Option-implied Risk-neutral Densities Research Articles

Measuring investors’ risk aversion in China’s stock market

Our paper measures investors’ aversion to downside risk in China’s stock market by comparing two estimates of probability density functions for asset prices. The dynamics of the risk aversion indicator justifies the theoretical argument that the option-implied risk-neutral density incorporates information about investors’ risk preferences which is not captured in the historical data. Our risk aversion indicator sheds light on the investment timing of market practitioners: when the degree of risk aversion is particularly low but about to rise, there will be positive abnormal returns for most market sectors in the near future.

Risk-Neutral Density Estimation: Looking at the Tails

Previous estimation results of risk-neutral densities explain in rather general terms that the tails of the resulting distribution “look fat,” and a way has to be found to model the tails of the estimated distribution. The author uses deep out-of-the-money S&amp;P 500 index options to examine model mispricing of the tails of daily estimated risk-neutral densities. Out-of-sample tests show that model mispricing increases as one moves farther into the tails of the distribution. Across most moneyness groups, model mispricing increases as the option reaches maturity. The author compares two curve-fitting methods that have been proposed in the literature to estimate risk-neutral densities. The first method interpolates with a fourth-order spline and attaches tails from the general extreme value distribution (Figlewski 2010). The second method extends the available implied volatility space by balancing smoothness and fit of the estimated risk-neutral density (Jackwerth 2004). Fitting a fourth-order spline produces a closer fit to the observed implied volatilities. Examining the ability to replicate the implied volatility with the complete estimated option-implied risk-neutral density by looking at mean root-mean-square error, the method by Jackwerth (2004) resulted in lower in- and out-of-sample model mispricing, except for the deepest out-of-the-money put options. <b>TOPICS:</b>Tail risks, options <b>Key Findings</b> • This article compares two methods from the curve-fitting literature to estimate option-implied risk-neutral densities and looks at the accuracy to recover implied volatilities. • Model mispricing, measured by the root-mean-square error, increases for deeper out-of-the-money options. • Model mispricing increases as the option reaches its maturity across most out-of-sample moneyness groups.

Subclavian steal syndrome due to dialysis fistula corrected with subclavian artery stenting

Previous estimation results of risk-neutral densities explain in rather general terms that the tails of the resulting distribution “look fat,” and a way has to be found to model the tails of the estimated distribution. The author uses deep out-of-the-money S&amp;P 500 index options to examine model mispricing of the tails of daily estimated risk-neutral densities. Out-of-sample tests show that model mispricing increases as one moves farther into the tails of the distribution. Across most moneyness groups, model mispricing increases as the option reaches maturity. The author compares two curve-fitting methods that have been proposed in the literature to estimate risk-neutral densities. The first method interpolates with a fourth-order spline and attaches tails from the general extreme value distribution (Figlewski 2010). The second method extends the available implied volatility space by balancing smoothness and fit of the estimated risk-neutral density (Jackwerth 2004). Fitting a fourth-order spline produces a closer fit to the observed implied volatilities. Examining the ability to replicate the implied volatility with the complete estimated option-implied risk-neutral density by looking at mean root-mean-square error, the method by Jackwerth (2004) resulted in lower in- and out-of-sample model mispricing, except for the deepest out-of-the-money put options. <b>TOPICS:</b>Tail risks, options <b>Key Findings</b> • This article compares two methods from the curve-fitting literature to estimate option-implied risk-neutral densities and looks at the accuracy to recover implied volatilities. • Model mispricing, measured by the root-mean-square error, increases for deeper out-of-the-money options. • Model mispricing increases as the option reaches its maturity across most out-of-sample moneyness groups.

Implied risk aversion and pricing kernel in the FTSE 100 index

This paper studies the estimation of the pricing kernel and explains the pricing kernel puzzle found in the FTSE 100 index. We use prices of options and futures on the FTSE 100 index to derive the risk neutral density (RND). The option-implied RND is inverted by using two nonparametric methods: the implied-volatility surface interpolation method and the positive convolution approximation (PCA) method. The actual density distribution is estimated from the historical data of the FTSE 100 index by using the threshold GARCH (TGARCH) model. The results show that the RNDs derived from the two methods above are relatively negatively skewed and fat-tailed, compared to the actual probability density, that is consistent with the phenomenon of “volatility smile.” The derived risk aversion is found to be locally increasing at the center, but decreasing at both tails asymmetrically. This is the so-called pricing kernel puzzle. The simulation results based on a representative agent model with two state variables show that the pricing kernel is locally increasing with the wealth at the level of 1 and is consistent with the empirical pricing kernel in shape and magnitude.

Crash risk and risk neutral densities

“Crash risk” has been one of the major focuses in the recent asset pricing literature. Motivated by the recent literature that suggests an increase in crash risk since Fall 2008 and the recent troubles in the Euro zone, we use EUR/USD FX options for January 2, 2008 to March 18, 2015 to study option-implied risk-neutral densities (RND). We find that RND, especially higher moments, has superior explanatory power in predicting and explaining crash risk and its risk premiums. Furthermore, the higher moments of RND co-move closely with macroeconomic variables. Consistently, we find RND moments outperform the implied volatility from the Black–Scholes model.

Can We Really Discard the Forecasting Ability of Risk-Neutral Distributions?

This paper analyzes the forecasting ability of option-implied Risk-Neutral densities (RNDs) for three US indexes, S&P 500, Nasdaq 100 and Russell 2000 and for a long series (from 1996 to 2015) encompassing two major crisis. Traditional tests rely on restrictive assumptions (mainly normality and independence). In order to overcome these assumptions, we calculate block-bootstrap-based critical values. Different to existent literature, our results conclude failure to reject their forecasting ability, being these results consistent across different forecast horizons, methodologies and indexes considered. We also analyze the fit of the tails of the RNDs separately, finding that they tend to overestimate the frequency of occurrence of events in the left tail, providing a good fitting for the right tail.

Eatimating risk aversion, Risk-Neutral and Real-World Densities using Brasilian Real currency options

This paper uses the Liu et al. (2007) approach to estimate the optionimplied Risk-Neutral Densities (RND), real-world density (RWD), and relative risk aversion from the Brazilian Real/US Dollar exchange rate distribution. Our empirical application uses a sample of exchange-traded Brazilian Real currency options from 1999 to 2011. Our estimated value of the relative risk aversion is around 2.7, which is in line with other articles for the Brazilian Economy. Our out-of-sample results showed that the RND has some ability to forecast the Brazilian Real exchange rate, but when we incorporate the risk aversion, the out-of-sample performance improves substantially.

The Option-Implied Density of the S&amp;P500: What Drives Market Uncertainty and Prediction?

We study the information content of the option-implied risk-neutral density (RND) of the S&P 500 Index. At the extraction level, we address the problem of estimating the tails of the distribution where option prices are observable. We then contribute further to the literature which has yet to incorporate cross-moment dynamics in modeling the RND. We model the extracted time-series of RND moments in a vector autoregression with exogenous variables framework. The RND moment interactions and the financial and economic factors affecting these moments are studied. Findings of significant cross-moment dynamics as well as impact from financial factors on the RNDs echo recent studies which recommend including cross-moment relations for forecasting the RND. We show that the forward-looking information in the predicted RND is useful for predicting market price turning points. The research has implications for traders as well as macroeconomic and financial economists who take an interest in the evolution of option and the underlying index prices.

Estimating Relative Risk Aversion, Risk-Neutral and Real-World Densities Using Brazilian Real Currency Options

Building Risk-Neutral Densities (RND) from options data can provide market-implied expectations about the future behavior of a financial variable. And market expectations on financial variables may influence macroeconomic policy decisions. It can be useful also for corporate and financial institutions decision making. This paper uses the Liu et all (2007) approach to estimate the option-implied Risk-neutral densities from the Brazilian Real/US Dollar exchange rate distribution. We then compare the RND with actual exchange rates, on a monthly basis, in order to estimate the relative risk-aversion of investors and also obtain a Real-world density for the exchange rate. We are the first to calculate relative risk-aversion and the option-implied Real World Density for an emerging market currency. Our empirical application uses a sample of Brazilian Real/US Dollar options traded at BM&F-Bovespa from 1999 to 2011. The RND is estimated using a Mixture of Two Log-Normals distribution and then the real-world density is obtained by means of the Liu et al. (2007) parametric risk-transformations. The relative risk aversion is calculated for the full sample. Our estimated value of the relative risk aversion parameter is around 2.7, which is in line with other articles that have estimated this parameter for the Brazilian Economy, such as Araujo (2005) and Issler and Piqueira (2000). Our out-of-sample evaluation results showed that the RND has some ability to forecast the Brazilian Real exchange rate. Abe et all (2007) found also mixed results in the out-of-sample analysis of the RND forecast ability for exchange rate options. However, when we incorporate the risk aversion into RND in order to obtain a Real-world density, the out-of-sample performance improves substantially, with satisfactory results in both Kolmogorov and Berkowitz tests. Therefore, we would suggest not using the “pure” RND, but rather taking into account risk aversion in order to forecast the Brazilian Real exchange rate.

Extraindo Densidades Neutras ao Risco de Opções de Taxas de Juros Brasileiras

A construção de densidades neutras ao risco a partir de dados de opções é uma forma de extrair as expectativas do mercado de uma variável financeira. Este artigo usa uma amostra de opções de IDI de 1998 até 2009 e estima a densidade neutral ao risco implícita em opções para a taxa de juros de curto prazo brasileira, usando três métodos: Shimko, Mistura de duas Log-Normais e Distribuição Beta generalizada do tipo dois. A avaliação dentro da amostra mostra que a Mistura de Normais possui uma melhor aderência aos dados de opções do que os outros métodos no período de análise. O formato da distribuição log-normal parece aderir bem à dinâmica das taxas de juros brasileiras. Também foi calculada a assimetria implícita nas distribuições neutras ao risco, e foi mostrado como ela poderia ter fornecido sinais dos futuros movimentos das taxas de juros em 2002 e 2003. De uma forma geral, as densidades neutras ao risco mostraram-se uma ferramenta útil para extração de expectativas de mercado sobre futuros desenvolvimentos da política monetária.

Are We Extracting the True Risk Neutral Density from Option Prices? A Question with No Easy Answer

In this paper we raise a question on the reliability of option implied risk neutral density. We prove that given any number of options, there exist numerous risk neutral densities which are piecewise constant, have only two values, either a lower bound or an upper bound on the true risk neutral density, and price all these options correctly. Similar results are proved with respect to the true risk neutral density's derivatives. These results show how difficult it is to ensure that the risk neutral density we extract from option prices is the true one and how large estimation errors can be.

Do Stochastic Volatility and/or Jumps Improve the Consistency of Risk-Neutral Distributions Implicit in the S and P 500 Index Option Market and the Cash Index Market

This paper examines the relative importance of stochastic volatility and price jumps in option pricing by exploiting the differences in option-based and index return-based risk-neutral densities. As evinced by the persistent smirked volatility smile, this study extracts the risk-neutral densities by exploiting the information embedded in the implied volatilities that are smoothed by a kernel regression. On the other hand, a canonical valuation approach is adopted to identify the risk-neutral density from the observed index returns. Statistical tests and implied risk aversion are constructed to investigate the differences between two risk-neutral densities. The 30-day S&P 500 options under stochastic volatility are found efficiently priced. By using a longer horizon of underlying returns, however, we are able to partly reconcile the differences between the index and option-implied risk-neutral densities after adding a jump component to the index dynamics. Option investors are found more risk averse than stock traders except for the 30-day jump-diffusion with stochastic volatility. The finding may help illustrate the puzzle that implicit volatilities are greater than subsequent realized volatilities.

Interpreting Implied Risk-Neutral Densities: The Role of Risk Premia*

This paper examines differences between risk-neutral and objective probability densities of future interest rates. The identification and quantification of these differences are important when risk-neutral densities (RNDs), such as option-implied RNDs, are used as indicators of actual beliefs of investors. We employ a multi-factor essentially affine modeling framework applied to German time-series and cross-section term structure data in order to identify both the risk-neutral and the objective term structure dynamics. We find important differences between risk-neutral and objective distributions due to risk premia in bond prices. Moreover, the estimated premia vary over time in a quantitatively significant way, which implies that the differences between the objective and the risk-neutral distributions also vary over time. We therefore conclude that one should be cautious in interpreting RNDs in terms of expectations. The method used in this paper provides an alternative approach to identifying objective probabilities of future interest rates.

Normality tests of option-implied risk-neutral densities: evidence from the small Finnish market

This study provides an empirical analysis of option-implied risk-neutral densities. The normality of the risk-neutral density is statistically assessed testing restrictions regarding the skewness and kurtosis of the implied distribution and by applying the traditional test approach involving a comparison of the pricing errors calculated using alternative models. It is found that both the approaches give similar results, whereas the former method has the advantage that the significance of the estimated parameters can be statistically tested. Using data from the small Finnish market, the normality of the distributions is soundly rejected as expected based on the theoretical framework by Damodaran [ J. Financ. Quant. Anal. 20 (1985) 423].

Interpreting Implied Risk-Neutral Densities: The Role of Risk Premia

This paper examines differences between risk-neutral and objective probability densities of future interest rates. The identification and quantification of these differences are important when risk-neutral densities (RNDs), such as option-implied RNDs, are used as indicators of actual beliefs of investors. We employ a multi-factor essentially affine modeling framework applied to German time-series and cross-section term structure data in order to identify both the risk-neutral and the objective term structure dynamics. We find important differences between risk-neutral and objective distributions due to risk premia in bond prices. Moreover, the estimated premia vary over time in a quantitatively significant way, which implies that the differences between the objective and the risk-neutral distributions also vary over time. We conclude that one should be cautious in interpreting RNDs as representing the true expectations of market participants. The method used in this paper provides an alternative approach to identifying probabilities of future interest rates.

Do option markets correctly price the probabilities of movement of the underlying asset?

We answer this question by comparing the risk-neutral density estimated in complete markets from cross-section of S&P 500 option prices to the risk-neutral density inferred from the time series density of the S&P 500 index. If investors are risk-averse, the latter density is different from the actual density that could be inferred from the time series of S&P 500 returns. Naturally, the observed asset returns do not follow the risk-neutral dynamics, which are therefore not directly observable. In contrast to the existing literature, we avoid making any assumptions on investors’ preferences, by comparing two risk-adjusted densities, rather than a risk-adjusted density from option prices to an unadjusted density from index returns. Our only maintained hypothesis is a one-factor structure for the S&P 500 returns. We propose a new method, based on an empirical Girsanov's change of measure, to identify the risk-neutral density from the observed unadjusted index returns. We design four different tests of the null hypothesis that the S&P 500 options are efficiently priced given the S&P 500 index dynamics, and reject it. By adding a jump component to the index dynamics, we are able to partly reconcile the differences between the index and option-implied risk-neutral densities, and propose a peso-problem interpretation of this evidence.