In this paper, we study a game with positive or plus infinite expectation and determine the optimal proportion of investment for maximizing the limit expectation of growth rate per attempt. With this objective, we introduce a new pricing method in which the price is different from that obtained by the Black-Scholes formula for a European option.
This paper focuses on multi-stage coordination for a population of thermostatically controlled loads (TCL). Each load maximizes the individual utility in response to an energy price, while the coordinator determines the price to maximize the social welfare subject to a peak energy constraint. The coordination problem is formulated as a dynamic Stackelberg game. The Stackelberg solution is derived using an indirect approach: we first obtain an upper bound of the Stackelberg game, then we prove that this upper bound is attainable. Sufficient conditions for the optimal solution are characterized.…
The pricing, hedging, optimal exercise and optimal cancellation of game or Israeli options are considered in a multi-currency model with proportional transaction costs. Efficient constructions for optimal hedging, cancellation and exercise strategies are presented, together with numerical examples, as well as probabilistic dual representations for the bid and ask price of a game option.
Expanding the ideas of the author's paper 'Nonexpansive maps and option pricing theory' (Kibernetica 34:6 (1998), 713-724) we develop a pure game-theoretic approach to option pricing, by-passing stochastic modeling. Risk neutral probabilities emerge automatically from the robust control evaluation. This approach seems to be especially appealing for incomplete markets encompassing extensive, so to say untamed, randomness, when the coexistence of infinite number of risk neutral measures precludes one from unified pricing of derivative securities. Our method is robust enough to be able to accommo…
This paper is devoted to a high-dimensional mixed leadership stochastic differential game on a finite horizon in feedback information mode, where the control variables enter into the diffusion term of state equation. A verification theorem for the feedback Stackelberg-Nash equilibrium is obtained by using a system of coupled and fully nonlinear parabolic partial differential equations. We apply the verification theorem to deal with a dynamic innovation and pricing decision problem where the buyer acts as the leader in the pricing decisions and the dynamic model is stochastic. Via the solutions…
We develop an option pricing model based on a tug-of-war game. This two-player zero-sum stochastic differential game is formulated in the context of a multi-dimensional financial market. The issuer and the holder try to manipulate asset price processes in order to minimize and maximize the expected discounted reward. We prove that the game has a value and that the value function is the unique viscosity solution to a terminal value problem for a parabolic partial differential equation involving the non-linear and completely degenerate infinity Laplace operator.
Today's queueing network systems are more rapidly evolving and more complex than those of even a few years ago. The goal of this paper is to study customers' behavior in an unobservable Markovian M/M/1 queue where consumers have to choose between two strategic decisions about information acquisition before joining or not the queue. According to their decision, customers decide to give up the service (balk the system) or to join the queue. We study the Nash equilibrium strategies: we compute the equilibrium and we prove its existence and uniqueness. Based on this result, we consider the problem…
We analyse a non-cooperative strategic game among two ride-hailing platforms, each of which is modeled as a two-sided queueing system, where drivers (with a certain patience level) are assumed to arrive according to a Poisson process at a fixed rate, while the arrival process of passengers is split across the two providers based on QoS considerations. We also consider two monopolistic scenarios: (i) each platform has half the market share, and (ii) the platforms merge into a single entity, serving the entire passenger base using their combined driver resources. The key novelty of our formulati…
This paper investigates design of noncooperative games from an optimization and control theoretic perspective. Pricing mechanisms are used as a design tool to ensure that the Nash equilibrium of a fairly general class of noncooperative games satisfies certain global objectives such as welfare maximization or achieving a certain level of quality-of-service (QoS). The class of games considered provide a theoretical basis for decentralized resource allocation and control problems including network congestion control, wireless uplink power control, and optical power control. The game design proble…
We consider a basic resource allocation game, where the players' strategy spaces are subsets of $R^m$ and cost/utility functions are parameterized by some common vector $u\in R^m$ and, otherwise, only depend on the own strategy choice. A strategy of a player can be interpreted as a vector of resource consumption and a joint strategy profile naturally leads to an aggregate consumption vector. Resources can be priced, that is, the game is augmented by a price vector $λ\in R^m_+$ and players have quasi-linear overall costs/utilities meaning that in addition to the original costs/utilities, a play…
Solar Renewable Energy Certificate (SREC) markets are a market-based system that incentivizes solar energy generation. A regulatory body imposes a lower bound on the amount of energy each regulated firm must generate via solar means, providing them with a tradeable certificate for each MWh generated. Firms seek to navigate the market optimally by modulating their SREC generation and trading rates. As such, the SREC market can be viewed as a stochastic game, where agents interact through the SREC price. We study this stochastic game by solving the mean-field game (MFG) limit with sub-population…
In 2002, Benjamin Jourdain and Claude Martini discovered that for a class of payoff functions, the pricing problem for American options can be reduced to pricing of European options for an appropriately associated payoff, all within a Black-Scholes framework. This discovery has been investigated in great detail by Sören Christensen, Jan Kallsen and Matthias Lenga in a recent work in 2020. In the present work we prove that this phenomenon can be observed in a wider context, and even holds true in a setup of non-linear stochastic processes. We analyse this problem from both probabilistic and ana…
A multi-player competitive Dynkin stopping game is constructed. Each player can either exit the game for a fixed payoff, determined a priori, or stay and receive an adjusted payoff depending on the decision of other players. The single period case is shown to be "weakly unilaterally competitive". We present an explicit construction of the unique value at which Nash and optimal equilibria are attained. Multiple period generalisations are explored. The game has interpretations in economic and financial contexts, for example, as a consumption model with bounded resources. It also serves as a star…
Many efficient algorithms have been designed to recover Nash equilibria of various classes of finite games. Special classes of continuous games with infinite strategy spaces, such as polynomial games, can be solved by semidefinite programming. In general, however, continuous games are not directly amenable to computational procedures. In this contribution, we develop an iterative strategy generation technique for finding a Nash equilibrium in a whole class of continuous two-person zero-sum games with compact strategy sets. The procedure, which is called the double oracle algorithm, has been su…
