A new simple square root option pricing model

This study derives a simple square root option pricing model using a general equilibrium approach in an economy where the representative agent has a generalized logarithmic utility function. Our option pricing formulae, like the Black–Scholes model, do not depend on the preference parameters of the utility function of the representative agent. Although the Black–Scholes model introduces limited liability in asset prices by assuming that the logarithm of the stock price has a normal distribution, our basic square root option pricing model introduces limited liability by assuming that the square root of the stock price has a normal distribution. The empirical tests on the S&P 500 index options market show that our model has smaller fitting errors than the Black–Scholes model, and that it generates volatility skews with similar shapes to those observed in the marketplace. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark     

Inferring information from the S&P 500, CBOE VIX,and CBOE SKEW indices

This paper compares the information extracted from the S&P 500, CBOE VIX, and CBOE SKEW indices for the S&P 500 index option pricing. Based on our empirical analysis, VIX is a very informative index for option prices. Whether adding the SKEW or the VIX term structure can improve the option pricing performance depends on the model we choose. Roughly speaking, the VIX term structure is informative for some models, while the SKEW is very noisy and does not contain much important information for option prices. This paper also extends Zhang et al. (2017, J Futures Markets, 37, 211–237) into three typical affine models.     

The cost of carry model and regime shifts in stock index futures markets: An empirical investigation

This article examines empirically the dynamic relationship between spot and futures prices in stock index futures markets employing a class of nonlinear, regime‐switching‐vector‐equilibrium‐correction models, which is novel in this context. Using data for the S&P 500 and the FTSE 100 over the post‐1987 crash period, it is shown that a long‐run relationship between spot and futures prices exists, which implies mean reversion of the basis. After providing strong evidence against the hypothesis of linear dynamics in the relationship under investigation, regime‐switching‐vector‐equilibrium‐correction models for spot and futures price movements are developed and shown to capture well the time‐series properties of our data, consistent with a large theoretical and empirical literature. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:603–624, 2000.     

VIX option pricing

Substantial progress has been made in developing more realistic option pricing models for S&P 500 index (SPX) options. Empirically, however, it is not known whether and by how much each generalization of SPX price dynamics improves VIX option pricing. This article fills this gap by first deriving a VIX option model that reconciles the most general price processes of the SPX in the literature. The relative empirical performance of several models of distinct interest is examined. Our results show that state‐dependent price jumps and volatility jumps are important for pricing VIX options. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:523–543, 2009     

Modeling nonlinear dynamics of daily futures price changes

The purpose of this article is to characterize linear and nonlinear serial dependence in daily futures price changes. The daily prices of four futures are included in this study: (i) S&P 500; (ii) Japanese yen; (iii) Deutsche mark; and (iv) Eurodollar. Our major empirical findings are: (i) Based on the results of nonlinearity tests (that is, the BDS, the Q², and the TAR-F tests), we found all futures price changes contain nonlinearity in the series; (ii) a GARCH model can explain the source of nonlinearity for three out of four series; (iii) a threshold autoregressive model and autoregressive volatility model can adequately represent nonlinear dynamics of S&P 500 series; and (iv) deterministic chaos is not evident in the scaled residuals from the nonlinear time series models. Hence we favor a statistical time series approach to represent the data-generating mechanism of futures price changes. © 1999 John Wiley & Sons, Inc. Jrl Fut Mark 19: 325–351, 1999     

Cash trading and index futures price volatility

This study examines the effect of cash market liquidity on the volatility of stock index futures. Two facets of cash market liquidity are considered: (1) the level of liquidity trading proxied by the expected New York Stock Exchange (NYSE) trading volume and (2) the noise composition of trading proxied by the average NYSE trading commission cost. Under the framework of spline–GARCH with a liquidity component, both the quarterly average commission cost and the quarterly expected NYSE volume are negatively associated with the ex ante daily volatility of S&P 500 and NYSE composite index futures. Conversely, liquidity and noise trading in the cash market both dampen futures price volatility, ceteris paribus. This negative association between secular cash trading liquidity and daily futures price volatility is amplified during times of market crisis. These results retain statistical significance and materiality after controlling for bid–ask bounce of futures prices and volume of traded futures contracts. This study establishes empirical evidence to affirm the conventional prediction of a liquidity–volatility relationship: the liquidity effect is secular and persistent across markets. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:465–486, 2011     

Hedging under the influence of transaction costs: An empirical investigation on FTSE 100 index options

The Black–Scholes (BS; F. Black & M. Scholes, 1973) option pricing model, and modern parametric option pricing models in general, assume that a single unique price for the underlying instrument exists, and that it is the mid‐ (the average of the ask and the bid) price. In this article the authors consider the Financial Times and London Stock Exchange (FTSE) 100 Index Options for the time period 1992–1997. They estimate the ask and bid prices for the index, and show that, when substituted for the mid‐price in the BS formula, they provide superior option price predictors, for call and put options, respectively. This result is reinforced further when they .t a non‐parametric neural network model to market prices of liquid options. The empirical .ndings in this article suggest that the ask and bid prices of the underlying asset provide a superior fit to the mid/closing price because they include market maker's, compensation for providing liquidity in the market for constituent stocks of the FTSE 100 index. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:471–494, 2007     

Unveiling the influence of COVID-19 on the online retail market: A comprehensive exploration

The global economy has experienced significant disruptions due to the COVID-19 pandemic, profoundly impacting production and consumption patterns. This paper examines the effect of the pandemic on consumer goods prices, focusing on a sample of hand sanitizers, soap, disinfectants, and baking yeasts. The selection of these products was motivated by multiple and comprehensive criteria, where we considered various factors such as changes in consumption patterns during the pandemic, the relevance of these products in household consumption, consumer behavior, availability of historical price data, their significance in daily domestic life, and their utility in the prevention of the disease. Employing a time-series analysis, the study investigates the relationship between product prices and the S&P 500 index, as well as the presence of structural breaks in the time series. The results reveal a significant correlation between all prices in the sample and the S&P 500 index, indicating the index's efficacy as an indicator of the overall economic state. Moreover, the study identifies a single structural break in each price time series, coinciding with the onset of the COVID-19 pandemic. Notably, hand sanitizers, soap, and disinfectants exhibit particularly substantial price jumps near the time of the break. These findings offer crucial insights into the impact of the COVID-19 pandemic on consumer prices, which can guide policy decisions aimed at mitigating the pandemic's economic consequences.     

The performance of VIX option pricing models: Empirical evidence beyond simulation

We examine the pricing performance of VIX option models. Such models possess a wide‐range of underlying characteristics regarding the behavior of both the S&P500 index and the underlying VIX. Our tests employ three representative models for VIX options: Whaley ( 1993 ), Grunbichler and Longstaff ( 1996 ), Carr and Lee ( 2007 ), Lin and Chang ( 2009 ), who test four stochastic volatility models, as well as to previous simulation results of VIX option models. We find that no model has small pricing errors over the entire range of strike prices and times to expiration. In particular, out‐of‐the‐money VIX options are difficult to price, with Grunbichler and Longstaff's mean‐reverting model producing the smallest dollar errors in this category. Whaley's Black‐like option model produces the best results for in‐the‐money VIX options. However, the Whaley model does under/overprice out‐of‐the‐money call/put VIX options, which is opposite the behavior of stock index option pricing models. VIX options exhibit a volatility skew opposite the skew of index options. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark31:251–281, 2011     

Pricing efficiency of the S&P 500 index market: Evidence from the Standard & Poor's Depositary Receipts

Standard & Poor's Depositary Receipts (SPDRs) are exchange traded securities representing a portfolio of S&P 500 stocks. They allow investors to track the spot portfolio and better engage in index arbitrage. We tested the impact of the introduction of SPDRs on the efficiency of the S&P 500 index market. Ex‐post pricing efficiency and ex‐ante arbitrage profit between SPDRs and futures were also examined. We found an improved efficiency in the S&P 500 index market after the start of SPDRs trading. Specifically, the frequency and length of lower boundary violations have declined since SPDRs began trading. This result is consistent with the hypothesis that SPDRs facilitate short arbitrage by simplifying the process of shorting the cash index against futures. Tests of pricing efficiency comparing SPDRs and futures suggested that index arbitrage using SPDRs as a substitute for program trading in general results in losses. Although short arbitrages earn a small profit on average, gains are statistically insignificant. A trade‐by‐trade investigation showed that prices are instantaneously corrected after the presence of mispricing signals, introducing substantial risk in arbitraging. Evidence in general supported pricing efficiency between SPDRs and the S&P 500 index futures—both ex‐post and ex‐ante. © 2002 Wiley Periodicals, Inc. Jrl Fut Mark 22:877–900, 2002     

Pricing and hedging S&P 500 index options with Hermite polynomial approximation: empirical tests of Madan and Milne's model

The universal use of the Black and Scholes option pricing model to value a wide range of option contracts partly accounts for the almost systematic use of Gaussian distributions in finance. Empirical studies, however, suggest that there is an information content beyond the second moment of the distribution that must be taken into consideration.This article applies a Hermite polynomial-based model developed by Madan and Milne (1994) to an investigation of S&P 500 index option prices from the CBOE when the distribution of the underlying index is unknown. The model enables us to incorporate the non-normal skewness and kurtosis effects empirically observed in option-implied distributions of index returns. Out-of-sample tests confirm that the model outperforms Black and Scholes in terms of pricing and hedging. © 1999 John Wiley & Sons, Inc. Jrl Fut Mark 19: 735–758, 1999     

Empirical tests of canonical nonparametric American option‐pricing methods

Alcock and Carmichael (2008, The Journal of Futures Markets, 28, 717–748) introduce a nonparametric method for pricing American‐style options, that is derived from the canonical valuation developed by Stutzer (1996, The Journal of Finance, 51, 1633–1652). Although the statistical properties of this nonparametric pricing methodology have been studied in a controlled simulation environment, no study has yet examined the empirical validity of this method. We introduce an extension to this method that incorporates information contained in a small number of observed option prices. We explore the applicability of both the original method and our extension using a large sample of OEX American index options traded on the S&P100 index. Although the Alcock and Carmichael method fails to outperform a traditional implied‐volatility‐based Black–Scholes valuation or a binomial tree approach, our extension generates significantly lower pricing errors and performs comparably well to the implied‐volatility Black–Scholes pricing, in particular for out‐of‐the‐money American put options. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:509–532, 2010     

Conditional Volatility and the GARCH Option Pricing Model with Non‐Normal Innovations

On the basis of the theory of a wedge between the physical and risk‐neutral conditional volatilities in Christoffersen, P., Elkamhi, R., Feunou, B., & Jacobs, K. (2010), we develop a modification of the GARCH option pricing model with the filtered historical simulation proposed in Barone‐Adesi, G., Engle, R. F., & Mancini, L. (2008). The one‐day‐ahead conditional volatilities under physical and risk‐neutral measures are the same in the previous model, but should have been allowed to be different. Using extensive data on S&P 500 index options, our approach, which employs one‐day‐ahead risk‐neutral conditional volatility estimated from the cross‐section of the option prices (in contrast to the existing GARCH option pricing models), maintains theoretical consistency under conditional non‐normality, and improves the empirical performances. Remarkably, the risk‐neutral volatility dynamics are stable over time in this model. In addition, the comparison between the VIX index and the risk‐neutral integrated volatility economically validates our approach. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 33:1–28, 2013     

The CBOE S&P 500 three‐month variance futures

In this article, we study the market of the Chicago Board Options Exchange S&P 500 three‐month variance futures that were listed on May 18, 2004. By using a simple mean‐reverting stochastic volatility model for the S&P 500 index, we present a linear relation between the price of fixed time‐to‐maturity variance futures and the VIX². The model prediction is supported by empirical tests. We find that a model with a fixed mean‐reverting speed of 1.2929 and a daily‐calibrated floating long‐term mean level has a good fit to the market data between May 18, 2004, and August 17, 2007. The market price of volatility risk estimated from the 30‐day realized variance and VIX² has a mean value of −19.1184. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:48–70, 2010     

Do S&P 500 index options violate the martingale restriction?

The study tests Longstaff's martingale restriction on S&P 500 index options over the period 1990–1994. Assuming the S&P index follows a lognormal distribution results in systematic violations of the martingale restriction, the implied index value from options consistently overestimating the market value. Adopting a generalized distribution, allowing for nonnormal third and fourth moments, produces economically insignificant rejections of the martingale restriction. A simulation analysis supports the empirical results from the lognormal model in the presence of nonnormal skewness and kurtosis. Overall, the results support the conclusion that the no-arbitrage assumption coupled with the generalized distribution offers a good working model for S&P index options over the period studied. © 1999 John Wiley & Sons, Inc. Jrl Fut Mark 19: 499–521, 1999     

The limits to stock index arbitrage: Examining S&P 500 futures and SPDRS

This study examines factors affecting stock index spot versus futures pricing and arbitrage opportunities by using the S&P 500 cash index and the S&P 500 Standard and Poor's Depository Receipt (SPDR) Exchange‐Traded Fund (ETF) as “underlying cash assets.” Potential limits to arbitrage when using the cash index are the staleness of the underlying cash index, trading costs, liquidity (volume) issues of the underlying assets, the existence of sufficient time to execute profitable arbitrage transactions, short sale restrictions, and the extent to which volatility affects mispricing. Alternatively, using the SPDR ETF as the underlying asset mitigates staleness and trading cost problems as well as the effects of volatility associated with the staleness of the cash index. Minute‐by‐minute prices are compared over different volatility levels to determine how these factors affect the limits of S&P 500 futures arbitrage. Employing the SPDR as the cash asset examines whether a liquid tradable single asset with low trading costs can be used for pricing and arbitrage purposes. The analysis examines how long mispricing lasts, the impact of volatility on mispricing, and whether sufficient volume exists to implement arbitrage. The minute‐by‐minute liquidity of the futures market is examined using a new transaction volume futures database. The results show that mispricings exist regardless of the choice of the underlying cash asset, with more negative mispricings for the SPDR relative to the S&P 500 cash index. Furthermore, mispricings are more frequent in high‐ and mid‐volatility months than in low‐volatility months and are associated with higher volume during high‐volatility months. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:1182–1205, 2008     

Delivery horizon and grain market volatility

We study the difference in the volatility dynamics of CBOT corn, soybeans, and oats futures prices across different delivery horizons via a smoothed Bayesian estimator. We find that futures price volatilities in these markets are affected by inventories, time to delivery, and the crop progress period and that there are important differences in the effects across delivery horizons. We also find that price volatility is higher before the harvest starts in most cases compared to the volatility during the planting period. These results have implications for hedging, options pricing, and the setting of margin requirements. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 30:846–873, 2010     

Causality in the VIX futures market

This study examines the price‐discovery function and information efficiency of a fast growing volatility futures market: the Chicago Board of Option Exchange VIX futures market. A linear Engle–Granger cointegration test with an error correction mechanism (ECM) shows that during the full sample period, VIX futures prices lead spot VIX index, which implies that the VIX futures market has some price‐discovery function. But a modified Baek and Brock nonlinear Granger test detects bi‐directional causality between VIX and VIX futures prices, suggesting that both spot and futures prices react simultaneously to new information. Quarter‐by‐quarter investigations show that, on average, the estimated parameters are not significantly different from zero, thus providing further evidence supporting information efficiency in the VIX futures market. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

Clustering in the futures market: Evidence from S&P 500 futures contracts

We document trade price clustering in the futures markets. We find clustering at prices of x.00 and x.50 for S&P 500 futures contracts. While trade price clustering is evident throughout time to maturity of these contracts, there is a dramatic change when the S&P 500 futures contract is designated a front‐month contract (decrease in clustering) and a back‐month contract (increase in clustering). We find that trade price clustering is a positive function of volatility and a negative function of volume or open interest. In addition, we find a high degree of clustering in the daily opening and closing prices, but a lower degree of clustering in the settlement prices. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:413–428, 2004     

A CONSISTENT PRICING MODEL FOR INDEX OPTIONS AND VOLATILITY DERIVATIVES

We propose a flexible framework for modeling the joint dynamics of an index and a set of forward variance swap rates written on this index. Our model reproduces various empirically observed properties of variance swap dynamics and enables volatility derivatives and options on the underlying index to be priced consistently, while allowing for jumps in volatility and returns. An affine specification using Lévy processes as building blocks leads to analytically tractable pricing formulas for volatility derivatives, such as VIX options, as well as efficient numerical methods for pricing of European options on the underlying asset. The model has the convenient feature of decoupling the vanilla skews from spot/volatility correlations and allowing for different conditional correlations in large and small spot/volatility moves. We show that our model can simultaneously fit prices of European options on S&P 500 across strikes and maturities as well as options on the VIX volatility index.