Empirical pricing kernels?
Joshua V. Ronberg a,* and Robert F. Engle bbe ud to
a Federal Rerve Bank of New York, New York, NY 10045, USA
b Stern School of Business, New York University, New York, NY 10012, USA
阻遏的意思
August 2001
____________________________________________________________________________ Abstract
This paper investigates the empirical characteristics of investor risk aversion over equity return states by estimating a time-varying pricing kernel, which we call the empirical pricing kernel (EPK). We estimate the EPK on a monthly basis from 1991 to 1995, using S&P 500 index option data and a stochastic volatility model for the S&P 500 return process. We find that the EPK exhibits counter cyclical risk aversion over
S&P 500 return states. We also find that hedging performance is significantly improved when we u hedge ratios bad the EPK rather than a time-invariant pricing kernel.bec商务英语中级
技术交流英文
JEL classification: G12, G13, C50
Keywords: Pricing kernels; Risk aversion; Derivatives; Hedging
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?We are grateful for comments from David Bates, Mark Broadie, Peter Bossaerts, Peter Carr, Ravi Bansal, Steve Figlewski, Joel Hasbrouck, Gur Huberman, Jens Jackwerth, Bruce Lehmann, Jo Lopez, Jayendu Patel, Mark Rubinstein, William Schwert, Allan Timmermann, two anonymous referees at the Journal of Financial Economics, and minar participants at New York University, Columbia University, Indiana University, Georgetown University, Northwestern University, Boston University, Ca Western University, the University of Iowa, the Federal Rerve Board of Governors, and the Federal Rerve Bank of New York. The authors also appreciate the comments of participants at the Conference on Risk Neutral and Objective Probability Distributions, the 1998 Western Finance Association meetings, the 1998 Computational and Quantitative Finance conference, and the 1997 Time Series Analysis of High Frequency Data Conference. The authors wish to thank David Hait for rearch assistance.
*Corresponding author. Tel: + 1-212-720-6317; fax: + 1-212-720-1773.
E-mail address: berg@ny.frb (J.V. Ronberg).
1. Introduction
The ast pricing kernel summarizes investor preferences for payoffs over different states of the world. In the abnce of arbitrage, all ast prices can be expresd as the expected value of the product of the pricing kernel and the ast payoff. Thus, the pricing kernel, when it is ud with a probability model for the states, gives a complete description of ast prices, expected returns, and risk premia.
黄油手
In this paper, we estimate the pricing kernel using current ast prices and a predicted ast payoff density. We define the empirical pricing kernel (EPK) as the preference function that provides the “best fit” to ast prices, given the forecast payoff density. By estimating the EPK at a quence of points in time, we can obrve and model the dynamic structure of the pricing kernel itlf. From this analysis, we obtain improved option pricing relations, hedging parameters, and a better understanding of the pattern of risk premia.
We estimate the EPK each month from 1991 to 1995, using S&P 500 index option data and a stochastic volatility model for the S&P 500 return process. We find substantial evidence that the pricing kernel exhibits counter cyclical risk aversion over S&P 500 return states. Empirical risk aversion is positively correlated with indicators of recession (widening of credit spreads) and negatively correlated with indicators of expansion (steepening of term structure slope).
We develop an option hedging methodology to compare the accuracy of veral pricing kernel specifications. Our tests measure relative performance in hedging out-of-the-money S&P 500 put options using at-the-money S&P 500 put options and the S&P 500 index portfolio. We find that hedge ratios formed using a time-varying pricing kernel reduce hedge portfolio volatility more than hedge ratios bad on a time-invariant pricing kernel.
Although there is a large literature on pricing kernel estimation using aggregate consumption data,
problems with impreci measurement of aggregate consumption can weaken the empirical results of the papers. Hann and Singleton (1982, 1983) postulate that the pricing kernel is a power function of aggregate U.S. consumption. They u maximum-likelihood estimation and the generalized method of moments to estimate the pricing kernel. Chapman (1997) us functions of co
nsumption and its lags as pricing kernel state variables, and he specifies the pricing kernel function as an orthogonal polynomial expansion. Hann and Jagannathan (1991) derive bounds for the mean and standard deviation of the consumption-bad pricing kernel in terms of the mean and standard deviation of the market portfolio excess returns.
Recently, Ait-Sahalia and Lo (2000) have ud option data and historical returns data to non-parametrically estimate the pricing kernel projected onto equity return states. This technique avoids the u of aggregate consumption data or a parametric pricing kernel specification. Along similar lines, Jackwerth (2000) nonparametrically estimates the “risk aversion function” using option data and historical returns data.
Ait-Sahalia and Lo (2000) and Jackwerth (2000) estimate investor expectations about future return probabilities by smoothing a histogram of realized returns over the past four years. Implicitly, the papers assume that investors form probability beliefs by equally weighting events over the prior four years and disregarding previous events. For example, using a four-year window, the October 1987 stock market crash influences probability beliefs until October 1991. In November 1991, the crash no longer has an effect on beliefs.
The assumptions are inconsistent with evidence from the stochastic volatility modeling literature—e.g., Bollerslev, Chou, and Kroner (1992)—indicating that future state probabilities depend more on the recent events than long-ago events, but that long-ago events still have some predictive power. Misspecification of state probabilities induces error in the estimation of the pricing kernel, since the denominator of the state-price-per-unit probability is incorrectly measured.
In Ait-Sahalia and Lo (2000) and Jackwerth (2000), state prices and probabilities are averaged over
time, so their estimates are perhaps best interpreted as a measure of the average pricing kernel over the sample period. Since the sample periods ud are at least one year in length, neither paper detects time variation at less than an annual frequency. Average pricing kernels are also limited in their ability to price and hedge asts on an ongoing basis, since asts are correctly priced only when risk aversion and state probabilities are at their average level. As noted by Ait-Sahalia and Lo (2000), “In contrast, the kernel SPD estimator is consistent across time [emphasis added] but there may be some dates for which the SPD estimator fits the cross ction of option prices poorly and other dates for which the SPD estimator performs very well.”
The remainder of the paper is organized as follows. Section 2 describes the theory and previous res
earch related to the pricing kernel. Section 3 prents the empirical pricing kernel estimation technique, EPK specification, and hedge ratio specification. Section 4 describes the data ud for estimation, and Section 5 prents the estimation results. Section 6 contains the hedging test results, and Section 7 concludes the paper.
2. Theory and previous rearch
Our initial discussion of ast pricing kernel theory and previous rearch introduces veral pricing kernel specifications and discuss some potential estimation problems. We then consider the characteristics of pricing kernel projections.
2.1. The ast pricing kernel
The ast pricing kernel is also known as the stochastic discount factor, since it is a state-dependent
function that discounts payoffs using time and risk preferences. Campbell, Lo, and MacKinlay (1997) and Cochrane (2001) provide comprehensive treatments of the role of the pricing kernel in ast pricing. Other related papers include Ross (1978), Harrison and Kreps (1979), Hann and Richard (1987), and Hann and Jagannathan (1991).
In the abnce of arbitrage, the current price of an ast equals the expected pricing-kernel-weighted payoff:
[]1+=t t t t X M E P (1)
where P t is the current ast price, M t is the ast pricing kernel, and X t+1 is the ast payoff in one period. In Lucas’s (1978) consumption-bad ast pricing model, the pricing kernel is equal to the intertemporal marginal rate of substitution, so M t = U’(C t+1)/U’(C t ). Under the assumption of power utility, the pricing kernel is M t = e -ρ(C t+1/C t )-γ, with a rate of time preference of ρ and a level of relative risk aversion of γ.
One of the basic characteristics of the pricing kernel is its slope, and standard risk-aversion measures are usually functions the pricing kernel slope. For example, the Arrow-Pratt (1964, 1965) measure of absolute risk aversion is the negative of the ratio of the derivative of the pricing kernel to the pricing kernel. The Arrow-Pratt measure of relative risk aversion is absolute risk aversion multiplied by current consumption:
[])(/)(11'1+++−=t t t t t t C M C M C γ. (2)
longlongagoGenerally, the pricing kernel will depend not only on current and future consumption, but also on all
joker延时喷剂variables that affect marginal utility. In the habit persistence models of Abel (1990), Constantinides (1990), or Campbell and Cochrane (1999), the pricing kernel depends on both past and current consumption.
Eichenbaum, Hann, and Singleton (1988) let the pricing kernel depend on leisure, while Startz (1989) us durable goods purchas. Bansal and Viswanathan (1993) specify the pricing kernel as a function of the equity market return, the Treasury bill yield, and the term spread.
布拉德皮特 僵尸When the pricing kernel is a function of multiple state variables, the level of risk aversion can also
fluctuate as the variables change. Campbell (1996) shows that a habit persistence utility function exhibits time-varying relative risk aversion, where relative risk aversion is decreasing in the amount that consumption exceeds the habit (the surplus consumption ratio). In Campbell’s (1996) model, we obrve decreas in relative risk aversion during economic expansions when consumption is high relative to the habit.
Furthermore, we obrve increas in relative risk aversion during economic contractions when consumption falls clor to the habit. In contrast, the power utility function exhibits relative risk aversion that is time-invariant.特点英语
To investigate the characteristics of investor preferences, many rearchers have ud Eq. (1) as an identifying equation for the pricing kernel. For example, Hann and Singleton (1982) identify the pricing kernel with an unconditional version of this equation:
[]1)/(01−=+t t t M P P E . (3)
Hann and Singleton (1982), using an approach followed in many subquent papers, specify the aggregate consumption growth rate as a pricing kernel state variable. They measure consumption using data from the National Income and Products Accounts (NIPA).
the hor whispererHowever, measurement error in the NIPA
consumption data can po a significant problem.
