Smooth convergence in the binomial model

In this article, we consider a general class of binomial models with an additional parameter λ. We show that in the case of a European call option the binomial price converges to the Black–Scholes price at the rate 1/n and, more importantly, give a formula for the coefficient of 1/n in the expansion of the error. This enables us, by making special choices for λ, to prove that convergence is smooth in Tian’s flexible binomial model and also in a new center binomial model which we propose. Ken Palmer was supported by NSC grant 93-2118-M-002-002.     

Multivariate binomial approximations for asset prices with nonstationary variance and covariance characteristics

In this article, we suggest an efficient method of approximatinga general, multivariate log-normal distribution by a multivariatebinomial process. There are two important features of such multivariatedistributions. First, the state variables may have volatilitiesthat change over time. Second, the two or more relevant statevariables involve may covary with each other in a specifiedmanner, with a time-varying covariance structure. We discussthe asymptotic properties of the resulting processes and showhow the methodology can be used to value a complex, multipleexerciseable option whose payoff depends on the prices of twoassets.     

Optimal consumption and investment for markets with random coefficients

We consider an optimal investment and consumption problem for a Black–Scholes financial market with stochastic coefficients driven by a diffusion process. We assume that an agent makes consumption and investment decisions based on CRRA utility functions. The dynamic programming approach leads to an investigation of the Hamilton–Jacobi–Bellman (HJB) equation which is a highly nonlinear partial differential equation (PDE) of the second order. By using the Feynman–Kac representation, we prove uniqueness and smoothness of the solution. Moreover, we study the optimal convergence rate of iterative numerical schemes for both the value function and the optimal portfolio. We show that in this case, the optimal convergence rate is super-geometric, i.e., more rapid than any geometric one. We apply our results to a stochastic volatility financial market.     

Integration of emerging stock markets with global stock markets

Financial integration for emerging economies should be seen as a long-term objective. In this paper, we examine stock market integration among five selected emerging stock markets (Brazil, China, Mexico, Russia and Turkey) and developed markets of the US, UK and Germany. The bounds testing approach to cointegration and error-correction modeling are used on monthly data from January 2001 to December 2014 to determine the short-run and long-run relationship between emerging stock market returns and the returns of the developed stock markets. The results show evidence of the existence of short-run integration among stock markets in emerging countries and the developed markets. However, the long-run coefficients for stock market returns in all emerging countries show a significant relationship only with Germany stock market return. The empirical findings in this study have important implications for academicians, international investors, and policymakers in emerging markets.     

Duality theory for robust utility maximisation

In this paper, we present a duality theory for the robust utility maximisation problem in continuous time for utility functions defined on the positive real line. Our results are inspired by – and can be seen as the robust analogues of – the seminal work of Kramkov and Schachermayer (Ann. Appl. Probab. 9:904–950, 1999). Namely, we show that if the set of attainable trading outcomes and the set of pricing measures satisfy a bipolar relation, then the utility maximisation problem is in duality with a conjugate problem. We further discuss the existence of optimal trading strategies. In particular, our general results include the case of logarithmic and power utility, and they apply to drift and volatility uncertainty.

     

Moody's correlated binomial default distributions for inhomogeneous portfolios

This paper generalizes Moody's correlated binomial default distribution for homogeneous (exchangeable) credit portfolios, which was introduced by Witt, to the case of inhomogeneous portfolios. We consider two cases of inhomogeneous portfolios. In the first case, we treat a portfolio whose assets have uniform default correlation and non-uniform default probabilities. We obtain the default probability distribution and study the effect of inhomogeneity. The second case corresponds to a portfolio with inhomogeneous default correlation. Assets are categorized into several different sectors and the inter-sector and intra-sector correlations are not the same. We construct the joint default probabilities and obtain the default probability distribution. We show that as the number of assets in each sector decreases, inter-sector correlation becomes more important than intra-sector correlation. We study the maximum values of the inter-sector default correlation. Our generalization method can be applied to any correlated binomial default distribution model that has explicit relations to the conditional default probabilities or conditional default correlations, e.g. Credit Risk⁺, implied default distributions. We also compare some popular CDO pricing models from the viewpoint of the range of the implied tranche correlation.     

Large sample validity of the binomial distribution for lives with unequal mortality rates

Abstract

Let us consider a group of n lives which are observed during some time or age interval. Suppose that the following conditions are satisfied: 1. The probability of death within the interval considered has the same value q for each person of the group.

2. These lives represent statistically independent observations (with respect to mortality).

     

Duality for pathwise superhedging in continuous time

Finance and Stochastics - We provide a model-free pricing–hedging duality in continuous time. For a frictionless market consisting of $d$  risky assets with continuous price...     

No-arbitrage criteria for financial markets with transaction costs and incomplete information

This note deals with criteria of absence of arbitrage opportunities for an investor acting in a market with frictions and having a limited access to the information flow. We develop a mathematical scheme covering major models of financial markets with transaction costs and prove several results including a criterion for the robust no-arbitrage property and a hedging theorem.      

A binomial approximation for two-state Markovian HJM models

This article develops a lattice algorithm for pricing interest rate derivatives under the Heath et al. (Econometrica 60:77–105, 1992) paradigm when the volatility structure of forward rates obeys the Ritchken and Sankarasubramanian (Math Financ 5:55–72) condition. In such a framework, the entire term structure of the interest rate may be represented using a two-dimensional Markov process, where one state variable is the spot rate and the other is an accrued variance statistic. Unlike in the usual approach based on the Nelson-Ramaswamy (Rev Financ Stud 3:393–430) transformation, we directly discretize the heteroskedastic spot rate process by a recombining binomial tree. Further, we reduce the computational cost of the pricing problem by associating with each node of the lattice a fixed number of accrued variance values computed on a subset of paths reaching that node. A backward induction scheme coupled with linear interpolation is used to evaluate interest rate contingent claims.     

An adjusted binomial model for pricing Asian options

We propose a model for pricing both European and American Asian options based on the arithmetic average of the underlying asset prices. Our approach relies on a binomial tree describing the underlying asset evolution. At each node of the tree we associate a set of representative averages chosen among all the effective averages realized at that node. Then, we use backward recursion and linear interpolation to compute the option price.     

Prediction markets for foresight

To navigate turbulent business environments, organizations have to develop foresight capacities that enable them to anticipate probable futures, respond rapidly to emerging changes, and support future oriented action. However, there are remaining barriers that impede a wider implementation of foresight. In particular, the necessities to deal with the future, anticipate change, enhance participation and reduce costs and complexity call for new methods to improve current foresight activities. In this paper, we introduce prediction markets to the field of foresight. Prediction markets are a structured approach to collect and aggregate information from groups and have recently gained attention in forecasting. Prediction markets go beyond simple forecasting and can contribute to foresight by providing advantages in terms of continuous and real-time information aggregation, motivation of participation and information revelation as well as cost-efficiency and scalability. We suggest four promising fields of application for prediction markets to enhance foresight: (1) continuous forecasting and environmental scanning, (2) combining with deliberative approaches, (3) continuous idea generation and (4) expert identification. We conclude by considering prediction markets as a nascent and promising method for foresight and advocate for further research.     

On option pricing in binomial market with transaction costs

Option replication is studied in a discrete-time framework with proportional transaction costs. The model represents an extension of the Cox-Ross-Rubinstein binomial option-pricing model to cover the case of proportional transaction costs for one risky asset with different interest rates on bank credit and deposit. Contingent claims are supposed to be 2-dimensional random variables. Explicit formulas for self-financing strategies are obtained for this case.Received: March 2004, Mathematics Subject Classification (2000): 62P05JEL Classification: G11, G13The authors are grateful to an anonymous referee for numerous helpful comments and to Yulia Romaniuk for final corrections. The paper was partially supported by grant NSERC 264186.     

Are the China-related stock markets segmented with both world and regional stock markets?

Implementing the Capital Asset Pricing Model framework, this study investigates the integration of three China-related stock markets, namely, the A-, B- and H-share markets, with both the Hong Kong stock market and the world market. An analysis of market segmentation versus integration using the Jorion and Schwartz model suggests that the A-share market was a segmented market during the period 1995–2004. However, evidence of a higher-level integration between the A- and B-share markets, and the A-share and Hong Kong stock markets is found in the sub-period tests. The hypothesis that the B- and H-share markets are becoming increasingly integrated with the world stock market is not supported.     

Secondary markets with changing preferences

If consumer valuations change over time, then secondary‐market frictions may raise monopoly profits and cause durability to be distorted away from the cost‐minimizing level. A monopolist who favors such frictions overinvests in durability, but planned obsolescence instead may be preferred when market frictions exist but a monopolist wishes they did not. Evidence from the book market is presented.     

International diversification with frontier markets

We provide an analysis of frontier market equities with respect to world market integration and diversification. Principal component results reveal that frontier markets exhibit low levels of integration. In contrast with developed and emerging markets, frontier markets offer no indication of increasing integration through time. Furthermore, individual frontier market countries do not exhibit consistent rates of changing integration. Structural break tests identify breakpoints in integration, as well as integration dynamics across countries. We show that frontier markets have low integration with the world market and thereby offer significant diversification benefits.