Closed-form transformations from risk-neutral to real-world distributions

Risk-neutral and real-world densities are derived from option prices and risk assumptions, and are compared with historical densities obtained from time series. Two parametric risk-transformations are used to convert risk-neutral densities into real-world densities. Both transformations are estimated by maximizing the likelihood of observed index levels, for two parametric density families. Results for the FTSE-100 index show that parametric densities derived from option prices have more explanatory power than historical densities and higher likelihoods than densities estimated by spline methods. A combination of parametric real-world and historical densities provides the preferred predictive densities.     

The information content of implied volatilities and model-free volatility expectations: Evidence from options written on individual stocks

We measure the volatility information content of stock options for individual firms using option prices for 149 US firms and the S&P 100 index. We use ARCH and regression models to compare volatility forecasts defined by historical stock returns, at-the-money implied volatilities and model-free volatility expectations for every firm. For 1-day-ahead estimation, a historical ARCH model outperforms both of the volatility estimates extracted from option prices for 36% of the firms, but the option forecasts are nearly always more informative for those firms that have the more actively traded options. When the prediction horizon extends until the expiry date of the options, the option forecasts are more informative than the historical volatility for 85% of the firms. However, at-the-money implied volatilities generally outperform the model-free volatility expectations.     

Predictable dynamics in implied volatility surfaces from OTC currency options

Recent empirical studies report predictable dynamics in the volatility surfaces that are implied by observed index option prices, such as those prescribed by general equilibrium models. Using an extensive data set from the over-the-counter options market, we document similar predictability in the factors that capture the daily variation of surfaces implied by options on 25 different foreign exchange rates. We proceed to demonstrate that simple vector autoregressive specifications for the factors can help produce accurate out-of-sample forecasts of the systematic component of the surface at short horizons. Profitable delta-hedged positions can be set up based on these forecasts; however, profits disappear when typical transaction costs are taken into account and when trading rules on wide segments of the surface are sought.     

Forecasting currency volatility: A comparison of implied volatilities and AR(FI)MA models

We compare forecasts of the realized volatility of the pound, mark and yen exchange rates against the dollar, calculated from intraday rates, over horizons ranging from one day to three months. Our forecasts are obtained from a short memory ARMA model, a long memory ARFIMA model, a GARCH model and option implied volatilities. We find intraday rates provide the most accurate forecasts for the one-day and one-week forecast horizons while implied volatilities are at least as accurate as the historical forecasts for the one-month and three-month horizons. The superior accuracy of the historical forecasts, relative to implied volatilities, comes from the use of high frequency returns, and not from a long memory specification. We find significant incremental information in historical forecasts, beyond the implied volatility information, for forecast horizons up to one week.     

A new approach to forecasting exchange rates

Building on purchasing power parity theory, this paper proposes a new approach to forecasting exchange rates using the Big Mac data from The Economist magazine. Our approach is attractive in three aspects. Firstly, it uses easily-available Big Mac prices as input. These prices avoid several potential problems associated with broad price indexes, such as the consumer price index used in conventional PPP studies. Secondly, this approach provides real-time exchange-rate forecasts at any forecast horizon. These high-frequency forecasts could be appealing to those who want up-to-date exchange-rate forecasts. Finally, as our forecasts are obtained through a simulation procedure, estimation uncertainty is made explicit in our framework that provides the entire distribution of exchange rates, not just a single point estimate. Using exchange rates of six major currencies to illustrate the approach, we compare the Big Mac forecasts with those derived from a random walk and the CPI and find some support for our approach, especially at longer term horizons.     

Recovering the real-world density and liquidity premia from option data

In this paper, we develop a methodology for simultaneous recovery of the real-world probability density and liquidity premia from observed S&P 500 index option prices. Assuming the existence of a numéraire portfolio for the US equity market, fair prices of derivatives under the benchmark approach can be obtained directly under the real-world measure. Under this modelling framework, there exists a direct link between observed call option prices on the index and the real-world density for the underlying index. We use a novel method for the estimation of option-implied volatility surfaces of high quality, which enables the subsequent analysis. We show that the real-world density that we recover is consistent with the observed realized dynamics of the underlying index. This admits the identification of liquidity premia embedded in option price data. We identify and estimate two separate liquidity premia embedded in S&P 500 index options that are consistent with previous findings in the literature.     

Inferring volatility dynamics and risk premia from the S&P 500 and VIX markets

We estimate a flexible affine model using an unbalanced panel containing S&P 500 and VIX index returns and option prices and analyze the contribution of VIX options to the model’s in- and out-of-sample performance. We find that they contain valuable information on the risk-neutral conditional distributions of volatility at different time horizons, which is not spanned by the S&P 500 market. This information allows enhanced estimation of the variance risk premium. We gain new insights on the term structure of the variance risk premium, present a trading strategy exploiting these insights, and show how to improve S&P 500 return forecasts.     

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.     

The jump component of S&P 500 volatility and the VIX index

Much research has investigated the differences between option implied volatilities and econometric model-based forecasts. Implied volatility is a market determined forecast, in contrast to model-based forecasts that employ some degree of smoothing of past volatility to generate forecasts. Implied volatility has the potential to reflect information that a model-based forecast could not. This paper considers two issues relating to the informational content of the S&P 500 VIX implied volatility index. First, whether it subsumes information on how historical jump activity contributed to the price volatility, followed by whether the VIX reflects any incremental information pertaining to future jump activity relative to model-based forecasts. It is found that the VIX index both subsumes information relating to past jump contributions to total volatility and reflects incremental information pertaining to future jump activity. This issue has not been examined previously and expands our understanding of how option markets form their volatility forecasts.     

Option augmented density forecasts of market returns with monotone pricing kernel

Basic financial theory indicates that the ratio of the conditional density of the future value of a market index and the corresponding risk neutral density should be monotone, but a sizeable empirical literature finds otherwise. We therefore consider an option augmented density forecast of the market return obtained by transforming a baseline density forecast estimated from past excess returns so as to monotonize its ratio with a risk neutral density estimated from current option prices. To evaluate our procedure, we compare baseline and option augmented monthly density forecasts for the S&P 500 index over the period 1997–2013. We find that monotonizing the pricing kernel leads to a modest improvement in the calibration of density forecasts. Supplementary results supportive of this finding are given for market indices in France, Germany, Hong Kong, Japan and the UK.     

Forecasting metal prices: Do forecasters herd?

We analyze more than 20,000 forecasts of nine metal prices at four different forecast horizons. We document that forecasts are heterogeneous and report that anti-herding appears to be a source of this heterogeneity. Forecaster anti-herding reflects strategic interactions among forecasters that foster incentives to scatter forecasts around a consensus forecast.     

Retrieving risk neutral moments and expected quadratic variation from option prices

This paper derives exact formulas for retrieving risk neutral moments of future payoffs of any order from generic European-style option prices. It also provides an exact formula for retrieving the expected quadratic variation of the stock market implied by European option prices, which nowadays is used as an estimate of the implied volatility, and a formula approximating the jump component of this measure of variation. To implement the above formulas to discrete sets of option prices, the paper suggests a numerical procedure and provides upper bounds of its approximation errors. The performance of this procedure is evaluated through a simulation and an empirical exercise. Both of these exercises clearly indicate that the suggested numerical procedure can provide accurate estimates of the risk neutral moments, over different horizons ahead. These can be in turn employed to obtain accurate estimates of risk neutral densities and calculate option prices, efficiently, in a model-free manner. The paper also shows that, in contrast to the prevailing view, ignoring the jump component of the underlying asset can lead to seriously biased estimates of the new volatility index suggested by the Chicago Board Options Exchange.     

Testing the martingale restriction for option implied densities

This paper contributes to the forecasting literature by presenting a new evaluation method for density and probability estimates. This procedure is particularly well suited for analyzing time series of forecasts implied from option prices, although the results are very general and can be applied also outside this framework. A small scale simulation study documents that valid and accurate inference can be drawn for option implied densities with the proposed method. The new testing procedure is demonstrated in an empirical application on density estimates implied from $/£ currency options.     

VOLATILITY FORECASTS,TRADING VOLUME,AND THE ARCH VERSUS OPTION‐IMPLIED VOLATILITY TRADE‐OFF

We investigate empirically the role of trading volume (1) in predicting the relative informativeness of volatility forecasts produced by autoregressive conditional heteroskedasticity (ARCH) models versus the volatility forecasts derived from option prices, and (2) in improving volatility forecasts produced by ARCH and option models and combinations of models. Daily and monthly data are explored. We find that if trading volume was low during period t?1 relative to the recent past, ARCH is at least as important as options for forecasting future stock market volatility. Conversely, if volume was high during period t?1 relative to the recent past, option‐implied volatility is much more important than ARCH for forecasting future volatility. Considering relative trading volume as a proxy for changes in the set of information available to investors, our findings reveal an important switching role for trading volume between a volatility forecast that reflects relatively stale information (the historical ARCH estimate) and the option‐implied forward‐looking estimate.     

A general closed form option pricing formula

A new method to retrieve the risk-neutral probability measure from observed option prices is developed and a closed form pricing formula for European options is obtained by employing a modified Gram–Charlier series expansion, known as the Gauss–Hermite expansion. This expansion converges for fat-tailed distributions commonly encountered in the study of financial returns. The expansion coefficients can be calibrated from observed option prices and can also be computed, for example, in models with the probability density function or the characteristic function known in closed form. We investigate the properties of the new option pricing model by calibrating it to both real-world and simulated option prices and find that the resulting implied volatility curves provide an accurate approximation for a wide range of strike prices. Based on an extensive empirical study, we conclude that the new approximation method outperforms other methods both in-sample and out-of-sample.

     

Pricing options on scenario trees

We examine valuation procedures that can be applied to incorporate options in scenario-based portfolio optimization models. Stochastic programming models use discrete scenarios to represent the stochastic evolution of asset prices. At issue is the adoption of suitable procedures to price options on the basis of the postulated discrete distributions of asset prices so as to ensure internally consistent portfolio optimization models. We adapt and implement two methods to price European options in accordance with discrete distributions represented by scenario trees and assess their performance with numerical tests. We consider features of option prices that are observed in practice. We find that asymmetries and/or leptokurtic features in the distribution of the underlying materially affect option prices; we quantify the impact of higher moments (skewness and excess kurtosis) on option prices. We demonstrate through empirical tests using market prices of the S&P500 stock index and options on the index that the proposed procedures consistently approximate the observed prices of options under different market regimes, especially for deep out-of-the-money options.     

Dispersion of Financial Analysts' Earnings Forecasts and the (Option Model) Implied Standard Deviations of Stock Returns

This study examines whether the information implied by simultaneous levels of option and stock prices (specifically, the implied standard deviation of returns) reflects other contemporaneously available information. The independent contemporaneous measure considered is the observed dispersion (across several financial analysts), at a point in time, in the forecasts of earnings per share for a given firm. The results indicate that implied standard deviations clearly reflect the contemporaneous dispersion in analysts' forecasts incrementally, i.e., beyond the information contained in the historical time series of returns.     

Risk premia in option markets

Risk premia are related to price probability ratios or for continuous time pure jump processes the ratios of jump arrival rates under the pricing and physical measures. The variance gamma model is employed to synthesize densities with risk premia seen as the ratio of the three parameters. The premia are shown to be mean reverting, predictable, focused on crashes at shorter horizons and rallies at the longer horizon. Predicted premia may be used to adjust physical parameters to develop option prices based on time series data.     

Model-based earnings forecasts vs. financial analysts' earnings forecasts

Existing accounting-based forecasting models of earnings either do not fully consider information that is contained in stock prices or use an ad hoc specification that is not based on rigorous valuation theory. In this paper, we develop an earnings forecasting model built on the theoretical linkages between future earnings and stock prices as well as a number of accounting fundamental variables. We find that our model-based forecasts of earnings are in general less biased and more accurate than both existing model-based forecasts and analysts' consensus forecasts, at both shorter and longer horizons. We also show that the accuracy of both model-based forecasts and financial analysts' forecasts depend on firm-specific characteristics such as firm size and industry membership.     

Market imperfections and the information content of implied and realized volatility

The information content of option implied volatility and realized volatility under market imperfections are studied in the context of GARCH modeling and volatility forecasts of Taiwan stock market (TAIEX) returns. Consistent with most studies, we find that the Taiwan implied volatility index (TVIX) calculated from the TAIEX option prices contains most of the information, and that White's [White, H., 2000. A reality check for data snooping. Econometrica 68, 1097–1126] reality check test cannot reject the null hypothesis that the TVIX provides the best forecast. Possibly due to market imperfections, however, the incremental information content of realized volatility as well as daily returns cannot be ruled out. Finally, we also find that the information is found only in the most recent TVIX, indicating information is being efficiently impounded on the TAIEX option prices. This finding suggests that appropriately designed derivative products can alleviate the problems caused by market imperfections.