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.     

The implied volatility smirk

This paper provides an industry standard on how to quantify the shape of the implied volatility smirk in the equity index options market. Our local expansion method uses a second-order polynomial to describe the implied volatility–moneyness function and relates the coefficients of the polynomial to the properties of the implied risk-neutral distribution of the equity index return. We present a formal, two-way representation of the link between the level, slope and curvature of the implied volatility smirk and the risk-neutral standard deviation, skewness and excess kurtosis. We then propose a new semi-analytical method to calibrate option-pricing models based on the quantified implied volatility smirk, and investigate the applicability of two option-pricing models.     

Option valuation with long-run and short-run volatility components

This paper presents a new model for the valuation of European options, in which the volatility of returns consists of two components. One is a long-run component and can be modeled as fully persistent. The other is short-run and has a zero mean. Our model can be viewed as an affine version of Engle and Lee [1999. A permanent and transitory component model of stock return volatility. In: Engle, R., White, H. (Eds.), Cointegration, Causality, and Forecasting: A Festschrift in Honor of Clive W.J. Granger. Oxford University Press, New York, pp. 475–497], allowing for easy valuation of European options. The model substantially outperforms a benchmark single-component volatility model that is well established in the literature, and it fits options better than a model that combines conditional heteroskedasticity and Poisson–normal jumps. The component model's superior performance is partly due to its improved ability to model the smirk and the path of spot volatility, but its most distinctive feature is its ability to model the volatility term structure. This feature enables the component model to jointly model long-maturity and short-maturity options.     

A closed-form GARCH option valuation model

This paper develops a closed-form option valuation formula fora spot asset whose variance follows a GARCH(p, q) process thatcan be correlated with the returns of the spot asset. It providesthe first readily computed option formula for a random volatilitymodel that can be estimated and implemented solely on the basisof observables. The single lag version of this model containsHeston's (1993) stochastic volatility model as a continuous-timelimit. Empirical analysis on S&P500 index options showsthat the out-of-sample valuation errors from the single lagversion of the GARCH model are substantially lower than thead hoc Black-Scholes model of Dumas, Fleming and Whaley (1998)that uses a separate implied volatility for each option to fitto the smirk/smile in implied volatilities. The GARCH modelremains superior even though the parameters of the GARCH modelare held constant and volatility is filtered from the historyof asset prices while the ad hoc Black-Scholes model is updatedevery period. The improvement is largely due to the abilityof the GARCH model to simultaneously capture the correlationof volatility, with spot returns and the path dependence involatility.     

Realizing smiles: Options pricing with realized volatility

We develop a discrete-time stochastic volatility option pricing model exploiting the information contained in the Realized Volatility (RV), which is used as a proxy of the unobservable log-return volatility. We model the RV dynamics by a simple and effective long-memory process, whose parameters can be easily estimated using historical data. Assuming an exponentially affine stochastic discount factor, we obtain a fully analytic change of measure. An empirical analysis of Standard and Poor's 500 index options illustrates that our model outperforms competing time-varying and stochastic volatility option pricing models.     

The Finite Moment Log Stable Process and Option Pricing

We document a surprising pattern in S&P 500 option prices. When implied volatilities are graphed against a standard measure of moneyness, the implied volatility smirk does not flatten out as maturity increases up to the observable horizon of two years. This behavior contrasts sharply with the implications of many pricing models and with the asymptotic behavior implied by the central limit theorem (CLT). We develop a parsimonious model which deliberately violates the CLT assumptions and thus captures the observed behavior of the volatility smirk over the maturity horizon. Calibration exercises demonstrate its superior performance against several widely used alternatives.     

The information content of ETF options

We examine the information content of China's Shanghai Stock Exchange (SSE) 50 ETF options introduced in 2015. Trading volume and implied volatilities of calls versus puts differ markedly: trading volume is consistently higher for calls, and implied volatility is higher for puts. Put-call volume and implied volatility ratios are not good predictors of future SSE 50 returns. Implied volatility follows a right-skewed smirk across strike prices, indicating a tendency among option traders to turn bullish and expect the stock market to recover from the June 2015 market crash. The options market dominates the price discovery process, with an average information leadership share of 67%. Our price discovery results persist during the COVID outbreak.     

Option pricing under time-varying risk-aversion with applications to risk forecasting

We present a two-factor option-pricing model, which parsimoniously captures the difference in volatility persistences under the historical and risk-neutral probabilities. The model generates an S-shaped pricing kernel that exhibits time-varying risk aversion. We apply our model for two purposes. First, we analyze the risk preference implied by S&P500 index options during 2001–2009 and find that risk-aversion level strongly increases during stressed market conditions. Second, we apply our model for Value-at-Risk (VaR) forecasts during the subprime crisis period and find that it outperforms several leading VaR models.     

How Integrated are Credit and Equity Markets? Evidence from Index Options

We study the extent to which credit index (CDX) options are priced consistent with S&P 500 (SPX) equity index options. We derive analytical expressions for CDX and SPX options within a structural credit-risk model with stochastic volatility and jumps using new results for pricing compound options via multivariate affine transform analysis. The model captures many aspects of the joint dynamics of CDX and SPX options. However, it cannot reconcile the relative levels of option prices, suggesting that credit and equity markets are not fully integrated. A strategy of selling CDX volatility yields significantly higher excess returns than selling SPX volatility.     

The role of stochastic volatility and return jumps: reproducing volatility and higher moments in the KOSPI 200 returns dynamics

This paper investigates the role of stochastic volatility and return jumps in reproducing the volatility dynamics and the shape characteristics of the Korean Composite Stock Price Index (KOSPI) 200 returns distribution. Using efficient method of moments and reprojection analysis, we find that stochastic volatility models, both with and without return jumps, capture return dynamics surprisingly well. The stochastic volatility model without return jumps, however, cannot fully reproduce the conditional kurtosis implied by the data. Return jumps successfully complement this gap. We also find that return jumps are essential in capturing the volatility smirk effects observed in short-term options.

Common asset pricing factors in volatilities and returns in futures markets

Factor-based asset pricing models have been used to explain the common predictable variation in excess asset returns. This paper combines means with volatilities of returns in several futures markets to explain their common predictable variation. Using a latent variables methodology, tests do not reject a single factor model with a common time-varying factor loading. The single common factor accounts for up to 53% of the predictable variation in the volatilities and up to 14% of the predictable variation in the means. S&P500 futures volatility predicted by the factor model is highly correlated with volatility implied in S&P500 futures options. But both the factor and implied volatilities are significant in predicting future volatility. In derivatives pricing, both implied volatility from options and factors extracted from asset pricing models should be employed.     

The Informational Role of Option Trading Volume in Equity Index Options Markets

This paper examines the dynamic relations between future price volatility of the S&P 500 index and trading volume of S&P 500 options to explore the informational role of option volume in predicting the price volatility. The future volatility of the index is approximated alternatively by implied volatility and by EGARCH volatility. Using a simultaneous equation model to capture the volume-volatility relations, the paper finds that strong contemporaneous feedbacks exist between the future price volatility and the trading volume of call and put options. Previous option volumes have a strong predictive ability with respect to the future price volatility. Similarly, lagged changes in volatility have a significant predictive power for option volume. Although the volume-volatility relations for individual volatility and volume terms are somewhat different under the two volatility measures, the results on the predictive ability of volume (volatility) for volatility (volume) are broadly similar between the implied and EGARCH volatilities. These findings support the hypothesis that both the information- and hedge-related trading explain most of the trading volume of equity index options.     

The role of time-varying jump risk premia in pricing stock index options

This paper examines out-of-sample option pricing performances for the affine jump diffusion (AJD) models by using the S&P 500 stock index and its associated option contracts. In particular, we investigate the role of time-varying jump risk premia in the AJD specifications. Our empirical analysis shows strong evidence in favor of time-varying jump risk premia in pricing cross-sectional options. We also find that, during a period of low volatility, the role of jump risk premia becomes less pronounced, making the differences across pricing performances of the AJD models not as substantial as during a period of high volatility. This finding can possibly explain poor pricing perfomances of the sophisticated AJD models in some previous studies whose sample periods can be characterized by low volatility.     

Explaining asset pricing puzzles associated with the 1987 market crash

The 1987 market crash was associated with a dramatic and permanent steepening of the implied volatility curve for equity index options, despite minimal changes in aggregate consumption. We explain these events within a general equilibrium framework in which expected endowment growth and economic uncertainty are subject to rare jumps. The arrival of a jump triggers the updating of agents' beliefs about the likelihood of future jumps, which produces a market crash and a permanent shift in option prices. Consumption and dividends remain smooth, and the model is consistent with salient features of individual stock options, equity returns, and interest rates.     

Mean-reversion properties of implied volatilities

In this paper, we present a new stylized fact for options whose underlying asset is a stock index. Extracting implied volatility time series from call and put options on the Deutscher Aktien index (DAX) and financial times stock exchange index (FTSE), we show that the persistence of these volatilities depends on the moneyness of the options used for its computation. Using a functional autoregressive model, we show that this effect is statistically significant. Surprisingly, we show that the diffusion-based stochastic volatility models are not consistent with this stylized fact. Finally, we argue that adding jumps to a diffusion-based volatility model help recovering this volatility pattern. This suggests that the persistence of implied volatilities can be related to the tails of the underlying volatility process: this corroborates the intuition that the liquidity of the options across moneynesses introduces an additional risk factor to the one usually considered.     

Modelling implied volatility with OLS and panel data models

The paper performs an empirical estimation of time-varying volatility using OLS regression. Error Components, and Dummy Variable models, by regressing the implied volatility on time to maturity, the strike price and a dummy. Both the daily OLS equations and the panel data model provide more accurate estimates of Black and Scholes option prices than the bench-mark standard deviation of log returns. FT-SE 100 Index European options are used for empirical analysis.     

Is the realized volatility good for option pricing during the recent financial crisis?

The contributions of this paper are threefold. The first contribution is the proposed logarithmic HAR (log-HAR) option-pricing model, which is more convenient compared with other option pricing models associated with realized volatility in terms of simpler estimation procedure. The second contribution is the test of the empirical implications of heterogeneous autoregressive model of the realized volatility (HAR)-type models in the S&P 500 index options market with comparison of the non-linear asymmetric GARCH option-pricing model, which is the best model in pricing options among generalized autoregressive conditional heteroskedastic-type models. The third contribution is the empirical analysis based on options traded from July 3, 2007 to December 31, 2008, a period covering a recent financial crisis. Overall, the HAR-type models successfully predict out-of-sample option prices because they are based on realized volatilities, which are closer to the expected volatility in financial markets. However, mixed results exist between the log-HAR and the heterogeneous auto-regressive gamma models in pricing options because the former is better than the latter in times of turmoil, whereas it is worse during the rather stable periods.     

Long-run risk in durable consumption

Durable consumption growth is persistent and predicted by the price-dividend ratio. This provides strong and direct evidence for the existence of a highly persistent expected component. Durable consumption growth is left-skewed and exhibits time-varying volatility. I model durable consumption growth as containing a persistent expected component and driven by counter-cyclical volatility, nondurable consumption as a random walk, and dividend growth as exposed to the expected component of durable consumption growth. Together with nonseparable Epstein-Zin preferences, the model demonstrates that long-run risk in durable consumption can explain major asset market phenomena. The model also generates an upward-sloping real term structure.     

Understanding the Implied Volatility Surface for Options on a Diversified Index

This paper describes a two-factor model for a diversified index that attempts to explain both the leverage effect and the implied volatility skews that are characteristic of index options. Our formulation is based on an analysis of the growth optimal portfolio and a corresponding random market activity time where the discounted growth optimal portfolio is expressed as a time transformed squared Bessel process of dimension four. It turns out that for this index model an equivalent risk neutral martingale measure does not exist because the corresponding Radon-Nikodym derivative process is a strict local martingale. However, a consistent pricing and hedging framework is established by using the benchmark approach. The proposed model, which includes a random initial condition for market activity, generates implied volatility surfaces for European call and put options that are typically observed in real markets. The paper also examines the price differences of binary options for the proposed model and their Black-Scholes counterparts. Mathematics Subject Classification: primary 90A12; secondary 60G30; 62P20 JEL Classification: G10, G13     

Can negative interest rates really affect option pricing? Empirical evidence from an explicitly solvable stochastic volatility model

The profound financial crisis generated by the collapse of Lehman Brothers and the European sovereign debt crisis in 2011 have caused negative values of government bond yields both in the USA and in the EURO area. This paper investigates whether the use of models which allow for negative interest rates can improve option pricing and implied volatility forecasting. This is done with special attention to foreign exchange and index options. To this end, we carried out an empirical analysis on the prices of call and put options on the US S&P 500 index and Eurodollar futures using a generalization of the Heston model in the stochastic interest rate framework. Specifically, the dynamics of the option’s underlying asset is described by two factors: a stochastic variance and a stochastic interest rate. The volatility is not allowed to be negative, but the interest rate is. Explicit formulas for the transition probability density function and moments are derived. These formulas are used to estimate the model parameters efficiently. Three empirical analyses are illustrated. The first two show that the use of models which allow for negative interest rates can efficiently reproduce implied volatility and forecast option prices (i.e. S&P index and foreign exchange options). The last studies how the US three-month government bond yield affects the US S&P 500 index.