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.     

基于齐次马氏域变方法的中国股市价格泡沫检验

本文使用间接度量方法构建了股市价格泡沫的理论模型,得到了股市价格泡沫所满足的行为方程。在该理论模型下,本文利用Johansen协整检验方法从上证指数1996年1月到2007年12月的数据中剔除其理性价值成份,并进而建立马氏域变模型对其泡沫情况进行分析,结果表明上证指数价格泡沫主要集中于1996年3月到1997年6月、1999年全年以及2006年11月到2007年12月。此外,本文还通过对深证指数的分析检验了沪深两市股价泡沫的联动性,最后根据文章分析结果,提出了相应的政策建议。     

Relative Responsiveness of Bargaining Solutions to Changes in Status-quo Payoffs

It is a well-known fact that several prominent bargaining solutions are responsive to changes in status-quo (i.e., disagreement or fallback) payoffs. When an agent’s status-quo payoff increases, his solution payoff either stays the same or increases. A fully general result for these solutions’ status-quo point ranking is impossible to establish. In this paper, using an important class of bargaining problems, a ranking of the relative status-quo point responsiveness of prominent bargaining solutions is obtained. Using the Constant Elasticity of Substitution class of bargaining problems, regardless of the concavity of the Pareto frontier and the level of increase in one’s status-quo payoff, we find the equal gains solution is the most responsive with respect to changes in status-quo payoffs, followed by the Nash solution. The equal sacrifice solutions is the least responsive, followed by the Kalai/Smorodinsky solution.

Approachability with bounded memory

We study Blackwell's approachability in repeated games with vector payoffs when the approaching player is restricted to use strategies with bounded memory: either strategies with bounded recall, or strategies that can be implemented by finite automata. Our main finding is that the following three statements are equivalent for closed sets. (i) The set is approachable with bounded recall strategies. (ii) The set is approachable with strategies that can be implemented with finite automata. (iii) The set contains a convex approachable set. Using our results we show that (i) there are almost-regret-free strategies with bounded memory, (ii) there is a strategy with bounded memory to choose the best among several experts, and (iii) Hart and Mas-Colell's adaptive learning procedure can be achieved using strategies with bounded memory.     

General equilibrium trade modelling with Canada–US transportation costs

Transportation costs are an important topic in international trade, but seldom have researchers paid attention to general equilibrium trade modelling with transportation costs and explored their relevant effects. This paper uses numerical general equilibrium trade model structures to simulate the impacts of transportation costs on welfare and trade for a Canada–US country pair case. We compare two groups of model structures: Armington assumption models and homogeneous goods models. Within these two groups of models, we also compare balanced trade structures to trade imbalance structures and production function transportation costs to iceberg transportation costs. Armington goods models generate more absolute welfare gains from transportation cost elimination than homogeneous goods models. Welfare gains under balanced trade structures are larger in production function transportation cost scenarios than in iceberg transportation cost scenarios, but under trade imbalance structures, welfare gains are greater under iceberg transportation cost scenarios. Canada's welfare gains in the iceberg transportation cost scenario are significantly larger than gains in the production function transportation cost scenario. On trade effects, homogeneous goods models generate more export and import gains, balanced trade structures have more trade variations, and iceberg transportation costs generate more trade effects.     

Collective rent seeking and division of labor

The article analyzes division of labor within multiple groups engaged in collective rent seeking through time. Each agent seeks membership in one of two subgroups within each group, conveniently labelled a leader-subgroup and a follower-subgroup. The within-group generated payoffs are used as input in the between-group competition for the other groups' payoffs and an external rent. Within-group egalitarian allocation, but not relative-effort allocation, alleviates leadership struggle if the rent is large or the group is moderately more decisive than the other groups. The group employing a more egalitarian sharing rule than the other groups alleviates its leadership struggle more successfully. The Folk theorem is used to show the conditions under which leadership struggle gets intensified or alleviated.     

Comparing Bargaining Solutions in the Shadow of Conflict: How Norms against Threats Can Have Real Effects

In many economic environments agents make costly and irreversible investments (in “guns”) that may enhance their respective threat payoffs but also shrink the utility possibilities set. In such settings, with variable threats and a variable utility possibilities set, it becomes possible to rank different bargaining solutions in terms of efficiency. We compare bargaining solutions within a class in which the influence of the threat point on the bargaining outcome varies across solutions. Under symmetry, we find that the solution in which the threat point is least influential—the equal sacrifice solution—Pareto-dominates the other solutions. Since the equal sacrifice solution puts the least weight on the threat point, norms against threats (that can be seen in many seemingly rhetorical pronouncements in adversarial relations) can mitigate some of the costs of conflict and therefore have efficiency-enhancing effects. Journal of Economic Literature Classification Numbers: C72, C78, D30, D70, D72, K42.     

Consulting experts with vested interests: The power of benchmarking

A decisionmaker gets independent advice from two experts, who can be of two different types. Experts are risk-neutral and prefer certain policies irrespective of the state of nature, so to induce information-sharing the decisionmaker must reward truthful experts. I show that, in this environment, a heterogeneous committee has no informational advantage over a single expert: a decision rule that encourages one expert to be honest immediately makes honesty less attractive for the other. With a homogenous committee, the higher is the correlation between the experts’ signals, the more the decisionmaker is willing to pay to secure independent advice.Received: April 2004, Accepted: Accepted: January 2005, JEL Classification: D71, D82Lars Frisell: I thank David Austen-Smith, Tim Feddersen, Johan Lagerlöf, Jan Potters, Ken Shotts, Jonas Vlachos, Jörgen W. Weibull, an anonymous referee, and especially Karl Wärneryd for helpful comments. Financial support from the Jan Wallander and Tom Hedelius Foundation and the Sweden-America Foundation is gratefully acknowledged.     

Collective rent seeking and division of labor

The article analyzes division of labor within multiple groups engaged in collective rent seeking through time. Each agent seeks membership in one of two subgroups within each group, conveniently labelled a leader-subgroup and a follower-subgroup. The within-group generated payoffs are used as input in the between-group competition for the other groups' payoffs and an external rent. Within-group egalitarian allocation, but not relative-effort allocation, alleviates leadership struggle if the rent is large or the group is moderately more decisive than the other groups. The group employing a more egalitarian sharing rule than the other groups alleviates its leadership struggle more successfully. The Folk theorem is used to show the conditions under which leadership struggle gets intensified or alleviated.