Volatility Function Research Articles

Asymptotic Equivalence of Continuously and Discretely Sampled Jump-Diffusion Models

We establish the Local Asymptotic Normality (LAN) property for a class of parametric jump-diffusion processes with state-dependent intensity and known volatility function sampled at high-frequency. We prove that the inference problem about the drift and jump parameters is adaptive with respect to parameters in the volatility function that can be consistently estimated.

Variation-based tests for volatility misspecification

We provide a simple and easy to use goodness-of-fit test for the misspecification of the volatility function in diffusion models. The test uses power variations constructed as functionals of discretely observed diffusion processes. We introduce an orthogonality condition which stabilizes the limit law in the presence of parameter estimation and avoids the necessity for a bootstrap procedure that reduces performance and leads to complications associated with the structure of the diffusion process. The test has good finite sample performance as we demonstrate in numerical simulations.

A defaultable HJM modelling of the Libor rate for pricing Basis Swaps after the credit crunch

A great deal of recent literature discusses the major anomalies that have appeared in the interest rate market following the credit crunch in August 2007. There were major consequences with regard to the development of spreads between quantities that had remained the same until then. In particular, we consider the spread that opened up between the Libor rate and the OIS rate, and the consequent empirical evidence that FRA rates can no longer be replicated using Libor spot rates due to the presence of a Basis spread between floating legs of different tenors. We develop a credit risk model for pricing Basis Swaps in a multi-curve setup. The Libor rate is considered here as a risky rate, subject to the credit risk of a generic counterparty whose credit quality is refreshed at each fixing date. A defaultable HJM methodology is used to model the term structure of the credit spread, defined through the implied default intensity of the contributing banks of the Libor corresponding to a chosen tenor. A forward credit spread volatility function depending on the entire credit spread term structure is assumed. In this context, we implement the model and obtain the price of Basis Swaps using a numerical scheme based on the Euler–Maruyama stochastic integral approximation and the Monte Carlo method.

A variant of K nearest neighbor quantile regression

Compared with local polynomial quantile regression, K nearest neighbor quantile regression (KNNQR) has many advantages, such as not assuming smoothness of functions. The paper summarizes the research of KNNQR and has carried out further research on the selection of k, algorithm and Monte Carlo simulations. Additionally, simulated functions are Blocks, Bumps, HeaviSine and Doppler, which stand for jumping, volatility, mutagenicity slope and high frequency function. When function to be estimated has some jump points or catastrophe points, KNNQR is superior to local linear quantile regression in the sense of the mean squared error and mean absolute error criteria. To be mentioned, even high frequency, the superiority of KNNQR could be observed. A real data is analyzed as an illustration.

An approximation formula for basket option prices under local stochastic volatility with jumps: An application to commodity markets

This paper develops a new approximation formula for pricing basket options in a local-stochastic volatility model with jumps. In particular, the model admits local volatility functions and jump components in not only the underlying asset price processes, but also the volatility processes. To the best of our knowledge, the proposed formula is the first one which achieves an analytical approximation for the basket option prices under this type of the models.Moreover, in numerical experiments, we provide approximate prices for basket options on the WTI futures and Brent futures based on the parameters through calibration to the plain-vanilla option prices, and confirm the validity of our approximation formula.

APPROXIMATING LOCAL VOLATILITY FUNCTIONS OF STOCHASTIC VOLATILITY MODELS: A CLOSED-FORM EXPANSION APPROACH

We propose a method for approximating equivalent local volatility functions of stochastic volatility models. Enlightened by the theory of generalized Wiener functionals proposed by Watanabe and Yoshida (1987, 1992), our key technique is to propose a closed-form expansion of conditional expectations involving marginal distributions generated by stochastic differential equations. A numerical test and an illustration of application are provided to demonstrate the efficiency of our approach.

RISK MARGIN QUANTILE FUNCTION VIA PARAMETRIC AND NON-PARAMETRIC BAYESIAN APPROACHES

AbstractWe develop quantile functions from regression models in order to derive risk margin and to evaluate capital in non-life insurance applications. By utilizing the entire range of conditional quantile functions, especially higher quantile levels, we detail how quantile regression is capable of providing an accurate estimation of risk margin and an overview of implied capital based on the historical volatility of a general insurers loss portfolio. Two modeling frameworks are considered based around parametric and non-parametric regression models which we develop specifically in this insurance setting. In the parametric framework, quantile functions are derived using several distributions including the flexible generalized beta (GB2) distribution family, asymmetric Laplace (AL) distribution and power-Pareto (PP) distribution. In these parametric model based quantile regressions, we detail two basic formulations. The first involves embedding the quantile regression loss function from the nonparameteric setting into the argument of the kernel of a parametric data likelihood model, this is well known to naturally lead to the AL parametric model case. The second formulation we utilize in the parametric setting adopts an alternative quantile regression formulation in which we assume a structural expression for the regression trend and volatility functions which act to modify a base quantile function in order to produce the conditional data quantile function. This second approach allows a range of flexible parametric models to be considered with different tail behaviors. We demonstrate how to perform estimation of the resulting parametric models under a Bayesian regression framework. To achieve this, we design Markov chain Monte Carlo (MCMC) sampling strategies for the resulting Bayesian posterior quantile regression models. In the non-parametric framework, we construct quantile functions by minimizing an asymmetrically weighted loss function and estimate the parameters under the AL proxy distribution to resemble the minimization process. This quantile regression model is contrasted to the parametric AL mean regression model and both are expressed as a scale mixture of uniform distributions to facilitate efficient implementation. The models are extended to adopt dynamic mean, variance and skewness and applied to analyze two real loss reserve data sets to perform inference and discuss interesting features of quantile regression for risk margin calculations.

Two-step estimation of the volatility functions in diffusion models with empirical applications

Abstract In this article, we develop a two-step estimation procedure for the volatility function in diffusion models. We firstly estimate the volatility series at sampling time points based on high-frequency data. Then, the volatility function estimator can be obtained by using the kernel smoothing method. The resulting estimators are presented based on high-frequency data, and are shown to be consistent and asymptotically normal. We also consider boundary issues and then propose two methods to handle them. The asymptotic normality of two boundary-corrected estimators is established under some suitable conditions. The proposed estimators are illustrated by Monte Carlo simulations and real data.

Nonparametric Regression Estimation for Multivariate Null Recurrent Processes

This paper discusses nonparametric kernel regression with the regressor being a \(d\)-dimensional \(\beta\)-null recurrent process in presence of conditional heteroscedasticity. We show that the mean function estimator is consistent with convergence rate \(\sqrt{n(T)h^{d}}\), where \(n(T)\) is the number of regenerations for a \(\beta\)-null recurrent process and the limiting distribution (with proper normalization) is normal. Furthermore, we show that the two-step estimator for the volatility function is consistent. The finite sample performance of the estimate is quite reasonable when the leave-one-out cross validation method is used for bandwidth selection. We apply the proposed method to study the relationship of Federal funds rate with 3-month and 5-year T-bill rates and discover the existence of nonlinearity of the relationship. Furthermore, the in-sample and out-of-sample performance of the nonparametric model is far better than the linear model.

Do Aussie markets smile? Implied volatility functions and determinants

If options are correctly priced, the interpretation of volatility in the Black–Scholes model (as identifying the volatility of the underlying asset) is violated. The empirical relation between the model ‘implied volatility’ and the degree to which the option is in-the-money (moneyness) has been reported as resembling a U-shape (or ‘smile’) for options on currencies (and more of a ‘smirk’ for options on equities). In this article, using multivariate time-series analysis and employing an impulse response function, we investigate the structural relationships and dynamics of the volatility smile in relation to the option liquidity, key features of the underlying asset and market momentum. Our findings confirm evidence of a number of biases in the Black–Scholes model consistent with Chou et al. (2011) in regard to liquidity in both the underlying and the option itself, and with Peña et al. (1999) as to the importance of the option time to maturity. As well as delineating such biases as they co-relate both with each other and with the underlying asset volatility and momentum, we find that the pronounced smile is related to the differential sensitivities of in-the-money and out-of-the-money options, which itself suggests an explanation for the characteristic smile shape.

Explicit form of approximate transition probability density functions of diffusion processes

A continuous-time diffusion process is very popular in modeling and provides useful tools to analyze particularly, but not restricted to, a variety of economic and financial variables. The transition probability density function (TPDF) of a diffusion process plays an important role in understanding and explaining the dynamics of the process. A new way to find closed-form approximate TPDFs for multivariate diffusions is proposed in this paper. This method can be applied to general multivariate time-inhomogeneous diffusion processes, as long as, roughly speaking, they have smooth drift and volatility functions. A diffusion process is said to be reducible if it can be converted into a unit diffusion process where the volatility is the identity matrix. We have established how to obtain the approximate TPDF of a reducible diffusion explicitly. When a diffusion process is not reducible, an explicit form of approximate TPDF can be obtained by using the results in Aït-Sahalia (2008) and Choi (2013). The TPDF expansion suggested here enables us to obtain a recursive formula for the coefficient of the approximate TPDF for a multivariate jump diffusion. Monte Carlo simulation studies of conducting maximum likelihood estimation (MLE) using our approximations provide convincing evidence that our TPDF expansion can be used for the MLE when the true TPDF is unavailable. We also applied our approximate TPDFs to option pricing. The differences between our option prices and those from the Extended Black–Scholes formula are shown to be quite small. This implies that our methods can be employed to price assets whose underlying state variables follow general diffusion models.

Affine Approximation for Moment Generating Function of Log-Normal Stochastic Volatility Model

While empirical studies have established that the log-normal stochastic volatility (SV) model is superior to its alternatives, the model does not allow for the analytical solutions available for affine models. To circumvent this, we show that the joint moment generating function (MGF) of the log-price and the quadratic variance (QV) under the log-normal SV model can be decomposed into a leading term, which is given by an exponential-affine form, and a residual term, whose estimate depends on the higher order moments of the volatility process. We prove that the second-order leading term is theoretically consistent with the expected values and covariance matrix of the log-price and the quadratic variance. We further extend this approach to the log-normal SV model with jumps. We use Fourier inversion techniques to value vanilla options on the equity and the QV and, by comparison to Monte Carlo simulations, we show that the second-order leading term is precise for the valuation of vanilla options. We generalize the affine decomposition to other non-affine stochastic volatility models with polynomial drift and volatility functions, and with jumps in the volatility process.

LET’S GET LADE: ROBUST ESTIMATION OF SEMIPARAMETRIC MULTIPLICATIVE VOLATILITY MODELS

We investigate a model in which we connect slowly time varying unconditional long-run volatility with short-run conditional volatility whose representation is given as a semi-strong GARCH(1,1) process with heavy tailed errors. We focus on robust estimation of both long-run and short-run volatilities. Our estimation is semiparametric since the long-run volatility is totally unspecified whereas the short-run conditional volatility is a parametric semi-strong GARCH(1,1) process. We propose different robust estimation methods for nonstationary and strictly stationary GARCH parameters with nonparametric long-run volatility function. Our estimation is based on a two-step LAD procedure. We establish the relevant asymptotic theory of the proposed estimators. Numerical results lend support to our theoretical results.

A generalized procedure for building trees for the short rate and its application to determining market implied volatility functions

One-factor no-arbitrage models of the short rate are important tools for valuing interest rate derivatives. Trees are often used to implement the models and fit them to the initial term structure. This paper generalizes existing tree building procedures so that a very wide range of interest rate models can be accommodated. It shows how a piecewise linear volatility function can be calibrated to market data and, using market data from days during the period 2004–2013, finds a best fit to cap prices.

Asymptotically distribution-free tests for the volatility function of a diffusion

This paper develops two tests for parametric volatility function of a diffusion model based on Khmaladze (1981)’s martingale transformation. The tests impose no restrictions on the functional form of the drift function and are shown to be asymptotically distribution-free. The tests are consistent against a large class of fixed alternatives and have nontrivial power against a class of root-n local alternatives. The paper also extends the tests of volatility to testing for joint specifications of drift and volatility. Monte Carlo simulations show that the tests perform well in finite samples. The proposed tests are then applied to testing models of short-term interest, using data of Treasury bill rate and Eurodollar deposit rate. The empirical results show that the commonly used CKLS volatility function of Chan et al. (1992) fits volatility function poorly and none of the parametric interest rate models considered in the paper fit data well.

Recovery of the local volatility function using regularization and a gradient projection method

This paper considers the problem of calibrating the volatility function using regularization technique and the gradient projection method from given option price data. It is an ill-posed problem because of at least one of three well-posed conditions violating. We start with the European option pricing problem. We formulate the problem by obtaining the integral equation from Dupire equation and provide a theory of identifying the local volatility function $\sigma(y,\tau)$ when the parameter $\mu\neq 0$, and then we apply regularization technique for volatility function retrieval problems. A projected gradient method is developed for recovering the volatility function. Numerical simulations are given to illustrate the feasibility of our method.

Variation-Based Tests for Volatility Misspecification

We provide a simple and easy to use goodness-of-fit test for the misspecification of the volatility function in diffusion models. The test uses power variations constructed as functionals of discretely observed diffusion processes. We introduce an orthogonality condition which stabilizes the limit law in the presence of parameter estimation and avoids the necessity for a bootstrap procedure that reduces performance and leads to complications associated with the structure of the diffusion process. The test has good finite sample performance as we demonstrate in numerical simulations.

KOSPI200 옵션의 내재변동성 추정

옵션가격의 결정에 있어서 실제 변동성은 사후에 알 수 있는 정보이므로 대용값으로 내재변동성을 가장 많이 사용하는데 본 연구에서는 동일한 기초자산을 가진 옵션의 잔존만기와 행사가격을 이용하여 내재변동성을 추정하고자 한다. KOSPI200 옵션 데이터와 서포트벡터회귀, 나무모형 및 회귀모형을 통해 모형의 설명력을 평균제곱근오차 (RMSE)와 평균절대오차 (MAE)를 사용하여 살펴보았다. 그 결과 서포트벡터회귀와 MART의 성능이 최소제곱회귀보다 우수한 것으로 나타났으며, 서포트벡터회귀와 MART의 성능은 거의 비슷하였다. Using the assumption that the price of a stock follows a geometric Brownian motion with constant volatility, Black and Scholes (BS) derived a formula that gives the price of a European call option on the stock as a function of the stock price, the strike price, the time to maturity, the risk-free interest rate, the dividend rate paid by the stock, and the volatility of the stock's return. However, implied volatilities of BS method tend to depend on the stock prices and the time to maturity in practice. To address this shortcoming, we estimate the implied volatility function as a function of the strike priceand the time to maturity for data consisting of the daily prices for KOSPI200 call options from January 2007 to May 2009 using support vector regression (SVR), the multiple additive regression trees (MART) algorithm, and ordinary least squaress (OLS) regression. In conclusion, use of MART or SVR in the BS pricing model reduced both RMSE and MAE, compared to the OLS-based BS pricing model.

Implied Volatility Function Approximation with Korean ELWs (Equity-Linked Warrants) via Gaussian Processes

A lot of researches have been conducted to estimate the volatility smile effect shown in the option market. This paper proposes a method to approximate an implied volatility function, given noisy real market option data. To construct an implied volatility function, we use Gaussian Processes (GPs). Their output values are implied volatilities while moneyness values (the ratios of strike price to underlying asset price) and time to maturities are as their input values. To show the performances of our proposed method, we conduct experimental simulations with Korean Equity-Linked Warrant (ELW) market data as well as toy data.

BEHAVIOR OF LONG-TERM YIELDS IN A LÉVY TERM STRUCTURE

We study the behavior of the long-term yield in a HJM setting for forward rates driven by Lévy processes. The long-term rates are investigated by examining continuously compounded spot rate yields with maturity going to infinity. In this paper, we generalize the model of Karoui et al. (1997) by using Lévy processes instead of Brownian motions as driving processes of the forward rate dynamics, and analyze the behavior of the long-term yield under certain conditions which encompass the asymptotic behavior of the interest rate model's volatility function as well as the variation of the paths of the Lévy process. One of the main results is that the long-term volatility has to vanish except in the case of a Lévy process with only negative jumps and paths of finite variation serving as random driver. Furthermore, we study the required asymptotic behavior of the volatility function so that the long-term drift exists.