Volatilities implied by price changes in the S&P 500 options and futures contracts

We develop a new volatility measure: the volatility implied by price changes in option contracts and their underlying. We refer to this as price-change implied volatility. We compare moneyness and maturity effects of price-change and implied volatilities, and their performance in delta hedging. We find that delta hedges based on a price-change implied volatility surface outperform hedges based on the traditional implied volatility surface when applied to S&P 500 future options.     

Sovereign rating actions and the implied volatility of stock index options

This paper examines the interaction between the equity index option market and sovereign credit ratings. S&P and Moody's signals exhibit strong impact on option-implied volatility while Fitch's influence is less significant. Moody's downgrades reduce the market uncertainty over the rated countries' equity markets. Strong causal relationships are found between movements in the option-implied volatility and all credit signals released by S&P and Fitch, but only actual rating changes by Moody's, implying differences in rating agencies' policies. The presence of additional ratings tends to reduce market uncertainty. The findings highlight the importance of rating information in the price discovery process and offer policy implications.     

Pricing and hedging contingent claims using variance and higher order moment swaps

This paper suggests perfect hedging strategies of contingent claims under stochastic volatility and random jumps of the underlying asset price. This is done by enlarging the market with appropriate swaps whose pay-offs depend on higher order sample moments of the asset price process. Using European options and variance swaps, as well as barrier options written on the S&P 500 index, the paper provides clear cut evidence that hedging strategies employing variance and higher order moment swaps considerably improves upon the performance of traditional delta hedging strategies. Inclusion of the third-order moment swap improves upon the performance of variance swap-based strategies to hedge against random jumps. This result is more profound for short-term out-of-the money put options.     

Pinning in the S&P 500 futures

We show that Standard & Poor's (S&P) 500 futures are pulled toward the at-the-money strike price on days when serial options on the S&P 500 futures expire (pinning) and are pushed away from the cost-of-carry adjusted at-the-money strike price right before the expiration of options on the S&P 500 index (anti-cross-pinning). These effects are driven by the interplay of market makers' rebalancing of delta hedges due to the time decay of those hedges as well as in response to reselling (and early exercise) of in-the-money options by individual investors. The associated shift in notional futures value is at least $115 million per expiration day.     

Volatility surfaces: theory,rules of thumb,and empirical evidence

Implied volatilities are frequently used to quote the prices of options. The implied volatility of a European option on a particular asset as a function of strike price and time to maturity is known as the asset's volatility surface. Traders monitor movements in volatility surfaces closely. In this paper we develop a no-arbitrage condition for the evolution of a volatility surface. We examine a number of rules of thumb used by traders to manage the volatility surface and test whether they are consistent with the no-arbitrage condition and with data on the trading of options on the S&P 500 taken from the over-the-counter market. Finally we estimate the factors driving the volatility surface in a way that is consistent with the no-arbitrage condition.     

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.     

Testing Greeks and price changes in the S&P 500 options and futures contract: A regression analysis

We use a regression model to test observed price changes with Greeks as regressors. Greeks are computed using implied volatility, price-change implied volatility and historical volatility. We find sufficient evidence to reject model Greeks as unbiased responses to underlying price as well as sufficient evidence that the American version of binomial model results in biased estimates of price changes. We use options on the S&P 500 futures contracts and their underlying. We also evaluate the frequency of “wrong signs.” Call prices and their underlying move in the opposite direction almost 10 percent of the time.     

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.     

Can we forecast the implied volatility surface dynamics of equity options? Predictability and economic value tests

We examine whether the dynamics of the implied volatility surface of individual equity options contains exploitable predictability patterns. Predictability in implied volatilities is expected due to the learning behavior of agents in option markets. In particular, we explore the possibility that the dynamics of the implied volatility surface of individual stocks may be associated with movements in the volatility surface of S&P 500 index options. We present evidence of strong predictable features in the cross-section of equity options and of dynamic linkages between the volatility surfaces of equity and S&P 500 index options. Moreover, time-variation in stock option volatility surfaces is best predicted by incorporating information from the dynamics in the surface of S&P 500 options. We analyze the economic value of such dynamic patterns using strategies that trade straddle and delta-hedged portfolios, and find that before transaction costs such strategies produce abnormal risk-adjusted returns.     

Market co-movement between credit default swap curves and option volatility surfaces

We analyze the co-movement between the Credit Default Index (CDX) curve and the S&P 500 index's option volatility surface. We connect the reduced-form no-arbitrage model with the Nelson-Siegel (N-S) model on hazard rate implied from the CDX curve, and identify the levels, slopes, and curvatures from these two markets via the Unscented Kalman Filter (UKF). We find that the changes in the level, slope, and curvature in the CDX curve and those in the volatility surface are correlated due to the bridge of the S&P 500 index return. Finally, the co-movement between the CDX curve and S&P 500 index's volatility surface become stronger after the late 2000s global financial crisis.     

Does social network sentiment influence the relationship between the S&P 500 and gold returns?

This study explored the relationship between investor sentiment (extracted from the StockTwits social network), the S&P 500 Index and gold returns. We investigated bilateral causality between gold prices and S&P 500 prices, the power of investor sentiment and gold returns to predict S&P 500 returns, and the influence of gold returns on S&P 500 volatility. We also considered whether the influence of sentiment varies according to the user's degree of experience. We considered the sentiment of messages that mentioned the S&P 500 Index and that users posted between 2012 and 2016. Granger causality analysis, ARIMA models and GARCH models were used for predicting S&P 500 Index returns and S&P 500 volatility. We observed a causal relationship between gold price and the S&P 500 Index. Our results also suggest that sentiment and gold returns predict S&P 500 Index returns. Finally, we observed that gold returns influence S&P 500 volatility and that the sentiment of experienced users affects S&P 500 returns.     

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.     

S&P 500 trading strategies and stock betas

This paper shows that S&P 500 stock betas are overstatedand the non-S&P 500 stock betas are understated becauseof liquidity price effects caused by the S&P 500 tradingstrategies. The daily and weekly betas of stocks added to theS&P 500 index during 1985-1989 increase, on average, by0.211 and 0.130. The difference between monthly betas of otherwisesimilar S&P 500 and non-S&P 500 stocks also equals 0.125during this period. Some of these increases can be explainedby the reduced nonsynchroneity of S&P 500 stock prices,but the remaining increases are explained by the price pressureor excess volatility caused by the S&P 500 trading strategies.I estimate that the price pressures account for 8.5 percentof the total variance of daily returns of a value-weighted portfolioof NYSE/AMEX stocks. The negative own autocorrelations in S&P500 index returns and the negative cross autocorrelations betweenS&P 500 stock returns provide further evidence consistentwith the price pressure hypothesis.     

New evidence on the implied-realized volatility relation

We consider the relation between the volatility implied in an option's price and the subsequently realized volatility. Earlier studies on stock index options have found biases and inefficiencies in implied volatility as a forecast of future volatility. More recently, Christensen and Prabhala find that implied volatility in at-the-money one-month OEX call options on the S&P 100 index in fact is an unbiased and efficient forecast of ex-post realized index volatility after the 1987 stock market crash. In this paper, the robustness of the unbiasedness and efficiency result is extended to a more recent period covering April 1993 to February 1997. As a new contribution, implied volatility is constructed as a trade weighted average of implied volatilities from both in-the-money and out-of-the-money options and both puts and calls. We run a horse race between implied call, implied put, and historical return volatility. Several robustness checks, including a new simultaneous equation approach, underscore our conclusion, that implied volatility is an efficient forecast of realized return volatility.     

The bias in Black-Scholes/Black implied volatility: An analysis of equity and energy markets

In this paper we examine the extent of the bias between Black and Scholes (1973)/Black (1976) implied volatility and realized term volatility in the equity and energy markets. Explicitly modeling a market price of volatility risk, we extend previous work by demonstrating that Black-Scholes is an upward-biased predictor of future realized volatility in S&P 500/S&P 100 stock-market indices. Turning to the Black options-on-futures formula, we apply our methodology to options on energy contracts, a market in which crises are characterized by a positive correlation between price-returns and volatilities: After controlling for both term-structure and seasonality effects, our theoretical and empirical findings suggest a similar upward bias in the volatility implied in energy options contracts. We show the bias in both Black-Scholes/Black implied volatilities to be related to a negative market price of volatility risk. JEL Classification G12 · G13     

International capital asset pricing: Evidence from options

We study the risk dynamics and pricing in international economies through a joint analysis of the time-series returns and option prices on three equity indexes underlying three economies: the S&P 500 Index of the United States, the FTSE 100 Index of the United Kingdom, and the Nikkei-225 Stock Average of Japan. We develop an international capital asset pricing model, under which the return on each equity index is decomposed into two orthogonal jump-diffusion components: a global component and a country-specific component. We apply separate stochastic time changes to the two components so that stochastic volatility can come from both global and country-specific risks. For each economy, we assign separate market prices for the two return risk components and the two volatility risk components. Under this specification, we obtain tractable option pricing solutions. Model estimation reveals several interesting insights. First, global and country-specific return and volatility risks show different dynamics. Global return movements contain a larger discontinuous component, and global return volatility is more persistent than the country-specific counterparts. Second, investors charge positive prices for global return risk and negative prices for volatility risk, suggesting that investors are willing to pay positive premiums to hedge against downside global return movements and upside volatility movements. Third, the three economies contain different risk profiles and also price risks differently. Japan contains the largest idiosyncratic risk component and smallest global risk component. Investors in the Japanese market also price more heavily against future volatility increases than against future market downfalls.     

The informational content of implied volatility

Implied volatility is widely believed to be informationallysuperior to historical volatility, because it is the 'market's'forecast of future volatility. But for S&P 1 00 index options,the most actively traded contract in the United States, we findimplied volatility to be a poor forecast of subsequent realizedvolatility. In aggregate and across subsamples separated bymaturity and strike price, implied volatility has virtuallyno correlation with future volatility, and it does not incorporatethe information contained in recent observed volatility.     

Learning minimum variance discrete hedging directly from the market

Option hedging is a critical risk management problem in finance. In the Black–Scholes model, it has been recognized that computing a hedging position from the sensitivity of the calibrated model option value function is inadequate in minimizing variance of the option hedge risk, as it fails to capture the model parameter dependence on the underlying price (see e.g. Coleman et al., J. Risk, 2001, 5(6), 63–89; Hull and White, J. Bank. Finance, 2017, 82, 180–190). In this paper, we demonstrate that this issue can exist generally when determining hedging position from the sensitivity of the option function, either calibrated from a parametric model from current option prices or estimated nonparametricaly from historical option prices. Consequently, the sensitivity of the estimated model option function typically does not minimize variance of the hedge risk, even instantaneously. We propose a data-driven approach to directly learn a hedging function from the market data by minimizing variance of the local hedge risk. Using the S&P 500 index daily option data for more than a decade ending in August 2015, we show that the proposed method outperforms the parametric minimum variance hedging method proposed in Hull and White [J. Bank. Finance, 2017, 82, 180–190], as well as minimum variance hedging corrective techniques based on stochastic volatility or local volatility models. Furthermore, we show that the proposed approach achieves significant gain over the implied BS delta hedging for weekly and monthly hedging.     

S&P 100 Index Option Volatility

Using transaction data on the S&P 100 index options, we study the effect of valuation simplifications that are commonplace in previous research on the timeseries properties of implied market volatility. Using an American-style algorithm that accounts for the discrete nature of the dividends on the S&P 100 index, we find that spurious negative serial correlation in implied volatility changes is induced by nonsimultaneously observing the option price and the index level. Negative serial correlation is also induced by a bid/ask price effect if a single option is used to estimate implied volatility. In addition, we find that these same effects induce spurious (and unreasonable) negative cross-correlations between the changes in call and put implied volatility.     

An approximate distribution of delta-hedging errors in a jump-diffusion model with discrete trading and transaction costs

We introduce a jump-diffusion model for asset returns with jumps drawn from a mixture of normal distributions and show that this model adequately fits the historical data of the S&P500 index. We consider a delta-hedging strategy (DHS) for vanilla options under the diffusion model (DM) and the proposed jump-diffusion model (JDM), assuming discrete trading intervals and transaction costs, and derive an approximation for the probability density function (PDF) of the profit-and-loss (P&L) of the DHS under both models. We find that, under the log-normal model of Black–Scholes–Merton, the actual PDF of the P&L can be well approximated by the chi-squared distribution with specific parameters. We derive an approximation for the P&L volatility in the DM and JDM. We show that, under both DM and JDM, the expected loss due to transaction costs is inversely proportional to the square root of the hedging frequency. We apply mean–variance analysis to find the optimal hedging frequency given the hedger's risk tolerance. Since under the JDM it is impossible to reduce the P&L volatility by increasing the hedging frequency, we consider an alternative hedging strategy, following which the P&L volatility can be reduced by increasing the hedging frequency.