Implied Volatility Data Research Articles

Simulation of Arbitrage-Free Implied Volatility Surfaces

We present a computationally tractable method for simulating arbitrage-free implied volatility surfaces. We illustrate how our method may be combined with a data-driven model based on historical SPX implied volatility data to generate dynamic scenarios for arbitrage-free implied volatility surfaces. Our approach conciliates static arbitrage constraints with a realistic representation of statistical properties of implied volatility co-movements.

Multiscale stochastic volatility for variance swaps with constant elasticity of variance

The variance swap is one of volatility derivatives popularly used for the risk management of financial instruments traded in volatile market. An appropriate choice of a volatility model is an important part of the risk management. One of desirable considerations should be given to the fact that the volatility varies on several characteristic time scales. Stochastic volatility models can reflect this feature by introducing multiscale volatility factors. However, pure stochastic volatility models cannot capture the whole volatility surface accurately although the model parameters have been calibrated to replicate the market implied volatility data for near at-the-money strikes. So, we choose a hybrid model of constant elasticity of variance type of local volatility and fast and slow scale stochastic volatility for evaluating the fair strikes of variance swaps. We obtain a closed-form solution formula for the approximate fair strike values of continuously sampled variance swaps and compute the solution. The theoretical formula is validated through Monte Carlo simulation. The predictability of the strike price movements is discussed in terms of the sensitive effects of the stochastic volatility and the elasticity of variance parameters for a given partial information about the underlying asset and volatility.

A Fractional Integration Analysis on Daily FX Implied Volatility: Long Memory Feature and Structural Changes

Purpose - The purpose of this paper is to analyze the dynamic factors of the daily FX implied volatility based on the fractional integration methods focusing on long memory feature and structural changes. Design/methodology/approach - This paper uses the daily FX implied volatility data of the EUR-USD and the JPY-USD exchange rates. For the fractional integration analysis, this paper first applies the basic ARFIMA-FIGARCH model and the Local Whittle method to explore the long memory feature in the implied volatility series. Then, this paper employs the Adaptive-ARFIMA-Adaptive-FIGARCH model with a flexible Fourier form to allow for the structural changes with the long memory feature in the implied volatility series. Findings - This paper finds statistical evidence of the long memory feature in the first two moments of the implied volatility series. And, this paper shows that the structural changes appear to be an important factor and that neglecting the structural changes may lead to an upward bias in the long memory feature of the implied volatility series. Research implications or Originality - The implied volatility has widely been believed to be the market’s best forecast regarding the future volatility in FX markets, and modeling the evolution of the implied volatility is quite important as it has clear implications for the behavior of the exchange rates in FX markets. The Adaptive-ARFIMA-Adaptive-FIGARCH model could be an excellent description for the FX implied volatility series

A two-step framework for arbitrage-free prediction of the implied volatility surface

In this study, we propose a two-step framework to predict the implied volatility surface (IVS) in a manner that excludes static arbitrage. First, we select features to represent the surface and predict them. Second, we use the predicted features to construct the IVS using a deep neural network (DNN) model by incorporating constraints that can prevent static arbitrage. We consider three methods to extract features from the implied volatility data: principal component analysis, variational autoencoder, and sampling the surface. We predict these features using the long short-term memory model. Additionally, we use a long time series of implied volatility data for S&P500 index options to train our models. We find that two feature construction methods (i.e. sampling the surface and variational autoencoders combined with DNN for surface construction) are the best performers in the out-of-sample prediction. Furthermore, both of them substantially outperform a popular regression model. We also find that the DNN model for surface construction not only removes static arbitrage but also significantly reduces the prediction error compared with a standard interpolation method.

A Hausman Test for Partially Linear Models with an Application to Implied Volatility Surface

This paper develops a test that helps assess whether the term structure of option implied volatility is constant across different levels of moneyness. The test is based on the Hausman principle of comparing two estimators, one that is efficient but not robust to the deviation being tested, and one that is robust but not as efficient. Distribution of the proposed test statistic is investigated in a general semiparametric setting via the multivariate Delta method. Using recent S&P 500 index traded options data from September 2009 to December 2018, we find that a partially linear model permitting a flexible “volatility smile” and an additive quadratic time effect is a statistically adequate depiction of the implied volatility data for most years. The constancy of implied volatility term structure, in turn, implies that option traders shall feel confident and execute volatility-based strategies using at-the-money options for its high liquidity.

Day-of-the-week effect of major currency pairs: new evidences from investors’ fear gauge

Extensive studies have been done on the day-of-the-week effect in currency markets using the spot or future prices. Surprisingly, no literature is available on the day-of-the-week effect, considering the gauge of investors’ fears, that is, implied volatility. In this study, we attempted to analyze the day-of-the-week effect in the context of implied volatility for the currency markets. An analysis of most heavily traded currency pairs was carried out using implied volatility data for 10 pairs with respect to spot rates. An empirical analysis was carried out using the compound of univariate generalized autoregressive conditional heteroscedasticity methods on each currency pair. The principal findings of this study are the presence of the Monday effect in the fear sentiments for all the currency pairs, denoting high positive returns with substantial value, and the Friday effect displaying negative returns. The variance of fear gauge observed for Mondays was positive and for Fridays was negative. However, the spot rate did not replicate the same effects as depicted by implied volatility. The analysis of the day-of-the-week effect of returns in the context of implied volatility validated the existence of the Monday effect and the Friday effect for the currency pairs under study.

Forecasting USD-BRL Currency Rate Volatility using Realized and Implied Volatilities Data

Abstract This article assesses the impact of exogenous variables in GARCH models, when applied to volatility forecasts for the Brazilian USD-BRL currency market. As exogenous variables, we used the realized variance, based on high frequency data, and the FXVol index, based on market implied volatility data. This is the first study to use the FXVol index and to investigate its effects on Brazilian foreign exchange volatility. The results indicate statistical significance of the superiority of the extended models when predicting volatility. We conclude that high frequency data and market implied volatility contain relevant information with respect to USD-BRL currency volatility. These find ings are relevant for hedgers, speculators and practitioners in general.

Nonlinear relationships amongst the implied volatilities of crude oil and precious metals

Unlike most prior studies, we rely on implied volatility data from the Chicago Board of Options Exchange to investigate the presence of cointegration and nonlinear causality between the global market of crude oil and the markets of precious metal (gold and silver) and gold-miner stocks. We apply nonlinear ARDL bound and symmetric and asymmetric nonlinear Granger causality tests to examine cointegration and short-run “lead-lag” relationships, respectively. For comparison purposes, we also apply the linear ARDL models. The results show that the nonlinear ARDL models successfully capture the long-term linkages between oil and precious metal markets, while the linear ARDL approach mostly fails to do so. Results from the nonlinear Granger causality test suggest a bidirectional and symmetric effect between crude oil and gold markets. Investment and policy implications are discussed.

Are there gains from using information over the surface of implied volatilities?

We investigate the out‐of‐sample predictability of implied volatility using the information over the implied volatility surface. We show that implied volatility surface is useful for the out‐of‐sample forecast of implied volatility up to 1 week ahead. Trading strategies based on the predictability of implied volatility could generate significant risk‐adjusted gains after controlling for transaction costs. Significant results also depend on the way of modeling implied volatility surface. We then calibrate a two‐factor stochastic volatility option pricing model to implied volatility data. Results show that implied volatility is better explained by both long‐ and short‐term variance factors.

The Hybrid Stochastic-Local Volatility Model with Applications in Pricing FX Options

This thesis presents our study on using the hybrid stochastic-local volatility model for option pricing. Many researchers have demonstrated that stochastic volatility models cannot capture the whole volatility surface accurately, although the model parameters have been calibrated to replicate the market implied volatility data for near at-the-money strikes. On the other hand, the local volatility model can reproduce the implied volatility surface, whereas it does not consider the stochastic behaviour of the volatility. To combine the advantages of stochastic volatility (SV) and local volatility (LV) models, a class of stochastic-local volatility (SLV) models has been developed. The SLV model contains a stochastic volatility component represented by a volatility process and a local volatility component represented by a so-called leverage function. The leverage function can be roughly seen as a ratio between local volatility and conditional expectation of stochastic volatility. The difficulty of implementing the SLV model lies in the calibration of the leverage function. In the thesis, we first review the fundamental theories of stochastic differential equations and the classic option pricing models, and study the behaviour of the volatility in the context of FX market. We then introduce the SLV model and illustrate our implementation of the calibration and pricing procedure. We apply the SLV model to exotic option pricing in the FX market and compare pricing results of the SLV model with pure local volatility and pure stochastic volatility models. Numerical results show that the SLV model can match the implied volatility surface very well as well as improve the pricing performance for barrier options. In addition, we further discuss some extensions of the SLV project, such as parallelization potential for accelerating option pricing and pricing techniques for window barrier options. Although the SLV model we use in the thesis is not entirely new, we contribute to the research in the following aspects: 1) we investigate the hybrid volatility modeling thoroughly from theoretical backgrounds to practical implementations; 2) we resolve some critical issues in implementing the SLV model such as developing a fast and stable numerical method to derive the leverage function; and 3) we build a robust calibration and pricing platform under the SLV model, which can be extended for practical uses.

Fractional Black–Scholes option pricing, volatility calibration and implied Hurst exponents in South African context

Background: Contingent claims on underlying assets are typically priced under a framework that assumes, inter alia, that the log returns of the underlying asset are normally distributed. However, many researchers have shown that this assumption is violated in practice. Such violations include the statistical properties of heavy tails, volatility clustering, leptokurtosis and long memory. This paper considers the pricing of contingent claims when the underlying is assumed to display long memory, an issue that has heretofore not received much attention.Aim: We address several theoretical and practical issues in option pricing and implied volatility calibration in a fractional Black–Scholes market. We introduce a novel eight-parameter fractional Black–Scholes-inspired (FBSI) model for the implied volatility surface, and consider in depth the issue of calibration. One of the main benefits of such a model is that it allows one to decompose implied volatility into an independent long-memory component – captured by an implied Hurst exponent – and a conditional implied volatility component. Such a decomposition has useful applications in the areas of derivatives trading, risk management, delta hedging and dynamic asset allocation.Setting: The proposed FBSI volatility model is calibrated to South African equity index options data as well as South African Rand/American Dollar currency options data. However, given the focus on the theoretical development of the model, the results in this paper are applicable across all financial markets.Methods: The FBSI model essentially combines a deterministic function form of the 1-year implied volatility skew with a separate deterministic function for the implied Hurst exponent, thus allowing one to model both observed implied volatility surfaces as well as decompose them into independent volatility and long-memory components respectively. Calibration of the model makes use of a quasi-explicit weighted least-squares optimisation routine.Results: It is shown that a fractional Black–Scholes model always admits a non-constant implied volatility term structure when the Hurst exponent is not 0.5, and that 1-year implied volatility is independent of the Hurst exponent and equivalent to fractional volatility. Furthermore, we show that the FBSI model fits the equity index implied volatility data very well but that a more flexible Hurst exponent parameterisation is required to fit accurately the currency implied volatility data.Conclusion: The FBSI model is an arbitrage-free deterministic volatility model that can accurately model equity index implied volatility. It also provides one with an estimate of the implied Hurst exponent, which could be very useful in derivatives trading and delta hedging.

A Dynamic Semiparametric Factor Model for Implied Volatility String Dynamics

A primary goal in modelling the implied volatility surface (IVS) for pricing and hedging aims at reducing complexity. For this purpose one fits the IVS each day and applies a principal component analysis using a functional norm. This approach, however, neglects the degenerated string structure of the implied volatility data and may result in a modelling bias. We propose a dynamic semiparametric factor model (DSFM), which approximates the IVS in a finite dimensional function space. The key feature is that we only fit in the local neighborhood of the design points. Our approach is a combination of methods from functional principal component analysis and backfitting techniques for additive models. The model is found to have an approximate 10% better performance than a sticky moneyness model. Finally, based on the DSFM, we devise a generalized vega-hedging strategy for exotic options that are priced in the local volatility framework. The generalized vega-hedging extends the usual approaches employed in the local volatility framework.

Short and Long Run Uncertainty

Uncertainty appears to have both a short-run and a long-run component, which we measure using firm and macro implied volatility data from options of 30 days to 10 years duration. We ask what may be driving uncertainty over these different time horizons, finding that oil price volatility is particularly important for short-run uncertainty, policy uncertainty is particularly important for long-run uncertainty, while currency volatility and CEO turnover appear to equally impact short- and long-run uncertainty. Examining a panel of over 4,000 firms from 1996 to 2013 we find that R&D is relatively more sensitive to long-run uncertainty than investment, and in turn investment is relatively more sensitive to long-run uncertainty than hiring. In a simulation model we investigate the channels underlying this pecking-order response to long-run uncertainty, and show that lower depreciation rates and higher adjustment costs lead R&D and investment to be more sensitive to longer-run uncertainty than hiring. Collectively, these results suggest that recent events that have raised long-run policy uncertainty may be particularly damaging to growth by reducing R&D and investment.

Fractional Black-Scholes Option Pricing, Volatility Calibration and Implied Hurst Exponents

This paper addresses several theoretical and practical issues in option pricing and implied volatility calibration in a fractional Black-Scholes market. In particular, we discuss how the fractional Black-Scholes model admits a non-constant implied volatility term structure when the Hurst exponent is not 0.5, and also that one-year implied volatility is independent of Hurst exponent and equivalent to fractional volatility. Building on these observations, we introduce a novel 8-parameter fractional Black-Scholes inspired, or FBSI, model. This deterministic volatility surface model is based on the fractional Black-Scholes framework and uses Gatheral’s (2004) SVI pamaterisation for the fractional volatility skew and a quadratic parameterisation for the Hurst exponent skew. The issue of arbitrage-free calibration for the FBSI model is addressed in depth and it is proven in general that any FBSI volatility surface will be free from calendar-spread arbitrage. The FBSI model is empirically tested on implied volatility data on a South African equity index as well as the USDZAR exchange rate. Results show that the FBSI model fits the equity index implied volatility data very well and that a more flexible Hurst exponent parameterisation is needed to accurately fit the USDZAR implied volatility surface data.

Psychological Barrier in Foreign Exchange Rate and Implied Volatility in Currency Exchange Option

In this paper, we examine the existence of the psychological barriers in three foreign exchange rate, won/dollar, euro/dollar, yen/dollar, and test that the psychological barriers effect to the implied volatilities of the FX options. For each exchange rate, the existence and spots of the psychological barriers are estimated from roughly 10 years data for each currency rate, and GARCH (1, 1) model was applied to observe the momentum effect about the mean and variance of the conditional returns, and the implied volatility of the FX-options for each currency rate near the psychological barriers. Since this effect is more clearly observed on the implied volatility data, this fact supports that psychological barriers affects to the price of the FX-options.

The hybrid stochastic-local volatility model with applications in pricing FX options

This thesis presents our study on using the hybrid stochastic-local volatility model for option pricing. Many researchers have demonstrated that stochastic volatility models cannot capture the whole volatility surface accurately, although the model parameters have been calibrated to replicate the market implied volatility data for near at-the-money strikes. On the other hand, the local volatility model can reproduce the implied volatility surface, whereas it does not consider the stochastic behaviour of the volatility. To combine the advantages of stochastic volatility (SV) and local volatility (LV) models, a class of stochastic-local volatility (SLV) models has been developed. The SLV model contains a stochastic volatility component represented by a volatility process and a local volatility component represented by a so-called leverage function. The leverage function can be roughly seen as a ratio between local volatility and conditional expectation of stochastic volatility. The difficulty of implementing the SLV model lies in the calibration of the leverage function. In the thesis, we first review the fundamental theories of stochastic differential equations and the classic option pricing models, and study the behaviour of the volatility in the context of FX market. We then introduce the SLV model and illustrate our implementation of the calibration and pricing procedure. We apply the SLV model to exotic option pricing in the FX market and compare pricing results of the SLV model with pure local volatility and pure stochastic volatility models. Numerical results show that the SLV model can match the implied volatility surface very well as well as improve the pricing performance for barrier options. In addition, we further discuss some extensions of the SLV project, such as parallelization potential for accelerating option pricing and pricing techniques for window barrier options. Although the SLV model we use in the thesis is not entirely new, we contribute to the research in the following aspects: 1) we investigate the hybrid volatility modeling thoroughly from theoretical backgrounds to practical implementations; 2) we resolve some critical issues in implementing the SLV model such as developing a fast and stable numerical method to derive the leverage function; and 3) we build a robust calibration and pricing platform under the SLV model, which can be extended for practical uses.

Constructing a South African Index Volatility Surface from Exchange Traded Data

The exchange traded index option market in South Africa has seen tremendous growth during the last couple of years. The biggest liquidity is in options on the near and middle Alsi future contracts. Alsi futures are listed future contracts on the FTSE/JSE Top40 index, the most important and tradable equity index in South Africa. OTC and listed options trade on a skew and most market makers have implemented their own proprietary skew generators. Clearing houses also use the volatility surface in estimating the initial margins for options. In this paper we show how to generate the implied volatility surface by fitting a quadratic deterministic function to implied volatility data from Alsi index options traded on Safex. This market is mostly driven by structured spread trades, and very few at-the-money options ever trade. It is thus difficult to obtain the correct at-the-money volatilities needed by the exchange for their mark-to-market and risk management processes. We further investigate the term structure of at-the-money volatilities and show how the at-the-money implied volatilities can be obtained from the same deterministic model. This methodology leads to a no-spread arbitrage and robust market related volatility surface that can be used by option traders and brokers in pricing structured option trades.

FROM SMILE ASYMPTOTICS TO MARKET RISK MEASURES

The left tail of the implied volatility skew, coming from quotes on out-of-the-money put options, can be thought to reflect the market's assessment of the risk of a huge drop in stock prices. We analyze how this market information can be integrated into the theoretical framework of convex monetary measures of risk. In particular, we make use of indifference pricing by dynamic convex risk measures, which are given as solutions of backward stochastic differential equations (BSDEs), to establish a link between these two approaches to risk measurement. We derive a characterization of the implied volatility in terms of the solution of a nonlinear PDE and provide a small time-to-maturity expansion and numerical solutions. This procedure allows to choose convex risk measures in a conveniently parametrized class, distorted entropic dynamic risk measures, which we introduce here, such that the asymptotic volatility skew under indifference pricing can be matched with the market skew. We demonstrate this in a calibration exercise to market implied volatility data.

Adjusting Volatility to Discrete Cash Dividends

This article studies the modelling of discrete cash dividends. We distinguish the escrowed dividend model from the stock price process jumping downwards at ex-dividend dates. We show how dividend jumps impact the second moment of the stock terminal distribution. This distinctive feature precludes usual implied volatility data from being directly input in such a type of model, since they are extracted from another pricing framework. A preliminary volatility adjustment to discrete cash dividends is then described. This global adjustment leads to a refinement of the local volatility surface. This refinement appears to be quite natural theoretically speaking. It proves also to be very efficient numerically, at least for the simplest products such as American-style options.

Estimation of latent factors for high-dimensional time series

This paper deals with the dimension reduction of high-dimensional time series based on common factors. In particular we allow the dimension of time series p to be as large as, or even larger than, the sample size n. The estimation of the factor loading matrix and the factor process itself is carried out via an eigenanalysis of a p £ p non-negative de¯nite matrix. We show that when all the factors are strong in the sense that the norm of each column in the factor loading matrix is of the order p1=2, the estimator of the factor loading matrix is weakly consistent in L2-norm with the convergence rate independent of p. This result exhibits clearly that the `curse' is canceled out by the `blessing' of dimensionality. We also establish the asymptotic properties of the estimation when factors are not strong. The proposed method together with their asymptotic properties are further illustrated in a simulation study. An application to an implied volatility data set, together with a trading strategy derived from the ¯tted factor model, is also reported.