Dynamic functional time-series forecasts of foreign exchange implied volatility surfaces

This paper presents static and dynamic versions of univariate, multivariate, and multilevel functional time-series methods to forecast implied volatility surfaces in foreign exchange markets. We find that dynamic functional principal component analysis generally improves out-of-sample forecast accuracy. Specifically, the dynamic univariate functional time-series method shows the greatest improvement. Our models lead to multiple instances of statistically significant improvements in forecast accuracy for daily EUR–USD, EUR–GBP, and EUR–JPY implied volatility surfaces across various maturities, when benchmarked against established methods. A stylised trading strategy is also employed to demonstrate the potential economic benefits of our proposed approach.     

Pure jump models for pricing and hedging VIX derivatives

Recent non-parametric statistical analysis of high-frequency VIX data (Todorov and Tauchen, 2011) reveals that VIX dynamics is a pure jump semimartingale with infinite jump activity and infinite variation. To our best knowledge, existing models in the literature for pricing and hedging VIX derivatives do not have these features. This paper fills this gap by developing a novel class of parsimonious pure jump models with such features for VIX based on the additive time change technique proposed in Li et al., 2016a, Li et al., 2016b. We time change the 3/2 diffusion by a class of additive subordinators with infinite activity, yielding pure jump Markov semimartingales with infinite activity and infinite variation. These processes have time and state dependent jumps that are mean reverting and are able to capture stylized features of VIX. Our models take the initial term structure of VIX futures as input and are analytically tractable for pricing VIX futures and European options via eigenfunction expansions. Through calibration exercises, we show that our model is able to achieve excellent fit for the VIX implied volatility surface which typically exhibits very steep skews. Comparison to two other models in terms of calibration reveals that our model performs better both in-sample and out-of-sample. We explain the ability of our model to fit the volatility surface by evaluating the matching of moments implied from market VIX option prices. To hedge VIX options, we develop a dynamic strategy which minimizes instantaneous jump risk at each rebalancing time while controlling transaction cost. Its effectiveness is demonstrated through a simulation study on hedging Bermudan style VIX options.     

Impact of volatility jumps in a mean-reverting model: Derivative pricing and empirical evidence

This paper investigates the critical role of volatility jumps under mean reversion models. Based on the empirical tests conducted on the historical prices of commodities, we demonstrate that allowing for the presence of jumps in volatility in addition to price jumps is a crucial factor when confronting non-Gaussian return distributions. By employing the particle filtering method, a comparison of results drawn among several mean-reverting models suggests that incorporating volatility jumps ensures an improved fit to the data. We infer further empirical evidence for the existence of volatility jumps from the possible paths of filtered state variables. Our numerical results indicate that volatility jumps significantly affect the level and shape of implied volatility smiles. Finally, we consider the pricing of options under the mean reversion model, where the underlying asset price and its volatility both have jump components.     

Which implied volatility provides the best measure of future volatility?

The volatility smile/skew phenomenon makes it unclear which implied volatility provides the best measure of the market volatility expectation over the remaining life of the option. Due to the high liquidity of at-the-money option and the low sensitivity of its implied volatility to the price error, the at-the-money implied volatility is often considered a good measure of future volatility. In this paper, we raise the question: is at-the-money implied volatility the best we can do? We provide in this paper an analytical rationale that the implied volatility from option with highest vega outperforms the at-the-money implied volatility in terms of forecasting ability, especially for long forecasting horizons. Our empirical findings are consistent with our theoretical argument.     

On implied volatility for options—Some reasons to smile and more to correct

We analyze the properties of the implied volatility, the commonly used volatility estimator by direct option price inversion. It is found that the implied volatility is subject to a systematic bias in the presence of pricing errors, which makes it inconsistent to the underlying volatility. We propose an estimator of the underlying volatility by first estimating nonparametrically the option price function, followed by inverting the nonparametrically estimated price. It is shown that the approach removes the adverse impacts of the pricing errors and produces a consistent volatility estimator for a wide range of option price models. We demonstrate the effectiveness of the proposed approach by numerical simulation and empirical analysis on S&P 500 option data.     

Dupire’s formulas in the Piterbarg option pricing model

In this paper we derive an expression for the local volatility of an underlying asset, given the prices of liquid European call options under the Piterbarg framework. The Piterbarg framework is a multi-curve derivative pricing model which extends the well known Black–Scholes–Merton model by relaxing the assumption of a risk-free interest rate, and includes collateral payments. The expressions for the local volatility is a function of the option price surface, and is then transformed to become a function of the implied volatility surface.     

The role of implied volatility in forecasting future realized volatility and jumps in foreign exchange,stock, and bond markets

We study the forecasting of future realized volatility in the foreign exchange, stock, and bond markets from variables in our information set, including implied volatility backed out from option prices. Realized volatility is separated into its continuous and jump components, and the heterogeneous autoregressive (HAR) model is applied with implied volatility as an additional forecasting variable. A vector HAR (VecHAR) model for the resulting simultaneous system is introduced, controlling for possible endogeneity issues. We find that implied volatility contains incremental information about future volatility in all three markets, relative to past continuous and jump components, and it is an unbiased forecast in the foreign exchange and stock markets. Out-of-sample forecasting experiments confirm that implied volatility is important in forecasting future realized volatility components in all three markets. Perhaps surprisingly, the jump component is, to some extent, predictable, and options appear calibrated to incorporate information about future jumps in all three markets.     

Modelling Regime‐Specific Stock Price Volatility*

Single‐state generalized autoregressive conditional heteroscedasticity (GARCH) models identify only one mechanism governing the response of volatility to market shocks, and the conditional higher moments are constant, unless modelled explicitly. So they neither capture state‐dependent behaviour of volatility nor explain why the equity index skew persists into long‐dated options. Markov switching (MS) GARCH models specify several volatility states with endogenous conditional skewness and kurtosis; of these the simplest to estimate is normal mixture (NM) GARCH, which has constant state probabilities. We introduce a state‐dependent leverage effect to NM‐GARCH and thereby explain the observed characteristics of equity index returns and implied volatility skews, without resorting to time‐varying volatility risk premia. An empirical study on European equity indices identifies two‐state asymmetric NM‐GARCH as the best fit of the 15 models considered. During stable markets volatility behaviour is broadly similar across all indices, but the crash probability and the behaviour of returns and volatility during a crash depends on the index. The volatility mean‐reversion and leverage effects during crash markets are quite different from those in the stable regime.     

Trading and non-trading period realized market volatility: Does it matter for forecasting the volatility of US stocks?

We study the potential merits of using trading and non-trading period market volatilities to model and forecast the stock volatility over the next one to 22 days. We demonstrate the role of overnight volatility information by estimating heterogeneous autoregressive (HAR) model specifications with and without a trading period market risk factor using ten years of high-frequency data for the 431 constituents of the S&P 500 index. The stocks’ own overnight squared returns perform poorly across stocks and forecast horizons, as well as in the asset allocation exercise. In contrast, we find overwhelming evidence that the market-level volatility, proxied by S&P Mini futures, matters significantly for improving the model fit and volatility forecasting accuracy. The greatest model fit and forecast improvements are found for short-term forecast horizons of up to five trading days, and for the non-trading period market-level volatility. The documented increase in forecast accuracy is found to be associated with the stocks’ sensitivity to the market risk factor. Finally, we show that both the trading and non-trading period market realized volatilities are relevant in an asset allocation context, as they increase the average returns, Sharpe ratios and certainty equivalent returns of a mean–variance investor.     

Recent developments in financial economics and econometrics: An overview

Research papers in empirical finance and financial econometrics are among the most widely cited, downloaded and viewed articles in the discipline of Finance. The special issue presents several papers by leading scholars in the field on “Recent Developments in Financial Economics and Econometrics”. The breadth of coverage is substantial, and includes original research and comprehensive review papers on theoretical, empirical and numerical topics in Financial Economics and Econometrics by leading researchers in finance, financial economics, financial econometrics and financial statistics. The purpose of this special issue on “Recent Developments in Financial Economics and Econometrics” is to highlight several novel and significant developments in financial economics and financial econometrics, specifically dynamic price integration in the global gold market, a conditional single index model with local covariates for detecting and evaluating active management, whether the Basel Accord has improved risk management during the global financial crisis, the role of banking regulation in an economy under credit risk and liquidity shock, separating information maximum likelihood estimation of the integrated volatility and covariance with micro-market noise, stress testing correlation matrices for risk management, whether bank relationship matters for corporate risk taking, with evidence from listed firms in Taiwan, pricing options on stocks denominated in different currencies, with theory and illustrations, EVT and tail-risk modelling, with evidence from market indices and volatility series, the economics of data using simple model free volatility in a high frequency world, arbitrage-free implied volatility surfaces for options on single stock futures, the non-uniform pricing effect of employee stock options using quantile regression, nonlinear dynamics and recurrence plots for detecting financial crisis, how news sentiment impacts asset volatility, with evidence from long memory and regime-switching approaches, quantitative evaluation of contingent capital and its applications, high quantiles estimation with Quasi-PORT and DPOT, with an application to value-at-risk for financial variables, evaluating inflation targeting based on the distribution of inflation and inflation volatility, the size effects of volatility spillovers for firm performance and exchange rates in tourism, forecasting volatility with the realized range in the presence of noise and non-trading, using CARRX models to study factors affecting the volatilities of Asian equity markets, deciphering the Libor and Euribor spreads during the subprime crisis, information transmission between sovereign debt CDS and other financial factors for Latin America, time-varying mixture GARCH models and asymmetric volatility, and diagnostic checking for non-stationary ARMA models with an application to financial data.     

Implied volatility surfaces and market activity over time

An impressive body of the literature has investigated the patterns of changes in implied volatilities across strike prices and maturities. Although such studies try to explain the existence of the volatility skew and term structure, they remain silent about the evolution of the volatility surface as time goes by and market variables move. Relying on a technique of signal processing called Independent Component Analysis, we extract volatility modes that account for most of the variations in the shape of the surface. We then relate the magnitude of volatility changes along those modes to market activity.     

Efficient predictability of stock return volatility: The role of stock market implied volatility

This study examines the predictability of stock market implied volatility on stock volatility in five developed economies (the US, Japan, Germany, France, and the UK) using monthly volatility data for the period 2000 to 2017. We utilize a simple linear autoregressive model to capture predictive relationships between stock market implied volatility and stock volatility. Our in-sample results show there exists very significant Granger causality from stock market implied volatility to stock volatility. The out-of-sample results also indicate that stock market implied volatility is significantly more powerful for stock volatility than the oil price volatility in five developed economies.     

A comparative goodness-of-fit analysis of distributions of some Lévy processes and Heston model to stock index returns

In this paper, we investigate the goodness-of-fit of three Lévy processes, namely Variance-Gamma (VG), Normal-Inverse Gaussian (NIG) and Generalized Hyperbolic (GH) distributions, and probability distribution of the Heston model to index returns of twenty developed and emerging stock markets. Furthermore, we extend our analysis by applying a Markov regime switching model to identify normal and turbulent periods. Our findings indicate that the probability distribution of the Heston model performs well for emerging markets under full sample estimation and retains goodness of fit for high volatility periods, as it explicitly accounts for the volatility process. On the other hand, the distributions of the Lévy processes, especially the VG and NIG distributions, generally improves upon the fit of the Heston model, particularly for developed markets and low volatility periods. Furthermore, some distributions yield to significantly large test statistics for some countries, even though they fit well to other markets, which suggest that properties of the stock markets are crucial in identifying the best distribution representing empirical returns.     

Multivariate rotated ARCH models

This paper introduces a new class of multivariate volatility models which is easy to estimate using covariance targeting, even with rich dynamics. We call them rotated ARCH (RARCH) models. The basic structure is to rotate the returns and then to fit them using a BEKK-type parameterization of the time-varying covariance whose long-run covariance is the identity matrix. This yields the rotated BEKK (RBEKK) model. The extension to DCC-type parameterizations is given, introducing the rotated DCC (RDCC) model. Inference for these models is computationally attractive, and the asymptotics are standard. The techniques are illustrated using data on the DJIA stocks.     

Does the option market produce superior forecasts of noise‐corrected volatility measures?

This paper assesses the robustness of the relative performance of spot‐ and options‐based volatility forecasts to the treatment of microstructure noise. Robustness of the results to the method of constructing option‐implied forecasts is also investigated. Using a test for superior predictive ability, model‐free implied volatility, which exploits information in the volatility ‘smile’, and at‐the‐money implied volatility, which does not, are both tested as benchmark forecasts of a range of alternative volatility proxies. The results provide compelling evidence against the model‐free forecast for three Dow Jones Industrial Average stocks, over a 2001–2006 evaluation period. In contrast, the at‐the‐money implied volatility forecast is given strong support for the three equities over this period. Neither benchmark is supported for the S&P500 index. Importantly, the main qualitative results are invariant to the method of noise correction used in measuring future volatility. Copyright © 2008 John Wiley & Sons, Ltd.     

Notes and Comments: An approximation of caplet implied volatilities in Gaussian models

Abstract It is market practice to quote interest rate derivatives traded “over the counter” in terms of their implied volatility. For this reason, the term structure of at-the-money cap volatilities as well as the volatility surface of at-the-money swaptions are directly observed. This paper analyzes the case of caps. Any analysis of these markets would most likely report two main facts. The first is that the level of the volatility is inversely related to the level of the interest rates. The second is that the term structure is either a decreasing or a humped function of maturity. For a reference, see Rebonato (2003) and Brigo and Mercurio (2001). Rebonato (2003) suggests that the structure of implied volatility is humped in periods of normal market conditions and decreasing when markets are “excited”. Interpreting and explaining such phenomena is indeed an interesting and important issue. Mathematics Subject Classification (2000): 91B70 Journal of Economic Literature Classification: E43, C13     

Implied volatility and the risk-free rate of return in options markets

We numerically solve systems of Black–Scholes formulas for implied volatility and implied risk-free rate of return. After using a seemingly unrelated regressions (SUR) model to obtain point estimates for implied volatility and implied risk-free rate, the options are re-priced using these parameters. After repricing, the difference between the market price and model price is increasing in time to expiration, while the effect of moneyness and the bid-ask spread are ambiguous. Our varying risk-free rate model yields Black–Scholes prices closer to market prices than the fixed risk-free rate model. In addition, our model is better for predicting future evolutions in model-free implied volatility as measured by the VIX.     

On the informational efficiency of S&P500 implied volatility

Implied volatility is often considered to represent a market's prediction of future volatility. If such a market was to generate efficient volatility forecasts, implied volatility should reflect all relevant conditioning information. The purpose of this paper is to determine whether a publicly available and commonly used implied volatility index, the VIX index (as published by the Chicago Board of Options Exchange) is in fact efficient with respect to a wide set of conditioning information. Results indicate that the VIX index is not efficient with respect to all elements in the information set that may be used to form volatility forecasts.     

Predicting volatility using the Markov-switching multifractal model: Evidence from S&P 100 index and equity options

In this paper, we evaluate the performance of the ability of Markov-switching multifractal (MSM), implied, GARCH, and historical volatilities to predict realized volatility for both the S&P 100 index and equity options. Some important findings are as follows. First, we find that the ability of MSM and GARCH volatilities to predict realized volatility is better than that of implied and historical volatilities for both the index and equity options. Second, equity option volatility is more difficult to be forecast than index option volatility. Third, both index and equity option volatilities can be better forecast during non-global financial crisis periods than during global financial crisis periods. Fourth, equity option volatility exhibits distinct patterns conditional on various equity and option characteristics and its predictability by MSM and implied volatilities depends on these characteristics. And finally, we find that MSM volatility outperforms implied volatility in predicting equity option volatility conditional on various equity and option characteristics.     

Testing for co-integration in vector autoregressions with non-stationary volatility

Many key macroeconomic and financial variables are characterized by permanent changes in unconditional volatility. In this paper we analyse vector autoregressions with non-stationary (unconditional) volatility of a very general form, which includes single and multiple volatility breaks as special cases. We show that the conventional rank statistics computed as in  and  are potentially unreliable. In particular, their large sample distributions depend on the integrated covariation of the underlying multivariate volatility process which impacts on both the size and power of the associated co-integration tests, as we demonstrate numerically. A solution to the identified inference problem is provided by considering wild bootstrap-based implementations of the rank tests. These do not require the practitioner to specify a parametric model for volatility, or to assume that the pattern of volatility is common to, or independent across, the vector of series under analysis. The bootstrap is shown to perform very well in practice.