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On Kernels and Sentiment


In his survey of asset pricing theory at the millenium, Campbell (2000) discusses traditional models in terms of the pricing kernel or stochastic discount factor (SDF). Towards the end of his survey, Campbell includes a discussion of behaviorally-based models. Notably, he does not couch this portion of his article in terms of the SDF. Yet, the SDF is a very general concept that applies whenever all pure arbitrage opportunities are fully exploited.

In this paper, I analyze the properties of the SDF in a behaviorally-based framework. The central question of the paper is conveyed by its title, and is as follows. How can the concept of sentiment be formally defined so as to identify the manner in which traders’ errors are manifest in the pricing kernel?

The central result of the paper is that the log-kernel can be decomposed into the sum of two terms, one term being sentiment and the other being an expression that only depends on economic fundamentals. Because the pricing kernel underlies the pricing of all assets, I analyze how traders’ errors impact the pricing of major asset classes, namely fixed income securities, options, mean-variance efficient portfolios, and the market portfolio.

The focal point of this paper is the impact of heterogeneity on the structure of the SDF. I make this the focal point because heterogeneity is the most common trait in the wide spectrum of studies that make up the literature in behavioral decision-making. This literature describes the manner in which individuals depart from the traditional rationality assumptions concerning Bayesian beliefs and expected utility maximization. In this respect, heterogeneity pervades the responses of subjects in prediction studies by Grether (1980) and De Bondt (1993) about the use of heuristics in place of Bayes rule.

It pervades the responses of the subjects in the studies on non-expected utility maximizing behavior (Kahneman and Tversky, 1979, 1992). And it pervades studies on individual risk tolerance (Barsky, Kimball, Miles, and Juster, 1997). I note that my formal model accommodates both heterogeneous beliefs and heterogeneous risk tolerance.

However, most of the paper emphasizes the role impact of heterogeneous beliefs on the kernel and asset prices. I suggest that the paper makes five distinct contributions. The first contribution of the paper is to establish the manner in which a behavioral-SDF is different from its traditional counterpart.

To address this issue, I note that traditional asset pricing models often assume that assets can be priced as if there were an expected utility maximizing representative trader who holds objectively correct beliefs. Under this assumption, the SDF can be interpreted as the ratio of the marginal utility of future consumption to the marginal utility of current consumption, across states of nature. Under standard concavity assumptions this implies that the SDF is a monotone decreasing function of consumption growth.

In contrast, a behavioral-SDF is not typically monotone decreasing. Instead, a behavioral-SDF typically exhibits a U-shaped smile pattern. The second contribution of the paper is to provide a general definition of sentiment. Most behavioral asset pricing models define a sentiment variable that measures the degree to which the means of noise traders’ subjective return distributions are too high (optimism) or too low (pessimism). See De Long, Shleifer, Summers, and Waldmann (1990), Barberis, Shleifer, and Vishny (1998) and Daniel, Hirshleifer, Subrahmanyan (1998).

I suggest that the focus on the mean error is overly narrow, and instead focus on the full stochastic process characterizing traders’ errors. This issue is key to understanding how these errors affect the SDF. In this respect, the SDF is a stochastic process. In the paper, I establish that the log-SDF can be decomposed into the sum of two terms. The first term is my sentiment variable. And the second term depends only on fundamentals. Because asset prices are obtained by integrating the product of an asset’s payoff and the SDF, the decomposition result serves to identify the formal channel through which sentiment affects all asset prices.

The third contribution of the paper is to argue that a U-shaped smile in the SDF gives rise to different asset pricing properties than a monotone decreasing pattern. To make this point concretely, I focus on option pricing. In particular, I develop some general equilibrium examples where there are two special cases. In the first case, traders are homogeneous, the log-SDF is monotone decreasing, and European options are priced by the Black-Scholes formula.

In the second case, beliefs are heterogeneous, the log-SDF is U-shaped, and option prices are no longer given by Black-Scholes. Instead, equilibrium option prices exhibit a volatility smile pattern. That is, the smile in the log-SDF manifests itself in terms of option volatility smile patterns.

Heterogeneity in beliefs is not the only reason for smile effects in option prices. Both heterogeneous risk tolerance and stochastic volatility can give rise to smiles. However, the latter smile effects are consistent with efficient prices, they are efficient smiles. The smile effects caused by heterogeneous beliefs are inefficient smiles. Although much of the analysis in the paper deals with option prices, I hasten to mention that the implications of an SDF-smile are not confined to options. I argue that the SDF-smile leads to a mean-variance frown, with the frown corresponding to an inverted-U in the relationship between the return to a risky mean-variance portfolio and aggregate consumption growth. In addition, the SDF-smile typically prevents the expectations hypothesis of the term structure of interest rates from holding.

The fourth contribution of the paper pertains to the characteristics of a representative trader who sets prices. There is a rich literature that goes back as far as Lintner (1969) analyzing the existence of a representative trader. See also Rubinstein (1974), Brennan and Kraus (1978), Cuoco and He (1994), and Basak (2000). This paper adds to that literature, by showing that the representative trader’s subjective probability density functions tend to be multi-modal and fat-tailed. In contrast, the representative trader in traditional models tends to hold objectively correct beliefs in which probability density functions are unimodal and do not have fat tails.

My representative trader is a market-based collage of the beliefs, errors, and risk tolerances of the individual traders. My framework serves as the theoretical basis for Shefrin (1999), who discusses the relationship between option smiles and various measures of sentiment.

In contrast, the recent literature in option pricing seeks to explain option smiles through a combination of stochastic volatility, jumps, and stochastic interest rates. Notably, this literature features a traditional representative trader who knows the objectively correct probabilities. See Bates (1996, 2000), Bakshi, Cao, and Chen (1997), David and Veronesi (2000), Jackwerth (2000), Ait-Sahalia and Lo (2000), and Pan (2001) who study smile effects when the underlying asset is the S&P 500. To be sure, nonzero sentiment stemming from heterogeneity can cause volatility and interest rates to be stochastic, as well as induce nonfundamentally-based jumps.

In this respect, the sentiment-based approach to option smiles has much in common with the above literature. Yet, there are serious differences. The major difference between, say Pan’s approach, and mine, boils down to the nature of the representative trader. My representative trader holds erroneous beliefs. Shefrin’s analysis of smiles in S&P 500 option prices, especially in regard to out-of-the-money puts, focuses on excessive pessimism by some traders. Pan’s representative trader holds objectively correct beliefs, and therefore is not excessively pessimistic about negative jumps. However, her representative trader is differentially averse to three kinds of shocks, diffusive, volatility, and jump. In particular, her representative trader is particularly averse to a large negative jump[2].

The final contribution of the paper is to identify conditions under which markets will aggregate heterogeneous beliefs in a manner that leads prices to be efficient. This is a nontrivial issue that both traditional asset pricing theorists and behavioral asset pricing theorists have tended to dodge. Traditional asset pricing theorists dodge the issue by assuming that all traders hold objectively correct beliefs, in which cases prices only reflect fundamentals and are therefore efficient. Behavioral asset pricing theorists dodge the issue by making assumptions that virtually guarantee that prices will be inefficient. They assume that securities are priced by a representative noise trader, or equivalently by identical noise traders who all commit the same error. In this case, there is no opportunity for errors to wash out in the aggregate. In my framework, traders can commit different errors from one another. Hence, in principle it is possible for these errors to cancel at the level of the market, and for prices to be efficient. Under what conditions will such cancellation occur?

The short answer is that they cancel if and only if the sentiment process is zero. My zero sentiment condition extends a result in Shefrin and Statman (1994). They establish that in a log-utility framework, prices are efficient if and only if the wealth-weighted mean trader error and error-wealth covariance sum to zero. Note that this condition implies that prices are efficient when the mean error is zero, and the error-wealth covariance is zero. When individual traders’ errors are nonzero, but the mean error is zero, then traders’ errors are said to be nonsystematic. When the error-wealth covariance is zero, then individual traders’ errors are distributed uniformly throughout the trading population rather than being concentrated among the most active traders.

The Shefrin-Statman efficiency condition is dynamic and stochastic. It may hold at some dates but not at others. Hence, prices can be efficient at some dates, but not at others. This is important, in that markets are not perpetually efficient as the efficient market school would have us believe, nor perpetually inefficient. In this respect, consider the 1998 collapse of hedge fund Long-Term Capital Management (LTCM), whose partners included Nobel laureates Myron Scholes and Robert Merton. In discussing this event for the television program NOVA, the late Nobel laureate Merton Miller commented: “Models they were using were based on normal behavior in the markets. When behavior got wild, no models were able to put up with it.” It seems to me that the models to which Miller refers are of the Black-Scholes variety, for which Scholes and Merton were acknowledged with a shared Nobel Prize. Before its collapse, LTCM had actually been quite successful. This initial success demonstrates that in some circumstances these models did work. But the wild behavior in 1998 to which Miller refers is evidence that these models lack the flexibility to accommodate mispricing stemming from nonzero sentiment. Despite recognizing wild behavior in markets, Miller rejected the idea that we should develop models to explain this wild behavior.

In contrast, I contend that we need to develop a general option pricing formula that coincides with Black-Scholes, or its stochastic volatility extension, when markets are “well-behaved,” and deviates from Black-Scholes when they are wildly-behaved. Indeed, that is what I do in the paper. I have organized the paper as follows. Section 2 reviews the literature on heterogeneity, from both a finance perspective and a behavioral perspective. Sections 3 and 4 contain the general framework and preliminary results. Section 5 introduces the formal definition of sentiment, and provides the central result in the paper, namely the log-SDF decomposition theorem. The next two sections, 6 and 7, describe the implications of nonzerosentiment for two classes of assets, risk-free securities of varying maturities and options. Section 8 provides examples to illustrate the main results in the paper.

Section 9 develops the implications of nonzero sentiment for the returns to mean-variance efficient portfolios. Section 10 describes the impact of nonzero sentiment on the return distribution of the market portfolio. Section 11 analyzes the manner in which heterogeneity in risk tolerance is different from heterogeneity in beliefs. Section 12 discusses some of the broader implications attached to the results in the paper. Additional discussion and some of the longer proofs are relegated to the appendix.

Prof. Hersh Shefrin

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