Equity Index Options Research Articles

The risk and return of equity and credit index options

We develop a structural credit risk model, which allows us to price equity/credit indices and their options through the asset dynamics of index constituents. We estimate the model via MLE and find that equity and credit index option prices are well explained out-of-sample. Contrary to recent empirical findings, the two option markets are not inconsistently priced through the lens of our model. Returns on both options, while extreme, do not indicate any evidence of mispricing. Our analysis suggests that jointly addressing the pricing of various instruments requires properly attributing three different sources of systematic risk: asset, variance, and jump risks.

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

ABSTRACTWe 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.

Construction and Hedging of Equity Index Options Portfolios

Construction and Hedging of Equity Index Options Portfolios

Practical Applications of Value for Equity Index Options: Expected—Not Realized—Volatility and the Distribution of Forecasts

Practical Applications of Value for Equity Index Options: Expected—Not Realized—Volatility and the Distribution of Forecasts

A Macrofinance Model for Option Prices: A Story of Rare Economic Events

We propose a macrofinance model that rationalizes robust features in equity index option markets. When rare disasters are followed by economic recoveries, the slope of the implied volatility term structure is positive in good times but turns negative in bad times. Additionally, implied volatility decreases with moneyness in bad times (volatility skew), whereas the shape becomes a smile in good times in the presence of rare economic booms. Our theory contributes to understanding the dynamics of the implied volatility surface yet keeping standard asset-pricing moments realistic. This paper was accepted by Gustavo Manso, finance. Funding: The authors are grateful to HEC Montreal, the University of Texas at Dallas, and particularly to the Canadian Derivatives Institute for generous financial support. Supplemental Material: The online appendix and data are available at https://doi.org/10.1287/mnsc.2022.4587 .

Value for Equity Index Options: Expected—Not Realized—Volatility and the Distribution of Forecasts

Models of option returns neglect the distribution of expected asset volatility, unfortunately for not only derivatives traders but also investors who monitor options as <i>fear gauges</i>. Six common GARCH (generalized autoregressive conditional heteroskedasticity) models afford estimates of the physical, rather than the risk-neutral, distribution of anticipated—instead of historical—volatility, as well as of volatility disagreement. This value specification covering nine global equity indexes and five expiries from 1 to 12 months fits implied volatilities closely, with sizeable and robust error-correction speeds out of sample, all else equal. Exploratory backtests of delta-neutral trading rules produce high Sharpe ratios and alphas, with modest drawdowns and skew.

Dark Matter in (Volatility and) Equity Option Risk Premiums

In their paper, “Dark Matter in (Volatility and) Equity Option Risk Premiums,” Bakshi, Crosby, and Gao ask a provocative question: is there dark matter embedded in volatility and equity options? They consider a theoretical approach that allows them to introduce the constructs of risk premiums on jumps crossing the strike and on local time. The treatment of jumps crossing the strike and local time is integral to their theory because their absence would be counterfactual from an empirical standpoint. They label such abstract uncertainties—driven by unspanned risk components—“dark matter” as these uncertainties can be hard to identify, but their presence is implied in options data, and the workings of dark matter can be economically influential. Their empirical exercises are based on weekly equity index options (the “weeklys”) in addition to the farther-dated (index and futures) options up to 88 days maturity.

GARCH-Based “Value” for Equity Index Options: Expected—Not Realized—Volatility and the Distribution of Forecasts

GARCH-Based “Value” for Equity Index Options: Expected—Not Realized—Volatility and the Distribution of Forecasts

Certificate pricing using Discrete Event Simulations and System Dynamics theory

The study proposes an innovative application of Discrete Event Simulations (DES) and System Dynamics (SD) theory to the pricing of a certain kind of certificates very popular among private investors and, more generally, in the context of wealth management. The paper shows how numerical simulation software mainly used in traditional engineering, such as industrial and mechanical engineering, can be successfully adapted to the risk analysis of structured financial products. The article can be divided into three macro-sections: in the first part a synthetic overview of the most widespread option pricing models in the quantitative finance branch is given to the readers together with the fundamental technical-instrumental background of the implemented DES and SD simulator. After dealing with some of the most popular models adopted for Equity and Equity index options, which are the most common underlying assets for the certificates structuring, we move, in the second part, to describe how the mathematical models can be integrated into a general simulation environment able to provide both DES and SD extensively used in the engineering field. The core stochastic differential equation (SDE) will therefore be translated, together with all its input parameters, into a visual block model which allows an immediate quantitative analysis of how market parameters and the other model variables can change over time. The possibility for the structurer to observe how the variables evolve day-by-day gives a strong sensitivity to evaluate how the price and the associated risk measures can be directly affected. The third part of the study compares the results obtained from the simulator designed by the authors with the more traditional pricing approaches, which consist in programming Matlab® codes for the numerical integration of the core stochastic dynamics through a Euler-Maruyama scheme. The comparison includes a price check using the Bloomberg® DLIB pricing module and a check directly against the valuation provided by the counterparty. In this section, real market cases will therefore be examined with a complete quantitative analysis of two of the most widespread categories of certificates in wealth management: Multi-asset Barrier Reverse Convertible with Issuer Callability and Multi-asset Express Certificate with conditional memory fixed coupon.

Stochastic Bounds for Reference Sets in Portfolio Analysis

A stochastic bound is a portfolio that stochastically dominates all alternatives in a reference portfolio set instead of a single alternative portfolio. An approximate bound is a portfolio that comes as close as possible to this ideal. To identify and analyze exact or approximate bounds, feasible approaches to numerical optimization and statistical inference are developed based on linear programming and subsampling. The use of reference sets and stochastic bounds is shown to improve investment performance in representative applications to enhanced benchmarking using equity industry rotation and equity index options combinations. This paper was accepted by Kay Giesecke, finance.

Delta hedging bitcoin options with a smile

We analyse robust dynamic delta hedging of bitcoin options using a set of smile-implied and other smile-adjusted deltas that are either model-free, in the sense that they are the same for every scale-invariant stochastic and/or local volatility model, or they are based on simple regime-dependent parameterisations of local volatility. These deltas are popular with option market makers in traditional assets because they are very easy to implement. Previous empirical research on dynamic delta hedging is based solely on equity index options, but analysis of our unique data on hourly historical bitcoin option prices reveals that bitcoin implied volatility curves behave very differently from those of equity index options. For call and put options with a wide range of moneyness and with synthetic constant maturities of 10, 20 and 30 days, we compare the dynamic hedging performance of different smile-adjusted deltas over two one-year periods. We also examine the use of the perpetual contract rather than the standard futures as hedging instrument because the basis risk for the perpetual is very much smaller than it is for calendar futures. Results are presented as testable statistics of hedging error variance ratios. In certain periods the use of smile-implied hedge ratios can significantly out-perform the simple Black–Scholes delta hedge, especially when using the perpetual swap as hedging instrument, where efficiency gains can exceed 30% for out-of-the-money puts, and reach an average of 15% when hedging short-term out-of-the money calls during periods when the implied volatility curve slopes upwards. The advantage of using the perpetual contract is especially evident during 2021, for the longer-term contracts for which the basis is still rather large.

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.

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.

COVID-19 and Market Expectations: Evidence from Option-Implied Densities

We compare risk-neutral densities from equity index options across several markets during the early phase of the COVID-19 pandemic. These densities reflect market expectations regarding its economic impact. The markets reacted abruptly and simultaneously initially, but with a marked time lag after ignoring the first clear warning signs for several weeks. As the crisis unfolds, option prices increasingly reflect differences in anticipated impact on and economic resiliency of these markets.

Macroeconomic Tail Risks and Asset Prices

AbstractI document that dividend growth and returns on the aggregate U.S. stock market are more correlated with consumption growth in bad economic times. In a consumption-based asset pricing model with a generalized disappointment-averse investor and small, IID consumption shocks, this feature results in a realistic equity premium despite low risk aversion. The model is consistent with the main facts about stock market risk premiums inferred from equity index options, remains tightly parameterized, and allows for analytical solutions for asset prices. An extension with non-IID dynamics accounts for excess volatility and return predictability, while preserving the model’s consistency with option moments.Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online.

Using the Volatility Risk Premium to Mitigate the Next Financial Crisis

Ten years have passed since the trough of the Global Financial Crisis, and the US equity market has experienced one of the longest stretches of ascent in history. Some investors have started questioning whether the US stock market is overvalued and if a recession is on the way. It is usually impossible to predict the next financial crisis. An investor’s best course of action may be to adjust the investment portfolio to be resilient against potential market headwinds. This article argues for utilizing the volatility risk premium (VRP), specifically option-selling VRP strategies, to mitigate the losses the portfolio may suffer from a future financial crisis. Such a VRP strategy, if implemented with out-of-the-money equity index options, can help investors cushion the losses from an equity market crash and recover more quickly than the broad equity market. Investors can utilize the VRP by itself or combine it with traditional equity to construct the most suitable investment strategies. This article further examines implementation choices of such strategies and stress tests their performance with four representative crises from the past three decades. <b>TOPICS:</b>Analysis of individual factors/risk premia, financial crises and financial market history, performance measurement <b>Key Findings</b> • The US equity market may be overvalued after the decade-old rally, and an investor’s best approach may be portfolio adjustment against potential market headwinds. • The volatility risk premium (VRP), specifically option-selling VRP strategies, can help investors mitigate losses the portfolio may suffer from a future financial crisis. • Two mechanisms help overlay VRP strategies: low delta options only initiate payouts after the index fall below strike prices; higher implied volatility (IV) translates into higher premium collection in crisis times.

Explaining CDS prices with Merton’s model before and after the Lehman default

We examine whether CDS prices around the Credit Crisis can be explained with Merton’s model. First we invert the model with market prices to reveal skewed volatility smiles over the whole 2005–2012 period. Then we calibrate the model to pre-Crisis data in two novel ways that allow for skewness, one based on equity-index options (MEIV) and the other on the sensitivity of CDS prices to equity volatility (MSKEW). In out-of-sample forecasts both calibrations match the in-Crisis peak of prices, but the second is better at capturing the systematic component of prices thereafter. Average CDS prices remain at twice their pre-Crisis level long after that event; the MSKEW calibration demonstrates that this is due to extra idiosyncratic risks, which are important for some firms but have negligible impact on others.

Give Credit Where Credit is Due: What Explains Corporate Bond Returns?

This paper explains the risk and returns of US corporate bond indices using a set of economically-motivated factors. In particular, I find that options markets explain a great deal of credit returns. Two particular features of corporate bonds generate option exposure. The first is that, in accordance with the Merton model, a corporate bond is economically equivalent to a short put option on a firm’s assets bundled with a risk-free bond. The second is that many corporate bonds include call provisions, which are basically options granted to the bond issuer. Thus, callable corporate bonds are positively exposed to firm asset values and negatively exposed to interest rates, firm volatility, and bond volatility. Using data spanning 21 years, I find that these identified risk factors explain between 60% and 76% of the return variability of the aggregate US investment grade corporate index, its sub-indices by maturity, and the aggregate US high yield index. I further decompose performance to identify systematic and idiosyncratic exposures. Systematic exposures compensate bond investors via the bond, equity, equity volatility, and bond volatility risk premia. Idiosyncratic exposures, on the other hand, provide risk without reward on average. Finally, I propose a Risk-Efficient Credit strategy that isolates the compensated risk premia by buying bonds and equities and by selling delta-neutralized equity index options and bond options. Risk-Efficient Credit strategies had similar or higher average returns than their corporate bond index counterparts, despite realizing between 15% and 48% lower volatility as well as attenuated drawdowns.

Cumulative Prospect Theory, Option Returns, and the Variance Premium

Abstract We develop a tractable equilibrium asset pricing model with cumulative prospect theory (CPT) preferences. Using GMM on a sample of U.S. equity index option returns, we show that by introducing a single common probability weighting parameter for both tails of the return distribution, the CPT model can simultaneously generate the otherwise puzzlingly low returns on both out-of-the-money put and out-of-the-money call options as well as the high observed variance premium. In a dynamic setting, probability weighting and time-varying equity return volatility combine to match the observed time-series pattern of the variance premium. Received May 30, 2017; editorial decision August 10, 2018 by Editor Andrew Karolyi.

The economic determinants of the implied volatility function for currency options

PurposeThe purpose of this paper is to ascertain the pattern of the implied volatility function for currency options traded on the National Stock Exchange of India (NSE), identify its potential determinants and to investigate any seasonality in the pattern.Design/methodology/approachThe paper examines four different specifications for the implied volatility smile of exchange-traded dollar-rupee options. These specifications are tested by running Ordinary Least Squares (OLS) regressions on a daily basis for all options over the entire sample period. Seven potential determinants for the shape of the volatility function are identified. Contemporaneous and lead-lag relationships between these determinants and the shape of the volatility function are examined using OLS and multivariate VAR. Impulse response functions are employed to test the strength and persistence of the lead-lag relations. Seasonality of the smile pattern is tested using OLS.FindingsThe study shows that the implied volatility function for dollar-rupee options is asymmetric and varies with the time to maturity of the option. Historical volatility, momentum and jumps in the exchange rate, time to maturity, traded volume of options and volatility in the stock market appear to Granger-cause the shape of the volatility smile. Feedback causality is observed from the shape of the smile to the volatility, momentum and jumps in the exchange rate and trading volume of currency options. A weak day-of-the-week effect is observed in the pattern of the volatility smile.Practical implicationsThe study sheds light on the potential determinants of the smile and highlights the predictive power of the smile which findings can be useful to market practitioners for pricing and hedging of dollar-rupee options. The study has strong practical implications during a period of increased volatility in the dollar-rupee pair.Originality/valueMost of the existing literature regarding implied volatility smiles has focused either on the volatility smile of US equity index options or that of major liquid currencies. There is a need for such studies in the context of options on emerging market currencies such as the Indian rupee which are characterized by thin trading and frequent central bank intervention and signaling. To the best of the author’s knowledge this study is the first to focus on the volatility smile of exchange-traded options on the US dollar–Indian rupee.