Volatility Smile Research Articles

VOLATILITY SMILE INTERPOLATION WITH RADIAL BASIS FUNCTIONS

The Radial Basis Functions (RBF) interpolation is a popular approximation technique used to smooth scattered data in various dimensions. This study uses RBF interpolation to interpolate the volatility skew of the S&P500 index options. The interpolated skews are used to construct the risk-neutral densities of the index and its local volatility surface. The RBF interpolation is contrasted throughout the study with the cubic spline interpolation. An analysis of the densities and the local volatility shows that RBF are an effective and practical tool for interpolating the implied volatility surface.

Asymptotic extrapolation of model-free implied variance: exploring structural underestimation in the VIX Index

We show that the VIX Index structurally underestimates model-free implied volatility because its implementation omits extrapolation of the volatility smile in the tails. We use the asymptotic behavior of the volatility surface to construct a correction term that is model-independent and only requires option prices at the two outermost strikes. We show how to apply this correction to the VIX Index ex-post as well as how to modify its implementation accordingly. Furthermore, we show that the degree of underestimation varies over time. For the S&P 500 Index and the DJIA Index the error is larger in periods of sustained low volatility. This cannot be observed for the Volatility-of-VIX Index.

Short-time implied volatility of additive normal tempered stable processes

Empirical studies have emphasized that the equity implied volatility is characterized by a negative skew inversely proportional to the square root of the time-to-maturity. We examine the short-time-to-maturity behavior of the implied volatility smile for pure jump exponential additive processes. An excellent calibration of the equity volatility surfaces has been achieved by a class of these additive processes with power-law scaling. The two power-law scaling parameters are β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}, related to the variance of jumps, and δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}, related to the smile asymmetry. It has been observed, in option market data, that β=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta =1$$\\end{document} and δ=-1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta =-1/2$$\\end{document}. In this paper, we prove that the implied volatility of these additive processes is consistent, in the short-time, with the equity market empirical characteristics if and only if β=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta =1$$\\end{document} and δ=-1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta =-1/2$$\\end{document}.

An empirical study on the characterization of implied volatility and pricing in the Chinese option market

This study investigates the problem of pricing model selection for SSE 50 ETF options in the Chinese market. Numerous studies identified a volatility smile in the implied volatility of options; thus, the BS model (BS-HV model), based on the assumption of constant volatility, was criticized. Stochastic volatility models, represented by the Heston model, have become popular owing to their ability to reflect non-constant volatility. This ability inspired us to address the controversial BS-HV model issue by combining the BS formulation with the method of generating the implied volatility surface using TPS interpolation (BS-IV model). In this study, we evaluate the pricing ability of the model from three perspectives: pricing error, volatility portrayal, and price trend prediction. We find that though the historical volatility of the BS-HV model cannot reflect the implicit information of the options market price, the longer the interval of its moving average, the smaller the pricing error of the test set. The BS-IV model can accurately predict the price trend but has the largest pricing error. The Heston model demonstrates average capability in the three aspects of volatility. Therefore, the merits of each model should be combined when pricing options, and the long-range BS-HV model should be used as a reference when pursuing small errors. The BS-IV model can be combined with the previous day's options closing price to obtain a reasonable options price if the trend prediction is important.

The Study on the Presence of Volatility Smile in NSE Indices

If James Taylor had been an options trader, he might well have not written that second line. Options "smile" at traders, but that smile is not a source of happiness. It is in fact a source of considerable confusion and misunderstanding. In this paper, we explore how this confusion manifests in the pattern of different implied volatilities for different options on the same stock, a pattern commonly known as the volatility smile.

Empirical Asset Pricing with Functional Factors

Abstract We propose a methodology to use functional factors in empirical asset pricing models. We establish conditions under which it is possible to recover linear beta pricing. The proposed estimation approach allows us to use high-dimensional functional curves, such as the term structure of interest rates or the implied volatility smile, as factors. This framework enables the estimation of functional factor loadings as well as risk premium parameters of factor models. We derive estimation algorithms and establish the asymptotic consistency and normality of the parameter estimates. In an empirical application, we show that the implied variance smile of the S&P500 is a potential pricing factor for momentum-sorted portfolios. In particular, a positive risk premium is earned by the convexity of the implied variance curve.

Average Type Smile in Posed Smile of Individuals Visiting Department of Orthodontics of a Tertiary Care Hospital: A Descriptive Cross-sectional Study.

New concepts and methods have been developed in orthodontics with patients' increased concern on esthetics. Among the various parameters of smile, smile type, smile arc, and smile symmetry are essential for a beautiful smile. The objective of this study was to find out the prevalence of average type in posed smiles of individuals visiting the department of orthodontics. This descriptive cross-sectional study was conducted from 1st April 2021 to 30th June 2021 at a tertiary care hospital. Ethical clearance was obtained from the Institutional Review Committee (Reference number: 1, CH No. 30, 2077/2078). A convenience sampling technique was used to collect a total of 413 samples of posed smile photographs. Data was collected and analysed using Microsoft Excel 2013. Point estimate at 95% Confidence Interval was calculated along with frequency and percentage. Among 413 individuals, the average type of smile was found in 193 (46.73%) (42.12-51.74 at 95% Confidence Interval). Out of 87 average smile type males, most of them 51 (58.62%) had non consonant smile arcs and 47 (54.02%) had asymmetric type smiles. On the contrary, the majority of females 76 (71.70%) had a consonant arc smile while 63 (59.43%) had an asymmetric smile. The prevalence of the average type of smile was higher in our study as in other published literatures. dental esthetics; dental photography; orthodontics; smiling.

Does the world smile together? A network analysis of global index option implied volatilities

This study uses directed acyclic graph and spillover index models to find significant evidence of both implied volatility contagion and spillovers. First, we observe strong regional clustering among the implied volatility smiles of global markets. European and American options markets form a separate contemporaneous contagion cluster from markets in the Asia-Pacific region. However, no market is completely independent from the markets in the other two regions in the long run. The European index options markets demonstrate the strongest implied volatility smile contagion. Second, we observe the heterogeneity across different markets in terms of implied volatility spillover, and extreme market conditions, such as crises, seem to intensify the spillover effects. The trend in the short-run underlying index return, the implied volatility of at-the-money options, and the interest rate term spread are key determinants of implied volatility spillovers.

A Discretely Formulated Option Pricing Model That, Absent Directness of Modeling of Volatility, Embeds the ‘Volatility Smile’

A Discretely Formulated Option Pricing Model That, Absent Directness of Modeling of Volatility, Embeds the ‘Volatility Smile’

Keep on Smiling: Market Imbalance, Option Pricing, and the Volatility Smile

Keep on Smiling: Market Imbalance, Option Pricing, and the Volatility Smile

Saddle-Point Approach to Large-Time Volatility Smile

Saddle-Point Approach to Large-Time Volatility Smile

Pricing autocallables under local-stochastic volatility

This paper investigates the pricing of single-asset autocallable barrier reverse convertibles in the Heston local-stochastic volatility (LSV) model. Despite their complexity, autocallable structured notes are the most traded equity-linked exotic derivatives. The autocallable payoff embeds an early-redemption feature generating strong path- and model-dependency. Consequently, the commonly-used local volatility (LV) model is overly simplified for pricing and risk management. Given its ability to match the implied volatility smile and reproduce its realistic dynamics, the LSV model is, in contrast, better suited for exotic derivatives such as autocallables. We use quasi-Monte Carlo methods to study the pricing given the Heston LSV model and compare it with the LV model. In particular, we establish the sensitivity of the valuation differences of autocallables between the two models with respect to payoff features, model parameters, underlying characteristics, and volatility regimes. We find that the improved spot-volatility dynamics captured by the Heston LSV model typically result in higher prices, demonstrating the dependence of autocallables on the forward-skew and vol-of-vol risk. Moreover, we show that the parameters of the stochastic component of LSV models enable controlling for the autocallables price while leaving the fit to European options unaffected.

Accounting Transparency and the Implied Volatility Skew

Accounting Transparency and the Implied Volatility Skew

Intraday volatility smile: Effects of fragmentation and high frequency trading on price efficiency

In 2007, the European Markets in Financial Instruments Directive ended the national concentration rule. As a result, market fragmentation has accelerated across multiple trading venues. Spatial fragmentation might create opportunities and incentives for High Frequency arbitrageurs to fill the void left by the lack of Reg NMS type order routing requirements in Europe, without neglecting market integrity. This paper examines intra-day volatility and price efficiency through the metric of the normalized volatility ratio for the years 2006, 2012 and 2013 for Euronext Paris, BATS and Chi-X Europe. Our findings show that price determination remains inefficient at market opening due to the complexity of price discovery activity following a period of non-trading and heavy information releases. However, we demonstrate that an active participation of high-frequency traders significantly improves market efficiency at opening session.

Markovian approximations of stochastic Volterra equations with the fractional kernel

We consider rough stochastic volatility models where the variance process satisfies a stochastic Volterra equation with the fractional kernel, as in the rough Bergomi and the rough Heston model. In particular, the variance process is therefore not a Markov process or semimartingale, and has quite low Hölder-regularity. In practice, simulating such rough processes thus often results in high computational cost. To remedy this, we study approximations of stochastic Volterra equations using an N-dimensional diffusion process defined as solution to a system of ordinary stochastic differential equation. If the coefficients of the stochastic Volterra equation are Lipschitz continuous, we show that these approximations converge strongly with superpolynomial rate in N. Finally, we apply this approximation to compute the implied volatility smile of a European call option under the rough Bergomi and the rough Heston model.

Masseteric nerve transference with sling tendon for symmetry in the restoration of facial paralysis: An effective combination of dynamic and static correction

Masseteric nerve transposition has shown to be an effective technique for the treatment of facial paralysis. Due to its low morbidity at the donor site, its consistent anatomy, and one-stage operation, it constitutes a reliable option for both complete and incomplete facial paralysis. Masseteric nerve transference has proven to achieve good commissural excursion and dynamic symmetry. However, some patients with heavy faces and complete facial paralysis recover incompletely after surgery characterized by an asymmetrical smile with asymmetry at rest, affecting the cosmetic appearance. For these patients, we propose a novel combination of masseteric nerve transposition for the dynamic rehabilitation of the smile with a tendon sling suspension to create symmetry at rest. A detailed description of the surgical technique is given and results after using it with eight patients show good functional and aesthetic satisfaction. A combination of both dynamic and static techniques for complete facial paralysis rehabilitation may provide adequate symmetry with the contralateral healthy side both at rest and when smiling.

Detecting Jump Risk and Jump-Diffusion Model for Bitcoin Options Pricing and Hedging

In this paper, we conduct a fast calibration in the jump-diffusion model to capture the Bitcoin price dynamics, as well as the behavior of some components affecting the price itself, such as the risk of pitfalls and its ambiguous effect on the evolution of Bitcoin’s price. In addition, in our study of the Bitcoin option pricing, we find that the inclusion of jumps in returns and volatilities are significant in the historical time series of Bitcoin prices. The benefits of incorporating these jumps flow over into option pricing, as well as adequately capture the volatility smile in option prices. To the best of our knowledge, this is the first work to analyze the phenomenon of price jump risk and to interpret Bitcoin option valuation as “exceptionally ambiguous”. Crucially, using hedging options for the Bitcoin market, we also prove some important properties: Bitcoin options follow a convex, but not strictly convex function. This property provides adequate risk assessment for convex risk measure.

The Riemann–Liouville field and its GMC as [formula omitted], and skew flattening for the rough Bergomi model

We consider a re-scaled Riemann–Liouville (RL) process ZtH=∫0t(t−s)H−12dWs, and using Lévy’s continuity theorem for random fields we show that ZH tends weakly to an almost log-correlated Gaussian field Z as H→0. Away from zero, this field differs from a standard Bacry–Muzy field by an a.s.Hölder continuous Gaussian process, and we show that ξγH(dt)=eγZtH−12γ2V ar(ZtH)dt tends to a Gaussian multiplicative chaos (GMC) random measure ξγ for γ∈(0,1) as H→0. We also show convergence in law for ξγH as H→0 for γ∈[0,2) using tightness arguments, and ξγ is non-atomic and locally multifractal away from zero. In the final section, we discuss applications to the Rough Bergomi model; specifically, using Jacod’s stable convergence theorem, we prove the surprising result that (with an appropriate re-scaling) the martingale component Xt of the log stock price tends weakly to Bξγ([0,t]) as H→0, where B is a Brownian motion independent of everything else. This implies that the implied volatility smile for the full rough Bergomi model with ρ≤0 is symmetric in the H→0 limit, and without re-scaling the model tends weakly to the Black–Scholes model as H→0. We also derive a closed-form expression for the conditional third moment E((Xt+h−Xt)3|Ft) (for H>0) given a finite history, and E(XT3) tends to zero (or blows up) exponentially fast as H→0 depending on whether γ is less than or greater than a critical γ≈1.61711 which is the root of 14+12logγ−316γ2. We also briefly discuss the pros and cons of a H=0 model with non-zero skew for which Xt/t tends weakly to a non-Gaussian random variable X1 with non-zero skewness as t→0.

Smiles in delta

Fukasawa introduced in Fukasawa [The normalizing transformation of the implied volatility smile. Math. Finance, 2012, 22(4), 753–762] two necessary conditions for no butterfly arbitrage on a given implied volatility smile which require that the functions d 1 and d 2 of the Black–Scholes formula have to be decreasing. In this article, we characterize the set of smiles satisfying these conditions, using the parametrization of the smile in delta. We obtain a parametrization of the set of such smiles via one real number and three positive functions, which can be used by practitioners to calibrate a weak arbitrage-free smile. We also show that such smiles and their symmetric smiles can be transformed into smiles in the strike space by a bijection. Our result motivates the study of the challenging question of characterizing the subset of butterfly arbitrage-free smiles using the parametrization in delta.

Tempered stable processes with time-varying exponential tails

In this paper, we introduce a new time series model with a stochastic exponential tail. This model is constructed based on the Normal Tempered Stable distribution with a time-varying parameter. It captures the stochastic exponential tail, which generates the volatility smile effect and volatility term structure in option pricing. Moreover, the model describes the time-varying volatility of volatility and empirically indicates stochastic skewness and stochastic kurtosis in the S&P 500 index return data. We present a Monte-Carlo simulation technique for parameter calibration of the model for S&P 500 option prices and show that a stochastic exponential tail improves the calibration performance.