Volatility Function Research Articles

Jump‐robust testing of volatility functions in continuous time models

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Calibration of the temporally varying volatility and interest rate functions

In this study, we develop a calibration method of the temporally varying volatility and interest rate functions using the Black–Scholes (BS) partial differential equation and the observed market option prices with different strikes and expiries. The proposed method uses the piecewise linear interpolations between data points which are defined at the middle points of maturity dates. When we construct the volatility and interest rate, we use the exponential function so that the interpolated values are always positive. Numerical experiments with synthetic and real market data demonstrate the superior performance of the proposed method.

Computation of the unknown volatility from integral option price observations in jump–diffusion models

In this work we propose a simple and efficient algorithm to numerically approximate the time-dependent implied volatility for jump–diffusion models in option pricing that generalize the Black–Scholes equation. Here we use implicit–explicit difference schemes to compute the derivative part with fully implicit method and the integral term — in an explicit way. An average in time linearization of the diffusion term is applied, followed by a special decomposition of the unknown volatility function, which enables us to derive the implied volatility in an explicit form. Furthermore, the correctness of the algorithms is established. The presented numerical simulations demonstrate the capabilities of the current approach and confirm the robustness of the proposed methodology.

Semi-Analytical Solutions for Barrier and American Options Written on a Time-Dependent Ornstein–Uhlenbeck Process

In this article, we develop semi-analytical solutions for the barrier (perhaps, time-dependent) and American options written on the underlying stock that follows a time-dependent Ornstein–Uhlenbeck process with a lognormal drift. Semi-analytical means that given the time-dependent interest rate, continuous dividend and volatility functions, one need to solve a linear (for the barrier option) or nonlinear (for the American option) Volterra equation of the second kind (or a Fredholm equation of the first kind). After that, the option prices in all cases are presented as one-dimensional integrals of combination of the preceding solutions and Jacobi theta functions. We also demonstrate that computationally our method is more efficient than the backward finite difference method traditionally used for solving these problems, and can be as efficient as the forward finite difference solver while providing better accuracy and stability. TOPICS:Derivatives, options, statistical methods Key Findings ▪ For the first time the method of generalized integral transform, invented in physics for solving an initial-boundary value parabolic problem at [0, y(t)] with a moving boundary [y(t)], is applied to finance. ▪ Using this method, pricing of barrier and American options, where the underlying follows a time-dependent OU process (the Bachelier model with drift) are solved in a semi-analytical form. ▪ It is demonstrated that computationally this method is more efficient than the backward and even forward finite difference method traditionally used for solving these problems whereas providing better accuracy and stability.

Reconstruction of the local volatility function using the Black–Scholes model

In this paper, we propose a robust and accurate numerical algorithm to reconstruct a local volatility function using the Black–Scholes (BS) partial differential equation (PDE). Using the BS PDE and given market data, option prices at strike prices and expiry times, a time-dependent local volatility function is computed. The proposed algorithm consists of the following steps: (1) The time-dependent volatility function is computed using a recently developed method; (2) A Monte Carlo simulation technique is used to find the effective region which has a strong influence on option prices; and we partition the effective domain into several sub-regions and define a local volatility function based on the time-dependent volatility function on the sub-regions; and (3) Finally, we calibrate the local volatility function using the fully implicit finite difference method and the conjugate gradient optimization algorithm. We demonstrate the robustness and accuracy of the proposed local volatility reconstruction algorithm using manufactured volatility surface and real market data.

Patterns of 50 ETF Options Implied Volatility in China: On Implied Volatility Functions

The aim of this study is to examine the volatility smile based on the European options on Shanghai stock exchange 50 ETF. The data gives evidence of the existence of a well-known U-shaped implied volatility smile for the SSE 50 ETF options market in China. For those near-month options, the implied volatility smirk is also observed. And the implied volatility remains high for the short maturity and decreases as the maturity increases. The patterns of the implied volatility of SSE 50 ETF options indicate that in-the-money options and out-of-the-money options are more expensive relative to at-the-money options. This makes the use of at-the-money implied volatility for pricing out-of- or in-the-money options questionable. In order to investigate the implied volatility, the regression-based implied volatility functions model is considered employed to study the implied volatility in this study as this method is simple and easy to apply in practice. Several classical implied volatility functions are investigated in this paper to find whether some kind of implied volatility functions could lead to more accurate options pricing values. The potential determinants of implied volatility are the degree of moneyness and days left to expiration. The empirical work has been expressed by means of simple ordinary least squares framework. As the study shows, when valuing options, the results of using volatility functions are mixed. For far-month options, using at-the-money implied volatility performs better than other volatility functions in option valuation. For near-month options, the use of volatility functions can improve the valuation accuracy for deep in-the-money options or deep out-of-the-money options. However, no particular implied volatility function performs very well for options of all moneyness level and time to maturity.

Consistency of an Estimator for Change Point in Volatility of Financial Returns

A non parametric Auto-Regressive Conditional Heteroscedastic model for financial returns series is considered in which the conditional mean and volatility functions are estimated non-parametrically using Nadaraya Watson kernel. A test statistic for unknown abrupt change point in volatility which takes into consideration conditional heteroskedasticity, dependence, heterogeneity and the fourth moment of financial returns, since kurtosis is a function of the fourth moment is considered. The test is based on L2norm of the conditional variance functions of the squared residuals. A non-parametric change point estimator in volatility of financial returns is further obtained. The consistency of the estimator is shown theoretically and through simulation. An application of the estimator in change point estimation in volatility of United States Dollar/Kenya Shilling exchange rate returns data set is made. Through binary segmentation procedure, three change points in volatility of the exchange rate returns are estimated and further accounted for.

Fast reconstruction of time-dependent market volatility for European options

This paper presents a robust and fast numerical algorithm to reconstruct the implied volatility as a piecewise linear function of time. This is done from a set of market observations in the Black–Scholes world. We use a fully implicit finite difference scheme to solve the partial differential equations. To find the time-dependent volatility function, we minimize the cost function defined as the sum of the squared errors between the theoretical prices and the prices observed on the market. On the last time step, right before each maturity, we apply a decomposition of the numerical option value with respect to the volatility which increases the stability and the solvability of the problem considered. We employ a predictor–corrector technique due to the non-uniqueness of the volatility function minimizer. The paper is concluded with profound numerical experiments with synthetic and real market data.

W-shaped implied volatility curves and the Gaussian mixture model

The number of crossings of the implied volatility function with a fixed level is bounded above by the number of crossings of the risk-neutral density with the density of a log-normal distribution with the same mean as the forward price. It is bounded below by the number of convex payoffs priced equally by the two densities. We discuss the implications of these bounds for the implied volatility in the N-component Gaussian mixture model, with particular attention to the possibility of W-shaped smiles. We show that the implied volatility in this model crosses any level at most times. We show that a bimodal density need not produce a W-shaped smile, and a unimodal density can produce an oscillatory smile. We give monotonicity properties of the implied volatility in Gaussian mixtures under stochastic orderings of the location parameters and volatilities of the mixture components. For some of these results we make use of a novel convexity property of the Black-Scholes price at one strike with respect to the price at another strike. The combined constraints from density crossings and extreme strike asymptotics restrict the allowed shapes of the implied volatility. As an application we discuss a symmetric N = 3 Gaussian mixture model which generates three possible smile shapes: U-shaped, W-shaped and an oscillatory shape with two minima and two maxima.

Daily nonparametric ARCH(1) model estimation using intraday high frequency data

In this paper, the intraday high-frequency data are used to estimate the volatility function of daily nonparametric ARCH(1) model. A nonparametric volatility proxy model is proposed to achieve this objective. Under regular assumptions, the asymptotic distribution of the proposed estimator is established. The impact of different proxies on the estimation precision is also discussed. Simulation and empirical studies show that using the intraday high frequency data can significantly improve the estimation accuracy of the considered model. The idea of this article can be easily extended to other nonparametric or semiparametric ARCH/GARCH models.

Probabilistic machine learning for local volatility

The local volatility model is widely used for pricing and hedging financial derivatives. While its main appeal is its capability of reproducing any given surface of observed option prices – it provides a perfect fit – the essential component is a latent function that can be uniquely determined only in the limit of infinite data. To (re)construct this function, numerous calibration methods have been suggested that involve steps of interpolation and extrapolation, most often of parametric form and with point-estimate representations. We use probabilistic machine learning to look at the calibration problem in a probabilistic framework based on Gaussian processes. This immediately gives a way of encoding prior beliefs about the local volatility function and a hypothesis model that is highly flexible yet not prone to overfitting. Besides providing a method for calibrating a (range of) point estimate(s), we draw posterior inference from the distribution over local volatility. This leads to a better understanding of uncertainty associated with the model in general, and with the calibration in particular. Further, we infer dynamical properties of local volatility by augmenting the hypothesis space with a time dimension. Ideally, this provides predictive distributions not only locally, but also for entire surfaces forward in time. We apply our approach to S&P 500 market data.

NOISY HIGH FREQUENCY DATA-BASED ESTIMATION OF VOLATILITY FUNCTION WITH APPLICATIONS

Diffusion models have been widely used to describe the stochastic dynamics of the underlying economic variables. Renò ( 2008 ) introduced a nonparametric estimator of the volatility function, which is based on the estimation of quadratic variation between observations by means of realized variance. However, they may be misleading when one uses intraday data to implement directly the estimator, because intraday data display microstructure effects that could seriously distort the estimation. To filter out the impact of microstructure noise on the estimation of the volatility function, in this paper we propose an improved estimator when there is microstructure noise in the observed price. Also, we show that the proposed estimator has the same asymptotic properties as the Renò estimator when the step of discretization inclines to zero. Some simulations and empirical applications on Shanghai Stock Exchange data from March 3, 2002 to December 31, 2008 are used to illustrate the finite sample performance of the proposed estimator.

The effect of time-induced stress on financial decision making in real markets: The case of traffic congestion

We study the role of stress induced by time constraints on investor decision making in real financial markets. We use unexpected traffic congestion as a stress trigger. Our dependent variable is the slope of the implied volatility function (IVF) of options on Russian Trading System Index (RTSI) futures at the left-hand side of the volatility smile (cf. Bollen and Whaley, 2004). Controlling for relevant factors, we find that this slope at the opening of the main trading session is higher subsequent to morning traffic jams, suggesting that investors under stress assign higher weights to extreme loss scenarios. This effect is economically exploitable before transaction costs.

Brazilian stock market bubble in the 2010s.

Brazilian stock markets underwent a period of remarkable exuberance between early 2016 and March 2020, only to crash with the global turmoil related to health worries and oil prices. The Ibovespa index tripled its market value between a low point in January 2016 and its maximum in January 2020—by March 12, half those gains had been erased. Narratives about a bubble in Brazilian stocks before the global crash and its subsequent burst are plentiful in specialized media. In this paper, we explore this narrative from within the framework of strict local martingale financial bubbles. A key result in this literature states some financial asset price displays a bubble only if it follows a strict local martingale under the equivalent risk-neutral measure. A diffusion process is a strict local martingale if its volatility increases faster than linearly as its level grows. We first apply a nonparametric method to estimate the volatility function of Ibovespa daily prices, then fit a stochastic volatility model whose parameter values can discriminate the underlying price process as either a true martingale or a strict local martingale. Our results are negative towards the presence of a strict local martingale bubble in the Ibovespa index. Strict local martingale bubbles are related to a positive relationship between returns and volatility which does not seem present in the data at hand. We also performed a comparative analysis of the patterns found for the Ibovespa with the S&P500 index, spot Brent oil and gold prices.Electronic supplementary materialThe online version of this article (10.1007/s43546-020-00005-w) contains supplementary material, which is available to authorized users.

ESTIMATION OF VOLATILITY FUNCTIONS IN JUMP DIFFUSIONS USING TRUNCATED BIPOWER INCREMENTS

In this article, we introduce and analyze a new methodology to estimate the volatility functions of jump diffusion models. Our methodology relies on the standard kernel estimation technique using truncated bipower increments. The relevant asymptotics are fully developed, allowing for the time span to increase as well as the sampling interval to decrease, and accommodate both stationary and nonstationary recurrent processes. We evaluate the performance of our estimators by simulation and provide some illustrative empirical analyses.

Nonparametric Range-Based Double Smoothing Spot Volatility Estimation for Diffusion Models

We consider nonparametric spot volatility estimation for diffusion models with discrete high frequency observations. Our estimator is carried out in two steps. First, using the local average of the range-based variance, we propose a crude estimator of the spot volatility. Second, we use usual nonparametric kernel smoothing to reconstruct the volatility function from the crude estimator. By inference, we find such a double smoothing operation can effectively reduce the estimation error.

COLLOCATING VOLATILITY: A COMPETITIVE ALTERNATIVE TO STOCHASTIC LOCAL VOLATILITY MODELS

We discuss a competitive alternative to stochastic local volatility models, namely the Collocating Volatility (CV) framework, introduced in [L. A. Grzelak (2019) The CLV framework — A fresh look at efficient pricing with smile, International Journal of Computer Mathematics 96 (11), 2209–2228]. The CV framework consists of two elements, a “kernel process” that can be efficiently evaluated and a local volatility function. The latter, based on stochastic collocation — e.g. [I. Babuška, F. Nobile & R. Tempone (2007) A stochastic collocation method for elliptic partial differential equations with random input Data, SIAM Journal on Numerical Analysis 45 (3), 1005–1034; B. Ganapathysubramanian & N. Zabaras (2007) Sparse grid collocation schemes for stochastic natural convection problems, Journal of Computational Physics 225 (1), 652–685; J. A. S. Witteveen & G. Iaccarino (2012) Simplex stochastic collocation with random sampling and extrapolation for nonhypercube probability spaces, SIAM Journal on Scientific Computing 34 (2), A814–A838; D. Xiu & J. S. Hesthaven (2005) High-order collocation methods for differential equations with random inputs, SIAM Journal on Scientific Computing 27 (3), 1118–1139] — connects the kernel process to the market and allows the CV framework to be perfectly calibrated to European-type options. In this paper, we consider three different kernel process choices: the Ornstein–Uhlenbeck (OU) and Cox–Ingersoll–Ross (CIR) processes and the Heston model. The kernel process controls the forward smile and allows for an accurate and efficient calibration to exotic options, while the perfect calibration to liquid market quotes is preserved. We confirm this by numerical experiments, in which we calibrate the OU-CV, CIR-CV and Heston-CV frameworks to FX barrier options.

Volatility and Liquidity on High-Frequency Electricity Futures Markets: Empirical Analysis and Stochastic Modeling

This paper investigates the relationship between volatility and liquidity on the German electricity futures market based on high-frequency intraday prices. We estimate volatility by the time-weighted realized variance acknowledging that empirical intraday prices are not equally spaced in time. Empirical evidence suggests that volatility of electricity futures decreases as time approaches maturity, while coincidently liquidity increases. Established continuous-time stochastic models for electricity futures prices involve a growing volatility function in time and are thus not able to capture our empirical findings a priori. We demonstrate that incorporating increasing liquidity into the established models is key to model the decreasing volatility evolution.

Variance reduction approach for the volatility over a finite-time horizon

The volatility is a measure for the uncertainty of an asset’s return and is used to reflect the risk level of a financial asset. In this article, we consider the double kernel nonparametric estimator for the volatility function in a diffusion model over a finite-time span based on high frequency sampling data. Under the minimum conditions, the asymptotic mixed normality for the underlying estimator is derived. Moreover, the better finite-sample performance as variance reduction and even mean squared error reduction of the proposed estimator is verified through a Monte Carlo simulation study and an empirical analysis on overnight Shibor in China.

Bermudan option in Singapore Savings Bonds

The Singapore Savings Bonds (SSB) is a unique investment program offered by the Singapore government whereby retail investors can earn risk-free tax-free step-up interest closely matched to Treasury bond rates for up to 10 years and can redeem on any business day prior to maturity without any early redemption penalty. This study analyses the unique design of the SSB and provides a valuation of the Bermudan option for early redemption that is embedded in the SSB. The Black–Derman–Toy model is used to build the interest rate tree, and an iterative method is employed to avoid arbitrary specification of the pre-determined short rate volatility function. This bespoke Bermudan option can have changing strike prices over time. It also has a novel characteristic whereby the value of exercise to a buyer need not equal to the cost of being exercised to a seller. Better understanding of embedded options within government savings bonds leads to innovative designs that may encourage effective citizens’ savings.