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Arithmetic Asian or average price options deliver payoffs based on the average underlying price over a prespecified time period. Asian options are an important family of derivative contracts with a wide variety of applications in currency, equity, interest rate, commodity, energy, and insurance markets. We derive two analytical formulas for the value of the continuously sampled arithmetic Asian option when the underlying asset price follows geometric Brownian motion. We use an identity in law between the integral of geometric Brownian motion over a finite time interval [0, t ] and the state at time t of a one-dimensional diffusion process with affine drift and linear diffusion and express Asian option values in terms of spectral expansions associated with the diffusion infinitesimal generator. The first formula is an infinite series of terms involving Whittaker functions M and W . The second formula is a single real integral of an expression involving Whittaker function W plus (for some parameter values) a finite number of additional terms involving incomplete gamma functions and Laguerre polynomials. The two formulas allow accurate computation of continuously sampled arithmetic Asian option prices.

We study the cyclical properties of sales, regular price changes, and average prices paid by consumers (\"effective\" prices) using data on prices and quantities sold for numerous retailers across many US metropolitan areas. Inflation in the effective prices paid by consumers declines significantly with higher unemployment while little change occurs in the inflation rate of prices posted by retailers. This difference reflects the reallocation of household expenditures across retailers, a feature of the data which we document and quantify, rather than sales. We propose a simple model with household store-switching and assess its implications for business cycles and policymakers.

How have house prices evolved over the long run? This paper presents annual house prices for 14 advanced economies since 1870. We show that real house prices stayed constant from the nineteenth to the mid-twentieth century, but rose strongly and with substantial cross-country variation in the second half of the twentieth century. Land prices, not replacement costs, are the key to understanding the trajectory of house prices. Rising land prices explain about 80 percent of the global house price boom that has taken place since World War II. Our findings have implications for the evolution of wealth-to-income ratios, the growth effects of agglomeration, and the price elasticity of housing supply.

Nonlinear pricing and taxation complicate economic decisions by creating multiple marginal prices for the same good. This paper provides a framework to uncover consumers' perceived price of nonlinear price schedules. I exploit price variation at spatial discontinuities in electricity service areas, where households in the same city experience substantially different nonlinear pricing. Using household-level panel data from administrative records, I find strong evidence that consumers respond to average price rather than marginal or expected marginal price. This suboptimizing behavior makes nonlinear pricing unsuccessful in achieving its policy goal of energy conservation and critically changes the welfare implications of nonlinear pricing.

This article establishes six stylized facts about firms' export prices using detailed customs data on the universe of Chinese trade flows. First, across firms selling a given product, exporters that charge higher prices earn greater revenues in each destination, have bigger worldwide sales, and enter more markets. Second, firms that export more, enter more markets, and charge higher export prices import more expensive inputs. Third, across destinations within a firm-product, firms set higher prices in richer, larger, bilaterally more distant and overall less remote countries. Fourth, across destinations within a firm-product, firms earn bigger revenues in markets where they set higher prices. Fifth, across firms within a product, exporters with more destinations offer a wider range of export prices. Finally, firms that export more, enter more markets, and offer a wider range of export prices pay a wider range of input prices and source inputs from more origin countries. We propose that trade models should incorporate two features to rationalize these patterns in the data: more successful exporters use higher quality inputs to produce higher quality goods (stylized facts 1 and 2), and firms vary the quality of their products across destinations by using inputs of different quality levels (stylized facts 3, 4, 5, and 6).

In the 1988-2004 microdata collected by the U.S. Bureau of Labor Statistics for the Consumer Price Index, price changes are frequent (every 4-7 months, depending on the treatment of sale prices) and large in absolute value (on the order of 10%). The size and timing of price changes vary considerably for a given item, but the size and probability of a price change are unrelated to the time since the last price change. Movements in aggregate inflation reflect movements in the size of price changes rather than the fraction of items changing price, because of offsetting movements in the fraction of price increases and decreases. Neither leading time-dependent models (Taylor or Calvo) nor first-generation state-dependent models match all of these facts. Some second-generation state-dependent models, however, appear broadly consistent with the empirical patterns.

Product-harm crises are omnipresent in today's marketplace. Such crises can cause major revenue and market-share losses, lead to costly product recalls, and destroy carefully nurtured brand equity. Moreover, some of these effects may spill over to nonaffected competitors in the category when they are perceived to be guilty by association. The extant literature lacks generalizable knowledge on the effectiveness of different marketing adjustments that managers often consider to mitigate the consequences of such events. To fill this gap, the authors use large household-scanner panels to analyze 60 fast-moving consumer good product crises that occurred in the United Kingdom and the Netherlands and resulted in the full recall of an entire variety. The authors assess the effects of postcrisis advertising and price adjustments on the change in consumers' brand share and category purchases. In addition, they consider the extent to which the effects are moderated by two key crisis characteristics: the extent of negative publicity surrounding the event and whether the affected brand had to publicly acknowledge blame. Using the empirical findings, the authors provide context-specific managerial recommendations on how to overcome a product-harm crisis.

We propose and test a catering theory of nominal stock prices. The theory predicts that when investors place higher valuations on low-price firms, managers respond by supplying shares at lower price levels, and vice versa. We confirm these predictions in time-series and firm-level data using several measures of time-varying catering incentives. More generally, the results provide unusually clean evidence that catering influences corporate decisions, because the process of targeting nominal share prices is not well explained by alternative theories.

This paper provides new insight into the relationship between inflation and the setting of individual prices by examining a large data set of Mexican consumer prices covering episodes of both low and high inflation. When the annual rate of inflation is low (below 10%-15%), the frequency of price changes comoves weakly with inflation because movements in the frequency of price decreases and increases partly offset each other. In contrast, the average magnitude of price changes correlates strongly with inflation because it is sensitive to movements in the relative shares of price increases and decreases. When inflation rises beyond 10%-15%, few price decreases are observed and both the frequency and average magnitude are important determinants of inflation. I show that a menu-cost model with idiosyncratic technology shocks predicts the average frequency and magnitude of price changes well over a range of inflation similar to that experienced by Mexico.

This paper studies household beliefs during the recent US housing boom. The first part presents evidence from the Michigan Survey of Consumers. To characterize the heterogeneity in households' views about housing and the economy, a cluster analysis is performed on survey responses at different stages of the boom. The estimation always finds a small cluster of households that believe it is a good time to buy a house because house prices will rise further. The size of this \"momentum\" cluster strongly increased toward the end of the boom. The second part of the paper provides a simple search model of the housing market to show how a small number of optimistic investors can have a large effect on prices without buying a large share of the housing stock.