Academic Review of My Past Research

My research has been predominantly in the area of asset pricing, both theoretical and empirical. My products may seem to be in quite different areas, but they are all related to and based on the fundamental framework of asset pricing. It is my belief that, with asset pricing theory itself being a subfield of economics, it would not give a researcher the right perspective if he or she would focus exclusively on a narrow area of asset pricing. For this reason, I have intentionally worked on a few seemingly different things, which has unquestionably broadened my understanding and appreciation of economics in general and finance in particular. In what follows, I will divide the summary of my research along two lines: (i) arbitrage-based models and (ii) equilibrium-based ones. Under these two modeling approaches, I have done work studying both frictionless and frictional economies. In other words, I will divide my work into four categories.

Arbitrage-Based Models

Due to the early work by Ross and Harrison and Kreps, it is now well known that in a frictionless economy the law of one price (LOP) holds if and only if there is some admissible, "well-defined" pricing rule for future payoffs. Suppose that the LOP holds in such an incomplete-markets economy. Then, the (infinite) set of such admissible pricing rules captures all the information implicit in the observed market prices about the way in which future payoffs are priced by the market. In this sense, if we have two markets each supporting the LOP separately, comparing the sets of implied pricing rules of the two markets can then tell one the extent to which the two markets price similar payoffs consistently. This is the basic idea underlying my paper with Peter Knez, "Measurement of Market Integration and Arbitrage", where we use the mean-square distance between these two sets of implied pricing rules to define a measure of market integration and where we also provide an iterative procedure to estimate the value of the integration measure for a given pair of markets. In addition, the stronger, no-arbitrage condition is used to come up with a more restricted set of implied (positive) pricing rules. Consequently, a strong market integration measure is defined to reflect how differently two given markets may price not only marketed payoffs but also payoffs outside of their payoff spans. Our measurement theory can be applied to test market integration without relying on any parametric asset pricing models.

Following recent work by Hansen and Jagganathan, we can apply the arbitrage valuation approach in other directions as well. For example, in the existing literature on portfolio performance measurement, numerous performance measures have been proposed and used, in an ad hoc way, in practice. One natural question to ask is: how differently will those performance measures rank a given set of managed funds? Or, what properties does each of the measures have? To answer these and other potential questions, we really need to first answer the more fundamental question: what constitutes an admissible portfolio performance measure? In my paper with Peter Knez, "Portfolio Performance Measurement: Theory and Applications," we say a function is an admissible portfolio performance measure if it satisfies four minimal conditions: it assigns zero performance to each reference portfolio and it is linear, continuous and nontrivial. Such an admissible measure is shown to exist if and only if the securities market obeys the law of one price. A positive admissible measure is shown to exist if and only if there is no arbitrage. In this paper, we characterize the (infinite) set of admissible performance measures, and we conclude that performance evaluation is generally quite arbitrary. A mutual fund data set is also used to demonstrate how the measurement method so developed can be applied.

Another application of the arbitrage approach concerns empirical tests of factor pricing models. In my paper with Peter Knez, "A Pricing Operator-Based Testing Foundation for a Class of Factor Pricing Models," we first develop a cross-market version of factor pricing models, and show that exact factor pricing holds across two submarkets with respect to their {\em common factors} if and only if the unique pricing operator for the first submarket is equal to that for the other submarket with probability one. We define an APT measure as the squared distance between the two pricing operators. Then, testing whether this measure is zero is equivalent to testing exact factor pricing across the two submarkets. Since the estimation of this measure does not require parameterizing and extracting the underlying factors, one can test factor pricing models without knowing any factors. In addition, we present a randomization procedure that one can use to conduct a comprehensive investigation on the empirical robustness of factor pricing models.

Given the importance of the no-arbitrage assumption in asset pricing theory, it should be of great interest to see whether and to what extent this assumption is empirically violated. One of the ideal markets for such an investigation is perhaps the Treasury bond market, because for Treasury securities their future payments are known in advance and default-free and the payment schedule is also fixed. In my paper with Mark Fedenia, "Arbitrage in the Treasury Bond Market," we rely on the statement that on a perfect Treasury bond market, the absence of arbitrage and the time value of money principles hold if and only if there is a yield curve in which all the discount factors are positive and declining with the term maturity. We test this implication using default-free, non-callable and non-flower Treasury securities for the period 1950-1989. Even with bid-ask spreads taken into account, the strict no-arbitrage condition was persistently violated, with the 1960's and 1980's showing the most severe violations. When the mispricing of bonds is regressed on various bond attributes, it is found that a significant portion of the mispricing can be explained by bond liquidity, maturity and age.

The afore-mentioned papers all assume frictionless financial markets. To see whether these and other arbitrage-related characterizations may change in the presence of market frictions, many authors have recently tried to extend the arbitrage valuation theory to incorporate frictions. In the paper "Viable Costs and Equilibrium Prices in Frictional Securities Markets," I start with the following general setup. Suppose in some securities market the set of achievable payoffs, M, and its associated portfolio cost functional f are both convex. This cost system (M, f ) is said to be viable if some investor from a certain preferences class can obtain an optimal trade. I show that an arbitrary convex cost system (M, f ) is viable if and only if there is a strictly positive and continuous linear functional lying below f . This represents a complete characterization of viable asset prices or costs, since a large class of securities markets, with or without frictions, can be described by a convex cost system. Among other things, I generalize the Harrison and Kreps (1979) extension property and also demonstrate that in a frictional economy its cost system generally differs from its equilibrium price system. A cost system determines how much investors have to pay to obtain a payoff while a price system is what security issuers use to price securities. This result means that in the presence of frictions, the arbitrage valuation approach may assign asset prices that are not achievable in a given equilibrium and hence the arbitrage approach is no longer as effective for pricing as in the absence of market frictions.

The Equilibrium Models

The theme of my doctoral dissertation at Yale University was equilibrium-based asset pricing. Since then, I have continued the efforts to extend the existing asset pricing theory in three directions: (i) make consumer/investor preferences or utility functions capture more realistically important elements; (ii) incorporate market frictions; and (iii) enrich the modeling structure of an economic equilibrium. In doing so, I have mostly collaborated with my coauthor, Gurdip Bakshi, with whom I have had the most enjoyable, productive collaboration experience.

My first paper with Gurdip Bakshi, "Baby Boom, Population Aging and Capital Markets," is based on a chapter of my Ph.D. dissertation. The motivation for this study is the fact that in previous asset pricing models an investor’s age never played any role in affecting investment-consumption decisions, which is clearly counterfactual. Given the recent dramatic demographic changes in the U.S., it has become ever more important to understand the implications of aging for financial decision making. In the context of the life-cycle investment hypothesis, we first examine how demographic changes can affect capital markets. The hypothesis states that an investor's portfolio composition changes with the life cycle. More specifically, at an early stage an investor allocates relatively more wealth in housing and then switches to financial assets at a later stage. The hypothesis suggests that, when the average age goes up, the stock market should rise but the price of housing should drop. This prediction is strongly supported in the post-1945 period. The second hypothesis examined in the paper asserts that an investor's risk aversion increases with age. We test this hypothesis by estimating the Euler equation of the representative investor. In the post-1945 period, we find significant variation in risk aversion. We also find that a rise in average age predicts a rise in future risk premiums, which is consistent with our hypothesis. Fluctuations in annual risk premiums are at least partly attributable to demographic changes. Our findings are largely due to the joint effect

of the Baby Boom and the increasing life expectancy.

Also based on a chapter of my Ph.D. dissertation, my second paper with Gurdip, "The Spirit of Capitalism and Stock Market Prices," extends the existing modeling paradigm in a different direction. In existing economics, wealth is no more valuable than its implied consumption rewards. In real life, however, investors acquire wealth not just for its implied consumption, but for the resulting social status. Max M. Weber refers to this desire for wealth as the spirit of capitalism. We examine, both analytically and empirically, implications of Weber's hypothesis for consumption, savings, and stock prices. It is shown that when investors care about relative social status, propensity to consume and risk-taking behavior will all depend on social wealth standards and stock prices will be more volatile than otherwise. The spirit of capitalism thus seems to be a driving force behind stock market volatility and economic growth.

Recently, we have concentrated efforts on exploring the relative advantages of the Lucas-type pure-exchange economic framework, as opposed to the Breeden, Cox-Ingersoll-Ross and Merton production-economic modeling framework, for the purpose of asset pricing in general and contingent claims valuation in particular. The CIR-Merton production-economy framework has been by far the main paradigm for continuous-time asset pricing developments. In applying their framework, one typically needs to follow two steps. First, solve the Bellman equation in closed-form for the investor’s indirect utility of wealth as well as for the investor’s optimal consumption-portfolio policy; Second, rely on the analytical solution from the first step to solve the fundamental valuation PDE. However, since the Bellman equation is a highly non-linear PDE, it is extremely difficult, if possible at all, to solve it and finish the first step, unless one assumes, as is often done in the literature, constant investment opportunities or a log utility function for the representative agent. This difficulty hence severely limits the appeal of the production-economy approach to asset pricing. In my paper with Gurdip, "An Alternative Valuation Model for Contingent Claims," we study security valuation by adopting the continuous-time counterpart of a Lucas-type pure-exchange economy. The fundamental valuation equation derived in this context differs from its Cox-Ingersoll-Ross production-economy counterpart in an important way: it is expressed in terms of the direct utility function and an exogenous output process, whereas the latter in terms of the indirect utility function and the endogenous wealth process. Consequently, our exchange economy-based valuation approach avoids the need to solve the nonlinear Bellman equation and hence offers superior tractability. To substantiate this point, we apply the approach to derive closed-form solutions for bond, bond option, individual stock, and stock option prices, under a more general setting than allowable using the Cox-Ingersoll-Ross framework. The resulting interest rate and stock price dynamics have many desirable properties. Our stock option pricing model with stochastic volatility and stochastic interest rates is also shown to have the ability to reconcile certain puzzling empirical regularities such as the volatility smile.

The results in the above paper concerning security valuation in an exchange economy has many implications for both theory and empirical research in asset pricing. We have been, and will continue, exploring those implications in different directions. To give a few examples, in the paper "Inflation, Asset Prices, and the Term Structure of Interest Rates in Monetary Economies," Gurdip and I have taken advantage of the tractability of the exchange-economy setup, to introduce a role for money and to study properties of asset prices and interest rates (both real and nominal). It should be recognized that in monetary economies, the price level, inflation, asset prices, and the real and nominal interest rates all have to be determined simultaneously and in relation to each other. This link allows us

to relate in closed form each of the dependent entities to the underlying real and monetary variables. Among other features of such economies, inflation can be partially non-monetary and the real and nominal term structures can depend on fundamentally different risk factors. It is found that in one extreme, the process followed by the real term structure can be independent of that followed by its nominal counterpart, which is a surprise because, according to the Fisher identity, nominal interest rates should always be correlated with their real counterpart. This study has developed a tractable monetary asset pricing framework.

Extending this tractable monetary exchange-economy model to an international two-country context, we examine the joint equilibrium determination of nominal exchange rates, domestic and foreign interest rates, forward and futures currency prices, spot and futures currency option prices, in our paper "Equilibrium Valuation of Foreign Exchange Claims." Since all rates and prices are endogenously determined and empirically plausible, it guarantees the internal consistency of these price processes with a general equilibrium. Especially, closed-form valuation formulas are presented for exchange rate options and exchange rate futures options. Common to these formulas is that stochastic volatility and stochastic interest rates are admitted. Hedge ratios and other comparative statics are provided analytically. It is shown that most existing currency option models are included as special cases.

The exchange economy-based valuation approach not only makes security valuation more tractable but also renders empirical asset pricing tests easier to conduct. In testing the standard CAPM, the Ross APT, the Merton intertemporal CAPM, and the Breeden and Lucas consumption-based asset pricing, for example, researchers need to rely on either the market portfolio or the aggregate consumption or both. According to Roll, the market portfolio is intrinsically unobservable. Aggregate consumption data is simply not available at relatively high frequency (e.g. daily, weekly and any frequency less than monthly). Even though aggregate consumption data is available at the monthly and quarterly frequencies, it is not of useable quality. Attempts have been made, for instance, by Campbell, to avoid the dependence on consumption data. But, since the theory that Campbell relies on is still production-economy-based, he cannot reduce the testable pricing equations to a form free of the market portfolio. In the paper "Asset Pricing without Consumption or Market Portfolio Data," Gurdip and I further exploit the exchange-economy framework to first make the pricing results expressed only in terms of unidentified factors that affect future consumption/output growth opportunities. Those equilibrium pricing results are independent of both consumption and market portfolio data. Next, we solve for the term structure of interest rates in closed form, which allows us to write the endogenous term structure variables as well-behaved functions of the unidentified factor state variables. Consequently, we use these internal solutions to substitute out the unidentified factor state variables by the observable term structure variables (e.g., yields and yield spreads). This substitution results in an Euler equation and a multi-factor pricing relation that are given in terms of observable endogenous variables such as interest rates and yield spreads. In other words, unlike the existing practice of extracting factor-mimicking portfolios or choosing economic factors in an ad hoc manner, our theory specifies an exact manner in which the pricing factors should be chosen. The chosen pricing factors have to be internally consistent with the model. This treatment resulting from our model is particularly appealing, because each testable equilibrium pricing relation is expressed as a function of readily observable financial market variables. Finally, we conduct Euler equation-based tests of our model without using consumption or market portfolio data, and the empirical results are strongly supportive of the moment restrictions.

Substantial progress has been made in extending the Black-Scholes model to incorporate either stochastic interest rates or stochastic volatility. But, there is not yet any comprehensive empirical study demonstrating whether and by how much each generalized feature will improve option pricing and hedging performance. In "Option Pricing and Hedging Performance under Stochastic Volatility and Stochastic Interest Rates," Gurdip Bakshi, Charles Cao and I fill this gap by first relying on the exchange-economy valuation approach to develop a simplified closed-form option pricing model that has the said features. The model includes many known ones as special cases. Based on the model, both delta-neutral and single-instrument minimum-variance hedging strategies are derived analytically. Using S\&P 500 option prices, we then compare the pricing and hedging performance of this model with that of three existing ones that respectively allow for (i) constant volatility and constant interest rates (the Black-Scholes), (ii) constant volatility but stochastic interest rates, and (iii) stochastic volatility but constant interest rates. Overall, incorporating stochastic volatility and stochastic interest rates produces the best performance in pricing and hedging, with the remaining pricing and hedging errors no longer so systematically related to contract features. The second performer in the horse-race is the stochastic volatility model, followed by the stochastic interest rates model and then by the Black-Scholes.

I have also done research on equilibrium properties of frictional economies. The first question that motivated me venturing into this area concerns the booming phenomenon of financial innovations. In a perfect Modigliani-Miller world, security innovations will not matter. Then, why have there been so many financial innovations? What are the incentives? In the paper "Financial Innovation and Arbitrage Pricing in Frictional Economies," I make the point that in economies with trading frictions, the incentives to innovate come from, among other things, the sublinearity of the equilibrium price function. When decomposing existing securities and opening new markets, innovators serve i) an informational role by revealing the fundamental values of more payoffs; ii) a cost-saving role by making future consumption achievable at lower costs; (iii) a spanning-improving role by enabling investors to better undo the effect of frictions on risk sharing. In particular, they serve these economic functions even when the newly-introduced securities are redundant but not yet issued. I also demonstrate that the strength of the arbitrage valuation approach is severely limited in frictional economies.

I

Continuing with equilibrium modeling of frictional economies, Gurdip and I study, in the paper "Market Frictions and the Preferred Habitat Theory of the Term Structure of Interest Rates," economies in which (i) investors receive too much labor income in certain time-states and too little in other time-states; (ii) investors are not allowed to issue claims against future labor income; and (iii) holdings in default-free bonds and other securities are subject to lower bounds. In this type of economies with idiosyncratic income risks, incomplete markets and trading frictions, investors can only use bonds and other securities as a buffer against income fluctuations, but, due to frictions, it is generally impossible to achieve perfect risk sharing. Consequently, they develop strong preferences, or habitats, for certain maturity bonds. Each bond is then priced according to how much the investors with habitats matching the bond's maturity are willing to pay. Interest rates are thus determined by habitats. This habitat effect, however, weakens as the frequency of bond trading increases. We thus demonstrate that the Preferred Habitat Hypothesis of the term structure on interest rates can be supported by the presence of trading frictions and idiosyncratic income risks.

Concluding Remarks

In summary, my work has been focused on asset pricing, with a strong application orientation. I will continue my efforts in the same directions as reviewed above. Financial market implications of demographic changes still need to be addressed, and there are many policy implications in this regard. We don’t yet completely know what are investors’s optimal consumption-investment policies and their resulting impact on financial market prices when they save and invest not just for the implied consumption rewards but also for the social and political status implications. Next, there are other potential applications of the exchange-economy-based valuation approach. For example, in a production-economy framework stock prices are exogenously given and not endogenously determined, whereas in an exchange-economy setup stock prices are endogenously determined together with other security prices. This means that the exchange-economy framework is better suited to address individual stock valuation issues, a research topic that is of great implications not just for the theory of finance but also for financial practice.

Finally, my future research focus is still on the valuation of individual stocks and contingent claims.