The Relation between Physical and Risk‐neutral Cumulants

Variance swaps are natural instruments for investors taking directional bets on volatility and are often used for portfolio protection. The empirical observation on skewness research suggests that derivative professionals may also desire to hedge beyond volatility risk and there exists the need to hedge higher‐moment market risks, such as skewness and kurtosis risks. We study two derivative contracts – skewness swap and kurtosis swap – which trade the forward realized third and fourth cumulants. Using S&P 500 index options data from 1996 to 2005, we document the returns of these swap contracts, i.e., skewness risk premium and kurtosis risk premium. We find that the both skewness and kurtosis risk premiums are significantly negative.     

The information content of the S&P 500 index and VIX options on the dynamics of the S&P 500 index

Given that both S&P 500 index and VIX options essentially contain information about the future dynamics of the S&P 500 index, in this study, we set out to empirically investigate the informational roles played by these two option markets with regard to the prediction of returns, volatility, and density in the S&P 500 index. Our results reveal that the information content implied from these two option markets is not identical. In addition to the information extracted from the S&P 500 index options, all of the predictions for the S&P 500 index are significantly improved by the information recovered from the VIX options. Our findings are robust to various measures of realized volatility and methods of density evaluation. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

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.     

A Markov regime switching approach for hedging stock indices

In this paper we describe a new approach for determining time‐varying minimum variance hedge ratio in stock index futures markets by using Markov Regime Switching (MRS) models. The rationale behind the use of these models stems from the fact that the dynamic relationship between spot and futures returns may be characterized by regime shifts, which, in turn, suggests that by allowing the hedge ratio to be dependent upon the “state of the market,” one may obtain more efficient hedge ratios and hence, superior hedging performance compared to other methods in the literature. The performance of the MRS hedge ratios is compared to that of alternative models such as GARCH, Error Correction and OLS in the FTSE 100 and S&P 500 markets. In and out‐of‐sample tests indicate that MRS hedge ratios outperform the other models in reducing portfolio risk in the FTSE 100 market. In the S&P 500 market the MRS model outperforms the other hedging strategies only within sample. Overall, the results indicate that by using MRS models market agents may be able to increase the performance of their hedges, measured in terms of variance reduction and increase in their utility. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:649–674, 2004     

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     

VIX futures

VIX futures are exchange‐traded contracts on a future volatility index (VIX) level derived from a basket of S&P 500 (SPX) stock index options. The authors posit a stochastic variance model of VIX time evolution, and develop an expression for VIX futures. Free parameters are estimated from market data over the past few years. It is found that the model with parameters estimated from the whole period from 1990 to 2005 overprices the futures contracts by 16–44%. But the discrepancy is dramatically reduced to 2–12% if the parameters are estimated from the most recent one‐year period. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:521–531, 2006     

VIX futures pricing with conditional skewness

We develop a closed‐form VIX futures valuation formula based on the inverse Gaussian GARCH process by Christoffersen et al. that combines conditional skewness, conditional heteroskedasticity, and a leverage effect. The new model outperforms the benchmark in fitting the S&P 500 returns and the VIX futures prices. The fat‐tailed innovation underlying the model substantially reduced pricing errors during the 2008 financial crisis. The in‐ and out‐of‐sample pricing performance indicates that the new model should be a default modeling choice for pricing the medium‐ and long‐term VIX futures.     

Equity volatility,bond yields,and yield spreads

This study examines the impact of implied and contemporaneous equity market volatility on Treasury yields, corporate bond yields, and yield spreads over Treasuries. The CBOE VIX is the measure of implied volatility, and the measure of contemporaneous volatility is constructed using intraday squared S&P 500 returns. We find that bond yields and spreads respond to changes in equity market volatility in a manner consistent with a flight‐to‐quality effect. Both short‐ and long‐term Treasury yields fall in response to increases in implied volatility, and the yield curve flattens modestly. Yields on short‐term investment grade bonds fall in response to contemporaneous volatility shocks, while long‐term spreads on low‐quality issues widen. This indicates that investors “look ahead” in anticipation of changes in equity market volatility but respond more strongly to changes in contemporaneous market activity. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

Is volatility risk priced in the securities market? Evidence from S&P 500 index options

The authors examine whether volatility risk is a priced risk factor in securities returns. Zero‐beta at‐the‐money straddle returns of the S&P 500 index are used to measure volatility risk. It is demonstrated that volatility risk captures time variation in the stochastic discount factor. The results suggest that straddle returns are important conditioning variables in asset pricing, and investors use straddle returns when forming their expectations about securities returns. One interesting finding is that different classes of firms react differently to volatility risk. For example, small firms and value firms have negative and significant volatility coefficients, whereas big firms and growth firms have positive and significant volatility coefficients during high‐volatility periods, indicating that investors see these latter firms as hedges against volatile states of the economy. Overall, these findings have important implications for portfolio formation, risk management, and hedging strategies. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:617–642, 2007     

The Term Structure of VIX

In this study, we extend the Chicago Board Options Exchange volatility index, VIX, from 30‐day to any arbitrary time‐to‐maturity, and study the term structure of VIX. We propose new concepts of instantaneous and long‐term squared VIXs as the limits at the short and long ends of the term structure respectively. Modeling the volatility process with instantaneous and long‐term squared VIXs, we establish a parsimonious approach to capture information contained in the term structure of VIX. Our study provides an efficient setup to further study the pricing of VIX derivatives and their relation with S&P 500 options.     

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     

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.     

Intraday return predictability,portfolio maximisation,and hedging

We examine whether intraday Chinese return predictability is linked to optimal portfolio holding and hedging. We find that: (1) S&P500 futures returns only predict Chinese spot market returns in up to 5-minute of trading with predictability disappearing at higher frequencies of trade; (2) the portfolio weight is maximised at the 5-minute trading frequency, when predictability is the strongest; and (3) when predictability is the strongest, significantly less shorting of the futures is required to minimise risk when a long position is taken in the Chinese market.     

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     

Volatility and jump risk in option returns

We examine the importance of volatility and jump risk in the time-series prediction of S&P 500 index option returns. The empirical analysis provides a different result between call and put option returns. Both volatility and jump risk are important predictors of put option returns. In contrast, only volatility risk is consistently significant in the prediction of call option returns over the sample period. The empirical results support the theory that there is option risk premium associated with volatility and jump risk, and reflect the asymmetry property of S&P 500 index distribution.     

Publicly traded venture capital funds: implications for institutional “fund of funds” investors

Institutional investors supply the bulk of the funds which are used by venture capital investment firms in financing emerging growth companies. These investors typically place their funds in a number of venture capital firms, thus achieving diversification across a range of investment philosophy, geography, management, industry, investment life cycle stage and type of security. Essentially, each institutional investor manages a “fund of funds,” attempting through the principles of portfolio theory to reduce the risk of participating in the venture capital business while retaining the up-side potential which was the original source of attraction to the business. Because most venture capital investment firms are privately held limited partnerships, it is very difficult to measure risk adjusted rates of return on these funds on a continuous basis.In this paper, we use the set of twelve publicly traded venture capital firms as a proxy to develop insight regarding the risk reduction effect of investment in a portfolio of venture capital funds, i.e., a fund of funds. Measurements of weekly total returns for the shares of these funds are compared with similar returns on a set of comparably sized “maximum capital gain” mutual funds and the daily return of the S&P 500 Index. A comparison of returns on an individual fund basis, as well as a correlation of daily returns of these individual funds, were made. In order to adjust for any systematic bias resulting from the “thin market” characteristic of the securities of the firms being observed, the Scholes-Williams beta estimation technique was used to reduce the effects of nonsynchronous trading.The results indicate that superior returns are realized on such portfolios when compared with portfolios of growth-oriented mutual funds and with the S&P 500 Index. This is the case whether the portfolios are equally weighted (i.e., “naive”) or constructed to be mean-variant efficient, ex ante, according to the capital asset pricing model. When compared individually, more of the venture funds dominated the S&P Market Index than did the mutual funds and by much larger margins. When combined in portfolios, the venture capital funds demonstrated very low beta coefficients and very low covariance of returns among portfolio components when compared with portfolios of mutual funds. To aid in interpreting these results, we analyzed the discounts and premia from net asset value on the funds involved and compared them to Thompson's findings regarding the contribution of such differences to abnormal returns. We found that observed excess returns greatly exceed the level which would be explained by these differences.The implications of these results for the practitioner are significant. They essentially tell us that, while investment in individual venture capital deals is considered to have high risk relative to potential return, combinations of deals (i.e., venture capital portfolios) were shown to produce superior risk adjusted returns in the market place. Further, these results show that further combining these portfolios into larger portfolios (i.e., “funds of funds”) provides even greater excess returns over the market index, thus plausibly explaining the “fund of funds” approach to venture capital investment taken by many institutional investors.While the funds studied are relatively small and are either small business investment companies or business development companies, they serve as a useful proxy for the organized venture capital industry, despite the fact that the bulk of the funds in the industry are institutionally funded, private, closely held limited partnerships which do not trade continuously in an open market. These results demonstrate to investors the magnitude of the differences in risk adjusted total return between publicly traded venture capital funds and growth oriented mutual funds on an individual fund basis. They also demonstrate to investors the power of the “fund of funds” approach to institutional involvement in the venture capital business. Because such an approach produces better risk adjusted investment results for the institutional investor, it seems to justify a greater flow of capital into the business from more risk averse institutional investment sources. This may mean greater access to institutional funds for those seeking to form new venture capital funds. For entrepreneurs seeking venture capital funds for their young companies, it may also mean a lower potential cost of capital for the financing of business venturing. From the viewpoint of public policy makers interested in facilitating the funding of business venturing, it may provide insight regarding regulatory issues surrounding taxation and the barriers and incentives which affect venture capital investment.     

Is investor misreaction economically significant? Evidence from short‐ and long‐term S&P 500 index options

Several recent studies present evidence of investor misreaction in the options market. Although the interpretation of their results is still controversial, the important question of economic significance has not been fully addressed. Here this gap is addressed by formulating regression‐based tests to identify misreaction and its duration and constructing trading strategies to exploit the empirical patterns of misreaction. Regular S&P 500 index options and long‐dated S&P 500 LEAPS are used to find an underreaction that on average dissipates over the course of 3 trading days and an increasing misreaction that peaks after four consecutive daily variance shocks of the same sign. Option trading strategies based on these findings produce economically significant abnormal returns in the range of 1–3% per day. However, they are not profitable in the presence of transaction costs. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:717–752, 2005     

Effects of nondiscretionary trading on futures prices

This paper examines the effects of the nondiscretionary trading demands of volatility index (VIX) exchange-traded products (ETPs) issuers on the prices and volumes in the VIX futures. We find that the ETPs' informationless, mechanical rebalancing of futures positions to maintain the constant maturity of the index and the promised leverage ratios of the VIX ETPs have significantly positive predictive power for end-of-day futures returns. We also show that the impact on price has diminished through time from increased liquidity provided by hedge funds, and the “natural” hedging of the issuers' inverse products.     

Market Price of Risk: A Comparison among the United States,United Kingdom,Australia and Japan*

This study examines and compares the market price of risk of the S&P 500, FTSE 100, All Ordinaries, and Nikkei 225 markets from 1984 to 2009 in the framework of Intertemporal Capital Asset Pricing Model (ICAPM). We follow the Vector Autoregressive instrumental variable approach in identifying the risk and hedge components of market returns and argue that in the context of market integration, covariance with a world market portfolio is a better measure of market risk than conditional market variance. Evidence is documented in support of using covariance as a risk measure in explaining market risk premiums in the Australian and Japanese markets. CAY, the consumption wealth ratio from the US market is found to be a robust state variable that helps to explain both conditional variance and covariance processes in the four markets. The market prices of risk, after controlling for the hedging demands, are positive and significant with the United States having the highest price of risk. The results are confirmed using a series of robustness tests that include varying the sampling interval.     

Information content of volatility spreads

This study reexamines the determinants of volatility spreads and suggests a new forecast of future volatilities. Contrary to earlier volatility forecasts, the newly introduced forecast is applicable when investors are not risk‐neutral or when underlying returns do not follow a Gaussian probability distribution. This implies that the method is consistent with the presence of risk premia for other risks such as volatility risk. Using S&P 500 index options, we show that the new volatility forecast outperforms other volatility forecasts including risk‐neutral implied volatility and historical volatility in two aspects. First, the new forecast is superior to other estimates in terms of forecasting errors for future realized volatilities. Second, it is an unbiased estimator of future realized volatilities. This is shown using an encompassing regression analysis. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:533–558, 2010