Riding on the smiles

Using a data set of vanilla options on the major indexes we investigate the calibration properties of several multi-factor stochastic volatility models by adopting the fast Fourier transform as the pricing methodology. We study the impact of the penalizing function on the calibration performance and how it affects the calibrated parameters. We consider single-asset as well as multiple-asset models, with particular emphasis on the single-asset Wishart Multidimensional Stochastic Volatility model and the Wishart Affine Stochastic Correlation model, which provides a natural framework for pricing basket options while keeping the stylized smile–skew effects on single-name vanillas. For all models we give some option price approximations that are very useful for speeding up the pricing process. In addition, these approximations allow us to compare different models by conveniently aggregating the parameters, and they highlight the ability of the Wishart-based models to control separately the smile and the skew effects. This is extremely important from a risk-management perspective of a book of derivatives that includes exotic as well as basket options.     

The Valuation of Volatility Options

This paper examines the valuation of European- and American-style volatilityoptions based on a general equilibrium stochastic volatility framework.Properties of the optimal exercise region and of the option price areprovided when volatility follows a general diffusion process. Explicitvaluation formulas are derived in four particular cases. Emphasis is placedon the MRLP (mean-reverting in the log) volatility model which has receivedconsiderable empirical support. In this context we examine the propertiesand hedging behavior of volatility options. Unlike American options,European call options on volatility are found to display concavity at highlevels of volatility.     

Can negative interest rates really affect option pricing? Empirical evidence from an explicitly solvable stochastic volatility model

The profound financial crisis generated by the collapse of Lehman Brothers and the European sovereign debt crisis in 2011 have caused negative values of government bond yields both in the USA and in the EURO area. This paper investigates whether the use of models which allow for negative interest rates can improve option pricing and implied volatility forecasting. This is done with special attention to foreign exchange and index options. To this end, we carried out an empirical analysis on the prices of call and put options on the US S&P 500 index and Eurodollar futures using a generalization of the Heston model in the stochastic interest rate framework. Specifically, the dynamics of the option’s underlying asset is described by two factors: a stochastic variance and a stochastic interest rate. The volatility is not allowed to be negative, but the interest rate is. Explicit formulas for the transition probability density function and moments are derived. These formulas are used to estimate the model parameters efficiently. Three empirical analyses are illustrated. The first two show that the use of models which allow for negative interest rates can efficiently reproduce implied volatility and forecast option prices (i.e. S&P index and foreign exchange options). The last studies how the US three-month government bond yield affects the US S&P 500 index.     

On VIX futures in the rough Bergomi model

The rough Bergomi model introduced by Bayer et al. [Quant. Finance, 2015, 1–18] has been outperforming conventional Markovian stochastic volatility models by reproducing implied volatility smiles in a very realistic manner, in particular for short maturities. We investigate here the dynamics of the VIX and the forward variance curve generated by this model, and develop efficient pricing algorithms for VIX futures and options. We further analyse the validity of the rough Bergomi model to jointly describe the VIX and the SPX, and present a joint calibration algorithm based on the hybrid scheme by Bennedsen et al. [Finance Stoch., forthcoming].     

Double-jump diffusion model for VIX: evidence from VVIX

This paper studies the continuous-time dynamics of VIX with stochastic volatility and jumps in VIX and volatility. Built on the general parametric affine model with stochastic volatility and jumps in the logarithm of VIX, we derive a linear relationship between the stochastic volatility factor and the VVIX index. We detect the existence of a co-jump of VIX and VVIX and put forward a double-jump stochastic volatility model for VIX through its joint property with VVIX. Using the VVIX index as a proxy for stochastic volatility, we use the MCMC method to estimate the dynamics of VIX. Comparing nested models of VIX, we show that the jump in VIX and the volatility factor are statistically significant. The jump intensity is also stochastic. We analyse the impact of the jump factor on VIX dynamics.     

Expectations Hypothesis of the Term Structure of Implied Volatility: Evidence from Foreign Currency and Stock Index Options

Using a stochastic volatility option pricing model, we showthat the implied volatilities of at-the-money options are notnecessarily unbiased and that the fixed interval time-seriescan produce misleading results. Our results do not support theexpectations hypothesis: long-term volatilities rise relativeto short-term volatilities, but the increases are not matchedas predicted by the expectations hypothesis. In addition, anincrease in the current long-term volatility relative to thecurrent short-term volatility is followed by a subsequent decline.The results are similar for both foreign currency and the S&P500 stock index options.     

Turbocharging Monte Carlo pricing for the rough Bergomi model

The rough Bergomi model, introduced by Bayer et al. [Quant. Finance, 2016, 16(6), 887–904], is one of the recent rough volatility models that are consistent with the stylised fact of implied volatility surfaces being essentially time-invariant, and are able to capture the term structure of skew observed in equity markets. In the absence of analytical European option pricing methods for the model, we focus on reducing the runtime-adjusted variance of Monte Carlo implied volatilities, thereby contributing to the model’s calibration by simulation. We employ a novel composition of variance reduction methods, immediately applicable to any conditionally log-normal stochastic volatility model. Assuming one targets implied volatility estimates with a given degree of confidence, thus calibration RMSE, the results we demonstrate equate to significant runtime reductions—roughly 20 times on average, across different correlation regimes.     

A Comparison of Option Prices Under Different Pricing Measures in a Stochastic Volatility Model with Correlation

This paper investigates option prices in an incomplete stochastic volatility model with correlation. In a general setting, we prove an ordering result which says that prices for European options with convex payoffs are decreasing in the market price of volatility risk.As an example, and as our main motivation, we investigate option pricing under the class of q-optimal pricing measures. The q-optimal pricing measure is related to the marginal utility indifference price of an agent with constant relative risk aversion. Using the ordering result, we prove comparison theorems between option prices under the minimal martingale, minimal entropy and variance-optimal pricing measures. If the Sharpe ratio is deterministic, the comparison collapses to the well known result that option prices computed under these three pricing measures are the same.As a concrete example, we specialize to a variant of the Hull-White or Heston model for which the Sharpe ratio is increasing in volatility. For this example we are able to deduce option prices are decreasing in the parameter q. Numerical solution of the pricing pde corroborates the theory and shows the magnitude of the differences in option price due to varying q.JEL Classification: D52, G13     

Impact of macroeconomic announcements on implied volatility slope of SPX options and VIX

This paper examines the impact of macroeconomic announcements on the high-frequency behavior of the observed implied volatility skew of S&P 500 index options and VIX. We document that macroeconomic announcements affect VIX significantly and slope at a lesser extent. We also find evidence that good and bad announcements significantly and asymmetrically change implied volatility slope and VIX.     

Short-time at-the-money skew and rough fractional volatility

The Black–Scholes implied volatility skew at the money of SPX options is known to obey a power law with respect to the time to maturity. We construct a model of the underlying asset price process which is dynamically consistent to the power law. The volatility process of the model is driven by a fractional Brownian motion with Hurst parameter less than half. The fractional Brownian motion is correlated with a Brownian motion which drives the asset price process. We derive an asymptotic expansion of the implied volatility as the time to maturity tends to zero. For this purpose, we introduce a new approach to validate such an expansion, which enables us to treat more general models than in the literature. The local-stochastic volatility model is treated as well under an essentially minimal regularity condition in order to show such a standard model cannot be dynamically consistent to the power law.     

Mean-reversion properties of implied volatilities

In this paper, we present a new stylized fact for options whose underlying asset is a stock index. Extracting implied volatility time series from call and put options on the Deutscher Aktien index (DAX) and financial times stock exchange index (FTSE), we show that the persistence of these volatilities depends on the moneyness of the options used for its computation. Using a functional autoregressive model, we show that this effect is statistically significant. Surprisingly, we show that the diffusion-based stochastic volatility models are not consistent with this stylized fact. Finally, we argue that adding jumps to a diffusion-based volatility model help recovering this volatility pattern. This suggests that the persistence of implied volatilities can be related to the tails of the underlying volatility process: this corroborates the intuition that the liquidity of the options across moneynesses introduces an additional risk factor to the one usually considered.     

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.     

Equity-linked annuities with multiscale hybrid stochastic and local volatility

In recent times, hybrid underlying models have become an industry standard for the pricing of derivatives and other problems in finance. This paper chooses a hybrid stochastic and local volatility model to evaluate an equity-linked annuity (ELA), which is a sort of tax-deferred annuity whose credited interest is linked to an equity index. The stochastic volatility component of the hybrid model is driven by a fast mean-reverting diffusion process while the local volatility component is given by the constant elasticity of variance (CEV) model. Since contracts of the ELA usually have long maturities over 10 years, a slowly moving factor in the stochastic volatility of stock index is expected to play a significant role in the valuation of the ELA, and thus, it is added to the aforementioned model. Based on this multiscale hybrid model, an analytic approximate formula is obtained for the price of a European option in terms of the CEV probability density function and then the result is applied to the value of the point-to-point ELA. The formula leads to the dependence structure of the ELA price on the fast and slow scale stochastic volatility and the elasticity of variance.     

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.     

A jump diffusion model for VIX volatility options and futures

Volatility indices are becoming increasingly popular as a measure of market uncertainty and as a new asset class for developing derivative instruments. Although jumps are widely considered as a salient feature of volatility, their implications for pricing volatility options and futures are not yet fully understood. This paper provides evidence indicating that the time series behaviour of the VIX index is well approximated by a mean reverting logarithmic diffusion with jumps. This process is capable of capturing stylized facts of VIX dynamics such as fast mean-reversion at higher levels, level effects of volatility and large upward movements during times of market stress. Based on the empirical results, we provide closed-form valuation models for European options written on the spot and forward VIX, respectively.     

Validity of discrete-time stochastic volatility models in non-synchronous equity markets

The paper investigates the validity of versions of discrete-time stochastic volatility models for index series known to contain component stocks exhibiting non-synchronous trading. The efficient method of moments (EMM) is used to fit versions of the discrete-time stochastic volatility (SV) model. The EMM methodology confronts moment conditions generated by a score generator (SNP) that are valid by construction. The moment generator suggests non-linearity in the index series. The EMM construction shows that a classical discrete time stochastic volatility model is rejected. An extended model incorporating an asymmetric volatility specification validates all the moment scores. Option values from Black and Scholes (BS) and Monte Carlo simulations (MC) seem significantly different. The results suggest that BS does not price asymmetry adequately. Asymmetry suggests increased market risk inducing higher BS call prices and lower (higher) BS put pricing for ATM and OTM options (ITM) relative to MC.     

Sequential Monte Carlo for fractional stochastic volatility models

In this paper, we consider a fractional stochastic volatility model, that is a model in which the volatility may exhibit a long-range dependent or a rough/antipersistent behaviour. We propose a dynamic sequential Monte Carlo methodology that is applicable to both long memory and antipersistent processes in order to estimate the volatility as well as the unknown parameters of the model. We establish a central limit theorem for the state and parameter filters and we study asymptotic properties (consistency and asymptotic normality) for the filter. We illustrate our results with a simulation study and we apply our method to estimate the volatility and the parameters of a long-range dependent model for S& P 500 data.     

Option pricing under a stressed-beta model

Empirical studies have concluded that stochastic volatility is an important component of option prices. We introduce a regime-switching mechanism into a continuous-time Capital Asset Pricing Model which naturally induces stochastic volatility in the asset price. Under this Stressed-Beta model, the mechanism is relatively simple: the slope coefficient—which measures asset returns relative to market returns—switches between two values, depending on the market being above or below a given level. After specifying the model, we use it to price European options on the asset. Interestingly, these option prices are given explicitly as integrals with respect to known densities. We find that the model is able to produce a volatility skew, which is a prominent feature in option markets. This opens the possibility of forward-looking calibration of the slope coefficients, using option data, as illustrated in the paper.     

Local volatility of volatility for the VIX market

Following a trend of sustained and accelerated growth, the VIX futures and options market has become a closely followed, active and liquid market. The standard stochastic volatility models—which focus on the modeling of instantaneous variance—are unable to fit the entire term structure of VIX futures as well as the entire VIX options surface. In contrast, we propose to model directly the VIX index, in a mean-reverting local volatility-of-volatility model, which will provide a global fit to the VIX market. We then show how to construct the local volatility-of-volatility surface by adapting the ideas in Carr (Local variance gamma. Bloomberg Quant Research, New York, 2008) and Andreasen and Huge (Risk Mag 76–79, 2011) to a mean-reverting process.