General Payoff Research Articles

Perpetual American Options Under Lévy Processes

We consider perpetual American options, assuming that under a chosen equivalent martingale measure the stock returns follow a Lévy process. For put and call options, their analogues for more general payoffs, and a wide class of Lévy processes that contains Brownian motion, normal inverse Gaussian processes, hyperbolic processes, truncated Lévy processes, and their mixtures, we obtain formulas for the optimal exercise price and the fair price of the option in terms of the factors in the Wiener--Hopf factorization formula, i.e., in terms of the resolvents of the supremum and infimum processes, and derive explicit formulas for these factors. For calls, puts, and some other options, the results are valid for any Lévy process. We use Dynkin's formula and the Wiener--Hopf factorization to find the explicit formula for the price of the option for any candidate for the exercise boundary, and by using this explicit representation, we select the optimal solution. We show that in some cases the principle of the smooth fit fails and suggest a generalization of this principle.

Stochastic differential portfolio games

We study stochastic dynamic investment games in continuous time between two investors (players) who have available two different, but possibly correlated, investment opportunities. There is a single payoff function which depends on both investors’ wealth processes. One player chooses a dynamic portfolio strategy in order to maximize this expected payoff, while his opponent is simultaneously choosing a dynamic portfolio strategy so as to minimize the same quantity. This leads to a stochastic differential game with controlled drift and variance. For the most part, we consider games with payoffs that depend on the achievement of relative performance goals and/or shortfalls. We provide conditions under which a game with a general payoff function has an achievable value, and give an explicit representation for the value and resulting equilibrium portfolio strategies in that case. It is shown that non-perfect correlation is required to rule out trivial solutions. We then use this general result explicitly to solve a variety of specific games. For example, we solve a probability maximizing game, where each investor is trying to maximize the probability of beating the other's return by a given predetermined percentage. We also consider objectives related to the minimization or maximization of the expected time until one investor's return beats the other investor's return by a given percentage. Our results allow a new interpretation of the market price of risk in a Black-Scholes world. Games with discounting are also discussed, as are games of fixed duration related to utility maximization.

Stochastic differential portfolio games

We study stochastic dynamic investment games in continuous time between two investors (players) who have available two different, but possibly correlated, investment opportunities. There is a single payoff function which depends on both investors’ wealth processes. One player chooses a dynamic portfolio strategy in order to maximize this expected payoff, while his opponent is simultaneously choosing a dynamic portfolio strategy so as to minimize the same quantity. This leads to a stochastic differential game with controlled drift and variance. For the most part, we consider games with payoffs that depend on the achievement of relative performance goals and/or shortfalls. We provide conditions under which a game with a general payoff function has an achievable value, and give an explicit representation for the value and resulting equilibrium portfolio strategies in that case. It is shown that non-perfect correlation is required to rule out trivial solutions. We then use this general result explicitly to solve a variety of specific games. For example, we solve a probability maximizing game, where each investor is trying to maximize the probability of beating the other's return by a given predetermined percentage. We also consider objectives related to the minimization or maximization of the expected time until one investor's return beats the other investor's return by a given percentage. Our results allow a new interpretation of the market price of risk in a Black-Scholes world. Games with discounting are also discussed, as are games of fixed duration related to utility maximization.

Valuation of Investments in Real Assets with Implications for the Stock Prices

A general model for the valuation of natural resource investments is formulated and analyzed within a stochastic control theoretic framework. Using dynamic programming, the value of such an investment with a general payoff function is determined under the assumption that the commodity price process is given by a stochastic differential equation. The analysis results in closed form analytic solutions which can easily be computed and exhibits qualitatively different optimal behaviors, depending on parameter values. Implications for stocks and options are also considered.

Stochastic Differential Portfolio Games

We study stochastic dynamic investment games in continuous time between two investors (players) who have available two different, but possibly correlated, investment opportunities. There is a single payoff function which depends on both investors' wealth processes. One chooses a dynamic portfolio strategy in order to maximize this expected payoff while his opponent is simultaneously choosing a dynamic portfolio strategy so as to minimize the same quantity. This leads to a stochastic differential game with controlled drift and variance. For the most part, we consider games with payoffs that depend on the achievement of relative performance goals and/or shortfalls. We provide conditions under which a game with a general payoff function has an achievable value, and give an explicit representation for the value and resulting equilibrium portfolio strategies in that case. It is shown that nonperfect correlation is required to rule out trivial solutions. We then use this general result to explicitly solve a variety of specific games. For example, we solve a probability maximizing game, where each investor is trying to maximize the probability of beating the other's return by a given predetermined percentage. We also consider objectives related to the minimization or maximization of the expected time until one investor's return beats the other investor's return by a given percentage. Our results allow a new interpretation of the market price of risk in a Black-Scholes world. Games with discounting are also discussed as are games of fixed duration related to utility maximization.

Virtual implementation in incomplete information environments with infinite alternatives and types

This paper considers virtual implementation in both dominant strategy equilibrium and Bayesian strategy equilibrium for incomplete information environments with general payoff functions, any number of agents, and arbitrary sets of alternatives and types. We characterize virtual implementability in private-value models by giving necessary and sufficient conditions. We show that a social choice function is virtually implementable in Bayesian (resp. dominant strategy) equilibrium if and only if it is Bayesian (resp. dominant strategy) incentive compatible. We also show that incentive compatibility is the necessary and sufficient condition for a class of common-value economic environements where there are two or more alternatives that have no valuation externalities. In addition, our mechanism for transferable utility models satisfies budget balancing constraints.

Iterated games and arms race models

In an iterated game with averaging there is a difference between averaging in payoff space and averaging in strategy space, with the latter being more appropriate for an arms race model. We examine this difference in some detail for both general and specific nonlinear payoff functions.

The existence of good Markov strategies for decision processes with general payoffs

For countable-state decision processes (dynamic programming problems), a general class of objective functions is identified for which it is shown that good Markov strategies always exist. This class includes product and lim inf rewards, as well as practically all the classical dynamic programming expected payoff functions.

Remarks on a simple optimal control problem with monotone control functions

We consider the minimization of the mean-square deviation of a prescribed function from the class of monotone functions. Two problems are considered. The first problem places no restriction on the initial value of the controls, while the second problem assumes that all the control functions must start at a fixed initial value. Optimal controls are exhibited in both problems. Finally, we consider the situation with general payoff and dynamics and give the heuristic characterization of the value function for such problems.

The existence of value in differential games

Introduction Definition of the game $G$ The concept of value The value in the sense of Friedman Another formula for the Friedman upper and lower values The value in the sense of Fleming An estimate for $W^+-W^-$ Relaxed controls The existence of value under conditions (F1)-(F5) he existence of value without conditions (F1)-(F5) Games with general payoff Optimal strategies and saddle points Relationship with the work of Friedman Bibliography.

Some descriptive aspects of two-person non-zero-sum games

1 This research was supported in whole by the United States Air Force under Contract No. AF 49(638)-317 monitored by the Air Force Office of Scientific Research of the Air Research and Development Command. as s ported in whole by the It is to be noted that in G-type games a2 dominates a, and p2 dominates pi, and, hence, the outcome (a2, /82) may appear to be the rational joint choice. On the other hand, the outcome (al, /i) has a larger joint payoff than any other; further, since x1 > X4, the outcome (ai, /1) has a larger individual payoff than that associated with (a2, P/2). We regard (ac, P/) as the collaborative strategy, since only in this way may subjects maximize their return. Whether or not one may reasonably expect pairs of subjects to collaborate with one another is an open question. However, in an iterated version of such a game there are strong intuitive grounds for expecting the emergence of a collaborative strategy, especially when the number of trials becomes large.4 The specific game employed in the paper mentioned above is given by xl = 3, x2 = 0, xa = 5, and x4 = 1. Actually, three versions of this game, G1, G2, and G3, were used. The results reported in this paper have to do with three additional G-type games and with four non-G-type games. The non-Gtype games have the same general payoff