Index Options Research Articles

The complete Gaussian kernel in the multi-factor Heston model: Option pricing and implied volatility applications

In this paper, we propose two new representation formulas for the conditional marginal probability density of the multi-factor Heston model. The two formulas express the marginal density as a convolution with suitable Gaussian kernels whose variances are related to the conditional moments of price returns. Via asymptotic expansion of the non-Gaussian function in the convolutions, we derive explicit formulas for European-style option prices and implied volatility. The European option prices can be expressed as Black–Scholes style terms plus corrections at higher orders in the volatilities of volatilities, given by the Black–Scholes Greeks. The explicit formula for the implied volatility clearly identifies the effect of the higher moments of the price on the implied volatility surface. Further, we derive the relationship between the VIX index and the variances of the two Gaussian kernels. As a byproduct, we provide an explanation of the bias between the VIX and the volatility of total returns, which offers support to recently proposed methods to compute the variance risk premium. Via a series of numerical exercises, we analyse the accuracy of our pricing formula under different parameter settings for the one- and two-factor models applied to index options on the S&P500 and show that our approximation substantially reduces the computational time compared to that of alternative closed-form solution methods. In addition, we propose a simple approach to calibrate the parameters of the multi-factor Heston model based on the VIX index, and we apply the approach to the double and triple Heston models.

The necessity to activate long-term ETD in Korea

Purpose This paper aims to provide the necessity to activate long-term exchange-traded derivatives (ETD) in Korea. In the era of aging, low interest rates and low economic growth, the investment demand for long-term financial products, and its hedging demand have steadily increased. Unfortunately, long-term ETD do not trade in Korea, and this study presents political suggestions to invigorate long-term ETD based on overseas cases and empirical analysis. Specifically, this study suggests the necessity to activate exchange traded funds (ETFs) options, long-term Korea treasury bond futures and options and long-term Volatility Index of Korea Composite Stock Price Index future and options. The introduction of those long-term ETD not only contributes to providing long-term investment and hedging vehicles but also reduces market inefficiencies in the Korean industry of ETFs, bonds and structured products.

A Bayesian nonparametric approach to option pricing

Accurately modeling the implied volatility surface is of great importance to option pricing, trading and hedging. In this paper, we investigate the use of a Bayesian nonparametric approach to fit and forecast the implied volatility surface with observed market data. More specifically, we explore Gaussian Processes with different kernel functions characterizing general covariance functions. We also obtain posterior distributions of the implied volatility and build confidence intervals for the predictions to assess potential model uncertainty. We apply our approach to market data on the S&P 500 index option market in 2018, analyzing 322,983 options. Our results suggest that the Bayesian approach is a powerful alternative to existing parametric pricing models

A robust nonstandard finite difference scheme for pricing real estate index options

ABSTRACT Real estate assets can be used to store capital, generate income through rentals and can act as collateral for debt instruments. Common risk management mechanisms for real estate investments use portfolio diversification techniques, but these techniques require large amounts of capital to work effectively. Real estate index derivatives offer an alternative mechanism for managing risks associated with real estate investments. They also increase market liquidity by providing a path for individuals who do not own real estate assets to participate in the real estate market. We consider the problem of pricing real estate index derivative contracts. The market is incomplete, so it is completed by futures derivatives on the same real estate index. A dynamic hedging strategy is employed leading to a parabolic partial differential equation with coefficients which are dependent on time and the real estate index. We then construct a nonstandard finite difference method to price European and American real estate index options. The scheme utilizes complete cubic spline interpolants of the option prices at time-dependent backtrack points on the spatial grid. Bounds for the global error are theoretically established. Numerical experiments are carried out to illustrate the accuracy of the scheme.

Identifying underpricing in index options

This study analyses the occurrence and extent of underpricing of index options on the National Stock Exchange (NSE) of India. The study uses discriminant analysis to explain the incidence of underpricing, and regression analysis to explain the extent of underpricing. Further, the study proposes three sets of models to explain underpricing of options. The first set of models considers only the impact of moneyness on underpricing. The second set of models considers the impact of both moneyness and time to expiration on underpricing. The third set of models considers the impact of moneyness, time to expiration, and changes in returns and volatility of the underlying asset on underpricing. The results of the study would enable speculators to identify arbitrage opportunities by predicting the occurrence and magnitude of underpricing in index options, vis-à-vis trading costs, in terms of moneyness, time to expiration, returns of the underlying index, and the volatility of the underlying index. The results also identify the inefficiencies of options in the price discovery mechanism, which the exchange can try to rectify by provision of sufficient market demand and liquidity.

The traders' rule and long‐term options

AbstractStudies show that option traders prefer using the traders' rule to price short‐term options over more mathematically sophisticated models. This study uses Standard & Poor's 500 index option data to determine whether the traders' rule also outperforms mathematically more sophisticated models in pricing long‐term options. The results show that, while the traders' rule still enjoys smaller pricing errors for short‐term options, more mathematically sophisticated models perform better for long‐term options. This result may reflect the greater liquidity of short‐term options, which are far more actively traded than longer‐term options.

Option-Based Estimation of the Price of Coskewness and Cokurtosis Risk

AbstractWe show that the prices of risk for factors that are nonlinear in the market return can be obtained using index option prices. The price of coskewness risk corresponds to the market variance risk premium, and the price of cokurtosis risk corresponds to the market skewness risk premium. Option-based estimates of the prices of risk lead to reasonable values of the associated risk premia. An analysis of factor models with coskewness risk indicates that the new estimates of the price of risk improve the models’ performance compared with regression-based estimates.

Hedging strategy in emerging market: Application long straddle option in gold price index

This research was conducted to test the implementation of gold price index option contracts using the Black Scholes and GARCH models with a long straddle strategy. The testing is done by looking at the comparison of the results of the calculation from the historical volatility value and the GARCH volatility. The results of the study are displayed by looking at the comparison of the Average Mean-square Error (AMSE) percentage values of the two models. From the research that has been done, it shows that the Black Scholes model has a better gold price index option contract than the GARCH model for maturities of 1 month, 2 months and 3 months.This is shown from the AMSE value of call options and put options in the Black Scholes model which is always smaller than the GARCH model for each contract maturity period. In addition, the potential for maximum profit by implementing the long straddle strategy in gold price index option contracts in the range of 2008-2018 is 54.98 percent with an average profit potential of around 25-30 percent. JEL Classification: G11, G12 DOI: https://doi.org/10.26905/jkdp.v24i4.4666

Smile-Implied Hedging with Volatility Risk

Options can be dynamically replicated using model-free Greeks extracted from the volatility smile. However, smile-implied delta and delta-gamma hedging do not achieve minimum variance in the presence of price-volatility correlation, and these strategies have shown poor performance relative to the Black-Scholes benchmark. We propose a way to extend smile-implied option replication with volatility risk management. Large-scale evidence on S&P 500 index options indicates that smile-implied delta-gamma-vega hedging strategies outperform the Black-Scholes approach as well as more sophisticated option hedging frameworks including stochastic volatility and jumps.

The Chinese equity index options market

Using Bakshi et al. (2000), and Bakshi and Kapadia's (2003) methodology, this paper studies the Chinese equity index options market that has been developing since 2015. Empirical evidence shows that the market price of call (put) option is generally lower (higher) than their Black-Scholes prices with historical volatility. The prices of the options do not support the one-dimensional diffusion model properties. We find 61.79% (63.25%) of delta-hedged gains in call (put) options to be negative. The analysis of the non-zero delta-hedged gain suggests that the investors are mainly trading on additional volatility risk in the options market in China.

Are Index Option Returns Priced In The Stock Market? — New Insights From Factor Decompositions

This paper develops a new approach to better understand why risk factors constructed from index option returns appear to be priced in the stock market. We decompose an option-based factor into three components—an implied-volatility- driven component, a linear index-driven component, and a nonlinear index-driven component—and identify the component responsible for the observed beta-return relationship. Applying this method to the bear risk factor developed by Lu and Murray (2019), we show that the authors’ interpretation of the findings is incorrect, and the negative relationship between bear betas and expected stock returns should not be viewed as an instance of systematic risk premia. Instead, it poses a new asset pricing puzzle that is close in spirit to the previously documented low-risk anomalies. Together with the recent critique of Levi and Welch (2019) concerning the evidence on the downside risk premium, our findings suggest that the evidence is still missing on asymmetric market risk being priced in the cross-section of stocks.

The impact of net buying pressure on index options prices

AbstractThis study examines whether the demand for options, as measured by the net buying pressure of index options, explains the implied volatility structure created by options prices. We decompose the buying pressure into the direction‐motivated (i.e., delta‐informed) and the volatility‐motivated (i.e., vega‐informed) demand for options. After controlling for options traders' hedging demand, we find that both delta‐ and vega‐informed trading play significant roles in explaining changes in implied volatility. Foreign institutions are more directionally informed in index options trading than their domestic counterparts are. Domestic investors effectively implement volatility trading using put options.

Index option returns and generalized entropy bounds

I develop a new spectrum of moment bounds on the pricing kernel. They stem from the solution of an optimization problem that is complementary to Hansen and Jagannathan (1991) approach. Economically, they measure the discrepancy between what an optimizing agent could achieve if all assets (that are priced by the pricing kernel) were tradable and what she can actually achieve in the real-world market. Through the lens of these bounds, I examine leading macro-finance models using index option returns. I show, in a model-free fashion, the difficulty of several classes of models in meeting option-implied bounds. I highlight the unique information that my bounds provide compared with existing approaches.

Co-movement of volatility risk premium: evidence from single stock options market in India

ABSTRACT This study investigates the existence of commonality of volatility risk premium in the index and individual stock options. Using data from the National Stock Exchange (NSE) India, the study finds that the commonality relationship is both economically and statistically significant. The commonality relationship is robust, even after including the market-specific and stock-specific factors.

Using the short-lived arbitrage model to compute minimum variance hedge ratios: application to indices, stocks and commodities

The short-lived arbitrage model has been shown to significantly improve in-sample option pricing fit relative to the Black–Scholes model. Motivated by this model, we imply both volatility and virtual interest rates to adjust minimum variance hedge ratios. Using several error metrics, we find that the hedging model significantly outperforms the traditional delta hedge and a current benchmark hedge based on the practitioner Black–Scholes model. Our applications include hedges of index options, individual stock options and commodity futures options. Hedges on gold and silver are especially sensitive to virtual interest rates.

COVID-19 and market expectations: Evidence from option-implied densities

We compare risk-neutral densities from equity index options across several countries during the early phase of the COVID-19 pandemic. The initial reaction in all analyzed markets was late, abrupt and simultaneous. Only a few weeks later, densities started to differ across markets.

Diversification with options and structured products

Different from diversification of stocks, there are two strategies to diversify portfolios consisting of options: one is to combine options on single underlying stocks, and the other one is to buy an option based on the index of these stocks. In this paper we analyse which diversification strategy is optimal for classical rational investors with constant relative risk aversion. We employ the Black–Scholes model and the stochastic volatility model of Heston for generating the processes of underlying stocks as well as pricing the derivatives. The results are developed first for options and then extended to some important classes of structured financial products: capital protected notes, discount certificates and bonus certificates. We find that investors’ choices on the two diversification strategies differ noticeably, but in general for convex payoffs index options are preferable, whereas for concave payoffs a portfolio of single stock options has usually higher utility.

How Risky Are the Options? A Comparison with the Underlying Stock Using MaxVaR as a Risk Measure

This paper investigates the risk exposure for options and proposes MaxVaR as an alternative risk measure which captures the risk better than Value-at-Risk especially. While VaR is a measure of end-of-horizon risk, MaxVaR captures the interim risk exposure of a position or a portfolio. MaxVaR is a more stringent risk measure as it assesses the risk during the risk horizon. For a 30-day maturity option, we find that MaxVaR can be 40% higher than VaR at a 5% significance level. It highlights the importance of MaxVaR as a risk measure and shows that the risk is vastly underestimated when VaR is used as the measure for risk. The sensitivity of MaxVaR with respect to option characteristics like moneyness, time to maturity and risk horizons at different significance levels are observed. Further, interestingly enough we find that the MaxVar to VaR ratio is higher for stocks than the options and we can surmise that stock returns are more volatile than options. For robustness, the study is carried out under different distributional assumptions on residuals and for different stock index options.

The use of option prices to assess the skewness risk premium

ABSTRACT The aims of this study are twofold. First, to determine the sign and magnitude of the skewness risk premium (SRP) in the Italian index option market using two procedures: (i) skewness swap contracts, (ii) option trading strategies consisting of positions in options and their underlying assets. Second, to investigate the term structure of the SRP for 30, 60 and 90-day maturities to provide investors with a proper time horizon for profitable skewness trading strategies. Several results are obtained. First, the SRP, defined as the difference between the physical and the risk-neutral skewness, is positive and statistically and economically significant. These findings indicate that the SRP does exist, it is positive in sign, and it can be quantified. Second, the SRP is higher in magnitude for short-term maturity (€35 for the 30-day maturity) and lower for 60-day and 90-day maturities (both about €27). Third, skewness trading strategies confirm our finding of a positive and economically significant SRP. Fourth, a strategy that sells out-of-the-money puts is more profitable for medium-term maturities compared to short-term maturities. A strategy that takes a long position on out-of-the-money calls, and a short position on out-of-the-money puts, yields a higher return, if near-term options are used.

Predictability of OTC Option Volatility for Future Stock Volatility

This study explores the information content of the implied volatility inferred from stock index options in the over-the-counter (OTC) market, which has rarely been studied in the literature. Using OTC calls, puts, and straddles on the KOSPI 200 index, we find that implied volatility generally outperforms historical volatility in predicting future realized volatility, although it is not an unbiased estimator. The results are more apparent for options with shorter maturity. However, while implied volatility has strong predictability during normal periods, historical volatility is superior to implied volatility during a period of crisis due to the liquidity contraction of the OTC options market. This finding suggests that the OTC options market can play a role in conveying important information to predict future volatility.