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.     

THE MULTIVARIATE supOU STOCHASTIC VOLATILITY MODEL

Using positive semidefinite supOU (superposition of Ornstein–Uhlenbeck type) processes to describe the volatility, we introduce a multivariate stochastic volatility model for financial data which is capable of modeling long range dependence effects. The finiteness of moments and the second‐order structure of the volatility, the log‐ returns, as well as their “squares” are discussed in detail. Moreover, we give several examples in which long memory effects occur and study how the model as well as the simple Ornstein–Uhlenbeck type stochastic volatility model behave under linear transformations. In particular, the models are shown to be preserved under invertible linear transformations. Finally, we discuss how (sup)OU stochastic volatility models can be combined with a factor modeling approach.     

Branding in the era of digital (dis)intermediation

Marketing academics are keenly aware of the seismic shifts in today's marketing environment caused by digital (dis)intermediation. In this article, we discuss four types of digital (dis)intermediation, and how they affect branding activities of incumbents and new firms. First, we discuss digital transaction intermediation, a development that is closely tied to the rise of ecommerce retailers. A second type is digital transaction disintermediation associated with the rise of ecommerce D2C models. These first two types of digital (dis) intermediation are primarily top-down processes, where firms are developing new ways to sell their brands to consumers. The next two types of digital (dis) intermediation are of the bottom-up kind — the consumer is in the driver's seat. Digital marketing intermediation and the rise of D2C brand-building models is powered by crowdsourcing. A final development is digital marketing disintermediation, which is closely tied to the rise of C2C models. We present issues in need of future research for each type of digital (dis)intermediation. We conclude with an appeal that marketing takes the lead in developing overarching, indigenous theories of digital (dis)intermediation to make sense of the rapid changes in the marketplace.     

On the Existence of Finite-Dimensional Realizations for Nonlinear Forward Rate Models

We consider interest rate models of the Heath–Jarrow–Morton type, where the forward rates are driven by a multidimensional Wiener process, and where the volatility is allowed to be an arbitrary smooth functional of the present forward rate curve. Using ideas from differential geometry as well as from systems and control theory, we investigate when the forward rate process can be realized by a finite-dimensional Markovian state space model, and we give general necessary and sufficient conditions, in terms of the volatility structure, for the existence of a finite-dimensional realization. A number of concrete applications are given, and all previously known realization results (as far as existence is concerned) for Wiener driven models are included and extended. As a special case we give a general and easily applicable necessary and sufficient condition for when the induced short rate is a Markov process. In particular we give a short proof of a result by Jeffrey showing that the only forward rate models with short rate dependent volatility structures which generically possess a short rate realization are the affine ones. These models are thus the only generic short rate models from a forward rate point of view.     

The information content of implied volatility in agricultural commodity markets

In this article we compare the incremental information content of lagged implied volatility to GARCH models of conditional volatility for a collection of agricultural commodities traded on the New York Board of Trade. We also assess the relevance of the additional information provided by the implied volatility in a risk management framework. It is first shown that past squared returns only marginally improve the information content provided by the lagged implied volatility. Secondly, value‐at‐risk (VaR) models that rely exclusively on lagged implied volatility perform as well as VaR models where the conditional variance is modelled according to GARCH type processes. These results indicate that the implied volatility for options on futures contracts in agricultural commodity markets provides relevant volatility information that can be used as an input to VaR models. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:441–454, 2003     

A practitioner’s guide to Bayesian estimation of discrete choice dynamic programming models

This paper provides a step-by-step guide to estimating infinite horizon discrete choice dynamic programming (DDP) models using a new Bayesian estimation algorithm (Imai et al., Econometrica 77:1865?C1899, 2009a) (IJC). In the conventional nested fixed point algorithm, most of the information obtained in the past iterations remains unused in the current iteration. In contrast, the IJC algorithm extensively uses the computational results obtained from the past iterations to help solve the DDP model at the current iterated parameter values. Consequently, it has the potential to significantly alleviate the computational burden of estimating DDP models. To illustrate this new estimation method, we use a simple dynamic store choice model where stores offer ??frequent-buyer?? type rewards programs. Our Monte Carlo results demonstrate that the IJC method is able to recover the true parameter values of this model quite precisely. We also show that the IJC method could reduce the estimation time significantly when estimating DDP models with unobserved heterogeneity, especially when the discount factor is close to 1.     

Homogeneity and heterogeneity in stochastic models of brand choice behavior

Heterogeneity of consumers is one of the cornerstones of empirical findings and theories in marketing. It serves, for example, as the foundation for such areas as market segmentation and product differentiation. This paper attempts to trace and clarify the evolution over the last twenty years of the homogeneity assumptions in the area of stochastic models of brand choice behavior. In analyzing individual choice behavior by means of stochastic models, all individuals were often assumed to possess the same set of transition probabilities or follow the same stochastic process. However, empirical studies at the individual level indicate that individuals are actually non-homogeneous in those probabilities and processes. In this article we provide an analytical proof that if the behavior of individuals is specified to be homogeneous when it is not, wrong inferences about the type of stochastic process individuals follow and about the expected behavior of the total population will be drawn. Ways to remedy these problems by allowing for heterogeneity are reviewed. The implications of heterogeneity and our findings in the various application areas which utilize stochastic choice models are examined.     

Asymptotics of the price oscillations of a European call option in a tree model

It is well known that the price of a European vanilla option computed in a binomial tree model converges toward the Black-Scholes price when the time step tends to zero. Moreover, it has been observed that this convergence is of order 1/ n in usual models and that it is oscillatory. In this paper, we compute this oscillatory behavior using asymptotics of Laplace integrals, giving explicitly the first terms of the asymptotics. This allows us to show that there is no asymptotic expansion in the usual sense, but that the rate of convergence is indeed of order 1/ n in the case of usual binomial models since the second term (in     ) vanishes. The next term is of type   C ₂( n )/ n   , with   C ₂( n )  some explicit bounded function of n that has no limit when n tends to infinity.     

The new open economy macroeconomics of government debt

In this paper we introduce an overlapping generations structure of the Blanchard (J. Polit. Econ. 93 (1985) 121) type in a New Open Economy Macroeconomics (NOEM) model. This allows us to study a wider range of fiscal shocks compared to the traditional Mundell-Fleming (MF) and to the baseline Redux models. We show that a debt-financed tax cut appreciates the short-run exchange rate, but this result is reversed in the long run. A debt-financed increase of government spending, on the other hand, has ambiguous exchange rate effects. Our model also provides a bridge between the NOEM framework and the MF model.     

On American VIX options under the generalized 3/2 and 1/2 models

In this paper, we extend the 3/2 model for VIX studied by Goard and Mazur and introduce the generalized 3/2 and 1/2 classes of volatility processes. Under these models, we study the pricing of European and American VIX options, and for the latter, we obtain an early exercise premium representation using a free‐boundary approach and local time‐space calculus. The optimal exercise boundary for the volatility is obtained as the unique solution to an integral equation of Volterra type. We also consider a model mixing these two classes and formulate the corresponding optimal stopping problem in terms of the observed factor process. The price of an American VIX call is then represented by an early exercise premium formula. We show the existence of a pair of optimal exercise boundaries for the factor process and characterize them as the unique solution to a system of integral equations.     

A dynamic approach to the analysis of strategic alliances

The increasing trend in strategic alliance formation between major firms around the world, has prompted researchers from various disciplines to look at this phenomenon in great detail. In this paper, we review alternate approaches in the literature in this area. We then propose a non-linear dynamic approach to study the formation of competitive strategic alliances and contrast it with the traditional game-theoretic approach. The pros and cons of these two approaches are discussed with reference to a competitive alliance scenario. Dynamic models have significant managerial implications as they enable us to investigate ‘if-then’ type scenarios and project the impact of different strategies.     

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 study of factors affecting intersection crash frequencies using random-parameter multivariate zero-inflated models

Recent research demonstrates the appropriateness of multivariate regression models in crash count modelling when one specific type of crash counts needs to be analysed, since they can better handle the correlated issues in multiple crash counts. In this paper, a random-parameter multivariate zero-inflated Poisson (RMZIP) regression model is proposed as an alternative multivariate methodology for jointly modelling crash counts simultaneously. Using this RMZIP model, we are able to account for the heterogeneity due to the unobserved roadway geometric design features and traffic characteristics. Our formulation also has the merit of handling excess zeros in correlated crash counts, a phenomenon that is commonly found in practice. The Bayesian method is employed to estimate the model parameters. We use the proposed modelling framework to predict crash frequencies at urban signalized intersections in Tennessee. To investigate the model performances, three models – a fixed-parameter MZIP model, a random-parameter multivariate negative binomial (RMNB) model, and a random-parameter multivariate zero-inflated negative binomial (RMZINB) model – have been employed as the comparison methods. The comparison results show that the proposed RMZIP models provide a satisfied statistical fit with more variables producing statistically significant parameters. In other word, the RMZIP models have the potential to provide a fuller understanding of how the factors affect crash frequencies on specific roadway intersections. A variety of variables are found to significantly influence the crash frequencies by varying magnitudes. These variables result in random parameters and thereby their effects on crash frequencies are found to vary significantly across the sampled intersections.     

SEPARABLE TERM STRUCTURES AND THE MAXIMAL DEGREE PROBLEM

This paper discusses separablc term structure diffusion models in an arbitrage-free environment. Using general consistency results we exploit the interplay between the diffusion coefficients and the functions determining the forward curve. We introduce the particular class of polynomial term structure models. We formulate the appropriate conditions under which the diffusion for a quadratic term structure model is necessarily an Ornstein-Uhlenbeck type process. Finally, we explore the maximal degree problem and show that basically any consistent polynomial term structure model is of degree two or less.     

TRACTABLE ROBUST EXPECTED UTILITY AND RISK MODELS FOR PORTFOLIO OPTIMIZATION

Expected utility models in portfolio optimization are based on the assumption of complete knowledge of the distribution of random returns. In this paper, we relax this assumption to the knowledge of only the mean, covariance, and support information. No additional restrictions on the type of distribution such as normality is made. The investor’s utility is modeled as a piecewise‐linear concave function. We derive exact and approximate optimal trading strategies for a robust (maximin) expected utility model, where the investor maximizes his worst‐case expected utility over a set of ambiguous distributions. The optimal portfolios are identified using a tractable conic programming approach. Extensions of the model to capture asymmetry using partitioned statistics information and box‐type uncertainty in the mean and covariance matrix are provided. Using the optimized certainty equivalent framework, we provide connections of our results with robust or ambiguous convex risk measures, in which the investor minimizes his worst‐case risk under distributional ambiguity. New closed‐form results for the worst‐case optimized certainty equivalent risk measures and optimal portfolios are provided for two‐ and three‐piece utility functions. For more complicated utility functions, computational experiments indicate that such robust approaches can provide good trading strategies in financial markets.     

Co-evolution of global sourcing: The need to understand the underlying mechanisms of firm-decisions to offshore

In this introductory paper, we first discuss the emergence of global sourcing of business services and how these have been largely ignored in the IB field. Offshoring of business services has reached substantial proportions. Despite the radical growth, IB research on global sourcing is still in its infancy. Offshoring of business services represents a new type of internationalization. The offshoring of high-value services confronts companies with many of the challenges that are typical of an internationalization process in which well-known models and concepts of internationalization are applicable but also leave important unanswered questions requiring reconsideration and revision of these theoretical positions. Offshoring of business services is fundamentally different from outsourcing offshore of manufacturing activities. Using data from the Offshore Research Network (ORN), we track patterns of the emergence and diffusion of global sourcing of business services. On the basis of these new insights, we make a plea for a more encompassing, co-evolutionary perspective of global sourcing stressing the interactions between managerial intentionality, path-dependent experience and knowledge accumulation, as well as the institutional and selection forces. In particular, we develop a co-evolutionary offshore decision model integrating managerial intentionality, knowledge/experience and institutional and selection forces that explain the heterogeneous outcomes of offshoring. Although emergent outcomes of co-evolutionary dynamics are highly idiosyncratic, we identify in this paper some underlying mechanism that drive specific global sourcing patterns. Finally, we position the papers of this special issue in this co-evolutionary model.     

Distributed Decision Making: A Proposal of Support Through Cooperative Systems

In this paper we show that decision making processes evolve to be more and more distributed and asynchronous. In order to support decision-makers who are not at the same place at the same time we define cooperation processes and a set of models models able to support designers of Cooperative decision support framework. These models come from a more general architecture of a Cooperative Knowledge Based System and are based on the knowledge acquisition field. A definition of cooperative decision support framework is also given.     

Super‐replication in fully incomplete markets

In this work, we introduce the notion of fully incomplete markets. We prove that for these markets, the super‐replication price coincides with the model‐free super‐replication price. Namely, the knowledge of the model does not reduce the super‐replication price. We provide two families of fully incomplete models: stochastic volatility models and rough volatility models. Moreover, we give several computational examples. Our approach is purely probabilistic.     

INFINITE HORIZON INCOMPLETE MARKETS WITH A CONTINUUM OF STATES

In this paper we address existence of equilibria in an incomplete markets economy with countably many periods and a continuum of states at each node of the infinite tree. We consider two models: one where agents have to honor their commitments and another where default is allowed. In both models, marginal utility of income, at each node, is shown to be bounded, and we prove existence by taking finite-dimensional approximations and applying Fatou's lemma sequentially.     

Innovative and sustainable business models in the fashion industry: Entrepreneurial drivers,opportunities, and challenges

New and existing companies are looking for ways to thrive in a competitive environment with innovative business models while respecting society and avoiding actions that harm the planet. Trends such as circular economy, fair trade, lowsumerism, and sharing economy are some of the many emerging entrepreneurial approaches that address this issue, but there is still a gap between what theory argues and the levels of environmental and social sustainability realized when theory is put into practice. In fact, most research on the topic of sustainable business models is still exploratory and does not fully acknowledge these emerging approaches, whose definitions, boundaries, and defining characteristics are still somewhat vague. This study seeks to contribute to the understanding of the inner entrepreneurial dynamics of innovative sustainable business models. In particular, we focus on the fashion business, a resource-intensive industry in which opportunities to reduce environmental impacts and to innovate business models abound. The aim of our research is to investigate innovative business models in the fashion industry that have sustainability as their defining characteristic, especially in terms of value proposition. In order to do that, we combine a systematic review of the literature with empirical research comprised of six interviews with specialists in sustainability, business model innovation, and the fashion industry, along with eight case studies on innovative fashion startups we define as ‘born sustainable.’ As a result, we propose a synthesizing framework that discloses trends and drivers of innovative and sustainable business models in the fashion industry. We also highlight opportunities and challenges for researchers and entrepreneurs interested in this topic.