for the rough Bergomi model, which treats the volatility as a nonlinear function of the driving fractional Brownian motion.

Acquisition, Aggregation, and Sharing of Information in Sequential-Move Aggregative Games

We study quadratic payoff common-value sequential-move aggregative games with signaling. Applications include Stackelberg quantity-setting, public good contribution, common resource, gas emission, and manufacturer-retailer games. Whereas efficient aggregation of information occurs in large Stackelberg markets, it does not occur in the four remaining large markets. If all agents share their information with their followers, expected welfare increases in the Stackelberg and supply chain games and it decreases in the remaining three games. Finally, we characterize the conditions for payoffs and welfare to increase with more precise private or public information. We also provide robustness checks when the values are private. (JEL C70, D43, D82, D83, H41, Q21, Q54)

Optimal attention management: A tractable framework

A well-intentioned principal provides information to a rationally inattentive agent without internalizing the agent's cost of processing information. Whatever information the principal makes available, the agent may choose to ignore some. We study optimal information provision in a tractable model with quadratic payoffs where full disclosure is not optimal. We characterize incentive-compatible information policies, that is, those to which the agent willingly pays full attention. In a leading example with three states, optimal disclosure involves information distortion at intermediate costs of attention. As the cost increases, optimal information changes from downplaying the state to exaggerating the state.

The art of brevity

We analyze a class of sender-receiver games with quadratic payoffs, which includes the communication games in Alonso et al. (2008) and Rantakari (2008) as special cases, for which the sender’s or the receiver’s maximum expected payoff when players have access to arbitrary, mediated communication protocols is attained in one round of face-to-face, unmediated cheap talk. This result is based on the existence for these games of a communication equilibrium with an infinite number of partitions of the state space. We provide explicit expressions for the maximum expected payoff of the sender and the receiver, and illustrate its use by deriving new comparative statics of the quality of optimal communication. For instance, a shift in the underlying uncertainty that reduces expected conflict can worsen the quality of communication.

Information Preferences of Individual Agents in Linear-Quadratic-Gaussian Network Games

We consider linear-quadratic-Gaussian (LQG) network games in which agents have quadratic payoffs that depend on their individual and neighbors’ actions, and an unknown payoff-relevant state. An information designer determines the fidelity of information revealed to the agents about the payoff state to maximize the social welfare. Prior results show that full information disclosure is optimal under certain assumptions on the payoffs, i.e., it is beneficial for the average individual. In this paper, we provide conditions for general network structures based on the strength of the dependence of payoffs on neighbors’ actions, i.e., competition, under which a rational agent is expected to benefit, i.e., receive higher payoffs, from full information disclosure. We find that all agents benefit from information disclosure for the star network structure when the game is homogeneous. We also identify that the central agent benefits more than a peripheral agent from full information disclosure unless the competition is strong and the number of peripheral agents is small enough. Despite the fact that all agents expect to benefit from information disclosure ex-ante, a central agent can be worse-off from information disclosure in many realizations of the payoff state under strong competition, indicating that a risk-averse central agent can prefer uninformative signals ex-ante.

Stability Analysis of Nash Equilibrium for Two-Agent Loss-Aversion-Based Noncooperative Switched Systems

The stability property of the loss-aversion-based noncooperative switched systems with quadratic payoffs is investigated. In this system, each agent adopts the lower sensitivity parameter in the myopic pseudo-gradient dynamics for the case of losing utility than gaining utility, and both systems' dynamics and switching events (conditions) are depending on agents' payoff functions. Sufficient conditions under which agents' state converges towards the Nash equilibrium are derived in accordance with the location of the Nash equilibrium. In the analysis, the mode transition sequence and interesting phenomena which we call flash switching instants are characterized. Finally, we present several numerical examples to illustrate the properties of our results.

Aggregation of Information in Large Stackelberg Markets

We study quadratic payoff common-value sequential-move aggregative games with signaling. Applications include Stackelberg quantity-setting, public good contribution, common resource, gas emission, and manufacturer-retailer games. Whereas efficient aggregation of information occurs in large Stackelberg markets, it does not occur in the four remaining large markets. If all agents share their information with their followers, expected welfare increases in the Stackelberg and supply chain games and it decreases in the remaining three games. Whereas the manufacturer always has incentives to share its private information with its retailers, agents have incentives to share their private information with their follower(s) if the number of agents is two or the signals' precision is sufficiently low in the four remaining games. Finally, we characterize the conditions for payoffs and welfare to increase with more precise private or public information. More precise public information always reduces welfare in the supply chain game. Morever, the expected welfare decreases with the precision of the public signal at some intermediate (high) values of the precision of the follower's signal in the Stackelberg game (the remaining three games) when the number of agents is two. We also provide robustness checks when the values are private.

Nash Equilibrium Seeking in Quadratic Noncooperative Games Under Two Delayed Information-Sharing Schemes

In this paper, we propose non-model-based strategies for locally stable convergence to Nash equilibrium in quadratic noncooperative games where acquisition of information (of two different types) incurs delays. Two sets of results are introduced: (a) one, which we call cooperative scenario, where each player employs the knowledge of the functional form of his payoff and knowledge of other players’ actions, but with delays; and (b) the second one, which we term the noncooperative scenario, where the players have access only to their own payoff values, again with delay. Both approaches are based on the extremum seeking perspective, which has previously been reported for real-time optimization problems by exploring sinusoidal excitation signals to estimate the Gradient (first derivative) and Hessian (second derivative) of unknown quadratic functions. In order to compensate distinct delays in the inputs of the players, we have employed predictor feedback. We apply a small-gain analysis as well as averaging theory in infinite dimensions, due to the infinite-dimensional state of the time delays, in order to obtain local convergence results for the unknown quadratic payoffs to a small neighborhood of the Nash equilibrium. We quantify the size of these residual sets and corroborate the theoretical results numerically on an example of a two-player game with delays.

Dynamics of Strategy Distributions in a One-Dimensional Continuous Trait Space for Games with a Quadratic Payoff Function

Evolution of distribution of strategies in game theory is an interesting question that has been studied only for specific cases. Here I develop a general method to extend analysis of the evolution of continuous strategy distributions given a quadratic payoff function for any initial distribution in order to answer the following question—given the initial distribution of strategies in a game, how will it evolve over time? I look at several specific examples, including normal distribution on the entire line, normal truncated distribution, as well as exponential and uniform distributions. I show that in the case of a negative quadratic term of the payoff function, regardless of the initial distribution, the current distribution of strategies becomes normal, full or truncated, and it tends to a distribution concentrated in a single point so that the limit state of the population is monomorphic. In the case of a positive quadratic term, the limit state of the population may be dimorphic. The developed method can now be applied to a broad class of questions pertaining to evolution of strategies in games with different payoff functions and different initial distributions.

Ergodic Control-Coding Capacity of Stochastic Control Systems: Information Signalling and Hierarchical Optimality of Gaussian Systems

The control-coding (CC) capacity of dynamical decision models (DMs) is defined as the maximum amount of information transfer per unit time from its inputs to its outputs, called CC rate $R$ in bits/second, which is operational with the aid of a controller-encoder and a decoder, as in Shannon's mathematical theory of communication over noisy channels, with the encoder replaced by a controller-encoder [C. Kourtellaris and C. D. Charalambous, IEEE Trans. Inform. Theory, 64 (2018), pp. 4962--4992]. In the first part of the paper, data processing inequalities and information structures of optimal controllers-encoders are derived for controlled and uncontrolled information processes. Further, the ergodic theory of a Markov decision is applied to establish direct and converse CC theorems. In the second part of the paper, the problem of signalling information via a Gaussian decision model (G-DM) subject to a quadratic cost constraint to another Gaussian control system (G-CS) with a quadratic pay-off is investigated. A hierarchical decomposition and decentralized optimality of linear quadruple of strategies, \controller-encoder, decoder, controller\, is shown by construction as follows: (i) the controller-encoder simultaneously controls the output of the G-DM and encodes the state of the G-CS, and operates at the CC capacity of the G-DM; (ii) the decoder is optimal with respect to a mean square error (MSE) criterion; and (iii) the controller of the G-CS which acts on the decoder's information is optimal.

Optimal Attention Management: A Tractable Framework

A well-intentioned principal provides information to a rationally inattentive agent without internalizing the agent’s cost of processing information. Whatever information the principal makes available, the agent may choose to ignore some. We study optimal information provision in a tractable model with quadratic payoffs where full disclosure is not optimal. We characterize incentive-compatible information policies, that is, those to which the agent willingly pays full attention. In a leading example with three states, optimal disclosure involves information distortion at intermediate costs of attention. As the cost increases, optimal information abruptly changes from downplaying the state to exaggerating the state.

Optimizing the strategy of activities using numerical methods for determining equilibrium

The paper considers issues on the theoretical substantiation of options for choosing an optimal strategy to integrate an agricultural enterprise into the wholesale market by using methodological tools of the non-cooperative game theory. We have proposed modeling the behavior of an agrarian enterprise in the market by achieving a Nash equilibrium under various scenarios of competitors’ activities and volumes of information on market conditions. The methodology has been substantiated to apply the iterative algorithms to calculate equilibria in a general class of non-quadratic convex polyhedra in order to form the methodologies and construct algorithms for a behavior of agricultural enterprises in market activity. It was determined that decision-making occurs in parallel to the real conditions of activity of an agricultural enterprise in the wholesale market. The comprehensive application of numerical methods based on solving the optimization problems provides a smooth approach to the Nash equilibrium. A game can have multiple isolated Nash equilibria if players have non-quadratic payment functions when solving such problems. Based on the above, the results were determined of local convergence, since global results have strong constraints in non-quadratic problems. However, there is a connection with semi-global practical asymptotic stability if players have quadratic payoff functions. It has been shown that there is a shift in the convergence in proportion to the amplitudes of disturbance signals and the third derivative of payoff functions for non-quadratic payoff functions. This shift in the convergence corresponds to the shift in a numerical example. It has been determined that the learning strategy developed in accordance with the main provisions of the theory of games remains attractive if one has partial information on the state of the market. Application of the indicated action strategy provides a company with a possibility to improve its initial position by measuring its own payoff values only and not using estimates of potentially uncertain parameters. It has been proposed to use applied tools from the game theory to determine an optimal action strategy for an agricultural enterprise for its integration into the wholesale market of vegetable products

Some partially observed multi-agent linear exponential quadratic stochastic differential games

Some multi-agent stochastic differential games described by a stochastic linear system driven by a Brownian motion and having an exponential quadratic payoff for the agents are formulated and solved. The agents have either complete observations or partial observations of the system state. The agents act independently of one another and the explicit optimal feedback control strategies form a Nash equilibrium. In the partially observed problem the observations are the same for all agents which occurs in broadcast situations. The optimal control strategies and optimal payoffs are given explicitly. The method of solution for both problems does not require solving either Hamilton-Jacobi-Isaacs equations or backward stochastic differential equations.

Variance Premium, Downside Risk, and Expected Stock Returns

We decompose the quadratic payoff on a stock into its loss and gain components and measure the premia associated with their fluctuations, called the loss and gain quadratic risk premium (QRP) respectively. The loss QRP interprets as the premium paid for downside risk hedging, while the gain QRP reads as the premium received for upside risk compensation. Long-short portfolio strategies based on the loss or gain QRP yield monthly risk-adjusted expected excess returns of up to 2.8%. This cross-sectional predictability survives a battery of robustness checks, and is reinforced among stocks experiencing limits to arbitrage, information asymmetry, and demand for lottery.

Linear Stochastic Differential Games with State Dependent Fractional Brownian Motion

A two person, zero-sum, noncooperative stochastic differential game is formulated and solved for a multidimensional linear stochastic system that has a quadratic payoff and a linear state dependent scalar fractional Brownian motion noise process in the stochastic system. The strategies are restricted to be linear feedback. A Riccati equation is given that provides the optimal control strategies for the two players using a direct method and this Riccati equation is different from the one for the corresponding problem with an additive Brownian motion. The strategies are shown to form a Nash equilibrium. The method presented and applied here for the solution requires only some elementary notions from a stochastic calculus for a fractional Brownian motion.

Two-Stage Game Theoretic Modelling of Airline Frequency and Fare Competition

Airlines make capacity and fare decisions in a competitive environment. Capacity decisions, encompassing decisions about frequency of service and seats-per-flight, affect both the operating costs and revenues of airlines. These decisions have significant implications for the performance of the air transportation system as a whole. Capacity and fare decisions of different airlines are interdependent, both serving as tools in an airlines competitive arsenal. This interdependency motivates a game theoretic approach to modeling the decision process. Capacity (especially frequency) decisions are typically made months in advance of flight departure, with only an approximate knowledge of what fares will be, while fare decisions are made weeks to minutes ahead of flight departure. Several studies have stressed the need to develop two-stage game theoretic models to account for the sequential nature of these decisions, but there are very few analytical, computational, or empirical results available for such models. In this article (working paper available at link in [1]), we develop a two-stage frequency and fare competition model, demonstrate its tractability across a wide range of assumptions, and validate its predictions against observed airline behavior. We take the payoff function of an airline operating in a set of markets to be the sum of the differences between revenues and costs in those markets, with costs as a linear function of flight frequency. To compute revenue, we explore two commonly used multinomial logit models of market share. Frequency decisions are made in the first stage of the game, while fare decisions are made in the second stage. We begin our analysis with a simplified version of this game: two airlines competing in a single market, with no connecting passengers, infinite seating capacity, and the absence of a nofly alternative. Under these assumptions, for either market share model, we are able to prove that (1) the second-stage fare game always has unique pure strategy Nash equilibrium (PSNE), (2) first-stage payoffs for each airline are concave with respect to that airline's frequency strategy across plausible utility parameter ranges, and (3) first-stage payoffs for each airline are submodular functions in the overall frequency strategy space. As the game is two-player, (3) means that by changing the sign of one player's strategy space, we can trivially convert the game into a supermodular game. These results demonstrate that subgame-perfect PSNE is a credible and tractable solution concept for our model. In particular, the existence and uniqueness results indicate the suitability of PSNE as a solution concept for the second-stage game. Concave payoffs ensure that individual first-stage payoff maximization problems are efficiently solvable, and supermodularity ensures that several iterative learning dynamics converge to equilibrium [2]. We then relax each of the assumptions made in this simplified model by computationally solving the second stage fare game, generating equilibrium fare decisions and profits for every set of frequency decisions for integer daily frequency values ranging from 1 to 20, for various numbers of players, seats per flight, and values of utility parameters (including the nofly option). Then, we fit quadratic approximations to these profits as functions of the frequencies of all players. We find an excellent fit (R2 > 0.9) in all cases. Additionally, the signs of all estimated coefficients are consistent with submodularity and concavity properties demonstrated earlier. We show that for an N-player game with such concave and submodular quadratic payoff functions, the myopic best response heuristic, where each player optimizes its payoff against fixed opponent strategies iteratively, converges to a PSNE. To test the tractability and predictive validity of our model in practice, we apply it to a 4-airline, 11-airport network in the Western U.S, using publicly available airline operations data. We use the quadratic functions of airline frequency fitted above and additionally enforce the aircraft availability constraints, and solve for equilibrium iteratively using the myopic best response heuristic. To calibrate the 11 free quadratic payoff coefficients of this model, we use a stochastic gradient approximation algorithm to minimize the absolute errors between observed and predicted frequency strategies. In practice, the game convergences to equilibrium quickly, and the calibrated model's frequency predictions approximate observed behavior both in-sample and out-ofsample, suggesting that refinements of the model could be pursued for use in scenario analysis, forecasting, planning, and policy making.