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.     

Risk minimization in stochastic volatility models: model risk and empirical performance

In this paper the performance of locally risk-minimizing delta hedge strategies for European options in stochastic volatility models is studied from an experimental as well as from an empirical perspective. These hedge strategies are derived for a large class of diffusion-type stochastic volatility models, and they are as easy to implement as usual delta hedges. Our simulation results on model risk show that these risk-minimizing hedges are robust with respect to uncertainty and misconceptions about the underlying data generating process. The empirical study, which includes the US sub-prime crisis period, documents that in equity markets risk-minimizing delta hedges consistently outperform usual delta hedges by approximately halving the standard deviation of the profit-and-loss ratio.     

Dynamics of foreign exchange implied volatility and implied correlation surfaces

The prices of currency options expressed in terms of their implied volatilities and the implied correlations between foreign exchange rates at a given point in time depend on option delta and time to maturity. Implied volatilities and implied correlations likewise may thus be represented as a surface. It is well known that these surfaces exhibit both skew/smile features and term structure effects and their shapes fluctuate substantially over time. Using implied volatilities on three currency pairs as well as historical implied correlation values between them, we study the nature of these fluctuations by applying a Karhunen-Loève decomposition that is a generalization of a principal component analysis. We demonstrate that the largest share in the dynamics of these surfaces' fluctuations may be explained by exactly the same three factors, providing evidence of strong interdependences between implied correlation and implied volatility of global currency pairs.     

Static versus dynamic hedges: an empirical comparison for barrier options

We conduct an empirical comparison of static versus dynamic hedges of barrier options. Using more than five years of data, we compare a number of static hedges from the literature with dynamic hedges based on the local volatility model. The main result is that the variability of profit-and-loss distributions from certain static hedges is significantly smaller than that of dynamic hedges and robust to changing market scenarios. Furthermore, these static hedges are able to provide a robust tracking of barrier options’ sensitivities. This article reflects the authors’ personal opinion and not necessarily the opinion of their employers.     

Model-driven statistical arbitrage on LETF option markets

In this paper, we study the statistical properties of the moneyness scaling transformation, which adjusts the moneyness coordinate of the implied volatility smile in an attempt to remove the discrepancy between the IV smiles for levered and unlevered ETF options. We construct bootstrap uniform confidence bands which indicate that the implied volatility smiles are statistically different after moneyness scaling has been performed. An empirical application shows that there are trading opportunities possible on the LETF market. A statistical arbitrage type strategy based on a dynamic semiparametric factor model is presented. This strategy presents a statistical decision algorithm which generates trade recommendations based on comparison of model and observed LETF implied volatility surface. It is shown to generate positive returns with a high probability. Extensive econometric analysis of the LETF implied volatility process is performed including out-of-sample forecasting based on a semiparametric factor model and a uniform confidence bands' study. These provide new insights into the latent dynamics of the implied volatility surface. We also incorporate Heston stochastic volatility into the moneyness scaling method for better tractability of the model.     

Fast and realistic European ARCH option pricing and hedging

The behavior of the implied volatility surface for European options was analysed in detail by Zumbach and Fernandez for prices computed with a new option pricing scheme based on the construction of the risk-neutral measure for realistic processes with a finite time increment. The resulting dynamics of the surface is static in the moneyness direction, and given by a volatility forecast in the time-to-maturity direction. This difference is the basis of a cross-product approximation of the surface. The subsequent speed-up for option pricing is large, allowing the computation of Greeks and the delta replication strategy in simulations with the cost of replication and the replication risk. The corresponding premia are added to the option arbitrage price in order to compute realistic implied volatility surfaces. Finally, the cross-product approximation for realistic prices can be used to analyse European options on the SP500 in depth. The cross-product approximation is used to compute a mean quotient implied volatility, which can be compared with the full theoretical computation. The comparison shows that the cost of hedging and the replication risk premium have contributions to the implied volatility smile that are of similar magnitude to the contribution from the process for the underlying asset.     

EVIDENCE ON DELTA HEDGING AND IMPLIED VOLATILITIES FOR THE BLACK‐SCHOLES,GAMMA, AND WEIBULL OPTION PRICING MODELS

Modifying the distributional assumptions of the Black‐Scholes model is one way to accommodate the skewness of underlying asset returns. Simple models based on the compensated gamma and Weibull distributions of asset prices are shown to produce some improvements in option pricing. To evaluate these assertions, I construct and compare delta hedges of all S&P 500 options traded on the Chicago Board Options Exchange between September 2001 and October 2003 for the Weibull, Black‐Scholes, and gamma models. I also compare implied volatilities and their smiles (i.e., nonlinearities) among the three models. None of the three models improves over the others as far as delta hedging is concerned. Volatilities implied by all three models exhibit statistically significant smiles.     

Short-term asymptotics for the implied volatility skew under a stochastic volatility model with Lévy jumps

The implied volatility skew has received relatively little attention in the literature on short-term asymptotics for financial models with jumps, despite its importance in model selection and calibration. We rectify this by providing high order asymptotic expansions for the at-the-money implied volatility skew, under a rich class of stochastic volatility models with independent stable-like jumps of infinite variation. The case of a pure-jump stable-like Lévy model is also considered under the minimal possible conditions for the resulting expansion to be well defined. Unlike recent results for “near-the-money” option prices and implied volatility, the results herein aid in understanding how the implied volatility smile near expiry is affected by important features of the continuous component, such as the leverage and vol-of-vol parameters. As intermediary results, we obtain high order expansions for at-the-money digital call option prices, which furthermore allow us to infer analogous results for the delta of at-the-money options. Simulation results indicate that our asymptotic expansions give good fits for options with maturities up to one month, underpinning their relevance in practical applications, and an analysis of the implied volatility skew in recent S&P 500 options data shows it to be consistent with the infinite variation jump component of our models.     

Does Net Buying Pressure Affect the Shape of Implied Volatility Functions?

This paper examines the relation between net buying pressure and the shape of the implied volatility function (IVF) for index and individual stock options. We find that changes in implied volatility are directly related to net buying pressure from public order flow. We also find that changes in implied volatility of S&P 500 options are most strongly affected by buying pressure for index puts, while changes in implied volatility of stock options are dominated by call option demand. Simulated delta‐neutral option‐writing trading strategies generate abnormal returns that match the deviations of the IVFs above realized historical return volatilities.     

Option pricing using a binomial model with random time steps (A formal model of gamma hedging)

This paper provides a new option pricing model which justifies the standard industry implementation of the Black-Scholes model. The standard industry implementation of the Black-Scholes model uses an implicit volatility, and it hedges both delta and gamma risk. This industry implementation is inconsistent with the theory underlying the derivation of the Black-Scholes model. We justify this implementation by showing that these adhoc adjustments to the Black-Scholes model provide a reasonable approximation to valuation and delta hedging in our new option pricing model.     

From implied to spot volatilities

This paper is concerned with the relation between spot and implied volatilities. The main result is the derivation of a new equation which gives the dynamics of the spot volatility in terms of the shape and the dynamics of the implied volatility surface. This equation is a consequence of no-arbitrage constraints on the implied volatility surface right before expiry. We first observe that the spot volatility can be recovered from the limit, as the expiry tends to zero, of at-the-money implied volatilities. Then, we derive the semimartingale decomposition of implied volatilities at any expiry and strike from the no-arbitrage condition. Finally the spot volatility dynamics is found by performing an asymptotic analysis of these dynamics as the expiry tends to zero. As a consequence of this equation, we give general formulas to compute the shape of the implied volatility surface around the at-the-money strike and for short expiries in general spot volatility models.     

Robustly Hedging Variable Annuities With Guarantees Under Jump and Volatility Risks

Recent variable annuities offer participation in the equity market and attractive protection against downside movements. Accurately quantifying this additional equity market risk and robustly hedging options embedded in the guarantees of variable annuities are new challenges for insurance companies. Due to sensitivities of the benefits to tails of the account value distribution, a simple Black–Scholes model is inadequate in preventing excessive liabilities. A model which realistically describes the real world price dynamics over a long time horizon is essential for the risk management of the variable annuities. In this article, both jump risk and volatility risk are considered for risk management of lookback options embedded in guarantees with a ratchet feature. We evaluate relative performances of delta hedging and dynamic discrete risk minimization hedging strategies. Using the underlying as the hedging instrument, we show that, under a Black–Scholes model, local risk minimization hedging can be significantly better than delta hedging. In addition, we compare risk minimization hedging using the underlying with that of using standard options. We demonstrate that, under a Merton's jump diffusion model, hedging using standard options is superior to hedging using the underlying in terms of the risk reduction. Finally, we consider a market model for volatility risks in which the at‐the‐money implied volatility is a state variable. We compute risk minimization hedging by modeling at‐the‐money Black–Scholes implied volatility explicitly; the hedging effectiveness is evaluated, however, under a joint model for the underlying price and implied volatility. Our computational results suggest that, when implied volatility risk is suitably modeled, risk minimization hedging using standard options, compared to hedging using the underlying, can potentially be more effective in risk reduction under both jump and volatility risks.     

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.     

Asymmetric Volatility and Trading Activity in Index Futures Options

We examine the impact of option trading activity on implied volatility changes to returns in the index futures option market. Controlling for option moneyness, delta‐to‐option‐premium ratio, and liquidity, we find that net buying pressure, profit‐maximization behavior, and liquidity are interrelated and affect asymmetric responses of implied volatilities to returns. Implied volatilities of options with more liquidity, a higher exercise price, and a higher delta‐to‐option‐premium ratio have the most profound asymmetric response.     

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.     

“Life Expectancy in the Future: A Summary of a Discussion among Experts”, Robert B. Friedland,October 1998

Abstract

In this paper we present an econometric model of implied volatilities of S&;P500 index options. First, we model the dynamics the CBOE VIX index as a proxy for the general level of implied volatilities. We then describe a parametric model of the implied volatility surface for options with a term of up to two years. We show that almost all of the variation in the implied volatility surface can be explained by the VIX index and one or two other uncorrelated factors. Finally, we present a model of the dynamics of these factors.     

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.     

Implied volatility index for the Norwegian equity market

We introduce and evaluate the NOVIX - an implied volatility index for the Norwegian equity index OBX. NOVIX is created according to the VIX methodology. We compare the NOVIX to the German VDAX-NEW and the U.S. VIX and find that NOVIX has similar properties as these two indices. We also evaluate the VIX, VDAX-NEW and NOVIX in terms of volatility forecasting. As a benchmark model we use a precise HAR model of Corsi (2009) based on high-frequency data. All three implied volatility indices significantly improve daily, weekly and monthly forecasts of volatility of their underlying equity indices. This improvement is largest for the VIX, followed by VDAX-NEW and NOVIX.     

The implied volatility smirk

This paper provides an industry standard on how to quantify the shape of the implied volatility smirk in the equity index options market. Our local expansion method uses a second-order polynomial to describe the implied volatility–moneyness function and relates the coefficients of the polynomial to the properties of the implied risk-neutral distribution of the equity index return. We present a formal, two-way representation of the link between the level, slope and curvature of the implied volatility smirk and the risk-neutral standard deviation, skewness and excess kurtosis. We then propose a new semi-analytical method to calibrate option-pricing models based on the quantified implied volatility smirk, and investigate the applicability of two option-pricing models.     

Maturity cycles in implied volatility

The skew effect in market implied volatility can be reproduced by option pricing theory based on stochastic volatility models for the price of the underlying asset. Here we study the performance of the calibration of the S&P 500 implied volatility surface using the asymptotic pricing theory under fast mean-reverting stochastic volatility described in [8]. The time-variation of the fitted skew-slope parameter shows a periodic behaviour that depends on the option maturity dates in the future, which are known in advance. By extending the mathematical analysis to incorporate model parameters which are time-varying, we show this behaviour can be explained in a manner consistent with a large model class for the underlying price dynamics with time-periodic volatility coefficients.Received: December 2003, Mathematics Subject Classification (2000): 91B70, 60F05, 60H30JEL Classification: C13, G13Jean-Pierre Fouque: Work partially supported by NSF grant DMS-0071744.Ronnie Sircar: Work supported by NSF grant DMS-0090067. We are grateful to Peter Thurston for research assistance.We thank a referee for his/her comments which improved the paper.