EXISTENCE OF AN EQUILIBRIUM WITH DISCONTINUOUS PRICES, ASYMMETRIC INFORMATION, AND NONTRIVIAL INITIAL σ-FIELDS

In this paper, we solve the problems of optimization and equilibrium on a continuous-time financial market with discontinuous prices, in which agents have different random endowments and different information on the structure and future behavior of the prices. Our purpose is to go over and to extend the work of Pikovsky and Karatzas (1996) by using the theory developed by Amendinger (2000) about martingale representation theorems for initially enlarged filtrations, and to generalize the results in the case of discontinuous prices.     

DUAL REPRESENTATIONS FOR GENERAL MULTIPLE STOPPING PROBLEMS

In this paper, we study the dual representation for generalized multiple stopping problems, hence the pricing problem of general multiple exercise options. We derive a dual representation which allows for cash flows which are subject to volume constraints modeled by integer‐valued adapted processes and refraction periods modeled by stopping times. As such, this extends the works by Schoenmakers ( 2012 ), Bender ( 2011a ), Bender ( 2011b ), Aleksandrov and Hambly ( 2010 ), and Meinshausen and Hambly ( 2004 ) on multiple exercise options, which either take into consideration a refraction period or volume constraints, but not both simultaneously. We also allow more flexible cash flow structures than the additive structure in the above references. For example, some exponential utility problems are covered by our setting. We supplement the theoretical results with an explicit Monte Carlo algorithm for constructing confidence intervals for prices of multiple exercise options and illustrate it with a numerical study on the pricing of a swing option in an electricity market.     

Robust martingale selection problem and its connections to the no‐arbitrage theory

We analyze the martingale selection problem of Rokhlin in a pointwise (robust) setting. We derive conditions for solvability of this problem and show how it is related to the classical no‐arbitrage deliberations. We obtain versions of the Fundamental Theorem of Asset Pricing in models spanning frictionless markets, models with proportional transaction costs, and models for illiquid markets. In all these models, we also incorporate trading constraints.     

A FIRST‐ORDER BSPDE FOR SWING OPTION PRICING

We study an optimal control problem related to swing option pricing in a general non‐Markovian setting in continuous time. As a main result we uniquely characterize the value process in terms of a first‐order nonlinear backward stochastic partial differential equation and a differential inclusion. Based on this result we also determine the set of optimal controls and derive a dual minimization problem.     

NO-FREE-LUNCH EQUIVALENCES FOR EXPONENTIAL LÉVY MODELS UNDER CONVEX CONSTRAINTS ON INVESTMENT

We provide equivalence of numerous no-free-lunch type conditions for financial markets where the asset prices are modeled as exponential Lévy processes, under possible convex constraints in the use of investment strategies. The general message is the following: if any kind of free lunch exists in these models it has to be of the most egregious type, generating an increasing wealth. Furthermore, we connect the previous to the existence of the numéraire portfolio , both for its particular expositional clarity in exponential Lévy models and as a first step in obtaining analogues of the no-free-lunch equivalences in general semimartingale models, a task that is taken on in Karatzas and Kardaras (2007) .     

A FIRST‐ORDER BSPDE FOR SWING OPTION PRICING: CLASSICAL SOLUTIONS

In a companion paper, we studied a control problem related to swing option pricing in a general non‐Markovian setting. The main result there shows that the value process of this control problem can uniquely be characterized in terms of a first‐order backward stochastic partial differential equation (BSPDE) and a pathwise differential inclusion. In this paper, we additionally assume that the cash flow process of the swing option is left‐continuous in expectation. Under this assumption, we show that the value process is continuously differentiable in the space variable that represents the volume in which the holder of the option can still exercise until maturity. This gives rise to an existence and uniqueness result for the corresponding BSPDE in a classical sense. We also explicitly represent the space derivative of the value process in terms of a nonstandard optimal stopping problem over a subset of predictable stopping times. This representation can be applied to derive a dual minimization problem in terms of martingales.     

Pricing Options On Risky Assets In A Stochastic Interest Rate Economy1

This paper studies contingent claim valuation of risky assets in a stochastic interest rate economy. the model employed generalizes the approach utilized by Heath, Jarrow, and Morton (1992) by imbedding their stochastic interest rate economy into one containing an arbitrary number of additional risky assets. We derive closed form formulae for certain types of European options in this context, notably call and put options on risky assets, forward contracts, and futures contracts. We also value American contingent claims whose payoffs are permitted to be general functions of both the term structure and asset prices generalizing Bensoussan (1984) and Karatzas (1988) in this regard. Here, we provide an example where an American call's value is well defined, yet there does not exist an optimal trading strategy which attains this value. Furthermore, this example is not pathological as it is a generalization of Roll's (1977) formula for a call option on a stock that pays discrete dividends.     

A Counterexample to Several Problems In the Theory of Asset Pricing

We construct a continuous bounded stochastic process ( S _t,) _1E[0,1] which admits an equivalent martingale measure but such that the minimal martingale measure in the sense of Föllmer and Schweizer does not exist. This example also answers (negatively) a problem posed by Karatzas, Lehozcky, and Shreve as well as a problem posed by Strieker.     

Maximum Entropy in Option Pricing: A Convex‐Spline Smoothing Method

Applying the principle of maximum entropy (PME) to infer an implied probability density from option prices is appealing from a theoretical standpoint because the resulting density will be the least prejudiced estimate, as “it will be maximally noncommittal with respect to missing or unknown information.” 1 Buchen and Kelly (1996) showed that, with a set of well‐spread simulated exact‐option prices, the maximum‐entropy distribution (MED) approximates a risk‐neutral distribution to a high degree of accuracy. However, when random noise is added to the simulated option prices, the MED poorly fits the exact distribution. Motivated by the characteristic that a call price is a convex function of the option's strike price, this study suggests a simple convex‐spline procedure to reduce the impact of noise on observed option prices before inferring the MED. Numerical examples show that the convex‐spline smoothing method yields satisfactory empirical results that are consistent with prior studies. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:819–832, 2001     

Error analysis of finite difference and Markov chain approximations for option pricing

Mijatovi? and Pistorius proposed an efficient Markov chain approximation method for pricing European and barrier options in general one‐dimensional Markovian models. However, sharp convergence rates of this method for realistic financial payoffs, which are nonsmooth, are rarely available. In this paper, we solve this problem for general one‐dimensional diffusion models, which play a fundamental role in financial applications. For such models, the Markov chain approximation method is equivalent to the method of lines using the central difference. Our analysis is based on the spectral representation of the exact solution and the approximate solution. By establishing the convergence rate for the eigenvalues and the eigenfunctions, we obtain sharp convergence rates for the transition density and the price of options with nonsmooth payoffs. In particular, we show that for call‐/put‐type payoffs, convergence is second order, while for digital‐type payoffs, convergence is generally only first order. Furthermore, we provide theoretical justification for two well‐known smoothing techniques that can restore second‐order convergence for digital‐type payoffs and explain oscillations observed in the convergence for options with nonsmooth payoffs. As an extension, we also establish sharp convergence rates for European options for a rich class of Markovian jump models constructed from diffusions via subordination. The theoretical estimates are confirmed using numerical examples.     

OPTIMAL CONSUMPTION AND PORTFOLIO SELECTION WITH INCOMPLETE MARKETS AND UPPER AND LOWER BOUND CONSTRAINTS

We study an optimal consumption and portfolio selection problem for an investor by a martingale approach. We assume that time is a discrete and finite horizon, the sample space is finite and the number of securities is smaller than that of the possible securities price vector transitions. the investor is prohibited from investing stocks more (less, respectively) than given upper (lower) bounds at any time, and he maximizes an expected time additive utility function for the consumption process. First we give a set of budget feasibility conditions so that a consumption process is attainable by an admissible portfolio process. Also we state the existence of the unique primal optimal solutions. Next we formulate a dual control problem and establish the duality between primal and dual control problems. Also we show the existence of dual optimal solutions. Finally we consider the computational aspect of dual approach through a simple numerical example.     

Merton's portfolio optimization problem in a Black and Scholes market with non-Gaussian stochastic volatility of Ornstein-Uhlenbeck type

We study Merton's classical portfolio optimization problem for an investor who can trade in a risk-free bond and a stock. The goal of the investor is to allocate money so that her expected utility from terminal wealth is maximized. The special feature of the problem studied in this paper is the inclusion of stochastic volatility in the dynamics of the risky asset. The model we use is driven by a superposition of non-Gaussian Ornstein-Uhlenbeck processes and it was recently proposed and intensively investigated for real market data by Barndorff-Nielsen and Shephard (2001) . Using the dynamic programming method, explicit trading strategies and expressions for the value function via Feynman-Kac formulas are derived and verified for power utilities. Some numerical examples are also presented.     

Dynamic Minimization of Worst Conditional Expectation of Shortfall

In a complete financial market model, the shortfall-risk minimization problem at the terminal date is treated for the seller of a derivative security F . The worst conditional expectation of the shortfall is adopted as the measure of this risk, ensuring that the minimized risk satisfies certain desirable properties as the dynamic measure of risk, as proposed by Cvitanić and Karatzas (1999) . The terminal value of the optimized portfolio is a binary functional dependent on F and the Radon-Nikodym density of the equivalent local martingale measure. In particular, it is observed that there exists a positive number x * that is less than the replicating cost x_F of F , and that the strategy minimizing the expectation of the shortfall is optimal if the hedger's capital is in the range [ x *, x_F ].     

Optimal insurance under rank‐dependent utility and incentive compatibility

Bernard, He, Yan, and Zhou (Mathematical Finance, 25(1), 154–186) studied an optimal insurance design problem where an individual's preference is of the rank‐dependent utility (RDU) type, and show that in general an optimal contract covers both large and small losses. However, their results suffer from the unrealistic assumption that the random loss has no atom, as well as a problem of moral hazard that provides incentives for the insured to falsely report the actual loss. This paper addresses these setbacks by removing the nonatomic assumption, and by exogenously imposing the “incentive compatibility” constraint that both indemnity function and insured's retention function are increasing with respect to the loss. We characterize the optimal solutions via calculus of variations, and then apply the result to obtain explicitly expressed contracts for problems with Yaari's dual criterion and general RDU. Finally, we use numerical examples to compare the results between ours and Bernard et al.     

On the Pricing of Contingent Claims with Frictions

This paper studies the problem of pricing contingent claims in a market which has frictions in the form of costs, such as penalty functions corresponding to constraints. An arbitrage-free interval is identified, and a fair price based upon utility functions is proposed. It provides a framework to study incomplete markets that is simplier than the one related to constraints on portfolios introduced by Karatzas and Kou.     

Robust risk aggregation with neural networks

We consider settings in which the distribution of a multivariate random variable is partly ambiguous. We assume the ambiguity lies on the level of the dependence structure, and that the marginal distributions are known. Furthermore, a current best guess for the distribution, called reference measure, is available. We work with the set of distributions that are both close to the given reference measure in a transportation distance (e.g., the Wasserstein distance), and additionally have the correct marginal structure. The goal is to find upper and lower bounds for integrals of interest with respect to distributions in this set. The described problem appears naturally in the context of risk aggregation. When aggregating different risks, the marginal distributions of these risks are known and the task is to quantify their joint effect on a given system. This is typically done by applying a meaningful risk measure to the sum of the individual risks. For this purpose, the stochastic interdependencies between the risks need to be specified. In practice, the models of this dependence structure are however subject to relatively high model ambiguity. The contribution of this paper is twofold: First, we derive a dual representation of the considered problem and prove that strong duality holds. Second, we propose a generally applicable and computationally feasible method, which relies on neural networks, in order to numerically solve the derived dual problem. The latter method is tested on a number of toy examples, before it is finally applied to perform robust risk aggregation in a real‐world instance.     

Monte Carlo Evaluation of Greeks for Multidimensional Barrier and Lookback Options

In this paper, we consider the problem of the numerical computation of Greeks for a multidimensional barrier and lookback style options: the payoff function depends in a rather general way on the minima and maxima of the coordinates of the d -dimensional underlying asset process. Using Malliavin calculus techniques, we derive additional weights that enable computation of the Greeks using Monte Carlo simulations. Numerical experiments confirm the efficiency of the method. This work is a multidimensional extension of previous results (see Gobet and Kohatsu-Higa 2001 ).     

Bayesian estimation of an open economy DSGE model with incomplete pass-through

In this paper, we develop a dynamic stochastic general equilibrium (DSGE) model for an open economy, and estimate it on Euro area data using Bayesian estimation techniques. The model incorporates several open economy features, as well as a number of nominal and real frictions that have proven to be important for the empirical fit of closed economy models. The paper offers: i) a theoretical development of the standard DSGE model into an open economy setting, ii) Bayesian estimation of the model, including assessments of the relative importance of various shocks and frictions for explaining the dynamic development of an open economy, and iii) an evaluation of the model's empirical properties using standard validation methods.     

Pricing Via Utility Maximization and Entropy

In a financial market model with constraints on the portfolios, define the price for a claim C as the smallest real number p such that sup_π E[U(X_T^{x+p, π}?C)]≥ sup_π E[U(X_T^{x, π})], where U is the negative exponential utility function and X^{x, π} is the wealth associated with portfolio π and initial value x. We give the relations of this price with minimal entropy or fair price in the flavor of Karatzas and Kou (1996) and superreplication. Using dynamical methods, we characterize the price equation, which is a quadratic Backward SDE, and describe the optimal wealth and portfolio. Further use of Backward SDE techniques allows for easy determination of the pricing function properties.     

Modeling and supporting task-oriented group processes: Purposeful complex adaptive systems and Evolutionary Systems Design

Evolutionary Systems Design (ESD) is a universal general problem solving, formal modeling, design framework for purposeful complex adaptive systems (PCAS) and processes, i.e., task-oriented group processes. These processes constitute policy making, group decision, negotiation, and multiagent problem solving with human and/or artificial agents. ESD is also a framework for computer group support systems (GSS) that support these processes. The ESD general framework can be applied to define and solve specific problems. In this article the ESD framework is presented and illustrated by example. The article provides background for ESD computer implementations discussed in two other related articles (Lewis and Shakun 1996; Bui and Shakun 1996).