order flow transaction clock and normality of asset returns the goal of this paper is to show that normality of asset returns can be recovered through a stochastic time change clark 1973 addressed this issue by representing the price process as a subordinated process with volume as the lognormally distributed subordinator we extend clark s results and find the following i stochastic time changes are mathematically much less constraining than subordinators ii the cumulative number of trades is a better stochastic clock than the volume for generating virtually perfect normality in returns iii this clock can be modeled nonparametrically allowing both the time change and price processes to take the form of jump diffusions the relations among trading volume stock prices and price volatility the subject of empirical and theoretical studies over many years have lately received renewed attention with the increased availability of high frequency data a vast amount of research has focused on issues such as news arrivals volume and price changes or volatility moves usually outside any framework of general or even partial equilibrium is the normality of returns a key issue for example in the mean variance paradigm for portfolio choice or the recent study of the problems of risk management e g in value at risk verified at any time horizon the evidence accumulated from a number of studies that document the presence of leptokurtosis and skewness in the distribution of returns of a wide variety of financial assets suggests that the answer is no studies as early as for example fama 1965 showed that daily returns are more long tailed than the normal density with the distribution of returns approaching normality as the holding period is extended to one month in the same manner volatility smiles and other observed

a tale of two regimes theory and empirical evidence for a markov modulated jump diffusion model of equity returns and derivative pricing implications abstract we provide closed form solutions for a continuous time markov modulated jump diffusion model in a general equilibrium framework for options prices under a variety of jump diffusion specifications we further demonstrate that the two state model provides the leptokurtic return features volatility smile and volatility clustering observed empirically for the dow jones industrial average djia and its component stocks using 10 years of stock return data we confirm the existence of jump intensity switching and clustering illustrate transition probabilities and verify superior empirical fit over competing poisson style models

the variance gamma v g model for share market returns a new stochastic process termed the variance gamma process is proposed as a model for the uncertainty underlying security prices the unit period distribution is normal conditional on a variance that is distributed as a gamma variate its advantages include long tailedness continuous time specification finite moments of all orders elliptical multivariate unit period distributions and good empirical fit the process is pure jump approximable by a compound poisson process with high jump frequency and low jump magnitudes applications to option pricing show differential effects for options on the money compared to in or out of the money copyright 1990 by the university of chicago

premia for correlated default risk abstract using data on corporate default experience in the u s and market rates of cdx index and tranche swaps of various maturities we estimate reduced form models of correlated default timing in the cdx high yield and investment grade portfolios under actual and risk neutral probabilities the striking contrast between the estimated processes followed by the actual and risk neutral arrival intensities of defaults and between the parameters governing the actual and risk neutral dynamics of the risk neutral intensities indicates the presence of substantial default risk premia in cdx swap market rates the effects of risk premia on swap rates covary strongly across maturities and depend on general stock market volatility and several measures of credit spreads large moves in the effects of these premia on swap rates have natural interpretations in terms of economic and financial market developments during the sample period april 2004 to october 2007 our results suggest that a large portion of the movements in cdx swap market rates observed during the sample period may be caused by changing attitudes toward correlated default risk rather than changes in the economic factors affecting the actual risk of clustered defaults which ultimately governs swap payoffs

the economics of two sided markets at a local best buy a child places a new sony playstation 3 on the cashier s counter while the parents dig out their visa card the gaming system and the payment card may appear to have little connection other than this purchase however these two items share an important characteristic that is generating a series of economic insights and has important implications for strategic decision making and economic policymaking both video game systems and payment cards are examples of two sided markets broadly speaking a two sided market is one in which 1 two sets of agents interact through an intermediary or platform and 2 the decisions of each set of agents affects the outcomes of the other set of agents typically through an externality in the case of a video game system the intermediary is the console producer sony in the scenario above while the two sets of agents are consumers and video game developers neither consumers nor game developers will be interested in the playstation if the other party is not similarly a successful payment card requires both consumer usage and merchant acceptance where both consumers and merchants value each others participation many more products fit into this paradigm such as search engines newspapers and almost any advertisersupported media examples in which consumers often negatively value rather than positively value the participation of the other side as well as most software or title based operating systems and consumer electronics malls which seek retailers and consumers convention organizers which seek buyers and sellers dating services which seek men and women and the journal of economic perspectives which seeks content and readership all experience the economics of two sided markets the multi sided nature of many internet and high technology markets as well as

on the simple economics of advertising marketing and product design we propose a framework for analyzing transformations of demand such transformations frequently stem from changes in the dispersion of consumers valuations which lead to rotations of the demand curve in many settings profits are a u shaped function of dispersion high dispersion is complemented by niche production and low dispersion is complemented by mass market supply we investigate numerous applications including product design advertising marketing and sales advice and the construction of quality differentiated product lines we also suggest a new taxonomy of advertising distinguishing between hype which shifts demand and real information which rotates demand

exploring the sources of default clustering abstract we study the sources of corporate default clustering in the united states we reject the hypothesis that firms default times are correlated only because their conditional default rates depend on observable and latent systematic factors by contrast we find strong evidence that contagion through which the default by one firm has a direct impact on the health of other firms is a significant clustering source the amount of clustering that cannot be explained by contagion and firms exposure to observable and latent systematic factors is insignificant our results have important implications for the pricing and management of correlated default risk

platform competition in two sided markets many if not most markets with network externalities are two sided to succeed platforms in industries such as software portals and media payment systems and the internet must get both sides of the market on board accordingly platforms devote much attention to their business model that is to how they court each side while making money overall this paper builds a model of platform competition with two sided markets it unveils the determinants of price allocation and end user surplus for different governance structures profit maximizing platforms and not for profit joint undertakings and compares the outcomes with those under an integrated monopolist and a ramsey planner jel l5 l82 l86 l96

a dynamic global portfolio immunization strategy in the world of multiple interest rate changes a dynamic immunization and minimax theorem man 5 and khang 9 demonstrate that it is possible to immunize a portfolio of default free assets against unexpected interest rate changes so that at the end of the planning period the investor will realize at least the returns expected at pur chase however this immunization strategy is applicable for the case in which the change in unexpected interest rate occurs only once at the instant after the purchase of the asset obviously the case depicted above is not likely to resemble the real world situation in at least two respects first the interest rate change is likely to occur at any time and second the interest rate change is likely to occur many times during the investor s planning period bierwag 4 deals with the case in which the interest rate change may occur many times during the planning period each occurring at the instant after the beginning of each time period and shows that provided the interest rate changes are small the investor can at the end of the planning period realize the returns expected at purchase by constructing the portfolio whose duration at the begin ning of each period is equal to the length of the remaining planning period thus bierwag s 4 immunization theorem is a local theorem bierwag makes a conjecture that the theorem is not likely to hold true globally this paper shows that the strategy of keeping the portfolio duration calcu lated on the basis of the term structure observed at each moment of time equal to the length of the remaining planning period during the entire planning period guarantees at least the returns expected or promised at purchase regardless of when and how many times interest rate changes occur during the planning period and regardless of sizes of change in the interest rate structure thus the theorem proven in the paper is a global theorem in a very general setting as opposed to bierwag s 4 local theorem a corollary is that bierwag s unconditional im munization policy does not immunize unless all changes in the term structure are university of oregon the author expresses thanks to an anonymous referee whose careful reading of the paper resulted in an improvement

partial information about contagion risk self exciting processes and portfolio optimization abstract this paper compares two classes of models that allow for additional channels of correlation between asset returns regime switching models with jumps and models with contagious jumps both classes of models involve a hidden markov chain that captures good and bad economic states the distinctive feature of a model with contagious jumps is that large negative returns and unobservable transitions of the economy into a bad state can occur simultaneously we show that in this framework the filtered loss intensities have dynamics similar to self exciting processes besides we study the impact of unobservable contagious jumps on optimal portfolio strategies and filtering

leverage effect volatility feedback and self exciting market disruptions equity index volatility variation and its interaction with the index return can come from three distinct channels first index volatility increases with the market s aggregate financial leverage second positive shocks to systematic risk increase the cost of capital and reduce the valuation of future cash flows generating a negative correlation between the index return and its volatility regardless of financial leverage finally large negative market disruptions show self exciting behaviors this article proposes a model that incorporates all three channels and examines their relative contribution to index option pricing and stock option pricing for different types of companies

cyclical correlations credit contagion and portfolio losses abstract we model aggregate credit losses on large portfolios of financial positions contracted with firms subject to both cyclical default correlation and direct default contagion processes cyclical correlation is due to the dependence of firms on common economic factors contagion is associated with the local interaction of firms with their business partners we provide an explicit normal approximation of the distribution of portfolio losses we quantify the relation between the variability of global economic fundamentals strength of local firm interaction and the fluctuation of losses we find that cyclical oscillations in fundamentals dominate average losses while local interaction causes additional fluctuations of losses around their average the strength of the contagion induced loss variability depends on the complexity of the business partner network

new insights into smile mispricing and value at risk the hyperbolic model the authors investigate a new basic model for asset pricing the hyperbolic model which allows an almost perfect statistical fit of stock return data after a detailed introduction into the theory they use secondary market data to compare the hyperbolic model to the classical black scholes model the authors study implicit volatilities the smile effect and pricing performance exploiting the full power of the hyperbolic model they construct an option value process from a statistical point of view by estimating the implicit risk neutral density function from option data finally the authors present some new value at risk calculations leading to new perspectives to cope with model risk copyright 1998 by university of chicago press

tying and freebies in two sided markets abstract in two sided markets where platforms are constrained to set non negative prices tying can be deployed by platforms as a tool to introduce implicit subsidies for a monopoly this raises participation and benefits consumers on both sides in a duopoly tying on one side makes a platform more or less competitive on the other side depending on externalities tying may not be ex ante optimal while the competing platform may benefit from it the impact on consumers surplus depends on whether competition is softened or intensified on the profitable side moreover tying increases total welfare if network effects are strong

chicken and egg competition among intermediation service providers we analyze a model of imperfect price competition between intermediation service providers we insist on features that are relevant for informational intermediation via the internet the presence of indirect network externalities the possibility of using the nonexclusive services of several intermediaries and the widespread practice of price discrimination based on users identity and on usage efficient market structures emerge in equilibrium as well as some specific form of inefficient structures intermediaries have incentives to propose non exclusive services as this moderates competition and allows them to exert market power we analyze in detail the pricing and business strategies followed by intermediation services providers copyright 2003 by the rand corporation

changes of numeraire for pricing futures forwards and options a change of numeraire argument is used to derive a general option parity or equivalence result relating american call and put prices and to obtain new expressions for futures and forward prices the general parity result unifies and extends a number of existing results the new futures and forward pricing formulas are often simpler to compute in multifactor models than existing alternatives we also extend previous work by deriving a general formula relating exchange options to ordinary call options a number of applications to diffusion models including stochastic volatility stochastic interest rate and stochastic dividend rate models and jump diffusion models are examined a self financing portfolio is called a numeraire if security prices measured in units of this portfolio admit an equivalent martingale measure the most commonly used numeraire is the reinvested short rate process the corresponding equivalent martingale measure is the risk neutral measure geman el karoui and rochet 1995 show that other numeraires can simplify many asset pricing problems in this article we build on their results and using the reinvested asset price as the numeraire unify and extend the literature on option parity or equivalence results relating american call and put prices for asset and futures options the same numeraire change is used to obtain new pricing formulas for futures and forwards that are often simpler to compute in multifactor models finally we use a numeraire change to simplify exchange option pricing extending a similar result in geman el karoui and rochet to dividend paying assets the change of numeraire method is most intuitive in the context of foreign currency derivative securities as discussed by grabbe 1983 an american call option to buy 1 dm with dollar price process s for k dollars is equivalent to an american put option to sell k dollars with dm price process k s for a strike price of 1 dm the dollar price of the call must therefore equal the product of the current exchange rate s0 and the dm price of the put the call price is computed using the dollar value of a u s i am grateful to kerry back the editor and costis skiadas for their many helpful suggestions thanks also to peter demarzo matt jackson naveen khanna david marshall robert mcdonald jim moser phyllis payette and an anonymous referee for their comments this article subsumes schroder 1992

time changed levy processes and option pricing abstract the classic black scholes option pricing model assumes that returns follow brownian motion but return processes differ from this benchmark in at least three important ways first asset prices jump leading to non normal return innovations second return volatilities vary stochastically over time third returns and their volatilities are correlated often negatively for equities time changed levy processes can simultaneously address these three issues we show that our framework encompasses almost all of the models proposed in the option pricing literature and it is straightforward to select and test a particular option pricing model through the use of characteristic function technology

information management and pricing in platform markets we study platform markets in which the information about users preferences is dispersed first we show how the dispersion of information introduces idiosyncratic uncertainty about participation decisions and how the latter shapes the elasticity of the demands and the equilibrium prices we then study the effects on profits consumer surplus and welfare of platform design blogs forums conferences advertising campaigns post launch disclosures and other information management policies affecting the agents ability to predict participation decisions on the other side of the market

differentiated networks equilibrium and efficiency we consider a model of price competition in a duopoly with product differentiation and network effects the value of a good for a consumer is the sum of a common and an idiosyncratic component the first captures the vertical dimension of quality the second captures horizontal differentiation each consumer privately observes his own value for each good but cannot separate the common and the idiosyncratic component therefore he has incomplete information about the value of the goods for the other consumers after firms announce prices consumers choose simultaneously which network to join facing a coordination problem in the efficient allocation both networks are active and the firm with the highest expected quality has the largest market share to characterize the equilibrium allocation we derive necessary and sufficient conditions for uniqueness of the equilibrium of the coordination game played by consumers for given prices the equilibrium allocation differs from the efficient one for two reasons first the equilibrium allocation of consumers to the networks is too balanced since consumers fail to internalize network externalities second if access to the networks is priced by strategic firms then the product with the highest expected quality is also the most expensive this further reduces the asymmetry between market shares and therefore social welfare

multidimensional platform design successful platforms attract not just many users but also those of the right kind the right kind of user is one who can either be directly monetized or who differentially attracts other valuable users bonacich centrality on the network of user sorting with direct value of monetization captures this feedback loop and thus characterizes the value of user characteristics we use this value to determine optimal steady state platform design and reliable means for platforms to reach such a steady state we apply these results respectively to explain the dynamic growth strategy of social networks and urban development policies of cities

pricing and hedging in incomplete markets abstract we present a new approach for positioning pricing and hedging in incomplete markets that bridges standard arbitrage pricing and expected utility maximization our approach for determining whether an investor should undertake a particular position involves specifying a set of probability measures and associated floors which expected payoffs must exceed in order for the investor to consider the hedged and financed investment to be acceptable by assuming that the liquid assets are priced so that each portfolio of assets has negative expected return under at least one measure we derive a counterpart to the first fundamental theorem of asset pricing we also derive a counterpart to the second fundamental theorem which leads to unique derivative security pricing and hedging even though markets are incomplete for products that are not spanned by the liquid assets of the economy we show how our methodology provides more realistic bid ask spreads

supplying information to facilitate price discrimination the authors examine the incentive of a seller to allow potential buyers to acquire private information about their tastes for the seller s product improved private information for buyers enables the seller to segment the market and charge higher prices to high value buyers however improved information can also provide rents to buyers in a variety of settings this tradeoff is optimally resolved at one of two extremes either buyers are supplied with the best available knowledge of their tastes or no information is supplied by the seller copyright 1994 by economics department of the university of pennsylvania and the osaka university institute of social and economic research association

credit contagion in a network of firms with spatial interaction this contribution studies the effects of credit contagion on the credit risk of a portfolio of bank loans to this aim we introduce a model that takes into account the counterparty risk in a network of interdependent firms that describes the presence of business relations among different firms the location of the firms is simulated with probabilities computed using an entropy spatial interaction model by means of a wide simulation analysis we investigate the behavior of the model proposed and study the effects of default contagion on the loss distribution of a portfolio of bank loans

tying foreclosure and exclusion in recent years the leverage theory of tied good sales has faced heavy and influential criticism in an important sense though the models used by its critics are actually incapable of addressing the leverage theory s central concerns here the author reconsiders the leverage hypothesis and argues that tying can indeed serve as a mechanism for leveraging market power the mechanism through which this leverage occurs its profitability and its welfare implications are discussed in detail copyright 1990 by american economic association

a top down approach to multiname credit a multiname credit derivative is a security that is tied to an underlying portfolio of corporate bonds and has payoffs that depend on the loss due to default in the portfolio the value of a multiname derivative depends on the distribution of portfolio loss at multiple horizons intensity based models of the loss point process that are specified without reference to the portfolio constituents determine this distribution in terms of few economically meaningful parameters and lead to computationally tractable derivatives valuation problems however these models are silent about the portfolio constituent risks they cannot be used to address applications that are based on the relationship between portfolio and component risks for example constituent risk hedging this paper develops a method that extends these models to the constituents we use random thinning to decompose the portfolio intensity into a sum of constituent intensities we show that a thinning process which allocates the portfolio intensity to constituents uniquely exists and is a probabilistic model for the next to default we derive a formula for the constituent default probability in terms of the thinning process and the portfolio intensity and develop a semi analytical transform approach to evaluate it the formula leads to a calibration scheme for the thinning processes and an estimation scheme for constituent hedge sensitivities an empirical analysis for september 2008 shows that the constituent hedges generated by our method outperform the hedges prescribed by the gaussian copula model which is widely used in practice

time changed birth processes and multiname credit derivatives a credit investor such as a bank granting loans to firms or an asset manager buying corporate bonds is exposed to correlated corporate default risk a multiname credit derivative is a financial security that allows the investor to transfer this risk to the credit market in this paper we study the valuation and risk analysis of multiname derivatives to capture the complex economic phenomena that drive the pricing of these securities we introduce a time changed birth process as a probabilistic model of correlated event timing the self exciting property of a time changed birth process captures the feedback from events that is often observed in credit markets the stochastic variation of arrival rates between events captures the exposure of firms to common economic risk factors we derive a closed form expression for the distribution of a time changed birth process and develop analytically tractable pricing relations for a range of multiname derivatives valuation problems we illustrate our results by calibrating a tranche forward and option pricer to market rates of index and tranche swaps

the fine structure of asset returns an empirical investigation we investigate the importance of diffusion and jumps in a new model for asset returns in contrast to standard models we allow for jump components displaying finite or infinite activity and variation empirical investigations of time series indicate that index dynamics are devoid of a diffusion component which may be present in the dynamics of individual stocks this leads to the conjecture confirmed on options data that the risk neutral process should be free of a diffusion component we conclude that the statistical and risk neutral processes for equity prices are pure jump processes of infinite activity and finite variation

a maxmin policy for bond management a simple immunization problem is formulated as a maxmin optimal control problem and analytically solved by means of dynamic programming the optimal control law namely the immunization policy turns out to be quite different from any duration based immunization policy moreover it is seemingly able to discriminate between favourable and unfavourable changes in the yield curve

simulating risk contributions of credit portfolios the 2007 2009 financial turmoil highlighted the need for more active management of credit portfolios after measuring portfolio credit risk an important step toward active risk management is to measure risk contributions of individual obligors to the overall risk of the portfolio in practice value at risk is often used as a risk measure for credit portfolios and it can be decomposed into a sum of the risk contributions of individual obligors estimation of these risk contributions is computationally challenging mainly because they are expectations conditioned on a rare event in this paper we tackle this computational problem by developing a restricted importance sampling ris method for a class of conditional independence credit risk models where defaults of obligors are conditionally independent given an appropriately chosen random vector we propose fast estimators for risk contributions and their confidence intervals furthermore we study the incorporation of traditional importance sampling methods into the ris method to further improve its efficiency for the widely used gaussian copula model numerical examples show that the proposed method works well

dynamic price competition with network effects i consider a dynamic model of competition between two proprietary networks consumers die and are replaced with a constant hazard rate and firms compete for new consumers to join their network by offering network entry prices i derive a series of results pertaining to a existence and uniqueness of symmetric equilibria b monotonicity of the pricing function e g larger networks set higher prices c network size dynamics increasing dominance vs reversion to the mean and d firm value how it varies with network effects finally i apply my general framework to the study of termination charges in wireless telecommunications i consider various forms of regulation and examine their impact on firm profits and market share dynamics

two sided markets a progress report the paper provides a road map to the burgeoning literature on two sided markets and presents new results it identifies two sided markets with markets in which the structure and not only the level of prices charged by platforms matters the failure of the coase theorem is necessary but not sufficient for two sidedness the paper builds a model integrating usage and membership externalities that unifies two hitherto disparate strands of the literature emphasizing either form of externality and obtains new results on the mix of membership and usage charges when price setting or bargaining determine payments between end users

common failings how corporate defaults are correlated we test the doubly stochastic assumption under which firms default times are correlated only as implied by the correlation of factors determining their default intensities using data on u s corporations from 1979 to 2004 this assumption is violated in the presence of contagion or frailty unobservable explanatory variables that are correlated across firms our tests do not depend on the time series properties of default intensities the data do not support the joint hypothesis of well specified default intensities and the doubly stochastic assumption we find some evidence of default clustering exceeding that implied by the doubly stochastic model with the given intensities why do corporate defaults cluster in time several explanations have been explored first firms may be exposed to common or correlated risk factors whose co movements cause correlated changes in conditional default probabilities second the event of default by one firm may be contagious in that one such event may directly induce other corporate failures as with the collapse of penn central railway in 1970 third learning from default may generate default correlation for example to the extent that the defaults of enron and worldcom revealed accounting irregularities that could be present in other firms they may have had a direct impact on the conditional default probabilities of other firms our primary objective is to examine whether cross firm default correlation that is associated with observable factors determining conditional default probabilities the first channel on its own is sufficient to account for the degree of time clustering in defaults that we find in the data

credit risk model with contagious default dependencies affected by macro economic condition abstract we consider a credit risk model with two industrial sectors where defaults of corporations would be influenced by two factors the first factor represents the macro economic condition which would affect the default intensities of the two industrial sectors differently the second factor reflects the influences of the past defaults of corporations against other active corporations where such influences would affect the two industrial sectors differently a two layer markov chain model is developed where the macro economic condition is described as a birth death process while another markov chain represents the stochastic characteristics of defaults with default intensities dependent on the state of the birth death process and the number of defaults in two sectors although the state space of the two layer markov chain is huge the fundamental absorbing process with a reasonable state space size could capture the first passage time structure of the two layer markov chain thereby enabling one to evaluate the joint probability of the number of defaults in two sectors via the uniformization procedure of keilson this in turn enables one to value a variety of derivatives defined on the underlying credit portfolios in this paper we focus on a financial product called cdo and a related option

asset pricing under the quadratic class we identify and characterize a class of term structure models where bond yields are quadratic functions of the state vector we label this class the quadratic class and aim to lay a solid theoretical foundation for its future empirical application we consider asset pricing in general and derivative pricing in particular under the quadratic class we provide two general transform methods in pricing a wide variety of fixed income derivatives in closed or semi closed form we further illustrate how the quadratic model and the transform methods can be applied to more general settings i introduction we identify and characterize a class of term structure models where bond yields are quadratic functions of the markov process we label this class the quadratic class and aim to lay a solid theoretical foundation for its future empirical application we identify the necessary and sufficient conditions for the quadratic class and consider the general asset pricing problem under the quadratic framework in particular we propose two transform methods to price a wide variety of interest rate derivatives in closed or semi closed form we further illustrate how the pricing methods can be applied to more general settings examples include option pricing for currencies or stocks with quadratic stochastic volatilities our interests in the quadratic class come mainly from concerns on empirical application recent empirical research within the affine framework of duffie and kan 1996 indicates an inherent tension between i delivering good empirical performance in matching salient features of the interest rate data and ii excluding positive probabilities of having negative interest rates for example backus telmer and wu 1999 and dai and singleton 2000 find that incorporating gaussian state variables in the affine framework significantly increases the flexibility for model design and greatly improves its empirical performance in capturing the conditional dynamics of interest rates dai and singleton 2001

risk analysis of collateralized debt obligations collateralized debt obligations which are securities with payoffs that are tied to the cash flows in a portfolio of defaultable assets such as corporate bonds play a significant role in the financial crisis that has spread throughout the world insufficient capital provisioning due to flawed and overly optimistic risk assessments is at the center of the problem this paper develops stochastic methods to measure the risk of positions in collateralized debt obligations and related instruments tied to an underlying portfolio of defaultable assets it proposes an adaptive point process model of portfolio default timing a maximum likelihood method for estimating point process models that is based on an acceptance rejection resampling scheme and statistical tests for model validation to illustrate these tools they are used to estimate the distribution of the profit or loss generated by positions in multiple tranches of a collateralized debt obligation that references the cdx high yield portfolio and the risk capital required to support these positions

comparative advertising disclosing horizontal match information improved consumer information about horizontal aspects of products of similar quality leads to better consumer matching but also to higher prices so consumer surplus can go up or down while profits rise with enough quality asymmetry though the higher quality and hence larger firm s price falls with more information so both effects benefit consumers this occurs when comparative advertising is used against a large firm by a small one comparative advertising as it imparts more information therefore helps consumers although it also improves the profitability of the small firm overall welfare goes down because of the large loss to the attacked firm

immunization as a maxmin strategy the purpose of this paper is to broaden the analysis of immunization when it is viewed as amaxmin strategy several results are demonstrated which provide a new interpretation of maxmin strategy by using a more general theory of lagrangians we clarify the exact assumptions required for the duration strategy and the maxmin strategy to be equivalent in a world with taxes it is shown that in a world with taxes the immunization strategy requires the portfolio to satisfy a constraint about the bond to be included in addition to the duration constraint