Market pricing of executive stock options and implied risk preferences

When managers get to trade in options received as compensation, their trading prices reveal several aspects of subjective option pricing and risk preferences. Two subjective pricing models are fitted to show that executive stock option prices incorporate a subjective discount. It depends positively on implied volatility and negatively on option moneyness. Further, risk preferences are estimated using the semiparametric model of Aït-Sahalia and Lo (2000). The results suggest that relative risk aversion is just above 1 for a certain stock price range. This level of risk aversion is low but reasonable, and it may be explained by the typical manager being wealthy and having low marginal utility. Related to risk aversion, it is found that marginal rate of substitution increases considerably in states with low stock prices.     

The Recovery Theorem

We can only estimate the distribution of stock returns, but from option prices we observe the distribution of state prices. State prices are the product of risk aversion—the pricing kernel—and the natural probability distribution. The Recovery Theorem enables us to separate these to determine the market's forecast of returns and risk aversion from state prices alone. Among other things, this allows us to recover the pricing kernel, market risk premium, and probability of a catastrophe and to construct model‐free tests of the efficient market hypothesis.     

Empirical risk aversion functions-estimates and assessment of their reliability

In this paper, we examine investor's risk preferences implied by option prices. In order to derive these preferences, we specify the functional form of a pricing kernel and then shift its parameters until realized returns are best explained by the subjective probability density function, which consists of the ratio of the risk-neutral probability density function and the pricing kernel. We examine, alternatively, pricing kernels of power, exponential, and higher order polynomial forms. Using S&P 500 index options, we find surprising evidence of risk neutrality, instead of risk aversion, in both the power and exponential cases. When extending the underlying assumption on the specification of the pricing kernel to one of higher order polynomial functions, we obtain functions exhibiting ‘monotonically decreasing’ relative risk aversion (DRRA) and anomalous ‘inverted U-shaped’ relative risk aversion. We find, however, that only the DRRA function is robust to variation in sample characteristics, and is statistically significant. Finally, we also find that most of our empirical results are consistent, even when taking into account market imperfections such as illiquidity.     

A Comparison of Option Prices Under Different Pricing Measures in a Stochastic Volatility Model with Correlation

This paper investigates option prices in an incomplete stochastic volatility model with correlation. In a general setting, we prove an ordering result which says that prices for European options with convex payoffs are decreasing in the market price of volatility risk.As an example, and as our main motivation, we investigate option pricing under the class of q-optimal pricing measures. The q-optimal pricing measure is related to the marginal utility indifference price of an agent with constant relative risk aversion. Using the ordering result, we prove comparison theorems between option prices under the minimal martingale, minimal entropy and variance-optimal pricing measures. If the Sharpe ratio is deterministic, the comparison collapses to the well known result that option prices computed under these three pricing measures are the same.As a concrete example, we specialize to a variant of the Hull-White or Heston model for which the Sharpe ratio is increasing in volatility. For this example we are able to deduce option prices are decreasing in the parameter q. Numerical solution of the pricing pde corroborates the theory and shows the magnitude of the differences in option price due to varying q.JEL Classification: D52, G13     

An Equilibrium Model of Rare-Event Premia and Its Implication for Option Smirks

This article studies the asset pricing implication of impreciseknowledge about rare events. Modeling rare events as jumps inthe aggregate endowment, we explicitly solve the equilibriumasset prices in a pure-exchange economy with a representativeagent who is averse not only to risk but also to model uncertaintywith respect to rare events. The equilibrium equity premiumhas three components: the diffusive- and jump-risk premiums,both driven by risk aversion; and the "rare-event premium,"driven exclusively by uncertainty aversion. To disentangle therare-event premiums from the standard risk-based premiums, weexamine the equilibrium prices of options across moneyness or,equivalently, across varying sensitivities to rare events. Wefind that uncertainty aversion toward rare events plays an importantrole in explaining the pricing differentials among options acrossmoneyness, particularly the prevalent "smirk" patterns documentedin the index options market.     

Equilibrium asset pricing: with non-Gaussian factors and exponential utilities

We analyse the equilibrium asset pricing implications for an economy with single period return exposures to explicit non-Gaussian systematic factors, that may be both skewed and long-tailed, and Gaussian idiosyncratic components. Investors maximize expected exponential utility and equilibrium factor prices are shown to reflect exponentially tilted prices for non-Gaussian factor risk exposures. It is shown that these prices may be directly estimated from the univariate probability law of the factor exposure, given an estimate of average risk aversion in the economy. In addition, a residual form of the capital asset pricing model continues to hold and prices the idiosyncratic or Gaussian risks. The theory is illustrated on data for the US economy using independent components analysis to identify the factors and the variance gamma model to describe the probability law of the non-Gaussian factors. It is shown that the residual CAPM accounts for no more than 1% of the pricing of risky assets, while the exponentially tilted systematic factor risk exposures account for the bulk of risky asset pricing.     

A valuation algorithm for indifference prices in incomplete markets

A probabilistic iterative algorithm is constructed for indifference prices of claims in a multiperiod incomplete model. At each time step, a nonlinear pricing functional is applied that isolates and prices separately the two types of risk. It is represented solely in terms of risk aversion and the pricing measure, a martingale measure that preserves the conditional distribution of unhedged risks, given the hedgeable ones, from their historical counterparts.Received: 1 September 2003, Mathematics Subject Classification: 93E20, 60G40, 60J75JEL Classification: C61, G11, G13The second author acknowledges partial support from NSF Grants DMS 0102909 and DMS 0091946.     

Asset pricing and energy consumption risk

This paper proposes energy consumption in the US as a new measure for the consumption capital asset pricing model. We find that (i) industrial energy growth produces reasonable values for the relative risk aversion coefficient and the implied risk-free rate; (ii) compared to alternative consumption measures, industrial energy performs well in explaining the cross-sectional variation in stock returns with the lowest implied risk aversion and pricing errors; (iii) the industrial energy consumption risk model performs equally well as the Fama–French three-factor model in the cross-sectional asset pricing tests; and (iv) total energy consumption risk is priced in the presence of the Fama–French factor risks.     

What drives stochastic risk aversion?

This paper examines determinants of stochastic relative risk aversion in conditional asset pricing models. Novel time-series specification tests are proposed as direct extensions of Guo, Wang, and Yang (2013, JMCB)'s model using nonlinear state-space models with heteroskedasticity. I then establish the following facts. First, the surplus consumption ratio implied by the external habit formation model is the most important determinant of relative risk aversion. Second, the CAY of Lettau and Ludvigson (2001a) without a look-ahead bias and the short term interest rate explain part of relative risk aversion. Third, the estimated risk aversion from 1957Q2 to 2010Q3 is countercyclical and positive. Finally, the selected models explain part of the momentum and the financial distress premiums.     

The valuation of nonsystematic risks and the pricing of Swedish lottery bonds

Swedish government lottery bonds have coupon payments determinedby lottery. They offer a unique opportunity to study a securitywith uncertain payoffs having a known, observable distribution.The risk associated with the lotteries is idiosyncratic by constructionand should not command a risk premium in equilibrium. The bondsare traded in two forms, allowing us to evaluate the rewardsto bearing extra lottery risk. Despite its idiosyncratic nature,we find prices appear to reflect aversion to this risk. We evaluatethe empirical determinants of this differential pricing andpossible explanations for it.     

Addressing insurance price discrimination in an era of diversity,equity, and inclusion

The early 2020s diversity, equity, and inclusion movement has prompted debate about banning the use of suspect insurance pricing variables because they discriminate against protected classes, such as gender. This paper demonstrates how banning an insurance pricing variable currently used in insurance pricing models can result in regulatory adverse selection if the ban heterogeneously combines policyowners with different expected losses into the same risk class, contrary to risk-based pricing. The paper begins by describing several recent regulatory and judicial decisions to ban insurance pricing variables. It next describes the process used by insurers to set insurance prices, followed by a discussion of applicable insurance discrimination laws. Using a simple risk aversion model, the paper next examines whether a ban on gender-based auto insurance pricing in California in 2019 results in regulatory adverse selection. The paper concludes by describing possible alternative pricing variables available to auto insurers if gender-based pricing is banned.     

Generalized Disappointment Aversion and Asset Prices

We characterize generalized disappointment aversion (GDA) risk preferences that can overweight lower‐tail outcomes relative to expected utility. We show in an endowment economy that recursive utility with GDA risk preferences generates effective risk aversion that is countercyclical. This feature comes from endogenous variation in the probability of disappointment in the representative agent's intertemporal consumption‐saving problem that underlies the asset pricing model. The variation in effective risk aversion produces a large equity premium and a risk‐free rate that is procyclical and has low volatility in an economy with a simple autoregressive endowment‐growth process.     

Do prices reflect market fundamentals in real estate markets?

The purpose of this article is to examine prices on land and REIT shares for possible evidence of deviations from market fundamentals, the underlying economic forces. Models of market fundamentals are developed from the intertemporal capital asset pricing model so that risk aversion and a stochastic investment opportunity set can be incorporated in the analysis. The approach in this article is to compare ex post values of actual discounted cash flows with prices and to test whether the price series are unbiased predictors of the future discounted cash flows. Several tests of the relationship are presented, and the results suggest that prices of real estate investment trusts and prices of farm land do not always reflect fundamental value.     

Margin requirements and equilibrium asset prices

This paper studies the effect of margin requirements on asset prices and trading volume in a general equilibrium asset pricing model where Epstein-Zin investors differ in their degree of risk aversion. Under the assumptions of unit intertemporal elasticity of substitution and zero net supply of riskless assets, I show analytically that binding margin requirements do not affect stock prices. This result stands in contrast to previous partial equilibrium analysis where fixed margin requirements increase the volatility of stock prices. In this framework, binding margin requirements induce a fall in the riskless rate, increase its volatility, and increase stock trading volume.     

Extracting Information from Options Markets: Smiles, State–Price Densities and Risk Aversion

In this paper, recent techniques of estimating implied information from derivatives markets are presented and applied empirically to the French derivatives market. We determine nonparametric implied volatility functions, state–price densities and historical densities from a high–frequency CAC 40 stock index option dataset. Moreover, we construct an estimator of the risk aversion function implied by the joint observation of the cross–section of option prices and time–series of underlying asset value. We report a decreasing implied volatility curve with the moneyness of the option. The estimated relative risk aversion functions are positive and globally consistent with the decreasing relative risk aversion assumption.     

Asset pricing with stochastic differential utility

Asset pricing theory is presented with representative-agentutility given by a stochastic differential formulation of recursiveutility. Asset returns are characterized from general first-orderconditions of the Hamilton-Bellman-Jacobi equation for optimalcontrol. Homothetic representative agent recursive utility functionsare shown to imply that excess expected rates of return on securitiesare given by a linear combination of the continuous-time market-portfolio-basedcapital asset pricing model (CAPM) and the consumption-basedCAPM. The Cox, Ingersoll and Ross characterization of the termstructure is examined with a recursive generalization, showingthe response of the term structure to variations in risk aversion.Also, a new multicommodity factor-return model, as well as anextension of the 'usual' discounted expected value formula forasset prices, is introduced.     

Option Pricing Bounds and the Elasticity of the Pricing Kernel

In this paper we use power functions as pricing kernels to derive option-pricing bounds. We derive option pricing bounds given the bounds of the elasticity of the true pricing kernel. The bounds of the elasticity of the true pricing kernel are closely related to the bounds of the representative investor's coefficient of relative risk aversion. This methodology produces a tighter upper call option bound than traditional approaches. As a special case we show how to use the Black–Scholes formula to obtain option pricing bounds under the assumption of lognormality.     

Unconventional monetary policy announcements and risk aversion: evidence from the U.S. and European equity markets

ABSTRACT

This paper examines the role of unconventional monetary policy announcements on risk aversion – as proxied by the variance premium – by using panel data analysis. The objective of this empirical analysis is to investigate the risk-taking channel of monetary policy for the major European and U.S. equity markets by studying the impact that the announcements of an unconventional monetary policy has on market uncertainty and risk perception. By measuring the difference between risk-neutral and realised and conditional variance, we estimate the variance premium, which captures the impact that pricing concerns have on the prices of options. The empirical analysis indicates that easing monetary policies can significantly reduce the variance premium. In addition, we examine the risk premium structure across markets to determine the potential differences in investors’ risk aversion.     

Sharing rules and equilibrium in an international capital market under uncertainty

International capital market equilibrium is characterized for a world economy in which consumption preferences are defined multiplicatively over many commodities. It is shown that the set of relative asset prices under pure exchange in international capital markets depends on the real purchasing power of nominal payoffs under uncertainty and does not depend on the currency in which the nominal payoffs are denominated. A Sharpe-Lintner type international capital asset pricing model is derived as a special case. Proportional ad valorem commodity taxes and transportation costs are incorporated in the valuation model, interest rate parity and purchasing power parity are reinterpreted under uncertainty, and international differences in borrowing and lending are shown to reflect, in part, differences in risk aversion across countries.     

A Utility-Based Model of Common Stock Price Movements

This paper develops and tests a nonlinear utility-based econometric model of the temporal behavior of aggregate stock price movements based on a constant relative risk aversion utility function and an observable information set consisting of aggregate consumption, aggregate dividends, and past stock prices. The stochastic process derived from time-series analyses of consumption and dividends measured over annual intervals is used to derive and empirically test a closed-form solution for stock-price movements. The endogenization of discount rate changes in the utility-based model is shown to be more consistent with aggregate stock price movements over a twenty-year holdout period than constant discount rate models. The model is also used to estimate the representative investor's relative risk aversion. The estimate of 4.22 is consistent with that used by Grossman and Shiller in their perfect foresight model and is significantly higher than the relative risk aversion of 1.0 implied by logarithmic utility.