Volatility Smile Research Articles

A Leptokurtic Distribution Explains Volatility Skew and Smile

A Leptokurtic Distribution Explains Volatility Skew and Smile

Esthetic crown lengthening treatment on the maxillary anterior teeth

Background: Asymmetrical smile design due to uneven gingival margin heights of the anterior upper teeth can lead to aesthetic issues and inadequate crown restorations, necessitating crown lengthening treatment. Purpose: The objective of this case report is to demonstrate the outcomes of crown lengthening to achieve improved aesthetics and symmetric gingival margin heights. Case: A 16-year-old female patient sought treatment for decayed left maxillary anterior teeth. Clinical examination revealed a non-vital tooth (tooth 21) with decay and excessively low gingival margins, along with asymmetrical gingival heights between the incisors and canines. Radiographs indicated radiolucent mass in the pulp chamber. Case Management: Crown lengthening was performed on teeth 12, 11, 21, and 22, and apexification was conducted on tooth 21 using a customized fiber post and lithium disilicate crown. Follow-up evaluations showed no complaints, well-maintained crowns, and healthy gingiva. Conclusion: Crown lengthening is feasible for anterior teeth with low gingival margins, enhancing clinical crown length to support aesthetic and final restorations in maxillary anterior teeth.

VIX-managed portfolios

We propose a simple portfolio management strategy that gauges the leverage based on the observed implied volatility index (VIX). The strategy involves taking less risk when the cumulative previous-month VIX is high and more when it is low. We show that the strategy yields more stable weights and thus requires less rebalancing than comparable strategies based on realized volatility. As a result, it produces substantially higher spanning regression alphas when transaction costs are taken into account. We document this for ten equity factors, six classes of mean–variance efficient portfolios and 176 anomaly portfolios. We argue that the superior performance of the VIX-based strategy is driven by its ability to time volatility and tail risk simultaneously, resulting from the forward-looking nature of the information entailed in the index and the higher-order return moments embedded in the implied volatility smile.

Estimating time-varying factors’ variance in the string-term structure model with stochastic volatility

This paper presents a novel term structure of interest rate (TSIR) model with stochastic volatility and jumps (SVJ) that combines the market framework proposed by Brace et al. (1997) with the string-shock framework of Santa-Clara and Sornette (2001). In this model, the factors’ variance is estimated through the eigendecomposition of a variance–covariance matrix obtained with a measure of market volatility derived from out-of-the-money option prices and historical correlations of interest rates traded in the futures market. The stochastic evolution of the factors’ variance is governed by the 4/2 model developed by Grasselli (2017), including jumps. A novel method is employed to estimate the parameters of the SVJ model that minimizes the distance between the sample moments and the moments of a gamma distribution. The empirical application of the model in the Brazilian derivatives market demonstrates its effectiveness in accurately capturing the volatility smile of interest rate options.

The rough Hawkes Heston stochastic volatility model

AbstractWe study an extension of the Heston stochastic volatility model that incorporates rough volatility and jump clustering phenomena. In our model, named the rough Hawkes Heston stochastic volatility model, the spot variance is a rough Hawkes‐type process proportional to the intensity process of the jump component appearing in the dynamics of the spot variance itself and the log returns. The model belongs to the class of affine Volterra models. In particular, the Fourier‐Laplace transform of the log returns and the square of the volatility index can be computed explicitly in terms of solutions of deterministic Riccati‐Volterra equations, which can be efficiently approximated using a multi‐factor approximation technique. We calibrate a parsimonious specification of our model characterized by a power kernel and an exponential law for the jumps. We show that our parsimonious setup is able to simultaneously capture, with a high precision, the behavior of the implied volatility smile for both S&P 500 and VIX options. In particular, we observe that in our setting the usual shift in the implied volatility of VIX options is explained by a very low value of the power in the kernel. Our findings demonstrate the relevance, under an affine framework, of rough volatility and self‐exciting jumps in order to capture the joint evolution of the S&P 500 and VIX.

Interaction between Sovereign Quanto Credit Default Swap Spreads and Currency Options

This study analyzes the term structures of sovereign quanto credit default swap (CDS) spreads and currency options, which are driven by anticipated currency depreciation risk following sovereign credit default (Twin Ds). We develop consistent pricing models for these instruments using a jump-diffusion stochastic volatility model, which allows us to decompose the term structure into the risk components. We find a common risk factor between the intensity process of sovereign credit risk and the stochastic volatility of the exchange rate, and the depreciation risk mainly captures the dependence structure between these markets during periods of high market stress in the Eurozone countries. Depreciation risk is an important component of sovereign quanto CDS spreads and is evident in the negative slope of the volatility smile in the currency option market.

Multi-regime foreign exchange rate model: Calibration and pricing

To price exotic foreign exchange (FX) options, a model needs to be selected for FX spot rate dynamics. The classic approach of modelling spot rates with Black–Scholes framework makes inappropriate assumptions of constant drift and volatility, resulting in mispricing. In this article, we investigate multi-regime Black–Scholes (MRBS) model for FX rate, with regime-switching behaviour of drift and volatility governed by a Markov chain. We derive an analytic formula for European FX call options via Fourier transform and present a Monte Carlo simulation algorithm for FX barrier option pricing. Further, we propose a calibration strategy with penalty. Finally, the empirical study shows that volatility smile can be recovered by MRBS model; calibrating MRBS with penalty gives stable calibrated parameters and small out-of-sample mean squared errors; combined use of our calibration methods and pricing algorithms provides reasonable prices for FX barrier options. Thus, MRBS model, together with proposed calibration-with-penalty strategy and pricing algorithm, provides a promising approach for FX option pricing.

Realized Volatility Skewness and Kurtosis Implied by Index Option Prices

Realized Volatility Skewness and Kurtosis Implied by Index Option Prices

Hedging Interest Rate Options with Reinforcement Learning: an investigation of a heavy-tailed distribution

Purpose: The study intends to model an interest rate index option using a heavy-tailed distribution. The goal is to calculate the interest rate path-dependent option prices that are consistent with market data and to develop a reinforcement learning strategy to discretely hedge the position considering transaction costs. Methodology: This paper presents a mathematical framework to calculate the price of interest rate path-dependent options. The research adapted a Fourier cosine series formula to employ the characteristic function of the present value of the forward index, which is modeled as a variance-gamma process and uses deep Q-learning to hedge such options. Findings: There is market evidence that the implied volatility curve is not flat. The study demonstrated that the variance-gamma process generates an increasing volatility smile, which is consistent with market observations. Additionally, hedging results show that the path-dependent options generated from the variance-gamma process can be efficiently hedged with advanced Q-learning techniques. Research limitations/implications: The study comprised only the variance-gamma process. Other probability distributions, such as the Normal Inverse Gaussian model, should be investigated. Practical implications: This study reveals which type of probability distribution should be present in a pricing engine to be consistent with implied volatilities. The approach provided here can assist managers in evaluating and comprehending market pricing behavior as well as achieving discrete hedging with costs. Originality: The paper addressed the merging of a fast pricing method for the interest rate options with a heavy-tailed distribution and the discrete interest rate derivatives hedging with reinforcement learning.

Interpolation of missing swaption volatility data using variational autoencoders

AbstractAlbeit of crucial interest for financial researchers, market-implied volatility data of European swaptions often exhibit large portions of missing quotes due to illiquidity of the underlying swaption instruments. In this case, standard stochastic interpolation tools like the common SABR model cannot be calibrated to observed volatility smiles, due to data being only available for the at-the-money quote of the respective underlying swaption. Here, we propose to infer the geometry of the full unknown implied volatility cube by learning stochastic latent representations of implied volatility cubes via variational autoencoders, enabling inference about the missing volatility data conditional on the observed data by an approximate Gibbs sampling approach. Up to our knowledge, our studies constitute the first-ever completely nonparametric approach to modeling swaption volatility using unsupervised learning methods while simultaneously tackling the issue of missing data. Since training data for the employed variational autoencoder model is usually sparsely available, we propose a novel method to generate synthetic swaption volatility data for training and afterwards test the robustness of our approach on real market quotes. In particular, we show that SABR interpolated volatilities calibrated to reconstructed volatility cubes with artificially imputed missing values differ by not much more than two basis points compared to SABR fits calibrated to the complete cube. Moreover, we demonstrate how the imputation can be used to successfully set up delta-neutral portfolios for hedging purposes.

Constructing copulas using corrected Hermite polynomial expansion for estimating cross foreign exchange volatility

A finite order multivariate Hermite polynomial expansion, as an approximation of a joint density function, can handle complex correlation structures. However, it does not construct copulas, because the density function can take negative values. In this study, we propose a formulation of multivariate Hermite polynomial expansion suitable for the application of correction that recovers non-negativity and construct a copula. Several useful expressions for integrals with the copula are derived from this formulation. We also apply this copula to estimate the volatility smile of cross-currency pairs in the foreign exchange option market. In the numerical experiments, we compare the estimation results of the volatility smile of EUR–JPY, GBP–JPY, and AUD–JPY for the proposed and other copulas to confirm the validity of the proposed copula.

Option Pricing with Fractional Stochastic Volatilities and Jumps

Empirical studies suggest that asset price fluctuations exhibit “long memory”, “volatility smile”, “volatility clustering” and asset prices present “jump”. To fit the above empirical characteristics of the market, this paper proposes a fractional stochastic volatility jump-diffusion model by combining two fractional stochastic volatilities with mixed-exponential jumps. The characteristic function of the log-return is expressed in terms of the solution of two-dimensional fractional Riccati equations of which closed-form solution does not exist. To obtain the explicit characteristic function, we approximate the pricing model by a semimartingale and convert fractional Riccati equations into a classic PDE. By the multi-dimensional Feynman-Kac theorem and the affine structure of the approximate model, we obtain the solution of the PDE with which the explicit characteristic function and its cumulants are derived. Based on the derived characteristic function and Fourier cosine series expansion, we obtain approximate European options prices. By differential evolution algorithm, we calibrate our approximate model and its two nested models to S&P 500 index options and obtain optimal parameter estimates of these models. Numerical results demonstrate the pricing method is fast and accurate. Empirical results demonstrate our approximate model fits the market best among the three models.

Pricing and Hedging with Volatility Skew

Pricing and Hedging with Volatility Skew

The Effects of Option Trading Behavior on Option Prices

This paper investigates the relationship between option trading behavior and option pricing patterns. We argue that greater active trading in the options market due to investor overconfidence leads to higher volatility and larger discrepancies in option pricing, which may be captured by implied volatility spread and implied volatility skewness. Using two different measures of excess option trading, we find that trading activities are correlated in different ways with volatility, volatility spread, and volatility skewness. We also find that these relationships exist both over time and cross-sectionally. We suggest that options investors tend to chase “hot” stocks, as we find evidence of a positive relationship between option trading activities and past underlying equity returns. Heavier trading in the options market also tends to make out-of-the-money call options more (less) expensive than the at-the-money counterparts over time (cross-sectionally). Because trading activities do not predict future equity returns, investor overconfidence, and not informed trading, seems to be a more plausible explanation for our findings.

Harvesting the volatility smile in a large emerging market: A Dynamic Nelson–Siegel approach

AbstractWhile there is a large literature on modeling volatility smile in options markets, most such studies are eventually focused on the forecasting performance of the model parameters and not on the applicability of the models in a trading environment. Drawing on the analogy of volatility smile like a term structure in the context of interest rates in fixed‐income markets, we evaluate the performance of the Dynamic Nelson–Siegel (DNS) approach to modeling the dynamics of volatility smile in a trading environment against competing alternatives. Using model‐based mispricing as our sorting criterion, and deploying a trading strategy of going long the options in the upper deciles and going short the options in the lower deciles, we show that dynamic models consistently outperform their static counterparts, with the worst dynamic model outperforming the best static model in terms of the percentage of mean returns from the trading portfolios and the Sharpe ratio. Specifically, we find that the DNS model consistently outperforms all other competing specifications on most of our selected criteria.

Implied volatility smoothing at COVID-19 times

This work aims at studying the impact of the SARS-CoV-2 pandemic on the global financial markets. In particular, such impact is analysed through the changes of the shape of the implied volatility smile of the options written on several equity indexes and on several stocks. The implied volatility function is estimated using the market-based information of liquid options and applying a semi-parametric smoothing technique that exploits a kernel function and no-arbitrage conditions. Such approach is applied to an extensive set of data to study the evolution of the implied volatility functions through the months of the pandemic. We show, in several cases, a sudden and massive change in the shape of the implied volatility functions.

NEW CONTINUUM OF OPTIONS VOLATILITY SMILE; Modification, Verification and Its Application

This functional exploration inspects the instability smile patters and its determinants for file choices. To survey the determinants that cause the grin design, we utilized Dumas et al. (1998) equation which gave us the best fitted suggested instability. So, it tends to be utilized in three distinct models, for example, Black et al. (1973), Heston (1993) and the MSV model to concentrate on the grin shape. This paper means to summing up the various methodologies in deciding the suggested unpredictability for the choices. Considering point-by-point writing on the guess strategies, a mathematical methodology is made sense of. This paper gives a survey of: (1) the improvement of the choice estimating model by Black et al. (1973), and the resulting changes of this model after 1987 (2) the experimental confirmation of each of the three models; and (3) uses of these models to really look at the examples of grin among various business sectors. We infer that there exists a positive connection between suggested instability and cash ness and the unpredictability grin is less lopsided for call choices. Likewise, the chance to termination for a choice and authentic unpredictability are the significant determinant.

50. Perceptual Smile Asymmetry is Impacted by Five Smile Parameters

PURPOSE: Facial expressions are integral to nonverbal communication. Dynamic smile reconstruction is a goal of care, and the aim of surgery is to create symmetry. This project evaluates how much asymmetry is tolerated in smile perception and quantifies that asymmetry. METHODS: Fifty-six facial photographs were categorized based on increasing smile asymmetry. Emotrics software was used to compute asymmetry in five mouth metrics. A Qualtrics survey consisting of 56 cropped photographs was disseminated using Amazon’s Mechanical Turk (MTurk). Three questions accompanied each photograph: 1) Is the person smiling? 2) What emotion is being expressed? 3) How symmetric is the facial expression? ANOVA, chi-squared, and multivariable linear modeling were performed. RESULTS: A total of 618 survey responses were analyzed. Smile recognition was 100%, 78%, and 33% in the symmetric, mild-moderately asymmetric, and asymmetric categories (p<0.001). Happy was identified in 89% of symmetric smiles, compared to 15% of asymmetric smiles (p<0.001). Disgust was conveyed in 26% of asymmetric smiles (p<0.001). If a smile was recognized, then 65% were happy; if not recognized, then disgust was chosen most frequently (p<0.001). Multivariable linear models using asymmetry in five mouth metrics predicted: smile recognition, happy emotion identification, and symmetry score (p<0.001). A 1% increase in asymmetry of lower lip height and commissure position decreased symmetry score by 32- and 18-points, respectively. CONCLUSIONS: Survey respondents are unable to recognize severely asymmetric smiles as smiles. Severely asymmetric smiles convey disgust rather than positivity. Mild differences in lower lip and commissure positions contribute to asymmetry perception, and models incorporating five smile parameters—nasolabial fold angle, interlabial distance, commissure height, commissure position, and lower lip height—were able to predict smile asymmetry.

Carr and Wu’s (2020) framework in the oil ETF option market

This paper studies the information inferred from the Carr and Wu’s (2020) formula based on a new option pricing framework in the United States Oil Fund (USO) options. We first document the term structure and dynamics of the risk-neutral variance and covariance rates which lead to a “U”-shaped implied volatility smile with a positive curvature. We then investigate the return predictability of the innovations in the risk-neutral variance and covariance rates (DRNV and DRNC) and their term structures (TRNV and TRNC) and find that DRNC is a significant and robust predictor to forecast daily, weekly and monthly USO excess returns in both statistical and economic terms based on in-sample and out-of-sample tests.

The log‐moment formula for implied volatility

AbstractWe revisit the foundational Moment Formula proved by Roger Lee fifteen years ago. We show that in the absence of arbitrage, if the underlying stock price at time T admits finite log‐moments for some positive q, the arbitrage‐free growth in the left wing of the implied volatility smile for T is less constrained than Lee's bound. The result is rationalized by a market trading discretely monitored variance swaps wherein the payoff is a function of squared log‐returns, and requires no assumption for the underlying price to admit any negative moment. In this respect, the result can be derived from a model‐independent setup. As a byproduct, we relax the moment assumptions on the stock price to provide a new proof of the notorious Gatheral–Fukasawa formula expressing variance swaps in terms of the implied volatility.