Local volatility dynamic models

This paper is concerned with the characterization of arbitrage-free dynamic stochastic models for the equity markets when Itô stochastic differential equations are used to model the dynamics of a set of basic instruments including, but not limited to, the underliers. We study these market models in the framework of the HJM philosophy originally articulated for Treasury bond markets. The main thrust of the paper is to characterize absence of arbitrage by a drift condition and a spot consistency condition for the coefficients of the local volatility dynamics.     

Empirical performance of multifactor term structure models for pricing and hedging Eurodollar futures options

This article compares two one-factor, two two-factor, two three-factor models in the HJM class and Black's [Black, F. (1976). The pricing of commodity contracts. Journal of Financial Economics, 3, 167-179.] implied volatility function in terms of their pricing and hedging performance for Eurodollar futures options across strikes and maturities from 1 Jan 2000 to 31 Dec 2002. We find that three-factor models perform the best for 1-day and 1-week prediction, as well as for 5-day and 20-day hedging. The moneyness bias and the maturity bias appear for all models, but the three-factor models produce lower bias. Three-factor models also outperform other models in hedging, in particular for away-from-the-money and long-dated options. Making Black's volatility a square root or exponential function performs similar to one-factor HJM models in pricing, but not in hedging. Correctly specified and calibrated multifactor models are thus important and cannot be replaced by one-factor models in pricing or hedging interest rate contingent claims.     

Unspanned stochastic volatility and fixed income derivatives pricing

We propose a parsimonious ‘unspanned stochastic volatility’ model of the term structure and study its implications for fixed-income option prices. The drift and quadratic variation of the short rate are affine in three state variables (the short rate, its long-term mean and variance) which follow a joint Markov (vector) process. Yet, bond prices are exponential affine functions of only two state variables, independent of the current interest rate volatility level. Because this result holds for an arbitrary volatility process, such a process can be calibrated to match fixed income derivative prices. Furthermore, this model can be ‘extended’ (by relaxing the time-homogeneity) to fit any arbitrary term structure. In its ‘HJM’ form, this model nests the analogous stochastic equity volatility model of Heston (1993) [Heston, S.L., 1993. A closed form solution for options with stochastic volatility. Review of Financial Studies 6, 327–343]. In particular, if the volatility process is specified to be affine, closed-form solutions for interest rate options obtain. We propose an efficient algorithm to compute these prices. An application using data on caps and floors shows that the model can capture very well the implied Black spot volatility surface, while simultaneously fitting the observed term structure.     

The importance of the volatility risk premium for volatility forecasting

In this paper, we study the role of the volatility risk premium for the forecasting performance of implied volatility. We introduce a non-parametric and parsimonious approach to adjust the model-free implied volatility for the volatility risk premium and implement this methodology using more than 20 years of options and futures data on three major energy markets. Using regression models and statistical loss functions, we find compelling evidence to suggest that the risk premium adjusted implied volatility significantly outperforms other models, including its unadjusted counterpart. Our main finding holds for different choices of volatility estimators and competing time-series models, underlying the robustness of our results.     

Neural network volatility forecasts

We analyse whether the use of neural networks can improve ‘traditional’ volatility forecasts from time-series models, as well as implied volatilities obtained from options on futures on the Spanish stock market index, the IBEX-35. One of our main contributions is to explore the predictive ability of neural networks that incorporate both implied volatility information and historical time-series information. Our results show that the general regression neural network forecasts improve the information content of implied volatilities and enhance the predictive ability of the models. Our analysis is also consistent with the results from prior research studies showing that implied volatility is an unbiased forecast of future volatility and that time-series models have lower explanatory power than implied volatility. Copyright © 2008 John Wiley & Sons, Ltd.     

Forecasting volatility of the U.S. oil market

We examine the information content of the CBOE Crude Oil Volatility Index (OVX) when forecasting realized volatility in the WTI futures market. Additionally, we study whether other market variables, such as volume, open interest, daily returns, bid-ask spread and the slope of the futures curve, contain predictive power beyond what is embedded in the implied volatility. In out-of-sample forecasting we find that econometric models based on realized volatility can be improved by including implied volatility and other variables. Our results show that including implied volatility significantly improves daily and weekly volatility forecasts; however, including other market variables significantly improves daily, weekly and monthly volatility forecasts.     

FX volatility forecasts and the informational content of market data for volatility

The paper examines the medium-term forecasting ability of several alternative models of currency volatility. The data period covers more than eight years of daily observations, January 1991 to March 1999, for the spot exchange rate, 1- and 3-month volatility of the DEM/JPY, GBP/DEM, GBP/USD, USD/CHF, USD/DEM and USD/JPY. Comparing with the results of ‘pure’ time series models, the reported work investigates whether market implied volatility data can add value in terms of medium-term forecasting accuracy. This is done using data directly available from the marketplace in order to avoid the potential biases arising from ‘backing out’ volatility from a specific option pricing model. On the basis of the over 34 000 out-of-sample forecasts produced, evidence tends to indicate that, although no single volatility model emerges as an overall winner in terms of forecasting accuracy, the ‘mixed’ models incorporating market data for currency volatility perform best most of the time.     

Displaced lognormal volatility skews: analysis and applications to stochastic volatility simulations

We analyze the implied volatility skews generated by displaced lognormal diffusions. In particular, we prove the global monotonicity of implied volatility, and an at-the-money bound on the steepness of downward volatility skews, under displaced lognormal dynamics, which therefore cannot reproduce some features observed in equity markets. A variant, the displaced anti-lognormal, overcomes the steepness constraint, but its state space is bounded above and unbounded below. In light of these limitations on what features the displaced (anti-)lognormal (DL) can model, we exploit the DL, not as a model, but as a control variate, to reduce variance in Monte Carlo simulations of the CEV and SABR local/stochastic volatility models. For either use—as model, or as control variate—the DL’s parameters require estimation. We find an explicit formula for the DL’s short-expiry limiting volatility skew, which allows direct calibration of its parameters to volatility skews implied by market data or by other models.     

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.     

Dynamic volatility spillover effects between oil and agricultural products

The strong volatility spillover between crude oil and agricultural commodity markets reduces the diversification benefits and implies costly risk management process faced by portfolio managers and agricultural producers. This paper proposes a comprehensive study of their dynamic implied volatility spillover effects after the global financial crisis 2008–2009, while considering the transition between oil volatility's regimes. By using implied volatility, our analysis emphasizes on the forward-looking information that market traders usually convey in making decisions. We employ the generalized spillover indices within a fractionally integrated VAR model to capture the dynamic patterns of the volatility spillover effects alongside the Markov Switching Autoregressive model to extract the regimes of oil. Our results show new evidence that the net volatility spillover effect from crude oil to all agricultural commodities tends to decrease when crude oil remains in its low volatility regime. Conversely, this effect experiences an increasing trend when crude oil remains in its relatively high volatility regime. A dynamic strategy that combines oil and the most balanced agricultural commodity in terms of volatility transmission with oil (i.e., close-to-zero net volatility spillovers) depending on oil's regimes consistently outperforms the buy and hold strategy in terms of information ratio.     

General approximation schemes for option prices in stochastic volatility models

In this paper we develop a general method for deriving closed-form approximations of European option prices and equivalent implied volatilities in stochastic volatility models. Our method relies on perturbations of the model dynamics and we show how the expansion terms can be calculated using purely probabilistic methods. A flexible way of approximating the equivalent implied volatility from the basic price expansion is also introduced. As an application of our method we derive closed-form approximations for call prices and implied volatilities in the Heston [Rev. Financial Stud., 1993, 6, 327–343] model. The accuracy of these approximations is studied and compared with numerically obtained values.     

Option markets and implied volatility: Past versus present

Traders in the nineteenth century appear to have priced options the same way that twenty-first-century traders price options. Empirical regularities relating implied volatility to realized volatility, stock prices, and other implied volatilities (including the volatility skew) are qualitatively the same in both eras. Modern pricing models and centralized exchanges have not fundamentally altered pricing behavior, but they have generated increased trading volume and a much closer conformity in the level of observed and model prices. The major change in pricing is the sharp decline in implied volatility relative to realized volatility, evident immediately upon the opening of the CBOE.     

Deep learning volatility: a deep neural network perspective on pricing and calibration in (rough) volatility models

We present a neural network-based calibration method that performs the calibration task within a few milliseconds for the full implied volatility surface. The framework is consistently applicable throughout a range of volatility models—including second-generation stochastic volatility models and the rough volatility family—and a range of derivative contracts. Neural networks in this work are used in an off-line approximation of complex pricing functions, which are difficult to represent or time-consuming to evaluate by other means. The form in which information from available data is extracted and used influences network performance: The grid-based algorithm used for calibration is inspired by representing the implied volatility and option prices as a collection of pixels. We highlight how this perspective opens new horizons for quantitative modelling. The calibration bottleneck posed by a slow pricing of derivative contracts is lifted, and stochastic volatility models (classical and rough) can be handled in great generality as the framework also allows taking the forward variance curve as an input. We demonstrate the calibration performance both on simulated and historical data, on different derivative contracts and on a number of example models of increasing complexity, and also showcase some of the potentials of this approach towards model recognition. The algorithm and examples are provided in the Github repository GitHub: NN-StochVol-Calibrations.     

Contemporaneous intraday volume, option, and futures volatility transmissions across parallel markets

The primary objective of this article is to investigate volatility transmission across three parallel markets operating on the Sydney Futures Exchange (SFE), both within and out of sample. Half-hourly observations are sampled from transaction data for the share price index (SPI) futures, SPI futures options, and 90-day bank accepted bill (BAB) futures markets, and the analysis is carried out using the simultaneous volatility (SVL) system of equations as well as competing volatility models. The results confirm the poor ability of GARCH models to fit intraday data. This study also applies an artificial nesting procedure to evaluate the out-of-sample volatility forecasts. Implied volatility has very limited (if any) predictive power when evaluated in isolation, whereas the SVL model with implied volatility embedded provides incremental information relative to competing model forecasts.     

Assessing the performance of symmetric and asymmetric implied volatility functions

This study examines several alternative symmetric and asymmetric model specifications of regression-based deterministic volatility models to identify the one that best characterizes the implied volatility functions of S&P 500 Index options in the period 1996–2009. We find that estimating the models with nonlinear least squares, instead of ordinary least squares, always results in lower pricing errors in both in- and out-of-sample comparisons. In-sample, asymmetric models of the moneyness ratio estimated separately on calls and puts provide the overall best performance. However, separating calls from puts violates the put-call-parity and leads to severe model mis-specification problems. Out-of-sample, symmetric models that use the logarithmic transformation of the strike price are the overall best ones. The lowest out-of-sample pricing errors are observed when implied volatility models are estimated consistently to the put-call-parity using the joint data set of out-of-the-money options. The out-of-sample pricing performance of the overall best model is shown to be resilient to extreme market conditions and compares quite favorably with continuous-time option pricing models that admit stochastic volatility and random jump risk factors.     

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.     

Option pricing under fast-varying and rough stochastic volatility

Recent empirical studies suggest that the volatilities associated with financial time series exhibit short-range correlations. This entails that the volatility process is very rough and its autocorrelation exhibits sharp decay at the origin. Another classic stylistic feature often assumed for the volatility is that it is mean reverting. In this paper it is shown that the price impact of a rapidly mean reverting rough volatility model coincides with that associated with fast mean reverting Markov stochastic volatility models. This reconciles the empirical observation of rough volatility paths with the good fit of the implied volatility surface to models of fast mean reverting Markov volatilities. Moreover, the result conforms with recent numerical results regarding rough stochastic volatility models. It extends the scope of models for which the asymptotic results of fast mean reverting Markov volatilities are valid. The paper concludes with a general discussion of fractional volatility asymptotics and their interrelation. The regimes discussed there include fast and slow volatility factors with strong or small volatility fluctuations and with the limits not commuting in general. The notion of a characteristic term structure exponent is introduced, this exponent governs the implied volatility term structure in the various asymptotic regimes.     

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.     

Forecasting implied volatility in foreign exchange markets: a functional time series approach

We utilise novel functional time series (FTS) techniques to characterise and forecast implied volatility in foreign exchange markets. In particular, we examine the daily implied volatility curves of FX options, namely; Euro/United States Dollar, Euro/British Pound, and Euro/Japanese Yen. The FTS model is shown to produce both realistic and plausible implied volatility shapes that closely match empirical data during the volatile 2006–2013 period. Furthermore, the FTS model significantly outperforms implied volatility forecasts produced by traditionally employed parametric models. The evaluation is performed under both in-sample and out-of-sample testing frameworks with our findings shown to be robust across various currencies, moneyness segments, contract maturities, forecasting horizons, and out-of-sample window lengths. The economic significance of the results is highlighted through the implementation of a simple trading strategy.     

Does implied volatility provide any information beyond that captured in model-based volatility forecasts?

This paper contributes to our understanding of the informational content of implied volatility. Here we examine whether the S&P 500 implied volatility index (VIX) contains any information relevant to future volatility beyond that available from model based volatility forecasts. It is argued that this approach differs from the traditional forecast encompassing approach used in earlier studies. The findings indicate that the VIX index does not contain any such additional information relevant for forecasting volatility.